Next-to-next-to-leading order gravitational spin-squared potential via the effective field theory for spinning objects in the post-Newtonian scheme

TL;DR

This work delivers, for the first time, the next-to-next-to-leading order spin-squared interaction potential for generic compact binaries within the effective field theory of gravitating spinning objects in the post-Newtonian regime. The calculation systematically enumerates and evaluates 64 Feynman diagrams up to order G^3, employing nonrelativistic gravitational fields to manage the spin-squared sector's complexity and nonlinear couplings. The resulting NNLO potential, decomposed into time-derivative orders, completes the conservative dynamics up to 4PN for rapidly rotating objects, with plans to derive the equations of motion and Hamiltonians in forthcoming work. The approach reinforces the robustness and versatility of the EFT framework for spinning bodies and sets the stage for precise waveform modeling and EFT-matching to physical multipole moments relevant to gravitational-wave data analysis.

Abstract

The next-to-next-to-leading order spin-squared interaction potential for generic compact binaries is derived for the first time via the effective field theory for gravitating spinning objects in the post-Newtonian scheme. The spin-squared sector is an intricate one, as it requires the consideration of the point particle action beyond minimal coupling, and mainly involves the spin-squared worldline couplings, which are quite complex, compared to the worldline couplings from the minimal coupling part of the action. This sector also involves the linear in spin couplings, as we go up in the nonlinearity of the interaction, and in the loop order. Hence, there is an excessive increase in the number of Feynman diagrams, of which more are higher loop ones. We provide all the Feynman diagrams and their values. The beneficial nonrelativistic gravitational fields are employed in the computation. This spin-squared correction, which enters at the fourth post-Newtonian order for rapidly rotating compact objects, completes the conservative sector up to the fourth post-Newtonian accuracy. The robustness of the effective field theory for gravitating spinning objects is shown here once again, as demonstrated in a recent series of papers by the authors, which obtained all spin dependent sectors, required up to the fourth post-Newtonian accuracy. The effective field theory of spinning objects allows to directly obtain the equations of motion, and the Hamiltonians, and these will be derived for the potential obtained here in a forthcoming paper.

Figures (6)

• Figure 1: NNLO spin-squared one-graviton exchange Feynman diagrams.
• Figure 2: NNLO spin-squared two-graviton exchange Feynman diagrams.
• Figure 3: NNLO spin-squared one-loop Feynman diagrams.
• Figure 4: NNLO spin-squared cubic in $G$ Feynman diagrams with no loops.
• Figure 5: NNLO spin-squared cubic in $G$ one-loop Feynman diagrams.