On the logarithmic Love number of black holes beyond general relativity

TL;DR

This work shows that the leading logarithmic running of black-hole tidal Love numbers can be computed directly from the local structure of the perturbation equation via a master Frobenius-based formula, without solving the full perturbation. By applying this to scalar probes and odd-parity tensor perturbations in static, spherically symmetric spacetimes, the authors derive general criteria for when the logarithmic coefficient a0 vanishes or remains nonzero and demonstrate the results with explicit examples such as Schwarzschild-Tangherlini and Hayward black holes. They prove that perturbative deformations of Schwarzschild or RN metrics generically induce nonzero running beyond certain multipoles, while nonperturbative or carefully constructed metrics can evade this conclusion. The Hayward metric serves as a concrete nonperturbative check, and the analysis clarifies the crucial role of perturbativity in determining the presence of logarithmic running, with implications for tests of GR and Beyond-GR theories via Love numbers. Overall, the paper provides a practical, universal method to assess logarithmic running in BH perturbations and highlights when such running is guaranteed or can be avoided in modified gravity scenarios.

Abstract

Tidal Love numbers and other response coefficients of black holes sometimes exhibit a logarithmic dependence on scale, or 'running'. We clarify that this coefficient is directly calculable from the structure of the equation obeyed by the field perturbation, and requires no knowledge of the full solution. The derived formula allows us to establish some general results on the existence of logarithmic running. In particular, we show that any static and spherically symmetric spacetime that modifies the Schwarzschild or Reissner-Nordström solutions in a perturbative way must have non-zero logarithmic Love numbers. This applies for instance to all regular black hole metrics. On the other hand, our analysis highlights the importance of the perturbativity assumption: without it, we find explicit black hole solutions beyond general relativity with exactly zero running. We also illustrate the advantage of our method by recovering and extending the known results for the Hayward metric.