












Click here for a list of measurements (including updates) which have been released since the cutoff for inclusion in this set of averages
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  Reference 

τ(B_{d})  (1.530 ± 0.008) ps  HFAG  Oscillations/Lifetime 
Δm_{d}  (0.507 ± 0.004) ps^{−1}  HFAG  Oscillations/Lifetime 
A_{⊥}^{2} (CPodd fraction in B^{0}→ J/ψK* CP sample) 
0.233 ± 0.010 ± 0.005 
BaBar: PRD 76 (2007) 031102
N(BB)=232m 
0.195 ± 0.012 ± 0.008 
Belle: PRL 95 (2005) 091601
N(BB)=275m 

0.219 ± 0.009 
Average
χ^{2} = 4.3/1 dof (CL=0.04 ⇒ 2.1σ) 
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  Reference 
Charmonium:  N(BB)=465M  N(BB)=535M   
BaBar (PRD 79 (2009) 072009)
Belle (PRL 98 (2007) 031802) 
J/ψK_{S} (η_{CP}=1)  0.657 ± 0.036 ± 0.012  0.643 ± 0.038_{stat}  
J/ψK_{L} (η_{CP}=+1)  0.694 ± 0.061 ± 0.031  0.641 ± 0.057_{stat}  
J/ψK^{0}  0.666 ± 0.031 ± 0.013  0.642 ± 0.031 ± 0.017 
0.655 ± 0.0244
(0.022_{statonly}) 
CL = 0.62 
ψ(2S)K_{S} (η_{CP}=1)  0.897 ± 0.100 ± 0.036  0.718 ± 0.090 ± 0.031
N(BB)=657M 
0.798 ± 0.071
(0.067_{statonly}) 
BaBar (PRD 79 (2009) 072009)
Belle (PRD77 (2008) 091103(R)) 
χ_{c1}K_{S} (η_{CP}=1)  0.614 ± 0.160 ± 0.040      BaBar (PRD 79 (2009) 072009) 
η_{c}K_{S} (η_{CP}=1)  0.925 ± 0.160 ± 0.057    
J/ψK*^{0} (K*^{0} → K_{S}π^{0}) (η_{CP}= 12A_{⊥}^{2})  0.601 ± 0.239 ± 0.087    
All charmonium  0.687 ± 0.028 ± 0.012  0.650 ± 0.029 ± 0.018 
0.670 ± 0.023
(0.020_{statonly}) 
CL = 0.52 
χ_{c0}K_{S} (η_{CP}=+1) 
0.69 ± 0.52 ± 0.04 ± 0.07 ^{(*)}
N(BB)=383M 
    BaBar (PRD 80 (2009) 112001) 
J/ψK_{S}, J/ψ → hadrons (η_{CP}=+1) 
1.56 ± 0.42 ± 0.21 ^{(**)}
N(BB)=88M 
    BaBar (PRD 69 (2004) 052001) 
All charmonium (incl. χ_{c0}K_{S} etc.) 
0.691 ± 0.031
(0.028_{statonly}) 
0.650 ± 0.029 ± 0.018 
0.672 ± 0.023
(0.020_{statonly}) 
CL = 0.30 
^{(*)} The BABAR result on χ_{c0}K_{S} comes from the timedependent Dalitz plot analysis of B^{0} → π^{+}π^{−}K_{S}. The third uncertainty is due to the Dalitz model.
^{(**)} BaBar (PRD 69 (2004) 052001) uses "hadronic and previously unused muonic decays of the J/ψ". We neglect a small possible correlation of this result with the main BaBar result that could be caused by reprocessing of the data.
Including earlier sin(2β) ≡ sin(2φ_{1}) measurements using B_{d} → J/ψK_{S} decays:
Parameter: sin(2β) ≡ sin(2φ_{1})  

Experiment  Value  Reference  
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  0.673 ± 0.023 (0.020_{statonly})  CL = 0.40 
from which we obtain the following solutions for β ≡ φ_{1} (in [0, π])
β ≡ φ_{1} = (21.1 ± 0.9)°  or  β ≡ φ_{1} = (68.9 ± 0.9)° 
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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Constraint on the ρbarηbar plane: 
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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 (from the BaBar results) for the following averages.
Parameter: C=−A (if not stated otherwise)  

