Expansion of the helicity amplitude in terms of unitary representations of the lorentz group

Abstract

Expansions of one-particle helicity states with nonzero mass and mass equal to zero in terms of irreducible unitary representations (IUR) of the Lorentz group are considered. The expansion of helicity amplitude (HA) of particles with non-zero mass in terms of matrix elements of IUR of the Lorentz group is performed, regarding it as a function given over the double sheeted hyperboloid. It is shown that the scattering amplitude as well as the HA, in Lorentz representation, has the simple symmetry properties under space reflection and time reversal. In the HA expansion where the square momentum transfer enters an explicit function these amplitudes are free of kinematic singularities. We also obtain the integral representation for partial HA. This representation gives correct threshold behavior, exponential decrease at large values of total angular momentum J and defines the partial HA as an analytical function J in a definite half-plane.