Patterns in Direct-Channel Helicity Amplitudes Caused by Crossed-Channel Exchanges

Abstract

We discuss a set of patterns for Reggeon exchanges in elastic scattering. Each pattern corresponds to a particular set of $t$-channel quantum numbers, and allows only certain $s$-channel helicity amplitudes ${{f}_{cd;ab}}^{s}$ to be present to order ${s}^{\ensuremath{\alpha}}$. These dominant amplitudes are those such that $|c\ifmmode\pm\else\textpm\fi{}a|=n$, and $|d\ifmmode\pm\else\textpm\fi{}b|=m$, where $n$ and $m$ are integers, and the pattern is labeled by $n$,$m$ and the choice of + or - in each case. Any Regge residue may be constructed as a linear combination of the basic "-" patterns; any Regge residue which vanishes sufficiently rapidly at $t=0$ may be constructed as a linear combination of the basic "+" patterns. Correct behavior for both $s$- and $t$-channel amplitudes at $t=0$ and $t$-channel thresholds is ensured by the formalism.