Continuum-extrapolated NNLO valence PDF of the pion at the physical point

We present lattice QCD calculations of valence parton distribution function (PDF) of pion em-ploying 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 π = 140 MeV at a lattice spacing a = 0 . 076 fm and m π = 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 nd , 4 th and 6 th 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.