# Restricted baby Skyrme-Maxwell theory in a magnetic medium: BPS
configurations and some properties

J. Andrade<EMAIL_ADDRESS>Rodolfo Casana<EMAIL_ADDRESS><EMAIL_ADDRESS>Departamento de Física, Universidade Federal do
Maranhão, 65080-805, São Luís, Maranhão, Brazil.  E. da Hora
<EMAIL_ADDRESS><EMAIL_ADDRESS>Coordenação do Curso de Bacharelado
Interdisciplinar em Ciência e Tecnologia,
Universidade Federal do Maranhão, 65080-805, São Luís, Maranhão, Brazil.  A.
C. Santos<EMAIL_ADDRESS>Departamento de Física, Universidade
Federal do Maranhão, 65080-805, São Luís, Maranhão, Brazil.

###### Abstract

We study the existence of BPS configurations in a restricted baby Skyrme-
Maxwell enlarged via the inclusion of a nontrivial magnetic permeability. In
order to attain such a goal, we use the Bogomol’nyi-Prasad-Sommerfield
prescription, which allows us to obtain the lower bound for the energy and the
BPS equations whose [electrically neutral] solutions saturate that bound.
During the energy minimization procedure, we find a differential constraint
which involves the self-dual potential, the superpotential itself and also the
magnetic permeability. In order to solve the BPS system, we focus our
attention on those solutions with rotational symmetry. For that, we fix the
magnetic permeability and select two BPS potentials which exhibit a similar
behavior near to the vacuum. We depict the resulting profiles and proceed to
an analytical description of the properties of the BPS magnetic field.
Furthermore, we consider some essential aspects of our model, such as the
conditions for the overall existence of the BPS solutions, and how the
permeability affects the magnetic flux. Finally, we present a family of exact
BPS solutions.

##  I Introduction

Topologically nontrivial structures are commonly described by means of those
time-independent solutions which come from highly nonlinear Euler-Lagrange
equations [1]. In such a context, the potential term which defines the vacuum
manifold of the respective theory not only introduces the nonlinearity itself,
but it is also expected to allow the spontaneous symmetry breaking mechanism
to occur (whose effects include the formation of a topological profile as a
result of the corresponding phase transition). The point is that highly
nonlinear equations of motion are typically quite hard to solve. However, this
issue can be circumvented via the minimization of the system’s total energy by
employing the Bogomol’nyi-Prasad-Sommerfield (BPS) prescription [2, 3]. The
implementation of such algorithm determines a specific expression for the
potential, but it also provides a lower bound for the energy (the BPS bound)
and the corresponding BPS equations whose solutions saturate that bound (and
therefore describe energetically stable configurations). In addition, it is
always possible to verify that the BPS equations are compatible with the
Euler-Lagrange equations, from which one concludes that the BPS profiles stand
for legitimate solutions of the model. In the literature, there are
alternative methods for the obtainment of such BPS configurations; see, for
instance, the study of the conservation of the energy-momentum tensor [4], the
on-shell procedure [5], and the strong-necessary conditions technique [6, 7,
8, 9, 10].

The full Skyrme model was proposed in 1961 as a generalized nonlinear sigma
theory defined in $(3+1)$-dimensions [11]. Its Lagrange density contains the
so-called Skyrme term (a quartic kinetic, i.e. a term of degree four in the
first-derivative of the scalar sector), the $\sigma$ term (a quadratic kinetic
one), and a potential which was originally proposed as an attempt to study the
pion mass. The Skyrme model can be interpreted as an effective low-energy
model of Quantum Chromodynamics which engenders stable solitonic structures,
so-called Skyrmions, which can be applied to the study some physical
properties of those hadrons and nuclei [12, 13, 14, 15, 16]. Phenomenological
applications of the gauged Skyrme model include not only the studies about the
electromagnetic transition strengths for light nuclei [17] and the spin
excitation energy of the nucleon [18], but also investigations on the energy
levels of a light nuclei $A=12$ [19], the proton and neutron properties in a
strong magnetic field [20] and, more recently, the electromagnetic transition
rates of C12 and O16 in rotational-vibrational models [21].

In this context, the study of the planar version of the Skyrme theory, known
as the baby Skyrme model [22], serves to the comprehension of many aspects of
the original $(3+1)$-dimensional scenario, including the conditions under
which it eventually accepts the implementation of the BPS prescription. The
baby Skyrme model in the absence of the $\sigma$-term, named the restricted
baby Skyrme model [23], supports a well-established BPS structure [24].
Furthermore, over the last years, the Skyrmions have also been used to
describe topological quantum Hall effect [25, 26, 27, 28, 29], in chiral
nematic liquid crystals [30, 31], superconductors [32], brane cosmology [33,
34, 35] and magnetic materials [36, 37], for instance.

Moreover, in order to investigate the electromagnetic properties of the baby
Skyrme model, it is necessary to couple it to an Abelian gauge field [38]. In
such a context, the BPS Skyrmions appear in a restricted baby Skyrme-Maxwell
model [39, 40, 41], and also occur when the Skyrme sector is minimally coupled
to the Chern-Simons term [42] and to the Maxwell-Chern-Simons action [43].
Additional results on the study of those BPS solutions in a Skyrme-Born-Infeld
scenario can be found in [44], while supersymmetric extensions of these
restricted gauged baby Skyrme theories are in the Refs. [45, 46, 47, 48, 49,
50, 51].

We now go a little bit further into this issue and consider how the
electromagnetic properties of a material medium affect the self-dual Skyrmions
which arise from a BPS restricted baby Maxwell-Skyrme model. Here, these
properties are studied via the introduction of a nonstandard function which
multiples the Maxwell term and therefore represents the magnetic permeability
of the medium.

To motivate our study, we highlight that enlarged models with a nontrivial
permeability have been considered with relative intensity in recent years. In
the context of scalar field theories, for instance, it is currently known that
the presence of such a permeability can be used to simulate geometrical
constrictions in the corresponding kinklike solutions [52], with the resulting
profiles mimicking experimental results and therefore clarifying the influence
of such a constriction on the magnetization in a magnetic material, see the
Ref. [53]. Moreover, inspired by an experimental investigation on the
possibility of controlling the domain wall polarity in a magnetic material in
the presence of an electric pulse [54], some authors have recently studied how
the presence of geometric constrictions influences the behavior of fermions in
a model with a nontrivial permeability, see the Ref. [55].

In order to present our results, this manuscript is organized as follows. In
the Section II, we introduce the restricted baby Maxwell-Skyrme model enlarged
via the inclusion of a nontrivial magnetic permeability. We present the
definitions and conventions which we adopt in our work. In the sequence, we
look for the BPS framework inherent to the generalized scenario via the
minimization of the its total energy by means of the BPS prescription. As a
result, it arises a differential constraint (which we call superpotential
equation) which relates the BPS potential to both the superpotential and the
nontrivial permeability. In view of such a constraint, we obtain not only the
BPS bound for the energy itself, but also the self-dual equations whose
solutions saturate it. We then particularize our work by focusing our
attention on the gauged Skyrmions in a planar context, from which we rewrite
the BPS equations in a rotationally symmetric form. The Sec. III is dedicated
to the BPS scenario and its solutions. Here, in view of the target space
inherent to a Skyrme-Maxwell scenario, we fix an specific analytical
expression for the permeability which then forces the gauge sector to assume a
nonusual shape. In this context, we consider two different scenarios based on
the “nature” of the superpotential, i.e. a first one in which the
superpotential is given by an exact expression, and a second case in which the
superpotential must be itself obtained numerically. In both cases, we work
with potentials which attain their vacuum values in the very same way, for the
sake of comparison. We then solve the two models numerically and depict the
corresponding profiles, wherefrom we identify how a noncanonical permeability
may give rise to BPS solutions with nonstandard shapes. We perform an
analytical study which explains the form which distinguishes the resulting
magnetic field. We also consider some basic aspects of our generalized theory
(in comparison to the standard case), such as the conditions under which BPS
solutions do exist, whether the nontrivial model is capable to reproduce the
BPS bound inherent to the ungauged baby Skyrme scenario and how a nontrivial
permeability affects the value of the magnetic flux calculated for small and
large electromagnetic coupling $g$. Next, we present some family of exact BPS
solution for the enlarged model. Finally, the Sec. IV brings a brief summary
and our perspectives regarding future contributions.

In this manuscript, we adopt the natural units system and
$\eta^{\mu\nu}=(+--)$ for the metric signature, for the sake of simplicity.

