Effective Lagrangian for Skyrmion physics.

Abstract

The effective chiral Lagrangian for low-energy pion-Skyrmion physics is considered as an expansion in powers of (\ensuremath{\partial}\ensuremath{\Phi}), where \ensuremath{\Phi} is the chiral field. The O((\ensuremath{\partial}\ensuremath{\Phi}${)}^{2}$) term is the nonlinear \ensuremath{\sigma}-model Lagrangian ${L}_{2}$. Two theoretical approaches to the higher-order terms are examined. The first involves integration over fermion degrees of freedom. The O((\ensuremath{\partial}\ensuremath{\Phi}${)}^{4}$) terms obtained this way, together with ${L}_{2}$, are found to be consistent with low-energy \ensuremath{\pi}\ensuremath{\pi} phase shifts---in particular, in the dominant \ensuremath{\rho} and \ensuremath{\sigma} channels. In the second approach, an effective pion Lagrangian is generated by integrating some simple chiral meson-field Lagrangians over the heavy mesons. At O((\ensuremath{\partial}\ensuremath{\Phi}${)}^{4}$) two terms are found: a term ${L}_{4}$,\ensuremath{\rho} of Skyrme form, with a model-independent coefficient determined by the \ensuremath{\rho} mass, and a non-Skyrme term ${L}_{4}$,\ensuremath{\sigma}, the coefficient of which is more model dependent and determined by the \ensuremath{\sigma} mass; both terms are compatible with the fermion integration results. It is concluded that the O((\ensuremath{\partial}\ensuremath{\pi}${)}^{4}$) terms are well established theoretically, and in good qualitative agreement with pion data. These terms do not, however, stabilize the soliton. Some discussion of the possible stabilizing effects of O((\ensuremath{\partial}\ensuremath{\pi}${)}^{6}$) terms is given.