Orbital Angular Momentum at Small $x$

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 $\alpha_s \ln^2 (1/x)$ with $\alpha_s$ 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-$N_c$ limit: \begin{align} L_{q + \bar{q}} (x, Q^2) = - \Delta \Sigma (x, Q^2) \sim \left(\frac{1}{x}\right)^{\frac{4}{\sqrt{3}} \, \sqrt{\frac{\alpha_s \, N_c}{2 \pi}} }, \ \ \ \ \ L_G (x, Q^2) \sim \Delta G (x, Q^2) \sim \left(\frac{1}{x}\right)^{\frac{13}{4 \sqrt{3}} \, \sqrt{\frac{\alpha_s \, N_c}{2 \pi}}} . \end{align}