First-principles completion of the De Rújula–Georgi–Politzer local-duality test of QCD in electroproduction: nonperturbative matrix elements $A_n$ and $B_{nk}$, the gluon sector at small $\xi$, and parameter-free predictions for $R$, the large-$\xi$ structure-function tail, and elastic form factors

In 1977 De Rújula, Georgi and Politzer showed that Bloom–Gilman local duality between the resonance region and the deep-inelastic scaling limit of electroproduction follows from the operator-product-expansion (OPE) moment predictions of QCD, and made this a quantitative test of asymptotic freedom at $1\lesssim Q^{2}\lesssim 16$ GeV$^{2}$, with a phenomenological fit of the proton's twist-two moments $A_{n}$ and a parametrized, undetermined higher-twist sector $B_{nk}$. The authors' own abstract acknowledged part of what the test still lacked: the existing $R=\sigma_{L}/\sigma_{T}$ data at large $\xi$ and $Q^{2}$ lay systematically above their twist-two prediction, the higher-twist sector $B_{nk}$ was parametrized, with the scale $M_{0}\approx 400$ MeV obtained from resonance-region data, and the twist-two moments $A_{n}$ came from a fit to scaling-region data; their companion paper ('Trouble with $\xi$ scaling?', Physical Review D 15 (1977) 2495) restated that the higher-twist corrections are negligible only for $W$ well above the proton mass. The subsequent literature identified the remaining gaps: a determined gluon distribution and controlled quark–gluon operator mixing below $\xi\lesssim 0.3$, higher-order perturbative input beyond the one-loop terms used in 1977, first-principles or independent determination of the $n$-dependent twist-four and higher matrix elements $B_{nk}(Q^{2})$, structure-function data at $\xi\gtrsim 0.8$, and a nonperturbative, first-principles determination of the proton matrix elements $A_{n}$ and of the quark/gluon separation. Later literature has delivered the perturbative orders (next-to-next-to-leading order (NNLO) splitting functions, five-loop $\beta$ function), the phenomenological gluon distribution with full singlet mixing, precision separated structure functions and extensive duality verification (Jefferson Lab), and the first physical-continuum lattice determinations of the nucleon gluon distribution. What no later work supplies is the problem's open core: a first-principles determination of the $n$-dependent higher-twist coefficients $B_{nk}$ and of the complete set of twist-two proton moments, enabling parameter-free local-duality predictions of the resonance-region structure functions, of $R$ at large $\xi$, and of the elastic form factors through the threshold moment relations, tested against precision data.

Background

In inclusive electron–proton scattering, a virtual photon of four-momentum squared $Q^{2}$ and energy transfer $\nu$ strikes the proton, and the cross section is written in terms of structure functions, conventionally $F_{2}(x,Q^{2})$ (historically $\nu W_{2}$) and $F_{1}(x,Q^{2})$, where $x=Q^{2}/(2M\nu)$ is the Bjorken variable and $M$ the proton mass. At large $Q^{2}$ these functions depend approximately only on $x$ ("scaling"), while asymptotic freedom — the decrease of the QCD coupling $\alpha_{s}(Q^{2})$ at large $Q^{2}$ — predicts small logarithmic violations of scaling. The light-cone operator product expansion (OPE) organizes this: moments of the structure functions are sums of matrix elements of local quark and gluon operators of increasing "twist" (dimension minus spin), each multiplied by perturbatively computable coefficient functions. Twist-two operators give the leading logarithmic evolution (via their anomalous dimensions, which govern $Q^{2}$-dependence), while twist-four and higher operators give power corrections suppressed by $M_{0}^{2}/Q^{2}$ for some hadronic scale $M_{0}$, and enhanced by powers of the moment index $n$. Because the proton mass is not negligible at moderate $Q^{2}$, one works with the Nachtmann variable $\xi = 2x/(1+\sqrt{1+4x^{2}M^{2}/Q^{2}})$ (introduced by Nachtmann in 1973), whose moments $\int d\xi\, \xi^{n}F$ project onto operators of definite spin and resum the kinematic target-mass corrections.

Two empirical phenomena in the moderate-$Q^{2}$ data motivated this framework. "Precocious scaling" is the observation that approximate scaling behavior sets in at unexpectedly low $Q^{2}$. Bloom–Gilman duality (quark–hadron duality), observed by Bloom and Gilman in 1970–71, is the fact that the resonance-region structure function, averaged over an appropriate energy interval, approximately equals the smooth deep-inelastic scaling curve at the same $\xi$; it is called "local" duality when the equality holds resonance by resonance rather than only after averaging over the whole resonance region. The ratio $R=\sigma_{L}/\sigma_{T}$ of longitudinal to transverse virtual-photon absorption cross sections isolates the longitudinal structure function $F_{L}$, which leading-twist (twist-two) QCD predicts to be nonzero but small at large $Q^{2}$.

