BPS Skyrme Model Research Articles

BPS skyrmions of generalized Skyrme model in higher dimensions

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 ≥ 1, a modified generalized hedgehog ansatz is used where we assign an integer ni 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-invariant model. The properties of the higher dimensional version of both submodels are studied and it is found that the BPS Skyrmions with B ≥ 1 exist in the first submodel but there is only B = 1 BPS Skyrmion in the second submodel. We also study the higher dimensional version of self-duality conditions in terms of strain tensor eigenvalues and find that, in general, the scale-invariant model has a stronger self-duality condition than the BPS Skyrme model.

Incompressible topological solitons

We discover a new class of topological solitons. These solitons can exist in a space of infinite volume like, e.g., $\mathbb{R}^n$, but they cannot be placed in any finite volume, because the resulting formal solutions have infinite energy. These objects are, therefore, interpreted as totally incompressible solitons. As a first, particular example we consider (1+1) dimensional kinks in theories with a nonstandard kinetic term or, equivalently, in models with the so-called runaway (or vacummless) potentials. But incompressible solitons exist also in higher dimensions. As specific examples in (3+1) dimensions we study Skyrmions in the dielectric extensions both of the minimal and the BPS Skyrme models. In the the latter case, the skyrmionic matter describes a completely incompressible topological perfect fluid.

Bogomolny equations for the BPS Skyrme models with impurity

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.

BPS Skyrme neutron stars in generalized gravity

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 Starobinsky model using other models of nuclear matter, but there are some surprising discrepancies. The “Newtonian mass” relevant for the surface redshift, e.g., results larger than the ADM mass in our model, in contrast to other investigations. This difference is related to the particularly high stiffness of nuclear matter described by the BPS Skyrme model and offers an interesting possibility to distinguish different models of nuclear matter within generalized gravity.

Background fields and self-dual Skyrmions

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 dielectric Skyrme model

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=±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 show that the |B|=1 BPS sector continues to exist for certain generalizations of the model like, for instance, after its coupling to a specific version of the BPS Skyrme model, i.e., the addition of the sextic term and a particular potential.

From the Sakai-Sugimoto model to the generalized Skyrme model

We derive the generalized Skyrme model as a low-energy effective model of the Sakai-Sugimoto model. The novelty with the past is the presence of the sextic term equal to the topological charge squared. This term appears when the $\omega$ meson, and the tower of states on top of it, are integrated out. We claim that, in the small 't Hooft coupling limit, the instanton is well described by a Skyrmion arising from the low energy effective Lagrangian of the Sakai-Sugimoto model. The sextic term plays a dominant role in this limit. Moreover, when a pion mass term is added we recover the BPS Skyrme model in the small 't Hooft coupling limit.

Bogomolny equations in certain generalized baby BPS Skyrme models

By using the concept of strong necessary conditions (CSNCs), we derive Bogomolny equations and Bogomol’nyi–Prasad–Sommerfield (BPS) bounds for two certain modifications of the baby BPS Skyrme model: the nonminimal coupling to the gauge field and the k-deformed ungauged 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 appears to be a very useful tool. We also find certain localized solutions of these Bogomolny equations.

“Half a proton” in the Bogomol’nyi-Prasad-Sommerfield Skyrme model

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 (skyrmions). 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 the target space. We demonstrate here that in the BPS Skyrme model such solutions exist that a fraction of its charge and energy densities is localised, and the remaining part can be far away, not interacting with the localised part.

A unified approach to nuclei: The BPS Skyrme Model

We present a concrete model of a low energy effective field theory of QCD, the well-known Skyrme Model. Specifically, we will work with the BPS submodel in order to describe the binding energies of nuclei. This BPS Skyrme model is characterized by having a saturated bound for the energy proportional to the baryon number of the nuclei. After presenting this classical result, we will proceed with a semi-classical quantization of the coordinates of spin and isospin. Then, with the further inclusion of the Coulomb interaction as well as a small explicit breaking of the isospin symmetry, we finally calculate the binding energies of nuclei, where an excellent agreement has been found for the nuclei with high baryon number. Besides this, we also apply this model to the study of some thermodynamic properties and to neutron stars.

Bogomolny equation for the BPS Skyrme model from strong necessary conditions

We present a systematic tool for derivation of the Bogomolny equation for the BPS Skyrme model. Furthermore, we find a generalization of the Bogomolny equation to the case corresponding with a non-zero value of the external pressure. The method is based on the concept of strong necessary conditions and can be applied to any Skyrme-like theory.

Time-dependent Bogomolny-Prasad-Sommerfeld skyrmions

An extended version of the Bogomolny-Prasad-Sommerfeld (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 are investigated. Then, the frequency and half-lifes of the radial oscillations of the constructed time-dependent solutions are determined.

Composite BPS Skyrmions from an Exact Isospin Symmetry Breaking

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.

Gravitating BPS Skyrmions

The BPS Skyrme model has many exact analytic solutions in flat space. We generalize the model to a curved space or spacetime and find that the solutions can only be BPS for a constant time-time component of the metric tensor. We find exact solutions on the curved spaces: a 3-sphere and a 3-hyperboloid; and we further find an analytic gravitating Skyrmion on the 3-sphere. For the case of a nontrivial time-time component of the metric, we suggest a potential for which we find analytic solutions on anti-de Sitter and de Sitter spacetimes in the limit of no gravitational backreaction. We take the gravitational coupling into account in numerical solutions and show that they are well approximated by the analytic solutions for weak gravitational coupling.

