Conformal mapping, algebraic parametrization and two-variable expansion of many-threshold scattering amplitudes

Abstract

The possibilities to approximate many-threshold scattering amplitudes by algebraic functions are studied. We confine ourselves to functions with Riemann surfaces of genus zero or of identical, branch structure in any sheets, respectively, and write down the corresponding threshold parametrizations. For more than two thresholds, these parametrizations have a much simpler analytic structure than the series recently proposed in the context of uniformization of scattering amplitudes by automorphic functions. Moreover, they possess rational representations in one or two algebraic functions, respectively, and thus also provide a more simple method for numerical extrapolations. We give a complete analysis of the corresponding compact Riemann surfaces in terms of uniformizing functions and also discuss the relations to the method of accelerated convergence expansions.