The Skyrme model and its generalisations provide a conceptually appealing field-theory basis for the description of nuclear matter and, after its coupling to gravity, also of neutron stars. In particular, a specific Skyrme submodel, the so-called Bogomol'nyi-Prasad-Sommerfield (BPS) Skyrme model, allows both for an exact field-theoretic and a mean-field treatment of neutron stars, as a consequence of its perfect fluid property. A pure BPS Skyrme model description of neutron stars, however, only describes the neutron star core, by construction. Here we consider different possibilities to extrapolate a BPS Skyrme neutron star at high baryon density to a description valid at lower densities. In the exact field-theoretic case, a simple effective description of the neutron star crust can be used, because the exact BPS Skyrme neutron star solutions formally extend to sufficiently low densities. In the mean-field case, on the other hand, the BPS Skyrme neutron star solutions always remain above the nuclear saturation density and, therefore, must be joined to a different nuclear physics equation of state already for the outer core. We study the resulting neutron stars in both cases, facilitating an even more complete comparison between Skyrmionic neutron stars and neutron stars obtained from other approaches, as well as with observations.
We show that the standard Skyrme model without pion mass term can be expressed as a sum of two BPS submodels, i.e., of two models whose static field equations, independently, can be reduced to first order equations. Further, these first order (BPS) equations have nontrivial solutions, at least locally. These two submodels, however, cannot have common solutions. Our findings also shed some light on the rational map approximation. Finally, we consider certain generalisations of the BPS submodels.
Using the BPS Lagrangian method we show that all known BPS submodels of the generalized Skyrme model, with a particular ansatz for the fields content, can be devided into three groups based on the (effective) number of derivative-terms in the BPS submodels. We are able to derive rigorously the Bogomolny's equations of those BPS submodels. The resulting Bogomolny's equations, along with possible constraint equations, are in general forms in which some of the known BPS submodels may contain other possible non-trivial (non-vacuum) solutions then the ones found in the literature. Furthermore, we d…
We construct the $\mathcal{N}=2$ supersymmetric extensions of different models containing a Chern-Simons term and study their BPS structure. We find that the coupling of the Chern-Simons term to the general gauged nonlinear $σ$-model allows for general potentials depending on the Kähler metric and preserving two supersymmetries. We construct from supersymmetry the potentials that allow for BPS structures in different $\mathbb{C}P^{N}$ models coupled to Chern-Simons and Maxwell terms. We also couple the Chern-Simons term to a higher-derivative field theory (the baby Skyrme model) and study its …
We show that the BPS Skyrme model, as well as its (2+1) dimensional baby version (restricted), can be coupled with an impurity in the BPS preserving manner. The corresponding Bogomolny equations are derived.
We study the coupling of nuclear matter described by the BPS Skyrme model to generalized gravity. Concretely, we consider the Starobinsky model which provides the leading-order correction to the Einstein-Hilbert action. Static solutions describing neutron stars are found both for the full field theory and for the mean-field approximation. We always consider the full Starobinsky model in the nonperturbative approach, using appropriately generalized shooting methods for the numerical neutron star calculations. Many of our results are similar to previous investigations of neutron stars for the St…
Two recently found coupled BPS submodels of the Skyrme model are further analyzed. Firstly, we provide a geometrical formulation of the submodels in terms of the eigenvalues of the strain tensor. Secondly, we study their thermodynamical properties and show that the mean-field equations of state coincide at high pressure and read $p=\barρ/3$. We also provide evidence that matter described by the first BPS submodel has some similarity with a Bose-Einstein condensate. Moreover, we show that extending the second submodel to a non-BPS model by including certain additional terms of the full Skyrme m…
Using the concept of strong necessary conditions (CSNC), we derive Bogomolny equations and BPS bounds for two modifications of the gauged baby BPS Skyrme model: the nonminimal coupling to the gauge field and k-deformed model. In particular, we study, how the Bogomolny equations and the equation for the potential, reflect these two modifications. In both examples, the CSNC method shows to be a very useful tool.
