Persistence of the NSS–vertex-correction relation at finite wave vector and frequency

Determine whether and how the relation between normal-state subtraction and the vertex-corrected electromagnetic response persists at finite wave vector and frequency beyond the static long-wavelength limit for the massive-Dirac superconductor with s-wave pairing.

Background

The paper establishes, for a massive-Dirac superconductor with s-wave pairing at zero temperature, that normal-state subtraction combined with the collective-mode vertex correction produces a gauge-invariant electromagnetic response in the static long-wavelength limit. In particular, the vertex correction cancels the longitudinal response while its transverse component vanishes, so the Meissner weight agrees with the normal-state-subtracted result.

The authors explicitly leave unresolved whether this correspondence continues to hold when the external perturbation has finite spatial wave vector and finite frequency. Resolving this issue would determine the range of validity of normal-state subtraction as an effective gauge-invariant prescription outside the limit analyzed in the paper.