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

We present a detailed study of $B\to K_1(1270)\gamma$ and $B\to K_1(1400) \gamma$decays\mathrm{.}Usingthelight-conesumruletechnique,wecalculatethe$B\to K_{1A} (1^3P_1)$and$B\to K_{1B} (1^1P_1)$tensorformfactors,$T_1^{K_{1A}}(0)$and$T_1^{K_{1B}}(0)$, where the contributions are included up to the first order in $m_{K_1}\mathrm{/}{m}_b$. We resolve the sign ambiguity of the $K_1(1270)$–$K_1(1400)$ mixing angle ${\theta }_{K_1}$ 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 $B\to K_1\gamma$ and ${\tau }^{-}\to K_1^{-}(1270){\nu }_{\tau }$, we find that ${\theta }_{K_1}=-(34\pm 13{)}^{\circ }$ is favored. In contrast to $B\to K^{\ast }\gamma$, the hard-spectator contribution suppresses the $B\to K_1(1270)\gamma$ and $B\to K_1(1400)\gamma$ 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 $R_{K_1}={\cal B}(B\to K_1(1400)\gamma)/{\cal B}(B\to K_1(1270)\gamma)$canofferabetterdeterminationforthe$\theta_{K_1}$ and consequently the theoretical uncertainties can be reduced.