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

Authors: Rodolfo Casana; Guillermo Lazar; Lucas Sourrouille
Journal: Advances in High Energy Physics
 (2018)

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## 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 specificAnsatz has led us to the explicit determination for the dielectric function associated with the Higgs sector to be proportional to λϕ2λ-2, λ>1. 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 λ, every model supports effective compact vortices. The true compacton solutions arising for λ=∞ 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.


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## Body

## 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 ask-theories arose initially as effective cosmological models for inflationary evolution [1]. Later the k-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 k-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 (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 O(3) 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 O(3) sigma model with both Maxwell and the Chern-Simons terms was studied in [48, 49].The existence of vortex solutions with compact-like profiles ink-generalized Abelian Maxwell-Higgs model was studied in [50, 51] and in k-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 ω(ϕ), composing the generalized term ω(ϕ)Dμϕ2, which only can be proportional to λϕ2λ-2 for λ>1 (a similar result was obtained in [53]). Secondly, we use the usual vortexAnsatz 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 λ and only in the limit λ→∞ 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 He4. 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:(1)L=-Gϕ4FμνFμν+ωϕDμϕ2-Vϕ.The nonstandard dynamics are introduced by two nonnegative functions G(ϕ) and ω(ϕ) depending of the Higgs field. The Greek index runs from 0 to 2. The vector Aμ is the electromagnetic field, the Maxwell strength tensor is Fμν=∂μAν-∂νAμ, and Dμϕ defines the covariant derivative of the Higgs field ϕ;(2)Dμϕ=∂μϕ-ieAμϕ.The function V(ϕ) is a self-interacting scalar potential.From (1), the gauge field equation reads(3)∂νGFνμ=eωJμ,where ωJμ is the conserved current density, that is, ∂μ(ωJμ)=0, and Jμ is the usual current density(4)Jμ=iϕDμϕ∗-ϕ∗Dμϕ.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 law(5)∂kG∂kA0=2e2ωA0ϕ2and the respective Ampère law(6)ϵkj∂jGB=eωJk.It is clear from the Gauss law thateωJ0 stands for the electric charge density, so that the total electric charge of the configurations is(7)Q=2e2∫d2xωA0ϕ2 which is shown to be null (Q=0) by integration of the Gauss law under suitable boundary conditions for the fields at infinity; that is, A0→0, ϕ→cte, and G(ϕ) 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,A0=0, which satisfies identically the Gauss law (5). With the choice A0=0, the static and electrically neutral configurations are described by the Ampère law (6) and the reduced equation for the Higgs field(8)DkωDkϕ-12B2∂G∂ϕ∗-∂ω∂ϕ∗Dkϕ2-∂V∂ϕ∗=0.To implement the BPS formalism, we first establish the energy for the static field configuration in the gaugeA0=0, so it reads(9)E=∫d2xG2B2+ωDkϕ2+V.To proceed, we need the fundamental identity(10)Diϕ2=D±ϕ2±eBϕ2±12ϵik∂iJk,where D±ϕ=D1ϕ±iD2ϕ. With it, the energy (9) is written as being(11)E=∫d2xG2B2+Vϕ+ωD±ϕ2±eωBϕ2±12ωϵik∂iJk.We observe that the termωϵik∂iJk 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 ofω(ϕ) 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 ω(ϕ). The starting point is the following expression:(12)ωϵik∂iJk=ϵik∂iωJk-ϵik∂iωJk.By considering ω to be an explicit function of ϕ2, after some algebraic manipulations, the last term ϵik(∂iω)Jk becomes expressed as(13)ϵik∂iωJk=ϕ2∂ω∂ϕ2ϵik∂iJk+2eBϕ4∂ω∂ϕ2,which, after being substituted in (12), allows obtaining(14)ω+ϕ2∂ω∂ϕ2ϵik∂iJk=ϵik∂iωJk-2eBϕ4∂ω∂ϕ2.At this point, we establish the function ω to satisfy the following condition:(15)ω+ϕ2∂ω∂ϕ2=λω,λ>0,whose solutions provide the explicit functional form of ω(ϕ) to be(16)ωϕ=λϕ2λ-2v2λ-2,guaranteeing that the vacuum expectation value of the Higgs field is ϕ=v.With the key condition (15), (14) allows expressing the term ωϵik∂iJk in the following way:(17)ωϵik∂iJk=1λϵik∂iωJk-2ev2λ-1ϕ2λv2λB.By putting the expression (17) in the energy (11), it becomes(18)E=∫d2xG2B2+Vϕ+ωD±ϕ2±ev2ϕ2λv2λB±12λϵik∂iωJk. We now manipulate the two first terms in such a form that the energy can be written as(19)E=∫d2xG2B∓2VG2+ωD±ϕ2±B2GV+ev2ϕ2λv2λ±12λϵik∂iωJk.With 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,(20)2GV+ev2ϕ2λv2λ=ev2. Consequently, the self-dual potential V(ϕ) becomes(21)Vϕ=1GUλϕ,where we have defined the potential Uλ(ϕ) given by(22)Uλϕ=e2v421-ϕ2λv2λ2.We can note that, for λ=1, it becomes the self-dual potential of the Maxwell-Higgs model.Hence, the energy (77) reads as(23)E=∫d2x±ev2B+ωD±ϕ2+G2B∓2UλG2±12λϵik∂iωJk. 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,(24)E≥±ev2∫d2xB=ev2Φ,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](25)D±ϕ=0,B=±ev2G1-ϕ2λv2λ.The function G(ϕ) must be a function providing a finite magnetic field such that B(x→→∞)→0 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 configurations(26)εBPS=2UλG+λϕ2λ-2v2λ-2Dkϕ2;it will be finite and positive-definite for λ>0. We here also require the function G(ϕ) yielding a finite BPS energy density such that εBPS(x→→∞)→0 is sufficiently rapid to provide a finite total energy.


