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$B \to K_1 γ$ Decays in the Light-Cone QCD Sum Rules

We present a detailed study of $B\to K_1(1270) γ$ and $B\to K_1(1400) γ$ decays. Using the light-cone sum rule technique, we calculate the $B\to K_{1A} (1^3P_1)$ and $B\to K_{1B} (1^1P_1)$ tensor form factors, $T_1^{K_{1A}}(0)$ and $T_1^{K_{1B}}(0)$, where the contributions are included up to the first order in $m_{K_1}/m_b$. We resolve the sign ambiguity of the $K_1(1270)$--$K_1(1400)$ mixing angle $θ_{K_1}$ by defining the signs of decay constants, $f_{K_{1A}}$ and $f_{K_{1B}}^\perp$. From the comparison of the theoretical calculation and the data for decays $B\to K_1 γ$ and $τ^-\to K_1^-(1270)ν_τ$, we find that $θ_{K_1}=-(34\pm 13)^\circ$ is favored. In contrast to $B\to K^* γ$, the hard-spectator contribution suppresses the $B\to K_1(1270) γ$ and $B\to K_1(1400) γ$ branching ratios slightly. The predicted branching ratios are in agreement with the Belle measurement within the errors. We point out that a more precise measurement for the ratio $R_{K_1}={\cal B}(B\to K_1(1400)γ)/{\cal B}(B\to K_1(1270)γ)$ can offer a better determination for the $θ_{K_1}$ and consequently the theoretical uncertainties can be reduced.