Table of Contents Author Guidelines Submit a Manuscript
Advances in High Energy Physics
Volume 2018 (2018), Article ID 4281939, 20 pages
https://doi.org/10.1155/2018/4281939
Research Article

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

1Departamento de Física, Universidade Federal do Maranhão, Campus Universitário do Bacanga, 65080-805 São Luís, MA, Brazil
2Coordenação do Curso de Licenciatura em Ciências Naturais, Universidade Federal do Maranhão, Campus III, 65700-000 Bacabal, MA, Brazil
3IMDEA Nanociencia, Calle de Faraday, 9, Cantoblanco, 28049 Madrid, Spain
4IFISUR, 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 SCOAP3.

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 [622]; 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 [2527]. 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 (NLσM) [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 [4045]. 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.

Figure 1: The profiles (a) and (b) come from generalized Maxwell-Higgs model (1) with and . Observe that (indigo lines) represents the usual MH model and the true compacton solution is given by (black lines).
Figure 2: The magnetic field (a) and the BPS energy density (b) come from generalized Maxwell-Higgs model (1) with and . Observe that (indigo lines) represents the usual MH model and the true compacton solution is given by (black lines). The convention for the color of the lines is the same given in Figure 1.
2.2. Maxwell-Higgs Compactons for

By considering the profile , in the limit , the potential (22) acquires the following form: where is the Heaviside function.

The BPS equations (31), in the limit , are written as

The boundary conditions for compacton solutions areThe radial distance is the value where the profile reaches the vacuum value and the gauge field profile becomes null.

The solutions (for ) of the compacton BPS equations (38) provide analytical profiles for the Higgs and gauge field, where the radial distance is given by The magnetic field and BPS energy density of the Maxwell-Higgs compacton are

The numerical solutions (for all model analyzed in the manuscript) were performed using the routines for boundary value problems of the software Maple 2015. We have chosen the upper signals in BPS equations (31). We have fixed , the winding number , and calculated the numerical solutions for some finite values of . The profiles for the Higgs and gauge fields are given in Figure 1 and the correspondent ones for the magnetic field and the self-dual energy density are depicted in Figure 2.

The numerical analysis shows that for sufficiently large but finite values of , the profiles are very similar to compacton solution ones; we have named them Maxwell-Higgs effective compact vortices. The true Maxwell-Higgs compacton is formed when (see black line profiles in Figures 1 and 2).

3. The Born-Infeld-Higgs Case

The Born-Infeld theory [57, 58] is on nonlinear electrodynamics that was introduced to remove the divergence of the electron self-energy. It is the only completely exceptional nonlinear electrodynamics because to the absence of shock waves and birefringence in its propagation properties [59, 60]. Concerning topological defects in the Born-Infeld-Higgs model, vortex solutions were found in [61]. One generalization of BIH model was firstly done in [52] but no self-dual solutions were found. On the other hand, the self-dual or BPS topological vortex solutions were found in a generalized Born-Infeld-Higgs model introduced in [62].

The Lagrangian density of our ()-dimensional theory is written aswhere we go to consider the function given by (16). We have also defined the following functions: The generalized potential , a nonnegative function, inherits its structure from the function , which is restricted by the condition , so . The Born-Infeld parameter provides modified dynamics for both scalar and gauge fields further enriching the family of possible models.

From the action (43) the gauge field equation of motion reads asWe are interested in stationary solutions, so the Gauss law becomesSimilarly to what happens in Maxwell-Higgs model, the field configurations are electrically neutral; therefore we go to work in the gauge .

Consequently, at static regime, in the gauge , the Ampère law is given byand the Higgs field equation reads as In the last two equations reads as

The energy of the system, in static regime and in the gauge , is given byand will be nonnegative whenever the condition is satisfied.

To proceed with the BPS formalism, we use the identities (10) and (17) such that (50) becomes We have introduced the potential given by (22) with the aim of obtaining the term proportional to the magnetic field .

The Bogomol’nyi procedure would be complete if we require the third row in (51) to be null, so we obtainThis provides a relation between the functions , , and . We here clarify that (52) is not arbitrary because, as we will observe later, in the BPS limit it becomes equivalent to the condition on the diagonal components of the energy-momentum tensor : , proposed by Schaposnik and Vega [63] to obtain self-dual configurations.

