Oblique corrections when $m_W \neq m_Z \cos{\theta_W}$ at tree level

The parametrization of the oblique corrections through $S$, $T$, and $U$ -- later extended by $V$, $W$, and $X$ -- is a convenient way of comparing the predictions for various electroweak observables at the one-loop level between the Standard Model and its extensions. That parametrization assumes that the extensions under consideration have ${SU(2)\times U(1)}$ gauge symmetry \emph{and} the tree-level relation $m_W = m_Z \cos{\theta_W}$ between the Weinberg angle and the gauge-boson masses. In models where that relation does not hold at the Lagrangian level, the parameter $T$ is not ultraviolet-finite, making the parametrization inadequate. We present expressions that parametrize the difference of the various predictions of two models with $m_W \neq m_Z \cos{\theta_W}$ in terms of oblique parameters. The parameter $T$ does not play a role in those expressions. Conveniently, they may be reached from the ones that were derived for models with tree-level $m_W = m_Z \cos{\theta_W}$, by performing a simple substitution for $T$. We also discuss the difficulties in using oblique parameters when comparing a model with $m_W \neq m_Z \cos{\theta_W}$ to the Standard Model. Finally, we compute the relevant five oblique parameters $S$, $U$, $V$, $W$, and $X$ in the SM extended by both, hypercharge $Y=0$ and $Y=1$, triplet scalars.