Anomalous decay $f_1(1285)\to \rho \gamma$ and related processes

We work out the low-energy expansion of the anomalous $f_1(1285)\to \rho \gamma$ decay amplitude by using the Nambu–Jona-Lasinio model with $U(2)\times U(2)$ chiral symmetric four-quark interactions in the one-quark-loop approximation. The related processes $f_1(1285)\to \omega \gamma$, $a_1(1260)\to \omega \gamma$, and $a_1(1260)\to \rho \gamma$, are also considered. An effective meson Lagrangian responsible for $f_1\rho \gamma$, $f_1\omega \gamma$, $a_1\rho \gamma$ and $a_1\omega \gamma$ interactions is found. The predicted radiative decay widths, ${\mathrm{\Gamma }}_{f_1\to {\rho }^0\gamma }=311\mathrm{keV}$, ${\mathrm{\Gamma }}_{f_1\to \omega \gamma }=34.3\mathrm{keV}$, ${\mathrm{\Gamma }}_{a_1\to {\rho }^0\gamma }=26.8\mathrm{keV}$, ${\mathrm{\Gamma }}_{a_1\to \omega \gamma }=238\mathrm{keV}$, allow an experimental test of the hypothesis that $f_1(1285)$ and $a_1(1260)$-mesons have a quark-antiquark nature. At present, only the $f_1(1285)\to \rho \gamma$ decay has been measured. Our result is in remarkably good agreement with the recent data of CLAS Collaboration ${\mathrm{\Gamma }}_{f_1\to {\rho }^0\gamma }=453\pm 177\mathrm{keV}$, but disagrees with the Particle Data Group-based estimate of ${\mathrm{\Gamma }}_{f_1\to {\rho }^0\gamma }=1326\pm 313\mathrm{keV}$. The calculations presented require a minimum of theoretical input, and are shown to be consistent with the nonrenormalization theorems of QCD.

Figure 1

The one-quark-loop contribution to the radiative decay amplitude $f_{1\alpha }(l)\to {\rho }_{\beta }(p){\gamma }_{\gamma }(q)$. The first diagram corresponds to the Lorentz tensor ${\tilde{T}}^{\alpha \beta \gamma }(p,q)$, the second one to the tensor ${\tilde{T}}^{\alpha \gamma \beta }(q,p)$. External lines represent the $f_1$ axial-vector field with polarization vector ${\epsilon }_{\alpha }(l)$, the vector $\rho$-meson field with polarization vector ${\epsilon }_{\beta }^{*}(p)$, and electromagnetic field $A_{\gamma }$ with polarization vector ${\epsilon }_{\gamma }^{*}(q)$.