On the introduction of Lorentz poles into the unequal-mass scattering amplitude

Abstract

We suggest a new type of kinematical decomposition of the unequal-mass scattering amplitude. We introduce two noncommuting, nondisjunct Poincare groups,P(+) andP(−), both of them are subgroups of theP1 ⊗P2 direct-product group, whereP1 andP2 are the Poincare groups of the one-particle transformations for the first and the second particle of the two-particle states, respectively. The groupP(+) is identical to the group of the two-particle Poincare transformations. Our first decomposition for the scattering amplitude is a double expansion with respect to the representations of both theP(+) andP(−) groups, simultaneously. The second proposal of ours is the partial-wave analysis not of the centre-of-mass states but of the «equal velocity» states, in which the individual particles move with the same velocity. Our expansions are valid for anys andt. In the equal-mass case they give the usual Lorentz-pole decomposition att=0. The formalism seems to be adequate for understanding the meaning of the «spectrum generating group» in the unequal-mass case. The variables of the expansion functions are unambiguously defined by the kinematical variables, and have branch points only at the thresholds and pseudothresholds, in opposition to other approaches.