Multiple Gluon Exchange Webs

Abstract

Webs are weighted sets of Feynman diagrams which build up the logarithms of correlators of Wilson lines, and provide the ingredients for the computation of the soft anomalous dimension. We present a general analysis of multiple gluon exchange webs (MGEWs) in correlators of semi-infinite non-lightlike Wilson lines, as functions of the exponentials of the Minkowski cusp angles, $α_{ij}$, formed between lines $i$ and $j$. We compute a range of webs in this class, connecting up to five Wilson lines through four loops, we give an all-loop result for a special class of diagrams, and we discover a new kind of relation between webs connecting different numbers of Wilson lines, based on taking collinear limits. Our results support recent conjectures, stating that the contribution of any MGEW to the soft anomalous dimension is a sum of products of polylogarithms, each depending on a single cusp angle, and such that their symbol alphabet is restricted to $α_{i j}$ and $1 - α_{i j}^2$. Finally, we construct a simple basis of functions, defined through a one-dimensional integral representation in terms of powers of logarithms, which has all the expected analytic properties. This basis allows us to compactly express the results of all MGEWs computed so far, and we conjecture that it is sufficient for expressing all MGEWs at any loop order.

Figures (13)

• Figure 1: An example of a multiple gluon exchange diagram connecting five Wilson lines at five loops; it is part of the (1,2,3,3,1) web. The lines meet at a local effective vertex representing the hard interaction. For this diagram $\Theta_D [ \left\{ s_k, t_k \right\} ] \, = \, \theta(t_1 > s_2) \, \theta(s_3 > t_2 > s_4) \, \theta(t_4 > s_5 > t_3)$.
• Figure 2: The two-loop crossed graph connecting two Wilson lines, and the corresponding effective vertex graph containing a double emission vertex $V_2$ on each of the two lines.
• Figure 3: The (1,2,2,1) web, connecting four Wilson lines at three loops.
• Figure 4: The (1,2,2,1) web in the effective vertex formalism.
• Figure 5: Effective vertex diagram for the (1,1,1,3) web.