Vector resonant relaxation and statistical closure theory. II. One-loop closure.

We use stellar dynamics as a testbed for statistical closure theory. We focus on the process of "Vector Resonant Relaxation," a long-range, nonlinear, and correlated relaxation mechanism that drives the reorientation of stellar orbital planes around a supermassive black hole. This process provides a natural setting to evaluate the predictive power of generic statistical closure schemes for dynamical correlation functions, in the fully nonlinear and nonperturbative regime. We develop a numerical scheme that explicitly implements the seminal "Martin-Siggia-Rose" formalism at one-loop order via an iterative fixed-point approach, thereby improving upon the bare order from the "Direct Interaction Approximation." Using this framework, we quantitatively validate the ability of the formalism to predict (i) the two-point two-time correlation function, (ii) the renormalized three-point interaction vertex, and (iii) the three-point three-time correlation function. These predictions are compared to direct measurements from numerical simulations. We conclude by discussing the limitations of this approach and presenting possible future venues.