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Lattice QCD determination of the Collins-Soper kernel in the continuum and physical mass limits

Jin-Xin Tan^1,2, Zhi-Chao Gong^1,2, Jun Hua^3,4,*, Xiangdong Ji⁵, Xiangyu Jiang⁶, Hang Liu⁷, Andreas Schäfer^8,9, Yushan Su⁵, Han-Zhang Wang^1,10 et al.

Wei Wang^1,11,†, Yi-Bo Yang^6,12,13,14, Jun Zeng¹⁵, Jian-Hui Zhang¹⁶, Jia-Lu Zhang^2,1, and Qi-An Zhang ^17,‡

¹State Key Laboratory of Dark Matter Physics, Key Laboratory for Particle Astrophysics and Cosmology (MOE), Shanghai Key Laboratory for Particle Physics and Cosmology, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
²Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 201210, People’s Republic of China
³State Key Laboratory of Nuclear Physics and Technology, Institute of Quantum Matter, South China Normal University, Guangzhou 510006, People’s Republic of China
⁴Guangdong Basic Research Center of Excellence for Structure and Fundamental Interactions of Matter, Guangdong Provincial Key Laboratory of Nuclear Science, Guangzhou 510006, People’s Republic of China
⁵Department of Physics, University of Maryland, College Park, Maryland 20742, USA
⁶CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
⁷Department of Physics, Shanghai Normal University, Shanghai 200234, People’s Republic of China
⁸Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
⁹Department of Physics, National Taiwan University, Taipei, Taiwan 106, People’s Republic of China
¹⁰Zhiyuan College, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
¹¹Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of Sciences, Huizhou 516000, Guangdong Province, People’s Republic of China
¹²School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, People’s Republic of China
¹³International Centre for Theoretical Physics Asia-Pacific, Beijing/Hangzhou, People’s Republic of China
¹⁴School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China
¹⁵College of Physics and Electronic Engineering, Hainan Normal University, Haikou 571158, Hainan, People’s Republic of China
¹⁶School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen 518172, People’s Republic of China
¹⁷School of Physics, Beihang University, Beijing 102206, People’s Republic of China

^*Contact author: junhua@scnu.edu.cn
^†Contact author: wei.wang@sjtu.edu.cn
^‡Contact author: zhangqa@buaa.edu.cn

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Phys. Rev. D 113, 054505 – Published 5 March, 2026

DOI: https://doi.org/10.1103/pry5-7729

Abstract

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 ( $b_{⊥}$ ) 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 $b_{⊥}$ -unexpanded matching kernel, we obtain the CS kernel at the continuum, chiral, and infinite-momentum limits. Our results are determined up to $b_{⊥} \sim 1 fm$ , with controllable uncertainties, and agree with perturbative QCD at small $b_{⊥}$ 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.

Physics Subject Headings (PhySH)

Transverse momentum dependent distribution

Article Text

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Phys. Rev. D 113, 054505– Published 5 March, 2026

Vol. 113, Iss. 5 — 1 March 2026

Received 10 December 2025
Accepted 6 February 2026

DOI: https://doi.org/10.1103/pry5-7729

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society

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Lattice QCD determination of the Collins-Soper kernel in the continuum and physical mass limits

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