We consider a version of the Skyrme model where both the kinetic term and the Skyrme term are multiplied by field-dependent coupling functions. For suitable choices, this "dielectric Skyrme model" has static solutions saturating the pertinent topological bound in the sector of baryon number (or topological charge) $B=\pm 1$ but not for higher $|B|$. This implies that higher charge field configurations are unbound, and loosely bound higher skyrmions can be achieved by small deformations of this dielectric Skyrme model. We provide a simple and explicit example for this possibility. Further, we s…
In this paper, we search for the BPS skyrmions in some BPS submodels of the generalized Skyrme model in five-dimensional spacetime using the BPS Lagrangian method. We focus on the static solutions of the Bogomolny's equations and their corresponding energies with topological charge $B>0$ is an integer. We consider two main cases based on the symmetry of the effective Lagrangian of the BPS submodels, i.e. the spherically symmetric and non-spherically symmetric cases. For the spherically symmetric case, we find two BPS submodels. The first BPS submodels consist of a potential term and a term pro…
We study several deformations of the Skyrme model in three dimensions with self-dual sectors of arbitrary baryonic charge. We show that, for a family of background metrics as well as for a family of field dependent couplings, the model has one BPS sector, which may have any topological charge. We also study the gravitating case, where there are infinite BPS sectors, provided that a cosmological constant is added to the model.
We study radial vibrations of spherically symmetric skyrmions in the BPS Skyrme model. Concretely, we numerically solve the linearised field equations for small fluctuations in a skyrmion background, both for linearly stable oscillations and for (unstable) resonances. This is complemented by numerical solutions of the full nonlinear system, which confirm all the results of the linear analysis. In all cases, the resulting fundamental excitation provides a rather accurate value for the Roper resonance, supporting the hypothesis that the BPS Skyrme model already gives a reasonable approximate des…
In this work we consider the higher dimensional Skyrme model, with spatial dimension $d > 3$, focusing on its BPS submodels and their corresponding features. To accommodate the cases with a higher topological degree, \(B\geq 1\), a modified generalized hedgehog ansatz is used where we assign an integer \(n_i\) for each rotational plane, resulting in a topological degree that proportional to product of these integers. It is found via BPS Lagrangian method that there are only two possible BPS submodels for this spherically symmetric ansatz which shall be called as BPS Skyrme model and scale-inva…
The baby Skyrme model is a well-known nonlinear field theory supporting topological solitons in two space dimensions. In the limit where the term quadratic in derivatives (the "sigma model term") vanishes some additional structure emerges. The resulting ("extreme" or "restricted" or "BPS") baby Skyrme model has exact soliton solutions saturating a BPS bound which exists for this restricted model. Further, the restricted model has infinitely many symmetries and infinitely many conservation laws. Here we consider the gauged version of the restricted baby Skyrme model with gauge group U(1) and th…
We consider the Skyrme model in the near-BPS limit. The BPS part is made of the sextic term plus a potential and the deformation is made of the standard massive Skyrme model controlled by a small parameter $ε\ll1$. In order to keep the perturbation under theoretical and computational control, we find a model for which BPS Skyrmions have compact support, henceforth denoted as compactons, and the spherically symmetric $B=1$ Skyrmion represents the most stable solution. We use the $ε$-expansion scheme to systematically calculate the corrections to the energy and compare with the exact numerical c…
A new type of gauged BPS baby Skyrme model is presented, where the derivative term is just the Schroers current (i.e., gauge invariant and conserved version of the topological current) squared. This class of models has a topological bound saturated for solutions of the pertinent Bogomolnyi equations supplemented by a so-called superpotential equation. In contrast to the gauged BPS baby Skyrme models considered previously, the superpotential equation is linear and, hence, completely solvable. Furthermore, the magnetic flux is quantized in units of $2π$, which allows, in principle, to define thi…
We consider the baby Skyrme model in a physically motivated limit of reaching the restricted or BPS baby Skyrme model, which is a model that enjoys area-preserving diffeomorphism invariance. The perturbation consists of the kinetic Dirichlet term with a small coefficient $ε$ as well as the standard pion mass term, with coefficient $εm_1^2$. The pions remain lighter than the soliton for any $ε$ and therefore the model is physically acceptable, even in the $ε\to 0$ limit. The version of the BPS baby Skyrme model we use has BPS solutions with Gaussian tails. We perform full numerical computations…
We consider a restricted baby Skyrme-Maxwell scenario enlarged via the inclusion of a nontrivial magnetic permeability. We then proceed with the minimization of its total energy by means of the Bogomol'nyi-Prasad-Sommerfield (BPS) prescription, from which we get that the self-dual potential now depends on the magnetic permeability itself. As a result, we obtain not only the lower bound for the energy, but also the self-dual equations whose solutions saturate that bound. In such a context, we focus our attention on those time-independent gauged skyrmions with radial symmetry and no electric cha…
An extended version of the BPS Skyrme model that admits time-dependent solutions is discussed. Initially, by introducing a power law at the original potential term of the BPS Skyrme model the existence, stability and structure of the corresponding solutions is investigated. Then, the frequencies and half-lifes of the radial oscillations of the constructed time-dependent solutions are determined.
