Short: How can I use Wilson lines to compute Bremsstrahlung? I'm particularly interested in QED, as a simple example.
Long:
I recently learned what is a Wilson line, in the simplest sense of $$W[C] = \mathrm{Tr}_R\left[\mathcal{P}\exp\left(i \int_C A \right)\right],$$ with $A$ the connection one-form and $C$ some (possibly open) path in spacetime. After searching a little online for why people are interested in these sorts of objects (the main one apparently being the gauge invariance of Wilson loops), I found someone stating Wilson lines can be used to recover information about Bremsstrahlung, for example.
I got interested in this, and decided to try to make some explicit computations in quantum electrodynamics to try to see if the (Liénard–)Larmor formula or something similar pops up. The motivation for using QED is merely because it is relatively easy, and I can compute the path integral involved in $\langle W[C]\rangle$ exactly. My first attempts were at an inertial motion (as a warm up), and then a uniformly accelerated charge (as a main interest). The calculations involved a lot of regularization, in particular a Pauli–Villars cutoff $\Lambda$ and a total path proper time $T$. My final result for the accelerated case was then $$\log \langle W[C] \rangle = - \frac{e^2 a^3 T^2}{16 \pi^2 \Lambda} - \frac{e^2 \Lambda T}{8 \pi^2} \log\left(\frac{\Lambda}{a}\right) - \frac{e^2}{16 \pi^2} \Lambda T (\gamma_E - 3 + i \pi) + \text{finite terms},$$ up to possible mistakes by me, of course.
I don't really see the Larmor formula in here. None of the terms has a dependence on the combination $e^2 a^2$, for example. This makes me question whether I'm on the right track to get this sort of result.
Upon reading more on this topic, I've seen people mentioning that in SCFTs ($\mathcal{N} = 4$ SYM and such), calculations seem to use a cusp to get a Bremsstrahlung function, and then plug that into a generalization of the Larmor formula. However, since I have no experience with SCFTs, navigating this literature has been challenging.
Hence, here I come. How can I use Wilson lines to compute Bremsstrahlung and reproduce the Larmor formula, for example? I'm guessing the tricks involve the cusp because it is easier than computing a whole accelerated Wilson line, in which case the problem is to understand how the Bremsstrahlung function appears as it does in each of these places (cusp, small angle limit, Larmor formula, etc).
I'm trying to formulate this question in a more standard manner, but resource recommendations are extremely welcome if they do not require a lot of previous knowledge about gauge theory and SCFT.
