A triplet gauge boson with hypercharge one

A vector boson $W_{1}^{\mu}$ with the quantum numbers $\left(\boldsymbol{3},1\right)$ under $SU\left(2\right)_{L}\times U(1)_{Y}$ could in principle couple with the Higgs field via the renormalizable term $W_{1}^{\mu*}HD_{\mu}H$. This interaction is known to affect the $T$ parameter and, in so doing, it could potentially explain the recent CDF measurement of the W-boson mass. As it is often the case with vectors, building a viable model with a $W_{1}$ gauge boson is non-trivial. In this work I will describe two variations of a minimal setup containing this field; they are based on an extended $SO(5)\times SU\left(2\right)\times U(1)$ electroweak group. I will nevertheless show that interactions such as $W_{1}^{\mu*}H\partial_{\mu}H$ are never generated in a Yang-Mills theory. A coupling between $W_{1}$, $H$ and another Higgs doublet $H^{\prime}$ is possible though. Finally, I will provide an explicit recipe for the construction of viable models with gauge bosons in arbitrary representations of the Standard Model group; depending on the quantum numbers, they may couple to pairs of Standard Model fermions, or to a Standard Model fermion and an exotic one.