Completing and validating the first-principles asymptotic QCD prediction of exclusive processes (elastic form factors, threshold structure functions, two-cluster electron-positron production, fixed-angle scattering): nonperturbative inputs, higher-order coefficient functions, Landshoff suppression, higher-twist onset, and confrontation with data

In 1979 Parisi proposed that the asymptotic behavior of exclusive processes at large momentum transfer in QCD — elastic form factors, the threshold behavior of structure functions at fixed final-state mass, production of two low-mass clusters in $e^{+}e^{-}$ annihilation, and fixed-angle elastic scattering — is computable from an operator expansion for the exclusive amplitude itself: operators of minimal twist and anomalous dimension dominate, the coefficient functions are perturbatively calculable, and only a few nonperturbative constants (such as the pion decay constant $f_\pi$ and the proton matrix element $f_{\mathrm{p}}=\langle0|O_{3}|\mathrm{p}\rangle$ of a three-quark operator) enter as inputs. The program left open the gaps that define this problem: the conjectural gauge-invariant extension of the expansion in a gauge theory; the undetermined nonperturbative constants; the suppression of independent-parton (Landshoff) scattering by soft-gluon radiation, established at first only qualitatively; the possible dominance of twist-4 terms in fixed-mass limits; and the lack of experimental verification of the predicted powers, logarithms, and absolute normalizations. The follow-up literature settled the structural conjectures — factorization into distribution amplitudes with ERBL evolution (Efremov-Radyushkin; Lepage-Brodsky) and resummed Sudakov suppression of Landshoff terms (Botts-Sterman; Li-Sterman) — and lattice QCD is converging on the pion-sector inputs, with the full two-loop pion form factor prediction completed only in 2024-2025 and the two-loop nucleon distribution-amplitude evolution only in 2025. What remains open, in the language of these later developments, is the completion and validation of the program: first-principles baryon-sector inputs, next-to-next-to-leading-order predictions for baryon form factors, a quantitative crossover scale for Landshoff suppression, a first-principles delineation of the leading-twist domain against twist-4 contributions, and experimental tests at momentum transfers large enough to probe the asymptotic regime.

Background

Exclusive processes are reactions in which the quantum numbers and momenta of all final-state particles are fixed — elastic and transition form factors (the amplitudes for a hadron to absorb momentum transfer $q$, with $q^{2}=-Q^{2}$ spacelike, without breaking up) and fixed-angle elastic scattering between two hadrons — in contrast to inclusive observables, in which unobserved final states are summed over. Because the final state is specified, the amplitude is a coherent sum over the valence Fock states (the quark and gluon configurations of the hadron), which makes exclusive amplitudes among the most demanding tests of quantum chromodynamics (QCD), the theory of the strong interaction. The physical hope is asymptotic freedom: the QCD running coupling $\alpha_s(Q^{2})$ decreases only logarithmically with the momentum scale, so at large momentum transfer the short-distance parts of the amplitude should become perturbatively computable, leaving long-distance matrix elements as the only inputs.

Three prior results framed the problem. First, dimensional counting rules — power-law falloffs obtained by counting the elementary fields in each external hadron, such as $F_\pi\sim 1/Q^{2}$ for the pion form factor, $F_1^{\mathrm{p}}\sim 1/Q^{4}$ for the leading proton form factor, and $\mathrm{d}\sigma/\mathrm{d}t\sim|t|^{-10}$ for fixed-angle proton-proton scattering at momentum transfer $t$ — predict the powers but neither the logarithmic corrections nor the absolute normalizations. Second, for inclusive deep-inelastic scattering the light-cone operator product expansion (OPE) organizes amplitudes in local operators of increasing twist (twist = mass dimension minus spin; each unit of twist above the minimum costs a power of $1/Q$) with anomalous dimensions $\gamma$ (the exponents governing the logarithmic scaling of renormalized matrix elements); the same analysis implies that soft-gluon radiation gives a fast colored particle an elastic form factor suppressed by a double logarithm, $\sim\exp[-(2C_F/\beta_0)\ln Q^{2}\ln\ln Q^{2}]$ with $C_F$ the quark Casimir and $\beta_0$ the first coefficient of the beta function — the Sudakov effect. Third, a dynamical competitor exists: at Born level, large-angle hadron-hadron scattering receives not only the single hard scattering in which all valence partons participate in one short-distance collision (the counting-rule picture) but also the Landshoff mechanism, in which spatially separated valence quarks scatter independently off the same target; independent scattering has a milder power falloff (for example $|t|^{-8}$ rather than $|t|^{-10}$ in $pp$), so the counting-rule prediction can dominate only if QCD dynamics suppresses it.

In this setting Parisi proposed in 1979 that the asymptotic behavior of exclusive processes — form factors, the threshold behavior of structure functions (the inclusive deep-inelastic response) at fixed final-state invariant mass $W$, the production of two low-mass clusters (low-mass jet pairs) in $e^{+}e^{-}$ annihilation, and elastic scattering — follows from an operator expansion for the exclusive amplitude itself; schematically, in the language of the follow-up literature,

$$J(p-p')\;\xrightarrow[q^{2}\to\infty]{}\;\sum_{a,b}C_{a,b}\bigl[\alpha(q^{2})\bigr]\,\left(1/q^{2}\right)^{\frac{\tau_{a}+\tau_{b}}{2}-1}\,(\ln q^{2})^{\gamma_{a}+\gamma_{b}}\,O_{a}(p)\,|0\rangle\langle0|\,O_{b}(p'),$$

