Mean field and interacting Dirac QFT: channels and spinors

Question

I am dealing with a QFT of Dirac fermions with an interaction term

$$L_I=\bar\psi\psi\bar\psi\psi=\psi^\dagger\gamma^0\psi\psi^\dagger\gamma^0\psi,$$

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?

Answer 1

Mean field approximation of interacting Dirac QFT has been first considered by Nambu more than 60 years ago in 1961, which is dubbed NJL model (see: Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345). You can refer to NJL paper for the details of mean field approximation and decoupling. As for the focused channel, NJL paper concentrates on the $<\bar\psi \psi>$ channel, which is inspired by the BCS model concentrating on the Cooper pair channel (the title of NJL paper is "Dynamical model of elementary particles based on an analogy with superconductivity").

Note that your Lagrangian lacks the chiral/axial symmetry when left-handed Dirac spinor and right-hande Dirac spinor transform differently under chiral/axial rotation $U_A(1)$. The NJL model involves an additional Lagrangian term $$L_I=\bar\psi \gamma_5\psi\bar\psi \gamma_5\psi$$ to preserve chiral/axial symmetry, provided there is no bare mass term.

The chiral/axial symmetry is broken by the dynamical symmetry breaking process when $\bar\psi \psi$ acquires a non-zero vaccume expectation value $$<\bar\psi \psi> \neq 0$$ which dynamically induces non-zero masses for Dirac fermions.