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

In the following, with loss of generality, we have chosenG(ϕ)=1 (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 ω(ϕ) 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 vortexAnsatz [24](27)ϕr,θ=vgreinθ,Aθr=-ar-ner,with n=±1,±2,±3,… standing for the winding number of the vortex solutions.The profilesg(r) and a(r) are regular functions describing solutions possessing finite energy and obeying the boundary conditions,(28)g0=0,a0=n,g∞=1,a∞=0.UnderAnsatz (27), the magnetic field reads as(29)Br=-1erdadr.The correspondent quantized magnetic flux is given by(30)Φ=∫d2xB=2πen,as expected.The BPS equations (25) are written as(31)g′=±agr,-a′r=±e2v21-g2λ.The upper (lower) signal corresponds to the vortex (antivortex) solution with winding number n>0 (n<0).The self-dual energy density (26) is expressed by(32)εBPS=e2v41-g2λ2+2λv2g2λ-2agr2.It will be finite and positive-definite for λ≥1.The total energy of the self-dual solutions is given by the lower bound (24),(33)EBPS=±ev2ΦB=±2πv2n;it is proportional to the winding number of the vortex solution, as expected.The behavior ofg(r) and a(r) near the boundaries can be easily determined by solving the self-dual equations (31) around the boundary conditions (28). Then, for r→0, the profiles behave as(34)gr≈Cnrn+⋯ar≈n-e2v22r2+⋯,where the constant Cn>0 is computed numerically.On the other hand, whenr→∞, they behave as(35)gr≈1-C∞re-mr,ar≈mC∞re-mr; the constant C∞ is determined numerically and m, the self-dual mass, is given by(36)m=ev2λ,reminding us that ev2 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 λ observed in Figures 1 and 2.Figure 1
The profilesg(r) (a) and a(r) (b) come from generalized Maxwell-Higgs model (1) with G(g)=1 and ω(g)=λg2λ-2. Observe that λ=1 (indigo lines) represents the usual MH model and the true compacton solution is given by λ=∞ (black lines).
(a) (b)Figure 2
The magnetic fieldB(r) (a) and the BPS energy density εBPS(r) (b) come from generalized Maxwell-Higgs model (1) with G(g)=1 and ω(g)=λg2λ-2. Observe that λ=1 (indigo lines) represents the usual MH model and the true compacton solution is given by λ=∞ (black lines). The convention for the color of the lines is the same given in Figure 1.
(a) (b)


### 2.2. Maxwell-Higgs Compactons forλ=∞

By considering the profile0≤g(r)<1, in the limit λ→∞, the potential (22) acquires the following form:(37)U∞g=e2v42Θ1-g, where Θ(1-g) is the Heaviside function.The BPS equations (31), in the limit λ→∞, are written as(38)g′=±agr,-a′r=±e2v2Θ1-g.The boundary conditions for compacton solutions are(39)g0=0,a0=n,gr=1,ar=0rc≤r<∞.The radial distance rc<∞ is the value where the profile g(r) reaches the vacuum value and the gauge field profile a(r) becomes null.The solutions (forn>0) of the compacton BPS equations (38) provide analytical profiles for the Higgs and gauge field,(40)g∞r=rrcnexp⁡n21-r2rc2Θrc-r+Θr-rc,a∞r=n1-r2rc2Θrc-r, where the radial distance rc is given by(41)rc=2nev. The magnetic field and BPS energy density of the Maxwell-Higgs compacton are(42)B∞r=ev2Θrc-r,εBPS∞r=e2v4Θrc-r.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 e=v=1, the winding number n=1, and calculated the numerical solutions for some finite values of λ. 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 ofλ, 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 λ=∞ (see black line profiles in Figures 1 and 2).


