Expansions of the Unequal-Mass Scattering Amplitude in Terms of Poincaré Representations and Complex Angular Momentum at Zero Energy

Abstract

The expansion of the unequal-mass scattering amplitude in terms of Poincar\'e-group representations was considered for positive and zero values of $s$, the squared total four-momentum. The usual singularity problem at $s=0$ was avoidable, but it turned out that the relevant variable is not $j$, the total angular momentum, but a quantity nonsingularly related to the Poincar\'e-invariant ${W}_{\ensuremath{\mu}}{W}^{\ensuremath{\mu}}$ even at $s=0$. The notion of complex angular momentum and signature was reexamined, and some modification of the old formalism seemed useful. The results are perfectly compatible with dispersion relations and with the requirements of Regge behavior. In an appendix a theorem is proved for the expansion of a class of functions which are not square-integrable, but have Regge behavior with respect to unitary E(2) representations (that is, for Fourier-Bessel expansions).