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

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 counterterms in the worldline effective theory. The counterterms 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 counterparts. As anticipated, the leftover 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 order, 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.

Figure 1

Topology contributions at $\mathcal{O}(G^3)$ to the Lagrangian. The first two diagrams are UV divergent, starting at 3PN order, while the third diagram is finite at this order. All of these topologies are divergent at 4PN order. The first diagram is UV while the third is IR divergent. The second one has both IR and UV poles.

Figure 2

The first nonlinear topology leading to IR divergences away from the static limit. The spurious poles occur when the propagators are Taylor expanded in powers of $p_0/|\mathit{p}|$.

Figure 3

Feynman diagram for the contribution to the radiation-reaction force due to the tail effect. See Ref. [17] 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 diagram, 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 counterterms required to remove divergences to 4PN order. The black square represents an insertion of either ${\mathcal{O}}_{a\dot{v}}$ or ${\mathcal{O}}_V$ (see text). Mirror images are also needed.

Figure 6

Diagram depicting a contribution due to the tail effect to the conservative sector of the radiation-reaction force coming from the $A^2\sigma$ coupling. Following Ref. [39], we use a single wavy line to represent the vector mode $\mathit{A}$, while the double wavy line accounts for the tensor $\sigma$. The total binary’s mass-energy and moments of the pseudo-stress-energy tensor are described by the black and grey blobs, respectively. All permutations of the fields must be considered.

Figure 7

Near-zone Feynman diagram at $\mathcal{O}(G^2)$ involving vector (single wavy) and tensor (double wavy) modes, associated with the far-zone conservative contributions in Fig. 6, at the same order. Mirror images must be added.

Figure 8

Near-zone Feynman diagram at $\mathcal{O}(G^3)$ associated with the far-zone conservative contributions in Fig. 6, at the same order. The dashed line represents the scalar mode. Mirror images must be added.