Model Independent Bound on the Unitarity Triangle from CP Violation in B → π+π− and B → ψKS

We derive model independent lower bounds on the CKM parameters (1− ρ̄) and η̄ as functions of the mixing-induced CP asymmetry S in B → ππ and sin 2β fromB → ψKS. The bounds do not depend on specific results of theoretical calculations for the penguin contribution to B → ππ. They require only the very conservative condition that a hadronic phase, which vanishes in the heavy-quark limit, does not exceed 90 in magnitude. The bounds are effective if − sin 2β ≤ S ≤ 1. Dynamical calculations indicate that the limits on ρ̄ and η̄ are close to their actual values. PACS numbers: 11.30.Er, 12.15.Hh, 13.25.Hw In the standard model CP violation is described by a single complex phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix of quark mixing. After the first observation of CP violation with KL → ππ decays in 1964 [1], the standard model scenario has passed a crucial test with the recent discovery of a large CP asymmetry in B → ψKS decays [2,3] in agreement with earlier predictions [4]. Beyond this qualitative confirmation of standard model CP violation, efforts are now increasingly directed towards precision studies, which could reveal the existence of new physics. A large amount of experimental data continues to be collected at current and future B-meson facilities, both at dedicated ee colliders and at hadron machines. To be useful for precision tests of weak interactions the observables must not depend on poorly understood hadronic quantities. Among the rather few observables that are essentially free of such theoretical uncertainties is the timedependent CP asymmetry in B → ψKS, which measures sin 2β (where β is one of the angles of the CKM unitarity triangle). Of comparable interest is the corresponding CP asymmetry in B → ππ decays. However, a well-known problem here are penguin contributions in the decay amplitude, which introduce hadronic physics into the CP asymmetry and spoil a straightforward interpretation in terms of weak phases. A method to eliminate hadronic input based on an isospin analysis [5] appears to be very difficult for a practical implementation [6,7]. In this Letter we show how theoretically clean information on CKM parameters can be obtained from the mixing-induced CP asymmetries in B → ψKS and B → ππ, in spite of the penguin contributions in the second case. We derive lower bounds on the unitarity triangle parameters (1− ρ̄) and η̄ [8] that depend only on observables of CP violation in these channels. The time-dependent CP asymmetry in B → ππ decays is defined by A CP (t) = B(B(t) → π+π−)− B(B̄(t) → ππ) B(B(t) → ππ) +B(B̄(t) → ππ) = −S sin(∆mBt) + C cos(∆mBt) (1) where S = 2 Imξ 1 + |ξ|2 C = 1− |ξ|2 1 + |ξ|2 ξ = e −2iβ e −iγ + P/T e+iγ + P/T (2) In terms of the Wolfenstein parameters ρ̄ and η̄ the CKM phase factors read e = ρ̄± iη̄ √ ρ̄ + η̄ e = (1− ρ̄) − η̄ − 2iη̄(1− ρ̄) (1− ρ̄) + η̄ (3) The penguin-to-tree ratio P/T can be written as P T = re √ ρ̄ + η̄ (4) The real parameters r and φ defined in this way are pure strong interaction quantities without further dependence on CKM variables.