The $x\to1$ limit of the proton $d/u$ parton distribution ratio and the valence-quark dynamics it discriminates

The ratio $d(x,Q^2)/u(x,Q^2)$ of the proton's down- and up-quark valence parton distributions in the limit $x\to1$ (Bjorken- $x$ approaching unity, where a single valence quark carries nearly all of the nucleon's momentum) is a classic discriminator of nonperturbative valence-quark dynamics: approximate spin-flavor SU(6) symmetry predicts $d/u\to1/2$ , scalar-diquark dominance predicts $d/u\to0$ , perturbative-QCD quark-counting rules predict $d/u\to1/5$ , and Dyson–Schwinger-equation treatments based on dynamical chiral symmetry breaking predict $F_2^n/F_2^p\to0.4$ – $0.5$ . Because no free neutron target exists, $d/u$ at large $x$ must be inferred from deuteron, tritium, and helium-3 deep-inelastic scattering data through nuclear, off-shell, and higher-twist corrections that remain model dependent. Modern extractions disagree ( $d/u(x\to1)=0.09\pm0.03$ in CJ15 versus $0.23\pm0.09$ from in-medium correlations), and the newest measurements (MARATHON 2022; Jefferson Lab Hall C 2025) and global fits (CJ26, JAM 2026) still leave the $x\to1$ limit extrapolation-dominated. The open problem is to determine the limiting value of $d/u$ — with controlled uncertainties — at a level sufficient to discriminate among the numerically distinct candidate limits.

Background

In deep-inelastic scattering (DIS), an electron scatters off a nucleon by exchanging a virtual photon of momentum transfer squared $Q^2$ ; the Bjorken scaling variable $x=Q^2/(2M\nu)$ is interpreted, in a frame where the nucleon moves infinitely fast, as the fraction of the nucleon's momentum carried by the struck quark. The structure function $F_2$ measures the quark momentum-weighted charge-squared sum over parton distribution functions (PDFs), the probability densities $f_i(x)$ for quarks of type $i$ : $F_2^p\approx x[(4/9)u+(1/9)d]$ for the proton at leading twist, with $u(x)$ and $d(x)$ the up- and down-quark PDFs. Charge symmetry (isospin) relates neutron and proton PDFs, $u^n=d^p$ , $d^n=u^p$ , so the neutron structure function isolates the proton's $d$ quark: at large $x$ , where sea quarks are negligible, $F_2^n/F_2^p\to(1+4r)/(4+r)$ with $r=d/u$ . Positivity of PDFs gives the Nachtmann inequality $1/4\le F_2^n/F_2^p\le4$ .

The $x\to1$ behavior is a half-century-old discriminator of the proton's valence structure. An SU(6)-symmetric quark model (SU(6) is the approximate spin-flavor symmetry classifying the nucleon's three-quark wavefunction, with valence content $uud$ ) gives $F_2^n/F_2^p=2/3$ , i.e. $d/u\to1/2$ . Suppression of the spin-1 relative to the spin-0 diquark pair (a diquark is a correlated quark pair inside the nucleon; the color-magnetic hyperfine interaction that splits the $N$ and $\Delta$ masses suppresses the $ud$ spin-1 pair) drives $F_2^n/F_2^p\to1/4$ , i.e. $d/u\to0$ — the behavior SLAC data circa 1970 appeared to show as $x\to1$ . Perturbative QCD quark-counting rules (helicity conservation with two hard gluon exchanges) instead give $F_2^n/F_2^p\to3/7$ , i.e. $d/u\to1/5$ . Dyson–Schwinger equation (DSE) treatments — continuum functional methods solving the QCD equations of motion for the quark–gluon vertex and quark propagator — find that non-pointlike diquark correlations are a natural consequence of dynamical chiral symmetry breaking (D $\chi$ SB, the spontaneous generation of a constituent-quark mass scale in the light-quark sector), and predict $F_2^n/F_2^p\to0.4$ – $0.5$ .

