Fixed-quadrupole static tidal response of Schwarzschild black holes in a cubic Weyl effective field theory

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Original abstract

Static Love numbers of four-dimensional Schwarzschild black holes vanish in general relativity. We study how the fixed-quadrupole static tidal solution is modified by the parity-even cubic Weyl operator in the gravitational effective field theory. Working perturbatively in $ε_{\rm e}=λ_{\rm e}(Λr_s)^{-4}$, we construct the reduced quadratic radial action for static even-parity $\ell=2$ perturbations, order-reduce the higher-derivative equations, and solve the resulting boundary-value problem directly in metric variables. The order-$ε_{\rm e}$ equations reduce to a first-order two-dimensional inhomogeneous system for $X_0$ and $X_K$, with $X_2$ fixed by an algebraic constraint. Horizon regularity leaves one constant, but matching to infinity shows that this freedom only renormalizes the applied tidal branch. After removing this tidal renormalization, the decaying branch is unambiguous. Calibrating the spatial sector at fixed $\ell=2$ against the associated-Legendre branches $P_2^{2}$ and $Q_2^{2}$, we obtain a fixed-quadrupole response amplitude $Δ(B/A)=-2400ε_{\rm e}$. Equivalently, the scalar fixed-$\ell$ quotient gives $Δk_{2,\rm sc}^{\rm fix}=-20ε_{\rm e}$. The second number is a scalar fixed-$\ell$ conversion of the metric branch ratio, not, by itself, the analytically continued, gauge-invariant electric Love number. A comparison with canonical Teukolsky-based Love numbers requires an additional continuation in $\ell$ and a precise map of normalizations. The result should therefore be viewed as a reproducible metric-sector benchmark for the cubic Weyl EFT, complementary to gauge-invariant master-equation approaches.

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