Click here for a list of measurements (including updates) which have been released since the cutoff for inclusion in this set of averages
Measurements related to the CKM angle β ≡ φ_{1}:
Measurements related to the CKM angle α ≡ φ_{2}:
Legend: if not stated otherwise,
We use Combos v3.20 (homepage, manual) for the rescaling of the experimental results to common sets of input parameters.
The experimental results have been rescaled to a common set of input parameters (see table below).
Parameter  Value  Ref. / Comments 

τ(B_{d})  (1.527 ± 0.008) ps  HFAG  Oscillations/Lifetime 
Δm_{d}  (0.508 ± 0.004) ps^{−1}  HFAG  Oscillations/Lifetime 
A_{⊥}^{2} (CPodd fraction in B^{0}→ J/ψK* CP sample) 
0.245 ± 0.015 ± 0.004 (note: acceptancecorrected central value; the uncorrected value is: 0.230) 
BABAR: PRD 71 (2005) 032005 
0.195 ± 0.012 ± 0.008  Belle: PRL 95 (2005) 091601  
0.217 ± 0.010  Average 
Additional note on commonly treated (correlated) systematic effects:
We obtain for sin(2β) ≡ sin(2φ_{1}) in the different decay modes:
Parameter: sin(2β) ≡ sin(2φ_{1})  

Mode  BABAR  Belle  Average  Ref. / Comments 
Charmonium:  N(BB)=227m  N(BB)=386m   
BABAR: PRL 94 (2005) 161803
BELLECONF0569 (hepex/0507037) 
J/ψK_{S}, ψ(2S)K_{S}, χ_{c1}K_{S}, η_{C}K_{S} (η_{CP}=1)  0.75 ± 0.04_{stat}  0.668 ± 0.047_{stat} (J/ψK_{S} only)  
J/ψK_{L} (η_{CP}=+1)  0.57 ± 0.09_{stat}  0.619 ± 0.069_{stat}  
J/ψK*^{0} (K*^{0} → K_{S}π^{0})  0.96 ± 0.32_{stat}    
All charmonium  0.722 ± 0.040 ± 0.023  0.652 ± 0.039 ± 0.020 
0.685 ± 0.032
(0.028_{statonly}) 
CL = 0.27 
Including earlier sin(2β) ≡ sin(2φ_{1}) measurements using B_{d} → J/ψK_{S} decays:
Parameter: sin(2β) ≡ sin(2φ_{1})  

Experiment  Value  Ref. / Comments  
ALEPH  0.84 ^{+0.82}_{−1.04} ± 0.16  PL B492 (2000) 259274  
OPAL  3.2 ^{+1.8}_{−2.0} ± 0.5  EPJ C5 (1998) 379388  
CDF (full Run I)  0.79 ^{+0.41}_{−0.44}(stat+syst)  PRD 61 (2000) 072005 
we find the only slightly modified average:
Parameter: sin(2β) ≡ sin(2φ_{1})  

All charmonium  0.687 ± 0.032 (0.028_{statonly})  CL = 0.48 
from which we obtain the following solutions for β ≡ φ_{1} (in [0, π])
β ≡ φ_{1} = (21.7 ^{+1.3}_{−1.2})°  or  β ≡ φ_{1} = (68.3 ^{+1.2}_{−1.3})° 
Plots:
Average of sin(2β) ≡ sin(2φ_{1}) from all experiments. 
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Averages of sin(2β) ≡ sin(2φ_{1}) and C=A from the B factories. 
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eps.gz png 

Constraint on the ρbarηbar plane: 
eps.gz png 
eps.gz png 
Constraining the Unitarity Triangle (ρ, η):
Visit the CKMfitter and UTfit sites for results on global CKM fits using different fit techniques and input quantities. 
Historically the experiments determined λ for the charmonium modes; more recently the parameters C = −A = (1−λ^{2})/(1+λ^{2}) are being used, as they are in all other timedependent CP analyses. We recompute C from λ for the following averages.
Parameter: C=−A (if not stated otherwise)  

Mode  BABAR  Belle  Average  Ref. / Comments 
Charmonium  N(BB)=227m  N(BB)=386m   
BABAR: PRL 94 (2005) 161803
BELLECONF0569 (hepex/0507037) 
J/ψK_{S}, ψ(2S)K_{S}, χ_{c1}K_{S}, η_{C}K_{S} 
λ = 0.950 ± 0.031 ± 0.013
C = 0.051 ± 0.033 ± 0.014 
C = 0.021 ± 0.034_{stat} (J/ψK_{S} only)  
J/ψK_{L}  C = −0.049 ± 0.039_{stat}  
J/ψK*^{0} (K*^{0} → K_{S}π^{0})    
All charmonium  −0.010 ± 0.026 ± 0.036 
0.026 ± 0.041
(0.020_{statonly}) 
CL = 0.31 
Digression and plots:
Constraining C_{J/ψ Ks} from
A_{CP}(B^{+} → J/ψ K^{+})
and A_{SL}:
As suggested by Y. Nir, one can obtain a powerful SM constraint on λ = q/pAbar/A via the relations A_{SL} = (1−q/p^{4})/(1+q/p^{4}) and A_{CP}(B^{+} → J/ψ K^{+}) = (Abar/A^{2}−1)/(Abar/A^{2}+1), where A_{SL} denotes the CP asymmetry in semileptonic B decays, and A_{CP}(B^{+} → J/ψ K^{+}) is the CPviolating charge asymmetry measured in B^{+} → J/ψ K^{+} decays. We take the A_{SL} result from the HFAG oscillation group, A_{SL} = −0.0026 ± 0.0067, and average the A_{CP}(B^{+} → J/ψ K^{+}) results from BABAR, Belle and CLEO, to find A_{CP}(B^{+} → J/ψ K^{+}) = −0.007 ± 0.019. These give q/p = 1.0013 ± 0.0034 and Abar/A = 0.993 ± 0.018, and hence λ_{indirect} = 0.994 ± 0.018, corresponding to C = 0.006 ± 0.018. Discussion: the amplitude relation between neutral and charged B → J/ψ K decays has been found by FleischerMannel to hold up to negligible corrections of the order O(λ^{3}). However, the identification of λ, measured through the C coefficient in B^{0} → J/ψ K^{0}, with q/pAbar/A assumes ΔΓ_{Bd}=0. The systematic error on C from a width difference ΔΓ_{Bd}/Γ_{Bd}~0.02 has been estimated by BABAR to be 0.0009. 
Parameter: sin(2β^{eff}) ≡ sin(2φ_{1}^{eff})  