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

In the following, with loss of generality, we have chosenG(ϕ)=1 (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 ω(ϕ) 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 vortexAnsatz [24](27)ϕr,θ=vgreinθ,Aθr=-ar-ner,with n=±1,±2,±3,… standing for the winding number of the vortex solutions.The profilesg(r) and a(r) are regular functions describing solutions possessing finite energy and obeying the boundary conditions,(28)g0=0,a0=n,g∞=1,a∞=0.UnderAnsatz (27), the magnetic field reads as(29)Br=-1erdadr.The correspondent quantized magnetic flux is given by(30)Φ=∫d2xB=2πen,as expected.The BPS equations (25) are written as(31)g′=±agr,-a′r=±e2v21-g2λ.The upper (lower) signal corresponds to the vortex (antivortex) solution with winding number n>0 (n<0).The self-dual energy density (26) is expressed by(32)εBPS=e2v41-g2λ2+2λv2g2λ-2agr2.It will be finite and positive-definite for λ≥1.The total energy of the self-dual solutions is given by the lower bound (24),(33)EBPS=±ev2ΦB=±2πv2n;it is proportional to the winding number of the vortex solution, as expected.The behavior ofg(r) and a(r) near the boundaries can be easily determined by solving the self-dual equations (31) around the boundary conditions (28). Then, for r→0, the profiles behave as(34)gr≈Cnrn+⋯ar≈n-e2v22r2+⋯,where the constant Cn>0 is computed numerically.On the other hand, whenr→∞, they behave as(35)gr≈1-C∞re-mr,ar≈mC∞re-mr; the constant C∞ is determined numerically and m, the self-dual mass, is given by(36)m=ev2λ,reminding us that ev2 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 λ observed in Figures 1 and 2.Figure 1
The profilesg(r) (a) and a(r) (b) come from generalized Maxwell-Higgs model (1) with G(g)=1 and ω(g)=λg2λ-2. Observe that λ=1 (indigo lines) represents the usual MH model and the true compacton solution is given by λ=∞ (black lines).
(a) (b)Figure 2
The magnetic fieldB(r) (a) and the BPS energy density εBPS(r) (b) come from generalized Maxwell-Higgs model (1) with G(g)=1 and ω(g)=λg2λ-2. Observe that λ=1 (indigo lines) represents the usual MH model and the true compacton solution is given by λ=∞ (black lines). The convention for the color of the lines is the same given in Figure 1.
(a) (b)


## 2.2. Maxwell-Higgs Compactons forλ=∞

By considering the profile0≤g(r)<1, in the limit λ→∞, the potential (22) acquires the following form:(37)U∞g=e2v42Θ1-g, where Θ(1-g) is the Heaviside function.The BPS equations (31), in the limit λ→∞, are written as(38)g′=±agr,-a′r=±e2v2Θ1-g.The boundary conditions for compacton solutions are(39)g0=0,a0=n,gr=1,ar=0rc≤r<∞.The radial distance rc<∞ is the value where the profile g(r) reaches the vacuum value and the gauge field profile a(r) becomes null.The solutions (forn>0) of the compacton BPS equations (38) provide analytical profiles for the Higgs and gauge field,(40)g∞r=rrcnexp⁡n21-r2rc2Θrc-r+Θr-rc,a∞r=n1-r2rc2Θrc-r, where the radial distance rc is given by(41)rc=2nev. The magnetic field and BPS energy density of the Maxwell-Higgs compacton are(42)B∞r=ev2Θrc-r,εBPS∞r=e2v4Θrc-r.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 e=v=1, the winding number n=1, and calculated the numerical solutions for some finite values of λ. 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 ofλ, 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 λ=∞ (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 (2+1)-dimensional theory is written as(43)L=β21-R+ωϕDμϕ2-Wϕ,where we go to consider the function ωϕ given by (16). We have also defined the following functions:(44)R=1+Gϕ2β2FμνFμν,Wϕ=β21-Vϕ. The generalized potential Wϕ, a nonnegative function, inherits its structure from the function V(ϕ), which is restricted by the condition 0<V(ϕ)≤1, so W(ϕ)>0. The Born-Infeld parameter β 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 as(45)∂νGRFνμ=eωJμ.We are interested in stationary solutions, so the Gauss law becomes(46)∂jGR∂jA0=2ωe2A0ϕ2.Similarly to what happens in Maxwell-Higgs model, the field configurations are electrically neutral; therefore we go to work in the gauge A0=0.Consequently, at static regime, in the gaugeA0=0, the Ampère law is given by(47)ϵkj∂jGRB-eωJk=0and the Higgs field equation reads as(48)0=ωDjDjϕ+∂jwDjϕ-∂ω∂ϕ∗Djϕ2-B22R∂G∂ϕ∗-∂W∂ϕ∗. In the last two equations R reads as(49)R=1+Gβ2B21/2.The energy of the system, in static regime and in the gaugeA0=0, is given by(50)E=∫d2xβ2R-V+ωDkϕ2and will be nonnegative whenever the condition R≥V is satisfied.To proceed with the BPS formalism, we use the identities (10) and (17) such that (50) becomes(51)E=2π∫d2x±ev2B+ωD±ϕ2±12λϵik∂iωJk+R2GGRB∓2Uλ2+β2R-V-12GB2R-RGUλ. We have introduced the potential Uλ(ϕ) given by (22) with the aim of obtaining the term proportional to the magnetic field ev2B.The Bogomol’nyi procedure would be complete if we require the third row in (51) to be null, so we obtain(52)V+RUλβ2G=12R+12R.This provides a relation between the functions V, G, and R. 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 Tμν: T11+T22=0, 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,(53)E=∫d2xωD±ϕ2+R2GGRB∓2Uλ2±ev2B±12λϵik∂iωJk.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 bound(54)E≥ev2Φ,with Φ being the total magnetic flux. Such a lower bound is saturated when the fields satisfy the BPS or self-dual equations(55)D±ϕ=0,(56)GRB=±ev21-ϕ2λv2λ.By using the BPS equations in (52) we compute the self-dual potential V(ϕ),(57)V=1R=1-2Uλβ2G. This way the second BPS equation (56) becomes(58)B=±ev2GV1-ϕ2λv2λ. By using the BPS equations in (50) we find that the BPS energy density is given by(59)εBPS=2UλGV+λϕ2λ-2v2λ-2Dkϕ2; it will be positive-definite λ>0.


