Covariant Amplitude Decomposition in Relativistic Fermion Scattering Problems

A parametrization of on-mass-shell relativistic fermion scattering amplitudes by a set of 4 covariant amplitudes is proposed, which in the non-relativistic limit turn to coefficients of the matrix amplitude decomposition over the unity and Pauli matrices, and in the ultra-relativistic limit - to symmetrized helicity amplitudes. In the general relativistic case, the covariant amplitudes express as spurs of the matrix amplitude supplemented by γ-matrix factors not exceeding 3rd degree. Algebraic computation of such spurs provides a comparatively short and fully covariant approach for calculation of fermion scattering processes, allowing account for all polarization observables. For extension of the method to problems of two-fermion scattering, when permitted are both ways of transition 1 → 3, 2 → 4 and 1 → 4, 2 → 3, relativistic on-mass-shell Fierz relations interconnecting the two possible definitions of transition amplitudes are derived, under simplifying assumptions of equal fermion masses and scattering elasticity. Eigenfunctions of the on-shell Fierz relations are constructed, and advantages of their use for automatic account for contributions from cross-diagrams are demonstrated with the example of Moller scattering