Two-loop soft anomalous dimension matrix and resummation at next-to-next-to-leading poles

We extend the resummation of dimensionally regulated amplitudes to next-to-next-to-leading poles. This requires the calculation of two-loop anomalous dimension matrices for color mixing through soft gluon exchange. Remarkably, we find that they are proportional to the corresponding one-loop matrices. Using the color-generator notation, we reproduce the two-loop single-pole quantities ${\mathbf{H}}^{(2)}$ introduced by Catani for quark and gluon elastic scattering. Our results also make possible threshold and a variety of other resummations at next-to-next-to-leading logarithm. All of these considerations apply to $2\to n$ processes with massless external lines.

Figure 1

One-loop diagrams that contribute to ${\Gamma }_{S_{\mathrm{f}}}^{(1)}$.

Figure 2

Color basis $\{c_1,c_2\}$ for the four-quark process.

Figure 3

Two-loop diagrams that contribute to ${\Gamma }_{W_{\mathrm{f}}}^{(2)}$.

Figure 4

(a)–(c): Pairs of 3E diagrams.