Chiral-Invariant Regularization and Renormalization of the Axial-Vector Current

The regularization and renormalization procedures for perturbation calculations involving the axial-vector current of the $\sigma$ model are considered. It is shown that, even in the absence of electromagnetism, special care must be exercised when introducing fermion regulator fields into the Lagrangian so that the axial-vector Ward-Takahashi identity is not violated in the process. A generalized method of regularization is proposed in which the regulator part of the Lagrangian satisfies chiral invariance and which reduces to the usual one when applied to quantities other than the axial-vector current. By virtue of the chiral-invariant character of this method, a subtraction procedure can be introduced for the axial-vector vertex which eliminates unrenormalizable cutoff-dependent terms. After this subtraction, the regularized axial-vector vertex of the $\sigma$ model satisfies the usual Ward-Takahashi identity and is renormalizable by rescaling of the external wave functions.