### 3.1. Born-Infeld-Higgs Effective Compact Vortices forλ 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 G(ϕ)=1 in the Lagrangian density (43).The searching for vortex solutions is made by means of the vortexAnsatz introduced in (27). Thus, the BPS equations (55) and (56) read as(60)g′=±agr,-a′r=±e2v21-g2λ1-e2v41-g2λ2/β2.The behavior of the profilesg(r) and a(r) when r→0 is determined by solving the self-dual equations (60), so we have(61)gr≈Cnrn+⋯,ar≈n-e2v2β2β2-e2v4r2+⋯.Similarly, the behavior of the profiles for r→∞ is(62)gr≈1-C∞re-mr,ar≈mC∞re-mr, where m, the self-dual mass, is given by(63)m=ev2λ;it 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(64)εBPS=e2v41-g2λ21-e2v41-g2λ2/β2+2λv2g2λ-2agr2; it will be positive-definite and finite for λ≥1.


### 3.2. Born-Infeld-Higgs Compactons forλ=∞

In the limitλ→∞, the BPS equations (60) read as(65)g′=±agr,-a′r=±e2v2Θ1-g1-e2v4/β2,with the profiles g(r) and a(r) satisfying the boundary conditions (39).By solving the BPS compacton equations for the BIH model, we obtain also analytical solutions(66)g∞r=rrcnexp⁡n21-r2rc2Θrc-r+Θr-rc,a∞r=n1-r2rc2Θrc-r, where the radial distance rc now is given by(67)rc=2nev1-e2v4β21/4.The magnetic field and BPS energy density profiles of the Born-Infeld-Higgs compacton are(68)B∞r=ev21-e2v4β2-1/2Θrc-r,εBPS∞r=e2v41-e2v4β2-1/2Θrc-r.In order to compute the numerical solutions we choose the upper signs in (60), e=1, v=1, β=3/5, and winding number n=1. Similarly to the MH model, the effective compacton behavior appears for sufficiently large values of λ; see Figures 3 and 4. The true Born-Infeld-Higgs compactons arising for λ=∞ also are depicted (see black line profiles) in Figures 3 and 4.Figure 3
The profilesg(r) (a) and a(r) (b) come from the generalized Born-Infeld-Higgs model (43) with G(g)=1 and ω(g)=λg2λ-2. Observe that λ=1 (indigo lines) represents the usual BIH model and the true compacton solution is given by λ=∞ (black lines).
(a) (b)Figure 4
The magnetic fieldB(r) (a) and the BPS energy density εBPS(r) (b) come from the generalized Born-Infeld-Higgs model (43) with G(g)=1 and ω(g)=λg2λ-2. Observe that λ=1 (indigo lines) represents the usual BIH model and the true compacton solution is given by λ=∞ (black lines). The convention for the color of the lines is the same given in Figure 3.
(a) (b)


## 3.1. Born-Infeld-Higgs Effective Compact Vortices forλ 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 G(ϕ)=1 in the Lagrangian density (43).The searching for vortex solutions is made by means of the vortexAnsatz introduced in (27). Thus, the BPS equations (55) and (56) read as(60)g′=±agr,-a′r=±e2v21-g2λ1-e2v41-g2λ2/β2.The behavior of the profilesg(r) and a(r) when r→0 is determined by solving the self-dual equations (60), so we have(61)gr≈Cnrn+⋯,ar≈n-e2v2β2β2-e2v4r2+⋯.Similarly, the behavior of the profiles for r→∞ is(62)gr≈1-C∞re-mr,ar≈mC∞re-mr, where m, the self-dual mass, is given by(63)m=ev2λ;it 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(64)εBPS=e2v41-g2λ21-e2v41-g2λ2/β2+2λv2g2λ-2agr2; it will be positive-definite and finite for λ≥1.


## 3.2. Born-Infeld-Higgs Compactons forλ=∞

In the limitλ→∞, the BPS equations (60) read as(65)g′=±agr,-a′r=±e2v2Θ1-g1-e2v4/β2,with the profiles g(r) and a(r) satisfying the boundary conditions (39).By solving the BPS compacton equations for the BIH model, we obtain also analytical solutions(66)g∞r=rrcnexp⁡n21-r2rc2Θrc-r+Θr-rc,a∞r=n1-r2rc2Θrc-r, where the radial distance rc now is given by(67)rc=2nev1-e2v4β21/4.The magnetic field and BPS energy density profiles of the Born-Infeld-Higgs compacton are(68)B∞r=ev21-e2v4β2-1/2Θrc-r,εBPS∞r=e2v41-e2v4β2-1/2Θrc-r.In order to compute the numerical solutions we choose the upper signs in (60), e=1, v=1, β=3/5, and winding number n=1. Similarly to the MH model, the effective compacton behavior appears for sufficiently large values of λ; see Figures 3 and 4. The true Born-Infeld-Higgs compactons arising for λ=∞ also are depicted (see black line profiles) in Figures 3 and 4.Figure 3
The profilesg(r) (a) and a(r) (b) come from the generalized Born-Infeld-Higgs model (43) with G(g)=1 and ω(g)=λg2λ-2. Observe that λ=1 (indigo lines) represents the usual BIH model and the true compacton solution is given by λ=∞ (black lines).
(a) (b)Figure 4
The magnetic fieldB(r) (a) and the BPS energy density εBPS(r) (b) come from the generalized Born-Infeld-Higgs model (43) with G(g)=1 and ω(g)=λg2λ-2. Observe that λ=1 (indigo lines) represents the usual BIH model and the true compacton solution is given by λ=∞ (black lines). The convention for the color of the lines is the same given in Figure 3.
(a) (b)