##  II The restricted gauged baby Skyrme model in a magnetic medium: The BPS
structure

We begin by presenting the $(2+1)$-dimensional restricted gauged baby Skyrme
model enlarged via the inclusion of an a priori arbitrary function which
represents a nontrivial magnetic permeability, the corresponding Lagrangian
function reading

$L=E_{0}\int d^{2}\mathbf{x}\,{\mathcal{L}}\text{,}$ (1)

where the factor $E_{0}$ sets the energy scale of the model (which will be
taken as $E_{0}=1$ hereafter). The Lagrangian density is

$\mathcal{L}=-\frac{G}{4g^{2}}F_{\mu\nu}F^{\mu\nu}-\frac{\lambda^{2}}{4}(D_{\mu}\vec{\varphi}\times
D_{\nu}\vec{\varphi})^{2}-V\text{.}$ (2)

Here, the first term stands for Maxwell’s action now multiplied by a magnetic
permeability function $G\equiv G(\varphi_{n})$ (this explicit dependence on
the quantity $\varphi_{n}=\hat{n}\cdot\vec{\varphi}$ will be clarified later
during the implementation of the BPS formalism). In the internal space,
$\hat{n}$ represents an unitary vector which defines a preferred direction,
while the Skyrme field $\vec{\varphi}=(\varphi_{1},\varphi_{2},\varphi_{3})$
is given as a triplet of real scalar fields constrained to satisfy
$\vec{\varphi}\cdot\vec{\varphi}=1$ and therefore describing a spherical
surface with unitary radius. Moreover,
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field strength tensor and

$D_{\mu}\vec{\varphi}=\partial_{\mu}\vec{\varphi}+A_{\mu}\hat{n}\times\vec{\varphi}$
(3)

stands for the usual covariant derivative of the Skyrme field. The third term
brings the self-interacting potential $V=V(\varphi_{n})$ which promotes the
spontaneous breaking of the internal symmetry. At the same time, both
$\lambda$ and $g$ are coupling constants inherent to the model (which we
assume to be nonnegative from now on). Moreover, the Skyrme field and the
function $G$ are dimensionless, while the gauge field, the electromagnetic
constant $g$ and the Skyrme one $\lambda$ have mass dimensions equal to $1$,
$1$, and $-1$, respectively.

Now, beyond the motivations previously cited in the Section I, we also point
out that the idea based on the inclusion of a nontrivial permeability has also
been widely used in the context of gauged models, with different purposes: for
instance, the Refs. [56, 57] applied it to describe a bag model similar to the
MIT [58] and SLAC [59] bag models, while some authors have implemented the
same idea to study peculiar properties of gauged vortices; see the Refs. [60,
61, 62]. In addition, in the Refs. [63, 64, 65, 66], a nontrivial permeability
was used in connection with the AdS/CFT correspondence. More recently, it was
also employed to study both the presence of electrically charged structures in
a multi-field scenario [67], and the arising of internal structures in Abelian
gauge field models generated by both an electric point charge [68] and an
electric dipole [69] when immersed in a medium controlled by scalar fields. In
particular, recent studies [70, 71, 72, 73] on dielectric Skyrme models were
investigated in view of their possible connections with the binding energies
of nuclei.

Here, it is worthwhile to clarify that the Lagrange density (2) must not be
considered as a trivial generalization of the model investigated in the Ref.
[39] once that, as we demonstrate below, the magnetic permeability composes
the differential constraint involving both the self-dual potential and the
superpotential, which provides support for the existence of the BPS structure.
Consequently, it is possible to modify the vacuum structure of the effective
model by conveniently choosing the form of the magnetic permeability, from
which configurations with different shapes and features may occur, for
instance.

It is instructive to write down the Gauss law for time-independent
configurations which comes from (2), i.e

$\partial_{i}\left(G\partial^{i}A^{0}\right)=-g^{2}\lambda^{2}A_{0}\left(\hat{n}\cdot\partial_{i}\vec{\varphi}\right)^{2}.$
(4)

The point here is that $A^{0}=0$ stands for a legitimate gauge choice, given
that it solves the Gauss law (4) identically. Thus, we conclude that the
stationary configurations we study in this manuscript are electrically neutral
(i.e. present no electric field and electric charge).

Instead of studying the solutions of the second-order Euler-Lagrange
equations, we focus our attention on those BPS configurations that minimize
the theory’s total energy. Here, we achieve such a goal via the implementation
of the BPS procedure whose starting point is the stationary energy density of
the model (2),

$\varepsilon=\frac{G}{2g^{2}}B^{2}+\frac{\lambda^{2}}{2}Q^{2}+V\text{,}$ (5)

where we have already implemented $A_{0}=0$ and $Q$ defined by

$\displaystyle Q$ $\displaystyle=$
$\displaystyle\vec{\phi}\cdot(D_{1}\vec{\phi}\times D_{2}\vec{\phi})$ (6)
$\displaystyle=$
$\displaystyle\vec{\varphi}\cdot(\partial_{1}\vec{\varphi}\times\partial_{2}\vec{\varphi})+\epsilon_{ij}A_{i}(\hat{n}\cdot\partial_{j}\vec{\phi}).$

where the term
$\vec{\varphi}\cdot(\partial_{1}\vec{\varphi}\times\partial_{2}\vec{\varphi})$
is related to the topological charge of the Skyrme field by means of

$\deg\left[\vec{\varphi}\right]=-\frac{1}{4\pi}\int
d^{2}\mathbf{x\leavevmode\nobreak\
}\vec{\varphi}\cdot(\partial_{1}\vec{\varphi}\times\partial_{2}\vec{\varphi})=k\text{,}$
(7)

where $k\in\mathbb{Z\setminus}\left\\{0\right\\}$.

We now establish the boundary conditions to be satisfied by the fields to
ensure the existence of finite energy configurations. For this, the energy
density (5) must be zero at the vacuum, i.e. when
$|\mathbf{x}|\rightarrow\infty$. Consequently, the magnetic field $B$, $Q$ and
the potential $V$ must satisfy the following boundary conditions:

$\lim_{|\mathbf{x}|\rightarrow\infty}\sqrt{G}B=0,\;\;\lim_{|\mathbf{x}|\rightarrow\infty}Q=0,\;\;\text{and}\;\;\lim_{|\mathbf{x}|\rightarrow\infty}V=0\text{.}$
(8)

The total energy $E$ is defined as the integral of the energy density (5), so
that the implementation of the BPS formalism allows us to write

$\displaystyle E$ $\displaystyle=$ $\displaystyle\int
d^{2}\mathbf{x}\left[\frac{\left(GB\pm\lambda^{2}g^{2}\mathcal{W}\right)^{2}}{2Gg^{2}}+\frac{\lambda^{2}}{2}\left(Q\pm\frac{\partial\mathcal{W}}{\partial\varphi_{n}}\right)^{2}\right.$
(9) $\displaystyle\hskip
28.45274pt\mp\lambda^{2}B\mathcal{W}\mp\lambda^{2}Q\frac{\partial\mathcal{W}}{\partial\varphi_{n}}$
$\displaystyle\hskip
28.45274pt\left.-\frac{\lambda^{4}g^{2}}{2G}\mathcal{W}^{2}-\frac{\lambda^{2}}{2}\left(\frac{\partial\mathcal{W}}{\partial\varphi_{n}}\right)^{2}+V\right]\text{,}\quad$

where we have introduced the superpotential function
$\mathcal{W}\equiv\mathcal{W}(\varphi_{n})$. Moreover, it proves useful to
define the BPS potential $V(\varphi_{n})$ as

$V=\frac{\lambda^{4}g^{2}}{2G}\mathcal{W}^{2}+\frac{\lambda^{2}}{2}\left({\frac{\partial\mathcal{W}}{\partial\varphi_{n}}}\right)^{2}\text{,}$
(10)

from that one gets that the third row of the Eq. (9) vanishes. It is
worthwhile to note that the Eq. (10) also explains why $G$ is supposed to
depend on $\varphi_{n}$ only, i.e., given the superpotential
$\mathcal{W}=\mathcal{W}(\varphi_{n})$ and to maintain the dependence of the
potential ${V}$ on $\varphi_{n}$ (which stands for a necessary condition to
the formation of gauged Skyrmions), we must choose $G=G(\varphi_{n})$.
Furthermore, when the Skyrme field assumes its vacuum configuration (supposed
to be $\varphi_{n}\rightarrow 1$, as usual), the potential vanishes, and the
Eq. (10) leads to

