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A<sc>bstract</sc> We develop a general expression for weighted cross sections in leptonic annihilation to hadrons based on time-ordered perturbation theory (TOPT). The analytic behavior of the resulting integrals over spatial momenta can be analyzed in the language of Landau equations and infrared (IR) power counting. For any infrared-safe weight, the cancellation of infrared divergences is implemented locally at the integrand level, and in principle can be evaluated numerically in four dimensions. We go on to show that it is possible to eliminate unphysical singularities that appear in time-ordered perturbation theory for arbitrary amplitudes. This is done by reorganizing TOPT into an equivalent form that combines classes of time orderings into a “partially time-ordered perturbation theory”. Applying the formalism to leptonic annihilation, we show how to derive diagrammatic expressions with only physical unitarity cuts. 

A bstract We describe the implementation of infrared subtractions for two-loop QCD corrections to quark-antiquark annihilation to electroweak final states. The subtractions are given as form-factor integrands whose integrals are known. The resulting subtracted amplitudes are amenable to efficient numerical integration. Our procedure is based on the universality of infrared singularities and requires a relatively limited set of subtractions, whose number grows as the number of two-loop diagrams, rather than with the number of singular regions of integration. 

In this closing talk, I review some of the lessons we've learned about quantum chromodynamics,and reflect on what we may hope to learn in the coming years. 

Heavy quarkonium production at high transverse momentum( p_T p T )in hadronic collisions is explored in the QCD factorization approach. Wefind that the leading power in the 1/p_T 1 / p T expansion is responsible for high p_T p T regime, while the next-to-leading power contribution is necessary forthe low p_T p T region. We present the first numerical analysis of the scale evolutionof coupled twist-2 and twist-4 fragmentation functions (FFs) for heavyquarkonium production and demonstrate that the QCD factorizationapproach is capable of describing the p_T p T spectrum of hadronic J/\psi J / ψ production at the LHC. 

We report the current understanding of heavy quarkonium production at high transverse momentum ( p T ) in hadronic collisions in terms of QCD factorization. In this presentation, we highlight the role of subleading power corrections to heavy quarkonium production, which are essential to describe the p T spectrum of quarkonium at a relatively lower p T . We also introduce prescription to match QCD factorization to fixed-order NRQCD factorization calculations for quarkonium production at low p T .