Search for $\mathrm{W^\prime}$ decaying to a vector-like quark and a top or bottom quark in the all-jets final state

A search for a heavy $\mathrm{W^\prime}$ boson resonance decaying to one B or T vector-like quark and a top or bottom quark, respectively, is presented. The analysis is performed using proton-proton collision data collected from the year 2016 to 2018 with the CMS detector at the LHC. The data correspond to an integrated luminosity of 137$~\mathrm{fb^{-1}}$ at a center-of-mass energy of $13~\mathrm{TeV}$. Both decay channels result in a final state with a top quark, a Higgs or Z boson, and a b quark, each produced with significant energy. The all-hadronic decays of both the Higgs boson and the top quark are considered. The final-state jets, some of which correspond to merged decay products of a boosted top quark and a Higgs or Z boson, are selected using jet substructure techniques, which help to suppress standard model backgrounds. A $\mathrm{W^\prime}$ boson signal would appear as a narrow peak in the invariant mass distribution of these three jets. No significant deviation in data with respect to the standard model background predictions is observed. The $95\%$ confidence level upper limits on the cross section for $\mathrm{W^\prime}$ boson production and the subsequent decay into a top quark, Higgs or Z boson, and b quark are set as a function of the $\mathrm{W^\prime}$ boson mass, for several vector-like quark mass hypotheses. We exclude a $\mathrm{W^\prime}$ boson in this channel with a mass below 3.2$~\mathrm{TeV}$ given benchmark model assumptions, and additionally present limits based on generalizations of these assumptions. This is the first search for a $\mathrm{W^\prime}$ boson in this channel using the full 2016 to 2018 CMS data.