This paper derives a single, closed-form expression for the dynamical tidal response of a non-rotating (Schwarzschild) black hole. Love numbers measure how an object deforms in an external tidal field. For black holes the static (zero-frequency) Love numbers are known to vanish, but their dynamical (frequency-dependent) counterparts are not. The authors present a formula for the full dynamical response F̄_{ℓ,s} that holds for every multipole ℓ and field spin s and that organizes the frequency dependence to all orders.

The central formula can be written compactly as F̄_{ℓ,s}/(4πR_S^{2ℓ+1}) = Φ_{ℓ,s}(ȳ) − (1/2) η Φ'_{ℓ,s}(ȳ). Here R_S is the Schwarzschild radius, η = i ω R_S encodes the wave frequency ω, and the argument ȳ is ȳ = −(1/2) η^2 τ. The novel piece is τ, a ‘‘dressed’’ logarithm: τ = log(R_S/R) − 2 Σ_{k≥2} ζ_k η^{k−1}. The ζ_k are Riemann zeta values — special numbers that appear in many expansions — and they form a universal tower that the authors identify as the gravitational analogue of the Newtonian (Coulomb) scattering phase.

At a high level the result follows a clean factorization. The response splits into three ingredients: a hard matching coefficient fixed by absorption at the horizon, an anomalous-dimension rate set by the near-horizon (near-zone) physics, and a dressed logarithm coming from long-range (far-zone) interactions. The function Φ_{ℓ,s} is the leading-log solution of the renormalization-group flow that controls how tidal coefficients change with scale. Lifting the running log to τ resums an infinite tower of zeta-value corrections. The monopole (ℓ = 0) is a special case with a Möbius-type form; multipoles ℓ ≥ 1 share a single exponential kernel determined by explicit coefficients the authors give.

Why this matters: the formula explains an intriguing pattern of zeta values previously seen in high-order computations and packages that pattern into a compact, scheme-independent form. It provides a route to generate dynamical Love numbers to arbitrarily high orders and unifies scalar, electromagnetic, and gravitational perturbations. The authors verify their closed form against existing calculations (including scalar results to O(G^9) and gravitational results to O(G^{11})) and by independent shell effective-field-theory computations that reach O(G^{15}). They also check the dissipative part of the response against absorption probabilities derived from known asymptotic amplitudes.