where $O_{a}$ are local interpolating operators of twist $\tau_{a}$ and anomalous dimension $\gamma_{a}$ and the coefficient functions $C_{a,b}$ are perturbatively calculable. Minimal twist then dominates, and only a few nonperturbative constants enter: the pion form factor becomes proportional to $\alpha_s(Q^{2})f_\pi^{2}/Q^{2}$, where $f_\pi$ is the pion decay constant measured in pion decay; the proton form factor is built from a three-quark operator $O_{3}$ with normalization $f_{\mathrm{p}}=\langle0|O_{3}|\mathrm{p}\rangle$ and anomalous dimension $\gamma_{O_3}$; analogous formulas, each with its own few nonperturbative constants, were given for threshold structure functions and two-cluster $e^{+}e^{-}$ production, and for fixed-angle $pp$ scattering. Parisi's derivation was for renormalizable theories without gauge invariance, and he conjectured its gauge-invariant extension in QCD through a limiting anomalous dimension $\gamma_{\mathrm{p}}^{\mathrm{GI}}=\lim_{n\to\infty}\gamma_{n}$ built from the twist-2 spin-$n$ anomalous dimensions, whose observed growth $\gamma_{n}\sim\ln n$ implies a divergent, Sudakov-like suppression of the isolated-parton elastic form factor $F_{\mathrm{q}}$; on this basis he argued that Landshoff contributions are suppressed by powers of the Sudakov-suppressed single-parton form factor $F_{\mathrm{q}}(t)$, and noted that twist-4 (next-to-leading-twist, suppressed by $1/Q^{2}$ relative to twist-2) contributions may dominate some fixed-mass limits, so the range of $Q^{2}$ in which the leading-twist formulas hold must be delineated. The nonperturbative constants were left undetermined, and experimental verification of the absolute predictions was posed as part of the program.

The modern framework that grew out of this proposal factorizes exclusive amplitudes into hadron distribution amplitudes — the light-cone wave functions of the valence Fock state, defined as gauge-invariant matrix elements of nonlocal light-ray quark operators closed by Wilson lines (path-ordered gauge parallel transporters) — convoluted with perturbatively calculable hard-scattering kernels; the distribution amplitudes evolve in $Q^{2}$ by the Efremov-Radyushkin-Brodsky-Lepage (ERBL) equation, whose eigenvalues are the anomalous dimensions of the same local operators. In this language the problem is the completion and validation of the 1979 program.

Problem Statement

As posed by Parisi's 1979 program, the problem is to make the asymptotic behavior of exclusive processes at large momentum transfer in QCD fully computable and validated. For (a) elastic and transition form factors, (b) the threshold behavior of structure functions at fixed final-state invariant mass $W^{2}$, (c) the production of two low-mass clusters in $e^{+}e^{-}$ annihilation, and (d) fixed-angle elastic scattering, determine from first principles — via the operator expansion in minimal-twist operators with perturbatively calculable coefficient functions, together with the Sudakov suppression of independent-parton (Landshoff) scattering — the complete asymptotic formulas: the power laws, the logarithmic factors generated by anomalous dimensions, the angular dependences, and the absolute normalizations expressed through the few nonperturbative constants of the theory (such as the pion decay constant $f_\pi$ and proton matrix elements of three-quark operators); and establish (i) the quantitative domain of validity of these formulas — the onset of leading-twist dominance over twist-4 contributions in fixed-mass limits, and the momentum-transfer scale at which Sudakov suppression renders Landshoff contributions negligible — and (ii) their consistency with experiment at the largest accessible momentum transfers. The process classes (a)-(d) and the goals (i)-(ii) form one coherent program rather than independent tasks: the settled factorization framework is the language in which the nonperturbative inputs are defined, the validity-domain questions (i) delimit where the asymptotic formulas apply, and the experimental test (ii) checks the resulting absolute predictions. Within this program the established results — the factorization of exclusive amplitudes into distribution amplitudes with ERBL evolution, the two-loop pion electromagnetic form factor, and the Sudakov suppression of the Landshoff terms — enter as known inputs that are not themselves to be re-derived, and the numerical conditions below are evaluated at the ten-percent level: a subleading contribution is treated as negligible once it falls below ten percent of the corresponding leading term. The quantities this program is to determine, all jointly required, are the baryon-sector contributions — the first-principles baryon-sector inputs and the next-to-next-to-leading-order prediction for the baryon form factors, which give the nucleon elastic and transition form factors (the baryonic part of (a)) and the baryon normalization of the fixed-angle scattering in (d) — together with the two crossover scales of (i) at that level (for the required observables: the momentum transfers at which the twist-4 contribution to the nucleon electromagnetic form factor falls below ten percent of its twist-2 term, and at which the Landshoff contribution to fixed-angle nucleon scattering falls below ten percent of its single-hard-scattering term; the fixed-mass threshold-structure-function twist-4 onset named in (i) is supplied by the established framework and is not itself a required computation) and, as a distinct experimental-reach requirement beyond those scales, the confrontation of the absolute predictions with experiment at the largest accessible momentum transfers together with a forecast of the momentum transfers at which a decisive asymptotic test — a measurement in the regime where the leading-twist formulas hold at this ten-percent level — would become possible. The mesonic parts of (a) (the pion electromagnetic form factor), the threshold structure functions (b), the two-cluster $e^{+}e^{-}$ production (c), and the Sudakov suppression of the Landshoff terms in (d) rest on the established framework and enter as known inputs rather than as separate required formula derivations.

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