Pion Valence-Quark Generalized Parton Distribution at Physical Pion Mass

We present the first lattice-QCD $x$-dependent pion valence-quark generalized parton distribution (GPD) calculated directly at physical pion mass using the Large-Momentum Effective Theory (LaMET) with next-to-next-to-leading order perturbative matching correction. We use clover fermions for the valence action on $2+1+1$ flavors of highly improved staggered quarks (HISQ), generated by MILC Collaboration, with lattice spacing $a \approx 0.09$fm and box size $L \approx 5.5$fm; the pion two-point measurements number up to $O(10^6)$ with boost momentum 1.73GeV. The pion valence distribution is renormalized in hybrid scheme with Wilson-line mass subtraction at large distances in coordinate space, followed by a procedure to match it to the $\overline{\text{MS}}$ scheme. We focus on the zero-skewness limit, where the GPD has a probability-density interpretation in the longitudinal Bjorken $x$ and the transverse impact-parameter distributions. We take the integral of our GPD functions to generate leading moment so that we can make comparisons with past lattice-QCD and experimental determinations of the pion form factors and found consistent agreement among them. We predict the higher GPD moments and reveal $x$-dependent tomography of the pion for the first time using lattice QCD.