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 $E\left(|\mathbf{p}|\right)$ are rotationally invariant and monotonic in $|\mathbf{p}|,$ 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.