Universal Closed Form for Dynamical Love Numbers of Black Holes

Black hole static Love numbers vanish, but their dynamical counterparts do not. We present the scheme-independent dynamical response ${\bar{F}}_{\mathrm{\ell },s}$ of a Schwarzschild black hole in closed form, to all orders, and for every spin $s$ and multipole $\mathrm{\ell }$. The result is ${\bar{F}}_{\mathrm{\ell },s}\mathrm{/}4\pi R_S^{2\mathrm{\ell }+1}={\mathrm{\Phi }}_{\mathrm{\ell },s}(\bar{y})-\frac{1}{2}\eta {\mathrm{\Phi }}_{\mathrm{\ell },s}^{\mathrm{\prime }}(\bar{y})$ with $\bar{y}=-\frac{1}{2}{\eta }^2\tau$ and $\eta =i\omega R_S$. Here ${\mathrm{\Phi }}_{\mathrm{\ell },s}$ is simply the leading-log solution to the renormalization group equation, but lifting the running logarithm to $\tau =\log (R_S\mathrm{/}R)-2{\sum }_{k\ge 2}{\zeta }_k{\eta }^{k-1}$ resums it to all orders. This tower of Riemann zeta values is the Newtonian phase in disguise: it originates from the same far-zone $\mathrm{\Gamma }(1-\eta )$ that governs long-range scattering, and is universal across multipole and spin. Our result exhibits a factorization pinned to three ingredients: the hard matching coefficient at the horizon, the anomalous dimension in the near zone, and the dressed log in the far zone. Using shell effective field theory, we independently verify our formula for scalar, electromagnetic, and gravitational perturbations, reaching $\mathcal{O}(G^{15})$.