Trouble with boson loops in Skyrmion physics

The possibility is raised that the O((∂φ${)}^4$) (stability) terms in the Skyrme Lagrangian may arise as quantum corrections, starting from an underlying bosonic field theory. Specifically, these terms are calculated from a derivative expansion of the one-loop effective action of the linear σ model, which is the simplest relevant theory. The calculated terms are logarithmically divergent in the chiral limit $m_{\pi }$ ${\mathrm{}}^2$→0. This physically incorrect result is traced to a failure of the derivative expansion caused by nonanalyticity of the Green’s functions at zero momentum (i.e., unitarity thresholds). The difficulties of handling ‘‘light’’ particle loops in the effective-Lagrangian approach to soliton physics are discussed. It is suggested that, in the absence of a systematic procedure for calculating unitarity corrections to effective Lagrangians, it may be more fruitful to settle for an approximate theory, in which the soliton Lagrangian is enlarged to include heavier mesons, and used only at the tree level.