This page features:

	Implementation of the Beneke-Buchalla-Neubert-Sachrajda (BBNS) calculations in CkmFitter
	Implementation of the Fleischer-Mannel (FM) and Neubert-Rosner (NR) bounds in CkmFitter

Fits of the BBNS calculation to the branching ratios and  direct CP asymmetries of charged and neutral B-decays to pipi and Kpi final states have been performed. Confidence  levels are presented in the rhobar-etabar and sin2alpha-sin2beta planes.

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

We present three different fits here:

• A global fit of BBNS to B->pipi/Kpi branching ratios

The fit inputs are B-> pipi/Kpi branching ratios measured by BABAR, Belle and CLEO and the CKM element |Vcb| (See the input page and hep-ph/0104062). If not stated otherwise, all branching ratios are understood as CP-averaged branching ratios. We used here the averages taken from BBNS where the experimental errors have been symmetrised:

BR(B0 -> pi+pi-)	( 4.4 +/- 0.9)*10^-6
BR(B+ -> pi+pi0)	( 5.6 +/- 1.5)*10^-6
BR(B0 -> K+ pi-)	(17.2 +/- 1.5)*10^-6
BR(B+ -> K+ pi0)	(12.1 +/- 1.7)*10^-6
BR(B+ -> K0 pi+)	(17.2 +/- 2.5)*10^-6
BR(B0 -> K0 pi0)	(10.3 +/- 2.5)*10^-6
|Vcb|	(40.75 +/- 0.4 +/- 2.0)*10^-3

The best fit gives a chi^2 value of 1.82 with 4 degrees of freedom and is found at (rhobar,etabar)=(0.05,3.81). The best fits prefer values for gamma around 90 degrees.  The confidence levels obtained show also good agreement  with those found in the standard CKM fit:

Constraints in the rhobar-etabar plane from a combined fit of the BBNS predictions of the B->pipi/Kpi branching ratios. Also shown are the single constraints at 5% CL from the standard CKM fit, using the constraints from |Vub/Vcb|, |epsilon_K|, Deltamd and Deltams. Given in addition are the >5% and >90% CL regions obtained from a combined CKM fit (not including the charmless branching ratios and sin2beta).

• A global fit of BBNS to B->pipi/Kpi branching ratios and direct CP asymmetries ACP

The constraints obtained in the previous exercise are symmetric under the mapping eta -> -eta. Direct CP asymmetries carry also information on the sign of the CKM angle gamma and hence are sensitive to the sign of eta. In addition to the CP averaged branching ratios used in the previous exercise also direct CP asymmetries of the self-tagging B->Kpi modes are used in this exercise.

We define the direct CP asymmetry ACP as:

ACP  =  (BR(Bbar -> fbar) - BR(B -> f)) / (BR(Bbar -> fbar) + BR(B -> f))

and used averaged results from BABAR, Belle and CLEO:

ACP(B0->K+-pi-+)	-0.109 +- 0.078
ACP(B0->K+-pi0)	-0.095 +- 0.118
ACP(B0->K0 pi-+)	-0.046 +- 0.136

At present, the statistical errors of these measurements are too large to provide a significant test of the FA. The biggest effect of direct CP violation is expected in B0->pi0pi0. However, the fact that the measured direct CP asymmetries are small is consistent with the predictions of the FA. The best fit gives a chi^2 of 2.0 (7 degrees of freedom). Negative values of eta are slightly preferred. This is due to the fact that the BABAR measurement of ACP(B0->K+pi-)=-0.19+/-0.10+/-0.03, though consistent with zero, prefers negative asymmetries, whereas BBNS predicts mainly positive ACP values for positive gamma. It should be noted that the most recent, preliminary BABAR result, ACP(B0->K+pi-)= -0.07+/-0.08+/-0.02, shows even better agreement with BBNS. However, this value was not included in the fit yet.

Constraints in the rhobar-etabar plane from a fit of the BBNS predictions of the B->pipi/Kpi branching ratios, and direct CP-asymmetries to the experimental values, measured in self-tagging B->Kpi modes.

• A global fit of BBNS to B->pipi/Kpi branching ratios in combination with the standard CKM fit

To test the consistency between the constraints obtained from the BBNS fit to the B->pipi/Kpi branching ratios and those given by the standard CKM fit a combined fit has been performed. Again, a small chi^2 of 2.09 (8 degrees of freedom) is found indicating that FA and the standard CKM fit are consistent at the present stage.

Constraints in the rhobar-etabar plane from a combined fit of the BBNS predictions of the B->pipi/Kpi branching ratios, and using |Vub/Vcb|, |epsilon_K|, Deltamd and Deltams. Given in addition are the >5% and >90% CL regions obtained from a combined CKM fit not including the charmless branching ratios.

The measurement of the time-dependent asymmetry in B->pi+pi- provides information on sin(2alpha) if the penguin P over tree T amplitude ratio is known. Within BBNS the ratio P/T is calculated. Under the assumption that the FA is an appropriate description one can translate the measurement of Spipi into a constraint in the rhobar-etabar plane.

We present here the constraints in the rhobar-etabar and sin2alpha-sin2beta plane from the measurement of Spipi within BBNS. In addition, constraints are shown if the sin2beta world average is used.

Constraints in the rhobar-etabar plane from the parameter Spipi, measured by BABAR in a time-dependent B0->pi+pi- analysis, and using the BBNS calculation of the P/T ratio. Theoretical errors as quoted in hep-ph/0104110 (BBNS) have been neglected. Three examples of possible CKM triangles are shown.
See figure caption above for more details. The world average of sin2beta has been additionally used in the fit.
Constraints in the sin2alpha-sin2beta plane from the parameter Spipi. See figure caption above for more details.

Compared to the experimental error of the Spipi measurement from BABAR, the theoretical error quoted in the hep-ph/0104110 (BBNS) is much smaller and was neglected in this exercise.

Less prone to theoretical uncertainties but also less predictive are the bounds obtained on the CP-averaged ratios R (FM) and R* (NR), defined by

Their respective theoretical predictions have been implemented in CkmFitter. It is assumed for the following plots that the strong phases involved be small, in accordance with the results of the FA. All theoretical parameters used are taken from Ref. hep-ph/0104110.

The tree-to-penguin ratio entering the prediction of R* can be constrained experimentally by means of the ratio

which introduces correlations due to the common denominator with R*. The same is true when one combines R and R* in a single fit. These correlations are properly introduced in CkmFitter.

We shall annotate that the immediate fit input of the ratios implies the use of asymmetric statistical errors with complicated probability density functions (see Appendix C in hep-ph/0104062). This procedure can be circumvented by fitting the branching fractions rather than their ratios, and adjusting simultaneously a free varying scale which is common to both quantities entering the respective ratios.

We present the results in the rhobar-etabar plane:

Constrained from R* in the rhobar-etabar plane. The colors correspond to confidence levels given by the scale on the rhs of the figure. Indicated by the black shades is the region determined by theoretical uncertainties. Also shown is the constraint from the global CKM fit,< including the world average of sin2beta.
Constrained from R* and R in the rhobar-etabar plane. See figure caption above above for more details.
Constrained from R*, R and |Vub/Vcb| in the rhobar-etabar plane. See figure caption above above for more details.