Mode  BaBar  Belle  Average  Reference 
Charmonium  N(BB)=465M  N(BB)=535M   
BaBar (PRD 79 (2009) 072009)
Belle (PRL 98 (2007) 031802) 
J/ψK_{S}  0.026 ± 0.025 ± 0.016  0.001 ± 0.028_{stat}  
J/ψK_{L}  −0.033 ± 0.050 ± 0.027  −0.045 ± 0.033_{stat}  
J/ψK^{0}  0.016 ± 0.023 ± 0.018  −0.018 ± 0.021 ± 0.014 
−0.003 ± 0.020
(0.016_{statonly}) 
CL = 0.32 
ψ(2S)K_{S}  0.089 ± 0.076 ± 0.020  −0.039 ± 0.069 ± 0.053
N(BB)=657M 
0.032 ± 0.060
(0.051_{statonly}) 
BaBar (PRD 79 (2009) 072009)
Belle (PRD77 (2008) 091103(R)) 
χ_{c1}K_{S}  0.129 ± 0.109 ± 0.025      BaBar (PRD 79 (2009) 072009) 
η_{c}K_{S}  0.080 ± 0.124 ± 0.029    
J/ψK*^{0} (K*^{0} → K_{S}π^{0})  0.025 ± 0.083 ± 0.054    
All charmonium  0.024 ± 0.020 ± 0.016  −0.018 ± 0.025 (0.020_{statonly}) 
0.004 ± 0.019
(0.014_{statonly}) 
CL = 0.38 
χ_{c0}K_{S} (η_{CP}=+1)  −0.29 ^{+0.53}_{−0.44} ± 0.03 ± 0.05 ^{(*)}      BaBar (PRD 80 (2009) 112001) 
All charmonium (incl. χ_{c0}K_{S}) 
0.023 ± 0.025
(0.020_{statonly}) 
−0.018 ± 0.025 (0.020_{statonly}) 
0.004 ± 0.019
(0.014_{statonly}) 
CL = 0.51 
^{(*)} The BABAR result on χ_{c0}K_{S} comes from the timedependent Dalitz plot analysis of B^{0} → π^{+}π^{−}K_{S}. The third uncertainty is due to the Dalitz model.
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.
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})_{J/ψK*}  cos(2β) ≡ cos(2φ_{1})_{J/ψK*}  Correlation  Reference 

BaBar
N(BB)=88M 
−0.10 ± 0.57 ± 0.14  3.32 ^{+0.76} _{−0.96} ± 0.27  −0.37 (stat)  PRD 71, 032005 (2005) 
Belle
N(BB)=275M 
0.24 ± 0.31 ± 0.05 
0.56 ± 0.79 ± 0.11
[using Solution II] 
0.22 (stat)  PRL 95 091601 (2005) 
Average 
0.16 ± 0.28
χ^{2} = 0.3/1 dof (CL = 0.61 → 0.5σ) 
1.64 ± 0.62
χ^{2} = 4.7/1 dof (CL = 0.03 → 2.2σ) 
uncorrelated averages 
HFAG
See remark below table 

