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 αsπ2\alpha_s\pi^2 and αsln⁡2n\alpha_s\ln^2 n in the nn-th Mellin moment, including the running of the coupling - and stated them to be valid in the window Q2≫Λ2n2Q^2\gg\Lambda^2 n^2, while acknowledging that the π2\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-NN singularities and threshold logarithms was proven (Ciafaloni-Curci; Sterman; Catani-Trentadue), the timelike-spacelike π2\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 xx: the all-order subleading-logarithmic and renormalon structure of the Sudakov exponent, the leading Λ2/W2\Lambda^2/W^2-enhanced higher-twist contributions and their matrix elements, the predicted location of the twist-two/higher-twist crossover as W2=Q2(1−x)/xW^2=Q^2(1-x)/x approaches Λ2\Lambda^2, and its confrontation with resonance-region and fixed-W2W^2 data.

Background

Perturbative quantum chromodynamics (QCD) computes hard scattering as an expansion in the running coupling αs(Q2)\alpha_s(Q^2), which decreases logarithmically with the momentum transfer Q2Q^2 and becomes strong at a scale Λ∼200\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 F2(x,Q2)F_2(x,Q^2) as functions of the Bjorken variable x=Q2/(2p⋅q)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 QQ. Collinear factorization expresses both cross sections as convolutions of perturbatively calculable coefficient functions with universal parton distribution functions (PDFs). Near the kinematical boundary - x→1x\to1 in DIS, or partonic threshold τ=Q2/s→1\tau=Q^2/s\to1 in Drell-Yan - the coefficient functions receive corrections that are large even when αs\alpha_s is small, because soft (low-energy) and collinear gluon emission is forced by phase space. In the nn-th Mellin moment (moments of F2F_2 or of the cross section in xx or τ\tau, with ln⁡n∼ln⁡1/(1−x)\ln n\sim\ln 1/(1-x)) these corrections appear as powers of αsln⁡2n\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 π2\pi^2 terms. At fixed-target energies the formal O(αs)\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 Cdis(q2,n)C_{\mathrm{dis}}(q^2,n) and CDY(Q2,n)C_{\mathrm{DY}}(Q^2,n), which resum the dominant αsln⁡2n\alpha_s\ln^2 n terms including the running of the coupling, and for the Drell-Yan correction factor R(α,n)R(\alpha,n), given by the product of the squared ratio of timelike to spacelike form factors - which exponentiates the leading αsπ2\alpha_s\pi^2 effect - and the ratio CDY/CdisC_{\mathrm{DY}}/C_{\mathrm{dis}}. The note stated the domain of validity as the window Q2≫Λ2n2Q^2\gg\Lambda^2 n^2, neglected the π2\pi^2 pieces inside the CC-factors, treated only quark channels (initial-state gluons neglected), and acknowledged that the π2\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-NN singularities [αs(Q2)ln⁡N]p[\alpha_s(Q^2)\ln N]^p (with NN the Mellin moment, our nn) 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 π2\pi^2 exponentiation on firm ground; Eynck, Laenen and Magnea extended the exponentiation to all threshold constants in the DIS and MS‾\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 Λ2/W2\Lambda^2/W^2 rather than Λ2/Q2\Lambda^2/Q^2, where W2=Q2(1−x)/xW^2=Q^2(1-x)/x is, up to the target mass, the invariant mass of the final-state hadronic system. As x→1x\to1 at fixed WW, 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 nn or insufficient Q2Q^2, a twist-dominance boundary distinct from the note's perturbative-validity window Q2≫Λ2n2Q^2\gg\Lambda^2 n^2. Gardi, Korchemsky, Ross and Tafat described the 'dramatic breakdown' of the OPE at large xx caused by the narrow final-state jet of invariant mass WW, 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 F2F_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 Q2/N∼Λ2Q^2/N\sim\Lambda^2, could not control the renormalon ambiguities (infrared ambiguities of the asymptotic perturbative series, of order powers of Λ/W\Lambda/W, which must cancel against higher twist) beyond the large-β0\beta_0 limit (β0\beta_0 is the first coefficient of the beta function governing the running of αs\alpha_s; the large-β0\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-xx, low-Q2Q^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 (Q2,x)(Q^2,x) plane (equivalently (Q2,n)(Q^2,n) with nn the Mellin moment, or (Q2,W2)(Q^2,W^2) with W2=Q2(1−x)/xW^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 Q2≫Λ2n2Q^2\gg\Lambda^2 n^2. The constructive parts of that conjecture - all-orders exponentiation of the αsln⁡2n\alpha_s\ln^2 n terms with running coupling, the exponentiation of the π2\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 W2∼Λ2W^2\sim\Lambda^2.
  2. The leading higher-twist contributions, parametrically of order Λ2/W2\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-W2W^2 analyses) that tests the predicted crossover and decides whether resummed twist-two plus explicitly computed power corrections suffices there.

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