Mode  BABAR N(BB)=209227m 
Belle N(BB)=386m 
Average  Ref. / Comments 
φK^{0}  0.50 ± 0.25 ^{+0.07}_{−0.04}  0.44 ± 0.27 ± 0.05  0.47 ± 0.19 CL=0.87 (0.2σ) 
BABAR: PRD 71 (2005) 091102(R)
BELLECONF0569 (hepex/0507037) 
η′K^{0}  0.36 ± 0.13 ± 0.03  0.62 ± 0.12 ± 0.04  0.50 ± 0.09 CL=0.16 (1.4σ) 
BABARCONF05/014 (hepex/0507087)
BELLECONF0569 (hepex/0507037) 
f_{0}K_{S}  0.95 ^{+0.23}_{−0.32} ± 0.10  0.47 ± 0.36 ± 0.08  0.75 ± 0.24 CL=0.32 (1.0σ) 
BABARCONF04/019 (hepex/0408095)
BELLECONF0569 (hepex/0507037) 
π^{0}K_{S}  0.35 ^{+0.30}_{−0.33} ± 0.04  0.22 ± 0.47 ± 0.08  0.31 ± 0.26 CL=0.82 (0.2σ) 
BABAR: PRD71 (2005) 111102
BELLECONF0569 (hepex/0507037) 
π^{0}π^{0}K_{S}  −0.84 ± 0.71 ± 0.08    −0.84 ± 0.71  BABARCONF05/020 (hepex/0508017) 
ωK_{S}  0.51 ^{+0.35}_{−0.39} ± 0.02  0.95 ± 0.53 ^{+0.12}_{−0.15}  0.64 ± 0.30 CL=0.49 (0.7σ) 
BABARPUB06/016 (hepex/0603040)
(submitted to PRL)
BELLECONF0569 (hepex/0507037) 
ρ^{0}K_{S}  0.17 ± 0.52 ± 0.26    0.17 ± 0.58  IN PREPARATION 
K^{+}K^{−}K^{0}
(excluding φK^{0}) 
0.41 ± 0.18 ± 0.07 ± 0.11_{CPeven}
(f_{CPeven}= 0.89 ± 0.08 ± 0.06 [moments]) 
0.60 ± 0.18 ± 0.04 ^{+0.19}_{−0.12}_{CPeven}
(f_{CPeven}= 0.93 ± 0.09 ± 0.05 [SU(2)]) 
0.51 ± 0.14 ^{+0.11}_{−0.08} CL=0.38 (0.9σ) (rescaled to average f_{CPeven}= 0.91 ± 0.07) 
BABARCONF05/002 (hepex/0507016)
BELLECONF0569 (hepex/0507037) 
K_{S}K_{S}K_{S}  0.63 ^{+0.28}_{−0.32} ± 0.04  0.58 ± 0.36 ± 0.08  0.61 ± 0.23 CL=0.92 (0.1σ) 
BABARCONF05/012 (hepex/0507052)
BELLECONF0569 (hepex/0507037) 
Naïve b→s penguin average  0.50 ± 0.06  CL=0.82 (0.2σ)  
Direct comparison of charmonium and spenguin averages (see comments below): CL=0.0092 (2.6σ) 
Please note that
The cosine coefficient:
Parameter: C=−A (if not stated otherwise)  

Mode  BABAR N(BB)=209227m 
Belle N(BB)=386m 
Average  Ref. / Comments 
φK^{0}  0.00 ± 0.23 ± 0.05  −0.14 ± 0.17 ± 0.07  −0.09 ± 0.14 CL=0.64 (0.5σ) 
BABAR: PRD 71 (2005) 091102(R)
BELLECONF0569 (hepex/0507037) 
η′K^{0}  −0.16 ± 0.09 ± 0.02  0.04 ± 0.08 ± 0.06  −0.07 ± 0.07 CL=0.14 (1.5σ) 
BABARCONF05/014 (hepex/0507087)
BELLECONF0569 (hepex/0507037) 
f_{0}K_{S}  −0.24 ± 0.31 ± 0.15  0.23 ± 0.23 ± 0.13  0.06 ± 0.21 CL=0.28 (1.1σ) 
BABARCONF04/019 (hepex/0408095)
BELLECONF0569 (hepex/0507037) 
π^{0}K_{S}  0.06 ± 0.18 ± 0.03  −0.11 ± 0.18 ± 0.08  −0.02 ± 0.13 CL=0.53 (0.6σ) 
BABAR: PRD71 (2005) 111102
BELLECONF0569 (hepex/0507037) 
π^{0}π^{0}K_{S}  0.27 ± 0.52 ± 0.13    0.27 ± 0.54  BABARCONF05/020 (hepex/0508017) 
ωK_{S}  −0.55 ^{+0.28}_{−0.26} ± 0.03  −0.19 ± 0.39 ± 0.13  −0.44 ± 0.23 CL=0.46 (0.7σ) 
BABARPUB06/016 (hepex/0603040)
(submitted to PRL)
BELLECONF0569 (hepex/0507037) 
ρ^{0}K_{S}  0.64 ± 0.41 ± 0.25    0.64 ± 0.48  IN PREPARATION 
K^{+}K^{−}K^{0}
(excluding φK^{0}) 
0.23 ± 0.13  0.06 ± 0.11 ± 0.07  0.15 ± 0.09 CL=0.36 (0.9σ) 
BABARCONF05/002 (hepex/0507016)
BELLECONF0569 (hepex/0507037) 
K_{S}K_{S}K_{S}  −0.10 ± 0.25 ± 0.05  −0.50 ± 0.23 ± 0.06  −0.31 ± 0.17 CL=0.25 (1.1σ) 
BABARCONF05/012 (hepex/0507052)
BELLECONF0569 (hepex/0507037) 
Naïve b→s penguin average  −0.04 ± 0.04  CL=0.25 (1.2σ) 
Compilation of results for −η×S ≈ sin(2β^{eff}) ≡ sin(2φ_{1}^{eff}) and C from spenguin decays. 
eps png 
eps png 
Same, but without π^{0}π^{0}K_{S} and ρ^{0}K_{S}, to allow closer inspection of the detail. 


Naïve spenguin averages 


Comparisons of averages in the different b→q qbar s modes 
eps png 
eps png 
Same, but without π^{0}π^{0}K_{S} and ρ^{0}K_{S}, to allow closer inspection of the detail. 


2D comparisons of averages in the different b→q qbar s modes.
Taken from the PDG 2005 review on "CP Violation in Meson Decays" by D.Kirkby and Y.Nir. * This plot (and the averages) assume no correlations between the S and C measurements in each mode. 
eps png 
The BABAR and Belle collaborations have performed measurements of sin(2β) & cos(2β) ≡ sin(2φ_{1}) & cos(2φ_{1}) in timedependent transversity analyses of the pseudoscalar to vectorvector decay B^{0}→ J/ψK*, where cos(2β) ≡ cos(2φ_{1}) enters as a factor in the interference between CPeven and CPodd amplitudes. In principle, this analysis comes along with an ambiguity on the sign of cos(2β) ≡ cos(2φ_{1}) due to an incomplete determination of the strong phases occurring in the three transversity amplitudes. BABAR resolves this ambiguity by inserting the known variation of the rapidly moving Pwave phase relative to the slowly moving Swave phase with the invariant mass of the Kπ system in the vicinity of the K*(892) resonance. The result is in agreement with the prediction obtained from squark helicity conservation. It corresponds to Solution II defined by Suzuki, which is the phase convention used for the averages given here.
Experiment  sin(2β) ≡ sin(2φ_{1})_{J/ψK*}  cos(2β) ≡ cos(2φ_{1})_{J/ψK*}  Correlation  Ref. / Comments 