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. 
BaBar
find a confidence level for cos(2β)>0 of 89%.
Note that due to the strong nonGaussian character of the BaBar measurement,
the interpretation of the average given above
has to be done with the greatest care.
We perform uncorrelated averages
(using the PDG prescription
for asymmetric errors).
The decays B_{d} → D^{(}*^{)}D^{(}*^{)}K_{S} are dominated by the b → ccbar s transition, and are therefore sensitive to 2β ≡ 2φ_{1}. However, since the final state is not a CP eigenstate, extraction of the weak phases is difficult. Browder et al. have shown that terms sensitive to cos(2β) ≡ cos(2φ_{1}) can be extracted from the analysis of B_{d} → D*D*K_{S} decays (with some theoretical input).
Analysis of the B_{d} → D*D*K_{S} decay has been performed by BaBar. and Belle.
The analyses proceed by dividing the Dalitz plot into two: m(D*^{+}K_{S})^{2} > m(D*^{−}K_{S})^{2} (η_{y} = +1) and m(D*^{+}K_{S})^{2} < m(D*^{−}K_{S})^{2} (η_{y} = 1). They then fit using a PDF where the timedependent asymmetry (defined in the usual way as the difference between the timedependent distributions of B^{0}tagged and B^{0}bartagged events, divided by their sum) is given by
A(Δt) = η_{y} (J_{c}/J_{0}) cos(Δm_{d}Δt) − [ (2J_{s1}/J_{0})sin(2β) + η_{y} (2J_{s2}/J_{0})cos(2β) ] sin(Δm_{d}Δt) 
The parameters J_{0}, J_{c}, J_{s1} and J_{s2} are the integrals over the halfDalitz plane m(D*^{+}K_{S})^{2} < m(D*^{−}K_{S})^{2} of the functions a^{2} + abar^{2}, a^{2}  abar^{2}, Re(abar a*) and Im(abar a*) respectively, where a and abar are the decay amplitudes of B^{0} → D*D*K_{S} and B^{0}bar → D*D*K_{S} respectively. The parameter J_{s2} (and hence J_{s2}/J_{0}) is predicted to be positive.
At present we do not apply a rescaling of the results to a common, updated set of input parameters.
Experiment  J_{c}/J_{0}  (2J_{s1}/J_{0})sin(2β) ≡ (2J_{s1}/J_{0})sin(2φ_{1})  (2J_{s2}/J_{0})cos(2β) ≡ (2J_{s2}/J_{0})cos(2φ_{1})  Correlation  Reference 

BaBar
N(BB)=230M 
0.76 ± 0.18 ± 0.07  0.10 ± 0.24 ± 0.06  0.38 ± 0.24 ± 0.05    PRD 74, 091101 (2006) 
Belle
N(BB)=449M 
0.60 ^{+0.25} _{−0.28} ± 0.08  −0.17 ± 0.42 ± 0.09  −0.23 ^{+0.43} _{−0.41} ± 0.13    PRD 76, 072004 (2007) 
Average 
0.71 ± 0.16
χ^{2} = 0.2 (CL=0.63 ⇒ 0.5σ) 
0.03 ± 0.21
χ^{2} = 0.3 (CL=0.59 ⇒ 0.5σ) 
0.24 ± 0.22
χ^{2} = 1.4 (CL=0.23 ⇒ 1.2σ) 
uncorrelated averages  HFAG 

