Matter one-loop logarithms and homogeneous TTNC scale response of Lifshitz black branes

TL;DR

This paper calculates the one-loop matter contributions to thermodynamics and scale response of Lifshitz black branes, revealing detailed coefficients and their behavior across different quantum fields and the relativistic limit.

Contribution

It provides explicit closed-form expressions for logarithmic coefficients and horizon contributions for scalar, Dirac, and Maxwell fields in Lifshitz black branes.

Findings

Derived closed expressions for heat-kernel and horizon contributions.

Identified gauge-field contact contributions in conical entropy.

Confirmed vanishing of source response in the relativistic limit.

Abstract

We compute the logarithmic one-loop matter contribution to the thermodynamics and homogeneous TTNC scale response of a four-dimensional Lifshitz black brane. The background is the neutral planar member of the analytic Einstein--Maxwell--dilaton Lifshitz black-brane family, while the quantum fields are treated as probes: a real scalar with arbitrary mass and nonminimal curvature coupling, a four-component Dirac spinor, and an Abelian Maxwell field with its Faddeev--Popov ghosts. For each spin sector, the logarithmic coefficient separates into a smooth radial heat-kernel contribution, governed by $\mathcal{C}_1$, and a horizon-localized conical contribution, governed by $\mathcal{W}_h$. The smooth coefficient controls the source-induced homogeneous projection of the TTNC Weyl Ward identity, whereas the logarithmic thermal entropy is controlled by $\mathcal{C}_{\rm therm}=\mathcal{C}_1+zL^2\mathcal{W}_h$. We give closed expressions for $\mathcal{C}_1$, $\mathcal{W}_h$, and $\mathcal{C}_{\rm therm}$ in the scalar, Dirac, and Maxwell probe sectors, identify the gauge-field contact contribution in the conical entropy, and verify that the smooth source response vanishes in the relativistic planar $z=1$ limit while the standard horizon heat-kernel entropy coefficient remains.