We demonstrate a new method of extracting parton distributions from lattice calculations. The starting idea is to treat the generic equal-time matrix element ${cal M} (Pz_3, z_3^2)$ as a function of the Ioffe time $ u = Pz_3$ and the distance $z_3$. The next step is to divide ${cal M} (Pz_3, z_3^2)$ by the rest-frame density ${cal M} (0, z_3^2)$. Our lattice calculation shows a linear exponential $z_3$-dependence in the rest-frame function, expected from the $Z(z_3^2)$ factor generated by the gauge link. Still, we observe that the ratio ${cal M} (Pz_3 , z_3^2)/{cal M} (0, z_3^2)$ has a Gaussian-type behavior with respect to $z_3$ for 6 values of $P$ used in the calculation. This means that $Z(z_3^2)$ factor was canceled in the ratio. When plotted as a function of $ u$ and $z_3$, the data are very close to $z_3$-independent functions. This phenomenon corresponds to factorization of the $x$- and $k_perp$-dependence for the TMD ${cal F} (x, k_perp^2)$. For small $z_3 leq 4a$, the residual $z_3$-dependence is explained by perturbative evolution, with $alpha_s/pi =0.1$.
تحميل البحث