[Paper Review] Instantaneous Bethe-Salpeter Equation: (Semi-)Analytical Solution

Wolfgang Lucha, Khin Maung Maung+1 more|ArXiv.org|Oct 9, 2000

Matrix Theory and Algorithms3 citations

TL;DR

This paper presents a semi-analytical solution to the instantaneous Bethe-Salpeter equation for fermion-antifermion bound states by transforming it into a matrix eigenvalue problem with explicitly algebraic matrices. For a linear confining potential, it derives an analytical bound-state mass formula, yielding M ≈ 1.696 GeV for λ = 0.2 GeV², within 2.4% of numerical results, demonstrating high accuracy with minimal computational cost.

ABSTRACT

The Bethe-Salpeter equation for bound states of a fermion-antifermion pair in the instantaneous approximation for the involved interaction kernel is converted into an equivalent matrix eigenvalue problem with explicitly (algebraically) given matrices.

Motivation & Objective

• To develop a (semi-)analytical method for solving the instantaneous Bethe-Salpeter equation for fermion-antifermion bound states.
• To reduce the complexity of solving the integral equation by converting it into a matrix eigenvalue problem with explicitly given matrices.
• To achieve high-accuracy results for bound-state masses using minimal computational resources, especially for low-dimensional systems.
• To validate the method by comparing analytical results with numerical solutions using large matrices.
• To explore the role of variational parameters and basis function expansions in controlling truncation errors and improving convergence.

Proposed method

• The Bethe-Salpeter equation is reformulated in momentum space using a linear confining potential V(r) = λr.
• The coupled integral equations for radial amplitudes Ψ₁ and Ψ₂ are reduced to a matrix eigenvalue problem via expansion in orthonormal basis states for ℓ = 0 and ℓ = 1.
• Explicit algebraic expressions are derived for all matrix elements in the Hamiltonian matrix, including kinetic, interaction, and coupling terms.
• Basis functions are chosen with a variational parameter μ to optimize convergence and accuracy.
• Matrix elements are computed using integrals involving Fourier-Bessel transforms of the potential and basis functions, with coefficients expressed algebraically.
• The resulting matrix is diagonalized to extract bound-state masses M, with analytical solutions possible for matrix sizes ≤ 4.

Experimental results

Research questions

• RQ1 Can the instantaneous Bethe-Salpeter equation for fermion-antifermion systems be solved analytically for low-dimensional matrix representations?
• RQ2 How accurate is the semi-analytical approach when compared to high-precision numerical solutions using large matrices?
• RQ3 What is the impact of basis function choice and variational parameter μ on the convergence and error control of the solution?
• RQ4 How do the algebraic matrix elements relate to the underlying physical structure of the interaction kernel and wave functions?
• RQ5 What analytical formula can be derived for the ground state mass in the massless case, and how does it compare to numerical benchmarks?

Key findings

• For the one-dimensional case, an analytical bound-state mass formula is derived: M = 4√(2λ/(3π)(2 + √5)).
• For λ = 0.2 GeV², this analytical formula yields M = 1.696 GeV, only 2.4% above the numerical result of 1.656 GeV obtained with a 15×15 matrix and N = 49.
• The method achieves high accuracy even with small matrices, demonstrating the effectiveness of the algebraic matrix formulation.
• For massive constituents, the analytical expression M² = 8m² + (8896/(315π))λ + (23/7)(128λ/(45πm))² is derived, valid for m ≠ 0.
• The consistency of matrix relations (e.g., ∑r c*ri c rj = δij) is verified with relative errors < 3% for 15×15 matrices and N = 49, confirming numerical stability.
• Numerical issues in symbolic computation (e.g., Mathematica 4.0) are identified, showing that low-precision arithmetic can yield spurious results, emphasizing the need for high-precision evaluation.