Then, the condition (52) allows writing the energy (50) in the Bogomol’nyi form,

Under suitable boundary conditions, the integration of the total derivative in (53) gives null contribution to the energy. Hence, it becomes clear that the energy possesses a lower boundwith being the total magnetic flux. Such a lower bound is saturated when the fields satisfy the BPS or self-dual equations

By using the BPS equations in (52) we compute the self-dual potential , This way the second BPS equation (56) becomes By using the BPS equations in (50) we find that the BPS energy density is given by it will be positive-definite .

3.1. Born-Infeld-Higgs Effective Compact Vortices for Finite

In [52] the existence of effective compact vortex solutions was explored but the self-dual ones were not found. In this section we show the existence of such self-dual effective compact solutions in BIH model. Without loss of generality, we perform the study by considering in the Lagrangian density (43).

The searching for vortex solutions is made by means of the vortex Ansatz introduced in (27). Thus, the BPS equations (55) and (56) read as

The behavior of the profiles and when is determined by solving the self-dual equations (60), so we haveSimilarly, the behavior of the profiles for is where , the self-dual mass, is given byit is exactly the same obtained for the generalized MH model analyzed in the previous section.

The BPS energy density for the self-dual vortices reads as it will be positive-definite and finite for .

3.2. Born-Infeld-Higgs Compactons for

In the limit , the BPS equations (60) read aswith the profiles and satisfying the boundary conditions (39).

By solving the BPS compacton equations for the BIH model, we obtain also analytical solutions where the radial distance now is given by

The magnetic field and BPS energy density profiles of the Born-Infeld-Higgs compacton are

In order to compute the numerical solutions we choose the upper signs in (60), , , , and winding number . Similarly to the MH model, the effective compacton behavior appears for sufficiently large values of ; see Figures 3 and 4. The true Born-Infeld-Higgs compactons arising for also are depicted (see black line profiles) in Figures 3 and 4.

Figure 3: The profiles (a) and (b) come from the generalized Born-Infeld-Higgs model (43) with and . Observe that (indigo lines) represents the usual BIH model and the true compacton solution is given by (black lines).
Figure 4: The magnetic field (a) and the BPS energy density (b) come from the generalized Born-Infeld-Higgs model (43) with and . Observe that (indigo lines) represents the usual BIH model and the true compacton solution is given by (black lines). The convention for the color of the lines is the same given in Figure 3.

4. The Chern-Simons-Higgs Case

In this section we apply the same formalism to construct self-dual solutions in the generalized Abelian Chern-Simons-Higgs model. Physics in two spatial dimensions is closely linked to CS theory, which contains theoretical novelties besides practical application in various phenomena of condensed matter, such as the physics of Anyons, and it is related to the fractional quantum Hall effect [64]. It can be found in extensive literature about CS theory that some of the pioneer papers concerning topological and nontopological solutions as well as relativistic and nonrelativistic models can be found in [6574]. Also a close connection between CS theory and supersymmetry exists. This connection was firstly demonstrated in [75], where from supersymmetric extension of CS model the specific potential for the Bogomol’nyi equations was found, which arise naturally.

The generalized Chern-Simons-Higgs model is described by the following Lagrangian density:with the function given by (16). The gauge field equation is to beand the Gauss law reads as It is clear that the electric charge density is whose integration performed via the Gauss law givesAs such it happens in usual CSH model, the electric charge is nonnull and proportional to the magnetic flux, so the field configurations always will be electrically charged.

These are the stationary points of the energy which for the static field configuration reads asFrom the static Gauss law, we obtain the relationwhich substituted in (73) leads to the following expression for the energy:

We now use the identities (10) and (17) in (73) such that the energy becomes

After some manipulation, the energy can be expressed almost in the Bogomol’nyi formWe observe that the Bogomol’nyi procedure will be complete if the term multiplying the magnetic field is a constant; that is, such that a condition allows determining the self-dual potential to beWe can see that, for , the -potential of the usual Chern-Simon-Higgs model is recovered.

Hence, the energy (77) reads as We see that under appropriated boundary conditions the total derivative gives null contribution to the energy. Then, the energy is bounded below by a multiple of the magnetic flux magnitude This bound is saturated by fields satisfying the first-order Bogomol’nyi or self-dual equations [32, 33]In order for the magnetic field to be nonsingular at origin, it is required that .

