Revisiting the matching of black hole tidal responses: A systematic study of relativistic and logarithmic corrections

The worldline effective field theory (EFT) gives a gauge-invariant definition of black hole conservative tidal responses (Love numbers), dissipation numbers, and their spin-0 and spin-1 analogs. In the first part of this paper we show how the EFT allows us to circumvent the source/response ambiguity without having to use the analytic continuation prescription. The source/response ambiguity appears if relativistic corrections to external sources overlap with the response. However, these corrections can be clearly identified and isolated using the EFT. We illustrate that by explicitly computing static one-point functions of various external fields perturbing the four-dimensional Schwarzschild geometry. Upon resumming all relevant Feynman diagrams, we find that the relativistic terms that may mimic the response actually vanish for static black holes. Thus, the extraction of Love numbers from matching the EFT and general relativity (GR) calculations is completely unambiguous, and it confirms previous results that the Love numbers vanish identically for all types of perturbations. We also study in detail another type of fine-tuning in the EFT, the absence of Love numbers' running. We show that logarithmic corrections to Love numbers do stem from individual loop diagrams in generic gauges, but cancel after all diagrams are summed over. In the particular cases of spin-0 and spin-2 fields the logarithms are completely absent if one uses the Kaluza-Klein metric decomposition. In the second part of the paper we compute frequency-dependent dissipative response contributions to the one-point functions using the Schwinger-Keldysh formalism. We extract black hole dissipation numbers by comparing the one-point functions in the EFT and GR. Our results are in perfect agreement with those obtained from a manifestly gauge-invariant matching of absorption cross sections.