In general, the constraints are presented in the parameter spaces (rhobar,etabar), the unitarity triangle angle alpha, the plane spanned by the observed coefficients of the sine and the cosine terms, (Spipi,Cpipi), as well as the relative strong phase vs. the penguin-to-tree ratio, (|P/T|,delta). It is the goal of the latter two representations to test theoretical approaches used to describe the dynamical structure of the transition amplitudes. A priori knowledge on the CKM phase, obtained from the standard CKM analysis is injected into these fits, either for comparison purposes, or as direct input to predict experimentally accessible observables or theoretical quantities.

We approach the problem following the hierarchical structure outlined below:

1. using as input Spipi and Cpipi as well the BR(B --> pipi) (all charges) and strong isospin symmetry SU(2) and neglecting electro-weak penguins.
2. using I. and B0 --> K+pi- together with SU(3) flavour symmetry and neglecting OZI-suppressed penguin annihilation topologies.
3. using I. and a phenomenological estimate of |P| by means of the decay rate of B+ --> K0pi+. The relative strong phase is unconstrained in this approach.
4. using I. and the prediction of the complex penguin-to-tree ratio, that is |P/T| and its phase, predicted by QCD Factorisation (BBNS).

Details about the analysis can be found in this preprint.

The theoretical description of hadronic charmless B-decays is under ongoing discussions. Therefore, one should consider  these fits as R&D studies when discussing confidence levels. As a consequence, all plots are  marked by an appropriate R&D logo:

#   world average B --> pipi,Kpi branching fractions (BABAR, Belle and CLEO [1][2][3]).

#   values of Cpipi and Spipi from BABAR and Belle.

#   for the standard CKM fit see the input page and hep-ph/0104062.

Confidence levels in the large (rhobar,etabar) plane obtained from the isospin analysis.
Confidence levels in the large (rhobar,etabar) plane obtained from SU(3).
Confidence levels in the large (rhobar,etabar) plane obtained from the SU(3) upper bound on |P|. Without predicting the relative strong phase, the sign information on etabar from the measurement of diret CP violation cannot be exploited. That is why this plot (as the first two) is symmetric with respect to etabar = 0.
Confidence levels in the large (rhobar,etabar) plane obtained using the QCD FA. There are strong constraints with these hypothesis. Due to the negative value of Cpipi, the etabar < 0 half plane is favoured. The comparison of the allowed regions obtained here (phases and magnitude of the tree and penguin amplitudes are predicted and their uncertainties are small with respect to the experimental errors) shows that the experimental accuracy is already competitive with the constraint from the standard CKM fit.

Predictions for Spipi and Cpipi within the four frameworks. The input values for rhobar and etabar are taken from the standard CKM fit. The relative large uncertainty of the QCD FA in Spipi is due to the weak sin2alpha constraint of the standard CKM fit. For comparison, the 1 sigma and 2 sigma contours for the experimental results from BABAR, Belle and their average are overlaid.

Constraints on the penguin-to-tree ratio and their relative strong phase delta. The shaded (coloured) region gives the CL obtained from a fit corresponding to the SU(2) approach. The CKM phase is constrained by means of the standard CKM fit. The extended contour indicates the 5% CL limit when only using Spipi and Cpipi. Also shown are the 5% CL contours corresonding to the three other approaches.

Confidence levels for alpha within the four frameworks. The CP and BRs experimental results in B --> h+h- decays are used. The radius (rhobar^2 + etabar^2)^(1/2) has been confined using in addition |Vub|. The prediction from the standard CKM fit which includes the WA of sin2beta is overlaid (dashed lines, hatched area). Significant constraints (up to the four-fold ambiguity) are obtained once the penguin-to-tree ratio is determined (the last two frameworks). Consistency with the Standard Model and comparable uncertainties are found for these cases.

Residual (alpha - alpha_eff) for the present (ICHEP'02) central values and errors as well as projected luminosities of BABAR, Belle and future B-factories (always keeping the central values unchanged). Note that due to finite binning only, the central peaks do not reach until CL=1. The full isospin analysis is employed, i.e., we assume the flavours of B0(B0bar) --> pi0pi0 are tagged. Moreover, we assume that direct CPV is absent (lambda_{pi0pi0}=1) leading to the symmetric solutions shown in the figure. When the flavour information for B0 --> pi0pi0 is missing (as currently the case), only the outer borders of the curves can be obtained from the isospin analysis while the inner structure remains unresolved because the strong phase of the tree amplitude is unconstrained. The inner structure reveals the remnants of the eight-fold ambiguity in the range alpha in {0,2pi}. For this setup of central values, only luminosities of the order of 10/ab allow to separate the mirror solutions.

Constraints on the branching fraction B0 --> pi0pi0 obtained from the measurements of B0 --> pi+pi-, B+ --> pi+pi0, B- --> pi-pi0 and Cpipi using the isospin relations (preprint). Given by the hatched area is the prediction from QCD FA: larger BR(B0 --> pi0pi0) require larger values of the unitarity triangle angle alpha. The upper plot uses Cpipi from BABAR, the lower plot uses Cpipi from Belle. The vertical arrow indicates the experimental upper limit at 90% CL found by BABAR. The horizontal arrows indicate the zone for which the CL is equal or better than 10%.