Small-x asymptotics of the gluon helicity distribution

We determine the small-x asymptotics of the gluon helicity distribution in a proton at leading order in perturbative QCD at large Nc. To achieve this, we begin by evaluating the dipole gluon helicity TMD at small x. In the process we obtain an interesting new result: in contrast to the unpolarized dipole gluon TMD case, the operator governing the small-x behavior of the dipole gluon helicity TMD is different from the operator corresponding to the polarized dipole scattering amplitude (used in our previous work to determine the small-x asymptotics of the quark helicity distribution). We then construct and solve novel small-x large-Nc evolution equations for the operator related to the dipole gluon helicity TMD. Our main result is the small-x asymptotics for the gluon helicity distribution: ΔG∼1xαhG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \varDelta G\sim {\left(\frac{1}{x}\right)}^{\alpha_h^G} $$\end{document} with αhG=1343αsNc2π≈1.88αsNc2π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\alpha}_h^G=\frac{13}{4\sqrt{3}}\sqrt{\frac{\alpha_s{N}_c}{2\pi }}\approx 1.88\sqrt{\frac{\alpha_s{N}_c}{2\pi }} $$\end{document}. We note that the power αhG is approximately 20% lower than the corresponding power αhq for the small-x asymptotics of the quark helicity distribution defined by Δq∼1xαhq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \varDelta q\sim {\left(\frac{1}{x}\right)}^{\alpha_h^q} $$\end{document} with αhq=43αsNc2π≈2.31αsNc2π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\alpha}_h^q=\frac{4}{\sqrt{3}}\sqrt{\frac{\alpha_s{N}_c}{2\pi }}\approx 2.31\sqrt{\frac{\alpha_s{N}_c}{2\pi }} $$\end{document} found in our earlier work.