The Skyrme model description of the spin-orbit force

We examine the derivation of the spin-orbit force from the Skyrme model. We find substantial agreement with phenomenological potentials. We show that a systematic and simultaneous treatment of all components of the kinetic energy introduces a minus sign relative to previous calculations for the iso-spin independent part of the interaction. We also explain the smallness of the iso-spin dependent part of the spin-orbit potential. The challenge of obtaining the NN interaction from a QCD inspired starting point has recently been given new impetus by results in the Skyrme model [1, 2]. Careful treatment of the non-linear nature of the model and of quantum corrections leads to significant central attraction, in substantial agreement with phenomenology. There is even a strong short range repulsion, though its detailed description certainly lies outside the scope of the Skyrme model with pions only. Combined with the well known Skyrme treatment of one pion exchange and the tensor force, all this yields a satisfactory picture of the static part of the NN interaction that gives a good description of the phase shifts. The next step is to consider the non-static parts of the interaction. Here we consider the spin-orbit force. The dominant part of the phenomenological spin-orbit interaction is iso-spin independent. Attempts to reproduce it from the Skyrme model have given the correct order of magnitude, but the wrong sign [3, 4]. In this note we point out that a careful treatment of the conversion from velocities to canonical momenta in fact reverses that sign and hence yields a spin-orbit interaction in substantial agreement with the data. We also show that a small iso-spin dependent spin-orbit force emerges naturally. The principal ingredients in the successful static NN interaction calculations are numerically “exact” solutions for the baryon number two (B = 2) system at finite ∗address after Sept. 1, 1993: Institüt für theoretische Physik III, Universität Erlangen-Nürnberg, D-91058 Erlangen, Germany