BABAR'04
N(BB)=88m 
−0.10 ± 0.57 ± 0.14  3.32 ^{+0.76}_{−0.96} ± 0.27  −0.37  PRD 71 (2005) 032005 
Belle'04
N(BB)=275m 
0.24 ± 0.31 ± 0.05  0.56 ± 0.79 ± 0.11
[using Solution II] 
+0.22  PRL 95 (2005) 091601 
Average  IN PREPARATION (CL = → xσ) 
IN PREPARATION (CL = → xσ) 
?  See remark below table 
Note that due to the strong nonGaussian character of the BABAR measurement (although the result is far positive, the confidence level for cos(2β)>0 is only 89%), the interpretation of the average given above has to be done with the greatest care.
Due to possible significant penguin pollution, both the cosine and the sine coefficients of the Cabibbosuppressed b → ccbar d decays are free parameters of the theory. Absence of penguin pollution would result in S_{ccbar d} = − η_{CP} sin(2β) ≡ − η_{CP} sin(2φ_{1}) and C_{ccbar d} = 0 for the CP eigenstate final states (η_{CP} = +1 for both J/ψπ^{0} and D^{+}D^{−}). For the nonCP eigenstates D*^{+−}D^{−+}, absence of penguin pollution (ie. no direct CP violation) gives A = 0, C_{+} = −C_{−} (but is not necessarily zero), S_{+} = 2 R sin(2β+δ)/(1+R^{2}) and S_{−} = 2 R sin(2β−δ)/(1+R^{2}). [With alternative notation, S_{+} = 2 R sin(2φ_{1}+δ)/(1+R^{2}) and S_{−} = 2 R sin(2φ_{1}−δ)/(1+R^{2})]. Here R is the ratio of the magnitudes of the amplitudes for B^{0} → D*^{+}D^{−} and B^{0} → D*^{−}D^{+}, while δ is the strong phase between them. If there is no CP violation of any kind, then S_{+} = −S_{−} (but is not necessarily zero). The vectorvector final state D*^{+}D*^{−} is a mixture of CPeven and CPodd; the longitudinally polarized component is CPeven. Note that in the general case of nonnegligible penguin contributions, the penguintree ratio and strong phase differences do not have to be the same for each helicity amplitude (likewise, they do not have to be the same for D*^{+}D^{−} and D*^{−}D^{+}).
At present we do not apply a rescaling of the results to a common, updated set of input parameters.
Experiment  S_{J/ψπ0}  C_{J/ψπ0} = −A_{J/ψπ0}  Correlation  Ref. / Comments  

BABAR'06
N(BB)=232m 
−0.68 ± 0.30 ± 0.04  −0.21 ± 0.26 ± 0.06  0.08  PRD 74 (2006) 011101  
Belle'04
N(BB)=152m 
−0.72 ± 0.42 ± 0.09  0.01 ± 0.29 ± 0.03  −0.12  PRL 93 (2004) 261801  
Average  −0.69 ± 0.25 CL=0.94 (0.1σ) 
−0.11 ± 0.20 CL=0.58 (0.6σ) 
uncorrelated averages  

eps png 
eps png 
Experiment  S_{D+D−}  C_{D+D−}  Correlation  Ref. / Comments 

BABAR'05
N(BB)=232m 
−0.29 ± 0.63 ± 0.06  0.11 ± 0.35 ± 0.06    PRL 95 (2005) 131802 
Belle  NOT YET AVAILABLE  NOT YET AVAILABLE     
Average  −0.29 ± 0.63  0.11 ± 0.36     
We convert Im(λ) = S/(1 + C) and λ^{2} = (1 − C)/(1 + C), taking into account correlations:
Experiment  S_{D*+D*−}  C_{D*+D*−}  Correlation  Ref. / Comments  

BABAR'05
N(BB)=227m 
−0.75 ± 0.25 ± 0.03  0.06 ± 0.17 ± 0.03  Table 
PRL 95 (2005) 151804
R_{T} = 0.125 ± 0.044 ± 0.007 

Belle'04
N(BB)=152m 
−0.75 ± 0.56 ± 0.10 ± 0.06[R_{T}]  0.26 ± 0.26 ± 0.05 ± 0.01[R_{T}]   
PLB 618 (2005) 34
R_{T} = 0.19 ± 0.08 ± 0.01 

Average^{(}*^{)}  −0.75 ± 0.23 CL=1.00 (0.0σ) 
0.12 ± 0.14 CL=0.52 (0.6σ) 
uncorrelated averages  

eps png 
eps png 
^{(}*^{)} Note that we have not preaveraged the CPodd fractions (and then accordingly rescaled the average sine coefficient). Since both data samples are independent, the results are (approximately) invariant under such a treatment, compared to the direct average that is performed here. Also: due to the lack of correlations coefficients, we have performed an uncorrelated average here.
Experiment  S_{+−}(D*^{+}D^{−})  C_{+−}(D*^{+}D^{−})  S_{−+}(D*^{−}D^{+})  C_{−+}(D*^{−}D^{+})  A(D*^{+−}D^{−+})  Correlations  Ref. / Comments 

BABAR'05
N(BB)=232m 
−0.54 ± 0.35 ± 0.07  0.09 ± 0.25 ± 0.06  −0.29 ± 0.33 ± 0.07  0.17 ± 0.24 ± 0.04    PRL 95 (2005) 131802  
Belle'04 N(BB)=152m (combined fully and partially rec. B decays) 
−0.55 ± 0.39 ± 0.12  −0.37 ± 0.22 ± 0.06  −0.96 ± 0.43 ± 0.12  0.23 ± 0.25 ± 0.06  0.07 ± 0.08 ± 0.04    PRL 93 (2004) 201802 
Average  −0.54 ± 0.27 CL=0.99 (0.0σ) 
−0.16 ± 0.17 CL=0.18 (1.3σ) 
−0.53 ± 0.27 CL=0.23 (1.2σ) 
0.20 ± 0.18 CL=0.87 (0.2σ) 
0.07 ± 0.09  uncorrelated averages 
Compilation of results for (left) sin(2β^{eff}) ≡ sin(2φ_{1}^{eff}) = −η_{CP}S and (right) C from timedependent b → ccbar d analyses with CP eigenstate final states. The results are compared to the values from the corresponding charmonium averages. 
eps png 
eps png 
B_{d} decays to final states such as Dπ^{0} are governed by the b → cubar d transitions. If one chooses a final state which is a CP eigenstate, ie. D_{CP}π^{0}, the usual timedependence formulae are recovered, with the sine coefficient sensitive to sin(2β) ≡ sin(2φ_{1}). Since there is no penguin contribution to these decays, there is even less associated theoretical uncertainty than for b → ccbar s decays like B_{d} → J/ψ K_{S}.
Bondar, Gershon and Krokovny have shown that when multibody D decays, such as D → K_{S}π^{+}π^{−} are used, a timedependent analysis of the Dalitz plot of the D decay allows a direct determination of the weak phase: β ≡ φ_{1}. Equivalently, both sin(2β) ≡ sin(2φ_{1}) and cos(2β) ≡ cos(2φ_{1}) can be measured. This information allows to resolve the ambiguity in the measurement of 2β ≡ 2φ_{1} from sin(2β) ≡ sin(2φ_{1}) alone.
Results of such an analysis are available from Belle. The decays B_{d} → Dπ^{0}, B_{d} → Dη, B_{d} → Dω, B_{d} → D*π^{0} and B_{d} → D*η are used. The daughter decays are D* → Dπ^{0} and D → K_{S}π^{+}π^{−}.
At present we do not apply a rescaling of the results to a common, updated set of input parameters.
Experiment  sin(2β) ≡ sin(2φ_{1})  cos(2β) ≡ cos(2φ_{1})  Reference / comments 

