CP-Asymmetry: Observables and Minimal Theoretical Input

Penguin contributions, being not negligible in general, can hide the information on the CKM angle α coming from the measurement of the timedependent B0(t) → π+π− CP-asymmetry. Nevertheless, we show that this information can be summarized in a set of simple equations, expressing α as a multi-valued function of a single theoretically unknown parameter, which conveniently can be chosen as a well-defined ratio of penguin to tree amplitudes. Using these exact analytic expressions, free of any assumption besides the Standard Model, and some reasonable hypotheses to constrain the modulus of the penguin amplitude, we derive several new upper bounds on the penguin-induced shift |2α − 2αeff |, generalizing the recent result of Grossman and Quinn. These bounds depend on the averaged branching ratios of some decays (π0π0, K0K0, Kπ) particularly sensitive to the penguin. On the other hand, with further and less conservative approximations, we show that the knowledge of the B → Kπ branching ratio alone gives sufficient information to extract the free parameter without the need of other measurements, and without knowing |Vtd| or |Vub|. More generally, knowing the modulus of the penguin amplitude with an accuracy of ∼ 30% might result in an extraction of α competitive with the experimentally more difficult isospin analysis. We also show that our framework allows to recover most of the previous approaches in a transparent and simple way, and in some cases to improve them. In addition we discuss in detail the problem of the various kinds of discrete ambiguities.