My understanding is that for a $U(1)$ gauge field $A_\mu$, the most general gauge-invariant kinetic term in the Lagrangian that can be written down which satisfies gauge invariance is something of the form $\Pi(p^2) (p^2 \eta_{\mu\nu} - p_\mu p_\nu) A^\mu A^\nu$ (with the usual Yang-Mills action coming from a constant $p$). Normally with Yang-Mills, there is a subtlety in inverting this equation to derive the propagator, as it has a zero eigenvalue. The usual way to fix this is to introduce a gauge-fixing term $\frac{1}{2\xi} (\partial_\mu A^\mu)^2$ to the Lagrangian. My question is essentially, are there any further subtleties that arise in our inversion when we have the general form of the propagator? In other words, can we simply introduce a term like $\frac{1}{2\xi}\Pi(p^2) (p_\mu A^\mu)^2$ as a gauge-fixing term and carry out the inversion the same as before?


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QFT books often jump straight to QED since this theory was one of the first figured out, and is also the first (so far as I'm aware) to produce new results matching experiment. This, however, leads to some mystery surrounding the procedure of gauge fixing because the machinery to gauge fix properly usually comes later on when non-Abelian gauge theories are considered.

Essentially, there are some worries about the gauge fixing function (for example see Gribov ambiguity), but these are usually ignored, especially for a first pass at the topic. To understand how all this comes to pass, however, you need to read about the Faddeev-Poppov method which is the means by which the gauge fixing term is actually added to the Lagrangian (it is not just tossed in by hand).

Any good book on QFT will discuss this in varying degrees of details, so feel free to choose whichever source you prefer.


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