Matrix algorithms for solving (in)homogeneous bound state equations

Area of Science:

• * Theoretical Particle Physics
• * Quantum Chromodynamics
• * Hadronic Bound State Physics

Background:

• * Functional approach to quantum chromodynamics (QCD) utilizes covariant integral equations, like the Bethe-Salpeter equation, to study hadronic bound states.
• * Solving these equations, particularly linear, homogeneous ones, can be computationally intensive, often requiring numerical solutions from other integral equations as input.
• * Inhomogeneous equations offer a complementary approach to extract off-shell information alongside bound-state properties.

Purpose of the Study:

• * To demonstrate the efficient numerical solution of both homogeneous and inhomogeneous Bethe-Salpeter equations.
• * To compare the efficiency and advantages of using the inhomogeneous equation versus the homogeneous equation for calculating hadronic properties.
• * To provide insights applicable to the study of baryons and other complex multi-quark systems.

Main Methods:

• * Application of well-established matrix algorithms for eigenvalue problems (homogeneous case).
• * Utilization of algorithms for solving linear systems (inhomogeneous case).
• * Numerical solution of both homogeneous and inhomogeneous Bethe-Salpeter equations.

Main Results:

• * Both homogeneous and inhomogeneous Bethe-Salpeter equations can be solved efficiently using matrix algorithms.
• * The inhomogeneous equation is found to be as efficient, or even more advantageous, than the homogeneous equation for mass spectrum calculations.
• * The methodology provides valuable insights for studying baryons and more complex hadronic systems.

Conclusions:

• * Efficient numerical methods exist for solving covariant integral equations in QCD.
• * The inhomogeneous Bethe-Salpeter equation presents a powerful and efficient tool for hadronic mass spectrum calculations.
• * This approach offers significant benefits for theoretical studies of multi-quark systems and beyond.