Highest-weight truncation, graded EFT structure, and renormalization of black hole Love numbers
Naman Kumar
TL;DR
The paper shows that the vanishing of static tidal Love numbers for four-dimensional black holes is a structural consequence of a near-zone highest-weight truncation enforced by horizon regularity. By weaving together an emergent $\mathrm{SL}(2,\mathbb{R})$ symmetry in the static near zone, the graded log- and multi-zeta structure in Shell EFT, and the on-shell RG framework from gravitational Raman scattering, it demonstrates that static invariants are forbidden while dynamical responses acquire a constrained, transcendental low-frequency expansion dominated by odd zeta values. The analysis extends from a scalar test field to full gravitational perturbations, highlighting a unified mechanism that explains the zero-sum rule and the nontrivial dynamical tidal response. The results provide a coherent EFT/storyline: horizon regularity fixes the static sector, whereas finite-frequency physics is governed by analytic structures in Gamma and hypergeometric functions, with the MST and Coulomb bases encoding the precise transcendental content.
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.