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Large-momentum effective theory’s asymptotic extrapolation vs the inverse problem

Jiunn-Wei Chen ^1,2, Xiang Gao ³, Jinchen He^4,5, Jun Hua ^6,7, Xiangdong Ji⁴, Andreas Schäfer ^8,9, Yushan Su ⁴, Wei Wang ¹⁰, Yi-Bo Yang^11,12,13,14 et al.

Jian-Hui Zhang ¹⁵, Qi-An Zhang ¹⁶, Rui Zhang⁵, and Yong Zhao ⁵

¹Department of Physics and Center for Theoretical Physics, National Taiwan University, Taipei, Taiwan 106
²Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan
³Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA
⁴Department of Physics, University of Maryland, College Park, Maryland 20850, USA
⁵Physics Division, Argonne National Laboratory, Lemont, Illinois 60439, USA
⁶Key Laboratory of Atomic and Subatomic Structure and Quantum Control (MOE), Guangdong Basic Research Center of Excellence for Structure and Fundamental Interactions of Matter, Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China
⁷Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Guangdong Provincial Key Laboratory of Nuclear Science, Southern Nuclear Science Computing Center, South China Normal University, Guangzhou 510006, China
⁸Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
⁹Department of Physics, National Taiwan University, Taipei, Taiwan 106
¹⁰State Key Laboratory of Dark Matter Physics, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
¹¹CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
¹²University of Chinese Academy of Sciences, School of Physical Sciences, Beijing 100049, China
¹³School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China
¹⁴International Centre for Theoretical Physics Asia-Pacific, Beijing/Hangzhou, China
¹⁵School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen 518172, China
¹⁶School of Physics, Beihang University, Beijing 102206, China

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Phys. Rev. D 113, 014509 – Published 28 January, 2026

DOI: https://doi.org/10.1103/fflw-qpcc

Abstract

Large-momentum effective theory is a physics-guided systematic expansion to calculate light-cone parton distributions, including collinear (PDFs) and transverse-momentum-dependent ones, at any fixed momentum fraction $x$ within a range of $[x_{\min}, x_{\max}]$ . It theoretically solves the ill-posed inverse problem that afflicts other theoretical approaches to collinear PDFs, such as short-distance factorizations. Recently, Dutrieux et al. raised practical concerns about whether current or even future lattice data will have sufficient precision in the subasymptotic correlation region to support an error-controlled extrapolation—and if not, whether it becomes an inverse problem where the relevant uncertainties cannot be properly quantified. While we agree that not all current lattice data have the desired precision to qualify for an asymptotic extrapolation, some calculations do, and more are expected in the future. We comment on the analysis and results in Dutrieux et al. and argue that a physics-based systematic extrapolation still provides the most reliable error estimates, even when the data quality is not ideal. In contrast, reframing the long-distance asymptotic extrapolation as a data-driven-only inverse problem with ad hoc mathematical conditioning could lead to unnecessarily conservative errors.

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Phys. Rev. D 113, 014509– Published 28 January, 2026

Vol. 113, Iss. 1 — 1 January 2026

Received 29 May 2025
Accepted 19 December 2025

DOI: https://doi.org/10.1103/fflw-qpcc

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.

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Large-momentum effective theory’s asymptotic extrapolation vs the inverse problem

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