It is well known that, because of the axial anomaly in QCD, mesons with $J^P=0^{-}$ are close to $SU(3{)}_{\mathrm{V}}$ eigenstates; the ${\eta }^{\prime }(958)$ meson is largely a singlet, and the $\eta$ meson an octet. In contrast, states with $J^P=1^{-}$ are flavor diagonal; e.g., the $\varphi (1020)$ is almost pure $\overline{s}s$. Using effective Lagrangians, we show how this generalizes to states with higher spin, assuming that they can be classified according to the unbroken chiral symmetry of $G_{\mathrm{fl}}=SU(3{)}_{\mathrm{L}}\times SU(3{)}_{\mathrm{R}}$. We construct effective Lagrangians from terms invariant under $G_{\mathrm{fl}}$ and introduce the concept of hetero- and homochiral multiplets. Because of the axial anomaly, only terms invariant under the $Z(3{)}_{\mathrm{A}}$ subgroup of the axial $U(1{)}_{\mathrm{A}}$ enter. For heterochiral multiplets, which begin with that including the $\eta$ and ${\eta }^{\prime }(958)$, there are $Z(3{)}_{\mathrm{A}}$ invariant terms with low mass dimension which cause states to mix according to $SU(3{)}_{\mathrm{V}}$ flavor. For homochiral multiplets, which begin with that including the $\varphi (1020)$, there are no $Z(3{)}_{\mathrm{A}}$ invariant terms with low mass dimension, and so states are diagonal in flavor. In this way, we predict the flavor mixing for the heterochiral multiplet with spin 1 as well as for hetero- and homochiral multiplets with spin 2 and spin 3.

• Received 2 October 2017

DOI:https://doi.org/10.1103/PhysRevD.97.091901

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Published by the American Physical Society