On the logarithmic Love number of black holes beyond general relativity
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-Nordstr\"om 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.