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We study quarkonium transport in the quark-gluon plasma by using thepotential nonrelativistic QCD (pNRQCD) effective field theory and the frameworkof open quantum systems. We argue that the coupling between quarkonium and thethermal bath is weak using separation of scales, so the initial density matrixof the total system factorizes and the time evolution of the subsystem isMarkovian. We derive the semiclassical Boltzmann equation for quarkonium byapplying a Wigner transform to the Lindblad equation and carrying out asemiclassical expansion. We resum relevant interactions to all orders in thecoupling constant at leading power of the nonrelativistic and multipoleexpansions. The derivation is valid for both weakly coupled and stronglycoupled quark-gluon plasmas. We find reaction rates in the transport equationfactorize into a quarkonium dipole transition function and a chromoelectricgluon distribution function. For the differential reaction rate, the definitionof the momentum dependent chromoelectric gluon distribution function involvesstaple-shaped Wilson lines. For the inclusive reaction rate, the Wilson linescollapse into a straight line along the real time axis and the distributionbecomes momentum independent. The relation between the two Wilson lines isanalogous to the relation between the Wilson lines appearing in the gluonparton distribution function (PDF) and the gluon transverse momentum dependentparton distribution function (TMDPDF). The centrality dependence of thequarkonium nuclear modification factor measured by experiments probes themomentum independent distribution while the transverse momentum dependence andmeasurements of the azimuthal angular anisotropy may be able to probe themomentum dependent one. We discuss one way to indirectly constrain thequarkonium in-medium real potential by using the factorization formula andlattice calculations. The leading quantum correction to the semiclassicaltransport equation of quarkonium is also worked out. The study can be easilygeneralized to quarkonium transport in cold nuclear matter, which is relevantfor quarkonium production in eA collisions in the future Electron-Ion Collider.

Quarkonium Semiclassical Transport in Quark-Gluon Plasma: Factorization and Quantum Correction