On the topological charge of $SO(2)$ gauged Skyrmions in $2+1$ and $3+1$ dimensions

The question of the dependence of the topological charge $q$ of a gauged Skyrmion, on the gauge field, is studied quantitatively. Two examples, both gauged with $SO(2)$ are studied and contrasted: i) The $O(3)$ model in $2+1$ dimensions, and ii) The $O(4)$ model in $3+1$ dimensions. In case i), where the (usual) Chern-Simons (CS) term is present, the value of $q$ changes sign, going through zero. This evolution is tracked by a parameter characterising the solutions in the given theory. In case ii), in which dimensions no CS density is available, the evolution of $q$ is not observed.


Introduction
The topological charge q of gauged Skyrmions presents peculiarities that are absent in gauged Higgs solitons. While in the latter case this quantity always equals the winding number of the Higgs field, in the case of Skyrmions the value of q in general may depart from the winding number, or the "baryon number".
The topological charge for the SO(2) gauged O(4) (Skyrme) model in 3 + 1 dimensions is defined in Ref. [1] and in [2,3], and, for the SO(2) gauged O(3) (planar Skyrme) model in 2 + 1 dimensions in Ref. [4]. The topological charges for the SO(D) gauged O(D + 1) sigma model on R D (D = 2, 3, 4) were defined in [5], and the soliton of the SO(3) gauged O(4) sigma model on R 3 was first reported in [6]. More detailed studies of the latter solitons were given in [7], while solitons of the SO(4) gauged O(5) sigma model on R 4 were reported in [8]. The most comprehensive and recent definitions of the generic topological charges of gauged Skyrmions are given in Appendix B of Ref. [9].
With the exception of the work in Ref. [1], where attention was focused on the decay of the baryon number 1 , the works of [2,3,4,5,6,8] are concerned with the construction of topologically stable solitons. Here, we are concerned with the SO(2) gauged O (3) and O(4) sigma models in 2 + 1 and 3 + 1 dimensions, respectively. But unlike in the works of Refs. [4,2,3], our focus here will be the evolution of the topological charge, e.g., its possible decay. 2 In odd dimensional spacetimes, where the Chern-Simons (CS) term is defined, one finds that the effect of this CS dynamics results in the mass/energy of the static solitons both to increase and to decrease with increasing global charge (electric charge and spin). This was clearly demonstrated in Section 4 of Ref. [13] in 2 + 1 dimensions, where the effect of Chern-Simons dynamics were studied also in other 2 + 1 dimensional models. This evolution of the mass/energy of the static solitons was tracked by a parameter characterising the solution in the given theory, which is strictly due to the CS dynamics. In the present work we show that this same dynamical effect results in the evolution of the value of the topological charge q, changing sign and passing through zero, in the given theory. Such a solution cannot be found in the absence of the CS term [13]. This is the main result of the first part of the present work, namely the study of the SO(2) gauged O(3) model, with emphasis on the issue of topological charge q. (A detailed study of generic SO(2) gauged Skyrmions is given in [10].) In even dimensional spacetimes however, no CS density is defined. There is nonetheless a class of Skyrme-CS (SCS) densities satisfying the properties of the usual CS density in all dimensions, including even dimensions. These are introduced in Ref. [9]. It happens that in the present example this quantity vanishes under the imposition of appropriate symmetries, and in the absence of any Chern-Simons type dynamics we are limited to studying the model considered in Refs. [2,3]. There, it was found that the mass/energy of the soliton of the gauged system increases monotonically with increasing electric (global) charge. Since no parameter characterising the solutions is available, the coupling strength of the Maxwell field, λ 0 in (11) below, was varied in [2]. This clearly changes the theory being considered, but since in any given theory the value of both the energy and the electric charge Q e depend on λ 0 , this enabled [2] the tracking of the energy with increasing Q e . Here by contrast, we consider the dependence of the topological charge on λ 0 and show that q remains constant. This paper is organised as follows. In Section 2 below we present our quantitative results on the SO(2) gauged Skyrmion in 2 + 1 dimensions, and in Section 3 our results on the SO(2) gauged Skyrmion in 3 + 1 dimensions. In Section 3 we summarise our results and point out to further developments.

