Conservative dynamics of binary systems to fourth Post-Newtonian order in the EFT approach II: Renormalized Lagrangian

TL;DR

The paper delivers a self-contained derivation of the conservative binary dynamics to 4PN within the EFT framework, providing an ambiguity-free renormalized Lagrangian computed entirely in the PN regime. By using dimensional regularization and the method of regions, it identifies and disentangles intermediate IR and UV poles, absorbing UV divergences into worldline counter-terms and removing spurious IR poles via zero-bin subtraction, while ensuring the final potential remains regulator-independent. The analysis reveals how near- and far-zone contributions cancel their unphysical divergences and how the 4PN dynamics acquire both local and nonlocal (tail) effects, with the nonlocal tail captured in a nonlocal-in-time term in the Lagrangian. The approach yields results consistent with previous independent methods (ADM, Fokker-action, etc.) but avoids ambiguity parameters and additional regulators, thereby sharpening the theoretical foundation for high-precision gravitational wave templates and pointing toward extensions to higher PN orders.

Abstract

We complete the derivation of the conservative dynamics of binary systems to fourth Post-Newtonian (4PN) order in the effective field theory (EFT) approach. We present a self-contained (ambiguity-free) computation of the renormalized Lagrangian, entirely within the confines of the PN expansion. While we confirm the final results reported in the literature, we clarify several issues regarding intermediate infrared (IR) and ultraviolet (UV) divergences, as well as the renormalization procedure. First, we properly identify the IR and UV singularities using (only) dimensional regularization and the method of regions, which are the pillars of the EFT formalism. This requires a careful study of scaleless integrals in the potential region, as well as conservative contributions from radiation modes due to tail effects. As expected by consistency, the UV divergences in the near region (due to the point-particle limit) can be absorbed into two counter-terms in the worldline effective theory. The counter-terms can then be removed by field redefinitions, such that the renormalization scheme-dependence has no physical effect to 4PN order. The remaining IR poles, which are spurious in nature, are unambiguously removed by implementing the zero-bin subtraction in the EFT approach. The procedure transforms the IR singularities into UV counter-parts. As anticipated, the left-over UV poles explicitly cancel out against UV divergences in conservative terms from radiation-reaction, uniquely determining the gravitational potential. Similar artificial IR/UV poles, which are intimately linked to the split into regions, are manifest at lower orders. Starting at 4PN, both local- and nonlocal-in-time contributions from the radiation region enter in the conservative dynamics. Neither additional regulators nor ambiguity-parameters are introduced at any stage of the computations.

Figures (8)

• Figure 1: Topologies contribution at ${\cal O}(G^3)$ to the Lagrangian. The first two diagrams are UV divergent, starting at 3PN, while the third diagram is finite at this order. All of these topologies are divergent at 4PN. The first diagram is UV while the third is IR divergent. The second one has both, IR and UV, poles.
• Figure 2: The first non-linear topology leading to IR divergences away from the static limit. The spurious poles occur when the propagators are Taylor expanded in powers of $p_0/|{\boldsymbol{p}}|$.
• Figure 3: Feynman diagram for the contribution to the radiation-reaction force due the tail effect. See Galley:2015kus for more details.
• Figure 4: Self-energy diagrams leading to scaleless integrals with logarithmic IR/UV poles at $G^2, G^3$ and $G^4$, respectively. Unlike the first, which does not depend on the mass of the companion, the other diagrams represent self-energy corrections to the gravitational interaction.
• Figure 5: Topologies for the two counter-terms required to remove divergences to 4PN. The black square represents an insertion of either ${\cal O}_{a\dot v}$ or ${\cal O}_V$ (see text). Mirror images are also needed.