  Sizable amount of heavy-quarkonium events can be produced through $W$-boson
decays at the LHC. Such channels will provide a suitable platform to study the
heavy-quarkonium properties. The "improved trace technology", which disposes
the amplitude ${\cal M}$ at the amplitude-level, is helpful for deriving
compact analytical results for complex processes. As an important new
application, in addition to the production of the lower-level Fock states
$|(Q\bar{Q'})[1S]>$ and $|(Q\bar{Q'})[1P]>$, we make a further study on the
production of higher-excited $|(Q\bar{Q'})>$-quarkonium Fock states
$|(Q\bar{Q'})[2S]>$, $|(Q\bar{Q'})[3S]>$ and $|(Q\bar{Q'})[2P]>$. Here
$|(Q\bar{Q'})>$ stands for the $|(c\bar{c})>$-charmonium,
$|(c\bar{b})>$-quarkonium and $|(b\bar{b})>$-bottomonium respectively. We show
that sizable amount of events for those higher-excited states can also be
produced at the LHC. Therefore, we need to take them into consideration for a
sound estimation.
