From Webs to Polylogarithms

TL;DR

This work develops a configuration-space, web-based framework to compute three-loop contributions to the multi-leg soft anomalous dimension in gauge theories, focusing on webs formed by multiple gluon exchanges between four Wilson lines. By organizing diagrams into subtracted webs (webs plus lower-order commutators), the authors expose a remarkably simple, factorized structure: the three-loop results can be written as sums of products of polylogarithms each depending on a single cusp angle, with a symbol alphabet restricted to α_ij and 1−α_ij^2. They establish that crossing symmetry is realized at the level of the symbol for subtracted webs, and propose a general conjecture (the alphabet/conjecture) that constrains the functional space of all such webs. Explicit three-loop results are provided for the 1-2-2-1 and 1-1-1-3 webs spanning four lines, demonstrating weight-5 pure functions and setting the stage for completing the full three-loop soft anomalous dimension. The findings illuminate the all-order structure of soft-gluon corrections and offer a framework potentially extensible to higher loops and more legs.

Abstract

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.

Figures (8)

• Figure 1: An example multiple-gluon-exchange diagram connecting four semi-infinite Wilson lines at seven loops. The 4-velocities associated with the directions of the Wilson lines in Minkowki space are indicated by $\beta_i$. The lines all meet at the origin, where there is a local effective vertex representing the hard interaction.
• Figure 2: Connected three-loop diagrams with 3- or 4-gluon vertices spanning four Wilson lines.
• Figure 3: The 1-1-1-2 web. Each of the two diagrams in this web ($A$ and $B$) contains two connected subdiagrams, one of which has a three-gluon vertex.
• Figure 4: One loop web, where the gluon is emitted between partons $i$ and $j$, whose kinematic part is given by eq. (\ref{['eq:Fijoneloop2']}).
• Figure 5: The analytic structure of the one-loop result in the complex $\alpha$ plane -- a logarithmic branch cut along the negative real axis -- shown together with a contour describing the values of $\alpha$ for real values of $\gamma$: the $\alpha\in (0,1)$ region corresponds to space-like kinematics (one incoming and one outgoing partons) where $\gamma$ varies between $-\infty$ and $-2$; next, the region of complex $\alpha$ with a positive imaginary part corresponds to the Euclidean region where $-2<\gamma<2$; and finally the region where $\alpha$ is near the branch cut, $\alpha=\alpha_r+{\rm i} \varepsilon$ with $\alpha_r\in (-1,0)$ and $\varepsilon>0$, corresponds to time-like kinematics.