Lattice QCD exploration of pseudo-PDFs

Abstract

We demonstrate a new method of extracting parton distributions from lattice calculations. The starting idea is to treat the generic equal-time matrix element $\mathcal{M}(P{z}_3,z_3^2)$ as a function of the Ioffe time $\nu =P{z}_3$ and the distance $z_3$. The next step is to divide $\mathcal{M}(P{z}_3,z_3^2)$ by the rest-frame density $\mathcal{M}(0,z_3^2)$. 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 $\mathcal{M}(P{z}_3,z_3^2)/\mathcal{M}(0,z_3^2)$ 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 $\nu$ 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 $\mathcal{F}(x,k_{\perp }^2)$. For small $z_3\le 4a$, the residual $z_3$-dependence is explained by perturbative evolution, with ${\alpha }_s/\pi =0.1$.

Comments: 12 pages, 18 figures, to appear in Phys. Rev. D