The Convergence Radius of the Chiral Expansion in the Dyson-Schwinger Approach

Abstract

We determine the convergence radius $m_{conv}$ for the expansion in the current quark mass using the Dyson-Schwinger (DS) equation of QCD in the rainbow approximation. Within a Gaussian form for the gluon propagator $D_{μν} ({\bf p}) \sim δ_{μν} χ^2 e^{- {{p^2} \over Δ}}$ we find that $m_{conv}$ increases with decreasing width $Δ$ and increasing strength $χ^2$. For those values of $χ^2$ and $Δ$, which provide the best known description of low energy hadronic phenomena, $m_{conv}$ lies around $2 Λ_{QCD}$, which is big enough, that the chiral expansion in the strange sector converges. Our analysis also explains the rather low value of $m_{conv} \approx 50 \dots 80 \ {\text MeV}$ in the Nambu--Jona-Lasinio model, which as itself can be regarded as a special case of the rainbow DS models, where the gluon propagator is a constant in momentum space.