$\lim_{\varphi_{n}\rightarrow 1}{\mathcal{W}}{(\varphi_{n})}=0\text{ \ \ and \
\ }\lim_{\varphi_{n}\rightarrow
1}\frac{\partial\mathcal{W}}{\partial\varphi_{n}}=0\text{,}$ (11)

i.e. the boundary conditions to be satisfied for the superpotential itself.
After using the explicit form of $Q$ given by Eq. (6), the total energy (9)
assumes the form

$\displaystyle E$ $\displaystyle=$ $\displaystyle\int
d^{2}\mathbf{x}\left[\frac{\left(GB\pm\lambda^{2}g^{2}\mathcal{W}\right)^{2}}{2Gg^{2}}+\frac{\lambda^{2}}{2}\left(Q\pm\frac{\partial\mathcal{W}}{\partial\varphi_{n}}\right)^{2}\right.$
(12)
$\displaystyle\left.\frac{}{}\mp\lambda^{2}\frac{\partial\mathcal{W}}{\partial\varphi_{n}}\vec{\varphi}\cdot(\partial_{1}\vec{\varphi}\times\partial_{2}\vec{\varphi})\mp\lambda^{2}\epsilon_{ij}\partial_{j}(\mathcal{W}A_{i})\right]\\!\\!.\quad$

The point is that, given the boundary conditions (11), the contribution due to
the total derivative $\epsilon_{ij}\partial_{j}(\mathcal{W}A_{i})$ appearing
in Eq. (12) vanishes. In this way, we can express the total energy as,

$E=\bar{E}+E_{bps}\text{,}$ (13)

where $\bar{E}$ represents the integral composed by the quadratic terms, i.e.

$\bar{E}=\int
d^{2}\mathbf{x}\left[\frac{\left(GB\pm\lambda^{2}g^{2}\mathcal{W}\right)^{2}}{2Gg^{2}}+\frac{\lambda^{2}}{2}\left(Q\pm\frac{\partial\mathcal{W}}{\partial\varphi_{n}}\right)^{2}\right]\\!\text{,}\quad$
(14)

and $E_{bps}$ defines the energy lower bound, which reads

$E_{bps}=\mp\lambda^{2}\int
d^{2}\mathbf{x}\left(\frac{\partial\mathcal{W}}{\partial\varphi_{n}}\right)\vec{\varphi}\cdot(\partial_{1}\vec{\varphi}\times\partial_{2}\vec{\varphi})>0\text{.}$
(15)

Given that $\bar{E}\geq 0$, the total energy (13) satisfies the typical BPS
inequality

$E\geq E_{bps}\text{,}$ (16)

from which we conclude that the energy lower bound is achieved when the fields
are such that $\bar{E}=0$, i.e. when they satisfy

$\displaystyle GB$ $\displaystyle=$ $\displaystyle\mp
g^{2}\lambda^{2}\mathcal{W}\text{,}$ (17) $\displaystyle Q$ $\displaystyle=$
$\displaystyle\mp\frac{\partial\mathcal{W}}{\partial\varphi_{n}}\text{,}$ (18)

which therefore stand for the self-dual equations inherent to the enlarged
model. The solutions of these equations also are classical solutions belonging
to an extended supersymmetric model [74, 75] whose bosonic sector would be
given by the Lagrangian density (2). Furthermore, some studies concerning the
gauged Skyrme model in the SUSY field theory context can be found, for
instance, in Refs. [76, 77, 78, 79].

The interested reader must note that, beyond multiplying the magnetic field
$B$ in the self-dual Eq. (17) (an expected fact given the way how the magnetic
permeability appears in the Lagrangian density (2)), the function $G$ also
composes the Eq. (10) relating the BPS potential $V(\varphi_{n})$ and the
superpotential $\mathcal{W}(\varphi_{n})$.

Moreover, the combination of Eqs. (5) and (10) together with the self-dual
ones (17) and (18) leads to the following expression for the BPS energy
density:

$\varepsilon_{bps}=\frac{G}{g^{2}}B^{2}+\lambda^{2}Q^{2}\text{,}$ (19)

which shows clearly the contribution of the magnetic permeability.

###  II.1 Rotationally symmetric BPS Skyrmions

Once we have developed the general BPS framework, we focus our investigation
on those solutions with rotational symmetry. In this sense, without loss of
generality, we set $\hat{n}=(0,0,1)$, wherefrom we get
$\varphi_{n}=\varphi_{3}$. As a consequence, the potential $V=V(\varphi_{3})$
now allows for the spontaneous breaking of the $SO(3)$ symmetry inherent to
the Skyrme-Maxwell model (2) that enables the occurrence of configurations
with a nontrivial topology as expected.

Moreover, in order to compare our results with the well-established ones, we
study time-independent solutions using the standard ansatz for the gauge field

$A_{i}=-\epsilon_{ij}\hat{x}_{j}\frac{Na\left(r\right)}{r}\text{,}$ (20)

and the Skyrme field

$\vec{\varphi}=\left(\begin{array}[]{c}\sin f\cos\left(N\theta\right)\\\ \sin
f\sin\left(N\theta\right)\\\ \cos f\end{array}\right)\text{,}$ (21)

where $r$ and $\theta$ are polar coordinates, $\epsilon_{ij}$ stands for the
Levi-Civita antisymmetric tensor (with $\epsilon_{12}=+1$) and
$\hat{x}_{i}=(\cos\theta,\sin\theta)$ represents the unit vector. Also, $N$ is
the winding number of the Skyrme field, while the profile functions $f(r)$ and
$a(r)$ are supposed to obey the boundary conditions which are known to support
the existence of regular solutions with finite energy,

$\displaystyle f\left(r=0\right)=\pi\text{ \ }$ and $\displaystyle\text{ \
}f\left(r\rightarrow\infty\right)\rightarrow 0\text{,}$ (22) $\displaystyle
a\left(r=0\right)=0$ and $\displaystyle\text{ \
}a^{\prime}\left(r\rightarrow\infty\right)\rightarrow 0\text{,}$ (23)

in which prime denotes the derivative with respect to the radial coordinate
$r$.

It is instructive to point out that the magnetic field in terms of the ansatz
reads

$B(r)=F_{21}=-\frac{N}{r}\frac{da}{dr}\text{.}$ (24)

In what follows, for the sake of convenience, we implement the field
redefinition

$h(r)=\frac{1}{2}\left(1-\cos f\right)\text{,}$ (25)

from where one gets that the new profile function $h(r)$ satisfies the
boundary conditions

$h\left(r=0\right)=1\text{ \ and}\text{ \
}h\left(r\rightarrow\infty\right)\rightarrow 0\text{.}$ (26)

In view of the Eq. (25), both $G$ and $\mathcal{W}$ become functions of $h$
only. In particular, the boundary conditions to be satisfied by
${\mathcal{W}}{(h)}$ can be summarized as

$\lim_{r\rightarrow 0}{\mathcal{W}}{(h)}={\mathcal{W}}_{0}\text{,}\ \
\lim_{r\rightarrow\infty}{\mathcal{W}}{(h)}=0\text{,}\ \
\lim_{r\rightarrow\infty}\frac{\partial\mathcal{W}}{\partial h}=0\text{,}$
(27)

where $\mathcal{W}_{0}>0$, whereas the two last ones correspond to those which
appear in Eq. (11).

The BPS energy given by Eq. (15) can be calculated explicitly, its value
reading

$E_{bps}=\mp 2\pi\lambda^{2}N{\mathcal{W}}_{0}>0\text{,}$ (28)

Here, the upper (lower) sign corresponds to $N<0$ ($N>0$).

The BPS equations (17) and (18) become

$B=-\frac{N}{r}\frac{da}{dr}=\mp\frac{\lambda^{2}g^{2}\mathcal{W}}{G}$ (29)
$\frac{\left(1+a\right)}{r}\frac{dh}{dr}=\pm\frac{1}{4N}\frac{\partial\mathcal{W}}{\partial
h}\text{,}$ (30)

respectively, where we have used the Eq. (24) for the magnetic field.

To summarize, these equations above describe a radially symmetric structure
whose total energy is given by Eq. (28). Further, the gauged Skyrmions emerge
as the numerical solutions of the BPS equations (29) and (30) obtained via the
boundary conditions (23) and (26).

In the next Sections, we demonstrate how the BPS framework introduced here can
be used to generate legitimate gauged Skyrmions in the presence of a
nontrivial magnetic permeability. Additionally, we also investigate some basic
properties of our enlarged model, in comparison to those presented by its
canonical version.

