QCD Factorization: matching hadrons to quarks and gluons with controllable approximations

QCD factorization has been developed to describe high energy hadronic scattering by consistently separating the strong interacting dynamics at the hadronic scale (~ fm) from those taking place at short-distance (< 0.1 fm), which are perturbatively calculable in QCD.  QCD factorization formalisms have been extremely successful in interpreting high energy experimental data with a large momentum transfer Q (> 2 GeV) from all facilities around the world, and have given us the confidence and the tools to discover the Higgs particle in proton-proton collisions and to search for the new physics.  However, the high energy probes are not sensitive to the complexity, mysteries, and rich dynamics of QCD at the Fermi scale, which are responsible for holding all hadrons together and their internal quark/gluon landscape.  In this talk, I will give an overview of QCD factorization and its recent developments, focusing on the factorization of hadronic observables with two distinguished scales: a hard scale to ensure a localized probe for quarks and gluons and a soft scale to be sensitive to the dynamics at the Fermi scale, which are naturally accessible at a lepton-hadron facility, like CEBAF and the future Electron-Ion Collider.  In addition, I will introduce QCD factorization beyond the leading power, as well as a joint factorization between QCD and QED to treat the collision induced QCD and QED radiations consistently,and demonstrate quantitatively their impact on interpreting high energy scattering data and our ability to probe the internal quark/gluon structure of a hadron.