Crossing Relations for Center-of-Mass- and Breit-System Canonical Amplitudes

Abstract

We derive the crossing relations for the canonical amplitudes, starting with the case of four particles of nonzero rest mass. We give the results for both the center-of-mass- and the Breit-frame amplitudes and compare two different points of view. The definition of the ``Breit system'' is adapted to a unified treatment of all the three types of momentum transfer. It is shown that the Breit-frame amplitudes need no preliminary adjustments of constraints or the definition of a ``generalized'' amplitude in the alternative point of view. We also show that the usual convention about the continuation of the covariant spinor amplitudes involve, for the particles to be crossed, a rotation π of the ``physical'' spin about the direction of the continued momentum. The corresponding results for the helicity and the transversity amplitudes are derived as corollaries. Finally, we discuss the case of amplitudes involving zero-mass particles and some remarks are added concerning the kinematic singularities. The definition of the canonical and the transversity amplitudes are compared and some useful Lorentz-transformation formulas are collected together. The transformations connecting the center of mass and the Breit frames are parametrized in terms of the invariants. The well-known invariant amplitudes for πN scattering are used to verify our canonical formula.