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

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 ${\Lambda }_{\mathrm{QCD}}/[x(1-x)P^z]$ to directly calculate PDFs $f(x)$ in a middle region of Bjorken $x\in [x_{\min }\sim {\Lambda }_{\mathrm{QCD}}/P^z,x_{\max }\sim 1-x_{\min }]$. SD-OPE applies perturbative QCD at small Euclidean distances $z$ to extract a range $[0,{\lambda }_{\max }]$ of leading-twist correlations, $h(\lambda =z{P}^z)$, 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 ${\lambda }_{\max }$. 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: $x\in [0,x_{\min }]$ and $[x_{\max },1]$. We use the recent results of the pion valence quark distribution from the ANL/BNL collaboration to demonstrate this point.

Comments: 10 pages, 5 figures