Orbital angular momentum at small $x$ in the large $N_c\&N_f$ limit

Abstract

We extend the small-$x$ analysis of the quark and gluon orbital angular momentum (OAM) distributions in the proton from the large-$N_c$ limit considered in our earlier works to the large-$N_c\&N_f$ limit, in which the numbers of quark colors $N_c$ and flavors $N_f$ are large with their ratio held fixed. Working in the double-logarithmic approximation (DLA), summing powers of $\alpha_s \ln^2(1/x)$ with $\alpha_s$ the strong coupling and $x$ the proton's momentum fraction carried by a parton, we correct the small-$x$ operator expression for the quark OAM distribution suggested earlier in \cite{Kovchegov:2024wjs} and relate both the quark and gluon OAM distributions to the impact-parameter moments of the polarized dipole amplitudes. We derive the large-$N_c\&N_f$ evolution equations for these moment amplitudes in the DLA; these include a new equation for the moment amplitude governing the quark OAM. We then solve these evolution equations numerically together with the helicity evolution for $N_f = 2,3,4,5,6$ and $N_c =3$. We find that, similar to the large-$N_c$ case, the OAM distributions share a common small-$x$ intercept with the helicity distributions, $L_{q+\bar{q}}(x,Q^2) \sim L_G(x,Q^2) \sim \Delta \Sigma (x,Q^2) \sim \Delta G(x,Q^2) \sim \left(\frac{1}{x} \right)^{\alpha_h}$ with the intercept $\alpha_h \approx {3.48 \sqrt{\alpha_s N_c/2\pi}}$ for $N_f = N_c = 3$ (cf.~\cite{Borden:2025ehe}): this result, along with $\alpha_h$ for other values of $N_f \neq 0$ that we studied, is smaller than the intercept of $3.66\sqrt{\alpha_s N_c/2\pi}$ found in the large-$N_c$ limit. We also compute the ratios of the OAM distributions to the helicity parton distribution functions as $x\to 0$, obtaining $L_{q+\bar{q}}(x,Q^2)/\Delta\Sigma(x,Q^2) \approx -1.01$ and $L_G(x,Q^2)/\Delta G(x,Q^2) \approx -1.94$ at $Q^2=10\, \mathrm{GeV}^2$, with both ratios being nearly independent of $N_f$.