Unstable-Particle Scattering and the Strip Approximation

We examine, in this paper, the problem of formulating a bootstrap calculation when one of the scattering particles is unstable. Having defined the unstable-particle scattering amplitude as an $S$-matrix pole residue, we go on to discuss its analytic structure and point out that it may be determined from the usual Landau rules. We conclude that although the instability of the external particle complicates the structure it does not do so too severely. Therefore, we are free to postulate that, in analogy with the stable case, the unstable-particle amplitude exhibits Regge asymptotic behavior. This assumption leads us to construct a strip approximation to the amplitude which is a crossing-symmetric superposition of Regge pole terms. We point out that this approximation exhibits, in some respects, satisfactory analytic structure. In particular it takes quite well into account certain anomalous threshold effects. It satisfies a quasi-Mandelstam representation which we use to explore the analytic structure of the corresponding partial-wave amplitudes and their continuation to arbitrary angular momentum. We use certain simple discontinuity formulas to obtain dynamical equations for the partial-wave amplitudes and are consequently able to construct, formally, a complete bootstrap scheme. Finally, we mention some difficulties and unsolved problems.