GPD-Based Contributions in Exclusive Reactions

• GPD-based contributions are a QCD framework that encodes the three-dimensional structure of hadrons by linking momentum and spatial distributions of partons.
• They factorize hard exclusive processes into a perturbative kernel and universal GPDs, measurable in reactions like DVCS and vector meson electroproduction.
• This approach complements Regge models by providing model-independent insights for nucleon tomography and guiding future experimental explorations at facilities such as the EIC.

Generalized Parton Distribution (GPD)-based contributions represent foundational mechanisms by which the internal quark-gluon structure of hadrons is encoded in hard exclusive reactions. The GPD framework extends the traditional parton distribution picture by encapsulating both momentum and spatial distributions, as well as correlations between partons, and it provides a rigorous link between inclusive deep inelastic scattering (DIS) and hard exclusive processes such as deeply virtual Compton scattering (DVCS) and exclusive meson electroproduction. In the context of high-energy hadronic and leptonic reactions, GPD-based approaches offer the only model-independent and QCD-consistent method for extracting information about the three-dimensional structure of the nucleon—specifically, via hard exclusive processes in kinematic regimes where factorization is valid and the handbag mechanism dominates.

1. Fundamental Principles of GPD Formalism in Exclusive Reactions

GPDs, typically denoted as $H(x, \xi, t)$, $E(x, \xi, t)$ and their helicity analogues, depend on the average longitudinal momentum fraction $x$, longitudinal momentum transfer (skewness) $\xi$, and overall momentum transfer squared $t$. In the regime of large photon virtuality $Q^2\gg \Lambda_\text{QCD}^2$, factorization theorems guarantee that hard exclusive processes such as $\gamma^*p\rightarrow Vp$ (with $V$ a vector meson) decompose into a perturbatively calculable hard kernel convoluted with universal GPDs and, for meson production, with a universal meson distribution amplitude. This separation is formalized in leading-twist QCD and underpins the theoretical predictive power of GPD-based contributions in hard exclusive electroproduction (Fradi, 2010).

The GPDs encode both the longitudinal parton distributions (reproducing PDFs in the forward limit $t\to 0$, $\xi\to 0$) and form factors (via sum rules over $x$), thus providing simultaneous access to both the spatial and momentum structure of hadrons. In exclusive vector meson electroproduction at high $Q^2$ and small $|t|$, the amplitude is dominated by the so-called “handbag” diagrams, where the virtual photon scatters off a single quark, which then re-hadronizes into the observed vector meson.

2. Kinematic Domains and Factorization Applicability

GPD-based contributions are controlled by kinematic constraints dictated by perturbative factorization. The theoretical validity of the factorization is secured in regimes where the photon virtuality $Q^2$ is large enough that the short-distance dynamics decouple from the long-distance, process-independent structure encoded in the GPDs, and the squared momentum transfer $|t|$ remains small enough to avoid the onset of Regge and non-handbag mechanisms (Fradi, 2010).

In exclusive vector meson production, this typically requires $Q^2\gtrsim~\mathrm{few}$ GeV$^2$ and $-t\ll Q^2$. In these regimes, the cross section is dominated by the longitudinally polarized photon contribution (leading twist), and the process is sensitive to the quark GPDs for light vector mesons ($\rho^0, \omega$) and to the gluon GPDs for heavy mesons (e.g., $J/\psi$) due to the dominance of gluon exchange at small $x$.

3. GPDs in Data Interpretation and Theoretical Models

Direct extraction of GPDs from experimental cross sections is non-trivial due to the convolution structure of the amplitude. The leading-twist QCD amplitude for exclusive meson electroproduction involves an integral over $x$ of the product of GPDs and hard-scattering kernels. Contemporary analyses perform global fits to exclusive production data using parameterizations of GPDs constrained by sum rules (e.g., reproducing the known electromagnetic form factors) and consistency with the inclusive PDF limit (Fradi, 2010).

