ExTRAPOLATION OF K→ππ DECAY AMPLITUDE

Abstract

We examine the uncertainties involved in the off-mass-shell extrapolation of the K → ππ decay amplitude with emphasis on those aspects that have so far been overlooked or ignored. Among them are initial-state interactions, choice of the extrapolated kaon field, and the relation between the asymptotic behavior and the zeros of the decay amplitude. In the inelastic region the phase of the decay amplitude cannot be determined by strong interaction alone and even its asymptotic value cannot be deduced from experiment. More a fundamental issue is intrinsic nonuniqueness of off-shell values of hadronic matrix elements in general. Though we are hampered with complexity of intermediate-energy meson interactions, we attempt to obtain a quantitative idea of the uncertainties due to the inelastic region and find that they can be much larger than more optimistic views portray. If large uncertainties exist, they have unfortunate implications in numerical accuracy of computation of the direct CP violation parameter ǫ ′ .

Dispersion relations and Omnès representations for $K\to \pi \pi$ decay amplitudes

“…The role of the inelastic FSI and ISI contributions in the dispersion relation was discussed in Ref. [12], assuming that the decay amplitude defined in Eq. (1) satisfies a Mandelstam representation.…”