Schwinger terms and quantum anomalies in self-consistent renormalization

Abstract

Using the self-consistent renormalization (SCR), a careful study of complicated tangle of problems associated with renormalizations, symmetries conservation, their breaking and anomalies is performed for some set of UV-divergent Feynman amplitudes (FAs) connected with mass-anysotropic AVV and AAA spinor triangles in the space-time n = 4. Most general quantum corrections (QCs) to the canonical Ward identities (WIs) and some nontrivial “daughter reduction identities” (DRIs) are obtained. The results are new both for a nondegenerate case and for the chiral case. For a nondegenerate case ( m 1 ≠ m 2 ≠ m 3, m l ≠ 0), the QCs are the zero degree homogeneous functions of masses and are expressed in terms of the Appel hypergeometric functions F 1. For the degenerate nonchiral case ( m 1 = m 2 = m 3 = m ≠ 0) these QCs either are equal to zero for vector WIs or reduce to the well-known Adler-Bell-Jackiw anomalies for axial-vector WIs. For the chiral case ( m = 0) and the chiral limit ( m → 0) the behaviour of the AVV- and AAA-amplitudes depends crucially on the discrete symmetry of these amplitudes in the cases m = 0 and m → 0. Schwinger terms contributions are non-zero only for the chiral case. They are responsible for the QCs to vector and axial-vector WIs which are the same up to sign. In this case “left-handed vector” current can be conserved and hence it can be more fundamental than vector or axial-vector currents. This may give some insight into why just the left-handed neutrino exists in Nature.