Neutron stars in the Bogomol'nyi-Prasad-Sommerfield Skyrme model: Mean-field limit versus full field theory

Using a solitonic model of nuclear matter, the BPS Skyrme model, we compare neutron stars obtained in the full field theory, where gravitational back reaction is completely taken into account, with calculations in a mean-field approximation using the Tolman-Oppenheimer-Volkoff approach. In the latter case, a mean-field-theory equation of state is derived from the original BPS field theory. We show that in the full field theory, where the energy density is non-constant even at equilibrium, there is no universal and coordinate independent equation of state of nuclear matter, in contrast to the mean-field approximation. We also study how neutron star properties are modified by going beyond mean field theory, and find that the differences between mean field theory and exact results can be considerable. Further, we compare both exact and mean-field results with some theoretical and phenomenological constraints on neutron star properties, demonstrating thus the relevance of our model even in its most simple version.

Baryon chemical potential and in-medium properties of BPS skyrmions

We continue the investigation of thermodynamical properties of the BPS Skyrme model. In particular, we analytically compute the baryon chemical potential both in the full field theory and in a mean-field approximation. In the full field theory case, we find that the baryon chemical potential is always exactly proportional to the baryon density, for arbitrary solutions. We further find that, in the mean-field approximation, the BPS Skyrme model approaches the Walecka model in the limit of high density - their thermodynamical functions as well as the equation of state agree in this limit. This fact allows to read off some properties of the $\omega$-meson from the BPS Skyrme action, even though the latter model is entirely based on the (pionic) $SU(2)$ Skyrme field. On the other hand, at low densities, at the order of the usual nuclear matter density, the equations of state of the two models are no longer universal, such that a comparison depends on some model details. Still, also the BPS Skyrme model gives rise to nuclear saturation in this regime, leading, in fact, to an exact balance between repulsive and attractive forces. The perfect fluid aspects of the BPS Skyrme model, which, together with its BPS properties, form the base of our results, are shown to be in close formal analogy with the Eulerian formulation of relativistic fluid dynamics. Within this analogy, the BPS Skyrme model, in general, corresponds to a non-barotropic perfect fluid.

Near BPS skyrmions and restricted harmonic maps

Motivated by a class of near BPS Skyrme models introduced by Adam, Sánchez-Guillén and Wereszczyński, the following variant of the harmonic map problem is introduced: a map φ:(M,g)→(N,h) between Riemannian manifolds is restricted harmonic if it locally extremizes E2 on its SDiff(M) orbit, where SDiff(M) denotes the group of volume preserving diffeomorphisms of (M,g), and E2 denotes the Dirichlet energy. It is conjectured that near BPS skyrmions tend to restricted harmonic maps in the BPS limit. It is shown that φ is restricted harmonic if and only if φ∗h has exact divergence, and a linear stability theory of restricted harmonic maps is developed, from which it follows that all weakly conformal maps are stable restricted harmonic. Examples of restricted harmonic maps in every degree class R3→SU(2) and R2→S2 are constructed. It is shown that the axially symmetric BPS skyrmions on which all previous analytic studies of near BPS Skyrme models have been based, are not restricted harmonic, casting doubt on the phenomenological predictions of such studies. The problem of minimizing E2 for φ:Rk→N over all linear volume preserving diffeomorphisms is solved explicitly, and a deformed axially symmetric family of Skyrme fields constructed which are candidates for approximate near BPS skyrmions at low baryon number. The notion of restricted harmonicity is generalized to restricted F-criticality where F is any functional on maps (M,g)→(N,h) which is, in a precise sense, geometrically natural. The case where F is a linear combination of E2 and E4, the usual Skyrme term, is studied in detail, and it is shown that inverse stereographic projection R3→S3≡SU(2) is stable restricted F-critical for every such F.

BPS Skyrmions as neutron stars

The BPS Skyrme model has been demonstrated already to provide a physically intriguing and quantitatively reliable description of nuclear matter. Indeed, the model has both the symmetries and the energy–momentum tensor of a perfect fluid, and thus represents a field theoretic realization of the “liquid droplet” model of nuclear matter. In addition, the classical soliton solutions together with some obvious corrections (spin–isospin quantization, Coulomb energy, proton–neutron mass difference) provide an accurate modeling of nuclear binding energies for heavier nuclei. These results lead to the rather natural proposal to try to describe also neutron stars by the BPS Skyrme model coupled to gravity. We find that the resulting self-gravitating BPS Skyrmions provide excellent results as well as some new perspectives for the description of bulk properties of neutron stars when the parameter values of the model are extracted from nuclear physics. Specifically, the maximum possible mass of a neutron star before black-hole formation sets in is a few solar masses, the precise value of which depends on the precise values of the model parameters, and the resulting neutron star radius is of the order of 10 km.

Thermodynamics of the BPS Skyrme model

One problem in the application of the Skyrme model to nuclear physics is that it predicts too large a value for the compression modulus of nuclear matter. Here we investigate the thermodynamics of the BPS Skyrme model at zero temperature and calculate its equation of state. Among other results, we find that classically (i.e. without taking into account quantum corrections) the compressibility of BPS Skyrmions is, in fact, infinite, corresponding to a zero compression modulus. This suggests that the inclusion of the BPS submodel into the Skyrme model Lagrangian may significantly reduce this too-large value, providing further evidence for the claim that the BPS Skyrme model may play an important role in the description of nuclei and nuclear matter.

Chiral Dirac–Born–Infeld solitons with SDiff symmetry

We propose a new DBI extension of a Skyrme type model which allows for BPS topological solitons with arbitrary value of the baryon charge. The model is built out of the baryon density squared and in the limit of small fields tends to the BPS Skyrme model. We consider some generalizations to higher dimensions and other K-deformed actions.