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Scattering Amplitudes, the Tail Effect, and Conservative Binary Dynamics at $O(G^4)$

Zvi Bern, Julio Parra-Martinez, Radu Roiban, Michael S. Ruf, Chia-Hsien Shen, Mikhail P. Solon, Mao Zeng

Abstract

We complete the calculation of conservative two-body scattering dynamics at fourth post-Minkowskian order, i.e. $O(G^4)$ and all orders in velocity, including radiative contributions corresponding to the tail effect in general relativity. As in previous calculations, we harness powerful tools from the modern scattering amplitudes program including generalized unitarity, the double copy, and advanced multiloop integration methods, in combination with effective field theory. The classical amplitude involves complete elliptic integrals, and polylogarithms with up to transcendental weight two. Using the amplitude-action relation, we obtain the radial action directly from the amplitude, and match the known overlapping terms in the post-Newtonian expansion.

Scattering Amplitudes, the Tail Effect, and Conservative Binary Dynamics at $O(G^4)$

Abstract

We complete the calculation of conservative two-body scattering dynamics at fourth post-Minkowskian order, i.e. and all orders in velocity, including radiative contributions corresponding to the tail effect in general relativity. As in previous calculations, we harness powerful tools from the modern scattering amplitudes program including generalized unitarity, the double copy, and advanced multiloop integration methods, in combination with effective field theory. The classical amplitude involves complete elliptic integrals, and polylogarithms with up to transcendental weight two. Using the amplitude-action relation, we obtain the radial action directly from the amplitude, and match the known overlapping terms in the post-Newtonian expansion.
Paper Structure (1 section, 12 equations, 2 figures, 3 tables)

This paper contains 1 section, 12 equations, 2 figures, 3 tables.

Table of Contents

  1. Amplitude in different regions.

Figures (2)

  • Figure 1: Complete set of generalized unitarity cuts for conservative contributions at $O(G^4)$. Exposed lines are on-shell. Thick lines represent massive scalars and thin lines are gravitons. The first eight cuts contain both (ppp) and (prr) contributions, while the last five, which have gravitons starting and ending on the same matter line, have no (ppp) contributions.
  • Figure 2: Sample diagrams at ${\cal O}(G^4)$. From left to right: a diagram with both (ppp) and (prr) contributions, a diagram with only (prr) contributions, a diagram containing iteration terms that cancel in the total amplitude.