We demonstrate a new method of extracting parton distributions from lattice calculations. The starting idea is to treat the generic equal-time matrix element M(P z3, z 2 3 ) as a function of the Ioffe time ν = P z3 and the distance z3. The next step is to divide M(P z3, z 2 3 ) by the rest-frame density M(0, z 2 3 ). Our lattice calculation shows a linear exponential z3-dependence in the rest-frame function, expected from the Z(z 2 3 ) factor generated by the gauge link. Still, we observe that the ratio M(P z3, z 2 3 )/M(0, z 2 3 ) has a Gaussian-type behavior with respect to z3 for 6 values of P used in the calculation. This means that Z(z 2 3 ) factor was canceled in the ratio. When plotted as a function of ν and z3, the data are very close to z3-independent functions. This phenomenon corresponds to factorization of the x-and k ⊥ -dependence for the TMD F(x, k 2 ⊥ ). For small z3 ≤ 4a, the residual z3-dependence is explained by perturbative evolution, with αs/π = 0.1.

We present the first exploratory lattice QCD calculation of the pion valence quark distribution extracted from spatially separated current-current correlations in coordinate space. We show that an antisymmetric combination of vector and axial-vector currents provides direct information on the pion valence quark distribution. Using the collinear factorization approach, we calculate the perturbative tree-level kernel for this current combination and extract the pion valence distribution. The main goal of this article is to demonstrate the efficacy of this general lattice QCD approach in the reliable extraction of parton distributions. With controllable power corrections and a good understanding of the lattice systematics, this method has the potential to serve as a complementary to the many efforts to extract parton distributions in global analyses from experimentally measured cross sections. We perform our calculation on an ensemble of 2+1 flavor QCD using the isotropicclover fermion action, with lattice dimensions 32 3 × 96 at a lattice spacing a = 0.127 fm and the quark mass equivalent to a pion mass mπ 416 MeV.