Belle'05
N(BB)=386m 
0.78 ± 0.44 ± 0.22  1.87 ^{+0.40}_{−0.53} ^{+0.22}_{−0.32}  hepex/0605023 (submitted to PRL) 
Note that direct measurements of phases often suffer from unusual and nonGaussian behaviour. Belle determine the sign of cos(2φ)_{1} to be positive at 98.3% confidence level.
Timedependent analyses of radiative b decays such as B^{0}→ K_{S}π^{0}γ, probe the polarization of the photon. In the SM, the photon helicity is dominantly lefthanded for b → sγ, and righthanded for the conjugate process. As a consequence, B^{0} → K_{S}π^{0}γ behaves like an effective flavor eigenstate, and mixinginduced CP violation is expected to be small  a simple estimation gives: S ~ −2(m_{s}/m_{b})sin(2β) ≡ −2(m_{s}/m_{b})sin(2φ_{1}) (with an assumption that the Standard Model dipole operator is dominant). Corrections to the above may allow values of S as large as 10% in the SM.
Atwood et al. have shown that (with the same assumption) an inclusive analysis with respect to K_{S}π^{0} can be performed, since the properties of the decay amplitudes are independent of the angular momentum of the K_{S}π^{0} system. However, if nondipole operators contribute significantly to the amplitudes, then the Standard Model mixinginduced CP violation could be larger than the expectation given above, and the CPV parameters may vary slightly over the K_{S}π^{0}γ Dalitz plot, for example as a function of the K_{S}π^{0} invariant mass.
An inclusive K_{S}π^{0}γ analysis has been performed by Belle using the invariant mass range up to 1.8 GeV/c^{2}. Belle also gives results for the K*(892) region: 0.8 GeV/c^{2} to 1.0 GeV/c^{2}. BABAR has measured the CPviolating asymmetries separately within and outside the K*(892) mass range: 0.8 GeV/c^{2} to 1.0 GeV/c^{2} is again used for K*(892)γ candidates, while events with invariant masses in the range 1.1 GeV/c^{2} to 1.8 GeV/c^{2} are used in the "K_{S}π^{0}γ (not K*(892)γ)" analysis.
We quote two averages: one for K*(892) only, and the other one for the inclusive K_{S}π^{0}γ decay (including the K*(892)). If the Standard Model dipole operator is dominant, both should give the same quantities (the latter naturally with smaller statistical error). If not, care needs to be taken in interpretation of the inclusive parameters; while the results on the K*(892) resonance remain relatively clean.
At present we do not apply a rescaling of the results to a common, updated set of input parameters.
Mode  Experiment  S_{Ksπ0γ}  C_{Ksπ0γ} = −A_{Ksπ0γ}  Correlation  Ref. / Comments 

K*(892)γ 
BABAR'05
N(BB)=232m 
−0.21 ± 0.40 ± 0.05  −0.40 ± 0.23 ± 0.04  −0.064  PRD 72 (2005) 051103 
Belle'05
N(BB)=386m 
0.01 ± 0.52 ± 0.11  −0.11 ± 0.33 ± 0.09  0.002  BELLECONF0570 (hepex/0507059)  
Average  −0.13 ± 0.32 CL=0.74 (0.3σ) 
−0.31 ± 0.19 CL=0.48 (0.7σ) 
  uncorrelated average  
K_{S}π^{0}γ
(incl. K*γ) 
BABAR'05
N(BB)=232m 
−0.06 ± 0.37  −0.48 ± 0.22  −0.066  PRD 72 (2005) 051103 
Belle'05
N(BB)=386m 
0.08 ± 0.41 ± 0.10  −0.12 ± 0.27 ± 0.10  0.004  BELLECONF0570 (hepex/0507059)  
Average  0.00 ± 0.28 CL=0.80 (0.3σ) 
−0.35 ± 0.17 CL=0.32 (1.0σ) 
  uncorrelated average  

eps png K*γ only: eps png K_{S}π^{0} only: eps png 
eps png K*γ only: eps png K_{S}π^{0} only: eps png 
Please note that at present we do not apply a rescaling of the results to a common, updated set of input parameters. Correlation due to common systematics are neglected in the following averages. We recall that we do NOT rescale (inflate) the errors due to measurement inconsistencies.
Experiment  S_{ππ}  C_{ππ} = −A_{ππ}  Correlation  Ref. / Comments  

BABAR'04
N(BB)=227m 
−0.30 ± 0.17 ± 0.03  −0.09 ± 0.15 ± 0.04  −0.016  PRL 95 (2005) 151803  
Belle'05
N(BB)=275m 
−0.67 ± 0.16 ± 0.06  −0.56 ± 0.12 ± 0.06  −0.09  PRL 95 (2005) 101801  
Average  −0.50 ± 0.12  −0.37 ± 0.10  −0.056  χ^{2} = 7.9 (CL=0.019 ⇒ 2.3σ)  

eps gif gif (high res) 
eps gif gif (high res) 
Digression and plots:
The following numerical exercises involve the SU(2) and SU(3) partners of the B_{d}→ π^{+}π^{−} decay. The relevant branching ratios and CPviolating charge asymmetries are taken from HFAG  Rare Decays.
Constraining α ≡ φ_{2}: using as input the measured C_{ππ} and S_{ππ} coefficients together with the present (HFAG) ππ branching fractions and CP asymmetries (including the direct CPasymmetry measurement for B^{0}→ π^{0}π^{0}), one can perform the GronauLondon isospin analysis. The plot on the right hand side uses the statistical interpretation of the CKMfitter analysis (Rfit) (see the UTfit pages for a Bayesian interpretation). Here, it does not include the Fiertz treatment of electroweak penguins for ππ. Including it would lead to a shift in α ≡ φ_{2} of approximately −2 deg. All other SU(2)breaking effects are also neglected. 
eps gif gif (high res) 
The Penguintotree ratio:
using as input the measured
C_{ππ} and S_{ππ} coefficients
together with the Wolfenstein parameters ρ and η
from the Global CKM fit using standard constraints,
one can infer module and phase of the complex penguin
to tree (P/T) ratio
in B_{d}→ π^{+}π^{−}
decays within the Standard Model.
Note that the definition of P/T is conventiondependent
(see, e.g.,
GroRos02).
One can choose to eliminate the charm quark in
the penguin loop using CKM unitarity, so that the amplitudes are
parameterized as follows:

The "Quasitwobody" (Q2B)
CP(t) analysis of B_{d}→ ρ^{+−}π^{−+} decays
(performed by Belle) assumes a narrow width approximation for the ρ meson.
The interference regions in the π^{+}π^{−}π^{0}
Dalitz plot are removed by kinematic cuts.
Dilution of the CP results
due to residual interference effects is not accounted
for in the systematic errors.
The Q2B analysis involves 5 different parameters, one of which
− the charge asymmetry A_{CP}(ρπ) − is timeindependent.
The timedependent decay rate is given by
Γ( B → ρ^{+−}π^{−+} (Δt) )
= (1 +− A_{CP}(ρπ)) e^{−Δt/τ}/8τ
× [1 + Q_{tag}(S_{ρπ}+−ΔS_{ρπ})sin(ΔmΔt)
− Q_{tag}(C_{ρπ}+−ΔC_{ρπ})cos(ΔmΔt)],
where Q_{tag}=+1(−1) when the tagging meson is a
B^{0} (B^{0}bar).
CP symmetry is violated if either one of the following conditions is true:
A_{CP}(ρπ)≠0, C_{ρπ}≠0 or S_{ρπ}≠0.
The first two correspond to CP violation in the decay, while the last
condition is CP violation in the interference of decay amplitudes
with and without B_{d} mixing. Note that the BABAR analysis
uses a full Dalitz plot approach and hence avoids the systematic effects
due to the Q2B approximation.
At present we do not apply a rescaling of the results due to the fit dependence on the B_{d} lifetime and oscillation frequency.
Experiment  A_{CP}(ρπ)  S_{ρπ}  C_{ρπ}  ΔS_{ρπ}  ΔC_{ρπ}  Correlations  Ref. / Comments 