##  III BPS Skyrmions in magnetic media

We now particularize our investigation by focusing our attention on some
effective models. Therefore, the results presented below can contribute to the
understanding of the electromagnetic properties of gauged Skyrmions by
studying their BPS-gauged versions. The point here is that exploring the
electromagnetic properties of the Skyrmions is commonly a rather complicated
work even in a non-BPS context, as shown by some currently available results,
see, for instance, the Refs. [80, 81] for correlated developments within the
standard gauged Skyrme model. In particular, it becomes clear that the
interaction with an Abelian gauge field plays a fundamental role concerning
the properties of baryons and atomic nuclei. In that regard, it is known, for
instance, that while the exact form of the low-energy Skyrme theory remains
unknown, its coupling to the electromagnetic sector is already fixed, see the
Refs. [82, 83, 84].

Concerning the (1+2)-dimensional case, Refs. [85, 86] have studied some
aspects of the magnetic properties arising in gauged BPS baby Skyrmions. This
way, the enlarged scenario plans to provide new results about BPS baby
Skyrmions immersed in a magnetic medium. As we have already argued, we intend
to identify the new effects produced on BPS Skyrmions due to a nontrivial
magnetic permeability. In particular, we look for the arising of Skyrmions
with internal structures. With such aimin mind, we choose the permeability as

$G(h)=\frac{1}{\left(\gamma-h^{2}\right)^{\beta}}\text{.}$ (31)

Here, $\gamma,\beta\in\mathds{R}$, with $\beta\geq 0$. Thus, the rotationally
symmetric version of the BPS equations (29) and (30) can be written in the
form

$N\frac{da}{dy}=\pm\lambda^{2}g^{2}\left(\gamma-h^{2}\right)^{\beta}\mathcal{W}\text{,}$
(32)
$\left(1+a\right)\frac{dh}{dy}=\pm\frac{1}{4N}\frac{d\mathcal{W}}{dh}\text{,}$
(33)

where we have introduced a new spatial coordinate $y$ defined by $y=r^{2}/2$.

Figure 1: Numerical solutions to both $h(y)$ (top) and $a(y)$ (bottom,
depicted in units of $\left|a_{\infty}\right|$) obtained from the Eqs. (40)
and (41) for $\gamma=0.50$ (red line), $\gamma=0.75$ (blue line),
$\gamma=0.90$ (orange line) and $\gamma=1.00$ (green line). The corresponding
usual profile (obtained via $G=1$) appears as the black line, for the sake of
comparison.

In what follows, we split our investigation into two different branches
according to the nature of the superpotential $\mathcal{W}(h)$. Within this
sense, in Sec. III.1, we first consider a particular scenario for which the
superpotential is given by an explicit function of $h(r)$, so that the Eq.
(10) provides the BPS potential in terms of $\mathcal{W}(h)$. Next, we use it
to solve the corresponding BPS equations numerically. As a second case, in
Sec. III.2, we select explicitly the functional form of $V(h)$, so that the
Eq. (10) becomes a differential equation for the superpotential
$\mathcal{W}(h)$, which, in general, must be solved numerically together with
the BPS equations. We then present some considerations about comparing the
resulting profiles emerging from these two contexts.

###  III.1 Analytical superpotential and its BPS solutions

In order to solve the BPS equations, we first consider the superpotential as

$\mathcal{W}(h)=\frac{h^{2}}{\lambda^{2}}\text{,}$ (34)

from which one gets $\mathcal{W}_{0}=\lambda^{-2}>0$ and the total energy of
the BPS configurations as $E_{bps}=2\pi\left|N\right|>0$, as expected, see the
Eq. (28). It is also clear that the superpotential above satisfies the
conditions given in the Eq. (27). This choice was motivated by the fact that
the superpotential (34) is known to support well-behaved Skyrmions which
attain their asymptotic values according to a Gaussian decay law, as explained
recently in the Refs. [42, 43, 44].

It is instructive to consider the potential $V(h)$ related to
$\mathcal{W}(h)=\lambda^{-2}h^{2}$. With this aim in mind, we write the
superpotential equation (10) as

$\mu^{2}U(h)=\frac{\lambda^{4}g^{2}}{2G}\mathcal{W}^{2}+\frac{\lambda^{2}}{8}\left({\frac{d\mathcal{W}}{dh}}\right)^{2}\text{,}$
(35)

where we have rescaled the potential $V(h)$ as $\mu^{2}U(h)$, for the sake of
comparison between our results and the standard ones. Solving the Equation
above for $U(h)$, we obtain

$U(h)=\frac{h^{2}}{2\mu^{2}\lambda^{2}}\left[1+\lambda^{2}g^{2}\left(\gamma-h^{2}\right)^{\beta}h^{2}\right]\text{,}$
(36)

where we have also considered the Eq. (31) for $G(h)$.

It is interesting to note that, in the limit
$h\left(r\rightarrow\infty\right)\rightarrow 0$, the generalized potential
above approaches the vacuum as

$U(h\rightarrow 0)\approx\frac{h^{2}}{2\mu^{2}\lambda^{2}}\text{,}$ (37)

i.e. in the very same way as its standard counterpartner. As a consequence, we
conclude that, despite the nontrivial expression which we have chosen for the
magnetic permeability, a superpotential of the form $\mathcal{W}(h)\propto
h^{2}$ leads to a potential which behaves as $U(h)\propto h^{2}$ in the
asymptotic region, and vice-versa (we return to such a conclusion later
below).

Now, in view of the Eq. (34), the BPS equations (32) and (33) assume the form

$N\frac{da}{dy}=\pm g^{2}\left(\gamma-h^{2}\right)^{\beta}h^{2}\text{,}$ (38)
$\left(1+a\right)\frac{dh}{dy}=\pm\frac{h}{2\lambda^{2}N}\text{,}$ (39)

via which we intend to investigate those gauged Skyrmions which behave
standardly at the boundaries and have a noncanonical profile for intermediate
values of $y$. As we clarify below, this type of configuration is directly
related to different values of $\gamma$, from which we work with fixed values
for the others parameters. In particular, we set $\beta=2$, $g=1$, $\lambda=1$
and $N=1$ (i.e. the lower signs in the BPS equations), for the sake of
simplicity.

Figure 2: Numerical solutions to the BPS magnetic field $B(y)$ obtained from
the Eqs. (40) and (41). Conventions as in the Fig. 1. Top: the profiles
(depicted in units of $B_{0}=B(y=0)$) for $\gamma=0.50$ and $\gamma=0.75$.
Bottom: the results for $\gamma=0.90$ and $\gamma=1.00$. The corresponding
standard solution again appears as the black line. Here, some of the
noncanonical profiles were normalized for the sake of visualization.

In view of these choices, the BPS Eqs. (38) and (39) reduce to

$\frac{da}{dy}=-h^{2}\left(\gamma-h^{2}\right)^{2}\text{,}$ (40)
$\left(1+a\right)\frac{dh}{dy}=-\frac{1}{2}h\text{,}$ (41)

which must be solved numerically via the implementation of a finite-difference
scheme together with the boundary conditions (23) and (26).

The Figure 1 shows the numerical solutions to both $h(y)$ (top) and $a(y)$
(bottom) for different values of $\gamma$. Here, the gauge profile function
$a(y)$ is depicted in units of $\left|a_{\infty}\right|$, with
$a_{\infty}=a(y\rightarrow\infty)$.

It is now clear how $\gamma$ affects the size of the core of $h(y)$ in an
inverse way, i.e. as the values of $\gamma$ increase, the resulting core
decreases. In addition, regarding the gauge profile function, it is worthwhile
to note that the solutions with $\gamma<1$ are characterized by the presence
of noncanonical plateaus which appear for intermediate values of the
coordinate $y$. In this sense, we point that our numerical investigation has
revealed that, for $\gamma>3$, both $h(y)$ and $a(y)$ tend to compactify as
$\gamma$ increases. Such a behavior is analogue to that already found in the
standard case ($G=1$) for increasing values of the coupling constant $g$.

In the Fig. 2, we show the numerical solutions to the BPS magnetic field
$B(y)$, from which it is possible to see how the shape of this field depends
on the value of $\gamma$ in a dramatic way. In the sequence, we proceed with
an analytical study of such a dependence, via which we clarify how the
aforecited plateaus give rise to the formation of nonstandard internal
structures which distinguish the behavior of the corresponding magnetic
sector.

