Universal Closed Form for Dynamical Love Numbers of Black Holes

Abstract

Black hole static Love numbers vanish, but their dynamical counterparts do not. We present the entire scheme-independent dynamical response $\bar{F}_{\ell,s}$ of a Schwarzschild black hole in closed form, to all orders in $G$, and for every spin $s$ and multipole $\ell$. The result is $\bar{F}_{\ell,s}/4πR_S^{2\ell+1}=Φ_{\ell,s}(\bar{y})-\tfrac12η\,Φ_{\ell,s}'(\bar{y})$ with $\bar{y}=-\tfrac12η^2τ$ and $η=iωR_S$. Here $Φ_{\ell,s}$ is simply the leading-log solution to the renormalization group equation, but lifting the running logarithm to $τ=\log(R_S/R)-2\sum_{k\ge2}ζ_k\,η^{k-1}$ resums it to all orders. This tower of Riemann zeta values is the Newtonian phase in disguise: it originates from the long-range $1/r$ potential in the far zone, and is universal across multipole and spin. Our result exhibits a factorization pinned to three ingredients: the strength of horizon absorption, the running rate in the near zone, and the dressed logarithm in the far zone. Using shell effective field theory, we independently verify our formula for scalar, electromagnetic, and gravitational perturbations, reaching $\mathcal O(G^{15})$.