Sign of the imaginary part of superfluid stiffness at μ=0 when Re(Δ̄Δ) ≤ 0

Ascertain, for the isotropic three-dimensional BCS-type non-Hermitian superconductor with constant s-wave pairing mean fields Δ and \bar{Δ} (not constrained to be Hermitian conjugates) at zero chemical potential μ=0, the sign of the imaginary part of the superfluid stiffness Q when Re(\bar{Δ}Δ) ≤ 0, where Q is defined as the second derivative of the action with respect to the vector potential evaluated at zero field.

Background

In the same 3D non-Hermitian BCS setting, the imaginary part of Q is linked to pumping/decay of Bogoliubov quasiparticles. For Re(\bar{Δ}Δ) > 0, the sign of Im[Q(μ=0)] follows the sign of Im(\bar{Δ}Δ).

However, when Re(\bar{Δ}Δ) ≤ 0, the integral expression contains competing contributions, and the sign of Im[Q(μ=0)] cannot be concluded analytically, leaving this sign determination unresolved.