Orbital Angular Momentum at Small $x$

Abstract

We revisit the problem of the small Bjorken-$x$ asymptotics of the quark and gluon orbital angular momentum (OAM) distributions in the proton utilizing the revised formalism for small-$x$ helicity evolution derived recently in a paper by Cougoulic, Kovchegov, Tarasov, and Tawabutr. We relate the quark and gluon OAM distributions at small $x$ to the polarized dipole amplitudes and their (first) impact-parameter moments. To obtain the $x$-dependence of the OAM distributions, we derive novel small-$x$ evolution equations for the impact-parameter moments of the polarized dipole amplitudes in the double-logarithmic approximation (summing powers of $α_s \ln^2(1/x)$ with $α_s$ the strong coupling constant). We solve these evolution equations numerically and extract the large-$N_c$, small-$x$ asymptotics of the quark and gluon OAM distributions, which we determine to be \[ L_{q+\bar{q}}(x, Q^2) \sim L_{G}(x,Q^2) \sim ΔΣ(x, Q^2) \sim ΔG(x,Q^2) \sim \left(\frac{1}{x}\right)^{3.66 \, \sqrt{\frac{α_s N_c}{2π}}},\] in agreement with an earlier work by Boussarie, Hatta, and Yuan within the precision of our numerical evaluation (here $N_c$ is the number of quark colors). We also investigate the ratios of the quark and gluon OAM distributions to their helicity distribution counterparts in the small-$x$ region.