By using the BPS equation in (75) we find that the energy density is given by it will be positive-definite .

4.1. Chern-Simons-Higgs Effective Compact Vortices for Finite

The BPS equations (82) read as

The behavior of and near the boundaries can be easily determined by solving the self-dual equations (84) around the boundary values (28). Thus, for , the profile functions behave aswhere the constant is determined only numerically.

On the other hand, when they behave as with the constant computed numerically and being the self-dual massWe observe that for , the mass becomes the same one of the usual self-dual Chern-Simons-Higgs bosons.

The BPS energy density for the self-dual vortices is given by and it will be positive-definite and finite for .

4.2. Chern-Simons-Higgs Compactons for

From (84) we obtain the BPS equations for the Chern-Simons-Higgs compactons with the boundary condition given in (39).

By solving the BPS compacton equations for , we obtain the analytic profiles The radial distance is calculated to be The magnetic field and BPS energy density of the Chern-Simons-Higgs compacton are

In order to compute the numerical solutions we choose the upper signs in (84), , , , and winding number . From the numerical analysis, we can see the appearance of the effective compacton behavior for not very large values of such that it is explicitly shown in Figures 5 and 6. An interesting feature of the Chern-Simons-Higgs effective compact vortices is the enhancement of the ring shape (inclusive for ), for increasing values of in the profiles for the magnetic field and the BPS energy density (see Figure 6). The analytic CSH compacton structures appearing in are represented by the black line profiles in Figures 5 and 6.

Figure 5: The profiles (a) and (b) come from the generalized Chern-Simons-Higgs model (69) with . Observe that (indigo lines) represents the usual CSH model and the true compacton solution is given by (vertical black lines representing the -Dirac function).
Figure 6: The magnetic field (a) and the BPS energy density (b) come from the generalized Chern-Simons-Higgs model (69) with . Observe that (indigo lines) represents the usual CSH model and the true compacton solution is given by (vertical black lines representing the -Dirac function). The convention for the color of the lines is the same given in Figure 5.

5. The Maxwell-Chern-Simons-Higgs Case

Electrically charged vortices were first found in the Abelian Higgs model by Paul and Khare [76], where the Chern-Simons term was included in the usual Maxwell-Higgs action. This was an ingenious manner to avoid the temporal gauge, , coupling the electric charge density to the magnetic field. This model has also been generalized by multiplying a dielectric (scalar) function in the Maxwell kinetic term [77, 78] yielding topological and nontopological solutions satisfying a Bogomol’nyi bound.

The generalized Maxwell-Chern-Simons-Higgs model is described by the following Lagrangian density: where the function is given by (16). The gauge field equation reads as the static Gauss law is

Similarly, the equation of motion of the Higgs field is

In the same way of the previous models, we are interested in time-independent soliton solutions that ensure the finiteness of the action (69). These are the stationary points of the energy which for the static field configuration reads as To proceed, we use the identities (10) and (17) such that the energy becomes After some algebraic manipulations, it can be expressed byAt this point, with the purpose for the total energy to have a lower bound proportional to the magnetic field, we chose the potential to bewhere in order for the vacuum expectation value of the Higgs field to be . Hence, the energy (99) reads asUnder appropriated boundary conditions on the fields, the integration of the total derivatives becomes null; then the total energy is bounded below by a multiple of the magnetic flux magnitude

The lower bound (102) is saturated by fields satisfying the first-order Bogomol’nyi or self-dual equations [32, 33]

The condition saturates the two last equations (105) and (106) so the self-dual solutions are obtained by solving the self-dual equationsand the Gauss law

The BPS energy density is

5.1. Maxwell-Chern-Simons-Higgs Effective Compact Vortices for Finite

By considering that and using Ansatz (27), the BPS equations (107) read asand the Gauss law (108) becomes

We analyze the behavior of the profiles and and at boundaries. This way, for , the profiles behave aswith the constants and being determined numerically for every .

The behavior at the origin for provides the boundary condition

On the other hand, for large values of (, they have the Abrikosov-Nielsen-Olesen behavior,with the constant being computed numerically and being the self-dual mass for , we recover self-dual mass of the usual Maxwell-Chern-Simons-Higgs bosons.