BABAR'04
N(BB)=213m 
−0.088 ± 0.049 ± 0.013  −0.10 ± 0.14 ± 0.04  0.34 ± 0.11 ± 0.05  0.22 ± 0.15 ± 0.03  0.15 ± 0.11 ± 0.03  Table  BABARCONF04/038, hepex/0408099 
Belle'04
N(BB)=152m 
−0.16 ± 0.10 ± 0.02  −0.28 ± 0.23 ^{+0.10}_{−0.08}  0.25 ± 0.17 ^{+0.02}_{−0.06}  −0.30 ± 0.24 ± 0.09  0.38 ± 0.18 ^{+0.02}_{−0.04}  Table  PRL 94 (2005) 121801 
Average  −0.102 ± 0.045  −0.13 ± 0.13  0.31 ± 0.10  0.09 ± 0.13  0.22 ± 0.10  Table  
Significance of CPV in the decay: Δχ^{2} = χ^{2}(Acp=C=0) − χ^{2} = 14.5 (CL = 0.00070, that is: 3.4σ) 
Digression and plots:
CP violation in the decay:
as shown by
Charles
it is convenient to transform the experimentally motivated
CP parameters A_{CP}(ρπ) and C_{ρπ}
into the physically motivated ones
Taking into account experimental
correlations,
one finds

eps gif gif (high res) 

Flavourcharge specific branching fractions:

The BABAR Collaboration has performed a full timedependent Dalitz plot analysis of the decay B_{d} → (ρπ)^{0} → π^{+}π^{−}π^{0}, which allows to simultaneously determine the complex decay amplitudes and the CPviolating weak phase α ≡ φ_{2}. The analysis follows the idea of Snyder and Quinn (1993). BABAR uses a model that consists of charged and neutral ρ(770) resonances and their radial excitations ρ(1450) and ρ(1700). No nonresonant contributions are found. BABAR determines 16 coefficients of the form factor bilinears from the fit to data. The unknown amplitude parameters, among which are the phases δ_{+−}=arg[A^{−+}A^{+−*}] and the UT angle α ≡ φ_{2}, are determined from a subsequent fit to the 16 bilinear coefficients.
Experiment  α ≡ φ_{2} (°)  δ_{+−} (°)  Ref. / Comments  

BABAR'04
N(BB)=213m 
113 ^{+27}_{−17} ± 6  −67 ^{+28}_{−31} ± 7  BABARCONF04/038, hepex/0408099  
Belle  NOT YET AVAILABLE    
Confidence levels for α ≡ φ_{2} (left hand plot) and δ_{+−} (right hand plot) as found by BABAR 
eps gif gif (high res) 
eps gif gif (high res) 
Note that Dalitz plot phases are nonGaussian quantities in general. Only marginal constraints are obtained beyond 2σ.
The vector particles in the pseudoscalar to vectorvector decay B_{d} → ρ^{+}ρ^{−} can have longitudinal and transverse relative polarization with different CP properties. The BABAR Collaboration determines the fraction of longitudinally polarized events with an angular analysis to be f_{long} = 0.978 ± 0.014^{+0.021}_{−0.029}, so that a perevent transversity analysis can be avoided and only the longitudinal CP parameters are determined. The Belle Collaboration determine the same quantity to be f_{long} = 0.941 ^{+0.034}_{−0.040} ±0.030
At present we do not apply a rescaling of the results to a common, updated set of input parameters.
Experiment  S_{ρρ,long}  C_{ρρ,long}  Correlation  Ref. / Comments 

BABAR'05
N(BB)=232m 
−0.33 ± 0.24 ^{+0.08}_{−0.14}
f_{long} = 0.978 ± 0.014^{+0.021}_{−0.029} 
−0.03 ± 0.18 ± 0.09  −0.042  PRL 95 (2005) 041805 
Belle'06
N(BB)=275m 
0.08 ± 0.41 ± 0.09
f_{long} = 0.941 ^{+0.034}_{−0.040} ± 0.030 
0.00 ± 0.30 ± 0.09  0.06  PRL 96 (2006) 171801 
Average  −0.21 ± 0.22  −0.03 ± 0.17  0.01 
Digression and plots:
The following numerical exercises involve the SU(2) partners of the B_{d} → ρ^{+}ρ^{−} decay. The relevant branching ratios, CPviolating charge asymmetries and fractions of longitudinal polarization are taken from HFAG  Rare Decays averages.
Constraining α ≡ φ_{2}:
Using as input the measured C_{ρρ,long} and S_{ρρ,long} coefficients together with the average (HFAG) ρρ branching fractions and longitudinal polarization fractions (including the limit on ρ^{0}ρ^{0}, for which the polarization is unknown), one can perform the GronauLondon isospin analysis (electroweak penguins can be taken into account, while other SU(2)breaking effects are usually neglected). Plots for confidence level representations of the P/T phase versus its module can be found on the corresponding CKMfitter and UTfit pages. 
The Penguintotree ratio:
Using as input the measured C_{ρρ,long} and S_{ρρ,long} coefficients together with the Wolfenstein parameters ρ and η using standard constraints, one can infer module and phase of the complex penguin to tree (P/T) ratio as done in the ππ case. Plots for confidence level representations of the P/T phase versus its module can be found on the corresponding CKMfitter and UTfit pages. 
Combined α ≡ φ_{2}
constraint from b → uubar d transitions:
Averaging the confidence level curves from the ππ and ρρ isospin analyses as well as the ρπ Dalitz plot analysis, leads to the combined constraint: 
eps gif 
Neutral B meson decays such as B_{d} → D^{+−}π^{−+}, B_{d} → D*^{+−}π^{−+} and B_{d} → D^{+−}ρ^{−+} provide sensitivity to γ ≡ φ_{3} because of the interference between the Cabibbofavoured amplitude (e.g. B^{0} → D^{−}π^{+}) with the doubly Cabibbosuppressed amplitude (e.g. B^{0} → D^{+}π^{−}). The relative weak phase between these two amplitudes is −γ ≡ −φ_{3} and, when combined with the B_{d}B_{d}bar mixing phase, the total phase difference is −(2β+γ) ≡ −(2φ_{1}+φ_{3}).
The size of the CP violating effect in each mode depends on the ratio of magnitudes of the suppressed and favoured amplitudes, e.g., r_{Dπ} = A(B^{0} → D^{+}π^{−})/A(B^{0} → D^{−}π^{+}). Each of the ratios r_{Dπ}, r_{D*π} and r_{Dρ} is expected to be about 0.02, and can be obtained experimentally from the corresponding suppressed charged B decays, (e.g., B^{+} → D^{+}π^{0}) using isospin, or from selftagging decays with strangeness (e.g., B^{0} → D_{s}^{+}π^{−}), using SU(3). In the latter case, the theoretical uncertainties are hard to quantify. The smallness of the r values makes direct extractions from, e.g., the D^{+−}π^{−+} system very difficult.
Both BABAR and Belle exploit partial reconstructions of D*^{+−}π^{−+} to increase the available statistics. Both experiments also reconstruct D^{+−}π^{−+} and D*^{+−}π^{−+} fully, and BABAR includes the mode D^{+−}ρ^{−+}. Additional states with similar quark content are also possible, but for vectorvector final states an angular analysis is required, while states containing higher resonances may suffer from uncertainties due to nonresonant or other contributions.
BABAR and Belle use different observables:
Here we convert the Belle results to express them in terms of a and c. Explicitly, the conversion reads:
Belle D*π (partial reconstruction):  a_{π}* = − (S^{+} + S^{−})/2 
c_{π}* = − (S^{+} − S^{−})/2  
Belle D*π (full reconstruction):  a_{π}* = + ( 2 R_{D*π} sin( 2φ_{1}+φ_{3} + δ_{D*π} ) + 2 R_{D*π} sin( 2φ_{1}+φ_{3} − δ_{D*π} ) )/2 
c_{π}* = + ( 2 R_{D*π} sin( 2φ_{1}+φ_{3} + δ_{D*π} ) − 2 R_{D*π} sin( 2φ_{1}+φ_{3} − δ_{D*π} ) )/2  
Belle Dπ (full reconstruction):  a_{π} = − ( 2 R_{Dπ} sin( 2φ_{1}+φ_{3} + δ_{Dπ} ) + 2 R_{Dπ} sin( 2φ_{1}+φ_{3} − δ_{Dπ} ) )/2 
c_{π} = − ( 2 R_{Dπ} sin( 2φ_{1}+φ_{3} + δ_{Dπ} ) − 2 R_{Dπ} sin( 2φ_{1}+φ_{3} − δ_{Dπ} ) )/2 
At present we do not rescale the results to a common set of input parameters. Also, common systematic errors are not considered.
Observable  BABAR  Belle  Average  Ref. / Comments  

