Soft gluons in logarithmic summations

We demonstrate that all the known single- and double-logarithm summations for a parton distribution function can be unified in the Collins–Soper resummation technique by applying soft approximations appropriate in different kinematic regions to real gluon emissions. Neglecting the gluon longitudinal momentum, we obtain the (double-logarithm) resummation for two-scale QCD processes, and the Balitsky–Fadin–Kuraev–Lipatov (single-logarithm) equation for one-scale processes. Neglecting the transverse momentum, we obtain the threshold (double-logarithm) resummation for two-scale processes, and the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (single-logarithm) equation for one-scale processes. If we keep the longitudinal and transverse momenta simultaneously, we derive a unified resummation for a large Bjorken variable x, and a unified evolution equation appropriate for both intermediate and small x. Received: 9 March 1999 / Revised version: 12 April 1999 / Published online: 3 August 1999