Orbital angular momentum small-\(x\,\)evolution: Exact results in the large- \(N_c\,\)limit
We construct an exact solution to the revised small-x orbital angular momentum (OAM) evolution equations derived in an earlier work. These equations are derived in the double logarithmic approximation (summing powers of \(\alpha_s \ln^2(1/x) \)with \(\alpha_s\,\)the strong coupling constant and \(x\,\)the Bjorken \(x\,\)variable) and the large-\(N_c\,\)limit, with \(N_c\,\)the number of quark colors. From our solution, we extract the small-\(x\), large-\(N_c\,\)expressions of the quark and gluon OAM distributions. Additionally, we determine the large-\(N_c\,\)small-\(x\,\)asymptotics of the OAM distributions to be the same as obtained in the small-\(x\,\)helicity evolution, which can be approximated as \(\alpha_h ≈3.66074\,\). This result is in complete agreement with the literature. Additionally, we calculate the ratio of the quark and gluon OAM distributions to the flavor-singlet quark and gluon helicity parton distribution functions respectively in the small-x region