Solving the Bars-Green equation for moving mesons in two-dimensional QCD

TL;DR

This paper numerically solves the Bars-Green equation for moving mesons in two-dimensional QCD, confirming gauge invariance and the behavior of wave functions at high boosts, extending previous stationary meson studies.

Contribution

It provides the first numerical solutions for moving mesons in the Bars-Green framework, verifying gauge invariance and wave function behavior in the 't Hooft model.

Findings

Wave functions approach light-cone form at high boosts

Agreement with light-cone gauge results confirms gauge invariance

Computed meson spectra and decay constants for various momenta

Abstract

The two-dimensional QCD in the large $N$ limit, generally referred to as the 't Hooft model, is numerically investigated in the axial gauge in a comprehensive manner. The corresponding Bethe-Salpeter equation for a bound $q\bar{q}$ pair, originally derived by Bars and Green in 1978, was first numerically tackled by Li and collaborators in late 1980s, yet only for the {\it stationary} mesons. In this paper, we make further progress by numerically solving the Bars-Green equation for {\it moving} mesons, ranging from the chiral pion to charmonium. By choosing several different quark masses, we computed the corresponding quark condensates, meson spectra and their decay constants for a variety of meson momenta, and found satisfactory agreement with their counterparts obtained using light-cone gauge, thus numerically verified the gauge and Poincar\'{e} invariance of the 't Hooft model. Moreover, we have explicitly confirmed that, as the meson gets more and more boosted, the large component of the Bars-Green wave function indeed approaches the corresponding 't Hooft light-cone wave function, while the small component of the wave function rapidly fades away.