The Construction of Gauge-Links in Arbitrary Hard Processes

TL;DR

The paper addresses the gauge-invariance of TMD distribution and fragmentation correlators by deriving a calculational scheme to construct the process-dependent gauge-links (Wilson lines) that connect field operators. By resumming collinear gluon exchanges and leveraging color-flow identities, it shows how the soft gauge structure is dictated by the hard subprocess and external parton content, yielding explicit gauge-link structures for 2→2 QCD processes. The main contributions include a practical, gauge-invariant method to obtain gauge-links, demonstrations across SIDIS, DY, and pp scattering channels, and a framework for gluonic pole cross sections relevant to single-spin asymmetries. This work clarifies how color flow, not fermion number, governs gauge-link topology and provides tools for analyzing SSA mechanisms and potential factorization issues in hadronic scattering.

Abstract

Transverse momentum dependent parton distribution and fragmentation functions are described by hadronic matrix elements of bilocal products of field operators off the light-cone. These bilocal products contain gauge-links, as required by gauge-invariance. The gauge-links are path-ordered exponentials connecting the field operators along a certain integration path. This integration path is process-dependent, depending specifically on the short-distance partonic subprocess. In this paper we present the technical details needed in the calculation of the gauge-links and a calculational scheme is provided to obtain the gauge-invariant distribution and fragmentation correlators corresponding to a given partonic subprocess.

Figures (7)

• Figure 1: The gauge-link structure in the correlator $\Phi$ in (a) SIDIS: $\mathcal{U}^{[+]}$ and (b) DY: $\mathcal{U}^{[-]}$.
• Figure 2: The correlators for the leading hard subprocesses in SIDIS, DY and $e^+e^-$-annihilation.
• Figure 3: Three examples of partonic scattering processes that contribute in proton-proton collisions: (a) a quark-quark scattering contribution; (b) a quark-gluon scattering contribution; (c) a gluon-gluon scattering contribution.
• Figure 4: The truncated Green's function $\Gamma_{(i_2j_2)(i_1j_1)}^{a_2a_1}(p',p_1,p_2,p_3,k_1,k_2,k_3)$.
• Figure 5: Two possible one-gluon insertions where the gluon momentum $p$ is collinear to $p^\prime$: (a) the insertion of the additional collinear gluon in the truncated amplitude involving the truncated Green's function $\Gamma_{(i_2j_2)(i_1j_1)}^{(\mu a)a_2a_1} (p;p'{-}p,p_1,p_3,p_3,k_1,k_2,k_3)$; (b) the insertion of the additional collinear gluon to the outgoing quark.