Mass Dimension 6 QED-Lagrangian Consider the QED Lagrangian
$$\mathcal{L}_{\text{QED}}=-\frac{1}{4} F^{\mu \nu} F_{\mu \nu} + \bar{\psi}(i D_{\mu} \gamma^\mu -m) \psi.$$
I need to extend the Lagrangian up to mass dimension 6, of course respecting all the symmetries/invariances of the theory.
My professor told me, that one can ignore pseudo-scalar terms such as $\bar{\psi}\gamma_5\psi$, since the theory has to be parity invariant.
But what about the product of the two pseudoscalars $$\Delta \mathcal{L} = \bar{\psi}\gamma_5\psi \bar{\psi}\gamma_5\psi.$$ Why can this not be a term in my Lagrangian? Is there a problem with one of the invariances?
 A: We are talking about chiral invariance, right?
First of all, the mass term
$$
m\bar{\psi} \psi
$$
breaks the chiral symmetry. So if your professor demands chiral invariance, then we are dealing with massless QED.
For massless QED, you can add a chiral symmetric mass dimension 6 term like (NJL 4-fermion interaction)
$$
\Delta \mathcal{L} = g (\bar{\psi}\psi \bar{\psi}\psi - \bar{\psi}\gamma_5\psi \bar{\psi}\gamma_5\psi).
$$
Note that


*

*The individual pseudoscalar-pseudoscalar-interaction term (second
term) is not chiral symmetric (no problem with local gauge $U(1)$ invariance and Lorentz invariance though). However, the aggregation of the scalar and pseudoscalar terms does respect the chiral symmetry.

*The mass dimension 6 4-fermion interactions are non-renormalizable. Hence a specific regularization regime is part and parcel of the model. 



On the other hand, if you forgo chiral symmetry, then a "complex" mass term is perfectly legit:
$$
m\bar{\psi} e^{\theta i\gamma_5} \psi = m\cos\theta \bar{\psi} \psi + m\sin\theta \bar{\psi} i\gamma_5\psi.
$$
See details here: Why is the Higgs $CP$ even?

Since you are considering mass dimension 6 terms, to be complete, don't miss out on mass dimension 5 terms like 
$$
i\bar{\psi}\gamma^\mu \gamma^\nu F_{\mu\nu} \psi, 
$$
and mass dimension 6 terms like 
$$
i\bar{\psi}\gamma^\mu \gamma^\nu \gamma^\rho F_{\mu\nu} D_\rho\psi.
$$