De Rújula, Georgi and Politzer ("DGP", Annals of Physics 103, 315 (1977)) gave this its QCD "demythification": they showed that local duality is a consequence of the OPE moment predictions, fitted the leading-twist evolution (one-loop anomalous dimensions and coefficient functions) to SLAC (Stanford Linear Accelerator Center) electroproduction data for $1\lesssim Q^{2}\lesssim 16$ GeV$^{2}$, used resonance-region moments to extract the higher-twist scale $M_{0}\approx 400$ MeV from terms of the form $(nM_{0}^{2}/Q^{2})^{k}B_{nk}(Q^{2})$, predicted $R=\sigma_{L}/\sigma_{T}$, and derived "threshold duality" relations expressing the elastic electromagnetic form factors $G_{E}$, $G_{M}$ as integrals of the scaling structure function over $\xi$ between the pion threshold and $\xi=1$, from which they extracted the proton magnetic form factor $G_{M}^{p}$ from large-$\xi$ resonance data. The whole construction, however, rested on inputs left undetermined in the analysis — some flagged by the authors themselves, others identified by the subsequent literature: the coefficients $B_{nk}$ were parametrized rather than computed; gluons were neglected below $\xi\lesssim 0.3$ and quark–gluon operator mixing (the mixing, under renormalization, of quark operators with gluon operators of the same spin, which couples the gluon distribution into the moment analysis) was ignored; the perturbative input was truncated at one loop with approximate coefficient functions; the fitted quark distribution $F(\xi)$ was unconstrained at $\xi\gtrsim 0.8$ where no data existed; the $R$ data at large $x$ and $Q^{2}$ sat systematically above the prediction; and the proton matrix elements $A_{n}$ came from fits, not from first principles. The open problem is to remove these crutches — to supply, from first principles or from data independent of the resonance region being tested, every input of the DGP local-duality construction, and to test the resulting parameter-free predictions.

Problem Statement

Complete the quantitative test of electroproduction local duality and precocious scaling formulated by De Rújula, Georgi and Politzer (1977), by replacing every phenomenological input of their analysis with a determination from first principles or from independent data, and confronting the resulting parameter-free predictions with precision measurements. The common scientific objective is a falsifiable, assumption-controlled test of the OPE moment basis of local duality (resonance-region averages equal to leading-twist moments plus controlled higher twist) in the moderate-$Q^{2}$ region; the six components below are the jointly necessary elements of that single test:

1. Existing published determinations of $R=\sigma_{L}/\sigma_{T}$ at large Nachtmann $\xi$ and $Q^{2}$ to test the twist-two longitudinal prediction (the 1977 SLAC data lay systematically above that prediction at $x\geq 0.67$, $Q^{2}\geq 8$ GeV$^{2}$).
2. A quantitative determination of the gluon distribution at small $\xi$ and a controlled inclusion of quark–gluon operator mixing, so the moment analysis extends below $\xi\lesssim 0.3$, where the 1977 analysis neglected gluons by assumption.
3. Higher-order perturbative input — anomalous dimensions, coefficient functions, and the $\beta$ function beyond the one-loop terms used in 1977 — so the theory side is controlled as $\alpha_{s}(Q^{2})$ grows at lower $Q^{2}$.
4. Determination of the twist-four and higher-twist operator matrix elements $B_{nk}(Q^{2})$, including their magnitudes, their $n$ dependence, and the higher-twist scale $M_{0}$ entering the terms $(nM_{0}^{2}/Q^{2})^{k}B_{nk}(Q^{2})$, from a nonperturbative, first-principles determination, so the power-suppressed terms are determined rather than parametrized.
5. Existing published structure-function data at large $\xi$ ($\xi\gtrsim 0.8$) in the scaling region, to test the predicted tail of the scaling function $F(\xi)$ and thereby verify or violate the elastic-form-factor predictions obtained from the threshold moment relations.
6. A nonperturbative, first-principles determination of the proton matrix elements $A_{n}$ and of the quark/gluon operator separation (the decomposition of each twist-two moment into its quark and gluon operator matrix elements, distinct from the gluon distribution and perturbative mixing of component 2), replacing phenomenological fits. The local-duality observables — resonance-region averages of the structure functions, $R(\xi,Q^{2})$ at large $\xi$, and the elastic form factors through the threshold $\xi$-moment relations — are predicted with quantified uncertainties from inputs obtained without using the data being predicted, and are either verified or quantitatively violated by those data. Each determination in components 4 and 6 carries a quantified uncertainty, and the prediction uncertainties are small enough to resolve the higher-twist correction being tested: they do not exceed the magnitude of the $(nM_{0}^{2}/Q^{2})^{k}B_{nk}$ term at the quoted $Q^{2}$.

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