Model Independent Bound on the Unitarity Triangle fromCPViolation inB→π+π−andB

Abstract

We derive model independent lower bounds on the CKM parameters (1 −ρ) andη as functions of the mixing-induced CP asymmetry S in B → π + π − and sin 2β from B → ψK S . The bounds do not depend on specific results of theoretical calculations for the penguin contribution to B → π + π − . 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 onρ andη are close to their actual values.

Penguin Pollution Estimates Relevant for the Extraction of

“…[21,22]. While not leading to a direct constraint on α, which is the main focus of this review, it offers an alternative way of using the same data as in the isospin analysis to obtain an information on the CKM unitarity triangle and is thus worth reviewing.…”

“…The error onη is thus small, at the order of a few percent. In the original proposal [21,22] the strong phase δ was taken to be small, so that cos δ → 1 with corrections only of second order in δ. These corrections are small if charming penguins are perturbative (in QCD factorization δ = 0.15 ± .25 [22]).…”

“…An interesting way of using 1/m b expansion to obtain precise constraints in the ρ, η plane was presented in Ref. [19,20]. A crucial observation by Buchalla…”

The Unitary Triangle Through $B\rightarrow \rho^{\pm} \pi^{\mp}$ Decays

“…In [9,10], it has been shown that the sensitivity ofη, in B d → π + π − modes, on the strong phase φ ππ is rather mild, since its dependence enters inη only at second order. Hence, using the lowest order result in φ ππ is most likely a very good approximation to the exact result (see [9,10]). In an analogous manner, the same argument holds for B d → ρ + ρ − channels.…”

Theory of hadronic B decays

“…• The Buchalla-Safir bound [24]. This assumes that |δ c | ≤ π/2, which leads to a lower bound on η as a function of S ππ .…”