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.

Fierz Identity to remove Dirac tensors in bilinears


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