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

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 $D_{\mu\nu} ({\bf p}) \sim \delta_{\mu\nu} \chi^2 e^{- {{p^2} \over \Delta}}$ we find that $m_{conv}$ increases with decreasing width $\Delta$ and increasing strength $\chi^2$. For those values of $\chi^2$ and $\Delta$, which provide the best known description of low energy hadronic phenomena, $m_{conv}$ lies around $2 \Lambda_{QCD}$, 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.