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

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