### Speaker

### Description

A vector boson $W_{1}$ with the quantum numbers $\left(3,1\right)$ under the electroweak group $SU\left(2\right)_{L}\times U(1)_{Y}$ could in principle couple

with the Higgs field via the renormalizable term $W_{1}^{\mu*}H D_{\mu}H$. This interaction is known to affect the $T$ parameter and, in so doing, it could potentially explain the 2022 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 talk 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 not generated in a Yang-Mills theory. A coupling between $W_{1}$, $H$ and another Higgs doublet $H^{\prime}$ is possible though.