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 → sℓ + ℓ − in the kinematical region q 2 > 4m 2 c , 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 were obtained numerically [1], we find that in the high q 2region the α s corrections to the matrix elements sℓ + ℓ − |O 1,2 |b calculated in the present paper lead to a decrease of the perturbative part 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%.

I compute the hard spectator interaction amplitude in B → ππ at NLO i.e. at O(α 2 s ). This special part of the amplitude, whose LO starts at O(α s ), is defined in the framework of QCD factorization. QCD factorization allows to separate the shortand the long-distance physics in leading power in an expansion in Λ QCD /m b , where the short-distance physics can be calculated in a perturbative expansion in α s .In this calculation it is necessary to obtain an expansion of Feynman integrals in powers of Λ QCD /m b . I will present a general method to obtain this expansion in a systematic way once the leading power is given as an input. This method is based on differential equation techniques and easy to implement in a computer algebra system.The numerical impact on amplitudes and branching ratios is considered. The NLO contributions of the hard spectator interactions are important but small enough for perturbation theory to be valid.

The approximate tree decays B À ! À 0 = À 0 may serve as benchmark channels for testing the various theoretical descriptions of the strong interaction dynamics in hadronic B meson decays. The ratios of hadronic and differential semileptonic B ! '=' decay rates at maximum recoil provide particularly clean probes of the QCD dynamics. We confront the recent next-to-next-to-leading order calculation in the QCD factorization framework with experimental data and find support for the factorization assumption. A detailed analysis of all tree-dominated B ! == decay modes seems to favor somewhat enhanced color-suppressed amplitudes, which may be accommodated in QCD factorization by a small value of the first inverse moment of the B meson light-cone distribution amplitude, B ' 250 MeV. Precise measurements of the semileptonic B ! ' spectrum could help to clarify this point.

I present an algorithm based on sector decomposition and Mellin-Barnes techniques to power expand Feynman integrals. The coefficients of this expansion are given in terms of finite integrals that can be calculated numerically. I show in an example the benefit of this method for getting the full analytic power expansion from differential equations by providing the correct ansatz for the solution. For method of regions the presented algorithm provides a numerical check, which is independent from any power counting argument. * Electronic address: volker.pilipp@itp.unibe.ch