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Heavy Black Hole Effective Theory

Poul H. Damgaard, Kays Haddad, Andreas Helset

TL;DR

This work introduces the Heavy Black Hole Effective Theory (HBET), an EFT for heavy masses interacting with gravity that mirrors HQET's separation of scales to isolate classical contributions in gravitational scattering. By restoring and tracking powers of ħ, HBET identifies which operators contribute classically at arbitrary loop orders and derives Lagrangians for both heavy scalars and fermions. The authors compute the 2→2 gravitational scattering amplitudes up to 2PM, decomposing results into spinless, spin-orbit, and spin-spin sectors and providing leading quantum corrections, while confirming consistency with known non-relativistic limits. The framework offers a principled path to higher-order and higher-spin calculations, and suggests connections to NRGR, on-shell methods, and the double-copy structure of gravitational amplitudes.

Abstract

We formulate an effective field theory describing large mass scalars and fermions minimally coupled to gravity. The operators of this effective field theory are organized in powers of the transfer momentum divided by the mass of the matter field, an expansion which lends itself to the efficient extraction of classical contributions from loop amplitudes in both the post-Newtonian and post-Minkowskian regimes. We use this effective field theory to calculate the classical and leading quantum gravitational scattering amplitude of two heavy spin-1/2 particles at the second post-Minkowskian order.

Heavy Black Hole Effective Theory

TL;DR

This work introduces the Heavy Black Hole Effective Theory (HBET), an EFT for heavy masses interacting with gravity that mirrors HQET's separation of scales to isolate classical contributions in gravitational scattering. By restoring and tracking powers of ħ, HBET identifies which operators contribute classically at arbitrary loop orders and derives Lagrangians for both heavy scalars and fermions. The authors compute the 2→2 gravitational scattering amplitudes up to 2PM, decomposing results into spinless, spin-orbit, and spin-spin sectors and providing leading quantum corrections, while confirming consistency with known non-relativistic limits. The framework offers a principled path to higher-order and higher-spin calculations, and suggests connections to NRGR, on-shell methods, and the double-copy structure of gravitational amplitudes.

Abstract

We formulate an effective field theory describing large mass scalars and fermions minimally coupled to gravity. The operators of this effective field theory are organized in powers of the transfer momentum divided by the mass of the matter field, an expansion which lends itself to the efficient extraction of classical contributions from loop amplitudes in both the post-Newtonian and post-Minkowskian regimes. We use this effective field theory to calculate the classical and leading quantum gravitational scattering amplitude of two heavy spin-1/2 particles at the second post-Minkowskian order.

Paper Structure

This paper contains 21 sections, 63 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Counting h-bar
  3. Counting at one-loop
  4. Counting at n-loops
  5. Heavy Quark Effective Theory
  6. Heavy Black Hole Effective Theory
  7. Long range 2-to-2 gravitational scattering amplitudes
  8. First Post-Minkowskian Order
  9. Second Post-Minkowskian Order
  10. Conclusion
  11. HQET Spinors
  12. Heavy Scalar Effective Theory
  13. Long range 2-to-2 electromagnetic scattering amplitudes
  14. Feynman rules
  15. Abelian HSET
  16. ...and 6 more sections

Figures (2)

  • Figure 1: Classical scattering of two particles at tree-level.
  • Figure 2: The one-loop Feynman diagrams containing non-analytic pieces that contribute to the classical scattering of two particles in GR. Solid lines represent fermions, wavy lines represent gravitons, and dashed lines represent the ghost field arising from working in the harmonic gauge Holstein:2008sx.