Large-Momentum Effective Theory's Asymptotic Extrapolation vs the Inverse Problem

TL;DR

Large-Momentum Effective Theory (LaMET) offers a systematic approach to calculate parton distributions at any momentum, addressing inverse problems in lattice QCD, with ongoing debates on data precision and error estimation.

Contribution

The paper clarifies LaMET's asymptotic extrapolation method, emphasizing its robustness over purely data-driven inverse problem approaches for estimating uncertainties.

Findings

Some lattice data are suitable for asymptotic extrapolation.

Systematic physics-based extrapolation provides reliable error estimates.

Re-framing as an inverse problem may lead to overly conservative errors.

Abstract

Large-Momentum Effective Theory (LaMET) 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 $[x_{\rm min}, x_{\rm max}]$. It theoretically solves the ill-posed inverse problem that afflicts other theoretical approaches to collinear PDFs, such as short-distance factorizations. Recently, arXiv:2504.17706 [1] raised practical concerns about whether current or even future lattice data will have sufficient precision in the sub-asymptotic 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 Ref. [1] 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, re-framing the long-distance asymptotic extrapolation as a data-driven-only inverse problem with ad hoc mathematical conditioning could lead to unnecessarily conservative errors.