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. 
From the above result and the assumption that J_{s2}>0, BaBar infer that cos(2β)>0 at the 94% confidence level.
Decays of the B_{s} meson via the b → ccbar s transition probe φ_{s}, a CP violating phase related to B_{s}–B_{s}bar mixing. An important difference with respect to the B_{d}–B_{d}bar system, is that the value of ΔΓ is predicted to significantly nonzero, allowing information on φ_{s} to be extracted without tagging the flavour of the decaying B meson. Within the Standard Model, φ_{s} is predicted to be very small, O(λ^{2}).
The vectorvector final state J/ψ φ contains mixtures of polarization amplitudes: the CPodd A_{⊥}, and the CPeven A_{0} and A_{}. These terms need to be disentangled, using the angular distributions, in order to extract φ_{s}, and their interference provides additional sensitivity. The sensitivity to φ_{s} depends strongly on ΔΓ, and less strongly on the perpendicularly polarized fraction, A_{⊥}^{2}.
In this discussion we make the approximation φ_{s} ≈ −2β_{s} where φ_{s} ≡ arg[ − M_{12} / Γ_{12} ] and 2β_{s} ≡ 2 arg[ − V_{ts}V_{tb}^{*} / V_{cs}V_{cb}^{*} ]. This is a reasonable approximation since, although the equality does not hold in the Standard Model, both are much smaller than the current experimental resolution, whereas new physics contributions add a phase φ_{NP} to φ_{s} and subtract the same phase from 2β_{s}, so that the approximation remains valid. 
Measurements of φ_{s} from B_{s} → J/ψ φ have been performed by CDF and D0.
CDF have carried out a flavourtagged, timedependent analysis of B_{s} → J/ψ φ using 2.8 fb^{−1} of data. They do not present a central value and its uncertainty, due to the highly nonGaussian shape of the likelihood function. Instead, they present a confidence region in the φ_{s}ΔΓ_{s} plane, from which they obtain φ_{s} ∈ [−0.56, −2.58] at the 68% confidence level. The consistency with the Standard Model expectation for (φ_{s},ΔΓ_{s}) is 7% (corresponding to 1.8σ). These results supercede results from a previous flavourtagged, timedependent analysis using 1.35 fb^{−1} of data, and an untagged analysis using 1.7 fb^{−1}. CDF results were updated with 4.2 fb^{−1} of data at FPCP2010.
D0 have performed a flavourtagged, timedependent analysis of B_{s} → J/ψ φ using 2.8 fb^{−1} of data. They perform a fit in which the strong phase differences δ_{0} and δ_{} (measured relative to δ_{⊥}; and denoted δ_{2} and δ_{1} in their analysis) are constrained to the equivalent values measured in B^{0} → J/ψ K*^{0} (see HFAG b → c), up to an error of π/5, allowing for SU(3) breaking effects. The results are given in the table below. They also obtain a 90% CL allowed interval φ_{s} ∈ [−1.20, +0.06]. These results supercede results from an untagged analysis using 1 fb^{−1}.
Averaging of the above results is being carried out by the HFAG oscillation group. For the latest results, see here. A combination has also been performed by the CDF and D0 collaborations.
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, eg. 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}. See e.g. Fleischer, NPB 659, 321 (2003).
Results of such an analysis are available from BaBar. 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 decay D* → Dπ^{0} is used. The CPeven D decay to K^{+}K^{−} is used for all decay modes, with the CPodd D decay to K_{S}ω also used in B_{d} → D^{(}*^{)}π^{0} and the additional CPodd D decay to K_{S}π^{0} also used in B_{d} → Dω.
BaBar have performed separate fits for the cases where the intermediate D^{(}*^{)} decays to CPeven and CPodd final states, since these receive different contributions fom subleading amplitudes in the Standard Model. Since the effects of these corrections are expected to be negligible (~0.02) compared to the current experimental uncertainty, they have also performed a fit with all decays combined.
Mode  Experiment  −sin(2β) ≡ −sin(2φ_{1})  C_{CP}  Correlation  Reference 

D^{(}*^{)}_{CP+} h^{0} 
BaBar
N(BB)=383M 
−0.65 ± 0.26 ± 0.06  −0.33 ± 0.19 ± 0.04  0.04 (stat)  PRL 99, 081801 (2007) 
D^{(}*^{)}_{CP−} h^{0}  −0.46 ± 0.46 ± 0.13  −0.03 ± 0.28 ± 0.07  −0.14 (stat)  
D^{(}*^{)} h^{0}  −0.56 ± 0.23 ± 0.05  −0.23 ± 0.16 ± 0.04  −0.02 (stat) 
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 both Belle and. BaBar. 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^{−}. Note that BaBar quote uncertainties due to the D decay model separately from other systematic errors, while Belle do not.
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})  λ  Correlations  Reference 

BaBar
N(BB)=383M 
0.29 ± 0.34 ± 0.03 ± 0.05  0.42 ± 0.49 ± 0.09 ± 0.13  1.01 ± 0.08 ± 0.02  (stat)  PRL 99, 231802 (2007) 
Belle
N(BB)=386M 
0.78 ± 0.44 ± 0.22  1.87 ^{+0.40} _{−0.53} ^{+0.22} _{−0.32}      PRL 97, 081801 (2006) 
Average 
0.45 ± 0.28
χ^{2} = 0.7 (CL=0.41 ⇒ 0.8σ) 
1.01 ± 0.40
χ^{2} = 3.2 (CL=0.07 ⇒ 1.8σ) 
  uncorrelated averages  HFAG 

