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This paper studies how a specific short-distance correction to gravity changes the way a non-rotating (Schwarzschild) black hole reacts to a static tidal field. In standard general relativity, the static tidal Love numbers of four-dimensional Schwarzschild black holes are zero. The authors add a parity-even cubic Weyl term to the action (a higher-curvature correction that can appear in effective field theory) and compute the resulting electric, static tidal response for every multipole number ℓ ≥ 2.
To do this they work directly with the metric (the gravitational field) and organize the angular dependence through L = ℓ(ℓ+1). They reduce the angular problem to exact radial actions and then simplify the three metric perturbation equations, by a perturbative order-reduction, into a constrained two-dimensional first-order system. Eliminating one field produces a single scalar equation whose homogeneous part is the same operator that governs static tides in general relativity. The cubic correction then acts as a source that can resonate with that operator.
Using a Frobenius expansion near the horizon and a Green-function analysis, the authors obtain a closed form for the gauge-invariant Zerilli–Moncrief logarithmic running coefficient: β_ZM_ℓ = ε_e · 7 L^2 (L−2)^2 (L−4)(L−6)/12, where ε_e parametrizes the small cubic coupling and L = ℓ(ℓ+1). The factor (L−6) explains why the quadrupole (ℓ = 2) is exceptional: it is the only electric multipole without logarithmic running. The paper also finds exact global metric solutions for ℓ = 2 (quadrupole) and ℓ = 3 (octupole). For the quadrupole they obtain a fixed-integer Zerilli–Moncrief branch ratio −2400 ε_e; this differs from the canonical analytically continued Love number k^E_2 = 448 ε_e because the two finite parts are defined in different ways.
For the octupole (ℓ = 3) the authors build the full horizon-regular solution and show explicitly how horizon logarithms cancel between two Green-function channels while logarithmic terms remain in the behaviour far from the hole. They then provide the translation between the Zerilli–Moncrief logarithm and the canonical normalization used in modified-Teukolsky calculations, and derive the implied running (scale dependence) of finite-size coefficients that appear in worldline effective actions. The canonical beta functions they obtain agree with previous modified-Teukolsky results, while the metric-action route gives the explicit radial mechanism and the full reconstructed metric.
Important caveats are that the work is perturbative in a small cubic coupling and restricted to static (zero-frequency), parity-even cubic corrections around non-rotating Schwarzschild black holes in four dimensions. Different normalization or subtraction conventions lead to different finite numbers for the same physical situation (hence the different quadrupole values mentioned above), and some parts of the analysis rely on analytic-continuation tools when separating local terms from the asymptotic response. The results therefore show how higher-curvature effects can produce tidal responses that vanish in pure general relativity, but the conclusions apply within the stated approximations and choice of cubic interaction.