Title: Short-distance breakdown of the Higgs mechanism and the robustness of the BCS theory for charged superconductors,
Abstract: 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.