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.