Hybrid nature of the abnormal solutions of the Bethe-Salpeter equation in the Wick-Cutkosky model

TL;DR

This paper investigates the abnormal solutions of the Bethe-Salpeter equation in the Wick-Cutkosky model, revealing their hybrid nature dominated by massless exchange particles and their similarity to hybrid states in QCD.

Contribution

It provides a detailed analysis of the abnormal solutions' Fock-space content and electromagnetic properties, highlighting their many-body, hybrid-like characteristics.

Findings

Abnormal solutions are dominated by massless exchange particles (over 90%).

Form factors confirm the many-body, hybrid nature of abnormal states.

Abnormal states resemble QCD hybrid states with multiple exchange particles.

Abstract

In the Wick-Cutkosky model, where two scalar massive constituents interact by means of the exchange of a scalar massless particle, the Bethe-Salpeter equation has solutions of two types, called "normal" and "abnormal". In the non-relativistic limit, the normal solutions correspond to the usual Coulomb spectrum, whereas the abnormal ones do not have non-relativistic counterparts -- they are absent in the Schr\"odinger equation framework. We have studied, in the formalism of the light-front dynamics, the Fock-space content of the abnormal solutions. It turns out that, in contrast to the normal ones, the abnormal states are dominated by the massless exchange particles (by 90 \% or more), what provides a natural explanation of their decoupling from the two-body Schr\"odinger equation. Assuming that one of the massive constituents is charged, we have calculated the electromagnetic elastic form factors of the normal and abnormal states, as well as the transition form factors. The results on form factors confirm the many-body nature of the abnormal states, as found from the Fock-space analysis. The abnormal solutions have thus properties similar to those of hybrid states, made here essentially of two massive constituents and several or many massless exchange particles. They could also be interpreted as the Abelian scalar analogs of the QCD hybrid states. The question of the validity of the ladder approximation of the model is also examined.