IR finiteness of the ghost dressing function from numerical resolution of the ghost SD equation

TL;DR

This paper numerically solves the ghost Schwinger-Dyson equation in Landau gauge, revealing two types of solutions for the ghost dressing function depending on the coupling, with lattice data supporting the regular, finite solution.

Contribution

It demonstrates the existence of two ghost dressing function solutions and shows the regular solution aligns with lattice data, depending on the coupling value.

Findings

Two types of ghost solutions: singular and regular.

Lattice data exclude the singular solution.

Regular solution matches lattice data at physical coupling.

Abstract

We solve numerically the Schwinger-Dyson (SD hereafter) ghost equation in the Landau gauge for a given gluon propagator finite at k=0 (alpha_gluon=1) and with the usual assumption of constancy of the ghost-gluon vertex ; we show that there exist two possible types of ghost dressing function solutions, as we have previously inferred from analytical considerations : one singular at zero momentum, satisfying the familiar relation alpha_gluon+2 alpha_ghost=0 between the infrared exponents of the gluon and ghost dressing functions(in short, respectively alpha_G and alpha_F) and having therefore alpha_ghost=-1/2, and another which is finite at the origin (alpha_ghost=0), which violates the relation. It is most important that the type of solution which is realized depends on the value of the coupling constant. There are regular ones for any coupling below some value, while there is only one singular solution, obtained only at a critical value of the coupling. For all momenta k<1.5 GeV where they can be trusted, our lattice data exclude neatly the singular one, and agree very well with the regular solution we obtain at a coupling constant compatible with the bare lattice value.