Large-Momentum Effective Theory vs. Short-Distance Operator Expansion: Contrast and Complementarity

TL;DR

This paper compares LaMET and SD-OPE methods for extracting PDFs from lattice QCD, highlighting their differences, complementarities, and the potential for combined phenomenological analysis, demonstrated with pion data.

Contribution

It clarifies the contrast and complementarity between LaMET and SD-OPE approaches for PDF extraction and proposes a combined phenomenological framework.

Findings

LaMET directly calculates PDFs in a middle x-region.

SD-OPE extracts correlations at small distances for Fourier transform.

Combined use improves PDF determination in all x-regions.

Abstract

Although equivalent in the infinite-momentum limit, large-momentum effective theory (LaMET) and short-distance operator product expansion (SD-OPE) are two different approaches to extract parton distribution functions (PDFs) from coordinate-space correlation functions in large-momentum hadrons. LaMET implements a momentum-space expansion in $\Lambda_{\rm QCD}/[x(1-x)P^z]$ to directly calculate PDFs $f(x)$ in a middle region of Bjorken $x\in [x_{\rm min}\sim \Lambda_{\rm QCD}/P^z, x_{\rm max}\sim 1-x_{\min}]$. SD-OPE applies perturbative QCD at small Euclidean distances $z$ to extract a range $[0,\lambda_{\rm max}]$ of leading-twist correlations, $h(\lambda=zP^z)$, corresponding to the Fourier transformation of PDFs. Similar to the quantum mechanical uncertainty principle, an incomplete leading-twist correlation cannot be readily converted to a momentum-space local distribution, and the methods to solve the ``inverse problem'' involve essentially modelling of the missing information beyond $\lambda_{\rm max}$. On the other hand, short-distance correlations, along with the expected end-point asymptotics, can be used to phenomenologically fit the PDFs in the LaMET-complementary regions: $x\in [0,x_{\rm min}]$ and $[x_{\rm max}, 1]$. We use the recent results of the pion valence quark distribution from the ANL/BNL collaboration to demonstrate this point.