Parton Distribution Functions in the Schwinger model from Tensor Network States

TL;DR

This work demonstrates direct computation of light-cone PDFs from Minkowski-space lattice gauge theory using tensor-network (TN) methods. By implementing the light-front Wilson line as a stepwise space–time evolution within a Hamiltonian framework and restricting to the physical subspace, the authors obtain continuum PDFs for the vector meson in the massive Schwinger model with controlled systematic errors. The approach yields physically sensible, antisymmetric PDFs with a peak near $\xi=0.5$ for various masses and shows good agreement with prior results while highlighting the method’s potential for quantum simulation and extension to higher dimensions and non-Abelian theories. The results establish TNs as a viable, controllable tool for dynamical, real-time observables in lattice gauge theories and point toward applications to more realistic QCD-like systems and quantum hardware implementations.

Abstract

Parton distribution functions (PDFs) describe the inner, non-perturbative structure of hadrons. Their computation involves matrix elements with a Wilson line along a direction on the light cone, posing significant challenges in Euclidean lattice calculations, where the time direction is not directly accessible. We propose implementing the light-front Wilson line within the Hamiltonian formalism using tensor network techniques. The approach is demonstrated in the massive Schwinger model (quantum electrodynamics in 1+1 dimensions), a toy model that shares key features with quantum chromodynamics. We present accurate continuum results for the fermion PDF of the vector meson at varying fermion masses, obtained from first-principle calculations directly in Minkowski space. Our strategy also provides a useful path for quantum simulations and quantum computing.

Figures (9)

• Figure 1: Light-cone matrix elements. a) Operators $\sigma^-$ and $\sigma^+$ from \ref{['eq:MatrixElements']} separated along a light front are calculated on the lattice by subsequent steps of a spatial evolution of the Wilson line (violet horizontal arrows) and time evolution steps (green vertical arrows). b) The light-cone structure that emerges in matrix elements; shown is the modulus of an even-to-even matrix element $\hat{\mathcal{M}}_{0,0} =$$\Bra{h} e^{i H {\frac{\Delta t\xspace}{2} t}}\xspace \prod_{k<\Delta z\xspace}\left(i\sigma_{k}^{z}\right) \sigma_{\Delta z\xspace}^{+} e^{-i H {_{0} \frac{\Delta t\xspace}{2} t}}\xspace \prod_{k'<0}\left(-i\sigma_{k'}^{z}\right) \sigma_{0}^{-} \Ket{h}$ for a light mass $\frac{m}{g} = 0.125$. $\hat{\mathcal{M}}_{0,0}$ is similar to $\mathcal{M}_{0,0}$ in \ref{['eq:MatrixElements']}, but with a fixed static charge to illustrate the emerging light-cone structure without enforcing a light-front direction by moving the static charge. Orange lines indicate the light-front $+$ and $-$ directions.
• Figure 2: Imaginary part of matrix element. The values for $x = 1000$ are discrete but shown as a line for clarity.
• Figure 3: PDF for different lattice spacings, obtained by a discrete Fourier transform. The values for $x=1000$ are shown as a dotted line for clarity. Solid line with error bands: extrapolation to the continuum limit at fixed volume. Dashed line: results from MoPerry (normalized).
• Figure 4: PDFs for different fermion masses $\tilde{m}\xspace = \frac{m \sqrt{\pi}}{g}$. The continuum limit $x \rightarrow \infty$ is taken with the volume $\tilde{V}\xspace = \tilde{m}\xspace \cdot \frac{N}{\sqrt{x}}$ kept fixed. The error bands denote the uncertainties due to the continuum extrapolations, and are smaller than the line width of the plot in most regions. For $\tilde{m}\xspace = 10$, instead, we estimate the errors due to finite $D$, $N_{\tau\xspace}\xspace$, and $x$, and take the infinite volume limit. The error band in this case includes all uncertainties, dominated mostly by the volume effects. See main text and Appendix \ref{['app:D']}supplemental for details.
• Figure 5: Total error of the PDF for $\tilde{m}\xspace = 10$ in fig. 4 in the main text and contributions from different error sources. The relative error at ${\xi}\xspace = \pm 0.5$ is $6\%$.