LaMET's Asymptotic Extrapolation vs. Inverse Problem

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.

Figures (4)

• Figure 1: An illustration of the difference between LaMET and SDF: LaMET in principle utilizes data at all $z$ to sharply resolve the momentum of a parton, though in practice they are usually truncated at $z_L\sim 1.0$ fm due to noisy data and extrapolated using physics-motivated asymptotic forms to $z=\infty$Ji:2020brr. Meanwhile, the SDF can only use data up to about $z_{\rm max}\sim 0.2-0.3$ fm (shaded blue region) to fit a model of the PDF, which constitutes an IP. Plotted are the hybrid-scheme nucleon transversity quasi-PDF matrix elements computed in Ref. LatticeParton:2022xsd. The shaded-red ellipse approximately indicates the sub-asymptotic region where the exponential decay starts to dominate. The concern raised in Dutrieux:2025jed is whether the data precision in this region allows for a reliable error estimate of the extrapolation; otherwise, it must be treated as an IP.
• Figure 2: An example of an IP in the SDF framework: reconstruction of the PDF from short-distance correlation functions at $z=0.26$ fm with proton momenta $P^z = 1.6,2.0,2.4,2.8,3.2$ GeV on the N203 ensemble, as reported in Ref. LatticeParton:2022xsd. The reconstruction is performed using the GPR method with the logRBF kernel proposed in Ref. Dutrieux:2024rem. To illustrate the model uncertainty arising from different hyperparameter choices, two sets of priors are compared: the red band corresponds to $l = 0.7 \approx \ln (2)$, $\sigma^2=16$ and prior mean $f(x) = 4$; the blue band corresponds to $l = 0.2 \approx \ln(1.25)$, $\sigma^2=2$ and prior mean $f(x) = 1$.
• Figure 3: An example of high-precision data in which we see clearly asymptotic exponential decay around $z=1$ fm.
• Figure 4: FT tests with the nucleon transversity quasi-PDF matrix elements from Ref. LatticeParton:2022xsd.