Tensor network simulations of quasi-GPDs in the massive Schwinger model

Abstract

Generalized Parton Distribution functions (GPDs) are off-diagonal light-cone matrix elements that encode the internal structure of hadrons in terms of quark and gluon degrees of freedom. In this work, we present the first nonperturbative study of quasi-GPDs in the massive Schwinger model, quantum electrodynamics in 1+1 dimensions (QED2), within the Hamiltonian formulation of lattice field theory. Quasi-distributions are spatial correlation functions of boosted states, which approach the relevant light-cone distributions in the luminal limit. Using tensor networks, we prepare the first excited state in the strongly coupled regime and boost it to close to the light-cone on lattices of up to 400 lattice sites. We compute both quasi-parton distribution functions and, for the first time, quasi-GPDs, and study their convergence for increasingly boosted states. In addition, we perform analytic calculations of GPDs in the two-particle Fock-space approximation and in the Reggeized limit, providing qualitative benchmarks for the tensor network results. Our analysis establishes computational benchmarks for accessing partonic observables in low-dimensional gauge theories, offering a starting point for future extensions to higher dimensions, non-Abelian theories, and quantum simulations.

Figures (6)

• Figure 1: Solution to \ref{['TH1']} for $\beta=0.1\sqrt{3}/\pi$ (blue), $\beta=\sqrt{3}/\pi$ (red), $\beta=10\sqrt{3}/\pi$ (black) using 13 Jacobi polynomials. See Grieninger:2024cdl for more details. The dashed lines are \ref{['ansatz']} plotted for the same $\beta$ values.
• Figure 2: Single pole GPD \ref{['eq:fullgpd']} for different values of the skewness $\xi$, as indicated in each panel, with $\beta=0.4$ and $M_0=1$. The red vertical lines delineating the DGLAP and ERBL regions are located at $x = \pm\xi$.
• Figure 3: Reggeized WKB GPD for the same parameter choices as in Fig. \ref{['fig:2part']}.
• Figure 4: The (unitless) energy and momentum dispersion relations as a function of rapidity $\chi$ with a Trotter step $\delta \chi = 0.05$, where $\left|\eta(\chi)\right\rangle = e^{i\mathbb K \chi}\left|\eta\right\rangle$. The top and bottom rows are the energy and momentum dispersions, respectively, while the left and right columns correspond to the dispersions for $m/g = 0.2$ and $m/g = 1.0$, respectively. The lattice energy expectation value is compared to the continuum result. The lattice momentum expectation value is compared with the continuum result as well as the lattice dispersion relation for an infinite spatial lattice, $\frac{1}{a} \sin(a P(\chi))$.
• Figure 5: Tensor network calculation of qPDFs for a light mass system (left column) and a heavy mass system (right column). Top row: Convergence of the qPDF to the lightcone for different values of the boost rapidity. Bottom row: 3D surface plots of the qPDFs in terms of the momentum fraction $x$ and the boost rapidity $\chi$.