In experimental studies such as those performed by CLAS, cross sections for exclusive vector meson production are interpreted through both GPD-based frameworks and alternative hadronic Regge models (e.g., the JML model). Whereas the Regge approach phenomenologically sums over exchanges of hadronic trajectories in the $t$ channel, the GPD formalism provides an essential link to the underlying partonic dynamics. The actual CLAS data (Fradi, 2010) overlay predictions both from the full Regge model and from GPD-based calculations, demonstrating that for $Q^2\gtrsim2$–$4.5$ GeV$^2$ the latter provides a successful description of the $Q^2$ and $t$ dependence—indicative of GPD dominance in this regime.

4. Limitations, Benchmarking, and Complementarity with Regge Models

While the GPD-based formalism is theoretically robust at high $Q^2$, in practice, experimental data often straddle a transition regime where non-perturbative hadronic contributions (Regge exchanges) compete with, or even dominate over, the partonic (GPD) mechanisms, particularly at low and moderate $Q^2$ or higher $|t|$. The CLAS proceedings explicitly state that the Reggeon exchange (JML) model still provides a globally consistent fit to the data across a wide range of $Q^2$ and $-t$ (Fradi, 2010).

Crucially, the GPD formalism does not specify the details of the $t$-channel exchange trajectories, which are the essence of Regge models. Benchmarks of GPD-based predictions against measured cross sections and their $Q^2, W, t$ dependences serve as a vital test of the scale at which the partonic picture becomes applicable. The CLAS data show that at photon virtualities up to $Q^2\sim4.5$ GeV$^2$, the Reggeon-exchange models continue to fit the overall kinematic dependences, suggesting that global GPD dominance sets in only gradually (Fradi, 2010).

5. Practical Extraction and Global Constraints

Current methodologies for extracting GPDs from experimental observables rely on global fits to exclusive cross-section data, typically using parameterized functional forms for GPDs that are constrained by PDFs, form factors, polynomiality, and positivity bounds. Models such as the double-distribution Ansatz, Regge-inspired parameterizations, and flexible functional fits are tuned to reproduce the $Q^2$ and $t$ dependence of vector meson production cross sections measured at experiments like CLAS (Fradi, 2010).

However, the inadequacy of available cross-section measurements alone for uniquely determining the full set of GPDs is a structural limitation. More exclusive observables—such as spin or polarization asymmetries, higher harmonics in azimuthal distributions, and processes with different final-state mesons—are necessary for comprehensive global fits.

6. Connection to Nucleon Tomography and Future Prospects

GPD-based contributions play a unique role in the program of nucleon tomography—the three-dimensional imaging of the nucleon in both momentum and position space. The Fourier transform of the $t$-dependence of GPDs encodes the transverse spatial distribution of partons as a function of their longitudinal momentum fraction, yielding a multidimensional picture of the nucleon's internal structure. The programmatic extraction and interpretation of GPDs from exclusive vector meson and deeply virtual Compton scattering processes remain a central goal of forthcoming electron–ion collider (EIC) experiments.

Systematic analyses in the GPD framework will ultimately provide not only robust determinations of parton distributions and form factors but also the decomposition of nucleon and nuclear spin, transverse parton imaging, and access to higher-order correlations.

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Table: Summary of GPD-based vs. Regge-based Contributions in Exclusive Vector Meson Electroproduction (Fradi, 2010)

Approach	Central Assumptions	Kinematic Dominance
GPD-based	Handbag dominance, QCD factorization, large $Q^2$	High $Q^2$, small $|t|$
Regge-based (JML)	t-channel meson, pomeron exchanges (phenomenological trajectories)	Low/moderate $Q^2$, larger $|t|$

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The general consensus, supported by CLAS results, is that GPD-based contributions dominate exclusive vector meson electroproduction at high $Q^2$ and low $|t|$, while Regge-based hadronic exchanges persist at lower $Q^2$ or larger $|t|$. The smooth transition observed in data underlines the necessity to benchmark both approaches and to interpret experimental measurements through the interplay of GPD and Regge phenomenology (Fradi, 2010).