Hard-pion calculation without current algebra

Abstract

Following the suggestions of earlier authors we explore the hypothesis that apart from fixing the scale of the weak axial-vector current, the main results of current algebra are largely implied in the concept of meson-pole dominance, which includes the famous PCAC principle. For definiteness we re-examine here the hard-pion analysis of the A 1 πϱ vertex by Schnitzer and Weinberg, not assuming anything about the commutator of two axial-vector currents but retaining all their other assumptions, specifically those about meson-pole dominance. We find, as expected, that these latter assumptions are powerful enough to determine the system up to the scale of the axial-vector current. As by-product of our analysis, we find a highly interesting corollary on Weinberg sum rules which allows us to exclude an otherwise permissible non-compact alternative to the usual SU(2) x SU(2) current algebra on mere covariance and positivity requirements.