Abstract

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}

Highlights

  • Are the quark and gluon components of the proton spin expressed in terms of the quark and gluon helicity distributions ∆Σ(x, Q2) = f ∆qf (x, Q2) + ∆qf (x, Q2) and ∆G(x, Q2)

  • 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 orbital angular momentum (OAM) distributions in the large-Nc limit: Lq+q(x, Q2) = −∆Σ(x, Q2) ∼

  • While the quark helicity distribution was related to the impact-parameter integrated fundamental polarized dipole amplitude [1], the quark OAM is related to the first impact parameter moment of this amplitude, as defined in eqs. (2.32)

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Summary

Introduction

Using the evolution equations constructed for the fundamental polarized dipole amplitude in [1, 6], in section 2.3 we construct and solve the evolution equations for the first impact parameter moment of the amplitude The consequences of this solution for quark OAM at small x are summarized, with the resulting small-x asymptotics of quark OAM distribution given by eq (2.50) (at large Nc and in DLA). Evolution equations for gluon helicity are a bit more involved than those for quark helicity [5]: the same applies to OAMs. the solution is employed in section 3.4 to derive the small-x asymptotics (3.74) of the gluon OAM distribution (in the DLA limit and at large Nc).

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