Domain of validity of threshold-resummed leading-twist QCD at large x: the crossover to higher-twist dominance in deep-inelastic scattering and Drell-Yan

Parisi's two-page 1980 note conjectured all-orders resummed formulae for the large perturbative corrections that dominate deep-inelastic scattering (DIS) and Drell-Yan moments near the kinematical boundary - corrections of order $\alpha_s\pi^2$ and $\alpha_s\ln^2 n$ in the $n$-th Mellin moment, including the running of the coupling - and stated them to be valid in the window $Q^2\gg\Lambda^2 n^2$, while acknowledging that the $\pi^2$ exponentiation was uncertain, initial-state gluon channels were neglected, and the boundary where higher-twist contributions overtake the Sudakov-suppressed leading twist was undetermined. The later literature resolved the constructive parts: exponentiation of large-$N$ singularities and threshold logarithms was proven (Ciafaloni-Curci; Sterman; Catani-Trentadue), the timelike-spacelike $\pi^2$ exponentiation was derived and verified at two loops (Magnea-Sterman; Eynck-Laenen-Magnea), the resummed Drell-Yan K factors passed two-loop and fixed-target phenomenology, and the gluon channels were resummed at next-to-leading power. The surviving open core is the domain of validity of the resummed leading-twist description at large $x$: the all-order subleading-logarithmic and renormalon structure of the Sudakov exponent, the leading $\Lambda^2/W^2$-enhanced higher-twist contributions and their matrix elements, the predicted location of the twist-two/higher-twist crossover as $W^2=Q^2(1-x)/x$ approaches $\Lambda^2$, and its confrontation with resonance-region and fixed-$W^2$ data.

Background

Perturbative quantum chromodynamics (QCD) computes hard scattering as an expansion in the running coupling $\alpha_s(Q^2)$, which decreases logarithmically with the momentum transfer $Q^2$ and becomes strong at a scale $\Lambda\sim 200$ MeV. Two benchmark processes are deep-inelastic scattering (DIS), in which a lepton scatters off a hadron and probes its structure functions $F_2(x,Q^2)$ as functions of the Bjorken variable $x=Q^2/(2p\cdot q)$, and the Drell-Yan process, in which a quark-antiquark pair from two colliding hadrons annihilates into a lepton pair of invariant mass $Q$. Collinear factorization expresses both cross sections as convolutions of perturbatively calculable coefficient functions with universal parton distribution functions (PDFs). Near the kinematical boundary - $x\to1$ in DIS, or partonic threshold $\tau=Q^2/s\to1$ in Drell-Yan - the coefficient functions receive corrections that are large even when $\alpha_s$ is small, because soft (low-energy) and collinear gluon emission is forced by phase space. In the $n$-th Mellin moment (moments of $F_2$ or of the cross section in $x$ or $\tau$, with $\ln n\sim\ln 1/(1-x)$) these corrections appear as powers of $\alpha_s\ln^2 n$, and for Drell-Yan the analytic continuation from the spacelike momentum of DIS to the timelike momentum of the annihilation converts the double logarithms of the quark Sudakov form factor (the elastic quark-photon vertex, whose large-momentum behavior is doubly logarithmic and exponentiated) into large $\pi^2$ terms. At fixed-target energies the formal $\mathcal{O}(\alpha_s)$ correction to the Drell-Yan K factor (the ratio of the full cross section to its naive parton-model value) is of order unity, so fixed-order perturbation theory loses predictive power. Parisi's 1980 note conjectured, by heuristic 'Russian school' soft-gluon methods, closed exponential formulae for the moment-space correction factors $C_{\mathrm{dis}}(q^2,n)$ and $C_{\mathrm{DY}}(Q^2,n)$, which resum the dominant $\alpha_s\ln^2 n$ terms including the running of the coupling, and for the Drell-Yan correction factor $R(\alpha,n)$, given by the product of the squared ratio of timelike to spacelike form factors - which exponentiates the leading $\alpha_s\pi^2$ effect - and the ratio $C_{\mathrm{DY}}/C_{\mathrm{dis}}$. The note stated the domain of validity as the window $Q^2\gg\Lambda^2 n^2$, neglected the $\pi^2$ pieces inside the $C$-factors, treated only quark channels (initial-state gluons neglected), and acknowledged that the $\pi^2$ exponentiation beyond leading order was uncertain, that the derivations were heuristic rather than rigorous, and that no numerical tests existed. The constructive parts of this program are now theorem-level. Ciafaloni and Curci proved that the large-$N$ singularities $[\alpha_s(Q^2)\ln N]^p$ (with $N$ the Mellin moment, our $n$) are calculable to all orders by a change of scale in the running coupling; Sterman and Catani-Trentadue derived all-orders threshold exponentiation from factorization; Magnea and Sterman constructed the exponentiated, infrared-finite ratio of timelike to spacelike Sudakov form factors, putting the $\pi^2$ exponentiation on firm ground; Eynck, Laenen and Magnea extended the exponentiation to all threshold constants in the DIS and $\overline{\mathrm{MS}}$ (modified minimal-subtraction) factorization schemes and verified it against exact two-loop results; Magnea (1991) and Rijken and van Neerven (1995) confirmed the resummed Drell-Yan K factors against two-loop and fixed-target phenomenology; and the gluon-initiated channels, which enter only at next-to-leading power (NLP) in the threshold expansion, have had their leading logarithms resummed as well. What the modern literature has not settled is exactly the limitation that bounds the note's validity window. Twist is the dimension-minus-spin classification of operators in the operator-product expansion (OPE): the leading-twist (twist-two) terms give the factorized, PDF-based description, while higher-twist (twist-four, ...) terms are suppressed by powers of $\Lambda^2/W^2$ rather than $\Lambda^2/Q^2$, where $W^2=Q^2(1-x)/x$ is, up to the target mass, the invariant mass of the final-state hadronic system. As $x\to1$ at fixed $W$, Sudakov suppression makes the twist-two contribution fall faster than the inverse-power behavior of higher-twist matrix elements, so higher twist must dominate for sufficiently large $n$ or insufficient $Q^2$, a twist-dominance boundary distinct from the note's perturbative-validity window $Q^2\gg\Lambda^2 n^2$. Gardi, Korchemsky, Ross and Tafat described the 'dramatic breakdown' of the OPE at large $x$ caused by the narrow final-state jet of invariant mass $W$, and conjectured that the dominant higher twist there is the mixing of twist-four operators with the leading twist. Gardi and Roberts showed that the leading-twist description of $F_2$ moments resummed to next-to-next-to-leading-logarithmic accuracy (NNLL) is consistent with existing data, with no experimental evidence for higher twist beyond it - but they explicitly excluded the region $Q^2/N\sim\Lambda^2$, could not control the renormalon ambiguities (infrared ambiguities of the asymptotic perturbative series, of order powers of $\Lambda/W$, which must cancel against higher twist) beyond the large-$\beta_0$ limit ($\beta_0$ is the first coefficient of the beta function governing the running of $\alpha_s$; the large-$\beta_0$ limit retains only this term), and lacked subleading logarithms beyond NNLL. Global PDF analyses, such as the CTEQ-Jefferson Lab fit of Accardi et al. and its most recent successor analysis of higher-twist uncertainties (Cerutti et al. 2025), still parametrize higher twist, target-mass and nuclear corrections phenomenologically in order to use large-$x$, low-$Q^2$ data. The quantitative location of the twist-two/higher-twist crossover therefore remains undetermined from first principles.

