From webs to polylogarithms

We compute a class of diagrams contributing to the multi-leg soft anomalous dimension through three loops, by renormalizing a product of semi-infinite non-lightlike Wilson lines in dimensional regularization. Using non-Abelian exponentiation we directly compute contributions to the exponent in terms of webs. We develop a general strategy to compute webs with multiple gluon exchanges between Wilson lines in configuration space, and explore their analytic structure in terms of alpha (ij) , the exponential of the Minkowski cusp angle formed between the lines i and j. We show that beyond the obvious inversion symmetry alpha (ij) -> 1/alpha (ij) , at the level of the symbol the result also admits a crossing symmetry alpha (ij) -> -alpha (ij) , relating spacelike and timelike kinematics, and hence argue that in this class of webs the symbol alphabet is restricted to alpha (ij) and . We carry out the calculation up to three gluons connecting four Wilson lines, finding that the contributions to the soft anomalous dimension are remarkably simple: they involve pure functions of uniform weight, which are written as a sum of products of polylogarithms, each depending on a single cusp angle. We conjecture that this type of factorization extends to all multiple-gluon-exchange contributions to the anomalous dimension.