Universality of Soft and Collinear Factors in Hard-Scattering Factorization

Universality in QCD factorization of parton densities, fragmentation functions, and soft factors is endangered by the process dependence of the directions of Wilson lines in their definitions. We find a choice of directions that is consistent with factorization and that gives universality between $e^{+}{e}^{-}$ annihilation, semi-inclusive deep-inelastic scattering, and the Drell-Yan process. Universality is only modified by a time-reversal transformation of the soft function and parton densities between Drell-Yan and the other processes, whose only effect is the known reversal of sign for $T$-odd parton densities such as the Sivers function. The modifications of the definitions needed to remove rapidity divergences with lightlike Wilson lines do not affect the results.

Figure 1

Sample virtual one-loop diagrams for $e^{+}{e}^{-}$ annihilation, with fragmentation considered in a spectator model.

Figure 2

Sample one-loop diagrams for SIDIS.

Figure 3

Sample one-loop diagrams for the Drell-Yan process.