## 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 [65–74]. Also a close connection between CS theory and supersymmetry exists. This connection was firstly demonstrated in [75], where from N=2 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:(69)L=κ4ϵμνρAμFνρ+ωϕDμϕ2-Vϕ,with the function ωϕ given by (16). The gauge field equation is to be(70)κ2ϵμαβFαβ-eωJμ=0,and the Gauss law reads as (71)κB=eωJ0. It is clear that the electric charge density is eωJ0 whose integration performed via the Gauss law gives(72)Q=∫d2xeωJ0=κ∫d2xB=κΦ.As 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 as(73)E=∫d2x-κA0B-e2ωA02ϕ2+ωDiϕ2+Vϕ.From the static Gauss law, we obtain the relation(74)A0=-κ2e2Bωϕ2,which substituted in (73) leads to the following expression for the energy:(75)E=∫d2xκ24e2B2ωϕ2+ωDiϕ2+Vϕ.We now use the identities (10) and (17) in (73) such that the energy becomes(76)E=∫d2xκ24e2B2ωϕ2+Vϕ+ωD±ϕ2±ev2ϕ2λv2λB±12λϵik∂iωJk.After some manipulation, the energy can be expressed almost in the Bogomol’nyi form(77)E=∫d2xκ24e21ϕ2ωB∓2eϕκωV2+ωD±ϕ2±12λϵik∂iωJk±BκeϕVω+ev2ϕ2λv2λ.We observe that the Bogomol’nyi procedure will be complete if the term multiplying the magnetic field is a constant; that is,(78)κeϕVω+ev2ϕ2λv2λ=ev2, such that a condition allows determining the self-dual potential V(ϕ) to be(79)Vϕ=λe4v6κ2ϕ2λv2λ1-ϕ2λv2λ2.We can see that, for λ=1, the ϕ6-potential of the usual Chern-Simon-Higgs model is recovered.Hence, the energy (77) reads as(80)E=∫d2x±ev2B±12λϵik∂iωJk+ωD±ϕ2+κ24e2ϕ2ωB∓2e2v2κ2λϕ2λv2λ2Uλ2. 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(81)E≥±ev2∫d2xB=ev2Φ. This bound is saturated by fields satisfying the first-order Bogomol’nyi or self-dual equations [32, 33](82)D±ϕ=0,B=±2e3v4κ2λϕ2λv2λ1-ϕ2λv2λ.In order for the magnetic field to be nonsingular at origin, it is required that λ>0.By using the BPS equation in (75) we find that the energy density is given by(83)εBPS=2λe4v6κ2ϕ2λv2λ1-ϕ2λv2λ2+λϕ2λ-2v2λ-2Dkϕ2; it will be positive-definite λ>0.


### 4.1. Chern-Simons-Higgs Effective Compact Vortices forλ Finite

The BPS equations (82) read as(84)g′=±agr,-a′r=±2e4v4κ2λg2λ1-g2λ.The behavior ofg(r) and a(r) near the boundaries can be easily determined by solving the self-dual equations (84) around the boundary values (28). Thus, for r→0, the profile functions behave as(85)gr≈Cnrn+⋯,ar≈n∓λe4v4nλ+1κ2Cn2λr2nλ+2+⋯,where the constant Cn>0 is determined only numerically.On the other hand, whenr→∞ they behave as(86)gr≈1-C∞re-mr,ar≈C∞mre-mr, with the constant C∞ computed numerically and m being the self-dual mass(87)m=2λe2v2κ.We observe that for λ=1, 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(88)εBPS=2λe4v6κ2g2λ1-g2λ2+2λv2g2λ-2agr2, and it will be positive-definite and finite for λ≥1.


