Quantum mass shift of the soliton in the Skyrme model.

Abstract

The quantum mass shift of the soliton in the Skyrme model has been calculated from all nonzero modes. The calculations were carried out using a method applied earlier to the sine-Gordon model. The mass shifts do not depend on the baryonic spin, therefore they are the same for the nucleon and the $\ensuremath{\Delta}$. Our model parameters are the pion decay constant ${F}_{\ensuremath{\pi}}$, with its experimental value, and the pionic mass ${m}_{\ensuremath{\pi}}$, once in the chiral limit ${m}_{\ensuremath{\pi}}=0$ and also with the experimental value. We justify taking for the Skyrme parameter $e$ the experimental value of ${g}_{\ensuremath{\rho}\ensuremath{\pi}\ensuremath{\pi}}$, the mesonic $\ensuremath{\rho}$-pion coupling constant. The results depend on the vibrational energy. In this case an energy cutoff must be introduced. Using renormalization considerations as in the nonlinear $\ensuremath{\sigma}$ model, we have chosen this cutoff to be $e{F}_{\ensuremath{\pi}}=1136$ MeV. Our results (for $e={g}_{\ensuremath{\rho}\ensuremath{\pi}\ensuremath{\pi}}=6.11$ and ${F}_{\ensuremath{\pi}}=186$ MeV) are ${M}_{N}=996$ MeV, ${M}_{\ensuremath{\Delta}}=1593$ MeV for ${m}_{\ensuremath{\pi}}=0$; ${M}_{N}=857$ MeV, ${M}_{\ensuremath{\Delta}}=1641$ MeV for ${m}_{\ensuremath{\pi}}=138$ MeV. The spin dependence and the energy contribution of the coupling term between vibrations and rotations are being considered here only in a qualitative way.