  We present the first numerical study of the ultraviolet dynamics of
non-asymptotically free gauge-fermion theories at large number of matter
fields. As testbed theories we consider non-abelian SU(2) gauge theories with
24 and 48 Dirac fermions on the lattice. For these number of flavors asymptotic
freedom is lost and the theories are governed by a gaussian fixed point at low
energies. In the ultraviolet they can develop a physical cutoff and therefore
be trivial, or achieve an interacting safe fixed point and therefore be
fundamental at all energy scales. We demonstrate that the gradient flow method
can be successfully implemented and applied to determine the renormalized
running coupling when asymptotic freedom is lost. Additionally, we prove that
our analysis is connected to the gaussian fixed point as our results nicely
match with the perturbative beta function. Intriguingly, we observe that it is
hard to achieve large values of the renormalized coupling on the lattice. This
might be an early sign of the existence of a physical cutoff and imply that a
larger number of flavors is needed to achieve the safe fixed point. A more
conservative interpretation of the results is that the current lattice action
is unable to explore the deep ultraviolet region where safety might emerge. Our
work constitutes an essential step towards determining the ultraviolet fate of
non asymptotically free gauge theories.
