Asymmetry A ψK S , sin 2 β , the Sign of ∆ M B and the Physics Beyond the Standard Model

We demonstrate that the sign of the B−B̄ mass difference ∆MB is irrelevant for the extraction of the angle β = φ1 of the unitarity triangle from the CP asymmetry AψKS . Only the weak phase in the mixing amplitude M12 matters. Consequently the extraction of sin 2β done by BaBar and Belle can only be affected by new weak phases in M12 in the extensions of the SM and possibly but unlikely by new physics contributions to the decay amplitude. The CP asymmetry in B d → ψKS AψKS(t) = Γ(B̄ d(t) → ψKS)− Γ(B 0 d(t) → ψKS) Γ(B̄ d(t) → ψKS) + Γ(B 0 d(t) → ψKS) (1) proposed by Bigi and Sanda [1] for the tests of CP violation in the B meson system in 1980 has been observed [2, 3]. Its value allows to extract sin 2β = 0.726± 0.037 , (2) where β = φ1 is one of the angles of the unitarity triangle. This result is in an impressive agreement with the Standard Model (SM) expectations [1, 4]. Yet (2) leads to a two fold ambiguity in the value of β βCKM = 23.3± 1.6 ◦, β̃ = π 2 − βCKM (3) with the second possibility inconsistent with the SM expectations. Measuring cos 2β will tell us which of these two solutions is chosen by nature. The first direct experimental result of BaBar [5] for cos 2β and other analyses [6, 7, 8] disfavour in fact the second solution in (3). In extracting the value given in (2) it has been assumed that the mass difference ∆MB = M1 −M2 > 0 with M1 and M2 denoting the masses of the neutral B meson eigenstates. As the sign of ∆MB is not known by itself, it is legitimate to ask what happens if ∆MB was assumed to be negative. In a recent article Bigi and Sanda [9] addressed the uniqueness of the sign in (2), reaching first the conclusion that it could not be determined irrespective of the sign of ∆MB and that with ∆MB < 0, the BaBar and Belle data imply sin 2β = −0.726±0.037. This would mean that in addition to (3) two additional solutions for the angle β exist. These findings, if correct, would weaken significantly the present believe that the BaBar and Belle data combined with the standard analysis of the unitarity triangle imply that the CKM matrix [10] is very likely the dominant source of CP violation observed in low energy experiments. In the first version of this note we challenged the conclusion of the authors of [9] by demonstrating that • The sign of ∆MB is irrelevant for the determination of sin 2β. • The only relevant quantity for this determination is the weak phase of the mixing amplitude M12. For a given phase convention of B 0 and B̄, that cancels in the final expression for the asymmetry anyway, the weak phase ofM12 is uniquely given in the SM and its possible extensions.