[Paper Review] Scattering states in Bethe-Salpeter equation

V. A. Karmanov, J. Carbonell|arXiv (Cornell University)|Dec 4, 2012

Quantum Chromodynamics and Particle Interactions6 references3 citations

TL;DR

This paper presents the first numerical solution of the Bethe-Salpeter equation for scattering states in Minkowski space with a ladder kernel, using a novel method to handle singularities in the propagators and kernel. The key contribution is the computation of the full off-shell scattering amplitude $ F_0(p_0, p; p_s) $, which enables accurate phase shifts, inelasticity above thresholds, and provides a foundation for form factor and three-body calculations.

ABSTRACT

The off-mass shell scattering amplitude, satisfying the Bethe-Salpeter equation for spinless particles in Minkowski space with the ladder kernel, is computed for the first time.

Motivation & Objective

• To compute the off-shell scattering amplitude in Minkowski space for spinless particles interacting via a ladder kernel, a task previously unachieved due to singularities.
• To develop a numerical method that correctly handles the complex singularities in the Bethe-Salpeter kernel and propagators, avoiding issues from Wick rotation and Euclidean space approximations.
• To provide a reliable off-shell amplitude for physical applications such as electromagnetic form factors and three-body Faddeev equations.
• To validate the method by reproducing known phase shifts and showing consistency with non-relativistic and Euclidean space results.

Proposed method

• A new method is developed to treat the four types of singularities in the Bethe-Salpeter equation: poles in the constituent propagators, logarithmic singularities from the exchange propagator, and singularities in the inhomogeneous term.
• Principal value (PV) integrals are regularized via subtraction techniques, removing singularities in the $ p'_0 $ and $ p' $ integrals.
• The $ p'_0 $ and $ p' $ integrals are split at singularity points, and variable transformations are applied to make each interval integrable numerically.
• The method uses the Nakanishi integral representation approach adapted for scattering states, ensuring correct analytic structure in Minkowski space.
• The solution is validated by comparing phase shifts with non-relativistic Schrödinger results and Euclidean space calculations, achieving 3–4 digit agreement.

Experimental results

Research questions

• RQ1 Can the off-shell scattering amplitude in Minkowski space be computed reliably for the Bethe-Salpeter equation with a ladder kernel?
• RQ2 How does the relativistic phase shift differ from the non-relativistic one at low energies, especially for strong coupling?
• RQ3 Does the method correctly reproduce the onset of inelasticity above the meson production threshold?
• RQ4 Can the full off-shell amplitude be computed and used for physical observables like form factors?

Key findings

• The off-shell scattering amplitude $ F_0(p_0, p; p_s) $ is computed for the first time in Minkowski space, with full dependence on $ p_0 $ and $ p $, providing a non-trivial structure with ridges and gaps from singularities.
• Phase shifts calculated from the on-shell amplitude $ F_0^{on} $ agree with Euclidean space results within 3–4 digits and show significant deviations from non-relativistic predictions, especially at higher coupling $ \alpha $.
• The imaginary part of the phase shift appears above the threshold $ p_s^* = \sqrt{m\mu + \mu^2/4} $, correctly capturing inelasticity due to exchange meson production.
• The scattering length $ a_0 $ diverges at critical coupling $ \alpha \approx 1.0 $ (BS) and $ \alpha \approx 0.8 $ (Schrödinger), signaling the onset of a bound state, with a sign change after the critical point.
• The unitarity condition is not imposed a priori but emerges naturally from the solution, confirming the method's consistency and accuracy.