Recently, machine learning was applied to extract both the normal and the anomalous components of the self-energy from photoemission data at the antinodal points in Bi- based cuprate high-temperature superconductors[Y. Yamajiet al.,arXiv:1903.08060]. It was argued that both components do show prominent peaks near 50 meV,which hold information about the pairing glue, but the peaks are hidden in the actual data, which measure onlythe total self-energy. We analyze the self-energy within an effective fermion-boson theory. We show that softthermal fluctuations give rise to peaks in both components of the self-energy at a frequency comparable tothe superconducting gap, while they cancel in the total self-energy, all irrespective of the nature of the pairingboson. However, in the quantum limitT→0 prominent peaks survive only for a very restricted subclass ofpairing interactions. We argue that the way to potentially nail down the pairing boson is to determine the thermalevolution of the peaks.