Electromagnetic logarithms in [formula omitted

Abstract

The B ¯ → X s ℓ + ℓ − decay rate is known at the next-to-next-to-leading order in QCD. It is proportional to α em ( μ ) 2 and has a ± 4 % scale uncertainty before including the O ( α em ln ( M W 2 / m b 2 ) ) electromagnetic corrections. We evaluate these corrections and confirm the earlier findings of Bobeth et al. Furthermore, we complete the calculation of logarithmically enhanced electromagnetic effects by including also QED corrections to the matrix elements of four-fermion operators. Such corrections contain a collinear logarithm ln ( m b 2 / m ℓ 2 ) that survives integration over the low dilepton invariant mass region 1 GeV 2 < m ℓ ℓ 2 < 6 GeV 2 and enhances the integrated decay rate in this domain. For the low- m ℓ ℓ 2 integrated branching ratio in the muonic case, we find B ( B → X s μ + μ − ) = ( 1.59 ± 0.11 ) × 10 −6 , where the error includes the parametric and perturbative uncertainties only. For B ( B → X s e + e − ) , in the current BaBar and Belle setups, the logarithm of the lepton mass gets replaced by angular cut parameters and the integrated branching ratio for the electrons is expected to be close to that for the muons.