Analytic calculation of two-loop QCD corrections to bto sell^+ell^- in the high q² region

We present our results for the NNLL virtual corrections to the matrix elements of the operators O_1 and O_2 for the inclusive process b\to s\ell^+\ell^- in the kinematical region q^2>4m_c^2, where q^2 is the invariant mass squared of the lepton-pair. This is the first analytic two-loop calculation of these matrix elements in the high q^2 region. We give the matrix elements as an expansion in m_c/m_b and keep the full analytic dependence on q^2. Making extensive use of differential equation techniques, we fully automatize the expanding of the Feynman integrals in m_c/m_b. In coincidence with an earlier work where the master integrals where calculated numerically (Ghinculov et al.) we find that in the high q^2 region the \alpha_s corrections to the matrix elements <s \ell^+ \ell^-|O_{1,2}|b> calculated in the present paper lead to a decrease of the q^2-spectrum by 10%-15% relative to the NNLL result in which these contributions are put to zero and reduce the renormalization scale uncertainty to ~2%.