Binary dynamics to second post-Newtonian order in scalar-tensor and Einstein-scalar-Gauss-Bonnet gravity from effective field theory

Using effective field theory methods, we compute in detail the Lagrangian for the conservative dynamics of compact binary systems, for spinless constituents and in the gravitationally bound case, in massless scalar-tensor and Einstein-scalar-Gauss-Bonnet gravity to the second post-Newtonian (2PN) order. We employ the Kaluza-Klein parametrization of the metric, and demonstrate that, also in this case, a significant reduction in the number of diagrams is achieved. Still, to the 2PN accuracy, an additional 39 diagrams involving scalar interactions must be added to the pure general relativistic result. Diagrams involving the Gauss-Bonnet term at the formal 1PN order are computed and shown to scale as a 3PN correction when observational constraints are taken into account. We also investigate at which order the additional $\mathcal{O}(G^3)$ topologies contribute. Finally, the results derived in this paper provide an essential step toward the correct derivation of the conservative dynamics at the 3PN level in scalar-tensor and Einstein-scalar-Gauss-Bonnet theories.

Figure 1

Topology for $\mathcal{O}(G^1)$ contributions to $S_{\mathrm{eff}}$. In this diagram, the solid vertical line represents a generic propagator, being either one of the four fields present in our EFT.

Figure 2

Topologies for $\mathcal{O}(G^2)$ contributions. In diagram (a), we have the first appearance of a quadratic worldline vertex, whereas diagram (b) is the first contribution that contains an internal three-graviton vertex.

Figure 3

Topologies for $\mathcal{O}(G^3)$ contributions. Diagrams (a)–(i) represent all the possible 2-loop diagrams that start to contribute, in principle, to the 2PN order, being also present in all subsequent orders.

Figure 4

Diagrams of $\mathcal{O}(G^1)$ contributing to the 0PN order. In (a), the dashed line represents a gravitational $\varphi$ propagator and yields the Newtonian potential. Diagram (b) provides a scalar field contribution, with a zigzag line representing the scalar $\mathrm{\Phi }$ mode.

Figure 5

Diagrams with $\mathcal{O}(G^1)$ topology that contribute to the 1PN order. Diagrams (a), (b), and (d) are pure GR contributions, while (c) and (e) are new ones stemming from the scalar field. Besides the dashed and zigzag lines already presented, wavy lines represent the propagator for the gravitational $A_i$ mode and crossed circles represent $\mathcal{O}(v^2)$ corrections in the propagator.

Figure 6

Diagrams of $\mathcal{O}(G^2)$ that contribute to the 1PN order. Diagram (a) gives a purely GR contribution while diagrams (a) and (b), which contain respectively the first appearance of a worldline scalar-graviton vertex and a scalar-scalar vertex, provide contributions from the scalar modes.

Figure 7

Diagrams of $\mathcal{O}(G^2)$ that, in principle, contribute to the 1PN order, but turn out to be vanishing at this level due to the use of the Kaluza-Klein decomposition. Diagram (a) involves only pure graviton modes, and is already present in GR, while diagrams (b) and (c) account for scalar interactions through the three scalar-scalar-graviton vertex.

Figure 8

Diagrams entering the $\mathcal{O}(G^1)$ sector of the 2PN order dynamics. In (i) and (ii), only corrections of $v$ to the worldlines are incorporated, while in diagrams (iii) and (iv), corrections of order $v^2$ and $v^4$ to the propagator, respectively, are also considered.

Figure 9

Diagrams of topology (a) entering the $\mathcal{O}(G^2)$ sector of the 2PN dynamics. Henceforth, only powers of $v$ beyond $v^0$ are explicitly displayed in the vertices. The nine diagrams throughout (i) to (ix) represent all the contributions from scalar interactions that must be added to the GR results of the corresponding sector.

Figure 10

Diagrams of topology (b) entering the $\mathcal{O}(G^2)$ sector of the 2PN dynamics. Thick wavy lines represent $\sigma$ mode propagators. Diagrams (i) and (iv) involve only pure graviton modes, and is already present in GR, while diagrams (ii), (iii), (v), and (vi) account for scalar interactions through the three scalar-scalar-graviton vertex, with gravitational mode being of the following types: $\sigma$ in (ii), $A$ in (iii), and $\varphi$ in (v) and (vi).

Figure 11

Diagrams of topology (a) entering the $\mathcal{O}(G^3)$ sector of the 2PN dynamics. These diagrams contain for the first time contributions from triple-worldline vertices made out of a scalar-graviton-graviton vertex, in (i), a scalar-scalar-graviton vertex, in (ii), and finally, a three-scalar vertex, in (iii).

Figure 12

Diagrams of topology (b) entering the $\mathcal{O}(G^3)$ sector of the 2PN dynamics. While in diagrams (i), (ii), (iii), and (iv), the worldlines exchange either one or two graviton modes, diagram (v) is the only one of a purely scalar nature.

Figure 13

Diagrams of topology (g) entering the $\mathcal{O}(G^3)$ sector of the 2PN dynamics. While in diagram (ii) both three vertices are of the scalar-scalar-graviton type, in diagrams (i) and (iii) at least one of the internal vertices contains only graviton modes.

Figure 14

Diagrams of topology (i) entering the $\mathcal{O}(G^3)$ sector of the 2PN dynamics. While diagram (i) involves two scalar-scalar-graviton vertices, diagram (ii) involves both a scalar-scalar-graviton and a three-graviton vertex.

Figure 15

Diagrams of topology (h) entering the $\mathcal{O}(G^3)$ sector of the 2PN dynamics. Diagram (a) involves both a scalar-scalar-graviton and a three-graviton vertex, while diagram (a) involves two scalar-scalar-graviton vertices.

Figure 16

Leading-order diagrams for the Gauss-Bonnet coupling. Here, we represent scalar-graviton vertices stemming from this new coupling by a square. While in diagram (a), the two scalar modes of the three scalar-scalar-graviton vextex is connected to the same worldline, in diagram (b), each scalar mode is connected to one of the two different worldlines.