Search for the Anomaly-Free Interactions

Abstract

By the use of the Pauli-Villars regularization method, we shall prove that an anomaly­ free theory can be constructed in both Abelian and non-Abelian cases, provided that the interactions always appear in an appropriate combination of the interactions with properly chosen coupling constants. In 1949, Fukuda and Miyamoto 1 > have found in the process of calculating the neutral meson decays that the Schwinger method 2 > and the Pauli-Villars reg­ ularization technique 3 > conflict with the Equivalence theorem , 4 > so that the perturbation calculations do not agree with the formal manipulation. 5 ) Recently this fact has been rediscovered 6 > in connection with the study of the validity of the PCAC hypothesis, and the extra terms that are found to be the source of the disagreement are named anomalous. The essence of this problem consists in the presence of the ill-defined field operator products. The s-separation 7 > and the regularization 8 > methods are the possible definitions of these products, but both methods lead to the anomalous terms. That is, we cannot exploit the am­ biguities present in the regularization method so as to avoid the disagreement. 9 >•*> We shall investigate the possibility of constructing the anomaly-free theory by introducing additional interactions. Let us start from the Abelian case. The Lagrangian is assumed to take the form where L(x) =(fJ(x) (iy'fJ'-m)cjJ(x) -g(j)(x)r'cf;(x)p'(x) - c(j)(x) irocfJ (x) Sp,vl1r:P'v (x) P 11 r: (x), p'v(x) =fJ'pv(x) -f}vp'(x). The vector and axial vector currents are defined by V'(x) =¢(x)r'cf;(x),