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

20 pages; references updated and minor revisions in the discussion; version to appear in JHEP

References in corpus (20)

• An Effective Field Theory of Gravity for Extended Objects
• The Effective Field Theorist's Approach to Gravitational Dynamics
• Post-Newtonian corrections to the motion of spinning bodies in NRGR
• Analytic Solutions of the Teukolsky Equation and their Low Frequency Expansions
• Tidal Deformation and Dissipation of Rotating Black Holes
• Spinning Black Holes Fall in Love
• Tidal Love Numbers of Kerr Black Holes
• On the Vanishing of Love Numbers for Kerr Black Holes
• The Tune of Love and the Nature(ness) of Spacetime
• Analytic Solutions of the Regge-Wheeler Equation and the Post-Minkowskian Expansion
• Tidal deformation of a slowly rotating material body. External metric
• Analytic Solutions of the Teukolsky Equation and their Properties
• Ladder Symmetries of Black Holes: Implications for Love Numbers and No-Hair Theorems
• Classical Effective Field Theory and Caged Black Holes
• Vanishing of black hole tidal Love numbers from scattering amplitudes
• Love symmetry
• Non-linearities in the tidal Love numbers of black holes
• Addressing issues in defining the Love numbers for black holes
• Rotating black holes experience dynamical tides
• Dynamical Tidal Response of Non-rotating Black Holes: Connecting the MST Formalism and Worldline EFT