Complete NNLO QCD Analysis of B̄ → Xₛ l⁺ l⁻ and Higher Order Electroweak Effects

We complete the next-to-next-to-leading-order QCD calculation of the branching ratio for B -> X_s l^+ l^- including recent results for the three-loop anomalous dimension matrix and two-loop matrix elements. These new contributions modify the branching ratio in the low-q^2 region, BR_ll, by about +2% and -2%, respectively. We furthermore discuss the appropriate normalization of the electromagnetic coupling alpha, and calculate the dominant higher order electroweak effects, showing that they affect BR_ll and the zero of the forward-backward asymmetry, q^2_0, by -5% to +3%, depending on the normalization scale adopted for alpha(mu_em). After introducing a few additional improvements in order to reduce the theoretical error, we perform a comprehensive study of the uncertainty. We obtain BR_ll(1 GeV^2 <= q^2 <= 6 GeV^2) = (1.702 +- 0.170) x 10^-6 and q^2_0 = (3.23 +- 0.12) GeV^2.