  Perturbatively renormalizable higher-derivative gravity in four space-time
dimensions with arbitrary signs of couplings has been considered. Systematic
analysis of the action with arbitrary signs of couplings in lorentzian flat
space-time for no-tachyons, fixes the signs. Feynman $+i\epsilon$ prescription
for these sign further grants necessary convergence in path-integral,
suppressing the field modes with large action. This also leads to a sensible
wick rotation where quantum computation can be performed. Running couplings for
these sign of parameters makes the massive tensor ghost innocuous leading to a
stable and ghost-free renormalizable theory in four space-time dimensions. The
theory has a transition point arising from renormalisation group (RG)
equations, where the coefficient of $R^2$ diverges without affecting the
perturbative quantum field theory. Redefining this coefficient gives a better
handle over the theory around the transition point. The flow equations pushes
the flow of parameters across the transition point. The flow beyond the
transition point is analysed using the one-loop RG equations which shows that
the regime beyond the transition point has unphysical properties: there are
tachyons, the path-integral loses positive definiteness, Newton's constant $G$
becomes negative and large, and perturbative parameters become large. These
shortcomings indicate a lack of completeness beyond the transition point and
need of a non-perturbative treatment of the theory beyond the transition point.
