Constraint from \sin 2\beta

The mixing induced CP asymmetry in the B^0 \rightarrow J/\psi K^0 decay allows one to determine the sine of the angle 2\beta almost without any hadronic uncertainties. In this case the CP asymmetry appears in the interference between amplitudes describing decays with and without mixing. The simplest case is when the final state is a specific CP state. The following time dependent asymmetry is studied:

\mathcal{A}=\frac{\Gamma(B^0 (t) \rightarrow J/\psi K^0) - \Gamma(\bar{B}^0 (t) \rightarrow J/\psi K^0)}{\Gamma(B^0 (t) \rightarrow J/\psi K^0) + \Gamma(\bar{B}^0 (t) \rightarrow J/\psi K^0)}=C \cos \Delta m_d t + S \sin \Delta m_d t

The decay is dominated by a single (tree level) amplitude (the same process can be described by a Penguin diagram which brings corrections at order \lambda ^4). It implies that C \simeq 0 and the expression becomes:

\mathcal{A}= - \eta_{CP} \sin 2\beta \sin \Delta m_d t

The constraint on \sin 2\beta is obtained combining the experimental measurement of the B^0 \rightarrow J/\psi K^0 decay to the data-driven determination of the theoretical error, obtained from the time-dependent asymmetry of B^0 \rightarrow J/\psi \pi^0 and the determination of decay rate and the C parameter of the time-dependent asymmetry in B^0 \rightarrow J/\psi K^0, following the prescription of hep-ph/0507290. The constraint on \sin 2\beta from B^0 decays to K^0 and higher charmonium states is not included, since the experimental information to determine the theoretical error on that is missing.

The experimental values we use are summarized in the Table of Inputs. The representation of this constraint in the (\bar{\rho},~\bar{\eta}) plane is given below.


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