Love beyond Einstein: Metric reconstruction and Love number in quadratic gravity using WEFT

TL;DR

This work generalizes the tidal Love-number analysis to four-dimensional quadratic-curvature gravity using worldline EFT (WEFT). By diagonalizing the higher-derivative sector into scalar $Φ$ and tensor $Σ$ modes and reconstructing the metric from tree-level one-point functions, the authors show that quadratic corrections induce nonzero but scale-invariant tidal responses, with no classical RG running. They derive Yukawa-deformed solutions for scalar perturbations and perform UV–IR matching to fix ultraviolet charges, linking Wilson coefficients to observable tidal coefficients. Overall, the study provides a robust EFT framework to test deviations from GR with gravitational waves and clarifies how higher-curvature terms imprint in black hole tidal responses, including distinct Yukawa scales and controlled PN behavior.

Abstract

We study tidal Love numbers of static black holes in four-dimensional quadratic theory of gravity, extending the result of GR. We use worldline effective field theory (WEFT) methods to compute metric perturbations from one-point functions, treating the higher-derivative terms perturbatively. We show that insertions of scalar fields on the worldline induce non-zero tidal tails, and the corresponding Love number displays no RG running. The same conclusion holds for the insertions of tensor fields. Furthermore, for scalar dipole perturbations, we derive a Yukawa-deformed Frobenius solution and match the asymptotic behavior to fix the UV charge, finding agreement with EFT predictions of Wilson coefficients. Our work demonstrates that quadratic higher-curvature corrections induce non-zero but scale-independent tidal responses, offering a robust EFT framework to test deviations from GR in gravitational wave observations.

Figures (5)

• Figure 1: A general figure of Pyramid-like diagrams appearing in the evaluation of the response coefficients.
• Figure 3: Schematic representation of diagrams that we compute to reconstruct the metric. The blob is made of various higher derivative terms, which are connected by propagators in the static gauge. The GR power counting is modified by the higher derivative vertices in the bulk.
• Figure 4: Feynman diagrams contributing to the computation of the one-point function of $\Sigma$ and $\Phi$ up to 1PN. We need to sum up these diagrams to perturbatively reconstruct the metric.
• Figure 5: Diagrammatic representation of Love number vertex. The cross denotes the insertion of the tidal field. The other line representing the response has a propagator associated with it (in the static limit).
• Figure 6: Diagrams contributing to the Love number for spin-0 perturbation. The wavy lines denote the insertion of re-defined fields-$\Sigma$ and $\Phi$ to the worldline. The cross denotes the insertion of the tidal field (external source).