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}