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

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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 gN₀ 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.