Oblique corrections when \(m_W \ne m_Z \cos {\theta _W}\) 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 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 \(m_W \ne m_Z \cos {\theta _W}\) 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 \(m_W \ne m_Z \cos {\theta _W}\) 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.