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. 
Interpretations:
Belle
determine the sign of cos(2φ)_{1} to be positive at 98.3% confidence level.
BaBar
favour the solution of β with cos(2β)>0 at 86% confidence level.
Note that the Belle measurement has strongly nonGaussian behaviour.
The interpretation of the
average given above has to be done with the greatest care.
We perform uncorrelated averages
(using the PDG prescription
for asymmetric errors).
Within the Standard Model, the b → s penguin transition carries approximately the same weak phase as the b → ccbar s amplitude used above to obtain sin(2β) ≡ sin(2φ_{1}). When this single phase dominates the decay to a (quasi)twobody CP eigenstate, the timedependent CP violation parameters should therefore by given by S = −η_{CP} × sin(2β^{eff}) ≡ −η_{CP} × sin(2φ_{1}^{eff}) and C ≡ −A = 0. The loop process is sensitive to effects from virtual new physics particles, which may result in deviations from the prediction that sin(2β^{eff}) ≡ sin(2φ_{1}^{eff}) (b → qqbar s) ∼ sin(2β) ≡ sin(2φ_{1}) (b → ccbar s).
Various different final states have been used by BaBar and Belle to investigate timedependent CP violation in hadronic b → s penguin transitions. These are summarised below. (Note that results from timedependent Dalitz plot analyses of B^{0} → K^{+}K^{−}K^{0} and B^{0} → π^{+}π^{−}K_{S} are also discussed in the next section — results for φK^{0}, ρ^{0}K_{S} and f_{0}K_{S} are extracted from these analyses. The third error, where given, is due to Dalitz model uncertainty.)
At present we do not apply a rescaling of the results to a common, updated set of input parameters. We take correlations between S and C into account where available, except if one or more of the measurements suffers from strongly nonGaussian errors. In that case, we perform uncorrelated averages (using the PDG prescription for asymmetric errors).
Mode  Experiment  sin(2β^{eff}) ≡ sin(2φ_{1}^{eff})  C_{CP}  Correlation  Reference 

φK^{0} 
BaBar
^{(*)}
N(BB)=465M 
0.26 ± 0.26 ± 0.03  −0.14 ± 0.19 ± 0.02    arXiv:0808.0700 
Belle
^{(*)}
N(BB)=657M 
0.90 ^{+0.09} _{−0.19}  −0.04 ± 0.20 ± 0.10 ± 0.02    FPCP 2010 preliminary  
Average ^{(*)}  0.56 ^{+0.16} _{−0.18}  −0.11 ± 0.15    HFAG  

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η′K^{0} 
BaBar
N(BB)=467M 
0.57 ± 0.08 ± 0.02  −0.08 ± 0.06 ± 0.02  0.03 (stat)  PRD 79 (2009) 052003 
Belle
N(BB)=535M 
0.64 ± 0.10 ± 0.04  0.01 ± 0.07 ± 0.05  0.09 (stat)  PRL 98 (2007) 031802  
Average  0.59 ± 0.07  −0.05 ± 0.05  0.04 
HFAG correlated average
χ^{2} = 0.9/2 dof (CL=0.63 ⇒ 0.5σ) 


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K_{S}K_{S}K_{S} 
BaBar
N(BB)=465M 
0.90 ^{+0.18} _{−0.20} ^{+0.03} _{−0.04}  −0.16 ± 0.17 ± 0.03  0.10 (stat)  CKM2008 preliminary 
Belle
N(BB)=535M 
0.30 ± 0.32 ± 0.08  −0.31 ± 0.20 ± 0.07    PRL 98 (2007) 031802  
Average  0.74 ± 0.17  −0.23 ± 0.13  0.06 
HFAG correlated average
χ^{2} = 2.7/2 dof (CL=0.26 ⇒ 1.1σ) 


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π^{0}K^{0} 
BaBar
N(BB)=467M 
0.55 ± 0.20 ± 0.03  0.13 ± 0.13 ± 0.03  0.06 (stat)  PRD 79 (2009) 052003 
Belle
N(BB)=657M 
0.67 ± 0.31 ± 0.08  −0.14 ± 0.13 ± 0.06  −0.04 (stat)  arXiv:0809.4366  
Average  0.57 ± 0.17  0.01 ± 0.10  0.02 
HFAG correlated average
χ^{2} = 2.0/2 dof (CL=0.37 ⇒ 0.9σ) 


