Conditions for renormalizability of quantum flavor dynamics

Abstract

The presence of a new anomaly, in addition to the Adler-Bell-Jackiw (ABJ) anomaly, is hinted at for gauge theories with ${\ensuremath{\gamma}}_{5}$ couplings. The ABJ anomaly is discussed first by translation of variables of a linearly divergent integral, then by dimensional regularization. Using the second method, the general non-Abelian case is considered in the presence of an overlapping divergence. A new anomaly is suggested which is not, in general, canceled by the usual restrictions because fermion masses are involved. Assuming no cancellation between different Feynman diagrams and current-algebra quark masses then leads to the conclusion that the standard model of quantum flavor dynamics (i.e., that of Glashow, Salam, and Weinberg) might be nonrenormalizable. Imposition of renormalizability would then imply that new dynamical constraints be met. Hence, only that part of the quark mass corresponding to the lepton mass in the same quark-lepton generation is generated by the electroweak interactions. The remaining mass comes presumably from the strong interaction for which the dynamical theory must therefore have at least some flavor dependence.