Structure functions for large $x$ and renormalization of Wilson loops

TL;DR

The paper develops a renormalization-group framework for Wilson loops whose paths partially lie on the light-cone and shows that their RG evolution encodes the evolution of parton distributions near the phase-space boundary (x→1). By computing the light-like Wilson loop W(C_S) to two loops and employing non-abelian exponentiation, the authors relate the resulting logarithmic structure to the cusp anomalous dimension, yielding a universal description of the large-x behavior via the splitting function near z→1. The work provides a concrete all-orders RG equation for W(C_S) that explains the origin of the L^3 and higher-logarithmic terms and demonstrates how soft and collinear dynamics factorize into a universal soft Wilson-loop factor, with a process-dependent finite piece. This framework offers a powerful method to resum large perturbative contributions in hard processes near the phase-space boundary and connects to heavy-quark effective theory through the geometry of Wilson lines.

Abstract

We discuss the relation between partonic distributions near the phase space boundary and Wilson loop expectation values calculated along paths partially lying on the light-cone. Due to additional light-cone singularities, multiplicative renormalizability for these expectation values is lost. Nevertheless we establish the renormalization group equation for the light like Wilson loops and show that it is equivalent to the evolution equation for the physical distributions. By performing a two-loop calculation we verify these properties and show that the universal form of the splitting function for large x originates from the cusp anomalous dimension of Wilson loops.

Figures (6)

• Figure 1: One-loop Feynman diagram contributing to the structure function of deep inelastic scattering. The gluon with momentum $k$ is emitted by the quark $p$ and absorbed by the quark $p'$ in the final state. We use solid line for quarks, curly lines for gluons, wavy lines for photons and dot-dashed line for the unitary cut.
• Figure 2: Integration path $C_S=\ell_1\cup\ell_2\cup\ell_3$ for the Wilson loop $W(C_S)$ corresponding to the structure function for large $x.$ The ray $\ell_1$ is along the time-like vector $v_\mu$ from $-\infty$ to $0;$ the segment $\ell_2$ is from point $0$ to $y$ along the light-cone; the ray $\ell_3$ is from the point $y$ to $-\infty$ along the vector $-v_\mu$. This path has two cusps at points $0$ and $y$ where the quark undergoes hard scattering.
• Figure 3: Integration path $C_T$ corresponding to the fragmentation function for large $x.$
• Figure 4: One-loop diagrams contributing to $W(C_S).$ Here, the double line represents the integration path in the Minkowski space as in fig. 2. The Feynman rules for these diagrams are given in the Appendix.
• Figure 5: Diagrams corresponding to fig. 4a for virtual and real gluon (see (\ref{['l1a']})). Due to a partial cancellation of Wilson lines, the sum of these two diagram gives the single contribution of fig. 4a. Similar cancellations hold for all diagrams.