Strategies for Fixing the CKM-angle γ and Obtaining Experimental Insights into the World of Electroweak Penguins

Using the SU(3) flavour symmetry of strong interactions, we propose strategies for extracting both the CKM-angle γ and the b̄ → ūus̄ tree-level amplitude T ′. We present also an approximate approach using the branching ratios for the modes B → πK, B d → π−K+, B̄ d → π+K− and B → ππ which should be rather promising from the experimental point of view. The quantities γ and T ′ determined this way may well be used as an input to control electroweak penguins in nonleptonic B-decays as has been discussed in previous work. Following these lines, we propose strategies for obtaining quantitative insights into the physics of the electroweak penguin operators and performing some consistency checks. As a by-product, we derive an upper bound of 6◦ for the uncertainty originating from electroweak penguins in the α-determination by means of B → ππ decays. Strategies for the determination of the angle γ in the unitarity triangle [1, 2] are among the central issues of present particle physics phenomenology. Although there are already methods on the market allowing an absolutely clean measurement of this quantity (see e.g. refs. [3]-[6]), they are quite challenging for experimentalists. An interesting approach to measure both weak and strong phases by using SU(3) triangle relations among B → {ππ, πK,KK̄} decays and making some plausible dynamical assumptions (neglect of annihilation topologies, etc.) was proposed last year by Gronau, Hernández, London and Rosner [7]-[12]. Unfortunately, similar to the situation arising in certain nonleptonic B-meson decays [13]-[18], electroweak penguins may have a considerable impact on this approach and may in particular preclude a clean determination of the CKM-angle γ [19, 20]. In order to eliminate the electroweak penguin contributions, Gronau et al. have constructed an amplitude quadrangle involving B → πK decays [20] that can be used in principle to extract γ. However, this approach is very difficult from the experimental point of view, since one diagonal of the quadrangle corresponds to the decay Bs → πη which is expected to have a very small branching ratio at the O(10−7) level. Recently, Deshpande and He have presented another SU(3)-based method [21] which uses the charged B-decays B− → {π−K̄0, π0K−, ηK−, π−π0} and is unaffected by electroweak penguins as well. Although this approach is more promising for experimentalists – the relevant branching ratios are O(10−5) – it suffers from η − η′−mixing and other SU(3)-breaking effects. In a recent publication [22], it has been shown that the b̄→ s̄ electroweak penguin amplitude (cu − cd)P ′ EW can be determined from B → πK decays having branching ratios O(10−5) if one uses the CKM-angle γ as one of the central inputs and makes some reasonable approximations. Since electroweak penguins are – in contrast to QCD penguins [23] – dominated to a good approximation by internal top-quark exchanges, the b̄ → s̄ electroweak penguin amplitude can be related to the b̄ → d̄ amplitude (cu − cd)PEW by using the SU(3) flavour symmetry of strong interactions. The knowledge of (cu − cd)PEW allows in particular to investigate whether the uncertainty ∆α in the Gronau–London–method [24] of measuring the CKM-angle α introduced by electroweak penguin effects is really as small as is expected from theoretical estimates [19, 20]. As we shall see below, an upper limit for this uncertainty is given by |∆α| < ∼ 6◦. In this letter we would like to discuss some other applications of the approach [22] which allow quantitative insights into the physics of the electroweak penguin operators and provide interesting tests of the Standard Model of electroweak interactions [25]. As the CKM-angle γ is one of the central ingredients of this method, let us begin our discussion by presenting a new strategy for extracting this quantity. To this end we consider the decays B → πK and B d → π−K+. If we apply the same nota-