Isospin constraints from/on \(B \to \pi\pi\)

Abstract

Abstract.

The Standard Model constraints on \(\alpha\) which can be derived from the \(B\rightarrow\pi\pi\) decays are revisited in some depth. As experimental inputs, the \(B^0\rightarrow\pi^ + \pi^-\), \(B^ + \rightarrow\pi^ + \pi^0\) decays complemented by the \(B^0\rightarrow\pi^0\pi^0\) decays, the CP parameters \(S_{\pi\pi}\) and \(C_{\pi\pi}\), and/or the value of \(\alpha\) as determined by the global CKM fit are used. The constraints discussed here are model independent in the sense that they rely only on Isospin symmetry, following the Gronau-London proposal. A new bound on \(\mathcal{B}^{00}\) and the function \(C_{00}(\mathcal{B}^{00})\) are introduced. While another bound applied to BABAR results is shown to imply that \(\cos(2\alpha_{\rm eff})\) is negative. The Grossman-Quinn bound is rediscussed. A close form expression is given for \(\alpha\) as a function of the measurements. Various scenarios for the future of the isospin analysis are explored. To probe the Standard Model the \((\mathcal{B}^{00},C_{00})\) plane is introduced.