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We report that, unlike most inclusive quarkonium-production processes, theproduction rate for p p -> J/psi+Upsilon+X dominantly depends on two notwell-known long-distance nonrelativistic QCD matrix elements,<O_8^{J/psi}(^3S_1)> and <O_8^{Upsilon}(^3S_1)> at leading order in the strongcoupling constant, which account for the transition probabilities of thecolor-octet spin-triplet heavy-quark-antiquark pairs c\bar{c}_8(^3S_1) -> J/psiand b\bar{b}_8(^3S_1) -> Upsilon, respectively. With the integrated luminosity~ 100 fb^{-1} at the center-of-momentum energy sqrt{s}=14 TeV we expect thatapproximately 1900 p p -> J/psi+Upsilon+X events can be observed at the CERNLarge Hadron Collider by tagging muon pairs, which are enough to probe to thecolor-octet mechanism. The forthcoming measurement may provide a usefulconstraint to resolve the decade-old puzzle for the polarization of promptJ/psi at the Fermilab Tevatron. If corresponding measured rate is significantlyless than the prediction, it may imply that the current values for thecolor-octet matrix elements are overestimated.

p p -> J/psi+Upsilon+X as a clean probe to the quarkonium production mechanism