Zero-momentum relaxation rate of the irrelevant scalar operator

Determine the lowest quasinormal frequency at zero spatial momentum, and hence the relaxation rate, of the dual operator \(\mathcal{O}_{-}\) in the perturbatively double-scalar-haired planar AdS\(_4\) black-hole backgrounds.

Background

The paper computes several hydrodynamic transport quantities, including the shear-viscosity ratio, bulk viscosity, and sound-attenuation coefficient. These quantities remain protected at their conformal values through the stated perturbative order because of the scaling properties of the background and the temperature-independent horizon scalar data.

The relaxation rate of the operator dual to the second scalar is explicitly distinguished from these protected quantities. It is associated with the lowest quasinormal mode at zero momentum, but its value is not calculated in the paper and is deferred to future work.

References

The relaxation rate of the dual operator \mathcal{O}{-} itself -- the lowest quasinormal frequency of \phi{-} at zero momentum -- is not of this protected type and is left for future work.

— Perturbative Double-Scalar Hair on Planar Black Holes in AdS$_{4}$ Einstein-Scalar Gravity  (2609.01332 - Yun, 1 Sep 2026) in Section 6.1, paragraph “Summary”