SO(2) gauged Skyrmions in + 1 dimensions
Ever since Schroers' construction of solitons [4] of the U(1) gauged planar Skyrme model, there has been considerable interest 3 in this area. The dynamics of the Abelian field in Ref. [4] was described by the Maxwell density, subsequently the latter was replaced by the Chern-Simons (CS) density [14,15,16]. More recently, the effect of Chern-Simons dynamics on the combined Maxwell-CS O(3) sigma model was studied in Ref. [13]. What was found there (in Section 4) is that the mass/energy of the soliton decreased with increasing global charges -electric charge and angular momentum -in some ranges of the parameters. This effect is absent, in the absence of the CS term in the Lagrangian, and is strictly a result of the Chern-Simons dynamics.
In the present note, we pursue further consequences of the presence of the Chern-Simons term. The new effect in question is the evolution of the topological charge due to the gauging of the the O(3) sigma model. While this is not strictly a dynamical effect, it is again predicated on the presence of a CS term in the Lagrangian. The precise mechanism of this is the role that the CS term plays in establishing the topological lower bound (Belavin inequalities). This, and other details will be exposed in a future extended work.
The Skyrme model on R 2 is described by the scalar φ a = (φ α , φ 3 ) , |φ a | 2 = 1 , and the SO(2) gauging prescription is 4 A µ being the SO(2) connection. The topological charge density for the SO(2) gauged Skyrme model is [5], [9], [10] independently of whether the dynamics is Maxwell or CS. In (3), ̺ 0 is the winding number density whose volume integral is the "baryon number", A i , i = 1, 2 , are the magnetic components of the Abelian connection, and v is a real constant. What is crucial here is that in the Maxwell gauged case the Belavin inequalities, which must be satisfied by the requirement of topological stability, force the constant v to be equal to one, v = 1, and as a consequence of the boundary condition φ 3 (∞) = 1, the contribution of the Abelian field to the integral of ̺ disappears. What remains is the integral of ̺ 0 , namely, the "baryon number". The situation is starkly different when the dynamics of the Abelian field is described by the Chern-Simons term. In that case, it follows from the Bogomol'nyi analysis, that the constant v in (3) can take any value, including v = 1, as a consequence of which the topological charge will depart from the "baryon number", quantitatively. It is our aim here to track the evolution of the topological charge, due to this mechanism.
To illustrate this mechanism briefly, consider the choice of potential V ≡ V [φ a ] in the Lagrangian. In the case of Maxwell dynamics, this potential is fixed by the constraints of the Belavin inequalities to be consistently with the boundary value The corresponding Bogomol'nyi analysis in the case of Chern-Simons dynamics, which was presented in [16] and in Appendix C of [10], leads to the potential (In (4)-(5), λ , η and κ are real constants.) The striking property of the potential (5) is that the (finite energy) condition V CS [φ a (∞)] = 0 can be satisfied for any value of the real constant v, since at spatial infinity This is precisely the situation where the topological charge will depart from the winding number n, namely from the "baryon number". In practice, v = 0 is the most convenient choice. The Chern-Simons-Skyrme (CS-S) model studied in Ref. [16], can be augmented by the Maxwell term without invalidating the topological inequalities, to yield the Maxwell-CS-S model described by the Lagrangian considered here. The Lagrangian (7) is not the minimal one that follows from the Belavin inequalities. It has been further augmented by the quartic kinetic Skyrme term with coupling τ . The reason for this is our requirement that after gauge decoupling, the system support Skyrmions with winding number n.
To analyse the system (7) numerically, we subject it to azimuthal symmetry where θ is the azimuthal angle and n is the winding number of the Skyrme scalar.
We have analysed the reduced the one dimensional Lagrangian descended from (7) subject to (8), (9), numerically. Also, subject to the above Ansatz, the volume integral of ̺ (as given by (3)), namely the topological charge, can be readily evaluated as 5 where a ∞ = a(∞). We see from the above relation that solutions characterised by a ∞ < 0 result in the dissipation of the topological charge. Thus, constructing solutions with a ∞ < 0, one can track the evolution of the topological charge, resulting also in the annihilation of q. The necessary handle on a ∞ needed for this results precisely from the same mechanism encountered in Ref. [13], where the (magnetic) quantity a ∞ changes with the (electric) b ∞ (= b(∞)). This is strictly a Chern-Simons (CS) effect, as the CS density features both the magnetic and electric fields.
