Why are these terms not present in the QED Lagrangian?

Question

I am working though some questions for my QFT/ QED exam and i am having trouble with the following question:

Explain why the following terms cannot be part of the Lagrangian of QED:

1. $-g(\bar{\psi}\psi)^2$
2. $\frac{1}{2}m^2A_\mu A^\mu$
3. $-\frac{1}{4}F_{\mu \nu}\Box F^{\mu \nu}$

My Answers:

1. I have no idea. An interaction term; it obeys the local $U(1)$ invariance.

2. This would suggest that the photon has mass, which we know it doesnt have.

3. I don't know if there are more or better reasons but, I think this term cannot be present as it does not represents the actual Lagrangian of electromagnetism. This term would implement a distorted version of it.

Are my answers correct or are there better arguments?

Answer 1

It is plausible for a term like $|g|\bar\psi\bar\psi\psi\psi$ to appear; this will be an interaction that eats two particles and spits back out two particles. But the question's $-g(\bar\psi\psi)^2$ will be that and an effective mass term. Now, renormalisation means that the allowed 2-particle interaction term will modify the mass spectrum, but we can impose that the renormalisation not alter the rest mass, whereas this term definitely will alter.

Depending upon your professor's choice of signs, if the potential from $-g(\bar\psi\psi)^2$ is considered to be negative semi-definite, then there is also no stable vacuum.

You should augment that the reason why photons cannot have mass is that the mass term singles out the Lorenz gauge as special, whereas the original Lagrangian is gauge invariant.

The $F_{\mu\nu}\square F^{\mu\nu}$ term means a theory that is 4th order derivative. The classical electrodynamics theory that incorporates radiation reaction by having a 3rd order derivative already gave Feynman enough of a headache, so imagine how bad 4th order will turn out to be.