Dynamical chiral symmetry breaking in quantum chromodynamics

We examine both dynamical chiral symmetry breaking and explicit breaking due to current quark masses in quantum chromodynamics (QCD). The renormalized current and constituent quark mass are defined. The quark self-energy $\Sigma \mathrm{}\left(p\right)={\Sigma }_D\mathrm{}\left(p\right)+{\Sigma }_E\mathrm{}\left(p\right)\mathrm{}$ has unique contributions from dynamical and explicit symmetry breaking. We determine the asymptotic behavior of ${\Sigma }_D\mathrm{}\left(p\right)\mathrm{}$ and ${\Sigma }_E\mathrm{}\left(p\right)\mathrm{}$ as $-p^2\to \infty$. Although the explicit symmetry breaking dominates in the region controlled by perturbation theory, the dynamical term, which receives contributions from instantons, dominates in the subasymptotic region. The dynamical term is often ignored in the calculations. We also discuss the possibility of a phase transition in QCD for massive quark systems. The structure of the chiral perturbation expansion for light quarks is found to have not only an essential singularity in the gauge-field coupling constant for $g^2\le 0$ but also a cut for $g^2\le 0$ in amplitudes for which the essential singularity is absent. We also calculate the second-order axial-vector renormalization for a quark with the result $g_A=1-\frac{g^2}{6{\pi }^2}$.