Regge Poles, Power Series, and a Crossing-Symmetric Watson-Sommerfeld Transformation

Abstract

It is shown that there exists a close relationship between the analytic properties of the partial-wave amplitude as a function of complex-angular momentum $l$ and those of the coefficients of expansions, other than the partial-wave expansion, as functions of the corresponding summation index, $\ensuremath{\nu}$. The case of power series in $z$, and in the Mandelstam variables $t$ and $u$ is studied in detail. We show how the $l$-plane Regge poles for $\mathrm{Re}l>\ensuremath{-}\frac{1}{2}$ determine all the $\ensuremath{\nu}$-plane poles for $\mathrm{Re}\ensuremath{\nu}>\ensuremath{-}\frac{1}{2}$ and vice versa. For the relativistic amplitude we write a representation consisting of three double power series in $s, t, \mathrm{and} u$. We establish the analytic properties of the expansion coefficients in the two index variables which are implied by Regge analyticity in the $l$ plane of each channel. This enables us to apply the Watson-Sommerfeld transformation twice and obtain a crossing-symmetric Regge-type representation which simultaneously displays the contributions of the Regge poles in all three channels.