Sum Rules inπΣScattering and theπΛΣandπΣΣCoupling Constants

Abstract

Following a procedure proposed recently by Atkinson, we derive sum rules for the $A$ and $B$ amplitudes for the reaction ${\pi }^{-}+{\Sigma }^{+}\to {\pi }^{+}+{\Sigma }^{-}$ by equating unsubtracted dispersion relations at $t=0\mathrm{}$ 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 $g_{\pi \Lambda {\Sigma }^2}$ and $g_{\pi \Sigma {\Sigma }^2}$. Choosing to fix $u=0\mathrm{}$, one obtains values of the coupling constants an order of magnitude larger than expected on the basis of, say, $\mathrm{SU}\left(3\right)\mathrm{}$. We argue that this is probably because of the large extrapolation to unphysical values of $cos\theta$ required in evaluating the fixed-$u$ dispersion relation for $u=0\mathrm{}$. Taking $u$ to be positive in such a way as to minimize the required extrapolations in angle, or choosing fixed $cos\theta =-1\mathrm{}$, one obtains results that are reasonably consistent with one another and with $\mathrm{SU}\left(3\right)\mathrm{}$, 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.