Orbital angular momentum at small x revisited

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 small-x helicity evolution derived recently in [1]. 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 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 ln²(1/x) with α_s the strong coupling constant). We solve these evolution equations numerically and extract the leading large-N_c, small-x asymptotics of the quark and gluon OAM distributions, which we determine to be $$L_{q+\overline{q}}\left(x,Q^2\right)~L_G\left(x,Q^2\right)~∆\mathrm{\Sigma }\left(x,Q^2\right)~∆G\left(x,Q^2\right)~{\left(\frac{1}{2}\right)}^{3.66\sqrt{\frac{{\alpha }_s{N}_c}{2\mathrm{\pi }}}},$$ \( {L}_{q+\overline{q}}\left(x,{Q}^2\right)\sim {L}_G\left(x,{Q}^2\right)\sim \Delta \Sigma \left(x,{Q}^2\right)\sim \Delta G\left(x,{Q}^2\right)\sim {\left(\frac{1}{2}\right)}^{3.66\sqrt{\frac{\alpha_s{N}_c}{2\uppi}}}, \)

in agreement with [2] 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.