Amplitude mode in Eliashberg superconductors

In the BCS theory of superconductivity the amplitude (Higgs) mode is degenerate with the pair-breaking threshold: its frequency coincides with $2\mathrm{\Delta }$, so it is an edge singularity rather than a true collective mode. This is why after a sudden weak perturbation at $t=0$ its oscillations decay algebraically as $1\mathrm{/}\sqrt{t}$. We show that the retardation effects due to the phonon dynamics in the strong-coupling theory lift this degeneracy. The Higgs mode detaches from the pair-breaking edge $2{\mathrm{\Delta }}_g$, set by the spectral gap, and becomes a sharp subgap excitation. We determine its frequency from the longitudinal pair susceptibility evaluated directly on the real frequency axis, using the Eilenberger equations for Migdal–Eliashberg theory with the Einstein phonon spectrum. The response of the mass renormalization supplies at least two fifths of the displacement, and possibly all of it: with it switched off, the binding falls below the resolution of the calculation. Its consequences, however, are significant because a percent-level shift transfers about a quarter of the spectral weight of the amplitude channel out of the two-quasiparticle continuum into a discrete energy level. What experiment sees is therefore not the displacement itself but its consequence: amplitude oscillations that persist instead of decaying as $1\mathrm{/}\sqrt{t}$.