Comment on “Towards exact solutions for the superconducting T c induced by electron-phonon interaction”

In a series of recent papers starting with Phys. Rev. B 103, 094501 (2021), Liu, Yang, Pan et al. claim to have determined the exact dressed electron-boson vertex for a number of very general interacting many-body problems by inverting certain Ward-Takahashi identities. Here, we point out that their Ward-Takahashi identities are missing terms which make the inversion of the identities impossible. One therefore cannot formulate exact self-contained integral equations for the dressed electron propagator. In addition, the proposed vertex expressions do not reproduce well-established results for the leading perturbative corrections.