  We calculate in perturbative QCD the transverse momentum broadening of vector
boson production in high energy nuclear collisions. We evaluate the effect of
initial-state parton multiple scattering for the production of the Drell-Yan
virtual photon and $W/Z$ bosons. We calculate both the initial- and final-state
multiple scattering effect for the production of heavy quarkonia and their
transverse momentum broadening in both NRQCD and Color Evaporation model of
quarkonium formation. We find that J/$\psi$ and $\Upsilon$ broadening in
hadron-nucleus collision is close to $2C_A/C_F$ times the corresponding
Drell-Yan broadening, which gives a good description of existing Fermilab data.
Our calculations are also consistent with RHIC data on J/$\psi$ broadening in
relativistic heavy ion collisions. We predict the transverse momentum
broadening of vector boson (J/$\psi$, $\Upsilon$, and $W/Z$) production in
relativistic heavy ion collisions at the LHC, and discuss the role of the
vector boson broadening in diagnosing medium properties.
