$B \to K^* \ell^+\ell^-$ in soft-collinear effective theory

We study the rare B decay $B \to K^* \ell^+ \ell^-$ using soft-collinear effective theory (SCET). At leading power in $1/m_b$, a factorization formula is obtained valid to all orders in $\alpha_s$. For phenomenological application, we calculate the decay amplitude including order $\alpha_s$ corrections, and resum the logarithms by evolving the matching coefficients from the hard scale ${\cal O}(m_b)$ down to the scale $\sqrt{m_b \Lambda_h}$. The branching ratio for $B \to K^* \ell^+ \ell^-$ is uncertain due to the imprecise knowledge of the soft form factors $\zeta_\perp (q^2)$ and $\zeta_\parallel (q^2)$. Constraining the soft form factor $\zeta_\perp (q^2=0)$ from data on $B \to K^* \gamma$ yields $\zeta_\perp (q^2=0)=0.32 \pm 0.02$. Using this input, together with the light-cone sum rules to determine the $q^2$dependence of $\zeta_\perp (q^2)$ and the other soft form factor $\zeta_\parallel (q^2)$, we eastimate the partially integrated branching ratio in the range $1~{GeV}^2 \le q^2 \le 7~{GeV}^2$ to be $(2.92^{+0.67}_{-0.61}) \times 10^{-7}$. We discuss how to reduce the form factor related uncertainty by combining data on $B \to \rho (\to \pi \pi) \ell \nu_\ell$ and $B\to K^* (\to K\pi) \ell^+\ell^-$. The forward-backward asymmetry is less sensitive to the input parameters. In particular, for the zero-point of the forward backward asymmetry in the standard model, we get $q_0^2=(4.07^{+0.13}_{-0.12})~{GeV}^2$. The scale dependence of $q_0^2$ is discussed in detail.