  We present numerically exact results from sign-problem free quantum Monte
Carlo simulations for a spin-fermion model near an $O(3)$ symmetric
antiferromagnetic (AFM) quantum critical point. We find a hierarchy of energy
scales that emerges near the quantum critical point. At high energy scales,
there is a broad regime characterized by Landau-damped order parameter dynamics
with dynamical critical exponent $z=2$, while the fermionic excitations remain
coherent. The quantum critical magnetic fluctuations are well described by
Hertz-Millis theory, except for a $T^{-2}$ divergence of the static AFM
susceptibility. This regime persists down to a lower energy scale, where the
fermions become overdamped and concomitantly, a transition into a $d-$wave
superconducting state occurs. These findings resemble earlier results for a
spin-fermion model with easy-plane AFM fluctuations of an $O(2)$ SDW order
parameter, despite noticeable differences in the perturbative structure of the
two theories. In the $O(3)$ case, perturbative corrections to the spin-fermion
vertex are expected to dominate at an additional energy scale, below which the
$z=2$ behavior breaks down, leading to a novel $z=1$ fixed point with emergent
local nesting at the hot spots [Schlief et al., PRX 7, 021010 (2017)].
Motivated by this prediction, we also consider a variant of the model where the
hot spots are nearly locally nested. Within the available temperature range in
our study ($T\ge E_F/200$), we find substantial deviations from the $z=2$
Hertz-Millis behavior, but no evidence for the predicted $z=1$ criticality.
