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Next-to-leading order gravitational spin-orbit coupling in an effective field theory approach

Michele Levi

TL;DR

This work computes the next-to-leading order spin-orbit interaction for a binary of spinning compact objects within an effective field theory framework. By employing nonrelativistic gravitational (NRG) fields and a detailed Feynman-rule set, the authors systematically evaluate one-, two-, and cubic-gravitational contributions, including nonstationary effects and spin supplementary-condition complexities. The resulting NLO spin-orbit Lagrangian is Legendre-transformed to a Hamiltonian and then related to canonical ADM results through noncanonical and canonical transformations, establishing precise equivalence with known Damour-DJS expressions. The analysis demonstrates the practicality and precision of EFT with NR gravity fields for high-order spin effects, setting the stage for future higher-order spin corrections in gravitational-wave physics.

Abstract

We use an effective field theory (EFT) approach to calculate the next to leading order (NLO) gravitational spin-orbit interaction between two spinning compact objects. The NLO spin-orbit interaction provides the most computationally complex sector of the NLO spin effects, previously derived within the EFT approach. In particular, it requires the inclusion of non-stationary cubic self-gravitational interaction, as well as the implementation of a spin supplementary condition (SSC) at higher orders. The EFT calculation is carried out in terms of the non-relativistic gravitational field parametrization, making the calculation more efficient with no need to rely on automated computations, and illustrating the coupling hierarchy of the different gravitational field components to the spin and mass sources. Finally, we show explicitly how to relate the EFT derived spin results to the canonical results obtained with the ADM Hamiltonian formalism. This is done using non-canonical transformations, required due to the implementation of covariant SSC, as well as canonical transformations at the level of the Hamiltonian, with no need to resort to the equations of motion or the Dirac brackets.

Next-to-leading order gravitational spin-orbit coupling in an effective field theory approach

TL;DR

This work computes the next-to-leading order spin-orbit interaction for a binary of spinning compact objects within an effective field theory framework. By employing nonrelativistic gravitational (NRG) fields and a detailed Feynman-rule set, the authors systematically evaluate one-, two-, and cubic-gravitational contributions, including nonstationary effects and spin supplementary-condition complexities. The resulting NLO spin-orbit Lagrangian is Legendre-transformed to a Hamiltonian and then related to canonical ADM results through noncanonical and canonical transformations, establishing precise equivalence with known Damour-DJS expressions. The analysis demonstrates the practicality and precision of EFT with NR gravity fields for high-order spin effects, setting the stage for future higher-order spin corrections in gravitational-wave physics.

Abstract

We use an effective field theory (EFT) approach to calculate the next to leading order (NLO) gravitational spin-orbit interaction between two spinning compact objects. The NLO spin-orbit interaction provides the most computationally complex sector of the NLO spin effects, previously derived within the EFT approach. In particular, it requires the inclusion of non-stationary cubic self-gravitational interaction, as well as the implementation of a spin supplementary condition (SSC) at higher orders. The EFT calculation is carried out in terms of the non-relativistic gravitational field parametrization, making the calculation more efficient with no need to rely on automated computations, and illustrating the coupling hierarchy of the different gravitational field components to the spin and mass sources. Finally, we show explicitly how to relate the EFT derived spin results to the canonical results obtained with the ADM Hamiltonian formalism. This is done using non-canonical transformations, required due to the implementation of covariant SSC, as well as canonical transformations at the level of the Hamiltonian, with no need to resort to the equations of motion or the Dirac brackets.

Paper Structure

This paper contains 12 sections, 116 equations, 4 figures.

Table of Contents

  1. Introduction
  2. EFT approach for binary inspiral in the PN approximation
  3. Self-gravitating spinning objects with an EFT approach
  4. Leading order spin-orbit potential
  5. Next-to-leading order spin-orbit interaction
  6. One-graviton exchange
  7. Two-graviton exchange
  8. Cubic self-gravitational interaction
  9. Next-to-leading order spin-orbit potential
  10. Conclusions
  11. Dimensional regularization and loop integrals
  12. PN canonical transformations

Figures (4)

  • Figure 1: LO spin-orbit interaction Feynman diagrams. The heavy solid lines represent the point particles worldlines. The oval gray and spherical black blobs represent the spin and mass couplings on the worldline, respectively. The solid and dashed lines represent the $\phi$ and $A_i$ fields, respectively. All diagrams must be included with their mirror images.
  • Figure 2: NLO spin-orbit interaction Feynman diagrams of one-graviton exchange. The double line represents the $\sigma_{ij}$ field. The encircled cross vertex corresponds to a propagator correction. All diagrams should be included with their mirror images.
  • Figure 3: Nonlinear NLO spin-orbit interaction Feynman diagrams of two-graviton exchange. These diagrams should be included together with their mirror images.
  • Figure 4: Nonlinear NLO spin-orbit interaction Feynman diagrams with a three-graviton vertex. The gray encircled cross vertex corresponds to the time dependent cubic vertex. These diagrams should be included together with their mirror images.