Two-Loop Anomalous-Dimension Matrix for Soft-Gluon Exchange

The resummation of soft-gluon exchange for QCD hard scattering requires a matrix of anomalous dimensions. We compute this matrix directly for arbitrary $2\to n$ massless processes for the first time at two loops. Using color-generator notation, we show that it is proportional to the one-loop matrix. This result reproduces all pole terms in dimensional regularization of the explicit calculations of massless $2\to 2$ amplitudes in the literature, and it predicts all poles at next-to-next-to-leading order in any $2\to n$ process that has been computed at next-to-leading order. The proportionality of the one- and two-loop matrices makes possible the resummation in closed form of the next-to-next-to-leading logarithms and poles in dimensional regularization for the $2\to n$ processes.

Figure 1

Two-loop diagrams involving three eikonal lines.