  We study the prethermal dynamics of an interacting quantum field theory with
a N-component order parameter and $O(N)$ symmetry, suddenly quenched in the
vicinity of a dynamical critical point. Depending on the initial conditions,
the evolution of the order parameter, and of the response and correlation
functions, can exhibit a temporal crossover between universal dynamical scaling
regimes governed, respectively, by a quantum and a classical prethermal fixed
point, as well as a crossover from a Gaussian to a non-Gaussian prethermal
dynamical scaling. Together with a recent experiment, this suggests that
quenches may be used in order to explore the rich variety of dynamical critical
points occurring in the non-equilibrium dynamics of a quantum many-body system.
We illustrate this fact by using a combination of renormalization group
techniques and a non-perturbative large-$N$ limit.
