RENORMALIZATION OF SIGMA-MODEL .2. FERMION FIELDS AND REGULARIZATION

The renormalization of the σ-model is discussed. The results presented here generalize a previous study of one of us (B.W.L.) who considered only the boson terms. It is emphasized that a consistent renormalization, namely one which is obtained by a mere redefinition of the bare constants of the Lagrangian, and by introducing chiral symmetric mass counter terms, can only be achieved by an appropriate choice of the regularization procedure. It is shown that this will be the case only if the same regularization can be applied to the chiral symmetric theory, i.e. the one where the scalar tadpoles are absent (in particular, the fermion mass is zero). As a consequence we show that one cannot regularize by adding fermion fields with the same chiral properties as the nucleon. In particular we exclude Bell and Jackiw's prescription as inconsistent. A new regularization procedure is proposed based on parity doublets. It is shown to lead to a consistent renormalization of the σ-model, in particular PCAC is obtained among renormalized quantities. This method, however, does not regularize the triangle graph discussed by Bell and Jackiw and Adler. We thus conclude that two types of regulators are needed in the σ-model when electromagnetism is superimposed. © 1969.