This paper studies how adding a massive scalar field to a specific higher-curvature modification of gravity—called scalar–Gauss–Bonnet (sGB) theory—changes the motion of two black holes as they spiral together. The authors compute the binary dynamics for nonspinning black holes using the post-Newtonian approximation, and for the first time they include corrections that mix the scalar mass with the higher-curvature coupling. Their main finding is that these scalar-mass corrections change the gauge-invariant binding energy of the binary by a small amount, tending to reduce it in the perturbative limit.

To reach this result the researchers solved the field equations and the two-body problem up to first post-Newtonian (1PN) order. Concretely they derived the equations of motion, the rules for transforming to the center-of-mass frame, and the binding energy for both circular and eccentric orbits. Their calculations use standard tools such as a post-Minkowskian expansion, near-zone and far-zone integrals, and an effective worldline description of scalarized black holes. Most intermediate steps apply to a wide range of scalar masses, but the final explicit formulas assume the scalar mass is small compared to the total binary mass and are expanded to quadratic order in that small ratio.

At a high level, sGB gravity adds a scalar field that couples to a particular quadratic curvature combination called the Gauss–Bonnet invariant. Giving the scalar a mass introduces a new length scale: the Compton wavelength of the scalar. Beyond that length scale the scalar profile is exponentially suppressed, so the mass changes how strongly the scalar affects widely separated binaries. In the equations the scalar-mass-dependent terms enter the binding energy with contributions of opposite signs at different orders, and in the perturbative limit those terms combine to produce an overall small decrease in binding energy. The paper finds the effect is largest for binaries with a large mass ratio between the components and for orbits with high eccentricity.