I am dealing with a QFT of Dirac fermions with an interaction term
with $\gamma^0$ a Dirac matrix and $\psi$, $\psi^\dagger$ a spinor field and its Hermitian conjugate. I believe that a mean-field approach would give good predictions for different mean values, and so I was looking at how could I decouple this four-Fermi term. As I understand, within mean-field theory there are typically three channels to do so: pairing, direct (Hartree) and exchange (Fock). However, the examples for fermions are usually done for terms like
$$c^\dagger c^\dagger c c.$$
For this simple case, I see very clearly the interpretation of the mean field channels. However, for the case of Dirac fermions, I am a bit unsure about how to interpret each channel, or even if they still exist. For example, I believe that the direct channel would include a term $\langle \psi\bar\psi\rangle$, which I am not sure at all that would exist (it is not a bilinear form, right?). So does the fact of working with Dirac spinors restrict in some way the channels that one can use when decoupling an interaction term, and more particularly, the one that I wrote above?