Factorization Theorem Relating Euclidean and Light-Cone Parton Distributions

Abstract

In a large-momentum nucleon state, the matrix element of a gauge-invariant Euclidean Wilson line operator accessible from lattice QCD can be related to the standard light-cone parton distribution function through the large-momentum effective theory (LaMET) expansion. This relation is given by a factorization theorem with a non-trivial matching coefficient. Using the operator product expansion we prove the large-momentum factorization of the quasi-parton distribution function in LaMET, and show that the more recently discussed Ioffe-time distribution approach also obeys an equivalent factorization theorem. Explicit results for the coefficients are obtained and compared at one-loop. Our proof clearly demonstrates that the matching coefficients in the $\overline{\rm MS}$ scheme depend on the large partonic momentum rather than the nucleon momentum.

Figures (4)

• Figure 1: One-loop Feynman diagrams for the quasi-PDF, spatial correlator and pseudo-PDFs. The first one is named "vertex", the second and third ones are named "sail", and the last one "tadpole". The standard quark self energy wavefunction is also included.
• Figure 2: The $\overline{\rm MS}$ scheme PDF $xf_{u-d}$ and the $\overline{\rm MS}$ quasi-PDF obtained from $x\, C^{\overline{\rm MS}}(p^z)\otimes f_{u-d}$, comparing results obtained with $p^z=yP^z$ and $p^z=P^z$.
• Figure 3: (Left) Comparison between the PDF $xf_{u-d}$ and the pseudo-PDF $x({\cal C}^{\overline{\rm MS}}\otimes f_{u-d})$ in the $\overline{\rm MS}$ scheme. The orange and blue bands indicate the results from varying the factorization scale $\mu=4\,{\rm GeV}$ by a factor of two. (Right) Same but now showing only central pseudo-PDF curves for different values of $z$.
• Figure 4: (left) Comparison between the light-cone time distribution $Q_{u-d}$ and spatial correlator from $({\cal C}^{\overline{\rm MS}}\otimes Q_{u-d})$ in the $\overline{\rm MS}$ scheme. The orange and blue bands indicate the results from varying the factorization scale $\mu=4\,{\rm GeV}$ by a factor of two. (right) Same but now showing only central spatial correlator curves for different values of $z$. The top panels show the real part, while bottom panels show the imaginary part.