ΔI=12rule in thePC-conserving nonleptonic weak interactions

Abstract

The question of the $delta$I = 1/2 selection rule in weak nonleptonic decays is studied. We assume that the weak amplitudes of the form obey unsubtracted dispersion relations in the momentum-transfer variable s = p/sub $alpha$/$sup 2$ and that they obey Holder's condition in s except at possible poles. Using unitarity, the determination of the weak amplitudes is transformed into solving a Hilbert problem. This in turn is transformed into a system of Fredholm's intergral equations. The number of solutions is discussed, and the solution which satisfies our assumptions is found to be unique and can expressed in terms of the strong interaction S-matrix elements. Weak amplitudes which possess a pole, namely the PC-conserving parity-violating $delta$I = 1/2 interactions (poles K$sup 0$ and anti K$sup 0$) and the parity-conserving $delta$I = 1/2 interactions (poles kappa$sup 0$ and anti kappa$sup 0$), are shown to be consistent with our assumptions. The other $delta$I amplitudes either vanish or do not satisfy unsubtracted dispersion relations. (AIP)