Problem Statement

Determine, quantitatively and within QCD, the domain in the $(Q^2,x)$ plane (equivalently $(Q^2,n)$ with $n$ the Mellin moment, or $(Q^2,W^2)$ with $W^2=Q^2(1-x)/x$) within which the threshold-resummed leading-twist description of deep-inelastic structure functions and Drell-Yan moments is reliable, and characterize the transition through which that description fails. The problem originates in Parisi's 1980 resummation conjecture, whose stated validity window is $Q^2\gg\Lambda^2 n^2$. The constructive parts of that conjecture - all-orders exponentiation of the $\alpha_s\ln^2 n$ terms with running coupling, the exponentiation of the $\pi^2$/timelike-spacelike effect, and the next-to-leading-power gluon channels - have been derived and verified in the later literature and are not part of this question. The open question, with the crossover location of item 3 as its central target and items 1, 2 and 4 supplying the analyses it rests on, is:

1. The all-order structure of the resummed Sudakov exponent beyond the logarithmic accuracy currently matched to fixed-order calculations (NNLL and above), together with its renormalon ambiguities on the approach to $W^2\sim\Lambda^2$.
2. The leading higher-twist contributions, parametrically of order $\Lambda^2/W^2$, in particular the dominant twist-four matrix elements and their mixing with the leading twist.
3. The resulting quantitative location of the crossover at which higher twist overtakes the Sudakov-suppressed twist-two term, predicted within QCD from the theory of items 1 and 2, with any phenomenological or non-perturbative input named and propagated into the stated uncertainty and with the crossover location itself not obtained by fitting it to data.
4. Confrontation with data in and near the transition region (resonance-region and fixed-$W^2$ analyses) that tests the predicted crossover and decides whether resummed twist-two plus explicitly computed power corrections suffices there.

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