In order to study the way $\gamma$ affects the shape of $B(y)$, we write this
field as

$B(y)=h^{2}\left(\gamma-h^{2}\right)^{2}\text{,}$ (42)

whose first derivative provides

$\frac{dB}{dy}=2h\left(\gamma-h^{2}\right)\left(\gamma-3h^{2}\right)\frac{dh}{dy}\text{.}$
(43)

Now, once the solution to the Skyrme profile function $h(y)$ is supposed to
vary monotonically from $1$ (at $y=0$) to $0$ (in the limit
$y\rightarrow\infty$, i.e. $h_{y}$ is always negative), one gets that the
condition $B^{\prime}(Y)=0$ provides the extreme points (note that we are
looking for intermediary values of $y$, from which we are here excluding both
the origin and the asymptotic limit)

$h(Y_{1})=h_{1}=\sqrt{\gamma}<1\text{,}$ (44)
$h(Y_{2})=h_{2}=\sqrt{\frac{\gamma}{3}}<1\text{,}$ (45)

where $0<Y_{1}<Y_{2}$.

At these points, the magnetic field assumes the values

$B_{1}=B\left(h_{1}\right)=0\text{,}$ (46)
$B_{2}=B\left(h_{2}\right)=\frac{4}{27}\gamma^{3}\text{,}$ (47)

respectively.

The first value, $B_{1}$, becomes a local minimum if $\gamma<1$, whereas
$B_{2}$ results in a local maximum if $\gamma<3$. Moreover, from Eq. (42), the
value of the magnetic field at the origin is given by

$B_{0}=B\left(y=0\right)=\left(\gamma-1\right)^{2}\text{.}$ (48)

In what follows, we use the Eqs. (46), (47) and (48) above to enumerate three
different pictures based on the values of $\gamma$. The interested reader can
apply the same prescription in order to describe additional configurations
with different $\gamma$. Here, it is important to emphasize that we are
considering intermediate values of $y$, i.e. we are excluding $y=0$ and those
values located in the asymptotic region $y\rightarrow\infty$.

Figure 3: Numerical solutions to the energy density of the BPS configurations,
i.e. $\varepsilon_{bps}(y)$ (depicted in units of
$\varepsilon_{bps,0}=\varepsilon_{bps}(y=0)$). Conventions as in the Fig. 1.
The profiles were normalized for the sake of visualization.

####  III.1.1 $\gamma=0$ case

The first picture is defined for $\gamma=0$, from which one gets that the BPS
equations (40) and (41) assume the form

$\frac{1}{r}\frac{da}{dr}=-h^{6}\text{,}$ (49)
$\frac{\left(1+a\right)}{r}\frac{dh}{dr}=-\frac{1}{2}h\text{,}$ (50)

which, in view of the redefinition $H(r)=\left[h(r)\right]^{3}$, can be
written as

$\frac{1}{r}\frac{da}{dr}=-H^{2}\text{,}$ (51)
$\frac{\left(1+a\right)}{r}\frac{dH}{dr}=-\frac{3}{2}H\text{.}$ (52)

In this case, despite the redefinition applied on the Skyrme profile function,
we note that the resulting Eqs. (51) and (52) can be obtained directly from
the general ones (29) and (30) for $G=1$, $g=N=1$ and $\lambda=\sqrt{1/3}$. We
then conclude that the a priori nontrivial case defined by $G(h)=h^{-4}$
stands for a merely redefinition of the usual case (defined by $G=1$) with a
different value of the coupling constant $\lambda$. As a consequence, we do
not expect significant changes to occur on the shape of the solutions,
especially on that of the magnetic sector. Therefore, in what follows, we
consider only the case with nonvanishing values of $\gamma$.

####  III.1.2 $0<\gamma<1$ case

A second picture occurs when $0<\gamma<1$. In this context, the solution (44)
is satisfied at some point $y=Y_{1}$. At this point, the magnetic field
vanishes (i.e. $B\left(y=Y_{1}\right)=0$, see the Eq. (46)), from which it is
reasonable to infer that the magnetic solution describes a centered lump
surrounded by a ring: the lump is positioned at the origin, its amplitude
being given by the Eq. (48) itself, while the radius of the ring is located at
some point $y=Y_{2}>Y_{1}$ (defined according to the Eq. (45)), the amplitude
of the ring standing for $B\left(y=Y_{2}\right)=\left(4/27\right)\gamma^{3}$,
see the previous Eq. (47).

We highlight how $\gamma$ determines the difference between these two
amplitudes: for $0<\gamma<0.75$, the magnitude of the centered lump is taller
than that of the ring (i.e.
$\left(\gamma-1\right)^{2}>\left(4/27\right)\gamma^{3}$). On the other hand,
when $\gamma=0.75$, the two magnitudes reach the very same value. Finally, for
$0.75<\gamma<1$, the amplitude of the ring is taller than that of the lump
positioned at $y=0$ (i.e.
$\left(\gamma-1\right)^{2}<\left(4/27\right)\gamma^{3}$).

The parameter $\gamma$ also controls the values of both $Y_{1}$ (i.e. the
point at which $B(y)$ vanishes) and $Y_{2}$ (the radius of the ring): as
$\gamma$ increases, the values of $h_{1}=\sqrt{\gamma}$ and
$h_{2}=\sqrt{\gamma/3}$ also increase and, once $h(y)$ varies monotonically
from $1$ to $0$, both $Y_{1}$ and $Y_{2}$ decrease (i.e. move toward the
origin).

####  III.1.3 $\gamma=1$ case

The case with $\gamma=1$ defines another picture, for which Eq. (44) holds at
the origin only (i.e. $h\left(y=0\right)=h_{1}=1$), which agrees with the
boundary condition (26). Therefore, the magnetic field vanishes at $y=0$,
which agrees with the result which comes from the Eq. (48) for $\gamma=1$. We
then conclude that the resulting magnetic profile stands for a single ring
whose radius is located at some point $y=Y_{2}$ (defined by
$h\left(y=Y_{2}\right)=h_{2}=\sqrt{1/3}$, see the Eq. (45)), its magnitude
being equal to $B\left(y=Y_{2}\right)=4/27$, see the Eq. (47).

As we said before, the very same prescription can be used by the reader to
describe additional configurations with different values of $\gamma$.

The following section explores a more complex context where the superpotential
is numerically determined. To achieve this goal, we consider the BPS and
superpotential equations parts of a self-dual system, which we solve using
numerical methods. Afterward, we compare the analytical and numerical
approaches by commenting on the main characteristics.

###  III.2 Numerical superpotential and its BPS solutions

Following the idea introduced in [39], we transform the Eq. (10) in a
differential equation for the superpotential $\mathcal{W}(h)$ which, together
with the BPS equations (29) and (30), forms a set of differential equations to
be solved for specific choices of both $V$ and $G$. Additionally, the boundary
conditions satisfied by the profiles $a(r)$, $h(r)$, and $\mathcal{W}(h)$
remain unaltered, from what it is possible to say that introducing a magnetic
permeability does not change the target space of the effective model, as
expected.

Table 1: Approximate numerical values used for the normalization of the noncanonical profiles obtained from the Eqs. (40) and (41). The standard values (with $\beta=0$) are $a_{\infty}=-0.632121$, $B_{0}=1$ and $\varepsilon_{bps,0}=2$, for the sake of comparison. $\gamma$ | $\quad a_{\infty}$ | $B_{0}$ | $\varepsilon_{bps,0}$
---|---|---|---
0.50 | $-0.079951$ | $0.2500$ | 1.2500
0.75 | $-0.135692$ | $0.0625$ | 1.0625
0.90 | $-0.215987$ | $0.0100$ | 1.0100
1.00 | $-0.283467$ | $0.0000$ | 1.0000

In order to continue, we now need to fix an specific expression for the
potential $U(h)$ itself. We then adopt an expression similar to the one which
appears in the asymptotic behavior exposed in the previous Eq. (37), that is

$U(h)=4h^{2},$ (53)

which is a power of the so-called old baby Skyrme potential $U_{o}(h)=2h$.
Then the superpotential equation (35) reads

$\frac{\lambda^{2}}{8}\left({\frac{d\mathcal{W}}{dy}}\right)^{2}+\left[\frac{\lambda^{4}g^{2}}{2}\left(\gamma-h^{2}\right)^{\beta}\mathcal{W}^{2}-4\mu^{2}h^{2}\right]\left({\frac{dh}{dy}}\right)^{2}=0\text{,}$
(54)

in terms of the coordinate $y=r^{2}/2$ and considering
$\mathcal{W}_{y}=\mathcal{W}_{h}h_{y}$.