In this way we obtain from (116) the boundary condition for when :

The BPS energy density of the self-dual vortices reads as being positive-definite and finite for .

5.2. Maxwell-Chern-Simons-Higgs Compactons for

From (110), the limit provides the BPS equation for the compacton configurations The compacton Gauss law obtained from (111) becomes

The compacton boundary conditions satisfied by the profiles , , and are The radial distance is the value where the profile reaches the vacuum value and the gauge field profile and scalar potential become null.

The system is solved analytically to be The radial distance is computed from where the function is the modified Bessel function of the first kind and order .

The magnetic field and BPS energy density of the Maxwell-Chern-Simons-Higgs compacton are

In order to compute the numerical solutions we choose the upper signs in (110), , , , and winding number . The profiles for the Higgs and gauge fields are given in Figure 7; the correspondent ones for the scalar potential and for the electric field are depicted in Figure 8. We can note again that an effective compact topological defect is formed for large values of . This feature can be seen from the magnetic field and BPS energy density profiles in Figure 9. Similarly to the previous studied models, the analytic MCSH compactons are formed for ; they are represented by the solid black lines in Figures 7, 8, and 9.

Figure 7: The profiles (a) and (b) come from generalized Maxwell-Chern-Simons-Higgs model (93) with and . Observe that (indigo lines) represents the usual MCSH model and the true compacton solution is given by (black lines).
Figure 8: The scalar potential (a) and the electric field (b) come from generalized Maxwell-Chern-Simons-Higgs model (93) with and . Observe that (indigo lines) represents the usual MCSH model and the true compacton solution is given by (black lines). The convention for the color of the lines is the same given in Figure 7.
Figure 9: The magnetic field (a) and the BPS energy density (b) come from generalized Maxwell-Chern-Simons-Higgs model (93) with and . Observe that (indigo lines) represents the usual MCSH model and the true compacton solution is given by (black lines). The convention for the color of the lines is the same given in Figure 7.

6. Remarks and Conclusions

We have found some generalized Abelian Higgs models whose BPS or self-dual equations give origin to both effective compact solutions and true compacton configurations. Our goal was obtained by means of a consistent implementation of the BPS formalism which besides providing the self-dual or BPS equations has also allowed finding the explicit form of the generalizing function (see (16)) which is parameterized by the positive parameter . Such a parameter determines explicitly new families of self-dual potential for every model and consequently characterizes their self-dual configurations. We draw attention to the importance to obtain self-dual effective compact and analyze true compacton configurations in Abelian Higgs. These models enhance the space of self-dual solutions which probably will imply interesting applications in physics and mathematics, for example, the construction of the respective supersymmetric extensions [7982].

For every model, we have studied the vortex solutions arising from the respective self-dual equations. The numerical analysis has shown that, for sufficiently large values of , the profiles (of the Higgs field, gauge field, magnetic field, and BPS energy density) are very similar to the ones of an effective compacton solution but still preserve a tail in their asymptotic decay. For every model, we have also analyzed the limit for arbitrary winding number . Our analysis has shown that, for , the analytical compacton structures arise in all models (see black line profiles in all figures in the manuscript).

Finally, we are considering the interesting challenge of looking for effective compact structures in gauge field models which engender monopoles or skyrmions, for example. Advances in this direction will be reported elsewhere.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

Rodolfo Casana and G. Lazar acknowledge CNPq, CAPES, and FAPEMA (Brazilian agencies) for their financial support. Lucas Sourrouille is supported by CONICET.

