Light-Cone Sum Rules for $B\to K\pi$ Form Factors and Applications to Rare Decays

We derive a set of light-cone sum rules relating the hadronic form factors relevant for $B\to K\pi\ell^+\ell^-$ decays to the $B$-meson light-cone distribution amplitudes (LCDAs). We obtain the sum rule relations for all $B\to K\pi$ form factors of (axial)vector and (pseudo)tensor $b\to s$ currents with a $P$-wave $K\pi$ system. Our results reduce to the known light-cone sum rules for $B\to K^*$ form factors in the limit of a single narrow-width resonance. We update the operator-product expansion for the underlying correlation function by including a more complete set of $B$-meson LCDAs with higher twists, and produce numerical results for all $B\to K^*$ form factors in the narrow-width limit. We then use the new sum rules to estimate the effect of a non-vanishing $K^*$ width in $B\to K^*$ transitions, and find that this effect is universal and increases the factorizable part of the rate of $B\to K^*X$ decays by a factor of $20\%$. This effect, by itself, goes in the direction of increasing the current tension in the differential $B\to K^*\mu\mu$ branching fractions. We also discuss $B\to K\pi$ transitions outside the $K^*$ window, and explain how measurements of $B\to K\pi\ell\ell$ observables above the $K^*$ region can be used to further constrain the $B\to K^*$ form factors.