Color-suppression of non-planar diagrams in bosonic bound states

TL;DR

This paper demonstrates that non-planar diagrams are significantly suppressed in a scalar QCD model of mesons due to color factors, supporting simplified truncation schemes in nonperturbative QCD calculations.

Contribution

It provides a numerical analysis of color suppression of non-planar diagrams in a scalar meson model using Bethe-Salpeter equations and light-front techniques.

Findings

Color suppression is large for N_c=3, validating rainbow-ladder truncations.

Cross-ladder effects have minimal impact on binding energies.

Suppression of non-planar diagrams affects wave functions and form factors.

Abstract

We study the suppression of non-planar diagrams in a scalar QCD model of a meson system in $3+1$ space-time dimensions due to the inclusion of the color degrees of freedom. As a prototype of the color-singlet meson, we consider a flavor-nonsinglet system consisting of a scalar-quark and a scalar-antiquark with equal masses exchanging a scalar-gluon of a different mass, which is investigated within the framework of the homogeneous Bethe-Salpeter equation. The equation is solved by using the Nakanishi representation for the manifestly covariant bound-state amplitude and its light-front projection. The resulting non-singular integral equation is solved numerically. The damping of the impact of the cross-ladder kernel on the binding energies are studied in detail. The color-suppression of the cross-ladder effects on the light-front wave function and the elastic electromagnetic form factor are also discussed. As our results show, the suppression appears significantly large for $N_c=3$, which supports the use of rainbow-ladder truncations in practical nonperturbative calculations within QCD.