^{1}

^{3}.

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.

In high energy physics, topological structures appear in a diversity of contexts and have been vastly studied over the years [

The simplest structures are kinks, which appear in

Potentials with extrema at infinity, similar to the ones we are going to study here, although inverted, also appear in classical mechanics [

By working in

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 [

The importance of vortices in high energy physics and in other areas of physics can be found in [

Topological structures may be studied with generalized models [

Noncanonical models considering defect structures were severely investigated over the years [

This work deals with a class of generalized Maxwell and Chern-Simons models that support vortex solutions in vacuumless systems. In Section

We deal with an action in

The equations of motion (

As was shown in [

The first example is given by the pair of functions

The function

For this model, the first-order equations (

Although (

The solutions

Before going further, we calculate the function

The magnetic field (left) and the energy density (right) for the solutions of (

Our second model arises from the pair of functions

The function

In this case, the first-order equations (

The solutions

In this case,

The magnetic field in (

In order to investigate the presence of vortices with the Chern-Simons dynamics, we consider the action

To start the investigation with the Chern-Simons dynamics, we consider the same

The function

The first-order equations (

The functions

We now turn our attention to the auxiliar function

The electric field (upper left), the magnetic field (upper right), the temporal gauge field component (bottom left), and the energy density (bottom right) for the solutions of (

We now present a new model, given by the functions

The function

To calculate our solutions, we consider the first-order equations (

The functions

In this case, the function

The electric field (upper left), the magnetic field (upper right), the temporal gauge component (bottom left), and the energy density (bottom right) for the solutions of (

In this work, we have investigated vortices in vacuumless systems with Maxwell and Chern-Simons dynamics. In both scenarios, we have studied the properties of the generalized models in the classes (

The behaviors of the potentials are different at

An interesting result is that the vortex solutions in vacuumless systems present a large tail that extends far away from the origin. The scalar field is asymptotically divergent and has infinite amplitude. Then, the solutions lose the locality. However the electric field, if it exists, the magnetic field, and the energy density are localized. This avoids the possibility of having infinite energies and fluxes. The flux is well defined and still works as a topological invariant. Unlike the kinks, we concluded that vortices in vacuumless systems do not require any special definition of the topological current to study its topological character.

We then discovered vortices with a new behavior, whose solutions present a long tail. We hope these results encourage new research in the area, stimulating the study of new models in this and other contexts. One can follow the direction of [

The data used to support the findings of this study are included within the article.

The author declares that there are no conflicts of interest.

We would like to thank Dionisio Bazeia and Roberto Menezes for the discussions that have contributed to this work. We would also like to acknowledge the Brazilian agency CNPq, research project 155551/2018-3, for the financial support.