Orbital angular momentum at small x in the large Nc&Nf limit
Abstract
We extend the small-x analysis of the quark and gluon orbital angular momentum (OAM) distributions in the proton from the large-Nc limit considered in our earlier works to the large-Nc&Nf limit, in which the numbers of quark colors Nc and flavors Nf are large with their ratio held fixed. Working in the double-logarithmic approximation (DLA), summing powers of αs ln2(1/x) with αs 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-Nc&Nf 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 Nf = 2,3,4,5,6 and Nc =3. We find that, similar to the large-Nc case, the OAM distributions share a common small-x intercept with the helicity distributions, Lq+q̅(x,Q2) ∼ LG(x,Q2) ∼ ΔΣ(x,Q2) ∼ ΔG(x,Q2) ∼ (1/x )αh with the intercept αh ≈ 3.48 √(αs Nc/2π) for Nf = Nc = 3 (cf.~\cite{Borden:2025ehe}): this result, along with αh for other values of Nf ≠ 0 that we studied, is smaller than the intercept of 3.66√(αs Nc/2π) found in the large-Nc limit. We also compute the ratios of the OAM distributions to the helicity parton distribution functions as x→ 0, obtaining Lq+q̅(x,Q2)/ΔΣ(x,Q2) = −1 and LG(x,Q2)/ΔG(x,Q2) = −2 at Q2=10 GeV2, with both ratios being independent of Nf.
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BibTeX
@misc{becker2026orbital,
title = {{Orbital angular momentum at small $x$ in the large $N_c\&N_f$ limit}},
author = {G. Zardo Becker and Yuri V. Kovchegov and Ming Li and Brandon Manley and Andrey Tarasov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.25120},
eprint = {2608.25120},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.25120},
}
RIS
TY - UNPB
TI - Orbital angular momentum at small x in the large N_c&N_f limit
AU - G. Zardo Becker
AU - Yuri V. Kovchegov
AU - Ming Li
AU - Brandon Manley
AU - Andrey Tarasov
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2608.25120
UR - https://arxiv.org/abs/2608.25120
AB - 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 α_s ln^2(1/x) with α_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+q̅(x,Q^2) ∼ L_G(x,Q^2) ∼ ΔΣ(x,Q^2) ∼ ΔG(x,Q^2) ∼ (1/x )^α_h with the intercept α_h ≈ 3.48 √(α_s N_c/2π) for N_f = N_c = 3 (cf.~\cite{Borden:2025ehe}): this result, along with α_h for other values of N_f ≠ 0 that we studied, is smaller than the intercept of 3.66√(α_s N_c/2π) found in the large-N_c limit. We also compute the ratios of the OAM distributions to the helicity parton distribution functions as x→ 0, obtaining L_q+q̅(x,Q^2)/ΔΣ(x,Q^2) = −1 and L_G(x,Q^2)/ΔG(x,Q^2) = −2 at Q^2=10 GeV^2, with both ratios being independent of N_f.
ER -