  In the classically unbroken phase, 3d $O(N)$ symmetric $\phi^4$ vector models
admit two equivalent descriptions connected by a strong-weak duality closely
related to the one found by Chang and Magruder long ago. We determine the exact
analytic renormalization dependence of the critical couplings in the weak and
strong branches as a function of the renormalization scheme (parametrized by
$\kappa$) and for any $N$. It is shown that for $\kappa=\kappa_*$ the two fixed
points merge and then, for $\kappa<\kappa_*$, they move into the complex plane
in complex conjugate pairs, making the phase transition no longer visible from
the classically unbroken phase. Similar considerations apply in 2d for the
$N=1$ $\phi^4$ theory, where the role of classically broken and unbroken phases
is inverted. We verify all these considerations by computing the perturbative
series of the 3d $O(N)$ models for the vacuum energy and for the mass gap up to
order eight, and Borel resumming the series. In particular, we provide
numerical evidence for the self-duality and verify that in renormalization
schemes where the critical couplings are complex the theory is gapped. As a
by-product of our analysis, we show how the non-perturbative mass gap at large
$N$ in 2d can be seen as the analytic continuation of the perturbative one in
the classically unbroken phase.
