A Lower Bound on sin 2β from Minimal Flavour Violation

We point out that there exists an absolute lower bound on sin 2β in all models with minimal flavour violation (MFV), that do not have any new operators beyond those present in the Standard Model and in which all flavour changing transitions are governed by the CKMmatrix with no new phases beyond the KM phase. This bound depends only on |Vcb|, |Vub/Vcb| and the hadronic parameters B̂K , FBd √ B̂d and ξ relevant for the CP-violating parameter ε and the B d,s − B̄ d,s mixings. Performing a simple scanning over the present ranges of these parameters we find sin 2β ≥ 0.34. We illustrate how this bound could become stronger when our knowledge of the parameters in question improves and when the upper bound on the B s − B̄ s mixing ((∆M)s) will be experimentally known. Provided the future accurate measurements of sin 2β through the CP asymmetry in B d(B̄ 0 d) → ψKS will confirm the low values recently reported by BaBar and Belle, there is a likely possibility that this class of models will be excluded. This would firmly imply the necessity of new CP-violating phases and/or new effective operators in the weak effective Hamiltonians forK−K̄ and B d,s−B̄ d,s mixings. We also point out that within the MFV models there exists also an absolute lower bound on the angle γ. We find sin γ ≥ 0.24. This lower bound could become stronger in the future.