Hard spectator interactions in B to pi pi at order alphas2

In the present thesis I discuss the hard spectator interaction
amplitude in B to pi pi at NLO i.e. at O(\alpha_s^2). This special
part of the amplitude, whose LO starts at O(alpha_s), is defined in
the framework of QCD factorization. QCD factorization allows to
separate the short- and the long-distance physics in leading power
in an expansion in Lambda/m_b, where the short-distance physics can
be calculated in a perturbative expansion in alpha_s. Compared to
other parts of the amplitude hard spectator interactions are
formally enhanced by the hard collinear scale sqrt{Lambda m_b},
which occurs next to the m_b-scale and leads to an enhancement of
alpha_s. From a technical point of view the main challenges of this
calculation are due to the fact that we have to deal with Feynman
integrals that come with up to five external legs and with three
independent ratios of scales. These Feynman integrals have to be
expanded in powers of Lambda/m_b. I will discuss integration by
parts identities to reduce the number of master integrals and
differential equations techniques to get their power expansions. A
concrete implementation of integration by parts identities in a
computer algebra system is given in the appendix. Finally I discuss
numerical issues like scale dependence of the amplitudes and
branching ratios. It will turn out that the NLO contributions of
the hard spectator interactions are important but small enough for
perturbation theory to be valid.