### 4.2. Chern-Simons-Higgs Compactons forλ=∞

From (84) we obtain the BPS equations for the Chern-Simons-Higgs compactons(89)g′=±agr,-a′r=2e4v4κ2δ1-g, with the boundary condition given in (39).By solving the BPS compacton equations forn>0, we obtain the analytic profiles(90)g∞r=rrcnΘrc-r+Θr-rc,a∞r=nΘrc-r. The radial distance rc is calculated to be(91)rc=nκe2v2. The magnetic field and BPS energy density of the Chern-Simons-Higgs compacton are(92)B∞r=nercδrc-r,εBPS∞r=nv2rcδrc-r.In order to compute the numerical solutions we choose the upper signs in (84), e=1, v=1, κ=1, and winding number n=1. From the numerical analysis, we can see the appearance of the effective compacton behavior for not very large values of λ 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 n=1), for increasing values of λ in the profiles for the magnetic field and the BPS energy density (see Figure 6). The analytic CSH compacton structures appearing in λ=∞ are represented by the black line profiles in Figures 5 and 6.Figure 5
The profilesg(r) (a) and a(r) (b) come from the generalized Chern-Simons-Higgs model (69) with ω(g)=λg2λ-2. Observe that λ=1 (indigo lines) represents the usual CSH model and the true compacton solution is given by λ=∞ (vertical black lines representing the δ-Dirac function).
(a) (b)Figure 6
The magnetic fieldB(r) (a) and the BPS energy density εBPS(r) (b) come from the generalized Chern-Simons-Higgs model (69) with ω(g)=λg2λ-2. Observe that λ=1 (indigo lines) represents the usual CSH model and the true compacton solution is given by λ=∞ (vertical black lines representing the δ-Dirac function). The convention for the color of the lines is the same given in Figure 5.
(a) (b)


## 4.1. Chern-Simons-Higgs Effective Compact Vortices forλ Finite

The BPS equations (82) read as(84)g′=±agr,-a′r=±2e4v4κ2λg2λ1-g2λ.The behavior ofg(r) and a(r) near the boundaries can be easily determined by solving the self-dual equations (84) around the boundary values (28). Thus, for r→0, the profile functions behave as(85)gr≈Cnrn+⋯,ar≈n∓λe4v4nλ+1κ2Cn2λr2nλ+2+⋯,where the constant Cn>0 is determined only numerically.On the other hand, whenr→∞ they behave as(86)gr≈1-C∞re-mr,ar≈C∞mre-mr, with the constant C∞ computed numerically and m being the self-dual mass(87)m=2λe2v2κ.We observe that for λ=1, 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(88)εBPS=2λe4v6κ2g2λ1-g2λ2+2λv2g2λ-2agr2, and it will be positive-definite and finite for λ≥1.


## 4.2. Chern-Simons-Higgs Compactons forλ=∞

From (84) we obtain the BPS equations for the Chern-Simons-Higgs compactons(89)g′=±agr,-a′r=2e4v4κ2δ1-g, with the boundary condition given in (39).By solving the BPS compacton equations forn>0, we obtain the analytic profiles(90)g∞r=rrcnΘrc-r+Θr-rc,a∞r=nΘrc-r. The radial distance rc is calculated to be(91)rc=nκe2v2. The magnetic field and BPS energy density of the Chern-Simons-Higgs compacton are(92)B∞r=nercδrc-r,εBPS∞r=nv2rcδrc-r.In order to compute the numerical solutions we choose the upper signs in (84), e=1, v=1, κ=1, and winding number n=1. From the numerical analysis, we can see the appearance of the effective compacton behavior for not very large values of λ 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 n=1), for increasing values of λ in the profiles for the magnetic field and the BPS energy density (see Figure 6). The analytic CSH compacton structures appearing in λ=∞ are represented by the black line profiles in Figures 5 and 6.Figure 5
The profilesg(r) (a) and a(r) (b) come from the generalized Chern-Simons-Higgs model (69) with ω(g)=λg2λ-2. Observe that λ=1 (indigo lines) represents the usual CSH model and the true compacton solution is given by λ=∞ (vertical black lines representing the δ-Dirac function).
(a) (b)Figure 6
The magnetic fieldB(r) (a) and the BPS energy density εBPS(r) (b) come from the generalized Chern-Simons-Higgs model (69) with ω(g)=λg2λ-2. Observe that λ=1 (indigo lines) represents the usual CSH model and the true compacton solution is given by λ=∞ (vertical black lines representing the δ-Dirac function). The convention for the color of the lines is the same given in Figure 5.
(a) (b)


