Lattice QCD Determination of the Collins-Soper Kernel in the Continuum and Physical Mass Limits

The Collins-Soper (CS) kernel governs the rapidity evolution of transverse-momentum-dependent (TMD) parton distributions, a cornerstone for QCD factorization and linking nucleon structure data across scales. Its nonperturbative behavior at large transverse separations ($b_{\perp}$) remains weakly constrained due to phenomenological model dependencies. We present a first-principles determination of the CS kernel at the continuum limit and physical pion mass from lattice QCD in the large-momentum effective theory framework. Using (2+1)-flavor configurations (lattice spacings $a \in[0.052, 0.105]$ fm, and pion mass $m_π \approx ( 136, 230, 300, 320)$ MeV), we simulating the nonlocal equal-time correlation function and extract the quasi-TMD wave functions. Taking into account systematic improvements including hypercubic smearing, nonperturbative renormalization, and a $b_{\perp}$-unexpanded matching kernel, we obtain the CS kernel at the continuum, chiral, and infinite-momentum limits. Our results are determined up to $b_{\perp} \sim 1$ fm, with controllable uncertainties, and agree with perturbative QCD at small $b_{\perp}$ and global TMD phenomenological extractions. We conduct a global analysis integrated with phenomenological fits and demonstrate the impact of our results on such fits. This work yields the most precise nonperturbative constraint on the CS kernel’s long-distance behavior from Lattice QCD, which not only bridges Lattice QCD, perturbation theory, and nucleon structure experiments for TMD studies, but also boosts the utility of our constraint for future global TMD analyses.

💡 Research Summary

This paper presents a landmark first-principles determination of the Collins-Soper (CS) kernel, a fundamental object in Quantum Chromodynamics (QCD) that governs the rapidity evolution of Transverse-Momentum-Dependent (TMD) parton distributions. Understanding the CS kernel is crucial for connecting nucleon structure data from experiments conducted at different energy scales, such as Semi-Inclusive Deep Inelastic Scattering (SIDIS) and Drell-Yan processes. While perturbatively calculable at short distances, its nonperturbative behavior at large transverse separations (b⊥) has previously been constrained only by model-dependent phenomenological extractions, leading to significant uncertainties.

The research overcomes the fundamental obstacle of direct light-cone operator calculation in Euclidean lattice QCD by employing the Large-Momentum Effective Theory (LaMET) framework. The core strategy involves computing “quasi-TMD wave functions” for a pion boosted to large momentum (Pz) on the lattice. According to LaMET factorization, the Pz-dependence of these Euclidean observables encodes the same rapidity evolution controlled by the CS kernel. By comparing results at different momenta, the kernel can be extracted nonperturbatively.

The calculation utilizes (2+1)-flavor gauge field configurations spanning multiple lattice spacings (a ∈