There is no free neutron target, so $F_2^n$ — and hence $d/u$ at large $x$ — must be extracted from light-nuclear targets. Deuterium data require nuclear corrections for Fermi motion and binding, the EMC effect (the observed modification of bound-nucleon structure functions), and possible off-shell deformation of the bound nucleon's PDFs; the tritium/helium-3 mirror-nuclei method of the MARATHON experiment exploits the near-cancellation of nuclear corrections in the $^3$ He/ $^3$ H ratio ( $^3$ He has two protons and one neutron; $^3$ H one proton and two neutrons); spectator-tagging experiments (BONuS) detect low-momentum recoil protons in deuterium to select scattering from nearly on-shell neutrons. At $x\to1$ all methods are squeezed: fixed beam energy forces small final-state invariant mass $W^2$ (entering the nucleon resonance region) and low $Q^2$ , so target-mass corrections (kinematic $M^2/Q^2$ corrections from the OPE) and higher-twist terms (nonperturbative $1/Q^2$ power corrections) entangle with the leading-twist PDFs. Global QCD fits (the CTEQ-Jefferson-Lab and JAM analyses) anchor $d/u$ at $x\lesssim0.7$ with Fermilab $W$ -boson charge-asymmetry data, but the $x\to1$ limit itself is unmeasurable directly and depends on the fit's PDF parametrization. The highest-precision data now reach $x\approx0.83$ (MARATHON, $F_2^n/F_2^p$ ) and $x\approx0.93$ (Hall C deuteron-to-proton $F_2$ ratio), yet as of the 2026 global fits the extrapolated limit still changes with the assumed nuclear and power-correction modeling, and the competing predictions — $d/u$ of $0$ , $1/5$ , $1/2$ , or DSE values near $0.2$ — remain undiscriminated.

Problem Statement

Determine the limiting behavior of the proton's down-to-up valence parton distribution ratio $d(x,Q^2)/u(x,Q^2)$ as $x\to1$ — equivalently, the $x\to1$ behavior of the neutron-to-proton structure function ratio $F_2^n/F_2^p$ , related at leading twist by $F_2^n/F_2^p\to(1+4r)/(4+r)$ with $r=d/u$ — at a quantitative level sufficient to separate the three numerically distinct limits $d/u\to0$ , $d/u\approx0.2$ , and $d/u\to1/2$ predicted by the candidate mechanisms of valence-quark dynamics: approximate SU(6) spin-flavor symmetry ( $d/u\to1/2$ ), scalar-diquark dominance ( $d/u\to0$ ), perturbative-QCD quark-counting rules ( $d/u\to1/5$ ), and dynamical-chiral-symmetry-breaking/Dyson–Schwinger dynamics ( $F_2^n/F_2^p\to0.4$ – $0.5$ ). These predictions are not all distinct in the observable reported here: the quark-counting value $d/u\to1/5$ corresponds to $F_2^n/F_2^p\to3/7\approx0.43$ , which lies inside the Dyson–Schwinger range $F_2^n/F_2^p\to0.4$ – $0.5$ (equivalently $d/u\approx0.17$ – $0.29$ ), so that perturbative QCD and Dyson–Schwinger dynamics share the single limit $d/u\approx0.2$ and are not separately resolved by this observable. A complete answer must (i) establish the limiting value, or a tightly constrained bound on it, at a stated reference scale, with uncertainties that are honestly propagated including the extrapolation beyond the measured region ( $x\gtrsim0.9$ ), and (ii) bring the nuclear corrections (Fermi motion, binding, off-shell deformation, EMC-effect isospin dependence in deuterium and $A=3$ targets) and the target-mass and higher-twist power corrections under sufficient control that the stated limit is a property of the free nucleon rather than an artifact of the extraction model or PDF parametrization — or, alternatively, derive the limit from QCD first principles with quantified systematics. The answer must state explicitly which of the candidate limiting values it excludes.

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