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ρ^{0}K_{S} 
BaBar
^{(*)}
N(BB)=383M 
0.35 ^{+0.26} _{−0.31} ± 0.06 ± 0.03  −0.05 ± 0.26 ± 0.10 ± 0.03    PRD 80 (2009) 112001 
Belle
^{(*)}
N(BB)=657M 
0.64 ^{+0.19} _{−0.25} ± 0.09 ± 0.10  −0.03 ^{+0.24} _{−0.23} ± 0.11 ± 0.10    PRD 79 (2009) 072004  
Average ^{(*)}  0.54 ^{+0.18} _{−0.21}  −0.06 ± 0.20    HFAG  

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ωK_{S} 
BaBar
N(BB)=467M 
0.55 ^{+0.26} _{−0.29} ± 0.02  −0.52 ^{+0.22} _{−0.20} ± 0.03  0.03 (stat)  PRD 79 (2009) 052003 
Belle
N(BB)=535M 
0.11 ± 0.46 ± 0.07  0.09 ± 0.29 ± 0.06  −0.04 (stat)  PRD 76 (2007) 091103(R)  
Average  0.45 ± 0.24  −0.32 ± 0.17  0.01 
HFAG correlated average
χ^{2} = 3.4/2 dof (CL=0.18 ⇒ 1.3σ) 


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f_{0}K^{0}  BaBar ^{(**)}  0.60 ^{+0.16} _{−0.18}  0.05 ± 0.16   
arXiv:0808.0700,
PRD 80 (2009) 112001 ^{(**)} 
Belle ^{(**)}  0.63 ^{+0.16} _{−0.19}  0.13 ± 0.17   
PRD 79 (2009) 072004
FPCP 2010 preliminary ^{(**)} 

Average  0.62 ^{+0.11} _{−0.13}  0.09 ± 0.12    HFAG  

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f_{2}K_{S} 
BaBar
^{(*)}
N(BB)=383M 
0.48 ± 0.52 ± 0.06 ± 0.10  0.28 ^{+0.35} _{−0.40} ± 0.08 ± 0.07  0.01 (stat)  PRD 80 (2009) 112001 
f_{X}K_{S} 
BaBar
^{(*)}
N(BB)=383M 
0.20 ± 0.52 ± 0.07 ± 0.07  0.13 ^{+0.33} _{−0.35} ± 0.04 ± 0.09  0.29 (stat)  PRD 80 (2009) 112001 
π^{0}π^{0}K_{S} ^{(****)} 
BaBar
N(BB)=227M 
−0.72 ± 0.71 ± 0.08  0.23 ± 0.52 ± 0.13  −0.02 (stat)  PRD 76 (2007) 071101 
φ K_{S} π^{0} 
BaBar
^{(***)}
N(BB)=465M 
0.97 ^{+0.03} _{−0.52}  −0.20 ± 0.14 ± 0.06    PRD 78 (2008) 092008 
π^{+} π^{−} K_{S} nonresonant 
BaBar
^{(*)}
N(BB)=383M 
0.01 ± 0.31 ± 0.05 ± 0.09  0.01 ± 0.25 ± 0.06 ± 0.05  −0.11 (stat)  PRD 80 (2009) 112001 
K^{+}K^{−}K^{0}
(excluding φK^{0}) 
BaBar
^{(*)}
N(BB)=465M 
0.86 ± 0.08 ± 0.03  −0.05 ± 0.09 ± 0.04    arXiv:0808.0700 
Belle
N(BB)=535M 
0.68 ± 0.15 ± 0.03^{+0.21} _{−0.13}_{CPeven}
(f_{CPeven}= 0.93 ± 0.09 ± 0.05 [SU(2)]) 
0.09 ± 0.10 ± 0.05  −0.00 (stat)  PRD 76 (2007) 091103(R)  
Average 
0.82 ± 0.07
χ^{2} = 1.0 (CL=0.31 ⇒ 1.0σ) 
0.01 ± 0.07
χ^{2} = 0.9 (CL=0.35 ⇒ 0.9σ) 
uncorrelated averages  HFAG  

