Probing Composite Structure and Spin-Orbit Coupling with GPDs in ${}^{4}$He

TL;DR

We develop a first-principles framework to express the GPDs of a spin-0 composite target in terms of the GPDs of spin-1/2 constituents, weighted by boost-invariant light-front Wigner distributions. The theory combines momentum and spin decompositions, a rest-frame dictionary, and symmetry constraints to produce a master integral for unpolarized quark GPDs that reveals a new $ΔL·S$ spin-orbit coupling contributing to GPD mixing. The phenomenology on a simple ${}^{4}$He model shows how orbital motion and spin–orbit effects smear and modulate constituent GPDs, offering potential experimental signatures in DVCS on light nuclei and a data-to-AI training framework for future analyses. The approach generalizes to GTMDs and supports AI-driven interpretation for upcoming EIC and ALERT measurements.

Abstract

In this work, we derive a relation between the generalized parton distributions (GPDs) of a spin-0 composite hadron and the GPDs of its spin-$\tfrac{1}{2}$ constituents. The method decomposes matrix elements of the composite parent into matrix elements of its constituents, weighted by the Wigner distributions of the constituents inside the composite parent. Exploiting the boost invariance of the light-front Wigner distributions, we derive the dictionary between rest-frame and boosted-frame kinematics and use it to perform a Pauli matrix decomposition of the spin degree of freedom. The constraints of unbroken rotational, parity, and time-reversal invariance lead to a master integral for the quark GPDs of the spin-0 composite target. We also identify a novel $Δ\vec{L}\cdot\vec{S}$ form of spin-orbit coupling responsible for GPD mixing in composite hadrons which is absent in the case of transverse momentum dependent parton distributions (TMDs) of composite hadrons. We then apply the framework to a composite ${}^{4}$He target with simple phenomenological models to identify possible experimental signatures of composite-structure effects in light nuclei. The framework we have constructed here can be readily extended to the case of generalized TMDs (GTMDs), while the phenomenological framework is applicable both to analyses of light nucleus data and as training input for AI-assisted applications.

Figures (5)

• Figure 1: Illustration of the decomposition \ref{['eq:matrix-decomp']}. The overall quark GPD of the scalar composite target is decomposed into four distinct components: the wave functions $\Psi^*, \Psi$ of the constituents, the scattering amplitude $\mathcal{S}$ of the spectators, and the quark matrix element of the active (connected) constituents. In the mean-field approximation, only one active constituent is considered at a time.
• Figure 2: Image (a) and (b) display the distribution of GPD $H_q$ and $E_q$, respectively, from the quark-target model defined by (\ref{['eq:quark-target-model']}) for $0<x<1$.
• Figure 3: Image (a) and (b) display the smearing of GPDs $H_q$ and $E_q$ found in $\mathcal{I}$ corresponding to (\ref{['eq:Gpd-unp']}) and (\ref{['eq:Gpd-pol']}), respectively. The red contour represents the maximum value of $x$ for a given $\vec{\Delta}_{\perp}$ value and the distribution goes to zero. The blue is for the critical value of $x$, where the distribution begins to go to zero.
• Figure 4: The angular dependence of (\ref{['eq:Gpd-unp']}) and (\ref{['eq:Gpd-pol']}) is displayed, which is obtained by integrating over the spatial impact parameter phase-space.
• Figure 5: Images (a) and (b) display the GPD for an unpolarized struck quark in a scalar ($\phi$), Helium-4 (${}^{4}$He) target, and working with the composite structure decomposition, for different spin-orbit coefficients ($\beta_i$). Image (b) displays the effects/role of spin-orbit coupling which is only apparent when $E_q \sim \order{H_q}$.