Pion amplitudes in a nonchiral world

Abstract

The popular strong PCAC (i.e., pion pole dominance of all Green's functions involving the divergence D of the axial vector current) and good chiral SU(2) × SU(2) symmetry framework has led to many successful predictions. It also, however, leads to some difficulties since, for example, it predicts the approximate vanishing of the K l 3 form factor ratio ξ = f − f + , of the π 0 → γγ decay amplitude F π (except in models with anamolous PCAC), and of the η → 3π decay amplitude A η . We therefore propose an alternate framework based on weak PCAC (i.e., pion pole dominance of physical matrix elements of D) and good SU(3). We show that a precise formulation of these older ideas can reproduce all of the good results of the chiral scheme and none of the bad ones. We assume the usual Gell-Mann current algebra and the (3, 3) + ( 3, 3) symmetry breaking Hamiltonian H′ = α 0 S 0 + α 8 S 8. We make, and motivate with reference to a composite picture of the hadrons, the following four assumptions: (1) approximate SU(3), (2) weak PCAC, (3) minimal subtractions, (4) Regge theory. In this framework, the dominance of a given Green's function T by the pion pole depends on the nature of the despersion relations it satisfies in variables for which the absorptive part is expressible in terms of physical matrix elements of D. An unsubtracted dispersion relation leads to pole dominance but a subtracted one introduces a constant (or worse) difference between T and the corresponding on-shell pion amplitude. From assumptions (2)–(4), we derive the following successful results of the chiral framework: the Adler consistency condition, the Adler-Weisberger relation, the Weinberg-Tomozawa pion-target scattering lengths, the Weinberg π-π scattering lengths, the Fubini one-pion production relations, the Weinberg-Callan-Treiman K l 4 form factors, and the K 3 π – K 2 π relations. The extra parameters (subtraction constants) encountered in applying weak PCAC, as determined by Regge theory, turn out not to contribute to these relations. Adding assumption (1), we obtain the further results: (α 0/ 2 α ∞)≃−0.17 , ξ ≅ −1, and an excellent simultaneous fit to the S-wave and P-wave hyperon decays. Here our extra parameters turn out to be present precisely in the places where they are required to insure the consistency of our approach. (For example, the σ terms in our scheme need not be small). None of these results are common to the chiral framework. We show further that F π and A η need not approximately vanish in our framework. We therefore conclude that our theory seems to provide a better description of the present empirical situation than does the chiral theory.