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?

  • $\begingroup$ You need to sharpen the question. A term like $๐œ“¯๐›พ_5๐œ“ ๐œ“¯๐›พ_5๐œ“ $ is allowed in QFT and is dimension 6, but it is not what we call QED, and I guess most likely it is not part of the effective action of QED either. $\endgroup$
    – Kphysics
    Nov 12, 2019 at 10:17
  • $\begingroup$ @Kostas Umm, since it is an allowed term, it must be there to get renormalizability, right? And we kinda have to call it QED because that would be the only renormalizable theory of a Dirac field with local $U(1)$ invariance. Correct me if I am mistaken. $\endgroup$
    – user87745
    Nov 12, 2019 at 13:02

1 Answer 1


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 $$ i\bar{\psi}\gamma^\mu \gamma^\nu \gamma^\rho F_{\mu\nu} D_\rho\psi. $$


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