Spinning U(1) gauged Skyrmions

We construct axially symmetric solutions of U(1) gauged Skyrme model. Possessing a nonvanishing magnetic moment, these solitons have also a nonzero angular momentum proportional to the electric charge.

features of small nuclei. The U (1) gauged Skyrme model was originally proposed by Callan and Witten to study the decay of the nucleons in the vecinity of a monopole [16]. Axially symmetric solutions of this model were constructed previously in [17], but the emphasis there was on the static properties of nucleons and not the calculation of its classical spin. We define our model in terms of the O(4) sigma model field φ a = (φ α , φ A ), α = 1, 2; A = 3, 4, satisfying the constraint |φ α | 2 + |φ A | 2 = 1, the Lagrangean of the Maxwell gauged Skyrme model is (up to an overall factor which we set equal to one) in terms of the Maxwell field strength F µν , and the covariant derivatives defined by the gauging prescription The energy-momentum tensor which follows from (1) is Here we note that the Skyrmion gauged with the purely magnetic U (1) field is a topologically stable soliton. This is stated in terms of topological lower bound on the static energy density functional of the purely magnetically gauged system, namely the T tt component of (3) with A t = 0, Defining the gauge invariant topological charge density as the gauge invariance of ̺ is manifest from (5), while it is easily checked that the finite energy conditions lead to the vanishing of the surface integral term in (6), as a result of which the topological is simple the volume integral of the first term, namely the winding number n or, Baryon charge. As was shown in [17] in detail, the energy density functional (4) is bounded from below by The ansatz.-In a cylindrical coordinate system, we parametrise the axially symmetric Maxwell connection as a(ρ, z) and b(ρ, z) corresponding to the electric and magnetic potentials, with n a positive integer -the winding number, and the polar parametrisation of the chiral field in terms of the two functions f (ρ, z) and g(ρ, z) as φ α = sin f sin g n α , φ 3 = sin f cos g, φ 4 = cos f, where ρ = |x α | 2 , α = 1, 2, and z = x 3 . In the following we will find it convenient instead to work with spherical coordinates (r, θ), i.e. ρ = r sin θ and z = r cos θ. After replacing this ansatz in (1), one finds the reduced lagrangeean The Euler-Lagrange equations arising from the variations of this Lagrangean have been integrated by imposing the following boundary conditions, which respect finite mass-energy and finite energy density conditions as well as regularity and symmetry requirements. We impose at infinity, and at the origin. For solutions with parity reflection symmetry (the case considered in this paper), the boundary conditions along the z-axis are and agree with the boundary conditions on the ρ-axis, except for g(r, θ = π/2) = π/2. It may appear from the boundary conditions (11)-(13) that the natural condition a| θ=0,π = n is not imposed. This is not done since its imposition in addition to (11)-(13) would be an an overdetermination. We have nontheless checked that a = n is satisfied on the z-axis by the numerical solutions.
The constant V appearing in (11) corresponds to the magnitude of the electric potential at infinity and has a direct physical relevance. In the pure Maxwell theory, one can set set V = 0 (or any other value) without any loss of generality. In the U(1) gauged Skyrme model, however, such a gauge transformation would render the whole configuration time-dependent.
Integration over all space of the energy density E yields the total mass-energy, E = T tt √ −gd 3 x. The total angular momentum is given by However, by using the field equations, the volume integral of the above quantity can be converted into a surface integral at infinity in terms of Maxwell potentials The field equations imply the asymptotic behaviour of the electric potential b ∼ V − Q/(2r) + O(1/r 2 ), the parameter Q corresponding to the electric charge of the solutions. Therefore the following relation holds which resembles the case of a monopole-antimonopole configuration in a YMH theory [7]. Note that the solutions discussed here possess also a magnetic dipole moment [17] which can be read from the asymptotics of the U (1) magnetic potential, A ϕ ∼ µ sin θ/r 2 .  Numerical solutions.-Subject to the above boundary conditions, we solve numerically the set of four Maxwell-Skyrme equations. The numerical calculations are performed by using the program CADSOL [18], based on the iterative Newton-Raphson method. As initial guess in the iteration procedure, we use the spherically symmetric regular solutions of the pure Skyrme model. The typical relative error is estimated to be lower than 10 −3 .  For a given Baryon number, the solutions depend on two continuos parameters, the values V of the electric potential at infinity and the Skyrme coupling constant κ. Here we consider solutions in the one baryon sector only, although similar results have been found for n > 1. The solutions with V = 0 have b = 0 and correspond to static dipoles discussed in [17]. A nonvanishing V leads to rotating regular configurations, with nontrivial functions f, g, a and b. Rotating solutions appear to exist for any value of κ. As we increase V from zero while keeping κ fixed, a branch of solutions forms. Along this branch, the total energy and the angular momentum increase continuously with V . The ration J/E increases also, but remains always smaller than one. At the same time, the numerical errors start to increase and we obtain large values for both E and J, and for some V max the numerical iterations fail to converge. An accurate value of V max is rather difficult to obtain, especially for large values of κ. Alternatively, we may keep fixed the magnitude of the electric potential at infinity and vary the parameter κ.
In Figure 1 we present the properties of typical branches of solutions. In Figure 1a, the angular momentum and the energy are parametrised by V for several fixed value of κ, while in Figure 1b these quantities are parametrised with κ for several fixed values of V , including V = 0 corresponding to the non-spinning soliton. The energy bound in the purely magnetically gauged case with V = 0 is not saturated, as is the case also for the ungauged Skyrmion. We expect likewise that this numerically constructed solution is topologically stable, but cannot estimate the energy excess above the lower bound analytically.
One can see from Figure 1b that, for a given value of κ, the energy of the spinning soliton is always smaller than the energy of the ungauged Skyrmion, but is larger than the energy of the corresponding nonspinning static gauged solution. The latter is gauged only with the magnetic field and minimises the energy functional, while the spinning system gauged with both the magnetic and the electric fields minimises the non-positive definite Lagrangian density, and the additional electric field does not feature in the topological lower bound. As a result, the spinning, electrically charged, solutions have higher energies than the static ones. The situation here is identical with that of the Julia-Zee dyon, in this respect.
In Figure 2a we plot the energy density E = T tt , and in Figure 2b the angular momentum density T ϕt of a typical n = 1 solution as a function of the coordinates ρ, z, for κ = 0.72, V = 0.067. We notice that the energy density ǫ = T tt does not exhibit any distinctly localised individual components, a surface of constant energy density being topologically a sphere. However, this is a deformed sphere such that the profiles of E = T tt (r, θ) versus r for each value of θ are distinct and non overlapping. It presents a peak on the symmetry axis, and the density profiles decrease monotonically with r.
Also, the electrically charged U (1) gauged Skyrmion rotates as a single object and the T ϕt -component of the energy momentum tensor associated with rotation presents a maximum in the z = 0 plane and no local extrema (see Figure 2b).
Conclusions.-We have presented here the first example of spinning solution residing in the one-soliton sector of the theory which has a topologically stable limit. These solutions of the U (1)-gauged Skyrme model carry mass, angular momentum, electric charge and a magnetic dipole momentum. The electric charge is induced by rotation and equals the total angular momentum.
Similar qualitative results have been found by adding to the Lagrangean (1), a self-interaction potential of the O(4) scalar field representing the pion mass. Nonzero pion masses lead to larger values for the energy and angular momentum.
Also, we have found that similar to the ungauged case, the spinning Skyrmions admit gravitating generalisations, which are currently under study. These solutions satisfy also the generic relation (16).