partially reconstructed N(BB)=232m 
fully reconstructed N(BB)=232m 
partially reconstructed N(BB)=386m 
fully reconstructed N(BB)=386m 

a_{D*π}  −0.034 ± 0.014 ± 0.009  −0.040 ± 0.023 ± 0.010  −0.041 ± 0.019 ± 0.017  −0.039 ± 0.020 ± 0.013  −0.037 ± 0.011
CL=0.96 (0.0σ) 
BABAR: PRD 71 (2005) 112003 (partially reco.) BABARPUB06/001 (hepex/0602049) (fully reco.) Belle: PRD 73, 092003 (2006) 
  COMBINED: −0.040 ± 0.014 ± 0.011  
c_{D*π}  −0.019 ± 0.022 ± 0.013
(lepton tags only) 
0.049 ± 0.042 ± 0.015
(lepton tags only) 
−0.007 ± 0.019 ± 0.017  −0.011 ± 0.020 ± 0.013  −0.006 ± 0.014 CL=0.41 (0.8σ) 

  COMBINED: −0.009 ± 0.014 ± 0.011  
a_{Dπ}    −0.010 ± 0.023 ± 0.007    −0.050 ± 0.021 ± 0.012  −0.030 ± 0.017
CL=0.24 (1.2σ) 

c_{Dπ}    −0.033 ± 0.042 ± 0.012
(lepton tags only) 
  −0.019 ± 0.021 ± 0.012  −0.022 ± 0.021
CL=0.78 (0.3σ) 

a_{Dρ}    −0.024 ± 0.031 ± 0.009      −0.024 ± 0.033  
c_{Dρ}    −0.098 ± 0.055 ± 0.018
(lepton tags only) 
    −0.098 ± 0.058 
Compilation of the above results. 
eps.gz png 
eps.gz png 
Averages of the D*π results. 
eps.gz png 
eps.gz png 
Digression:
Constraining 2β+γ ≡
2φ_{1}+φ_{3}:
The constraints can be tightened if one is willing to use theoretical input on the values of R and/or δ. One popular choice is the use of SU(3) symmetry to obtain R by relating the suppressed decay mode to B decays involving D_{s} mesons. For more information, visit the CKMfitter and UTfit sites. 
eps.gz png 
eps.gz png CL: eps.gz png 
eps.gz png 
eps.gz png CL: eps.gz png 
A theoretically clean measurement of the angle γ ≡ φ_{3} can be obtained from the rate and asymmetry measurements of B^{−} → D^{(}*^{)}_{CP}K^{(}*^{)−} decays, where the D^{(}*^{)} meson decays to CP even (D^{(}*^{)}_{CP+}) and CP odd (D^{(}*^{)}_{CP−}) eigenstates. The method benefits from the interference between the dominant b→cubar s transitions with the corresponding doubly CKMsuppressed b→ucbar s transition. It was proposed by Gronau, Wyler and Gronau, London (GLW).
BABAR and Belle use consistent definitions for A_{CP+−} and R_{CP+−}, where
A_{CP+−} = [Γ(B^{−} → D^{(}*^{)}_{CP+−}K^{(}*^{)}^{−}) − Γ(B^{+} → D^{(}*^{)}_{CP+−}K^{(}*^{)}^{+})] / Sum , 
R_{CP+−} = [Γ(B^{−} → D^{(}*^{)}_{CP+−}K^{(}*^{)}^{−}) + Γ(B^{+} → D^{(}*^{)}_{CP+−}K^{(}*^{)}^{+})] / [Γ(B^{−} → D^{(}*^{)0} K^{(}*^{)}^{−}) + Γ(B^{+} → D^{(}*^{)0}bar K^{(}*^{)}^{+})]. 
Experimentally, it is convenient to measure R_{CP+−} using double ratios, in which similar ratios for B^{−} → D^{(}*^{)} π^{(}*^{)}^{−} decays are used for normalization.
These observables have been measured so far for three D^{(}*^{)}K^{(}*^{)−} modes. Both Belle and BABAR use the CP even D decays to K^{+}K^{−} and π^{+}π^{−} in all three modes; both experiments also use only the D* → Dπ^{0} decay, which gives CP(D*) = CP(D). For CPodd D decay modes, Belle use K_{S}π^{0}, K_{S}φ and K_{S}ω in all three analyses, and also use K_{S}η in DK^{−} and D*K^{−} analyses. BABAR use K_{S}π^{0}, K_{S}φ and K_{S}ω for DK^{−} analysis.
At present we do not rescale the results to a common set of input parameters. Also, common systematic errors are not considered.
Mode  Experiment  A_{CP+}  A_{CP−}  R_{CP+}  R_{CP−}  Ref. / Comments 

