Matrix algorithms for solving (in)homogeneous bound state equations

TL;DR

This paper demonstrates efficient matrix algorithms for solving homogeneous and inhomogeneous Bethe-Salpeter equations in quantum chromodynamics, improving calculations of hadronic bound states and their properties.

Contribution

It introduces the application of matrix algorithms to solve Bethe-Salpeter equations, showing that inhomogeneous equations are more efficient for mass spectrum calculations.

Findings

Inhomogeneous equations are more efficient for mass spectrum calculations.

Matrix algorithms enable faster solutions of Bethe-Salpeter equations.

The approach aids in studying complex systems like baryons.

Abstract

In the functional approach to quantum chromodynamics, the properties of hadronic bound states are accessible via covariant integral equations, e.g. the Bethe-Salpeter equations for mesons. In particular, one has to deal with linear, homogeneous integral equations which, in sophisticated model setups, use numerical representations of the solutions of other integral equations as part of their input. Analogously, inhomogeneous equations can be constructed to obtain off-shell information in addition to bound-state masses and other properties obtained from the covariant analogue to a wave function of the bound state. These can be solved very efficiently using well-known matrix algorithms for eigenvalues (in the homogeneous case) and the solution of linear systems (in the inhomogeneous case). We demonstrate this by solving the homogeneous and inhomogeneous Bethe-Salpeter equations and find, e.g. that for the calculation of the mass spectrum it is more efficient to use the inhomogeneous equation. This is valuable insight, in particular for the study of baryons in a three-quark setup and more involved systems.