Dalitz plot analysis of the $B\to K\eta \gamma$ decays

Recently $B$ factories have published new results on the $B\to K\eta \gamma$ decays being inspired by the theoretical suggestion to search for new physics in $B\to P_1{P}_2\gamma$ decays. Using heavy-meson chiral perturbation theory, we find a mechanism which governs the amplitude in parts of the Dalitz plot where either $K$ or $\eta$ mesons are soft. The dominant contributions in these cases are coming from the nonresonant decay modes. We discuss also the $B\to K{\eta }^{\prime }\gamma$ Dalitz plot. Our partially integrated rates are in agreement with the experimental findings.

Figure 1

${\overline{B}}^0\to {\overline{K}}^0\eta \gamma$ phase space regions where soft pseudoscalars have energy below 1.2 GeV (0.8 GeV) in the light-gray (gray) region. The left corner corresponds to soft $\eta$ and the right one to soft $K$.

Figure 2

The left graph shows the leading contribution in the region of soft $\eta$. On the right, $K$ is soft. The left (right) graph governs the decay amplitude in the left (right) region in Fig. 1.

Figure 3

${\overline{B}}^0\to {\overline{K}}^0\eta \gamma$ spectra. Left panel: Photon spectrum in the region of $E_{\eta }<0.8\mathrm{GeV}$ (solid line), $E_{\eta }<1.0\mathrm{GeV}$ (dashed line), and $E_{\eta }<1.2\mathrm{GeV}$ (dotted line). Right panel: The same for soft $K$, $E_K<0.8$, 1.0, 1.2 GeV.

Figure 4

${\overline{B}}^0\to {\overline{K}}^0{\eta }^{\prime }\gamma$ spectra. Left panel: Photon spectrum in the region of $E_{{\eta }^{\prime }}<1.1\mathrm{GeV}$ (solid line), $E_{{\eta }^{\prime }}<1.2\mathrm{GeV}$ (dashed line), and $E_{{\eta }^{\prime }}<1.3\mathrm{GeV}$ (dotted line). Right panel: The same for soft $K$, $E_K<0.8$, 1.0, 1.2 GeV.