Singularities of massless planar diagrams, large-N_c mesons in 3+1 dimensions, and the 't Hooft model

TL;DR

This paper analyzes the singularities of planar diagrams in massless quark scattering, establishing a link between 3+1 and 1+1 dimensional systems, and explores implications for large-N_c mesons and confinement.

Contribution

It derives a general relation between singularities in 3+1 and 1+1 dimensions and classifies Landau surfaces, extending understanding of meson dynamics and confinement.

Findings

Identified conditions for singular surfaces in 3+1 dimensions.

Classified all Landau surfaces in 1+1 dimensions.

Connected singularities to the 't Hooft model and large-N_c mesons.

Abstract

We study the singular Landau surfaces of planar diagrams contributing to scattering of a massless quark and antiquark in 3+1 dimensions. In particular, we look at singularities which remain after integration with respect to the various angular degrees of freedom. We derive a general relation between these singularities and the singularities of quark- antiquark scattering in 1+1 dimensions. We then classify all Landau surfaces of the 1+1 dimensional system. Combining these results, we deduce that the singular surfaces of the angle- integrated 3+1 dimensional amplitude must satisfy at least one of three conditions, which we call the planar light-cone conditions. We discuss the extension of our results to non-perturbative processes by means of the non-perturbative operator product expansion. Our findings offer new insights into the connection between the 't Hooft model and large-N_c mesons in 3+1 dimensions and may prove useful in studies of confinement in relativistic meson systems.