Orbital Angular Momentum Small-$x$ Evolution: Exact Results in the Large-$N_c$ Limit

Abstract

We construct an exact solution to the revised small-$x$ orbital angular momentum (OAM) evolution equations derived recently, based on an earlier work. These equations are derived in the double logarithmic approximation (summing powers of ${\alpha }_s{\ln }^2(1/x)$ with ${\alpha }_s$ the strong coupling constant and $x$ the Bjorken $x$ variable) and the large-$N_c$ limit, with $N_c$ the number of quark colors. From our solution, we extract the small-$x$, large-$N_c$ expressions of the quark and gluon OAM distributions. Additionally, we determine the large-$N_c$ small-$x$ asymptotics of the OAM distributions to be \begin{align} \notag L_{q+\bar{q}}(x,Q^2) \sim L_G(x,Q^2) \sim \Delta \Sigma (x,Q^2) \sim \Delta G(x,Q^2) \sim \left(\frac{1}{x} \right)^{\alpha_h}, \end{align} with the intercept ${\alpha }_h$ the same as obtained in the small-$x$ helicity evolution, which can be approximated as ${\alpha }_h\approx 3.66074\sqrt{\frac{{\alpha }_s{N}_c}{2\pi }}$. This result is in complete agreement with the literature. Additionally, we calculate the ratio of the quark and gluon OAM distributions to the flavor-singlet quark and gluon helicity parton distribution functions respectively in the small-$x$ region.

Comments: 27 pages, 2 figures; v2: typos corrected, references added, appendix C modified, 22 pages, 2 figures. Signs corrected, discussion expanded; v4: typos corrected