Therefore, the Eq. (54) and the BPS ones (32) and (33 constitute a system of
differential equations which must be solved numerically according to the
boundary conditions (23), (26) and (27) in terms of the $y$-variable.

Below, to compare the numerical results with those obtained via the analytical
superpotential, we again fix $\beta=2$, $g=1$, $\lambda=1$, and $N=1$.
Moreover, we set $\mu^{2}=0.1$.

Figure 4 shows the numerical solutions to both $h(y)$(top) and $a(y)$
(bottom). We see that these profiles behave in the same general way as in the
previous case, including the arising of plateaus (which, as before, can be
understood as the origin of the formation of internal structures that
characterize the solution to the magnetic sector) in the solutions to the
gauge profile function for intermediary values of $y$.

The numerical solutions shown in Figure 5 depict the BPS magnetic field
$B(y)$. Again, the numerical behavior mimics the one obtained previously
(i.e., for a purely analytical superpotential), including the presence of
internal structures for intermediary $y$. In particular, the value of $B_{0}$
is controlled by $\gamma$ in the very same way as before, such as can be seen
from the Eq. (32) at $y=0$,

$B_{0}=\mp\lambda^{2}g^{2}\left(\gamma-1\right)^{\beta}\mathcal{W}_{0}.$ (55)

In this regard, the magnetic field at the origin always vanishes for
$\gamma=1$, independently of the value for $\mathcal{W}_{0}$.

###  III.3 Additional considerations

We now consider some attributes of our generalized model beyond its BPS
framework and the corresponding solutions. This section aims to demonstrate
that the standard Skyrme-Maxwell scenario’s basic properties remain unaltered
when adding a magnetic permeability. In order to perform such goal, we follow
the prescription previously stated in Ref. [39] for the Skyrme-Maxwell model.

####  III.3.1 On the existence of BPS solutions

We first consider the superpotential equation (10), where $V$, $G$, and
$\mathcal{W}$ are functions of $h$ only, i.e. (here,
$\mathcal{W}_{h}=d\mathcal{W}/dh$)

$V=\frac{\lambda^{4}g^{2}}{2G}\mathcal{W}^{2}+\frac{\lambda^{2}}{8}\left(\mathcal{W}_{h}\right)^{2}\text{,}$
(56)

which we solve for $\mathcal{W}_{h}$ as

$\mathcal{W}_{h}=\sqrt{\frac{8}{\lambda^{2}}V-4g^{2}\lambda^{2}\frac{\mathcal{W}^{2}}{G}}\text{,}$
(57)

from which we calculate $\mathcal{W}_{hh}$, i.e.

$\mathcal{W}_{hh}=\frac{4}{\mathcal{W}_{h}}\left[\frac{1}{\lambda^{2}}V_{h}-g^{2}\lambda^{2}\frac{\mathcal{W}}{G}\left(\mathcal{W}_{h}-\frac{\mathcal{W}}{2G}G_{h}\right)\right]\text{,}$
(58)

which reveals that $\mathcal{W}_{h}=0$ leads to a nonsingular
$\mathcal{W}_{hh}$ only provided that $V_{h}=0$ and $G_{h}=0$. In other words,
if $V_{h}\neq 0$ or $G_{h}\neq 0$, $\mathcal{W}_{h}=0$ produces a singularity,
and therefore a regular superpotential cannot be defined within the target
space, from which we conclude that the corresponding theory does not support
BPS solutions. As a consequence, also the Conjecture 2 as stated in the Ref.
[39] continues to hold even in the enlarged scenario defined in terms of a
nontrivial permeability, i.e., BPS solitons exist if and only if the
superpotential equation admits a well-defined solution on the whole target
space, with $\mathcal{W}_{h}=0$ in the corresponding open interval.

Figure 4: Numerical solutions to both $h(y)$ (top) and $a(y)$ (bottom,
depicted in units of $\left|a_{\infty}\right|$) obtained via the first-order
system formed by the Eqs. (32) and (33) and (54) for $\gamma=0.50$ (red line),
$\gamma=0.75$ (blue line), $\gamma=0.90$ (orange line) and $\gamma=1.00$
(green line). The corresponding usual profile (obtained via $G=1$) appears as
the black line, for the sake of comparison.

As before, one can always imagine the existence of a particular point
$h=h_{s}$ within the target space at which $\mathcal{W}_{h}(h=h_{s})=0$,
$V_{h}(h=h_{s})=0$, and $G_{h}(h=h_{s})=0$ simultaneously as an exception to
the aforecited Conjecture 2. In such a case, a well-defined superpotential
$\mathcal{W}(h)$ can be obtained from the superpotential equation. However,
given $\mathcal{W}_{h}=0$, the BPS Eq. (30) (which does not depend on the
magnetic permeability explicitly, i.e. is the very same one that appears in
the standard case) predicts a Skyrme profile function $h(r)$ with a
nonmonotonic behavior. The question here is that this same argument can be
applied to any arbitrary point $h=h_{s}$. In this sense, for $h_{s}$
sufficiently close to $1$, the corresponding solution reaches values that are
greater than the unity and therefore are outside the target space, which is
incompatible with the boundary conditions $h(r=0)=1$ and
$h(r\rightarrow\infty)\rightarrow 0$.

####  III.3.2 Bogomol’nyi bound for $g\rightarrow 0$

In Ref. [39], the authors argued that potentials of the type $V\sim
h^{\alpha}$ (with $\alpha>0$) allow for the existence of a global (i.e.
defined in the whole target space) solution $\mathcal{W}(h)$ for the standard
superpotential equation and, therefore, for the complete BPS scenario (with a
Bogomol’nyi bound and BPS solutions). This way, the point to be enlightened is
that if a BPS bound exists, it must attain the same value inherent to the
ungauged BPS baby Skyrme model in the limit of a vanishing electromagnetic
coupling constant $g$.

Figure 5: Numerical solutions to the BPS magnetic field $B(y)$ obtained via
the first-order system formed by the Eqs. (32) and (33) and (54). Conventions
as in the Fig. 1. Top: the profiles (depicted in units of $B_{0}$) for
$\gamma=0.50$ and $\gamma=0.75$. Bottom: the results for $\gamma=0.90$ and
$\gamma=1.00$. The corresponding standard solution again appears as the black
line. Here, some of the noncanonical profiles were normalized for the sake of
visualization.

We now verify whether such a convergence still holds even in our generalized
case. To attain this goal, we implement the prescription established by the
Ref. [39] and expand the second root, which appears on the right-hand side of
the Eq. (57) in a power series in $g^{2}$, via which, by considering only the
first relevant terms, we obtain

$\mathcal{W}_{h}\approx\frac{2}{\lambda}\sqrt{2V}\left(1-\frac{g^{2}\lambda^{4}}{4}\frac{\mathcal{W}^{2}}{VG}\right).$
(59)

In view of the expansion, we also consider the superpotential at second-order
in $g$, i.e.

$\mathcal{W}\approx\mathcal{W}^{(0)}+g^{2}\mathcal{W}^{(2)}\text{,}$ (60)

from which the Eq. (59) provides a set of differential equations for every
order in $g$ as

$\frac{\partial\mathcal{W}^{(0)}}{\partial
h}=\frac{2\sqrt{2}}{\lambda}\sqrt{V}\text{,}$ (61)
$\frac{\partial\mathcal{W}^{(2)}}{\partial
h}=-\frac{\sqrt{2}\lambda^{3}}{2}\frac{\left(\mathcal{W}^{\left(0\right)}\right)^{2}}{G\sqrt{V}}\text{.}$
(62)

In the sequence, by regarding the family of potentials $V(h)=h^{2\alpha}$
($\alpha>0$), we promptly integrate the Eq. (61) and obtain

$\mathcal{W}^{(0)}=\frac{2\sqrt{2}}{\lambda\left(\alpha+1\right)}h^{\alpha+1}\text{,}$
(63)

via which the Eq. (62) assumes the form

$\frac{\partial\mathcal{W}^{(2)}}{\partial
h}=-\frac{4\sqrt{2}\lambda}{\left(\alpha+1\right)^{2}}\frac{h^{\alpha+2}}{G}.$
(64)

The equation above clarifies that the solution for $\mathcal{W}^{(2)}$ depends
upon the magnetic permeability $G$, i.e. according to Eq. (60), the
superpotential $\mathcal{W}(h)$ echoes the presence of $G(h)$ starting from
the second-order in the electromagnetic coupling constant. In what follows, we
also consider $G(h)=(\gamma-h^{2})^{-2}$, from which we write Eq. (64) as

$\frac{\partial\mathcal{W}^{(2)}}{\partial
h}=-\frac{4\sqrt{2}\lambda}{\left(\alpha+1\right)^{2}}h^{\alpha+2}\left(\gamma-h^{2}\right)^{2}\text{,}$
(65)

whose solution reads

$\mathcal{W}^{(2)}=-\frac{4\sqrt{2}\lambda}{\left(\alpha+1\right)^{2}}h^{\alpha+3}\left(\frac{\gamma^{2}}{\alpha+3}-\frac{2\gamma
h^{2}}{\alpha+5}+\frac{h^{4}}{\alpha+7}\right)\text{,}$ (66)

which, together with the Eq. (63), leads to the solution for the
superpotential in the limit of sufficiently small $g$, i.e.