We have constructed such solutions with κ = 0, the relation of the asymptotic value of the electric function, b ∞ to the magnetic counterpart a ∞ being displayed in Figure 1. We know already from our results in Ref. [13] that due to this effect, the dependence of the mass/energy E on the global charges is non-standard, namely the dependence on the electric charge Q e and the angular momentum J displayed respectively (left and right panels) in Figure 2.
Here, we analyse further the relation of the topological charge q and the energy E. In Figure 3 (left panel) we represent the energy per unit winding number n versus the topological charge for three values of n for the same parameters as in Figure. 1. For small values of the coupling strength τ of the (quartic) Skyrme kinetic term, the minimum of the energy occurs at values of the topological charge around the winding number. However, when higher values of τ are considered, the minimum of the energy occurs at values clearly different from the winding number. This is  shown in Figure. 3 (right panel), where the minimum of the curve for n = 1, τ = 10 is located at q ≈ 0.556. Concerning the stability of these solutions, one would be tempted to state that most stable solution would correspond to that with least energy, which according to Figure. 3 (right panel) does not possess integer topological charge, in general. From Figure 3. (right panel), we see that in the absence of the Skyrme term (τ = 0) the energetically favoured configurations are those with topological charge equal to the winding number. But when the Skyrme term is introduced, with increasing values of the coupling strength τ , it appears that the energetically favoured solutions have topological charge progressively smaller than the winding number, the latter being the topological charge of the Skyrmion in the gauge decoupling limit.

SO(2) gauged Skyrmions in + 1 dimensions
The stationary solutions of this system were studied in Refs. [2] and [3]. In [2], it was seen that the mass of the electrically charged Skyrmion was larger than that of the electrically neutral one, while in [3] it was shown that these Skyrmions spin. That the mass of the electrically charged Skyrmion is always larger than the electrically neutral one was seen in [2] by observing that these energies both depend on the Maxwell coupling λ 0 (in (11) below), thus plotting both these energies versus λ 0 illustrates this fact, as seen in Figure 4 (left panel).
That the mass increases monotonically with increasing electric charge seen in [2] is not surprising, since we know from our work in Ref. [13] that to reverse this trend can be achieved only by the introduction of Chern-Simons (CS) dynamics, and, in 3 + 1 dimensions no (usual) CS density is defined. While there is a class of densities introduced in Ref. [9] which share the properties of Chern-Simons that can be defined in even dimensional spacetimes, the particular one that applies here, defined in terms of the Maxwell field and the O(6) Skyrme scalar, actually vanishes when subjected to axial symmetry. Hence, in the absence of any Chern-Simons dynamics, we are back to the study in Refs. [2,3].
What we have done here, additionally to tracking the increase of mass with increasing electric charge and angular momentum in Refs. [2,3], is to study the effect of the electromagnetic field on the topological charge 6 .
The studied SO(2) gauged Skyrmion system is described by the Lagrangian where φ a = (φ α , φ A ) ; α = 1, 2 ; A = 3, 4, with |φ a | 2 = 1, Skyrme scalars of the O(4) sigma model, with V = 1 − φ 4 the usual Skyrme potential. λ i (i = 0, 1, 2, 3), in (11) are real numbers parameterising the coupling constants of the model. Also, the gauging prescriptions for φ a is In the ungauged case, the topological charge density of the Skyrmions is The natural generalization of this expression in the presence of gauge fields is found by replacing partial derivatives with gauge derivatives, which, however, does not leads to a conserved charge. The conserved, gauge invariant generalization of (13) is the sum of two term 7 which can be can be written alternatively in a manifestly gauge invariant form As such, the topological charge q consists in the sum of two terms, encoding the contribution of ̺ G and W in (16) with q G the bare charge (as found by integrating directly the density (14)), and q W a compensating contribution (from (16)). Moreover, as seen from (15) the topological charge is still an integer, as long as there are no singularities and the surface terms from Ω i vanish. Imposing axial symmetry on the Maxwell connection A µ = (A i , A z .A 0 ), we have the Ansatz where a = a(ρ, z) , b = b(ρ, z), and, ρ 2 = |x i | 2 , i = 1, 2.
