SUM RULES IN PISIGMA SCATTERING AND PI ALPHA SIGMA AND PI SIGMA SIGMA COUPLING CONSTANTS

Following a procedure proposed recently by Atkinson, we derive sum rules for the A and B amplitudes for the reaction π-+Î<>+→π++Î<>- by equating unsubtracted dispersion relations at t=0 and at fixed u or fixed backward angle. Combining these with the already known superconvergence relation for the B amplitude, and assuming the sum rules can be saturated with known resonances, we obtain three equations for two unknown coupling constants gπΛÎ<>2 and gπÎÎ<>2. Choosing to fix u=0, one obtains values of the coupling constants an order of magnitude larger than expected on the basis of, say, SU(3). We argue that this is probably because of the large extrapolation to unphysical values of cosθ required in evaluating the fixed-u dispersion relation for u=0. Taking u to be positive in such a way as to minimize the required extrapolations in angle, or choosing fixed cosθ=-1, one obtains results that are reasonably consistent with one another and with SU(3), to within estimated uncertainties of 50% or more, resulting from experimental error in the resonance widths, large cancellations between the contributions of different resonances, and unknown nonresonant-background and high-energy contributions. © 1969 The American Physical Society.