  Renormalization group (RG) and resummation techniques have been used in
$N$-component $\phi^4$ theories at fixed dimensions below four to determine the
presence of non-trivial IR fixed points and to compute the associated critical
properties. Since the coupling constant is relevant in $d<4$ dimensions, the RG
is entirely governed by renormalization scheme-dependent terms. We show that
the known proofs of the Borel summability of observables depend on the
renormalization scheme and apply only in "minimal" ones, equivalent in $d=2$ to
an operatorial normal ordering prescription, where the $\beta$-function is
trivial to all orders in perturbation theory. The presence of a non-trivial
fixed point can be unambiguously established by considering a physical
observable, like the mass gap, with no need of RG techniques. Focusing on the
$N=1$, $d=2$ $\phi^4$ theory, we define a one-parameter family of
renormalization schemes where Borel summability is guaranteed and study the
accuracy on the determination of the critical exponent $\nu$ as the scheme is
varied. While the critical coupling shows a significant sensitivity on the
scheme, the accuracy in $\nu$ is essentially constant. As by-product of our
analysis, we improve the determination of $\nu$ obtained with RG methods by
computing three more orders in perturbation theory.
