The phenomenological representation of K-nucleon scattering and reaction amplitudes

Abstract An explicit parametrization of the K -N scattering and reaction amplitudes, satisfying the unitarity conditions and including the implications of timereversal invariance, is given in terms of the elements of a suitably-defined reaction (or K -) matrix. Plausible approximations then lead to the formulas of a “zero-range” theory, an extension of that introduced by Jackson et al. , in which all the low-energy K N cross sections are expressed in terms of two complex scattering amplitudes ( A 0 , A 1 ), a phase angle φ t and a real parameter ϵ. The zero-energy and 175 Mev/ c | K − - p data allow four possible sets ( a ±), ( b ±) of these parameters; the Coulomb-nuclear interference in K − - p scattering favors ( a +) and ( b +). The most important qualitative conclusion based on this formalism is that the data indicates that the KYN coupling is so strong that perturbative methods are in general quite inadequate for the calculation of K -particle processes. The relationship of pion-hyperon ( πY ) scattering cross sections with the K N data is discussed in detail and it is shown that the parameter sets ( a −) or ( b −) would require the existence of a πY resonant state not far below the K − - p threshold. After determination of expressions for the K N → πY amplitudes, the energy dependence of the Σ − Σ + and Λ Σ 0 ratios are discussed, together with some remarks on the K − p → Λππ cross section. The Coulomb corrections to the nuclear K -matrix elements are derived, as well as the modification of the corresponding T -matrix elements by the K − − K o mass difference through the unitarity condition. These corrections are far from negligible and lead to appreciable modification of the parameter sets ( a ±), ( b ±). Each of these sets fits the Σ − Σ + ratios at zero energy and in the range 100–200 Mev/ c ; the ( a +) and ( b −) sets require a strong upward cusp in this ratio at the K 0 -n threshold, the ( a −) and ( b +) sets a downward S-shaped cusp. Taking all the present data together, the ( a +) set appears favored. A brief discussion of the advantages and disadvantages of the K -matrix formalism is given in the Appendix.