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A Direct Algebraic Pathway to Hadronic Observables in the Contact Model

Jiayin Kang, Zanbin Xing, Lei Chang

TL;DR

This work addresses the challenge of computing hadronic observables in the contact-interaction (CI) framework without resorting to bound-state wave functions. It introduces a direct algebraic reformulation based on Fierz transformations, yielding projected amplitudes $g^H_a(Q)$ and $\mathcal{A}^H_a(P,Q)$ that satisfy closed homogeneous and inhomogeneous equations, with universal kernels $\Pi_{ba}(Q)$ and $\mathcal{T}_{cab}(P,Q)$. The method is demonstrated for the vector meson, obtaining masses, decay constants, and form factors from these projected amplitudes, and shows a unified, efficient mapping from the CI interaction to hadronic observables. The approach naturally extends to baryons and parton-structure quantities like PDFs and DAs, offering a compact framework for elastic and inelastic hadron structure within a single algebraic formulation.

Abstract

We present a novel algebraic framework for computing hadron properties directly within the contact interaction model. Utilizing Fierz transformations, the method recasts the Bethe-Salpeter dynamics into equations for a minimal set of \emph{projected amplitudes} for bound-state static properties and form factors, bypassing the conventional need for the meson wave function. This approach is fully demonstrated for the vector meson, enabling the direct extraction of its decay constants and form factors. The formalism provides a more efficient and unified pathway to hadron observables, with clear potential for extension to baryons and more sophisticated interactions.

A Direct Algebraic Pathway to Hadronic Observables in the Contact Model

TL;DR

This work addresses the challenge of computing hadronic observables in the contact-interaction (CI) framework without resorting to bound-state wave functions. It introduces a direct algebraic reformulation based on Fierz transformations, yielding projected amplitudes and that satisfy closed homogeneous and inhomogeneous equations, with universal kernels and . The method is demonstrated for the vector meson, obtaining masses, decay constants, and form factors from these projected amplitudes, and shows a unified, efficient mapping from the CI interaction to hadronic observables. The approach naturally extends to baryons and parton-structure quantities like PDFs and DAs, offering a compact framework for elastic and inelastic hadron structure within a single algebraic formulation.

Abstract

We present a novel algebraic framework for computing hadron properties directly within the contact interaction model. Utilizing Fierz transformations, the method recasts the Bethe-Salpeter dynamics into equations for a minimal set of \emph{projected amplitudes} for bound-state static properties and form factors, bypassing the conventional need for the meson wave function. This approach is fully demonstrated for the vector meson, enabling the direct extraction of its decay constants and form factors. The formalism provides a more efficient and unified pathway to hadron observables, with clear potential for extension to baryons and more sophisticated interactions.
Paper Structure (11 sections, 24 equations, 3 figures, 1 table)

This paper contains 11 sections, 24 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Equations for Physical Observables
  3. Homogeneous Equation: Masses and Decay Constants
  4. Inhomogeneous Equation: Form Factors and Transition Amplitudes
  5. Structure and Universality of the projected amplitudes
  6. Vector Meson Observables
  7. Basis Projected Amplitudes
  8. Numerical results
  9. summary
  10. Acknowledgments
  11. $\Pi_{ba}(Q)$ and $\mathcal{T}_{cab}(P,Q)$ in CI

Figures (3)

  • Figure 1: Scalar and pseudo scalar form factors. $F_{S,\{1,2\}}(Q^2)$ possess scalar meson pole while $F_{P}(Q^2)$ possesses pseudo scalar meson pole.
  • Figure 2: Vector and axial vector form factors. $F_{V,\{1,2,3\}}(Q^2)$ possess vector meson pole while $F_{A,\{1,2,3\}}(Q^2)$ possess axial vector meson pole. The form factor $F_{A,2}(Q^2)$ possesses an additional pseudo scalar meson pole.
  • Figure 3: Tensor form factors. $F_{T,\{1,2,3\}}(Q^2)$ possess vector meson poles while $F_{T,\{4,5\}}(Q^2)$ do not possess any meson pole.