Soft Wilson lines in soft-collinear effective theory

TL;DR

The paper analyzes soft gluon emissions near phase-space boundaries within soft-collinear effective theory (SCET), showing that soft effects factorize into soft Wilson-line operators K(η) whose analytic structure depends on the process. Through a two-step matching (SCET_I to SCET_II) and explicit one-loop computations, the authors derive the renormalization-group evolution of these soft operators for deep inelastic scattering, Drell-Yan, jet production in e+e−, and a toy pi-gamma form factor, revealing universal cusp-driven kernels and, in some cases, Brodsky-Lepage type kernels. The results unify the treatment of soft gluon emission across several high-energy processes and demonstrate how the soft sector can be cleanly separated and resummed within SCET. The work highlights the importance of iε prescriptions and path-ordering in determining the precise anomalous dimensions and supports the SCET framework as a practical tool for resummation near phase-space boundaries.

Abstract

The effects of the soft gluon emission in hard scattering processes at the phase boundary are resummed in the soft-collinear effective theory (SCET). In SCET, the soft gluon emission is decoupled from the energetic collinear part, and is obtained by the vacuum expectation value of the soft Wilson-line operator. The form of the soft Wilson lines is universal in deep inelastic scattering, in the Drell-Yan process, in the jet production from e+e- collisions, and in the gamma* gamma* -> pi0 process, but its analytic structure is slightly different in each process. The anomalous dimensions of the soft Wilson-line operators for these processes are computed along the light-like path at leading order in SCET and to first order in alpha_s, and the renormalization group behavior of the soft Wilson lines is discussed.

Figures (10)

• Figure 1: Attachment of soft gluons to (a) an incoming quark, (b) an outgoing quark, (c) an outgoing antiquark, and (d) an incoming antiquark.
• Figure 2: Forward Compton scattering amplitude in deep inelastic scattering. At the spacetime point 0, a quark $\chi$ from $-\infty$ is annihilated and a quark $\xi$ is produced and moves to $z$. At $z$, a quark $\xi$ is annihilated, and a quark $\chi$ is produced and moves to $\infty$.
• Figure 3: The description of the usoft Wilson lines in deep inelastic scattering. The usoft Wilson lines with $\xi$ (a) going to $-\infty$, (b) going to $\infty$, (c) the resultant usoft Wilson lines from (a) and (b).
• Figure 4: (a) The usoft Wilson lines in the Drell-Yan process at lowest order. (b) the configuration of the usoft Wilson lines for the Drell-Yan process at higher orders in $\alpha_s$, which is equal to that of deep inelastic scattering, hence giving the same radiative corrections.
• Figure 5: The Drell-Yan processes at order $\alpha_s$ in which a valence quark comes from the proton and (a) an antiquark, (b) a gluon from the antiproton.