Confinement and chiral-symmetry breakdown: Estimates ofFπand of effective quark masses

We illustrate how a field theory of confinement automatically leads to spontaneous breakdown of (flavor nonsinglet) chiral symmetry, with accompanying massless pions. This breakdown is described as a tunneling process involving a quark and virtual $q\overline{q}$ pairs (as in superconductivity), and is driven by the infrared singularities of the confinement mechanism. Effective quark masses (not to be confused with either current or constituent quark masses) are defined in terms of {${\gamma }_5$, $S^{-1\mathrm{}}\mathrm{}\left(p\right)\mathrm{}$} at zero momentum, where $S\left(p\right)\mathrm{}$ is the effective quark propagator (an entire function in most gauges), and these masses yield values for $F_{\pi }$. Aside from messy numbers of $O\left(1\right)\mathrm{}$ discussed in the text and leaving out short-distance corrections, $F_{\pi }$ is essentially $2\sqrt{3}a{\left(2\mathrm{}\pi \right)}^{-\frac{3}{2}}$, where $a\simeq 400$ MeV appears in the linearly rising potential $V\left(r\right)=a^{2}r$. We calculate chiral-breaking effects in four models, beginning with $d=2+1\mathrm{}$ QED for massless electrons, to introduce techniques [using Ward identities and the full Dyson equation for $S\left(p\right)\mathrm{}$] for dealing with confining field theories. Two other models are $d=3+1\mathrm{}$ propagator models of confinement and the fourth explicitly exhibits the tunneling to a $q\overline{q}$ pair mentioned above, in a theory with area-law confinement.