References

  1. C. Armendariz-Picon, T. Damour, and V. Mukhanov, “-inflation,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 458, no. 2-3, pp. 209–218, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  2. C. Armendariz-Picon and E. A. Lim, “Haloes of k-essence,” Journal of Cosmology and Astroparticle Physics, vol. 2005, no. 8, article 007, 2005. View at Publisher · View at Google Scholar
  3. V. Mukhanov and A. Vikman, “Enhancing the tensor-to-scalar ratio in simple inflation,” Journal of Cosmology and Astroparticle Physics, no. 02, p. 004, 2005. View at Google Scholar
  4. A. Sen, “Tachyon matter,” Journal of High Energy Physics, vol. 7, article 065, 2002. View at Publisher · View at Google Scholar
  5. N. Arkani-Hamed, H. C. Cheng, M. A. Luty, and S. Mukohyama, “Ghost condensation and a consistent infrared modification of gravity,” Journal of High Energy Physics, vol. 2004, no. 5, article 74, 2004. View at Publisher · View at Google Scholar
  6. D. Bazeia, E. da Hora, R. Menezes, H. P. de Oliveira, and C. dos Santos, “Generalized self-dual Chern-Simons vortices,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 81, no. 12, Article ID 125014, 2010. View at Publisher · View at Google Scholar · View at Scopus
  7. D. Bazeia, E. da Hora, R. Menezes, H. P. de Oliveira, and C. dos Santos, “Compactlike kinks and vortices in generalized models,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 81, no. 12, Article ID 125016, 2010. View at Publisher · View at Google Scholar · View at Scopus
  8. C. dos Santos and E. da Hora, “Domain walls in a generalized Chern-Simons model,” The European Physical Journal C, vol. 70, no. 4, pp. 1145–1151, 2010. View at Publisher · View at Google Scholar · View at Scopus
  9. C. dos Santos and E. da Hora, “Lump-like solitons in a generalized Abelian-Higgs Chern-Simons model,” The European Physical Journal C, vol. 71, p. 1519, 2011. View at Google Scholar
  10. C. dos Santos, “Compact solitons in an Abelian-Higgs Chern-Simons model,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 82, Article ID 125009, 2010. View at Publisher · View at Google Scholar
  11. D. Bazeia, E. da Hora, C. dos Santos, and R. Menezes, “BPS solutions to a generalized Maxwell-Higgs model,” The European Physical Journal C, vol. 71, no. 12, pp. 1–9, 2011. View at Publisher · View at Google Scholar · View at Scopus
  12. R. Casana, M. M. Ferreira Jr., and E. da Hora, “Generalized BPS magnetic monopoles,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 86, no. 8, Article ID 085034, 2012. View at Publisher · View at Google Scholar
  13. E. Babichev, “Global topological -defects,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 74, no. 8, Article ID 085004, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  14. E. Babichev, “Gauge k-vortices,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 77, Article ID 065021, 2008. View at Publisher · View at Google Scholar
  15. C. Adam, J. Sanchez-Guillen, and A. Wereszczynski, “-defects as compactons,” Journal of Physics A: Mathematical and General, vol. 40, no. 45, pp. 13625–13643, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  16. C. Adam, J. Sanchez-Guillen, and A. Wereszczynski, “Corrigendum: -defects as compactons [MR2387309],” Journal of Physics A: Mathematical and General, vol. 42, no. 8, 089801, 5 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  17. C. Adam, N. Grandi, J. Sanchez-Guillen, and A. Wereszczynski, “ fields, compactons and thick branes,” Journal of Physics A: Mathematical and General, vol. 41, no. 21, 212004, 7 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  18. C. Adam, N. Grandi, J. Sanchez-Guillen, and A. Wereszczynski, “Corrigendum: fields, compactons and thick branes [MR2442308],” Journal of Physics A: Mathematical and General, vol. 42, no. 15, 159801, 1 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  19. C. Adam, N. Grandi, P. Klimas, J. Sánchez-Guillén, and A. Wereszczyński, “Compact self-gravitating solutions of quartic fields in brane cosmology,” Journal of Physics A: Mathematical and Theoretical, vol. 41, no. 37, Article ID 375401, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  20. C. Adam, P. Klimas, J. Sánchez-Guillén, and A. Wereszczyński, “Compact gauge vortices,” Journal of Physics A: Mathematical and Theoretical, vol. 42, no. 13, Article ID 135401, 19 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  21. R. Casana and L. Sourrouille, “Self-dual soliton solutions in a Chern-Simons-CP(1) model with a nonstandard kinetic term,” Modern Physics Letters A, vol. 29, no. 23, Article ID 1450124, 1450124, 11 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  22. L. Sourrouille, “Self-dual soliton solution in a generalized Jackiw-Pi model,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 86, Article ID 085014, 2012. View at Publisher · View at Google Scholar
