Multipolar static tidal response of Schwarzschild black holes in cubic gravity: a metric-action derivation of tidal running

Abstract

Static tidal Love numbers of four-dimensional Schwarzschild black holes vanish in general relativity, whereas higher-curvature interactions can generate a nontrivial response. We investigate the parity-even cubic Weyl correction directly in metric variables and derive the electric, static response for every integer multipole $\ell \ge 2$. Organizing the angular reduction through $L=\ell (\ell +1)$, we obtain exact radial actions and show that perturbative order reduction converts the three metric equations into a constrained two-dimensional first-order system. Eliminating one field yields a scalar equation whose homogeneous operator is precisely the general-relativistic static tidal operator. A Frobenius and Green-function analysis gives the gauge-invariant Zerilli--Moncrief running coefficient ${\beta }_{\ell }^{\mathrm{ZM}}={\epsilon }_{\mathrm{e}}7L^2(L-2{)}^2(L-4)(L-6)/12$ and identifies the factor $L-6$ as the reason why the quadrupole is the unique physical electric multipole without logarithmic running. We solve the quadrupole exactly, obtaining the fixed-integer branch ratio $-2400{\epsilon }_{\mathrm{e}}$ and explaining why it differs from the analytically continued canonical Love number $k_2^E=448{\epsilon }_{\mathrm{e}}$. For the octupole, we construct the complete horizon-regular global metric solution and exhibit the cancellation of horizon logarithms between the two Green-function channels. Finally, we derive the normalization map to the canonical electric beta functions and the corresponding running of finite-size worldline coefficients. The canonical result agrees with the modified-Teukolsky calculation, while the metric-action approach reveals the radial mechanism behind the exceptional quadrupole and provides the full metric reconstruction.

33 pages, 8 tables; includes ancillary Wolfram Language files for symbolic verification and reproducibility