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Effective field theory approach to the gravitational two-body dynamics, at fourth post-Newtonian order and quintic in the Newton constant

Stefano Foffa, Pierpaolo Mastrolia, Riccardo Sturani, Christian Sturm

TL;DR

This work computes the missing $G_N^5$ contributions to the conservative two-body dynamics at $4$PN within an effective field theory (EFT) framework. By mapping each gravity amplitude at order $G_N^\\ell$ to a $(\ell-1)$-loop massless 2-point integral and applying integration-by-parts (IBP) with a basis of master integrals in dimensional regularization, the authors evaluate the 50 genuine $O(G_N^5)$ diagrams. The resulting complete 4PN Lagrangian is finite and matches existing results upon accounting for regularization and lower-order contributions, with explicit terms such as $\mathcal{L}_{49}=(32-3\pi^2)\frac{G_N^5 m_1^3 m_2^3}{r^5}$ and the total $\sum_a \mathcal{L}_a$ given by a specific combination of mass factors and powers of $r$. This EFT-based calculation provides a robust cross-check for the 4PN dynamics and demonstrates the feasibility of high-loop, classical-gravity calculations within the EFT paradigm.

Abstract

Working within the post-Newtonian (PN) approximation to General Relativity, we use the effective field theory (EFT) framework to study the conservative dynamics of the two-body motion at fourth PN order, at fifth order in the Newton constant. This is one of the missing pieces preventing the computation of the full Lagrangian at fourth PN order using EFT methods. We exploit the analogy between diagrams in the EFT gravitational theory and 2-point functions in massless gauge theory, to address the calculation of 4-loop amplitudes by means of standard multi-loop diagrammatic techniques. For those terms which can be directly compared, our result confirms the findings of previous studies, performed using different methods.

Effective field theory approach to the gravitational two-body dynamics, at fourth post-Newtonian order and quintic in the Newton constant

TL;DR

This work computes the missing contributions to the conservative two-body dynamics at PN within an effective field theory (EFT) framework. By mapping each gravity amplitude at order to a -loop massless 2-point integral and applying integration-by-parts (IBP) with a basis of master integrals in dimensional regularization, the authors evaluate the 50 genuine diagrams. The resulting complete 4PN Lagrangian is finite and matches existing results upon accounting for regularization and lower-order contributions, with explicit terms such as and the total given by a specific combination of mass factors and powers of . This EFT-based calculation provides a robust cross-check for the 4PN dynamics and demonstrates the feasibility of high-loop, classical-gravity calculations within the EFT paradigm.

Abstract

Working within the post-Newtonian (PN) approximation to General Relativity, we use the effective field theory (EFT) framework to study the conservative dynamics of the two-body motion at fourth PN order, at fifth order in the Newton constant. This is one of the missing pieces preventing the computation of the full Lagrangian at fourth PN order using EFT methods. We exploit the analogy between diagrams in the EFT gravitational theory and 2-point functions in massless gauge theory, to address the calculation of 4-loop amplitudes by means of standard multi-loop diagrammatic techniques. For those terms which can be directly compared, our result confirms the findings of previous studies, performed using different methods.

Paper Structure

This paper contains 12 sections, 55 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The method
  3. Amplitudes and Feynman Integrals
  4. Results and discussion
  5. Conclusion
  6. Master integrals
  7. Master integrals known in $d$ dimensions
  8. Master integrals known in $d = 3 + \varepsilon$ dimensions
  9. Results for all the amplitudes
  10. Evaluation of $\mathcal{A}_{33}$ and $\mathcal{A}_{50}$
  11. Amplitude 33
  12. Amplitude 50

Figures (3)

  • Figure 1: The diagrams contributing at order $G^5_N$. As in the EFT approach the massive objects are non-dynamical, the horizontal black lines have to be seen as classical sources, and not as propagators. Green solid lines stand for $\sigma$ field propagators, blue dashed lines for $\phi$ fields.
  • Figure 2: Four-loop 2-point topologies corresponding to the diagrams in fig.\ref{['diaG5']}.
  • Figure 3: The master integrals which appear in the calculation of the amplitudes in the set $\mathcal{A}_{II}$. The names of the diagrams follow refs. Baikov:2010hfLee:2011jtLee:2015eva.