  23. D. Bazeia, L. Losano, R. Menezes, and J. C. R. E. Oliveira, “Generalized global defect solutions,” The European Physical Journal C, vol. 51, no. 4, pp. 953–962, 2007. View at Publisher · View at Google Scholar · View at Scopus
  24. H. B. Nielsen and P. Olesen, “Vortex-line models for dual strings,” Nuclear Physics B, vol. 61, pp. 45–61, 1973. View at Publisher · View at Google Scholar · View at Scopus
  25. P.-O. Jubert, R. Allenspach, and A. Bischof, “Magnetic domain walls in constrained geometries,” Physical Review B: Condensed Matter and Materials Physics, vol. 69, no. 22, Article ID 220410, pp. 1–220410, 2004. View at Publisher · View at Google Scholar · View at Scopus
  26. A. Fert, V. Cros, and J. Sampaio, “Skyrmions on the track,” Nature Nanotechnology, vol. 8, no. 3, pp. 152–156, 2013. View at Publisher · View at Google Scholar · View at Scopus
  27. N. Romming, C. Hanneken, M. Menzel et al., “Writing and deleting single magnetic skyrmions,” Science, vol. 341, no. 6146, pp. 636–639, 2013. View at Publisher · View at Google Scholar · View at Scopus
  28. C. Adam, J. Sanchez-Guillen, and A. Wereszczynski, “A Skyrme-type proposal for baryonic matter,” Physics Letters B, vol. 691, p. 105, 2010. View at Google Scholar
  29. B. Hartmann, B. Kleihaus, J. Kunz, and I. Schaffer, “Compact boson stars,” Physics Letters B, vol. 714, no. 1, pp. 120–126, 2012. View at Publisher · View at Google Scholar · View at Scopus
  30. J. M. Speight, “Compactons and semi-compactons in the extreme baby Skyrme model,” Journal of Physics A: Mathematical and General, vol. 43, no. 40, 405201, 16 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  31. C. Adam, T. Romańczukiewicz, J. Sánchez-Guillén, and A. Wereszczyński, “Investigation of restricted baby Skyrme models,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 81, no. 8, 2010. View at Publisher · View at Google Scholar
  32. E. B. Bolgomol'nyi, “The stability of classical solutions,” Soviet Journal of Nuclear Physics, vol. 24, pp. 449–454, 1976. View at Google Scholar
  33. M. Prasad and C. Sommerfield, “Exact classical solution for the 't Hooft Monopole and the Julia-Zee Dyon,” Physical Review Letters, vol. 35, no. 12, p. 760, 1975. View at Publisher · View at Google Scholar
  34. R. Jackiw and E. J. Weinberg, “Self-dual Chern-Simons vortices,” Physical Review Letters, vol. 64, no. 19, pp. 2234–2237, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  35. R. Jackiw, K.-M. Lee, and E. J. Weinberg, “Self-dual Chern-Simons solitons,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 42, no. 10, pp. 3488–3499, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  36. C. Lee, K. Lee, and H. Min, “Self-dual Maxwell Chern-Simons solitons,” Physics Letters B, vol. 252, no. 1, pp. 79–83, 1990. View at Publisher · View at Google Scholar · View at Scopus
  37. A. A. Belavin and A. M. Polyakov, “Metastable states of two-dimensional isotropic ferromagnets,” JETP Letters, vol. 22, p. 245, 1975. View at Google Scholar
  38. R. Rajaraman, Solitons and Instantons, North-Holland, Amsterdam, 1982.
  39. W. J. Zakrzewski, Low-Dimensional Sigma Models, Hilger, Bristol, UK, 1989. View at MathSciNet
  40. B. J. Schroers, “Bogomol'nyi solitons in a gauged O(3)sigma model,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 356, no. 2-3, pp. 291–296, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  41. G. Nardelli, “Magnetic vortices from a nonlinear sigma model with local symmetry,” Physical Review Letters, vol. 73, no. 19, pp. 2524–2527, 1994. View at Publisher · View at Google Scholar · View at Scopus
