# Fierz identities to eliminate all vector and tensor Dirac matrices in effective operator (Weinberg)

In the paper titled "Baryon- and Lepton- Non-conserving processes" (prl, 1979) S. Weinberg used operator formalism in effective field theory to analyse beyond the standard model processes which violates Baryon and/or Lepton numbers. In particular for proton decay he listed these operators (image attached) which respects the Lorentz symmetry and are $$SU(3) \otimes SU(2) \otimes U(1)$$ invariant. In general one could have all possible Dirac Gamma matrices ($$\Gamma \in {1, \gamma^{\mu}, \gamma^5, \sigma_{\mu \nu}, \gamma^{\mu} \gamma^5}$$) sitting inside the bilinear. After listing the operators in a form where we don't have the Dirac gamma matrices or tensors, he says "Fierz transformations have been used to put the various Fermi interactions in the form of eq (1)-(6), and in particular, to eliminate all vector and tensor Dirac matrices".

Now I understand that one can use Fierz identities to write a bilinear as a linear sum of bilinears exploiting the completeness relation, but I don't see immediately how one can get rid of the Dirac vectors and tensors completely in that case. Could someone explicitly show how this can be done? Thanks in advance.