The mixing induced CP asymmetry, a_{J/Ψ K⁰}, in the B_d → J/Ψ K⁰ decays allows to determine the sine of the angle 2β almost without any hadronic uncertainties. In this case the CP asymmetry appears in the interference between amplitudes describing decays with and without mixing. The simples case is when the final state is a specific CP state. The following time dependent asymmetry is studied :

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 ~ λ⁴ ). It implies that C_f ~ 0 and the expression simplify to :

The constraint on sin2β is obtained combining the experimental measurement on B⁰ → J/Ψ K⁰ decays to the data-driven determination of the theoretical error, obtained from the time-dependent asymmetry of B⁰ → J/Ψ π⁰ and the determination of decay rate and the C parameter of the time-dependent asymmetry in B⁰ → J/Ψ K⁰, following the prescription of hep-ph/0507290. The constraint on sin2β from B⁰ decays to K⁰ and higher charmonium states is not included, since the experimental information to determine the theoretical error on that is missing. The world averages for the experimental observables (taken from HFAG when not specified) are summarized in the table below

Using these values, we obtain the following shapes for the theoretical error (top-left) and the value (top-right) of sin2β, shown also on the

(EPS) [JPG] theoretical error on sin2β ΔS = 0.000 ± 0.012	(EPS) [JPG] Distribution of sin2β from B0 → J/Ψ K0 sin2β = 0.668 ± 0.028
(EPS) [JPG] sin2β bound from B0 → J/Ψ K0 shown on the (ρ, η) plane