  42. M. Arai, M. Naganuma, M. Nitta, and N. Sakai, “Manifest supersymmetry for BPS walls in nonlinear sigma models,” Nuclear Physics. B. Theoretical, Phenomenological, and Experimental High Energy Physics. Quantum Field Theory and Statistical Systems, vol. 652, no. 1-3, pp. 35–71, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  43. J. M. Baptista, “Vortex equations in Abelian gauged sigma-models,” Communications in Mathematical Physics, vol. 261, no. 1, pp. 161–194, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  44. A. Alonso-Izquierdo, W. G. Fuertes, and J. M. Guilarte, “Two species of vortices in massive gauged non-linear sigma models,” Journal of High Energy Physics, vol. 2015, no. 2, 2015. View at Publisher · View at Google Scholar
  45. P. Mukherjee, “Magnetic vortices in a gauged O(3) sigma model with symmetry breaking self-interaction,” Physical Review D, vol. 58, p. 105025, 1998. View at Google Scholar
  46. P. K. Ghosh and S. K. Ghosh, “Topological and nontopological solitons in a gauged O(3) sigma model with Chern-Simons term,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 366, no. 1-4, pp. 199–204, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  47. P. Mukherjee, “On the question of degeneracy of topological solitons in a gauged O(3) non-linear sigma model with Chern-Simons term,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 403, no. 1-2, pp. 70–74, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  48. K. Kimm, K. Lee, and T. Lee, “Anyonic Bogomol’nyi solitons in a gauged O(3) sigma model,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 53, no. 8, pp. 4436–4440, 1996. View at Publisher · View at Google Scholar · View at Scopus
  49. J. Han and H.-S. Nam, “On the topological multivortex solutions of the self-dual Maxwell-Chern-Simons gauged O(3) sigma model,” Letters in Mathematical Physics, vol. 73, no. 1, pp. 17–31, 2005. View at Publisher · View at Google Scholar · View at Scopus
  50. D. Bazeia, E. da Hora, R. Menezes, H. P. de Oliveira, and C. dos Santos, “Compactlike kinks and vortices in generalized models,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 81, Article ID 125016, 2010. View at Publisher · View at Google Scholar
  51. D. Bazeia, L. Losano, M. A. Marques, R. Menezes, and I. Zafalan, “Compact vortices,” The European Physical Journal C, vol. 77, no. 2, article no. 63, 2017. View at Publisher · View at Google Scholar · View at Scopus
  52. D. Bazeia, E. da Hora, and D. Rubiera-Garcia, “Compact vortexlike solutions in a generalized Born-Infeld model,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 84, no. 12, Article ID 125005, 2011. View at Publisher · View at Google Scholar
  53. R. Casana and L. Sourrouille, “Self-Dual Configurations in a Generalized Abelian Chern-Simons-Higgs Model with Explicit Breaking of the Lorentz Covariance,” Advances in High Energy Physics, vol. 2016, Article ID 5315649, 2016. View at Publisher · View at Google Scholar · View at Scopus
  54. A. A. Abrikosov, “On the magnetic properties of superconductors of the second group,” Soviet Physics—JETP, vol. 5, p. 1174, 1957. View at Google Scholar
  55. A. Vilenkin and E. P. Shellard, Cosmic Strings and Other Topological Defects, Cambridge University Press, Cambridge, UK, 1994. View at MathSciNet
  56. R. Casana, M. M. Ferreira, E. Da Hora, and C. Dos Santos, “Analytical BPS Maxwell-Higgs vortices,” Advances in High Energy Physics, vol. 2014, Article ID 210929, 2014. View at Publisher · View at Google Scholar · View at Scopus
  57. B. Born and L. Infeld, “Foundations of the new field theory,” Proceedings of the Royal Society of London A, vol. 144, p. 425, 1935. View at Google Scholar
  58. P. A. M. Dirac, “An extensible model of the electron,” Proceedings of the Royal Society of London A, vol. 268, p. 57, 1962. View at Google Scholar
  59. G. Boillat, “Nonlinear electrodynamics: lagrangians and equations of motion,” Journal of Mathematical Physics, vol. 11, no. 3, pp. 941–951, 1970. View at Publisher · View at Google Scholar · View at Scopus
  60. G. W. Gibbons, “Born-Infeld particles and Dirichlet -branes,” Nuclear Physics. B. Theoretical, Phenomenological, and Experimental High Energy Physics. Quantum Field Theory and Statistical Systems, vol. 514, no. 3, pp. 603–639, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  61. K. Shiraishi and S. Hirenzaki, “Bogomol’nyi equations for vortices in Born-Infeld Higgs systems,” International Journal of Modern Physics A, vol. 6, no. 15, pp. 2635–2647, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  62. R. Casana, E. D. da Hora, D. Rubiera-Garcia, and C. D. Santos, “Topological vortices in generalized Born–Infeld–Higgs electrodynamics,” The European Physical Journal C, vol. 75, no. 8, article no. 380, 2015. View at Publisher · View at Google Scholar · View at Scopus
