Hard spectator interactions in B to pi pi at order alphas2
In the present thesis I discuss the hard spectator interaction\namplitude in B to pi pi at NLO i.e. at O(\\alpha_s^2). \nThis special part of\nthe amplitude, whose LO starts at O(alpha_s), is defined in\nthe framework of QCD factorization. QCD factorization allows to separate the \nshort- and the long-distance physics in leading power in an expansion\nin Lambda/m_b, where the short-distance physics can be\ncalculated in a perturbative expansion in alpha_s. Compared to\nother parts of the amplitude hard\nspectator interactions are formally enhanced by the hard collinear\nscale sqrt{Lambda m_b}, which occurs next to the m_b-scale and\nleads to an enhancement of alpha_s.\n\nFrom a technical point of view the main challenges of this calculation\nare due to the fact that we have to deal with Feynman\nintegrals that come with up to five external legs and with three \nindependent ratios of scales. These Feynman integrals have to be\nexpanded in powers of Lambda/m_b.\nI will discuss integration by parts identities to reduce the \nnumber of master integrals and differential equations techniques to\nget their power expansions. A concrete implementation of integration by\nparts identities in a computer algebra system is given in the\nappendix.\n\nFinally I discuss numerical issues like scale dependence of the\namplitudes and branching ratios. It will turn out that the NLO\ncontributions of the hard spectator interactions are important but\nsmall enough for perturbation theory to be valid.