Theorem on Invariant Amplitudes

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Abstract
It is shown that any basis of covariant polynomials for a two-particle scattering process yields invariant amplitudes free of kinematical singularities, provided (a) the total number of basis polynomials equals the number of spin space components of the scattering amplitude and (b) the polynomials of each of the two parity signatures are separately linearly independent at all points where three of the particle 4-momenta are linearly independent. This result allows one to directly identify good basis sets without going through the very tedious algebra involved in comparing them to the sets of Hepp and Williams. The latter are not useful for practical applications because the spinor indices belonging to different particles are coupled and these sets do not transform into themselves under the relevant discrete symmetry operations.