Theory of Low-EnergyπωScattering

Abstract

The $\ensuremath{\pi}\ensuremath{\omega}$ problem, as the simplest case of the pseudoscalar-meson and vector-meson system, is discussed from the standpoint of the $S$-matrix approach. A general procedure of constructing invariant amplitudes in spin and isospin space of the pseudoscalar-vector system is given, and for $\ensuremath{\pi}\ensuremath{\omega}$ scattering, a set of invariant amplitudes are conveniently chosen and their crossing properties are discussed. These amplitudes are expressed by one-dimensional representations which are derived from the Mandelstam representations by the Cini-Fubini technique. Partial-wave expansions as well as projections are done by the use of the Jacob-Wick helicity amplitudes. A prescription for calculating the driving forces from the exchange of particles is presented and applied to the exchange of the $\ensuremath{\rho}$ and $B$ mesons in states of the two possible quantum numbers ${J}^{P}={1}^{+} \mathrm{and} {2}^{\ensuremath{-}}$. The procedure consists of a zero-width approximation to the transition amplitudes in states of given $J$ and $L$, crossing-symmetric relations, and the one-dimensional dispersion representations of the invariant amplitudes. The relationship between the invariant amplitudes and the helicity amplitudes greatly facilitates this procedure. The $t$-channel reaction is also analyzed. A method of solution of the partial-wave dispersion relations is discussed based on a recently developed formalism, and is extended further to avoid the difficulty associated with the zeros of the driving forces. A systematic program to understand the quantum numbers of the $B$ meson as a $\ensuremath{\pi}\ensuremath{\omega}$ resonance is also discussed. The qualitative nature of the forces due to the $B$ exchange in states of each possible quantum number is briefly sketched. A model calculation which favors a ${2}^{\ensuremath{-}}$ state of the $B$ meson is presented.