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We investigate the presence of vortices in generalized Maxwell-Higgs models with a hidden sector. The model engenders

Topological structures appear in high energy physics in a variety of dimensions [

Vortices, in particular, were firstly investigated by Helmholtz in [

They were largely investigated in several different contexts, including the case of generalized models; see [

More recently, in [

Models with the

There are other interesting issues that suggest the study of topological structures in models described by the

Dark matter is one among several problems that cannot be solved inside the Standard Model of particle physics, so the enhancement of the

In order to introduce and investigate the

We work in

In our model, the visible and hidden sectors are coupled through

By varying the action with respect to the fields, we get that the equations of motion associated with the Lagrangian density (

Invariance under spacetime translations

We use (

Before doing that, however, we notice that the presence of the first-order equations (

Another feature of the above procedure, which can be seen from the potential in (

Let us now illustrate our findings with two distinct examples. Before doing so, we can simplify the problem by rescaling the fields as

Considering the hidden sector to be controlled by

We then go on and investigate how the hidden fields modify the visible sector. We take the choice

In Figure

The solutions

The presence of an internal structure in the quantities related to the visible sector motivated us to plot the magnetic field and the energy density in the plane. It can be seen in Figure

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

We now make a modification in the magnetic permeability of the hidden sector and suggest that it is driven by the function

We then investigate how the solutions (

The solutions

This model engenders a distinct behavior from the previous one in a qualitative manner, presenting a hole in magnetic field near the origin. The new features supported by our models motivated us to depict the magnetic field and the energy density for the solutions of (

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

In this work, we studied vortex structures in generalized Maxwell-Higgs models with visible and hidden sectors. The models considered in this work contain two additional functions,

We have chosen a specific form of the potential, which allowed for the Bogomol’nyi procedure to work out and for the presence of first-order differential equations that solve the equations of motion. The procedure has shown that the first-order equations of the hidden sector decouple from the visible one and could be solved independently. In this sense, the hidden sector acts as a source for the solutions in the visible sector. By taking specific forms for the magnetic permeabilities, we have found that the hidden charged scalar field affects the visible vortex configuration, generating an internal structure to it. In the first example, the magnetic field presents an apparent valley outside the origin that seems to connect two separated structures. The effect is less evident in the energy density, although it is also there. In the second example, the vortex presents a magnetic field with a hole around the origin. Surprisingly, the energies and fluxes of the hidden and visible vortices are fixed by the boundary conditions, and they are independent from each other. Moreover, they do not depend on the specific form we choose for the magnetic permeabilities to construct the system. As a particularly interesting result, we could find a specific function

Here we have discarded the coupling between the two electromagnetic strength tensors, so an issue to be further examined would be to add this kind of coupling. Another possibility is to generalize the covariant derivative terms, as suggested before in [

We are also thinking of enlarging the model to accommodate other symmetries, such as the

Another possibility is to use the same

We inform that the calculations in the work are analytical and numerical. The analytical calculations are fully explained in the text of the manuscript. The numerical calculations are also standard and consist in solving differential equations numerically. The numerical and analytical calculations that we did in the manuscript are all well established and can be checked by others with no further difficulties.

The authors declare that they have no conflicts of interest.

We would like to acknowledge the Brazilian agency CNPq for partial financial support. D. Bazeia acknowledges support from Grant 306614/2014-6, L. Losano acknowledges support from Grant 303824/2017-4, M. A. Marques acknowledges support from Grant 155551/2018-3, and R. Menezes acknowledges support from Grant 306504/2018-9.