Solving the Bethe-Salpeter Equation in Euclidean Space

TL;DR

This paper explores various methods to numerically solve the Bethe-Salpeter equation in Euclidean space, emphasizing the use of Dirac matrices and hyperspherical harmonics, and compares relativistic results with experimental data.

Contribution

It introduces a comprehensive approach using the complete Dirac matrix basis and hyperspherical harmonics for solving the Bethe-Salpeter equation with realistic interactions.

Findings

Complete Dirac basis improves numerical solutions.

Relativistic calculations align with experimental data.

Hyperspherical harmonics facilitate efficient numerical representation.

Abstract

Different approaches to solve the spinor-spinor Bethe-Salpeter (BS) equation in Euclidean space are considered. It is argued that the complete set of Dirac matrices is the most appropriate basis to define the partial amplitudes and to solve numerically the resulting system of equations with realistic interaction kernels. Other representations can be obtained by performing proper unitary transformations. A generalization of the iteration method for finding the energy spectrum of the BS equation is discussed and examples of concrete calculations are presented. Comparison of relativistic calculations with available experimental data and with corresponding non relativistic results together with an analysis of the role of Lorentz boost effects and relativistic corrections are presented. A novel method related to the use of hyperspherical harmonics is considered for a representation of the vertex functions suitable for numerical calculations.