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

...+6
Aug 2022

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

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