[Paper Review] Probing non-perturbative QCD through hadronic matrix elements extracted from exclusive hard processes

B. Pire, L. Szymanowski|ArXiv.org|Sep 1, 2009

Quantum Chromodynamics and Particle Interactions10 references3 citations

TL;DR

This paper investigates non-perturbative QCD through hadronic matrix elements extracted from exclusive hard processes, focusing on generalized parton distributions (GPDs), distribution amplitudes (DAs), and transition distribution amplitudes (TDAs). It highlights challenges in computing these quantities using lattice QCD and QCD sum rules, and critically evaluates the AdS/QCD approach, showing it fails to naturally reproduce key partonic features like Callan-Gross scaling and fermionic structure without fine-tuning.

ABSTRACT

QCD is the theory of strong interactions and non-perturbative methods have been developed to address the confinement property of QCD. Many experimental measurements probe the confining dynamics, and it is well-known that hard scattering processes allow the extraction of non perturbative hadronic matrix elements. To study exclusive hard processes, such as electromagnetic form factors and reactions like gamma* N -> gamma N', gamma* N -> pi N', gamma* gamma -> pi pi, antiproton proton ->gamma* pi in particular kinematics (named as generalized Bjorken regime), one introduces specific non-perturbative objects, namely generalized parton distributions (GPDs), distribution amplitudes (DA) and transition distribution amplitudes (TDA), which are Fourier transformed non-diagonal matrix elements of non-local operators on the light-cone. We review here a selected sample of exclusive amplitudes in which the quark and gluon content of hadrons is probed, and emphasize that much remains to be done to successfully compute their non-perturbative parts. We present some difficulties with respect to the application of the much publicized AdS-QCD approach to the calculation of these partonic quantities.

Motivation & Objective

• To extract and analyze non-perturbative hadronic matrix elements from exclusive hard scattering processes such as $̳^*N\to\gamma N'$ and $̳^*\gamma\to\pi\pi$.
• To assess the reliability and consistency of theoretical methods—lattice QCD and QCD sum rules—for computing proton and meson distribution amplitudes (DAs), especially given large discrepancies in results.
• To examine the applicability of the AdS/QCD correspondence to reproduce fundamental partonic features in deep inelastic scattering, such as scaling behavior and polarization dependence.
• To identify open theoretical challenges in computing transition distribution amplitudes (TDAs) for processes like $p\to\pi N'$, particularly the $t$-dependence and $ξ\to0$ limit.

Proposed method

• Uses factorization theorems in exclusive hard processes to extract non-perturbative matrix elements of non-local light-cone operators, such as $\bar{\psi}(z)\psi(0)$ and $\psi(z)\psi(z')\psi(0)$.
• Applies Fourier transforms to map position-space matrix elements to momentum-space distribution amplitudes (DAs), including twist-3 DAs for the proton and exotic hybrid mesons.
• Employs QCD sum rules and lattice QCD techniques to estimate normalization and functional forms of DAs, comparing results across methods.
• Analyzes the AdS/QCD correspondence by computing virtual photon scattering amplitudes on scalar and vector hadronic states to test scaling and polarization structure.
• Derives conditions under which the AdS/QCD model reproduces Callan-Gross scaling and fermionic behavior, requiring $\Delta_0 = 1$ for the initial state.
• Evaluates the role of conformal dimensions and the need for fine-tuning in AdS/QCD to match partonic features, highlighting non-predictivity.

Experimental results

Research questions

• RQ1 Can exclusive hard processes reliably extract non-perturbative hadronic matrix elements such as GPDs and DAs?
• RQ2 Why do lattice QCD and QCD sum rule results for proton DAs show large discrepancies, and how can theoretical uncertainties be consistently estimated?
• RQ3 Can the AdS/QCD correspondence reproduce the scaling behavior and polarization structure of deep inelastic scattering without fine-tuning?
• RQ4 What are the functional forms and symmetries of distribution amplitudes for exotic hybrid mesons with $J^{PC}=1^{-+}$?
• RQ5 How does the $t$-dependence of transition distribution amplitudes (TDAs) in $p\to\pi N'$ processes relate to the impact parameter structure of the pion cloud in the nucleon?

Key findings

• Lattice QCD and QCD sum rule estimates for the first moments of proton distribution amplitudes differ significantly: (0.40, 0.30, 0.30) on the lattice versus (0.56, 0.19, 0.23) from sum rules, with inconsistent theoretical uncertainties.
• The leading twist distribution amplitude for the $J^{PC}=1^{-+}$ hybrid meson is $\phi_1^H(u) = 30u(1-u)(1-2u)$, with $f_H \approx 50$ MeV from QCD sum rules, though no lattice result exists.
• The AdS/QCD model fails to reproduce the Callan-Gross relation without requiring $\Delta_0 = 1$, which violates standard constraints for electromagnetic form factors and lies at the unitarity bound.
• The scaling behavior of the structure functions in DIS is independent of the vector meson conformal dimension $\Delta_V$, depending only on $\Delta_0 = 1$ for the scalar initial state.
• The partonic picture in AdS/QCD requires adding scalar-to-scalar and scalar-to-vector amplitudes, leading to an infinite number of free parameters and breaking predictivity.
• No lattice calculations have yet determined the first $x_i$-moments of transition distribution amplitudes (TDAs) for $p\to\pi N'$, and the $\xi\to0$ limit remains unconstrained.