Black hole scattering and partition functions

• Published in: The journal of high energy physics 2022-10, Vol.2022 (10), p.39-27, Article 39
• Main Authors: Law, Y. T. Albert, Parmentier, Klaas
• Format: Article
• Language: English
• Subjects: Black Holes; Classical and Quantum Gravitation; Density of states; Elementary Particles; Energy; Free energy; Gravity; High energy physics; Ideal gas; Integrals; Mathematical analysis; Models of Quantum Gravity; Particle density (concentration); Partitions (mathematics); Physics; Physics and Astronomy; PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; Scalars; Scattering; Spacetime; String Theory; Thermal Field Theory
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP10(2022)039

A bstract When computing the ideal gas thermal canonical partition function for a scalar outside a black hole horizon, one encounters the divergent single-particle density of states (DOS) due to the continuous nature of the normal mode spectrum. Recasting the Lorentzian field equation into an effective 1D scattering problem, we argue that the scattering phases encode non-trivial information about the DOS and can be extracted by “renormalizing” the DOS with respect to a reference. This defines a renormalized free energy up to an arbitrary additive constant. Interestingly, we discover that the 1-loop Euclidean path integral, as computed by the Denef-Hartnoll-Sachdev formula, fixes the reference free energy to be that on a Rindler-like region, and the renormalized DOS captures the quasinormal modes for the scalar. We support these claims with the examples of scalars on static BTZ, Nariai black holes and the de Sitter static patch. For black holes in asymptotically flat space, the renormalized DOS is captured by the phase of the transmission coefficient whose magnitude squared is the greybody factor. We comment on possible connections with recent works from an algebraic point of view.