D_{CP}K^{−} 
BABAR'05
N(BB)=232m 
0.35 ± 0.13 ± 0.04  −0.06 ± 0.13 ± 0.04  0.90 ± 0.12 ± 0.04  0.86 ± 0.10 ± 0.05  PRD 73, 051105 (2006) 
Belle'06
N(BB)=275m 
0.06 ± 0.14 ± 0.05  −0.12 ± 0.14 ± 0.05  1.13 ± 0.16 ± 0.08  1.17 ± 0.14 ± 0.14  PRD 73, 051106 (2006)  
Average  0.22 ± 0.10  −0.09 ± 0.10  0.98 ± 0.10  0.94 ± 0.10  
D*_{CP}K^{−} 
BABAR'04
N(BB)=123m 
−0.10 ± 0.23 ^{+0.03}_{−0.04}    1.06 ± 0.26 ^{+0.10}_{−0.09}    PRD 71 (2005) 031102 
Belle'06
N(BB)=275m 
−0.20 ± 0.22 ± 0.04  0.13 ± 0.30 ± 0.08  1.41 ± 0.25 ± 0.06  1.15 ± 0.31 ± 0.12  PRD 73, 051106 (2006)  
Average  −0.15 ± 0.16  0.13 ± 0.31  1.25 ± 0.19  1.15 ± 0.33  
D_{CP}K*^{−} 
BABAR'05
N(BB)=232m 
−0.08 ± 0.19 ± 0.08  −0.26 ± 0.40 ± 0.12  1.96 ± 0.40 ± 0.11  0.65 ± 0.26 ± 0.08  PRD 72 (2005) 071103(R) 
Belle  NO RESULTS AVAILABLE (*)    
Average  −0.08 ± 0.21  −0.26 ± 0.42  1.96 ± 0.41  0.65 ± 0.27 
Compilation of the above results. 
eps.gz png 
eps.gz png 
CP+ only 
eps.gz png 
eps.gz png 
CP only 
eps.gz png 
eps.gz png 
Digression:
Constraining γ ≡ φ_{3}:
The rate ratios and asymmetries of the GLW method can be
expressed in terms of amplitude ratios and strong phase differences,
as well as the weak phase difference γ ≡ φ_{3}.
For the GLW observables, one has:
where r_{B} = A(b→u)/A(b→c) and δ_{B} = arg[A(b→u)/A(b→c)]. Only the weak phase difference γ ≡ φ_{3} is universal, while the other parameters depend on the decay process. Plots upcoming. 
A modification of the GLW idea has been suggested by Atwood, Dunietz and Soni, where B^{−} → DK^{−} with D → K^{+}π^{−} (or similar) and the charge conjugate decays are used. Here, the favoured (b→c) B decay followed by the doubly CKMsuppressed D decay interferes with the suppressed (b→u) B decay followed by the CKMfavored D decay. The relative similarity of the combined decay amplitudes enhances the possible CP asymmetry. BABAR and Belle use consistent definitions for A_{Kπ} and R_{Kπ}, where
A_{Kπ} = [Γ(B^{−} → [K^{+}π^{−}]_{D(*)}K^{(}*^{)}^{−}) − Γ(B^{+} → [K^{−}π^{+}]_{D(*)}K^{(}*^{)}^{+})] / [Γ(B^{−} → [K^{+}π^{−}]_{D(*)}K^{(}*^{)}^{−}) + Γ(B^{+} → [K^{−}π^{+}]_{D(*)}K^{(}*^{)}^{+})] , 
R_{Kπ} = [Γ(B^{−} → [K^{+}π^{−}]_{D(*)}K^{(}*^{)}^{−}) + Γ(B^{+} → [K^{−}π^{+}]_{D(*)}K^{(}*^{)}^{+})] / [Γ(B^{−} → [K^{−}π^{+}]_{D(*)}K^{(}*^{)}^{−}) + Γ(B^{+} → [K^{+}π^{−}]_{D(*)}K^{(}*^{)}^{+})] . 
(Some of) these observables have been measured so far for the D^{(}*^{)}K^{−} modes.
At present we do not rescale the results to a common set of input parameters. Also, common systematic errors are not considered.
Mode  Experiment  A_{Kπ}  R_{Kπ}  Ref. / Comments 

DK^{−}
D→Kπ 
BABAR'05
N(BB)=232m 
  0.013 ^{+0.011}_{−0.009}  PRD 72 (2005) 032004 
Belle'05
N(BB)=386m 
  0.000 ± 0.008 ± 0.001  BELLECONF0552 (hepex/0508048)  
Average    0.006 ± 0.006  
D*K^{−}
D* → Dπ^{0} D→Kπ 
BABAR'05
N(BB)=232m 
  −0.002^{+0.010}_{−0.006}  PRD 72 (2005) 032004 
Average    −0.002^{+0.010}_{−0.006}  
D*K^{−}
D* → Dγ D→Kπ 
BABAR'05
N(BB)=232m 
  0.011^{+0.018}_{−0.013}  PRD 72 (2005) 032004 
Average    0.011^{+0.018}_{−0.013}  
DK*^{−}
D→Kπ 
BABAR'05
N(BB)=232m 
−0.22 ± 0.61 ± 0.17  0.046 ± 0.031 ± 0.008  BABARPUB05/039 (hepex/0508001) 
Average  −0.22 ± 0.63  0.046 ± 0.032 
Compilation of the above results. 
eps.gz png 
Digression:
Constraining γ ≡ φ_{3}: As for the GLW method, the rate ratios and asymmetries of the ADS method can be expressed in terms of amplitude ratios and strong phase differences, as well as the weak phase difference γ ≡ φ_{3}. For the ADS observables, one has:
where r_{B} = A(b→u)/A(b→c) and δ_{B} = arg[A(b→u)/A(b→c)] as before. r_{D} and δ_{D} are the corresponding amplitude ratio and strong phase difference of the D meson decay amplitudes. The value of r_{D}^{2} is obtained from the ratio of the suppressedtoallowed branching fractions BR(D^{0} → K^{+}π^{−}) = (1.38 ± 0.11)×10^{−4} and BR(D^{0} → K^{−}π^{+}) = (3.80 ± 0.09)×10^{−2} [PDG 2004], respectively. With this it is found r_{D} = 0.0603 ± 0.0025. The strong phase is different, in general, for D and D* mesons. Bondar and Gershon have pointed out that there is an effective strong phase shift of π between the cases that D* is reconstructed in the Dπ^{0} and Dγ final states, which in principle allows γ ≡ φ_{3} to be measured using the ADS technique with B^{+−} → D* K^{+−} alone. Plots upcoming. 
As can be seen from the expressions above, the maximum size of the asymmetry, for given values of r_{B} and r_{D} is given by: A_{Kπ} (max) = 2r_{B}r_{D} / (r_{B}^{2}+r_{D}^{2}). Thus, sizeable asymmetries may be found also for B^{−} → D^{(}*^{)}π^{−} decays, despite the expected smallness (~0.01) of r_{B} for this case, providing sensitivity to γ ≡ φ_{3}. The observables have been measured by Belle in the Dπ^{−} mode.
Mode  Experiment  A_{Kπ}  R_{Kπ}  Ref. / Comments 

