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

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>x</a:mi> </a:math> within a range of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mo stretchy="false">[</c:mo> <c:msub> <c:mi>x</c:mi> <c:mi>min</c:mi> </c:msub> <c:mo>,</c:mo> <c:msub> <c:mi>x</c:mi> <c:mi>max</c:mi> </c:msub> <c:mo stretchy="false">]</c:mo> </c:math> . It theoretically solves the ill-posed inverse problem that afflicts other theoretical approaches to collinear PDFs, such as short-distance factorizations. Recently, Dutrieux 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 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 mathematical conditioning could lead to unnecessarily conservative errors.