We study various properties of a perturbed signum-Gordon model, which has been obtained through the dimensional reduction of the called `first BPS submodel of the Skyrme model'. This study is motivated by the observation that the first BPS submodel of the Skyrme model may be partially responsible for the good qualities of the rational map ansatz approximation to the solutions of the Skyrme model. We investigate the existence, stability and various properties of oscillons and other time-dependent states in this perturbed signum-Gordon model.
We consider a class of generalizations of the Skyrme model to five spacetime dimensions ($d=5$), which is defined in terms of an $O(5)$ sigma model. A special ansatz for the Skyrme field allows angular momentum to be present and equations of motion with a radial dependence only. Using it, we obtain: 1) everywhere regular solutions describing localised energy lumps (Skyrmions); 2) Self-gravitating, asymptotically flat, everywhere non-singular solitonic solutions (Skyrme stars), upon minimally coupling the model to Einstein's gravity; 3) both static and spinning black holes with Skyrme hair, the…
The BPS Skyrme model is a model containing an $SU(2)$-valued scalar field, in which a Bogomol'nyi-type inequality can be satisfied by soliton solutions. In this model, the energy density of static configurations is the sum of the square of the topological charge density plus a potential. The topological charge density is nothing else but the pull-back of the Haar measure of the group $SU(2)$ on the physical space by the field configuration. As a consequence, this energy expression has a high degree of symmetry: it is invariant to volume preserving diffeomorphisms both on physical space and on …
We show that a suitable background field can bring a non-BPS topological soliton into its BPS, self-dual, counterpart. As an example we consider Skyrmions in the minimal Skyrme model. We prove the triviality of the corresponding moduli space. This means that the resulting self-dual Skyrmion does statically interact with the background field. We also show that the originally self-dual Skyrmions (e.g. solutions of the BPS Skyrme model) can preserve the self-duality after a coupling with a background field. In this case, BPS Skyrmions can be effortless moved with respect to the background.
The Skyrme model is a low energy effective field theory of strong interactions where nuclei and baryons appear as collective excitations of pionic degrees of freedom. In the last years, there has been a revival of Skyrme's ideas and new related models, some of them with BPS bounds (topological lower energy bounds), have been proposed. It is the aim of this paper to review how they can be applied to the study of neutron stars allowing for a description by means of topological solitons. We will focus on different aspects as the equation of state or the mass-radius relation, where we find that hi…
We study the BPS Skyrme model with potentials breaking the isospin symmetry and analyse how properties of exact solitonic solutions depend on a form of the isospin breaking potential. In the case of the strong symmetry breaking a new topologic structure is observed which enables us to decompose a BPS skyrmion into a lower dimensional defect localised on a brane (kink). We investigate some thermodynamical properties of such solitons as well as the role of the symmetry breaking potential in the resulting mean-field equation of state.
First proposed in 2013 by Yagi and Yunes, the quasi-universal \emph{I-Love-Q relations} consist of a set of relations between the moment of inertia, the spin-induced quadrupole moment and the electric quadrupolar tidal deformability of neutron stars which are independent of the Equation of State (EoS) within an accuracy of $\sim1\%$. In this work, we show that these relations hold for different Skyrme-based nuclear matter EoS and also for the star-like solutions of different Einstein-BPS-Skyrme-models, some of which do not even present a barotropic equation of state. Further, other quasi-unive…
