Crossing equations and spin structure. II. FB → FB and BB → FF reactions

A study of the spin structure of two-body helicity amplitudes begun in a previous paper is extended to include general-mass reactions with two fermions and two bosons. The Trueman-Wick matrix joining the FB→FB and BB→FF channels (B=boson, F=fermion) is transformed into a direct sum of simple matrices Mᵢ. The latter are 2 × 2 matrices whose elements are simple polynomials in s and t. The new s- and t-channel amplitudes related by the Mᵢ are proportional to k.r. (kinematically regular, i.e., free from kinematic singularities and simple zeros) functions of s and t. Thus we obtain a complete set of new k.r. amplitudes for general-mass FB→FB reactions. As before, we can use the simple crossing algebra to expand the new k.r. amplitudes in terms of an elegant set of invariant amplitudes. The coefficients of these invariant amplitudes are simple functions of the crossing and scattering angles. Within a given spin class of reactions (e.g., those reactions with spins 1/2+0→J+0) these coefficients are also simple functions of the spins. The procedure is described by two detailed examples with FB→FB channel spins \textonehalf{} + 0 → \textonehalf{} + 0 and 1/2+0→3/2+0. We describe how one can achieve a general classification of invariant amplitudes based on the different kinematic roles played by different invariant amplitudes. The simplest possible connections between this classification scheme and observable dynamical situations are discussed.