Hard spectator interactions in B to pi pi at order alpha_s^2

TL;DR

This paper calculates the next-to-leading order hard spectator interaction amplitude in B to pi pi decays within QCD factorization, providing a systematic method for expanding Feynman integrals and analyzing their numerical impact.

Contribution

It introduces a systematic differential equation method for expanding Feynman integrals in powers of dotQCD/m_b at NLO in B to pi pi decays.

Findings

NLO hard spectator contributions are significant but manageable.

The method simplifies the expansion of Feynman integrals in QCD calculations.

Perturbation theory remains valid with these corrections.

Abstract

I compute 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 $\lqcd/m_b$, where the short-distance physics can be calculated in a perturbative expansion in $\alpha_s$. In this calculation it is necessary to obtain an expansion of Feynman integrals in powers of $\Lambda_\text{QCD}/m_b$. 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.