Ghost fields in particle physics

Question

In my particle physics lecture, ghost fields were briefly mentioned.

As far as I understand, these come up when computing cross sections by the path integral method, to compensate for equivalent contributions due to gauge freedom.

I'm still not sure how these equivalent contributions come up in the calculations. Say you want to compute the cross section of an electron anti-electron annihilation into a photon near a massive nucleus. I assume you would then take the sum of the path integrals over all possible Feynman diagrams corresponding to that interaction.

QED has charge symmetry (i.e. $U(1)$), so the physics stay the same if the two charges are swapped.

Where concretely do we have to introduce these ghost fields, and why can we not simply combine the Feynman diagrams that are graph-isomorphisms of each other into a single equivalence class and count the contribution of each equivalence class only once into the final probability amplitude.

Please keep the answers simple, as I'm not a physics major.

Answer 1

In a nutshell, the Grassmann-odd Faddeev-Popov ghosts fields appear from the exponentiation of the Faddeev-Popov determinant, i.e., when we write the determinant as a Gaussian integral over Grassmann-odd variables.

The Faddeev-Popov determinant can roughly be viewed as a Jacobian factor in the path integral that appears because the path integral variables are in general not "parallel" to the gauge-orbits and the gauge-fixing constraints.

Finally let us mention, that by working in a so-called unitary gauge, one can make all Faddeev-Popov ghosts decouple from the theory.