Short-distance breakdown of the Higgs mechanism and the robustness of the BCS theory for charged superconductors

Through the Higgs mechanism, the long-range Coulomb interaction eliminates the low-energy Goldstone phase mode in superconductors and transfers spectral weight all the way up to the plasma frequency. Here we show that the Higgs mechanism breaks down for length scales shorter than the superconducting coherence length while it stays intact, even at high energies, in the long-wavelength limit. This effect is a consequence of the composite nature of the Higgs field of superconductivity and the broken Lorentz invariance in a solid. Most importantly, the breakdown of the Higgs mechanism inside the superconducting coherence volume is crucial to ensure the stability of the BCS mean-field theory in the weak-coupling limit. We also show that changes in the gap equation due to plasmon-induced fluctuations can lead to significant corrections to the mean-field theory and reveal that changes in the density-fluctuation spectrum of a superconductor are not limited to the vicinity of the gap.