Flavor symmetry breaking and meson masses

The axial-vector Ward-Takahashi identity is used to derive mass formulas for neutral pseudoscalar mesons. Flavor symmetry breaking entails nonideal flavor content for these states. Adding that the ${\eta }^{\text{'}}$ is not a Goldstone mode, exact chiral-limit relations are developed from the identity. They connect the dressed-quark propagator to the topological susceptibility. It is confirmed that in the chiral limit the ${\eta }^{\text{'}}$ mass is proportional to the matrix element which connects this state to the vacuum via the topological susceptibility. The implications of the mass formulas are illustrated using an elementary dynamical model, which includes an Ansatz for that part of the Bethe-Salpeter kernel related to the non-Abelian anomaly. In addition to the current-quark masses, the model involves two parameters, one of which is a mass-scale. It is employed in an analysis of pseudoscalar- and vector-meson bound-states. While the effects of $\mathit{SU}(N_f=2)$ and $\mathit{SU}(N_f=3)$ flavor symmetry breaking are emphasized, the five-flavor spectra are described. Despite its simplicity, the model is elucidative and phenomenologically efficacious; e.g., it predicts $\eta \text{−}{\eta }^{\text{'}}$ mixing angles of $~-{15}^{°}$ and ${\pi }^0\text{−}\eta$ angles of $~1^{°}.$

Figure 1

An illustration of the nature of $K_A$; viz., the contribution to the Bethe-Salpeter kernel associated with the non-Abelian anomaly. All terms have the “hairpin” structure illustrated in the lower panel. No finite sum of such intermediate states is sufficient. Straight lines denote quarks, with $f_1$ and $f_2$ independent, and springs denote gluons.

Figure 2

Vector–pseudoscalar meson mass splitting calculated from our results in Tables and , namely, ${\check{m}}_H$ as a function of ${\overline{m}}_H$, where ${\check{m}}_H=m_{H^{*}}-m_H$ and ${\overline{m}}_H=(1/2)(m_{H^{*}}+m_H)$. For the purpose of this figure, we compared ϕ with the $\overline{s}s$ pseudoscalar described in connection with Eq. (63). The solid curve is $\mu /{\overline{m}}_H$ with $\mu =0.309$ GeV${}^2$.