  We analyze the possible quantum phase transition patterns occurring within
the $O(N) \times {\mathbb{Z}_2}$ scalar multi-field model at vanishing
temperatures in $(1+1)$-dimensions. The physical masses associated with the two
coupled scalar sectors are evaluated using the loop approximation up to second
order. We observe that in the strong coupling regime, the breaking $O(N) \times
{\mathbb{Z}_2} \to O(N)$, which is allowed by the
Mermin-Wagner-Hohenberg-Coleman theorem, can take place through a second-order
phase transition. In order to satisfy this no-go theorem, the $O(N)$ sector
must have a finite mass gap for all coupling values, such that conformality is
never attained, in opposition to what happens in the simpler ${\mathbb{Z}_2}$
version. Our evaluations also show that the sign of the interaction between the
two different fields alters the transition pattern in a significant way. These
results may be relevant to describe the quantum phase transitions taking place
in cold linear systems with competing order parameters. At the same time the
super-renormalizable model proposed here can turn out to be useful as a
prototype to test resummation techniques as well as non-perturbative methods.