  63. H. J. de Vega and F. A. Schaposnik, “Classical vortex solution of the Abelian Higgs model,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 14, no. 4, pp. 1100–1106, 1976. View at Publisher · View at Google Scholar
  64. Z. F. Ezawa, Quantum Hall effects, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, USA, 2nd edition, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  65. R. Jackiw and S.-Y. Pi, “Self-dual Chern-Simons solitons,” Progress of Theoretical Physics Supplement, vol. 107, pp. 1–40, 1992. View at Google Scholar
  66. V. D. Gerald, “Self-dual Chern-Simons theories,” Lecture Notes in Physics Monographs, vol. 36, 1995. View at Google Scholar
  67. V. D. Gerald, “Aspects of Chern-Simons theory,” https://arxiv.org/abs/hep-th/9902115.
  68. F. A. Schaposnik, “Vortices,” https://arxiv.org/abs/hep-th/0611028.
  69. P. A. Horvathy and P. Zhang, “Vortices in (abelian) Chern-Simons gauge theory,” Physics Reports, vol. 481, no. 5-6, pp. 83–142, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  70. J. Hong, Y. Kim, and P. Y. Pac, “Multivortex solutions of the abelian Chern-Simons-Higgs theory,” Physical Review Letters, vol. 64, no. 19, pp. 2230–2233, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  71. R. Jackiw and S.-Y. Pi, “Soliton solutions to the gauged nonlinear Schrödinger equation on the plane,” Physical Review Letters, vol. 64, no. 25, pp. 2969–2972, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  72. R. Jackiw and S.-Y. Pi, “Classical and quantal nonrelativistic Chern-Simons theory,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 42, no. 10, pp. 3500–3513, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  73. R. Jackiw and S.-Y. Pi, “Erratum: Classical and quantal nonrelativistic Chern-Simons theory (Physical Review D (1993) 48, 8, (3929)),” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 48, no. 8, p. 3929, 1993. View at Publisher · View at Google Scholar · View at Scopus
  74. R. Jackiw, K. Lee, and E. J. Weinberg, “Self-dual Chern-Simons solitons,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 42, no. 10, pp. 3488–3499, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  75. C. Lee, K. Lee, and E. J. Weinberg, “Supersymmetry and self-dual Chern-Simons systems,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 243, no. 1-2, pp. 105–108, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  76. S. K. Paul and A. Khare, “Self-dual factorization of the Proca equation with Chern-Simons term in 4K - 1 dimensions,” Physics Letters B, vol. 171, no. 2-3, pp. 244–246, 1986. View at Publisher · View at Google Scholar · View at Scopus
  77. P. K. Ghosh, “Bogomol'nyi equations of Maxwell-Chern-Simons vortices from a generalized Abelian Higgs model,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 49, no. 10, pp. 5458–5468, 1994. View at Publisher · View at Google Scholar · View at Scopus
  78. D. Bazeia, R. Casana, E. da Hora, and R. Menezes, “Generalized self-dual Maxwell-Chern-Simons-Higgs model,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 85, no. 12, Article ID 125028, 2012. View at Publisher · View at Google Scholar · View at Scopus
  79. E. Witten and D. Olive, “Supersymmetry algebras that include topological charges,” Physics Letters B, vol. 78, no. 1, pp. 97–101, 1978. View at Publisher · View at Google Scholar
  80. J. Edelstein, C. Núnez, and F. Schaposnik, “Supersymmetry and Bogomol'nyi equations in the Abelian Higgs model,” Physics Letters B, vol. 329, no. 1, pp. 39–45, 1994. View at Publisher · View at Google Scholar · View at Scopus
  81. W. G. Fuertes and J. M. Guilarte, “Self-dual solitons in supersymmetric Chern-Simons gauge theory,” Journal of Mathematical Physics, vol. 38, no. 12, pp. 6214–6229, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  82. Y. Isozumi, M. Nitta, K. Ohashi, and N. Sakai, “All exact solutions of a Bogomol'nyi-Prasad-Sommerfield equation,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 71, no. 6, 065018, 6 pages, 2005. View at Publisher · View at Google Scholar · View at MathSciNet