Crossing-Symmetric Expansions of Scattering Amplitudes, Threshold Behavior, and Asymptotics

A two-variable, explicitly crossing-symmetric expansion of the scattering amplitude is discussed for the two-body scattering of spinless particles (with arbitrary masses). It converges simultaneously in the physical regions of at least two channels, has the correct threshold behavior, and allows for amplitudes growing asymptotically as arbitrary powers of $s$ and $t$. The expansion is based on the representation theory of the group $O(2,1)\mathrm{}$ in a basis not corresponding to any subgroup reduction and making use of Lamé functions.