Asymptotic freedom and Goldstone realization of chiral symmetry

The feasibility of Goldstone realization of weakly probed chiral symmetries is examined in the case that strong interactions are described by an asymptotically free theory of zero-bare-mass quarks and gauge vector gluons. This investigation is restricted to finding solutions to the homogeneous Bethe-Salpeter equation for the symmetry-breaking part $G$ of the quark propagator $S$, $G\mathrm{}\left(p\right)\mathrm{}{\gamma }_5\propto \left\{S^{-1\mathrm{}}\mathrm{}\right(p),{\gamma }_5\}$, in the limit $p^2\to -\infty$. Renormalization-group techniques are extremely useful in this limit, and are used extensively. Their naive application implies the leading asymptotic behavior $G\mathrm{}\left(p\right)\mathrm{}\sim {\left(\ln p\right)}^{-A}$, where $A$ is a calculable positive constant. More importantly, it is shown that the Bethe-Salpeter kernel is well approximated by the ladder graph alone, with the effective coupling $g^2\mathrm{}\left(p\right)\mathrm{}\sim {\left(\ln p\right)}^{-1\mathrm{}}$, when the strong interactions are asymptotically free. Two solutions are found for $G$. The asymptotically dominant one, $G_{+}\mathrm{}\left(p\right)\mathrm{}\sim {\left(\ln p\right)}^{-A}$, is just what was predicted by straightforward renormalization-group analysis, and does not correspond to Goldstone realization of the symmetry. The other solution has much softer asymptotic behavior, $G_{-}\mathrm{}\left(p\right)\mathrm{}\sim p^{-2\mathrm{}}{\left(\ln p\right)}^A$. That this solution actually corresponds to the Goldstone mode is established by relating it, through the axial-vector Ward identity, to the Goldstone-boson-quark-antiquark vertex function, whose large-momentum limit is analyzed via the Wilson operator-product expansion.