Relations between Partially Conserved Axial-Vector Current, Axial-Vector Charge Commutation Relations, and Conspiracy Theory

Abstract

The Adler self-consistency condition is shown to be true without reference to currents, provided that the pion trajectory is a member of a conspiracy with $|M|\ensuremath{\geqq}1$. It is then shown from the Adler self-consistency condition that the antisymmetric part of the amplitude for forward scattering of low-energy low-mass pions off any target is equal to a universal constant multiplied by the isotopic spin of the target. Thus all restrictions which current algebra imposes on hadron scattering amplitudes can be derived from conspiracy theory; production amplitudes are not examined here. We next investigate the question whether a system which contains massless pions satisfying the Adler self-consistency condition possesses a conserved axial-vector current. Assuming that the usual Omnes-type equation can be solved, we find that it does. Finally, the commutation relations between total axial-vector charges are shown to follow from the conservation equation under very general assumptions.