Thermodynamic Consistency of Logarithmic Entropy Corrections on the Schwarzschild Branch of $f(\mathbb{Q})$ Gravity and a Superradiance No-Go Result

Abstract

We present a concise and explicit analysis of logarithmically corrected black-hole thermodynamics and scalar scattering on the Schwarzschild branch of symmetric teleparallel gravity. Treating the metric and the flat, torsion-free affine connection as independent variables, we formulate the relevant nonmetricity geometry and field equations and show that the linear branch is dynamically equivalent to general relativity up to a boundary term. The vacuum solution is therefore the Schwarzschild spacetime. The leading entropy is derived from both the Noether-charge method and the classical first law, after which a logarithmic correction is introduced. When the geometry and ADM mass are kept fixed, the geometric Hawking temperature remains unchanged, whereas the temperature defined through the corrected first law is modified. The apparent divergence of the heat capacity occurs outside the regime in which the logarithmic expansion is reliable and therefore cannot be interpreted as a physical phase transition. We also derive the scalar radial equation, effective potential, conserved Wronskian, and reflection-transmission relation. For a neutral scalar field on a static neutral background, no superradiant amplification occurs, and the entropy correction produces no first-order change in the scattering amplitudes unless a genuine semiclassical backreaction or an explicit rotating or charged black-hole solution is provided.