Allowed Domains in Theories of Broken Chiral Symmetry

We consider theories in which the explicit breaking of the chiral symmetry ${\mathrm{SU}}_3\times {\mathrm{SU}}_3$ of the energy density ${\Theta }_{00}$ is of the form ${\epsilon }_0{u}_0+{\epsilon }_8{u}_8$, where ${\epsilon }_0$, ${\epsilon }_8$ are real constants and $u_0$ and $u_8$ are scalar densities in the representation $(3,\overline{3})+(\overline{3},3)$. We propose an extension of this picture, in which the ${\mathrm{SU}}_3\times {\mathrm{SU}}_3$-invariant part of ${\Theta }_{00}$ has the form ${\overline{\Theta }}_{00}+{\epsilon }_9{u}_9$, where ${\overline{\Theta }}_{00}$ is $U_3\times U_3$ invariant and the ${\mathrm{SU}}_3\times {\mathrm{SU}}_3$-invariant scalar $u_9$ breaks $U_3\times U_3$ in a specific way. Specifically, it is assumed that the ninth axial charge $F_0^5$ transforms $u_9$ into a pseudoscalar $v_9$ according to $i[F_0^5,u_9]=\kappa v_9$, where $i[F_0^5,v_9]=-\kappa u_9$. (The parameter $\kappa$ labels the representation.) Given this group structure, the analysis of Okubo and Mathur may be extended to find allowed domains for the symmetry-breaking parameters ${\epsilon }_0$, ${\epsilon }_8$, ${\epsilon }_9$ and the vacuum expectation values $〈u_0〉$, $〈u_8〉$, and $〈u_9〉$. The positivity conditions now restrict $〈u_9〉$ so that the allowed domains occupy certain volumes in a three-dimensional space.