Coupled fermion gap and vertex equations for chiral-symmetry breakdown in QCD

The fermion gap equation for QCD is usually posed as a simple variant of the Baker-Johnson-Willey equation, with ad hoc cutoffs for infrared singularities in the running charge imposed as needed, and no confinement effects taken into account. While we too omit confinement effects, we replace the ad hoc cutoffs with a physically consistent picture of QCD, with a dynamically generated gluon mass $m$ serving as an infrared regulator, and study not only the fermion gap equation but also nonlinear vertex equations which determine the running charge. Fermion loops are included in the vertex equations, which allows us to study the dependence of ${\alpha }_s\left(q^2\right)$ on the fermion constituent mass $M$. As one might expect from the one-loop running charge, ${\alpha }_s\mathrm{}\left(0\right)\mathrm{}$ is an increasing function of $M$. When we use a standard form of the fermion gap equation and leave out all confinement effects, the relations between ${\alpha }_s\mathrm{}\left(0\right)\mathrm{}$ and $M$ which follow from the gap equation and the vertex equation are inconsistent with each other, for any value of the gluon mass $m$. Although our model is crude, this suggests that confinement plays an important role in chiral-symmetry breakdown in QCD. Furthermore, lack of confinement above the deconfining temperature may explain why the chiral-symmetry-restoration temperature is so close to the deconfining temperature.