  We investigate the nonequilibrium dynamics of spherical active Brownian
particles in three spatial dimensions that interact via a pair potential. The
investigation is based on a predictive local field theory that is derived by a
rigorous coarse-graining starting from the overdamped Langevin dynamics of the
particles. This field theory is highly accurate and applicable even for the
highest activities. It includes configurational order parameters and
derivatives up to infinite orders. We present also three finite reduced models
that result from the general field theory by suitable approximations and are
easier to apply. Furthermore, we use the general field theory and the simplest
one of the reduced models to derive analytic expressions for the
density-dependent mean swimming speed and the spinodal corresponding to the
onset of motility-induced phase separation of the particles, respectively. Both
of these results show a good agreement with recent findings described in the
literature. The analytic result for the spinodal yields also a prediction for
the associated critical point whose position has not been determined before.
