Three-Loop Corrections to the Soft Anomalous Dimension in Multileg Scattering

We present the three-loop result for the soft anomalous dimension governing long-distance singularities of multileg gauge-theory scattering amplitudes of massless partons. We compute all contributing webs involving semi-infinite Wilson lines at three loops and obtain the complete three-loop correction to the dipole formula. We find that nondipole corrections appear already for three colored partons, where the correction is a constant without kinematic dependence. Kinematic dependence appears only through conformally invariant cross ratios for four colored partons or more, and the result can be expressed in terms of single-valued harmonic polylogarithms of weight five. While the nondipole three-loop term does not vanish in two-particle collinear limits, its contribution to the splitting amplitude anomalous dimension reduces to a constant, and it depends only on the color charges of the collinear pair, thereby preserving strict collinear factorization properties. Finally, we verify that our result is consistent with expectations from the Regge limit.

Figure 1

Representative three-loop connected webs contributing to the soft anomalous dimension.

Figure 2

Representative nonconnected three-loop diagrams of webs which contribute to ${\mathrm{\Delta }}_4^{(3)}$.