Highest-weight truncation, graded EFT structure, and renormalization of black hole Love numbers

TL;DR

This paper explains why four-dimensional black holes have zero static Love numbers by revealing a common near-zone truncation mechanism that leads to a highest-weight representation, eliminating static invariants.

Contribution

It uncovers a unified near-zone truncation explanation for the vanishing static Love numbers and related structures in black hole perturbation theory and Shell EFT.

Findings

Static solutions form a highest-weight representation, truncating hypergeometric solutions.

The truncation eliminates static Wilson coefficients in the effective theory.

No negative-weight invariants exist in the static sector, explaining zero Love numbers.

Abstract

The static tidal Love numbers of four-dimensional black holes vanish identically, unlike their nontrivial dynamical response at finite frequency. Recent work has provided three complementary descriptions of this phenomenon: an emergent $\mathrm{SL}(2,\mathbb{R})$ organization of static near-zone perturbations, a graded logarithmic and multi-zeta structure in Shell Effective Field Theory (Shell EFT), and an on-shell matching framework based on gravitational Raman scattering with renormalization group (RG) running. We show that these features arise from a common near-zone truncation mechanism. For a massless scalar field, horizon regularity selects a unique static solution forming a highest-weight-type representation, truncating the hypergeometric solution to a finite polynomial and eliminating the independent decaying branch at large radius. This excludes a static Wilson coefficient in the effective theory. We demonstrate that the same truncation operates in the static Regge-Wheeler and Zerilli equations for four-dimensional Schwarzschild black holes. Analytic continuation of the horizon-regular solution to small frequency via the Coulomb-hypergeometric or Mano-Suzuki-Takasugi formalisms preserves this truncation as an anchoring condition for the renormalized angular momentum parameter. The resulting low-frequency expansion is controlled by Gamma and hypergeometric functions, generating a graded algebra of logarithms and odd Riemann zeta values. Within this structure no invariant of negative weight exists in the static sector, so the vanishing of the static Love number follows as a structural consequence. This explains the ``zero-sum'' rule of Shell EFT and why the self-induced RG flow in gravitational Raman scattering cannot generate a static invariant.