Advances in High Energy Physics

Volume 2019, Article ID 9406585, 16 pages

https://doi.org/10.1155/2019/9406585

## Vortices in Vacuumless Systems

Departamento de Física, Universidade Federal da Paraíba, 58051-970 João Pessoa, PB, Brazil

Correspondence should be addressed to M. A. Marques; moc.liamg@suehtam.mam

Received 15 January 2019; Accepted 18 March 2019; Published 3 April 2019

Academic Editor: Diego Saez-Chillon Gomez

Copyright © 2019 M. A. Marques. 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 investigate the presence of vortex solutions in potentials without vacuum state. The study is conducted considering Maxwell and Chern-Simons dynamics. Also, we use a first-order formalism that helps us to find the solutions and their respective electromagnetic fields and energy densities. As a bonus, we get to calculate the energy without knowing the explicit solutions. Even though the solutions present a large “tail” which goes far away from the origin, the magnetic flux remains a well defined topological invariant.

#### 1. Introduction

In high energy physics, topological structures appear in a diversity of contexts and have been vastly studied over the years [1, 2]. In spatial dimensions lower than three, the most known ones are kinks and vortices, which are static solutions of the equations of motion.

The simplest structures are kinks, which appear in spacetime dimensions by the action of scalar fields [3]. Kinks connect the minima of the potential and have a topological character that ensures its stability. However, it was shown in [4] that topological defects may arise in potentials without a vacuum state, whose minima are located at infinity. Regarding the kink in the vacuumless system, it is asymptotically divergent and has infinite amplitude. Nevertheless, it is stable and can be associated with a topological charge by using a special definition for the topological current [5]. Over the years, many papers have studied vacuumless topological defects in a diversity of contexts in high energy physics [6–14].

Potentials with extrema at infinity, similar to the ones we are going to study here, although inverted, also appear in classical mechanics [15]. In this scenario, if the energy is small enough, the motion is bounded. As the energy gets higher, the boundary values become far form each other, until the limit where they are infinitely separated. This limit distinguishes bounded and unbounded motion, so for sufficiently high values of the energy, the motion becomes unbounded. A similar situation happens in the interaction of a body with the gravitational potential, , which vanishes only at , when one calculates the escape velocity: the zero energy of the system describes the limit between bounded and unbounded motion. In high energy physics, vacuumless potentials arise in the massless limit of supersymmetric QCD due to nonperturbative effects [16]. They also appear in the cosmological context, where their energy densities could act as a cosmological constant that decreases slower than the densities of matter and radiation [17, 18].

By working in spacetime dimensions one can find vortices. The first relativistic model that supports these objetcs was studied in [19, 20], with the action of a complex scalar field coupled to a gauge field under the symmetry in Maxwell dynamics. These structures are electrically neutral and engender a quantized flux which is conserved and works as a topological invariant. Their equations of motion are of second order with couplings between the fields; thus, they are hard to be solved. To simplify the problem, the BPS formalism was developed in [21, 22] for this model, which allowed for the presence of first order equations and the energy without knowing the explicit form of the solutions.

Models with the gauge field governed by the Maxwell dynamics, however, are not the only ones which support vortices solutions. One can also investigate these structures with the dynamics of the gauge field governed by the Chern-Simons term [23–25]. In this case, the vortex presents a quantized flux, which also is topological invariant, and a quantized electric charge. The first studies of vortices in Chern-Simons dynamics are [26–28]; for more on this, see [29].

The importance of vortices in high energy physics and in other areas of physics can be found in [1, 2, 30]. For instance, they may appear during the cosmic evolution of our Universe [1] and in models that includes the so-called hidden sector, which is of interest in dark matter [31–34], by enlarging the symmetry to ; see [35–37]. Following this direction of enlarged symmetries, they are also present in models, with the addition of extra degrees of freedom to the vortex via the inclusion of a triplet scalar field, and in models, with the inclusion of a neutral scalar field that acts as a source to the internal structure of the vortex [38]. Other motivations come from the context of condensed matter, where they may emerge in superconductors and in magnetic materials as magnetic domains [39]. They may also appear in dipolar Bose-Einstein condensates, where the atoms interact as dipole-dipole, which leads to the presence of non standard vortex structures [40–42].

