Generalized Fierz identities

Abstract

Low energy weak interaction calculations with fermions frequently involve a superposition of quartic products of Dirac spinors, in which the order of the spinors is not the same in all the contributing terms. A common trick that is used to bring them to a uniform ordering is the Fierz transformation. We show that the standard Fierz transformation quoted in textbooks is one element of a class of transformations that exist between fermion amplitudes that are written with different, arbitrary reorderings, of the spinors. The explicit form of all such transformation matrices is given, and in addition to the usual Lorentz scalar quartic products, we consider the pseudoscalar ones. Moreover, we include the cases in which some, or all, of the spinors in the original amplitude appear in the transformed amplitudes in their Lorentz-invariant complex conjugate form. Such manipulations and formulas are useful in practical calculations, as we show with some examples.