On the analytic properties of chiral solitons in the presence of the $ω$--meson

Abstract

A thorough study is performed of the analytical properties of the fermion determinant for the case that the time components of (axial) vector fields do not vanish. For this purpose the non--Hermitian Euclidean Dirac Hamiltonian is generalized to the whole complex plane. The Laurent series are proven to reduce to Taylor series for the corresponding eigenvalues and --functions as long as field configurations are assumed for which level crossings do not occur. The condition that no level crossings appears determines the radius convergence. However, the need for regularization prohibits the derivation of an analytic energy functional because real and imaginary parts of the eigenvalues are treated differently. Consistency conditions for a Minkowski energy functional are extracted from global gauge invariance and the current field identity for the baryon current. Various treatments of the Nambu--Jona--Lasinio soliton are examined with respect to these conditions. Motivated by the studies of the Laurent series for the energy functional the Euclidean action is expanded in terms of the $ω$--field. It is argued that for this expansion the proper--time regularization scheme has to be imposed on the operator level rather than on an expression in terms of the one--particle eigenenergies. The latter treatment is plagued by the inexact assumption that the Euclidean Dirac Hamiltonian and its Hermitian conjugate can be diagonalized simultaneously. It is then evident that approaches relying on counting powers of the $ω$--field in the one--particle eigenenergies are