Soft Pions, Chiral Symmetry, and Phenomenological Lagrangians

Abstract

A method is devised for extracting the entire content of the joint assumptions of partially conserved axial-vector current (PCAC) and current algebra. This technique is used to derive a set of identities which generate all possible soft-pion theorems; moreover, the direct relationship between the PCAC-current-algebra approach and the so-called phenomenological Lagrangian approach to deriving these theorems is explicitly established. The technique used throughout clearly reveals all of the assumptions inherent in the phenomenological Lagrangian approach, and provides formal expressions for the correction terms to the soft-pion theorems following from the Lagrangian formalism. Particular attention is paid to clarifying the relationship between so-called "PCAC correction terms" and "$\Sigma$ terms," with special emphasis placed upon their role in the calculation of formulas for pion-pion scattering. It is argued that our results strongly suggest that the only appealing explanation for the success of the PCAC hypothesis is that the real world satisfies an approximate $\mathrm{SU}\left(2\right)\mathrm{}\otimes \mathrm{SU}\left(2\right)\mathrm{}$ symmetry; and, moreover, in a world in which such a symmetry is exact, the pion corresponds to a Goldstone boson.