Short-Range Repulsion and Broken Chiral Symmetry in Low-Energy Scattering

This paper is an essentially modified version of the chapter “Short-range repulsion” from the English edition [1] of the book “Dispersion theories of strong interactions at low-energies” (in the Russian edition [1] this chapter is absent). Unlike the English version, we have employed the concept of chiral symmetry and ist consequences for the low-energy characteristics of strong interactions as boundary conditions on the solution of dispersion equations. The introduction of the short-range repulsion “potentials” into the low-energy equations for lower partial waves makes it possible to eliminate the main difficulties of the purely elastic low-energy (Pele) approximation. There is then a possibility in principle of obtaining solutions with small s-wave scattering lengths and broad resonances. The use of threshold conditions resulting from chiral symmetry allows us (under certain additional conditions) to express the main resonance scattering parameters in terms of the pion decay characteristics. Formulas are presented, by means of one of which the ϱ meson mass m_ϱ is expressed in terms of the pion mass μ and the decay constant ƒ_π from the PCAC condition (Eq. (5.26) and the other (Eq. (5.27)) expresses the ϱ meson width Γ via μƒ_π and m_π and is a generalization of the well-known KSFR relation taking into account unitarity corrections. Similar results have been obtained for the Δ₃₃ resonance in pion-nucleon scattering. Thus, using the broken chiral symmetry approximation and unitarity dispersion equations for low-energy ππ and πN scattering we have obtained masses, life-time and coupling constants for p-wave resonances by specifying only the pion and nucleon masses, their life-times and the Fermi coupling constant. Reported at the Conference on the High Energy Physics in the Institute for Theoretical Physics, Kiev, the Ukrainian SSR, October 1969.