Looking through the QCD conformal window with perturbation theory

We study the conformal window of QCD using perturbation theory, starting from the perturbative upper edge and going down as much as we can towards the strongly coupled regime. We do so by exploiting the available five-loop computation of the $overline{{m MS}}$ $eta$-function and employing Borel resummation techniques both for the ordinary perturbative series and for the Banks-Zaks conformal expansion. Large-$n_f$ results are also used. We argue that the perturbative series for the $overline{{m MS}}$ $eta$-function is most likely asymptotic and non-Borel resummable, yet Borel resummation techniques allow to improve on ordinary perturbation theory. We find substantial evidence that QCD with $n_f=12$ flavours flows in the IR to a conformal field theory. Though the evidence is weaker, we find indications that also $n_f=11$ might sit within the conformal window. We also compute the value of the mass anomalous dimension $gamma$ at the fixed point and compare it with the available lattice results. The conformal window might extend for lower values of $n_f$, but our methods break down for $n_f<11$, where we expect that non-perturbative effects become important. A similar analysis is performed in the Veneziano limit.