$\displaystyle\left.\mathcal{W}(h)=\frac{2\sqrt{2}h^{\alpha+1}}{\lambda\left(\alpha+1\right)}\left[1-\frac{2\lambda^{2}g^{2}h^{2}}{\alpha+1}\left(\frac{\gamma^{2}}{\alpha+3}\right.\right.\right.$
$\displaystyle\left.\left.-\frac{2\gamma
h^{2}}{\alpha+5}+\frac{h^{4}}{\alpha+7}\right)\right].$ (67)

The evaluation of the above expression at $h=1$ provides the BPS bound, which
allows us to conclude that our enlarged scenario correctly reproduces the
bound inherent to the ungauged baby Skyrme case (the interested reader may
compare the above expression with the Eq. (112) of the Ref. [39]). The novelty
here appears in the leading correction for small $g$ (which is of order
$g^{2}$ and negative, as in the Skyrme-Maxwell case with $G=1$), i.e. a
magnetic permeability affects not only the general behavior of the
corresponding term (through the power of $h$) but also its value calculated at
$h=1$.

####  III.3.3 Magnetic flux

It is also interesting to clarify whether the presence of a nontrivial
magnetic permeability affects the value of the magnetic flux calculated for
small and large electromagnetic coupling $g$. In order to offer a response to
this question, we first observe that the magnetic flux can be expressed as

$\Phi_{B}=2\pi\int B(r)rdr=-2\pi Na_{\infty}\text{,}$ (68)

where we have used both the Eq. (24) for the magnetic field and the conditions
(23). Here, we have defined $a_{\infty}=a(r\rightarrow\infty)$.

In the sequence, we divide the Eq. (29) by the Eq. (30), from which we obtain

${\frac{da}{\left(1+a\right)}=g^{2}\lambda^{2}F_{h}\,dh}\text{,}$ (69)

where

$F_{h}=\frac{4\mathcal{W}}{\mathcal{W}_{h}G}\text{.}$ (70)

The Eq. (69) has the solution (here, $C$ stands for an integration constant)

$\ln\left[C\left(1+a\right)\right]=g^{2}\lambda^{2}F(h)\text{,}$ (71)

where

$F(h)=4\underset{0}{\overset{h}{\int}}\frac{\mathcal{W}\left(h^{\prime}\right)}{\mathcal{W}_{h}\left(h^{\prime}\right)G\left(h^{\prime}\right)}dh^{\prime}\text{.}$
(72)

In order to calculate the value of $C$, we evaluate the solution Eq. (71) at
$r=0$, which leads to

$C=e^{g^{2}\lambda^{2}F(1)}\text{,}$ (73)

where we have used the boundary conditions $h(r=0)=1$ and $a(r=0)=0$.

Figure 6: Numerical solutions to the energy density of the BPS configurations,
i.e. $\varepsilon_{bps}(y)$ (depicted in units of $\varepsilon_{bps,0}$).
Conventions as in the Fig. 4. The profiles were normalized for the sake of
visualization.

Moreover, at the vacuum $h=0$, the Eq. (27) predicts $\mathcal{W}(h=0)=0$,
from which it is reasonable to suppose that the potential behaves $V\sim
h^{2\alpha}$ (again with $\alpha>0$). In such a scenario, the superpotential
equation Eq. (56) suggests the adoption of $\mathcal{W}_{h}\sim h^{\alpha}$,
$\mathcal{W}\sim h^{\alpha+1}$, and
$G^{-1}\sim\Delta+\mathcal{O}\left(h^{2}\right)$ (with $\Delta$ constant; for
$G(h)=\left(\gamma-h^{2}\right)^{-2}$, one gets $\Delta=\gamma^{2}$), from
which the Eq. (70) leads to

$F_{h}\sim 4\Delta h\text{,}$ (74)

which indicates that not only $F_{h}$, but also $F(h)$ itself vanishes at
$h=0$.

Now, whether we evaluate the Eq. (71) in the asymptotic region
$r\rightarrow\infty$, we get that (here, we have used both $F(h=0)=0$ and the
Eq. (73) for the value of $C$)

$a_{\infty}=-1+e^{-g^{2}\lambda^{2}F(1)}\text{,}$ (75)

via which the magnetic flux $\Phi_{B}$ Eq. (68) assumes the form

$\Phi_{B}=2\pi N\left[1-e^{-g^{2}\lambda^{2}F(1)}\right]\text{,}$ (76)

which leads to the following expressions

$g\text{ small: }\Phi_{B}\sim 2\pi Ng^{2}\lambda^{2}F(1)\text{,}$ (77)
$g\text{ large: }\Phi_{B}\sim 2\pi N\text{,}$ (78)

which are the same ones that appear in the restricted baby Skyrme-Maxwell
theory (with $G=1$), see the eqs. (121) and (122) of the Ref. [39],
respectively. However, despite the same symbolic form, the value of the
magnetic flux for small $g$ in Eq. (77) is now influenced by the nontrivial
magnetic permeability via the definition of $F(h)$ given in Eq. (72).

####  III.3.4 Some exact BPS solutions

We end this manuscript by exploring whether our generalized model admits exact
solutions. In this sense, we use the prescription proposed in the Ref. [39] in
the context of the canonical Skyrme-Maxwell scenario, the starting-point being
the BPS Eqs. (29), (30), now rewritten in terms of the variable $y=r^{2}/2$ as

$Na_{y}=-\lambda^{2}g^{2}\frac{\mathcal{W}}{G}\text{,}$ (79)
$4Nh_{y}\left(1+a\right)=-\mathcal{W}_{h}\text{,}$ (80)

for $N>0$

Whether we introduce the new field $p=h^{2}$, the above first-order
expressions can be written as

$Na_{y}=-\lambda^{2}g^{2}\frac{\mathcal{W}}{G}\text{,}$ (81)
$Np_{y}\left(1+a\right)=-\mathcal{W}_{p}p\text{,}$ (82)

where both $\mathcal{W}$ and $G$ are now functions of $p$.

In addition, from the Eq. (81), one gets that

$-\frac{N}{\lambda^{2}g^{2}}a_{yy}=\frac{p_{y}}{G}\left(\mathcal{W}_{p}-\frac{\mathcal{W}}{G}G_{p}\right)\text{,}$
(83)

which can be combined with the eqs. (81) and (82) in order to give

$-Na_{yy}=\frac{a_{y}}{\left(1+a\right)}\frac{\mathcal{W}_{p}p}{\mathcal{W}}\left(\mathcal{W}_{p}-\frac{\mathcal{W}}{G}G_{p}\right)\text{.}$
(84)

In order to continue with our construction, it is now necessary to specify
both $G(p)$ and $\mathcal{W}(p)$. Then, for such an aim, we set the
superpotential as (with $\sigma\geq 1$)

$\mathcal{W}(p)=\mathcal{W}_{0}p^{\sigma},$ (85)

where $\mathcal{W}_{0}=\mathcal{W}(p(0))=\mathcal{W}(1)$. In addition, for the
magnetic permeability, we choose

$\displaystyle G(p)$ $\displaystyle=$ $\displaystyle G_{0}p^{1-\beta}\;\text{
(for $\sigma=1$)},$ (86) $\displaystyle G(p)$ $\displaystyle=$ $\displaystyle
G_{0}p^{\sigma}\exp\left(-\frac{\beta(p^{1-\sigma}-1)}{\sigma(1-\sigma)}\right)\;\text{(for
$\sigma>1$)},\quad$ (87)

with $G_{0}=G(p(0))=G(1)$. Here, we also have defined the parameter $\beta$ as

$\beta=\frac{\delta}{\mathcal{W}_{0}}.$ (88)

By substituting both the equations above in Eq. (84), we attain a differential
equation similar to that found in Ref. [39], i.e.

