Complete NNLO QCD Analysis of B -> X_s l^+ l^- and Higher Order Electroweak Effects

Abstract

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 +1% and -4%, respectively. We furthermore discuss the appropriate normalization of the electromagnetic coupling alpha and calculate the dominant higher order electroweak effects, showing that, due to accidental cancellations, they change BR_ll by only -1.5% if alpha(mu) is normalized at mu = O(m_b), while they shift it by about -8.5% if one uses a high scale normalization mu = O(M_W). The position of the zero of the forward-backward asymmetry, q_0^2, is changed by around +2%. 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) = (1.57 +- 0.16) x 10^-6 and q_0^2 = (3.76 +- 0.33) GeV^2 and note that the part of the uncertainty due to the b-quark mass can be easily reduced.