Nonrelativistic Dynamics of the Amplitude (Higgs) Mode in Superconductors

Despite the formal analogy with the Higgs particle, the amplitude fluctuations of the order parameter in weakly coupled superconductors do not identify a real mode with a Lorentz-invariant dynamics. Indeed, its resonance occurs at 2?0, which coincides with the threshold 2Egap for quasiparticle excitations that spoil any relativistic dynamics. Here we investigate the fate of the Higgs mode in the unconventional case where 2Egap becomes larger than 2?0, as due to strong coupling or strong disorder. We show that also in this situation, the amplitude fluctuations never identify a real mode at 2?0, since such a "bosonic" limit is always reached via strong mixing with the phase fluctuations, which dominate the low-energy part of the spectrum. Our results have direct implications for the interpretation of the subgap optical absorption in disordered superconductors.