The solutions constructed are typified by the asymptotic values where Q is the electric charge and b ∞ is a free constant characterising the given solution (the electrostatic potential). Also, in contrast with the solutions in 2 + 1 dimensions constructed in the previous Section, there is no freedom in the asymptotic value of the magnetic potential, with a ∞ = −n for all solutions. Again we are using the notation a ∞ = a(∞) and b ∞ = b(∞). The corresponding Ansatz for the O(4) scalar φ a is where R, S and T are functions of ρ and z, subject to the constraint R 2 + S 2 + T 2 = 1. Also, n α is the unit vector in the subplane (x 1 , x 2 ), with vorticity n.
Subject to symmetry (18)- (20), the topological charge density (15) (or (16)) reduces to the two dimensional density which, as expected, is a curl. In principle, the value of the integral of (21) could be different from the integral of ̺ 0 , since it encodes a contribution of the gauge potential. However, in contrast with the 2+1 dimensional system endowed with Chern-Simons dynamics 8 , studied in the previous Section, this contribution vanishes for regular configurations, and the integral of (21) is still equal to the baryon number, q = n, for any value of the varying the coupling constants, in particular for any λ 0 . At the same time, the contribution of the gauge invariant terms q G and q W to the topological charge is model dependent, as confirmed by our numerical results 9 . The dependence of the mass/energy E and of the electric charge Q e of the spinning Skyrmions on λ 0 is displayed in Figure 4 (left panel). As expected, for fixed coupling constants λ i and winding number n, E increases with increasing electrostatic potential b ∞ for all values of λ 0 . In the gauge decoupling limit λ 0 → ∞, these are the uncharged, spinning Skyrmions characterised by b ∞ , found in Ref [22].
The difference between the 'bare' charge q G and the 'compensating' one q W is displayed in Figure 4 (right panel) as a function of λ 0 . We see that the influence of the gauge field is to cause this quantity to have positive, zero and negative values in distinct theories. For large λ 0 , the contribution of q W is negligible, with q G → q. However, q G decreases with λ 0 , with q W dominating for small λ 0 . We note that for very small values of λ 0 , when the difference between q G and q W is vanishing (or even becomes negative, as seen from Figure 4 (right panel)), the energy of the Skyrmion (as seen in Figure 4 (left panel)), is finite.

Summary
We have studied two SO(2) gauged Skyrme systems, the O(3) and O(4) sigma models in 2+1 and 3 + 1 dimensions, respectively. In the first case, the system can be endowed with Chern-Simons dynamics, while in the second case, not. The main message here is to highlight the contrast when Chern-Simons dynamics is present or absent.
When Chern-Simons dynamics is present, two related phenomena are observed. i) The mass of the soliton can both increase and decrease with increasing global charge (electric or spin), and ii) The topological charge can evolve through positive to zero to negative values, for various solutions characterised by a ∞ , the asymptotic value of the magnetic field defining the topological charge. This evolution is contingent on the dependence of b ∞ , the asymptotic value of the electric field, this relation being dependent on the intertwining of the electric and magnetic fields in the definition of the Chern-Simons density. This is clearly demonstrated in the 2 + 1 dimensional system studied here, for the SO(2) gauged Skyrmion, in a model with both Maxwell and Chern-Simons dynamics, and both with and without a quartic kinetic Skyrme term.
Interestingly, the energetically favoured configurations appear to occur for smaller values of the topological charge, with progressively stronger coupling τ of the Skyrme term. Indeed, when τ = 0 the minimum energy configurations coincide with the winding number n (see Figure 3), while for very large values of τ the energetically favoured configuration tends to the n/2 (see quantitative details in [10].) In the 3 + 1 dimensional system studied, where there is no Chern-Simons dynamics, these phenomena are absent. The mass of the soliton increases monotonically with increasing the electric charge charge, and the topological charge is fixed. Moreover, changing the theory by varying a coupling constant does not lead to changing values of the topological charge. We suggest that the phenomena of non-standard mass-electric charge/spin 10 and evolving topological charge observed in the 2+1 dimensional example are absent in the 3+1 dimensional model here, because of the absence of Chern-Simons dynamics in the latter case.
The challenge is to supply an example of a gauged Skyrme model in 3 + 1 dimensions, where a Chern-Simons term is defined. Such examples are considered in Ref [9]. In the case of SO(2) gauging considered in this work, such a Skyrme-CS term was considered, but it turned out that that term vanished under symmetry imposition. The challenge thus is to consider gauging with a higher (appropriate) gauge group, but this is a subject of another investigation.