Orbital angular momentum at small x

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We determine the small Bjorken x asymptotics of the quark and gluon orbital angular momentum (OAM) distributions in the proton in the double-logarithmic approximation (DLA), which resums powers of αs ln2(1/x) with αs the strong coupling constant. Starting with the operator definitions for the quark and gluon OAM, we simplify them at small x, relating them, respectively, to the polarized dipole amplitudes for the quark and gluon helicities defined in our earlier works. Using the small-x evolution equations derived for these polarized dipole amplitudes earlier we arrive at the following small-x asymptotics of the quark and gluon OAM distributions in the large-Nc limit:1aLq+q¯xQ2=−ΔΣxQ2∼1x43αsNc2π,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ {L}_{q+\\overline{q}}\\left(x,{Q}^2\\right)=-\\Delta \\Sigma \\left(x,{Q}^2\\right)\\sim {\\left(\\frac{1}{x}\\right)}^{\\frac{4}{\\sqrt{3}}\\kern0.5em \\sqrt{\\frac{\\alpha_s\\kern0.5em {N}_c}{2\\pi }}}, $$\\end{document}1bLGxQ2∼ΔGxQ2∼1x1343αsNc2π.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ {L}_G\\left(x,{Q}^2\\right)\\sim \\Delta G\\left(x,{Q}^2\\right)\\sim {\\left(\\frac{1}{x}\\right)}^{\\frac{13}{4\\sqrt{3}}\\kern0.5em \\sqrt{\\frac{\\alpha_s\\kern0.5em {N}_c}{2\\pi }}}. $$\\end{document}