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 formula with a non-trivial matching coefficient. Using the operator product expansion we derive the large-momentum factorization of the quasi-parton distribution function in LaMET, and show that the more recently discussed pseudo-parton distribution approach also obeys an equivalent factorization formula. Explicit results for the coefficients are obtained and compared at one-loop. We also prove that the matching coefficients in the MS scheme depend on the large partonic momentum rather than the nucleon momentum.

Parton distributions from lattice data: the nonsinglet case

“…[21], and proved in Refs. [24,25] using the OPE, renormalized quasi-PDFs share the same IR behaviour as the renormalized light-cone PDFs. It follows that they can be written as…”

“…This was resolved in Ref. [25] by adding terms outside of the plus prescription in the matching coefficient. As noticed in Ref.…”

“…An alternative modification of the MS scheme that guarantees vector current conservation was derived in an updated version of Ref. [25]. This defines the so-called "ratio" scheme.…”

“…We find that the matrix elements obtained by this alternative procedure are compatible with the actual ones within statistical uncertainties, with relatively larger effects observed for small z/a in the imaginary part (up to O(5%)) and intermediate z/a in the real part (the real part is small there -thus, the observed absolute effects of around 0.2 can be a large percentage of the value). Apart from the scheme conversion truncation effects, the necessary perturbative ingredient of the approach is the matching between quasi-PDFs and light-cone distributions, also known to one loop [21,25,45]. Without knowing the two-loop formulae, it is difficult to estimate their size.…”

“…Following these novel ideas, a number of publications have appeared (see Refs. [8,9] for recent reviews), from both lattice and high-energy communities, addressing the main theoretical problems of the new lattice approach: the definition and renormalization of the non-local operators involved in the lattice simulation [10][11][12][13][14][15][16][17][18][19][20], the proof of the factorization theorem between PDFs and quasi-PDFs [7,[21][22][23][24][25], the computation of the matching coefficients relating latticecomputable quantities to PDFs in different renormalization schemes [21,22,[24][25][26][27][28][29][30][31][32][33]. Also, data coming from first lattice QCD simulations have started appearing and have gotten into already a relatively advanced stage over the last few years [13,[30][31][32][33][34][35][36][37][38][39][40][41][42]…”