Topological structures may be studied with generalized models [43, 44]. Vortices, in particular, firstly appeared in noncanonical models in [45, 46]. Since then, several works arised with other motivations. In the context of inflation, for instance, a model with a modified kinetic term was introduced in [47]. In this scenario, these models present distinct features from the standard case: they may not need a potential to drive the inflation. Moreover, generalized models were used in [48, 49] as a tentative to explain why the universe is accelerated at a late stage of its evolution.

Noncanonical models considering defect structures were severely investigated over the years [50–63]. Among the many investigations, a first-order formalism was developed for some classes of noncanonical models in [45, 46, 60, 64, 65]. However, only in [66] it was completely developed for any generalized model. An interesting fact is that compact structures, which were firstly presented in [67], are possible to appear as Maxwell and Chern-Simons vortices only if generalized models are considered; see [68, 69]. Noncanonical models also allow for the presence of vortices that share the same field configuration and energy density, known as twinlike models [70].

This work deals with a class of generalized Maxwell and Chern-Simons models that support vortex solutions in vacuumless systems. In Section 2, we investigate the properties of vortices with Maxwell dynamics, including its first-order formalism, and introduce two new models, one of them with analytical results. In Section 3 we conduct a similar investigation, however in the Chern-Simons scenario, also considering its first-order formalism, and we introduce two new models. Finally, in Section 4 we present our ending comments and conclusions.

#### 2. Maxwell-Higgs Models

We deal with an action in flat spacetime dimensions for a complex scalar field and a gauge field governed by the Maxwell dynamics. We follow the lines of [66] and write , with the Lagrangian density given by In the above equation, denotes the complex scalar field, is the gauge field, represents the electromagnetic strength tensor, stands for the covariant derivative, is the electric charge, and is the potential, which is supposed to present symmetry breaking. The function is dimensionless and, in principle, arbitrary. Nevertheless, it has to admit solutions with finite energy. It is straightforward to show that gives the standard case considered in [19]. One may vary the action with respect to the fields and to get the equations of motionwhere the current is and . Invariance under spacetime translations, , with constant, leads to the energy momentum tensorIn order to investigate vortex solutions in the model, we consider static configurations. As a consequence, the component of (2b) becomes an identity under the choice . This makes the electric field vanish, so the vortex is electrically uncharged. Since we are dealing with two spatial dimensions, we define the magnetic field as . In this case, the surviving components of the energy momentum tensor (3) areThe energy density is and the components define the stress tensor. We then take the usual ansatz for vortex solutionswhere and are the polar coordinates and is the vorticity. The functions and must obey the boundary conditionsIn the above equations, is a parameter that is involved in the symmetry breaking of the potential. Considering the ansatz (5a) and (5b), the magnetic field becomesBy integrating it all over the space one can show that the flux is given byTherefore, the magnetic flux is conserved and quantized by the vorticity . As one knows, it is possible to introduce the conserved topological currentin which the component plays the role of a topological charge density. By integrating this, one can see that the flux (8) plays an important role in the theory since it gives the topological charge of the system.

The equations of motion (2a) and (2b) with the ansatz (5a) and (5b) becomeMoreover, the components of the energy momentum tensor with the ansatz take the form

As was shown in [66], the stability against contractions and dilatations in the solutions requires the stressless condition. By setting , we get the first-order equationsThe pair of equations for the upper signs are related to the lower signs ones by the change . These equations are compatible with the equations of motion (10a) and (10b) if the potential and the function are constrained byFor , we have , which is the standard case firstly studied in [19]. This constraint shows that generalized models are required to study different potentials and their correspondent vortexlike solutions from the ones of the standard case. The first-order equations (12) also gives rise to the possibility of introducing an auxiliary function in the formso the energy density is written asBy integrating it all over the space, we get the energyThus, the energy of the stressless solutions may be calculated without knowing their explicit form. Below, by properly choosing and that satisfy the constraint in (13), we show new models that engender a set of minima of the potential at infinity. Thus, we have in (6). In order to prepare the model for numerical investigation, we work with dimensionless fields and consider unit vorticity, , which requires the upper signs in the first-order equations (12).

##### 2.1. First Model

The first example is given by the pair of functionsThe above potential does not present a vacuum state; that is the reason we call it vacuumless potential. However, since , we see the set of mimima of the potencial is located at infinity, which allows it to support vortex solutions. Its maximum is at , with . In Figure 1 we plot the above functions. We see that , which is the function that controls the kinetic term of the model, behaves similarly to the potential , having a maximum in the origin and its set of minima at infinity.