Infrared singularities in the null-plane bound-state equation when going to 1+1 dimensions

TL;DR

This paper demonstrates that the null-plane bound-state equation in 1+3 dimensions reduces to 't Hooft's equation in 1+1 dimensions, revealing a fundamental connection between these formulations in different dimensions.

Contribution

It establishes a detailed link between the null-plane bound-state equation in 3+1 dimensions and 't Hooft's equation in 1+1 dimensions, clarifying the dimensional reduction process.

Findings

The null-plane bound-state equation reduces to 't Hooft's equation in 1+1 dimensions.

The reduction involves an $x^+$-instantaneous interaction.

The paper discusses the reasons behind this dimensional coincidence.

Abstract

In this paper we first consider the null-plane bound-state equation for a $q \bar q$ pair in 1+3 dimensions and in the lowest-order Tamm-Dancoff approximation. Light-cone gauge is chosen with a causal prescription for the gauge pole in the propagator. Then we show that this equation, when dimensionally reduced to 1+1 dimensions, becomes 't Hooft's bound-state equation, which is characterized by an $x^+$-instantaneous interaction. The deep reasons for this coincidence are carefully discussed.