Logarithmic corrections to massive particle trajectories for the classical soft log theorems at leading order in $G$

We study the logarithmic corrections to the asymptotic trajectories of massive particles. These corrections enter the calculation of the soft-log sector of the radiative field at null infinity in four-dimensional gravitational scattering. We first isolate the part of the metric which enters the particle equations of motion. We show that the nonlinear source can only affect this sector through homogeneous modes on the hyperboloid, and that matching to the proposed soft tower implies that these modes vanish, except for the first mode. We show that this mode cannot affect the trajectory coefficients at leading order in G. We then invert the matter energy-momentum tensor on the particle worldlines to obtain the trajectory coefficients at leading order in $G$ and at every logarithmic order using Legendre polynomials. The first two coefficients reproduce the known results at orders $G$ and $G^2$. .