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Partonic distribution functions and amplitudes using tensor network methods

Zhong-Bo Kang, Noah Moran, Peter Nguyen, Wenyang Qian

TL;DR

The paper addresses the challenge of directly computing parton distribution functions (PDFs) and distribution amplitudes (DAs) from first principles, which are time-dependent light-cone objects. It introduces a uniform tensor-network framework that encodes hadrons as matrix product states and performs real-time evolution to extract PDFs and DAs within the 1+1D Nambu–Jona-Lasinio model, mapping fermionic degrees of freedom to spins via Jordan-Wigner. The authors demonstrate direct extraction of PDFs and DAs, achieving convergence up to 102 qubits and observing mass- and coupling-dependent behavior with qualitative agreement to perturbative and non-relativistic limits, validated against exact diagonalization and quantum circuit simulations. This work establishes tensor networks as a viable path to hadron-structure calculations from first principles and sets the stage for extensions to gauge theories, multiple flavors, and higher dimensions as computational methods and hardware advance.

Abstract

Calculations of the parton distribution function (PDF) and distribution amplitude (DA) are highly relevant to core experimental programs as they provide non-perturbative inputs to inclusive and exclusive processes, respectively. Direct computation of the PDFs and DAs remains challenging because they are non-perturbative quantities defined as light-cone correlators of quark and gluon fields, and are therefore inherently time-dependent. In this work, we use a uniform quantum strategy based on tensor network simulation techniques to directly extract these hadronic quantities from first principles using the matrix product state of the hadrons in the same setup. We present exemplary numerical calculations with the Nambu-Jona-Lasinio model in 1+1 dimensions and compare with available exact diagonalization and quantum circuit simulation results. Using tensor networks, we evaluate the PDF and DA at various strong couplings in the large-qubit limit, which is consistent with expectations at perturbative and non-relativistic limits.

Partonic distribution functions and amplitudes using tensor network methods

TL;DR

The paper addresses the challenge of directly computing parton distribution functions (PDFs) and distribution amplitudes (DAs) from first principles, which are time-dependent light-cone objects. It introduces a uniform tensor-network framework that encodes hadrons as matrix product states and performs real-time evolution to extract PDFs and DAs within the 1+1D Nambu–Jona-Lasinio model, mapping fermionic degrees of freedom to spins via Jordan-Wigner. The authors demonstrate direct extraction of PDFs and DAs, achieving convergence up to 102 qubits and observing mass- and coupling-dependent behavior with qualitative agreement to perturbative and non-relativistic limits, validated against exact diagonalization and quantum circuit simulations. This work establishes tensor networks as a viable path to hadron-structure calculations from first principles and sets the stage for extensions to gauge theories, multiple flavors, and higher dimensions as computational methods and hardware advance.

Abstract

Calculations of the parton distribution function (PDF) and distribution amplitude (DA) are highly relevant to core experimental programs as they provide non-perturbative inputs to inclusive and exclusive processes, respectively. Direct computation of the PDFs and DAs remains challenging because they are non-perturbative quantities defined as light-cone correlators of quark and gluon fields, and are therefore inherently time-dependent. In this work, we use a uniform quantum strategy based on tensor network simulation techniques to directly extract these hadronic quantities from first principles using the matrix product state of the hadrons in the same setup. We present exemplary numerical calculations with the Nambu-Jona-Lasinio model in 1+1 dimensions and compare with available exact diagonalization and quantum circuit simulation results. Using tensor networks, we evaluate the PDF and DA at various strong couplings in the large-qubit limit, which is consistent with expectations at perturbative and non-relativistic limits.
Paper Structure (13 sections, 14 equations, 4 figures, 2 tables)

This paper contains 13 sections, 14 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Theoretical setup and computational methods
  3. Parton distribution functions and amplitudes
  4. Nambu--Jona-Lasinio model in 1+1 dimensions
  5. Tensor network methods
  6. Density matrix renormalization group
  7. Time dependent variational principle
  8. Results
  9. Hadron state preparation
  10. Real-time evolution
  11. Parton distribution function and amplitudes
  12. Conclusion and outlook
  13. Hadron mass details obtained from tensor network simulations

Figures (4)

  • Figure 1: Simulation results for the PDF comparing tensor network, exact diagonalization, and quantum computing methods. We use $N = 10$, $m=0.7a^{-1}$, and $g=0.25$. Right plot shows the error of the PDF calculated using tensor network and quantum circuit simulation.
  • Figure 2: Simulation results for the PDF and DA over increasing number of qubits $N$. Here, we fix $m=0.7a^{-1}$ and $g=0.25$. The asymptotic of perturbative QCD for the DA is provided.
  • Figure 3: Simulation results for the PDF and DA at different values of strong coupling $g$. The mass is fixed at $m=0.5a^{-1}$ and total qubits are $N=102$.
  • Figure 4: Simulation results for the PDF and DA at different values of mass $m$. The strong coupling is fixed at $g=0.25$ and total qubits are $N=102$.