{
  "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 $α_s \\ln^2(1/x)$ 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, $L_{q+\\bar{q}}(x,Q^2) \\sim L_G(x,Q^2) \\sim ΔΣ(x,Q^2) \\sim ΔG(x,Q^2) \\sim \\left(\\frac{1}{x} \\right)^{α_h}$ with the intercept $α_h \\approx {3.48 \\sqrt{α_s N_c/2π}}$ for $N_f = N_c = 3$ (cf.~\\cite{Borden:2025ehe}): this result, along with $α_h$ for other values of $N_f \\neq 0$ that we studied, is smaller than the intercept of $3.66\\sqrt{α_s N_c/2π}$ found in the large-$N_c$ limit. We also compute the ratios of the OAM distributions to the helicity parton distribution functions as $x\\to 0$, obtaining $L_{q+\\bar{q}}(x,Q^2)/ΔΣ(x,Q^2) = -1$ and $L_G(x,Q^2)/ΔG(x,Q^2) = -2$ at $Q^2=10\\, \\mathrm{GeV}^2$, with both ratios being independent of $N_f$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.25120",
  "author_total": 5,
  "authors": [
    {
      "name": "G. Zardo Becker"
    },
    {
      "name": "Yuri V. Kovchegov"
    },
    {
      "name": "Ming Li"
    },
    {
      "name": "Brandon Manley"
    },
    {
      "name": "Andrey Tarasov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ebfn2twd80eep.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.25120",
  "fulltext_url": null,
  "id": "ebfn2twd80eep",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.25120",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "hep-ph"
    ],
    [
      "openalex",
      "3106|Quantum Chromodynamics and Particle Interactions"
    ]
  ],
  "openalex": "W7204491387",
  "pdf_url": "https://arxiv.org/pdf/2608.25120",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ebfn2twd80eep.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:18Z",
      "native_id": "2608.25120",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.25120"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:hep-ph",
      "subfield": "high-energy"
    }
  ],
  "title": "Orbital angular momentum at small $x$ in the large $N_c\\&N_f$ limit",
  "title_text": "Orbital angular momentum at small x in the large N_c&N_f limit",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:18Z",
  "url": "https://research.draconitic.com/paper/ebfn2twd80eep",
  "venue": "arXiv",
  "versions": []
}