$W$-Boson Exotic Decay into Three Charged Leptons at the LHC

We investigate the $W$ boson's exotic decay channel, $W \rightarrow \ell\ell\ell \nu$, at the LHC. Although the four-body final states suppress the decay branching ratio, the large production of $W$ bosons makes detecting and precisely measuring this decay probability entirely feasible. Our simulation study indicates that this tiny branching ratio can be measured with sub-percent precision at the HL-LHC. This decay channel can also constrain Standard Model extensions. Using the $L_\mu-L_\tau$ model as a benchmark, we find that the current bound on the gauge coupling for $Z'$ mass in the range of $[4,75]$ GeV can significantly improve.