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 $D_{\mu \nu }(\mathbf{p})\sim {\delta }_{\mu \nu }{\chi }^2{e}^{-\frac{p^2}{\Delta }}$ 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 $m_{conv}\approx 50\dots 80\text{M}eV$ 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.

Comments: 12 pages (REVTEX) + 4 postscript figures (uuencoded tar-compressed), DOE/ER/40561-140INT94-00-01