## 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, A0≠0, 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:(93)L=-Gϕ4FμνFμν+κ4ϵμνρAμFνρ+ωϕDμϕ2+Gϕ2∂μN∂μN-e2ωϕN2ϕ2-Vϕ, where the function ωϕ is given by (16). The gauge field equation reads as(94)∂νGFνμ+κ2ϵμαβFαβ-eωJμ=0; the static Gauss law is(95)∂kG∂kA0-κB=2e2ωA0ϕ2.Similarly, the equation of motion of the Higgs field is(96)0=DμωDμϕ+14FμνFμν-12∂μN∂μN∂G∂ϕ∗-∂ω∂ϕ∗Dμϕ2+2e2ωNϕ2+∂V∂ϕ∗.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(97)E=∫d2xG2B2+G2∂jA02+e2ωA02ϕ2+ωDjϕ2+G2∂jN2+e2ωN2ϕ2+Vϕ. To proceed, we use the identities (10) and (17) such that the energy becomes(98)E=∫d2xG2B2+Vϕ+G2∂jA02+G2∂jN2+e2ωA02ϕ2+e2ωN2ϕ2+ωD±ϕ2±ev2ϕ2λv2λB±12λϵik∂iωJk. After some algebraic manipulations, it can be expressed by(99)E=∫d2xωD±ϕ2+G2B∓2VG2+G2∂jA0±∂jN2+e2ωϕ2A0±N2±B2GV+ev2ϕ2λv2λ+κN∓∂jNG∂jA0±12λϵik∂iωJk.At this point, with the purpose for the total energy to have a lower bound proportional to the magnetic field, we chose the potential V(ϕ) to be(100)Vϕ=12Gev2-ev2ϕ2λv2λ-κN2,where C=λv2-2λ in order for the vacuum expectation value of the Higgs field to be ϕ=v. Hence, the energy (99) reads as(101)E=∫d2x±ev2B+ωD±ϕ2+G2B∓1Gev2-ev2ϕ2λv2λ-κN2+G2∂jA0±∂jN2+e2ωϕ2A0±N2∓∂jNG∂jA0±12λϵik∂iωJk.Under 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(102)E≥±ev2∫d2xB=ev2Φ.The lower bound (102) is saturated by fields satisfying the first-order Bogomol’nyi or self-dual equations [32, 33](103)D±ϕ=0,(104)B=±ev2G1-ϕ2λv2λ∓κNG,(105)∂jA0±∂jN=0,(106)A0±N=0.The conditionN=∓A0 saturates the two last equations (105) and (106) so the self-dual solutions are obtained by solving the self-dual equations(107)D±ϕ=0,B=±ev2G1-ϕ2λv2λ+κA0G,and the Gauss law(108)∂kG∂kA0-κB=2e2v2λϕ2λv2λA0.The BPS energy density is(109)εBPS=GB2+G∂jA02+λϕ2λ-2v2λ-2Djϕ2+2e2v2λϕ2λv2λA02.


### 5.1. Maxwell-Chern-Simons-Higgs Effective Compact Vortices forλ Finite

By considering thatG(ϕ)=1 and usingAnsatz (27), the BPS equations (107) read as(110)g′=±agr,-a′r=±e2v21-g2λ+eκA0and the Gauss law (108) becomes(111)1rrA0′′-κB=2λe2v2g2λA0.We analyze the behavior of the profilesg(r) and a(r) and A0(r) at boundaries. This way, for r→0, the profiles behave as(112)gr≈Cnrn+⋯,ar≈n-eev2+κA002r2+⋯,A0r≈A00+κev2+κA004r2+⋯,with the constants Cn>0 and A0(0) being determined numerically for every n.The behavior at the origin forA0(r) provides the boundary condition(113)A0′0=0.On the other hand, for large values ofr (r→∞), they have the Abrikosov-Nielsen-Olesen behavior,(114)gr≈1-C∞re-mr,(115)ar≈C∞mre-mr,(116)A0r≈-κκmeC∞re-mr,with the constant C∞ being computed numerically and m being the self-dual mass(117)m=12κ2+8λe2v2-κ2; for λ=1, we recover self-dual mass of the usual Maxwell-Chern-Simons-Higgs bosons.In this way we obtain from (116) the boundary condition for A0(r) when r→∞:(118)A0∞=0.The BPS energy density of the self-dual vortices reads as(119)εBPS=B2+A0′2+2v2λg2λ-2agr2+2e2v2λg2λA02, being positive-definite and finite for λ≥1.


### 5.2. Maxwell-Chern-Simons-Higgs Compactons forλ=∞

From (110), the limit λ→∞ provides the BPS equation for the compacton configurations(120)g′=±agr,-a′r=±e2v2Θ1-g+eκA0. The compacton Gauss law obtained from (111) becomes(121)A0′+κea-nr=0.The compacton boundary conditions satisfied by the profilesg(r), a(r), and A0(r) are(122)g0=0,a0=n,A0′0=0,gr=1,ar=0,A0r=0,rc≤r<∞. The radial distance rc<∞ is the value where the profile g(r) reaches the vacuum value and the gauge field profile a(r) and scalar potential A0(r) become null.The system is solved analytically to be(123)g∞r=rrcnexp⁡e2v2κ21-I0,κrI0,κrcΘrc-r+Θr-rc,a∞r=n1-rI1,κrrcI1,κrcΘrc-r,A0∞r=ev2κ-1+I0,κrI0,κrcΘrc-r. The radial distance rc is computed from(124)I0,κrc=rce2v2nκI1,κrc, where the function I(ν,x) is the modified Bessel function of the first kind and order ν.The magnetic field and BPS energy density of the Maxwell-Chern-Simons-Higgs compacton are(125)B∞r=ev2I0,κrI0,κrcΘrc-r,εBPS∞r=e2v4I0,κrI0,κrc2Θrc-r+e2v4I1,κrI0,κrc2Θrc-r.In order to compute the numerical solutions we choose the upper signs in (110), e=1, v=1, κ=-1, and winding number n=1. 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 λ. 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 λ=∞; they are represented by the solid black lines in Figures 7, 8, and 9.Figure 7
The profilesg(r) (a) and a(r) (b) come from generalized Maxwell-Chern-Simons-Higgs model (93) with G(g)=1 and ω(g)=λg2λ-2. Observe that λ=1 (indigo lines) represents the usual MCSH model and the true compacton solution is given by λ=∞ (black lines).
(a) (b)Figure 8
The scalar potentialA0(r) (a) and the electric field E(r) (b) come from generalized Maxwell-Chern-Simons-Higgs model (93) with G(g)=1 and ω(g)=λg2λ-2. Observe that λ=1 (indigo lines) represents the usual MCSH model and the true compacton solution is given by λ=∞ (black lines). The convention for the color of the lines is the same given in Figure 7.
(a) (b)Figure 9
The magnetic fieldB(r) (a) and the BPS energy density εBPS(r) (b) come from generalized Maxwell-Chern-Simons-Higgs model (93) with G(g)=1 and ω(g)=λg2λ-2. Observe that λ=1 (indigo lines) represents the usual MCSH model and the true compacton solution is given by λ=∞ (black lines). The convention for the color of the lines is the same given in Figure 7.
(a) (b)


