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 $α_{ij}$, the exponential of the Minkowski cusp angle formed between the lines $i$ and $j$. We show that beyond the obvious inversion symmetry $α_{ij}\to 1/α_{ij}$, at the level of the symbol the result also admits a crossing symmetry $α_{ij}\to -α_{ij}$, relating spacelike and timelike kinematics, and hence argue that in this class of webs the symbol alphabet is restricted to $α_{ij}$ and $1-α_{ij}^2$. 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.

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