Collins-Soper Kernel in the QCD Instanton Vacuum

TL;DR

This work develops a non-perturbative framework to extract the Collins-Soper kernel from the QCD instanton vacuum (ILM) by computing soft functions from Wilson loops in Euclidean space and performing analytic continuation to Minkowski space. The ILM yields a non-perturbative CS kernel $K_{\rm CS}$ whose large-$b_\perp$ behavior is logarithmic, and which, when combined with perturbative contributions via a $b^*$ scheme, agrees with recent lattice determinations and several phenomenological extractions. A key result is the factorization of the angular dependence as $K(\rho,b_\perp,\theta) \approx K_{\rm CS}(b_\perp/\rho) \; h(\theta)$, allowing straightforward Euclidean-to-Minkowski continuation and enabling lattice extractions of the CS kernel from Euclidean soft functions. The findings provide a principled, first-principles constraint on non-perturbative TMD evolution and offer a practical route to refine phenomenological models at large transverse separation.

Abstract

We outline a general framework for evaluating the non-perturbative soft functions in the QCD instanton vacuum. In particular, from the soft function we derive the Collins-Soper (CS) kernel, which drives the rapidity evolution of the transverse-momentum-dependent parton distributions. The resulting CS kernel, when supplemented with the perturbative contribution, agrees well with recent lattice results and some phenomenological parameterizations. Moreover, our CS kernel depends logarithmically on the large quark transverse separation, providing a key constraint on its phenomenological parametrization. Finally, a lattice calculation can be directly compared to our generic results in Euclidean signature, thus providing a new approach for evaluating the soft function and extracting the CS kernel by analytical continuation.

Figures (8)

• Figure 1: The Contours of the Wilson lines for the soft function of the cross section in (a) Drell-Yan, (b) electron-positron annihilation, and (c) SIDIS. Hadron $A$ is moving near the light-cone direction $n$, and hadron $B$ is moving near the light-cone direction $\bar{n}$Vladimirov:2014hla.
• Figure 2: Soft function
• Figure 3: (a) path $L_1$ is labeled by $v_1$ and $b_1$; (b) path $L_2$ is labeled by $v_2$ and $b_2$; (c) path $L_3$ is labeled by $v_3$ and $b_3$; (d) path $L_4$ is labeled by $v_4$ and $b_4$
• Figure 4: (a) Instanton liquid estimation of the Coulomb potential between two parallel Wilson lines, with the full calculation \ref{['V_inst']} and the weak field approximation \ref{['V_weak']}. We also plot their ratio in (b).
• Figure 5: (a) Instanton liquid estimation on CS kernel with the full calculation \ref{['K']} and weak field approximation \ref{['CS_weak']}. We also plot their ratio in (b).