Dπ^{−}
D→Kπ 
Belle'05
N(BB)=386m 
0.10 ± 0.22 ± 0.06  0.0035 ^{+0.0008}_{−0.0007} ± 0.0003  BELLECONF0552 (hepex/0508048) 
Average  0.10 ± 0.23  0.0035 ^{+0.0009}_{−0.0008} 
Another method to extract γ ≡ φ_{3} from the interference between B^{−} → D^{(}*^{)0} K^{−} and B^{−} → D^{(}*^{)0}bar K^{−} uses multibody D decays. A Dalitz plot analysis allows simultaneous determination of the weak phase difference γ ≡ φ_{3}, the strong phase difference δ_{B} and the ratio of amplitudes r_{B}. This idea was proposed by Giri, Grossman, Soffer and Zupan and the Belle Collaboration. The assumption of a D decay model results in an additional model uncertainty.
Results are available from both Belle and BABAR using B^{−} → D K^{−}, B^{−} → D*K^{−} and B^{−} → DK*^{−}. Belle use the D* decay to Dπ^{0} only, while BABAR also use Dγ, and take the effective strong phase shift into account. In all cases the decay D → K_{S}π^{+}π^{−} is used.
For the DK*^{−} mode, both collaborations use K^{*−} → K_{S}π^{−}; in this case some care is needed due to other possible contributions to the B^{−} → DK_{S}π^{−} final state. Belle assign an additional (model) uncertainty, while BABAR use an alternative parametrization [replacing r_{B} and δ_{B} with κr_{s} and δ_{s}, respectively] suggested by Gronau. Due to these different treatments, the averages for this mode should be treated with care.
If the values of γ ≡ φ_{3}, δ_{B} and r_{B} are obtained by directly fitting the data, the extracted value of r_{B} is biased (since it is positive definite by nature). Since the error on γ ≡ φ_{3} depends on the value of r_{B} some statistical treatment is necessary to correctly estimate the uncertainty. To obviate this effect, both experiments now use a different set of variables in the fits:
x_{+} = r_{B} cos( δ_{B}+γ ) ≡ r_{B} cos( δ_{B}+φ_{3} )  y_{+} = r_{B} sin( δ_{B}+γ ) ≡ r_{B} sin( δ_{B}+φ_{3} ) 
x_{−} = r_{B} cos( δ_{B}−γ ) ≡ r_{B} cos( δ_{B}−φ_{3} )  y_{−} = r_{B} sin( δ_{B}−γ ) ≡ r_{B} sin( δ_{B}−φ_{3} ) 
These parameters have the advantage of having (approximately) Gaussian distributions, and of having small statistical correlations. Some statistical treatment is necessary to convert these measurements into constraints on the underlying physical parameters γ ≡ φ_{3}, δ_{B} and r_{B} [BaBar do not obtain constraints on r_{B} and δ_{B} for the B^{−} → DK*^{−} decay due to the reparametrization described above]. Both experiments use frequentist procedures, though there are differences in the details.
The results below have three sets of errors, which are statistical, systematic, and model related uncertainties respectively. Note that the Belle results for (x_{+}, y_{+}, x_{−}, y_{−}) do not include model uncertainty. (See Appendix of Ref.) Since the model uncertainty should be 100% correlated between experiments, we average the values of (x_{+}, y_{+}, x_{−}, y_{−}) without including this source of error, and then add it back on to the averaged values.
Mode  Experiment  x_{+}  y_{+}  x_{−}  y_{−}  Correlations  Ref. / Comments 

DK^{−}
D→K_{S}π^{+}π^{−} 
BABAR'05
N(BB)=227m 
−0.13 ± 0.07 ± 0.03 ± 0.03  0.02 ± 0.08 ± 0.02 ± 0.02  0.08 ± 0.07 ± 0.03 ± 0.02  0.06 ± 0.09 ± 0.04 ± 0.04   
BABAR:
PRL 95, 121802 (2005); BABARCONF05/018 (hepex/0507101) Belle: hepex/0604054 (submitted to PRD) 
Belle'06
N(BB)=386m 
−0.14 ± 0.07 ± 0.02  −0.09 ± 0.09 ± 0.01  0.03 ^{+0.07}_{−0.08} ± 0.01  0.17 ^{+0.09}_{−0.12} ± 0.02  See Appendix of Ref.  
Average  −0.14 ± 0.05 ± 0.03  −0.03 ± 0.06 ± 0.02  0.05 ± 0.05 ± 0.02  0.11 ± 0.07 ± 0.04  UNCORRELATED AVERAGES  
D*K^{−}
D*→Dπ^{0} & D*→Dγ D→K_{S}π^{+}π^{−} 
BABAR'05
N(BB)=227m 
0.14 ± 0.09 ± 0.03 ± 0.03  0.01 ± 0.12 ± 0.04 ± 0.06  −0.13 ± 0.09 ± 0.03 ± 0.02  −0.14 ± 0.11 ± 0.02 ± 0.03    
Belle'06
N(BB)=386m 
0.03 ± 0.12 ± 0.01  0.01 ± 0.14 ± 0.01  −0.13 ^{+0.17}_{−0.15} ± 0.02  −0.34 ^{+0.17}_{−0.16} ± 0.03  See Appendix of Ref.  
Average  0.10 ± 0.07 ± 0.03  0.01 ± 0.09 ± 0.06  −0.13 ± 0.08 ± 0.02  −0.20 ± 0.09 ± 0.03  UNCORRELATED AVERAGES  
DK^{*−}
K^{*−} → K_{S}π^{−} D→K_{S}π^{+}π^{−} 
BABAR'05
N(BB)=227m 
−0.07 ± 0.23 ± 0.13 ± 0.03  −0.01 ± 0.32 ± 0.18 ± 0.05  −0.20 ± 0.20 ± 0.11 ± 0.03  0.26 ± 0.30 ± 0.16 ± 0.03    
Belle'06
N(BB)=386m 
−0.11 ^{+0.18}_{−0.17} ± 0.01  0.00 ± 0.16 ± 0.01  −0.78 ^{+0.25}_{−0.30} ± 0.03  −0.28 ^{+0.44}_{−0.34} ± 0.05  See Appendix of Ref.  
Average  −0.10 ± 0.15 ± 0.03  0.00 ± 0.15 ± 0.05  −0.46 ± 0.17 ± 0.03  0.05 ± 0.27 ± 0.03  UNCORRELATED AVERAGES 
Digression:
Constraining γ/φ_{3}: The measurements of x_{+,−} and y_{+,−} in the various D^{(}*^{)}K^{(}*^{)} decay modes can be used to place bounds on γ ≡ φ_{3}. Both experiments have done so using frequentist techniques.  
BABAR obtain
γ = (67 ± 28 ± 13 ± 11)° 
Belle obtain
φ_{3} = (53 ^{+15}_{−18} ± 3 ± 9)° 

The experiments also obtain values for the hadronic parameters  
r_{B} (DK^{−}) = 0.12 ± 0.08 ± 0.03 ± 0.04  δ _{B} (DK^{−}) = (104 ± 45 ^{+17}_{−21} ^{+16}_{−24})°  r_{B} (DK^{−}) = 0.16 ± 0.05 ± 0.01 ± 0.05  δ _{B} (DK^{−}) = (146 ^{+19}_{−20} ± 3 ± 23)° 
r_{B} (D*K^{−}) = 0.17 ± 0.10 ± 0.03 ± 0.03  δ _{B} (D*K^{−}) = (296 ± 41 ^{+14}_{−12} ± 15)°  r_{B} (D*K^{−}) = 0.18 ^{+0.11}_{−0.10} ± 0.01 ± 0.05  δ _{B} (D*K^{−}) = (302 ^{+34}_{−35} ± 6 ± 23)° 
.  .  r_{B} (DK*^{−}) = 0.56 ^{+0.22}_{−0.16} ± 0.04 ± 0.08  δ _{B} (DK*^{−}) = (243^{+20}_{−23} ± 3 ± 50 )° 
Note that the above results suffer an ambiguity: γ → γ + π ≡ φ_{3} → φ_{3} + π, δ → δ + π. We quote the result which is consistent with the Standard Model fit. 