A zero-mode quantization of the Skyrmion

In the semi-classical approach to the Skyrme model, nuclei are approximated by quantum mechanical states on a finite-dimensional space of field configurations; in zero-mode quantization this space is generated by rotations and isorotations. Here, simulated annealing is used to find the axially symmetric Skyrme configuration which extremizes the zero-mode quantized energy for the nucleon.


Introduction
The Skyrme model is an effective theory of pions and nucleons. It is a non-linear field theory in which nuclei correspond, classically, to topological soliton solutions, called skyrmions. The model is non-renormalizable and so the approach usually taken is to reduce to a finite-dimensional space of Skyrme configurations before quantizing. In this approach, quantummechanical states on a space of topological charge B Skyrme configurations model baryon number B nuclei at low energies.
The Skyrme Lagrangian is acted on invariantly by rotations and isorotations; this generates from a single Skyrme configuration a space of energy-degenerate configurations. This space is known as the space of zero modes and quantization on this space is called zero-mode quantization. This approach began with the seminal paper [1] where the nucleon and delta are constructed as quantum mechanical states on the zero-mode space generated from the classical minimum energy configuration of unit baryon number.
In this Letter, the spin-half, isospin-half quantum Hamiltonian is calculated on the zero-mode space generated from a general axially symmetric configuration and simulated annealing is used to find the Skyrme configuration that minimizes the energy of the lowest state. This approach has previously been considered for the zeromode space of a general spherical symmetric configuration [2,3]. In this space the quantum Hamiltonian reduces to a scalar, here it is a matrix. Aspects of our approach are also shared with a very recent paper [4] which appeared when our Letter was in preparation. The quantum Hamiltonian used in that paper is a scalar ansatz motivated by the spherical Hamiltonian used in [1] and differs from the Hamiltonian we derive here. In fact, we will see that this will not make a significant difference, the numerical results obtained for the nucleon in both papers do not differ much from each other or from what would be calculated using the classical minimum. However, the approach here is more direct than the approach described in [4] and can be generalized to higher charge nuclei.

Quantization procedure
Written in terms of the vector currents R μ = ∂ U U μ † of an SU(2) field U (x), the Skyrme model has the Lagrangian where m , F π π and e are parameters that are adjusted to fit experimental data. Using F / π 4e as our unit of energy and 2/eF π as our unit of length we obtain The skyrmion mass for a static field U s (x) can be derived from this Lagrangian and is We wish to quantize the rotational and isorotational degrees of freedom. Rather than acting on a specific Skyrme configuration, we want to consider the zero-mode space of fields generated from a general static configuration U s (x) by isorotation C and rotation D: Since rotation and isorotation are symmetries of the original Lagrangian, these configurations are all energy-degenerate. The effective Lagrangian on this restricted space of configurations can be calculated by allowing C and D to depend on time, giving L = −M + L rot , where L rot is the kinetic Lagrangian with the rotational and isorotational angular velocities ω and Ω given by and the moment of inertia tensors U ij , V ij and W ij given by the following integrals: where the σ i are the usual Pauli matrices.
In this Letter, we will restrict our discussion to axially symmetric skyrmion solutions. Numerical simulations indicate that there is also a reflection symmetry in the xy-plane and so the principal axes of inertia can be taken as the standard orthogonal axes, with U ij = V ij = W ij = 0 where i = j , and we can set U ii = U i and so forth. By inspection, cylindrical symmetry in the xy-plane will also mean (10) We can use axial symmetry to establish an additional identification between the normal moments of inertia (see Appendix A): Applying these restrictions to L rot (6) we get or, written as a sum of complete squares, The rotation and isorotation angular momentum vectors L and K canonically conjugate to ω and Ω are Note that L 3 = −K 3 : upon quantization, this is the axially symmetric condition expressed in operator form. Our approach will be to find the minimum energy skyrmion such that its energy eigenstate is also a null eigenstate of L 3 + K 3 . From (13) and (14), the Hamiltonian for the rotational and isorotational degrees of freedom can now be calculated: For a spin-n, isospin-n particle, the angular momentum operators L can be written as (2n + 1) × (2n + 1)-dimensional matrix representation Σ L 1 , Σ L 2 , Σ L 3 of SU(2), as can the isospin operators Σ K 1 , Σ K 2 , Σ K 3 of K for an isospin-n particle. If we define our quantum state using the |l, l 3 ⊗|k, k 3 basis, l, l 3 and k, k 3 being the quantum numbers for L and K, respectively, we can embed L and K into the resulting SO(3) L × SO(3) K direct product space: where I 2n+1 is the (2n + 1)-dimensional identity matrix.
We can now find the lowest energy nucleon state; first we insert the spin-half, isospin-half matrix representation of L and K into the Hamiltonian (15) to get: There are two eigenvectors of H which are also eigenvectors of L 3 + K 3 with eigenvalue zero: The first eigenvector is a spherically symmetric state and also has the lower energy since κ 1 and κ 2 are always positive; hence and is therefore the energy of an axially symmetric nucleon. We see that in the spherically symmetric case U 2 = U 3 = W 2 = V 2 , and E N reduces to the rotational energy formula obtained in [1]: The energies of all other eigenstates go to infinity in the spherically symmetric limit.
The energy, M + E N , of the quantum state would be difficult to extremize using gradient-based methods; instead, simulated annealing [5] is used to find the skyrmion configuration that minimizes this energy. Since the configuration is assumed to be axially symmetric, the cartoon method [6] is used. The configuration is annealed on a quarter-disk two-dimensional lattice Fig. 1. Plot of F π , as a function of e, for which the energy M + E N is equal to the nucleon mass 939 MeV; the pion mass parameter m π set to its experimental value of 138 MeV. Our results (bold circles) are compared with those obtained using the rigid body approach taken in [9] (solid line). Fig. 1, but with the pion mass parameter m π set to the value of 345 MeV suggested in [4]. Our results (bold circles) are compared with those obtained using the rigid body approach taken in [9] (solid line).

Fig. 2. As
with a radius of 250 lattice points and a lattice spacing of 0.06. A variant of the adaptive simulated annealing probability distribution [7] is used for the field perturbations; this seems to improve the speed of convergence and allows an exponential cooling schedule. The algorithm needs an initial configuration to perturb; any configuration of unit baryon number will suffice, and the ansatz given in [8] is probably the easiest to implement.
In Fig. 1 and Fig. 2, the results of our simulations are shown in comparison with results obtained using the rigid body approximation used in [9]. The pion mass m π was set to its experimental value of 138 MeV in Fig. 1, whereas in Fig. 2 it was set to the larger value of 345 MeV suggested in [4]; we see that the nucleon deformation only becomes noticeable at high values of the pion mass.

Discussion
Although it has appealing mathematical and physical properties, the Skyrme model has only been modestly successful in modeling nuclear dynamics. Perhaps one of the biggest challenges is that the zero-mode quantization of the classical minimum fails to give even the correct lowest energy state for many values of the baryon number [10,11]. It is possible that this is because inappropriate parameter values are used to calculate the classical minima: for example, in [12] it is suggested that using a non-zero, or even an unphysically large, pion mass will significantly affect the structure of higher charge skyrmions and alter the symmetries on which the quantization is based [13]. Another possibility is that the classical minimum is inappropriate and that the energy minimum of the effective quantum Hamiltonian should be used for zero-mode quantization. In this Letter this has been done in the simple case of a axial symmetric nucleon; the higher charge cases will be more challenging computationally but should follow in a similar way.