Improving Bounds on Penguin Pollution in B → ππ

In the presence of penguin contributions, the indirect CP asymmetry in B(t) → ππ measures sin(2α + 2θ), where 2θ parametrizes the size of the penguin “pollution.” We present a new upper bound on |2θ|, involving also the measurement of B → ππ, along with an upper bound on or a measurement of BR(B → ππ)+BR(B0 → ππ). The new bound is stronger than those previously discussed in the literature. Indeed, since the bound results from the requirement that the two isospin triangles close and have a common base, it is the most stringent bound possible on |2θ|, without separate measurements of B → ππ and B0 → ππ. [email protected] [email protected] [email protected] [email protected] Over the past decade or so, a great deal of attention has been focussed on CP violation in the B system. By measuring α, β and γ, the three interior angles of the unitarity triangle, it will be possible to test the standard model (SM) explanation of CP violation [1]. Indeed, the first measurements of β have already been reported [2], and it is hoped that we will soon have definitive evidence of CP violation in B decays. For the measurement of the angle α, a principal decay mode considered is B(t) → ππ. (The decays B(t) → ρπ → πππ [3] and B d,s(t) → KK̄ [4] can also be used to cleanly obtain α.) Unfortunately, this mode suffers from a wellknown problem: penguin contributions may be large [5], and their presence will spoil the clean extraction of α. This problem of penguin “pollution” can be eliminated with the help of an isospin analysis [6]. By measuring the rates for B → ππ and B/B0 → ππ, in addition to B(t) → ππ, the penguin contributions can be eliminated so that α can again be measured cleanly. However, the isospin analysis itself suffers from a potential practical complication: it requires separate measurements of BR(B → ππ) and BR(B0 → ππ). This may be a problem for several reasons. First, these branching ratios are expected to be smaller than B → ππ. Second, the presence of two π’s in the final state means that the reconstruction efficiency is also smaller. And third, in order to measure the two branching ratios individually, it will be necessary to tag the decaying B or B0 meson, which will further reduce the measurement efficiency. The upshot is that it may not be possible to measure either of these two branching ratios, or we may only have information (i.e. an actual measurement or an upper limit) on the sum of the branching ratios. In either case, a full isospin analysis cannot be carried out. But this then begs the question: assuming that we have, at best, only partial knowledge of the sum of BR(B → ππ) and BR(B0 → ππ), can we at least put bounds on the size of penguin pollution? To be more precise: in the presence of penguin amplitudes, the CP asymmetry in B(t) → ππ does not measure sin 2α, but rather sin(2α + 2θ), where 2θ parametrizes the effect of the penguin contributions. Is it possible to constrain θ? As demonstrated by Grossman and Quinn [7], the answer to this question is yes. They were able to show that |2θ| can be bounded even if we have only an upper limit on the sum of BR(B → ππ) and BR(B0 → ππ). Charles [8] also examined this question, and found an improvement to the Grossman-Quinn bound involving the direct asymmetry in B → ππ, as well as an independent bound involving different measurements. The main purpose of this Letter is to present a new bound on |2θ| which is an improvement on both the Grossman-Quinn and Charles bounds. In contrast to the earlier bounds, the new bound follows from the requirements that the two isospin triangles close and have a common base, making it the most stringent bound possible on |2θ|. Indeed, the new bound contains the two previous bounds as limiting cases. We also present the constraints on the sum of BR(B → ππ) and BR(B0 → ππ) which follow from the requirement of closure of the triangles. As we will show, if BR(B → ππ)/BR(B → ππ) is larger than one, as present experimental central values suggest, the branching ratios for B/B0 → ππ cannot be tiny. In this case, it may well be possible to carry out the full isospin analysis. Finally, we