Gaussian fluctuation corrections to the BCS mean field gap amplitude at zero temperature

The leading (Gaussian) fluctuation correction to the weak-coupling zero temperature BCS superconducting gap equation is computed. We find that the dominant contribution comes from the high energies and momenta (compared to the gap) and gives a correction smaller by the weak-coupling factor $gN_0$ than the mean-field terms. This correction is small due to cancellation of singular contributions from the amplitude and phase mode at high energies and momenta.