On the renormalization of multiparton webs

TL;DR

The paper integrates the renormalization of Wilson-line correlators with the diagrammatic web approach to soft-gluon exponentiation in multiparton amplitudes. It derives general renormalization constraints showing that all higher-order poles of webs are fixed by lower-order webs, with a finite soft anomalous-dimension matrix capturing the exponent's structure. Through explicit three-loop, four-leg examples, it confirms web-by-web satisfaction of these constraints and reveals how commutators generate subleading poles. A new conjecture on web-mixing matrices—a weighted column-sum rule—is proposed, alongside an exponential regulator to disentangle UV/IR effects, setting the stage for higher-loop determinations of the soft anomalous dimension.

Abstract

We consider the recently developed diagrammatic approach to soft-gluon exponentiation in multiparton scattering amplitudes, where the exponent is written as a sum of webs - closed sets of diagrams whose colour and kinematic parts are entangled via mixing matrices. A complementary approach to exponentiation is based on the multiplicative renormalizability of intersecting Wilson lines, and their subsequent finite anomalous dimension. Relating this framework to that of webs, we derive renormalization constraints expressing all multiple poles of any given web in terms of lower-order webs. We examine these constraints explicitly up to four loops, and find that they are realised through the action of the web mixing matrices in conjunction with the fact that multiple pole terms in each diagram reduce to sums of products of lower-loop integrals. Relevant singularities of multi-eikonal amplitudes up to three loops are calculated in dimensional regularization using an exponential infrared regulator. Finally, we formulate a new conjecture for web mixing matrices, involving a weighted sum over column entries. Our results form an important step in understanding non-Abelian exponentiation in multiparton amplitudes, and pave the way for higher-loop computations of the soft anomalous dimension.

Figures (15)

• Figure 1: The six 3-loop diagrams forming the 2-3-1 web in which three eikonal lines are linked by three gluon exchanges.
• Figure 2: One loop web, where the gluon is emitted between partons $i$ and $j$, whose kinematic part is given by eq. (\ref{['eq:Fijoneloop']}).
• Figure 3: Diagram ($3D$) from figure \ref{['3lsix']} showing the labels for gluon emissions used in eq. (\ref{['F3D1']}).
• Figure 4: The two-loop webs which contribute to $W_{(2,3,1)}$, labelled as in Gardi:2010rn.
• Figure 5: The two possible decompositions of diagram $3D$ into a product of a two-loop subdiagram times a one-loop one, corresponding to the two terms in (\ref{['3D_decomp']}). Note that the latter equation is expressed in terms of $(2g)$, while the actual subdiagram is $(2f)$. However, the two are related by $\mathcal{F}^{(2,-1)}(2f)=-\mathcal{F}^{(2,-1)}(2g)$.