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

Abstract

We compute the logarithmic one-loop matter contribution to the thermodynamics and homogeneous twistless torsional Newton--Cartan 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 $\calC_1$, and a horizon-localized conical contribution, governed by $\calW_h$. This separation identifies which part of the matter one-loop logarithm is visible in the boundary Lifshitz/TTNC Ward identity and which part instead belongs to horizon replica entropy. The smooth coefficient $\calC_1$ controls the source-induced homogeneous projection of the TTNC Weyl Ward identity, namely the smooth boundary-source contribution to the Ward combination $z\epsilon-2p$, whereas the thermodynamic logarithmic entropy is controlled by $\calC_{\rm therm}=\calC_1+zL^2\calW_h$. We give closed expressions for $\calC_1$, $\calW_h$, and $\calC_{\rm therm}$ for 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.