Continuum-extrapolated NNLO Valence PDF of Pion at the Physical Point

We present lattice QCD calculations of valence parton distribution function (PDF) of pion employing next-to-next-leading-order (NNLO) perturbative QCD matching. Our calculations are based on three gauge ensembles of 2+1 flavor highly improved staggered quarks and Wilson–Clover valance quarks, corresponding to pion mass $m_{\pi }=140$~MeV at a lattice spacing $a=0.076$~fm and $m_{\pi }=300$~MeV at $a=0.04,0.06$~fm. This enables us to present, for the first time, continuum-extrapolated lattice QCD results for NNLO valence PDF of the pion at the physical point. Applying leading-twist expansion for renormalization group invariant (RGI) ratios of bi-local pion matrix elements with NNLO Wilson coefficients we extract $2^{\mathrm{n}\mathrm{d}}$, $4^{\mathrm{t}\mathrm{h}}$ and $6^{\mathrm{t}\mathrm{h}}$ Mellin moments of the PDF. We reconstruct the Bjorken-$x$ dependence of the NNLO PDF from real-space RGI ratios using a deep neural network (DNN) as well as from momentum-space matrix elements renormalized using a hybrid-scheme. All our results are in broad agreement with the results of global fits to the experimental data carried out by the xFitter and JAM collaborations.