## 5.1. Maxwell-Chern-Simons-Higgs Effective Compact Vortices forλ Finite

By considering thatG(ϕ)=1 and usingAnsatz (27), the BPS equations (107) read as(110)g′=±agr,-a′r=±e2v21-g2λ+eκA0and the Gauss law (108) becomes(111)1rrA0′′-κB=2λe2v2g2λA0.We analyze the behavior of the profilesg(r) and a(r) and A0(r) at boundaries. This way, for r→0, the profiles behave as(112)gr≈Cnrn+⋯,ar≈n-eev2+κA002r2+⋯,A0r≈A00+κev2+κA004r2+⋯,with the constants Cn>0 and A0(0) being determined numerically for every n.The behavior at the origin forA0(r) provides the boundary condition(113)A0′0=0.On the other hand, for large values ofr (r→∞), they have the Abrikosov-Nielsen-Olesen behavior,(114)gr≈1-C∞re-mr,(115)ar≈C∞mre-mr,(116)A0r≈-κκmeC∞re-mr,with the constant C∞ being computed numerically and m being the self-dual mass(117)m=12κ2+8λe2v2-κ2; for λ=1, we recover self-dual mass of the usual Maxwell-Chern-Simons-Higgs bosons.In this way we obtain from (116) the boundary condition for A0(r) when r→∞:(118)A0∞=0.The BPS energy density of the self-dual vortices reads as(119)εBPS=B2+A0′2+2v2λg2λ-2agr2+2e2v2λg2λA02, being positive-definite and finite for λ≥1.


## 5.2. Maxwell-Chern-Simons-Higgs Compactons forλ=∞

From (110), the limit λ→∞ provides the BPS equation for the compacton configurations(120)g′=±agr,-a′r=±e2v2Θ1-g+eκA0. The compacton Gauss law obtained from (111) becomes(121)A0′+κea-nr=0.The compacton boundary conditions satisfied by the profilesg(r), a(r), and A0(r) are(122)g0=0,a0=n,A0′0=0,gr=1,ar=0,A0r=0,rc≤r<∞. The radial distance rc<∞ is the value where the profile g(r) reaches the vacuum value and the gauge field profile a(r) and scalar potential A0(r) become null.The system is solved analytically to be(123)g∞r=rrcnexp⁡e2v2κ21-I0,κrI0,κrcΘrc-r+Θr-rc,a∞r=n1-rI1,κrrcI1,κrcΘrc-r,A0∞r=ev2κ-1+I0,κrI0,κrcΘrc-r. The radial distance rc is computed from(124)I0,κrc=rce2v2nκI1,κrc, where the function I(ν,x) is the modified Bessel function of the first kind and order ν.The magnetic field and BPS energy density of the Maxwell-Chern-Simons-Higgs compacton are(125)B∞r=ev2I0,κrI0,κrcΘrc-r,εBPS∞r=e2v4I0,κrI0,κrc2Θrc-r+e2v4I1,κrI0,κrc2Θrc-r.In order to compute the numerical solutions we choose the upper signs in (110), e=1, v=1, κ=-1, and winding number n=1. 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 λ. 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 λ=∞; they are represented by the solid black lines in Figures 7, 8, and 9.Figure 7
The profilesg(r) (a) and a(r) (b) come from generalized Maxwell-Chern-Simons-Higgs model (93) with G(g)=1 and ω(g)=λg2λ-2. Observe that λ=1 (indigo lines) represents the usual MCSH model and the true compacton solution is given by λ=∞ (black lines).
(a) (b)Figure 8
The scalar potentialA0(r) (a) and the electric field E(r) (b) come from generalized Maxwell-Chern-Simons-Higgs model (93) with G(g)=1 and ω(g)=λg2λ-2. Observe that λ=1 (indigo lines) represents the usual MCSH model and the true compacton solution is given by λ=∞ (black lines). The convention for the color of the lines is the same given in Figure 7.
(a) (b)Figure 9
The magnetic fieldB(r) (a) and the BPS energy density εBPS(r) (b) come from generalized Maxwell-Chern-Simons-Higgs model (93) with G(g)=1 and ω(g)=λg2λ-2. Observe that λ=1 (indigo lines) represents the usual MCSH model and the true compacton solution is given by λ=∞ (black lines). The convention for the color of the lines is the same given in Figure 7.
(a) (b)


## 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 functionω(ϕ) (see (16)) which is parameterized by the positive parameter λ. 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 [79–82].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 ofλ, 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 λ→∞ for arbitrary winding number n. Our analysis has shown that, for λ=∞, 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.



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