$-Na_{yy}=\delta\,\frac{a_{y}}{1+a},$ (89)

whose solution for the gauge profile function $a(y)$ reads

$a(y)=-1+\frac{1}{C}\text{Li}^{-1}\left(\text{Li}(C)-\frac{C\delta}{N}y\right),$
(90)

expressed in terms of the logarithmic integral function Li, where the
parameter $C$ now is read

$\quad
C=\exp\left(\frac{\lambda^{2}g^{2}\mathcal{W}_{0}}{G_{0}\delta}\right).$ (91)

Immediately, we also solve the Eq. (79), which provides the following
expressions for $h(y)=\sqrt{p(y)}$, i.e.

$h(y)=\left(\frac{\ln[C(1+a(y))]}{\ln(C)}\right)^{\frac{1}{2\beta}},$ (92)

which holds for $\sigma=1$, while

$h(y)=\left[1+\frac{\sigma(1-\sigma)}{\beta}\ln\left(\frac{\ln\left[C\left(1+a\left(y\right)\right)\right]}{\ln\left(C\right)}\right)\right]^{\frac{1}{2(1-\sigma)}},$
(93)

holds when $\sigma>1$. Here, $a(y)$ is given by the Eq. (90).

The solutions for $\sigma\geq 1$ are extended profiles along all the radial
axis. Thus, the behavior in the limit $y\rightarrow\infty$ for the gauge field
reads

$a(y)\thickapprox a_{\infty}+\frac{1}{C}\exp\left(-\frac{C\delta}{N}y\right),$
(94)

where $a_{\infty}=-1+C^{-1}$. We therefore observe that the respective tail
follows a Gaussian-law decay. On the other hand, the behavior of the Skyrmion
profile function depends on the values of $\sigma$. This way, for $\sigma=1$,
such a function also presents a Gaussian-law decay, i.e.

$h(y)\thickapprox\left(\frac{1}{\ln\left(C\right)}\right)^{\frac{1}{2\beta}}\exp\left(-\frac{C\delta}{2\beta
N}y\right),$ (95)

while for $\sigma>1$, the respective tail follows a power-law decay,

$h(y)\approx\left(\frac{\beta N}{\delta
C\sigma(\sigma-1)}\right)^{\frac{1}{2(\sigma-1)}}y^{-\frac{1}{2(\sigma-1)}}.$
(96)

Table 2: Approximate numerical values used for the normalization of the noncanonical profiles obtained from the Eqs. (32), (33), and (54), for $\beta=2$, $g=1$, $\lambda=1$, $N=1$ and $\mu^{2}=0.1$. The standard values (with $\beta=0$) are $a_{\infty}=-0.679311$, $B_{0}=0.7057$ and $\varepsilon_{bps,0}=0.8$, for the sake of comparison. $\gamma$ | $\quad a_{\infty}$ | $B_{0}$ | $\varepsilon_{bps,0}$
---|---|---|---
0.50 | $-0.082308$ | $0.2189$ | 0.8000
0.75 | $-0.136245$ | $0.0551$ | 0.8000
0.90 | $-0.217839$ | $0.0087$ | 0.8000
1.00 | $-0.286781$ | $0.0000$ | 0.8000

It is important to note that, given our choices (85), (86), and (87) for
$\mathcal{W}(p)$ and $G(p)$, the factor
$\mathcal{W}^{-1}\mathcal{W}_{p}p\left(\mathcal{W}_{p}-G^{-1}\mathcal{W}G_{p}\right)$
which appears in the Eq. (84) can be reduced to a constant (in this case,
$\delta$). As a direct consequence, we arrive at the Eq. (89) (containing
$a(y)$ itself and its derivatives only), which also plays a central role in
the construction of our exact solutions.

##  IV Summary and perspectives

We have investigated BPS solitons inherent to a gauged baby Skyrme scenario
immersed in a magnetic medium. We have minimized the corresponding total
energy by implementing the BPS technique, from which we have verified that the
enlarged model also possesses a well-defined BPS structure. As expected, it
allows us to attain the self-dual equations and a lower bound for the total
energy. In such a context, we have clarified how the permeability enters the
differential relation between the superpotential and the corresponding BPS
potential, and also the self-dual equation which defines the magnetic field.
Consequently, the permeability may engender a magnetic field with an internal
structure, i.e. one that behaves in the standard way at the vacuum but
sometimes not at the origin. Besides, along the radial axis, its profile can
have a format different from the one found in the canonical gauged BPS Skyrme
model.

To solve the BPS system of differential equations, we have focused our
attention on those configurations possessing a rotational symmetry described
by the profile functions $h(r)$ and $a(r)$. Next, after choosing an analytical
expression for the permeability function (which includes the parameter
$\gamma\in\mathds{R}$), we have explored two different scenarios based on the
nature of the superpotential: In the first case, we set the superpotential as
an explicit function of $h(r)$, such that, together with the permeability, the
BPS potential be defined analuytically. In contrast, in the second situation,
we fix the particular form of the BPS potential and determine the
superpotential through the numerical solution of a system of differential
equations formed by the two BPS equations and the superpotential equation
itself.

For the sake of comparison, in both scenarios, we have worked with potentials
whose behaviors when approaching the vacuum are similar (here, both potentials
behave as $h^{2}$). We have then numerically solved the two scenarios and
depicted the corresponding profiles in the figs. 1, 2 and 3 (for the
analytical superpotential), and in the figs. 4, 5 and 6 (for the second case).
The resulting solutions have revealed how the profiles change with variations
on $\gamma$, giving rise to configurations with internal structures. In
particular, for the case with an exact superpotential, we have analyzed the
main features inherent to the shape of the magnetic field emerging within a
range of values of $\gamma$. This study has shown, for instance, the relation
between the amplitudes of the peaks (local maxima) inherent to the magnetic
profile, showing a format which differs dramatically from the one obtained in
the gauged BPS Skyrme model. Similarly, we could analyze the form of the
magnetic field for other values of $\gamma$.

Beyond the BPS framework and its solutions, we have investigated some basic
properties of our enlarged Skyrme-Maxwell scenario. For instance, we have
verified that it mimics some properties which the canonical model itself
satisfies. In particular, we have clarified that both the Conjecture 1 and
Conjecture 2 and the Corollary 1 stated in the Ref. [39] continue to hold. As
a second point, we have also discovered that our generalized theory correctly
reproduces the bound inherent to the ungauged baby Skyrme model in the limit
of a sufficiently small electromagnetic coupling constant $g$. In addition,
the effects of our generalization also appear at $g^{2}$-order. Subsequently,
in our third item, we have studied how permeability affects the value of the
magnetic flux $\Phi_{B}$ for both small and large values of $g$ by obtaining
the very same [symbolic] analytical expressions that appear in the standard
case. Thus, for small $g$, the permeability influences the magnetic flux. In
contrast, for large $g$, the flux $\Phi_{B}$ remains unaltered. Finally, we
have selected a class of functions defining the superpotential and magnetic
permeability which generate a family of noncompact solutions. Whereas the
gauge field and the magnetic one attain their vacuum values by following a
Gaussian-law decay, the tail of the Skyrmion’s profile follows a Gaussian-law
decay or a power-law one.

The results introduced in this manuscript aim to contribute to the
understanding of the electromagnetic properties of gauged Skyrmions by
studying its gauged BPS baby Skyrme versions. We now intend to apply the same
approach to other gauged versions of the restricted baby Skyrme enlarged
model, for instance, in the presence of the Chern-Simons’ or Born-Infeld’s
gauge fields. The results concerning these perspectives will be reported in
future contributions.

###### Acknowledgements.

The authors thank Prof. Dionísio Bazeia and Prof. Lukasz Stepien for
motivating discussions. This work was financed in part by the Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code
001, the Conselho Nacional de Pesquisa e Desenvolvimento Científico e
Tecnológico - CNPq and the Fundação de Amparo à Pesquisa e ao Desenvolvimento
Científico e Tecnológico do Maranhão - FAPEMA (Brazilian agencies). In
particular, J. A. thanks the full support from CAPES (via a PhD scholarship).
R. C. acknowledges the support from the grants CNPq/306724/2019-7,
CNPq/312155/2023-9, FAPEMA/Universal-01131/17, FAPEMA/Universal-00812/19 and
FAPEMA/APP-12299/22. E. H. thanks the support from the grant
CNPq/309604/2020-6. A. C. S. thanks the grants CAPES/88882.315461/2019-01 and
CNPq/150402/2023-6.

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