Threshold configurations in the presence of Lorentz violating dispersion relations

A general characterization of lower and upper threshold configurations for two particle reactions is determined under the assumptions that the single particle dispersion relations E(\p\) are rotationally invariant and monotonic in \p\, and that energy and momentum are conserved and additive for multiple particles. It is found that at a threshold the final particle momenta are always parallel and the initial momenta are always antiparallel. The occurrence of new phenomena not occurring in a Lorentz invariant setting, such as upper thresholds and asymmetric pair production thresholds, is explained, and an illustrative example is given.