Excited doubly heavy baryon production via $W^+$ boson decays

In this paper, decay widths of the doubly heavy baryons ($\Xi_{cc} ~\text{and} ~\Xi_{bc}$) production are theoritically calculated in the whole phase space through $W^+ \to \Xi_{cc}+ \bar{c}+\bar{s}$ and $W^+ \to \Xi_{bc}+ \bar{b}+\bar{s}$, within the framework of nonrelativistic QCD (NRQCD). Differential widths $d\Gamma/ds_{12}$, $d\Gamma/ds_{23}$, $d\Gamma/dcos\theta_{12}$, and $d\Gamma/dcos\theta_{13}$ are also given. In addition to the ordinary S wave contributions for the baryons, we specifically calculate P wave contributions as a comparison, namely the high excited states of the intermediate diquark, including $[^1P_1]$ and $[^3P_J]$ (with $J=0,1,2$) in both color anti-triplet state $\overline{\mathbf{3}}$ and color sextuplet state $\mathbf{6}$. It shows that the contribution from P wave is about one order lower than S wave. According to the results, we can expect plentiful events produced at the LHC, i.e., $3.69\times10^5$ $\Xi_{cc}$ events and $4.91\times10^4$ $\Xi_{bc}$ events per year.