Trying to understand the Photoelectric Effect from a QFT perspective has the potential to helpfully shatter a lot of illusions one may have built up from studying QM/QFT.
The cross-section can be calculated explicitly for the K-shell electrons, i.e. for a hydrogen atom - it's sometimes called the 'Stobbe' cross section, and it's not simple to derive (it's done in a 'modern' fashion in  and , it's done in an older though still similar manner in ).
Notice we are talking about an electron moving in the external electromagnetic field of the nucleus (the Coulomb potential), while also allowing a single quantized photon to interact with the electron moving in that field in an absorption process.
In other words, we are already talking about 'QFT' for a 'bound state' in an 'external field'.
Further we are talking about wave functions of single Hydrogen atom electrons in the initial state, and wave functions of a single electron in the continuous spectrum in the final state, where the process of calculating the cross sections involves using normalization factors for those continuous spectrum wave functions, a statement that should ring alarm bells since continuous spectrum wave functions are supposed to be non-normalizable and non-physical right?
Even worse, since we are talking about a transition involving a single photon, it means we are doing 'first order QED', which is 'proven' to be trivial at first order in QFT books.
First order QED? Bound states? External potentials? Everything is time-independent? Continuous spectrum wave functions which are normalized? Electron wave functions which are solutions of the Dirac/Schrodinger equation and not expanded in terms of creation and annihilation operators?
The applicability of some of these statements to 'QFT' are commonly denied/seriously-questioned even on this site, yet one of the most important problems in physics uses all of them.
But we are doing 'First order QED' using external fields, so that by-passes the usual 'first order' proof, and we are using wave function solutions of the Dirac equation (or Schrodinger equation, in the non-relativistic limit) to capture the effect of the external field, and we are using quantized EM fields to explain why the photon gets annihilated, and the single particle Dirac equation and it's stationary state solutions obviously apply because we can obviously measure the process, and if we've studied the continuous spectrum properly we know there's obviously no issue with normalizing them and physically interpreting the result. Further everything is time-independent, so in the derivation of the Fermi Golden Rule one does not even touch the matrix element of the interaction term, the interesting thing (given the unbelievable comments in the links above) about the Fermi Golden Rule is that since it involves the continuous spectrum one may find extra factors in the formula as one does in .
- Landau and Lifshitz, "Quantum Electrodynamics"
- Akhiezer and Berestetskii, "Quantum Electrodynamics"
- Heitler, "Quantum Theory of Radiation".