eps.gz png  eps.gz png  .  
Naïve b→s penguin average 
0.64 ± 0.04
χ^{2} = 26/23 dof (CL=0.28 ⇒ 1.1σ) 
−0.04 ± 0.03
χ^{2} = 16/23 dof (CL=0.84 ⇒ 0.2σ) 
uncorrelated averages  HFAG  
eps.gz png  eps.gz png  
Direct comparison of charmonium and spenguin averages (see comments below): χ^{2} = 0.6 (CL=0.43 ⇒ 0.8σ) 
^{(*)}
BaBar and Belle results for φK^{0} and
ρ^{0}K_{S} are determined from their
timedependent Dalitz plot analyses of
B^{0} → K^{+}K^{−}K^{0} and
B^{0} → π^{+}π^{−}K_{S},
respectively.
For the experimental results,
we quote Q2B parameters that are given in the respective references,
where possible.
(Belle have not reported Q2B S parameters from their
timedependent Dalitz plot analysis of
B^{0} → K^{+}K^{−}K_{S},
so we convert their results on φ_{1}.)
The averages of the directly fitted parameters
are more reliable than those of the Q2B parameters,
therefore we convert those results to give the averages quoted in the
table above.
BaBar results for f_{2}K_{S},
f_{X}K_{S} and
π^{+} π^{−} K_{S} nonresonant
are determined from their
timedependent Dalitz plot analysis of
B^{0} → π^{+}π^{−}K_{S}.
The BaBar results for K^{+}K^{−}K^{0}
are for the inclusive "high mass"
(m_{K+K−} > 1.1 GeV/c^{2}) region
in the timedependent Dalitz plot analysis.
Belle results for K^{+}K^{−}K^{0}
are from a previous analysis in which the inclusive state
(excluding φK^{0}) was treated as a quasitwobody mode.
^{(**)} BaBar and Belle results for f_{0}K^{0} are combinations of results from the two Dalitz plot analyses: B^{0} → f_{0}K^{0} with f_{0} → K^{+}K^{−}, and B^{0} → f_{0}K_{S} with f_{0} → π^{+}π^{−}. Note that Q2B parameters extracted from Dalitz plot analyses are constrained to lie within the physical boundary (S_{CP}^{2} + C_{CP}^{2} < 1), and consequently the obtained errors can be highly nonGaussian when the central value is close to the boundary. This is particularly evident in the BaBar results from B^{0} → f_{0}K_{S} with f_{0} → π^{+}π^{−}. These results must be treated with extreme caution. As above, we convert the averages of the directly fitted parameters from the timedependent Dalitz plot analyses back to the Q2B parameters given in the table above.
^{(***)} The BaBar results on φ K_{S} π^{0} come from a simultaneous angular analysis of B → φ K^{+} π^{−} and B → φ K_{S} π^{0}, where the angular parameters of the two decays modes are related since only (Kπ) resonances contribute to the final state. Note that Q2B parameters extracted in this way are constrained to lie within the physical boundary (S_{CP}^{2} + C_{CP}^{2} < 1), and consequently the obtained errors are highly nonGaussian when the central value is close to the boundary. The single uncertainty given for sin(2β^{eff}) in this result includes both statistical and systematic uncertainties.
^{(****)} We do not include a preliminary result from Belle on π^{0}π^{0}K_{S} that remains unpublished after more than two years.
Please note that
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 f_{2}K_{S}, f_{X}K_{S} π^{0}π^{0}K_{S}, π^{+} π^{−} K_{S} nonresonant and φ K_{S} π^{0} to allow closer inspection of the detail. 


Comparisons of averages in the different b→q qbar s modes 
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Same, but without f_{2}K_{S}, f_{X}K_{S} π^{0}π^{0}K_{S}, π^{+} π^{−} K_{S} nonresonant and φ K_{S} π^{0} to allow closer inspection of the detail. 


2D comparisons of averages in the different b→q qbar s modes. 
eps png 
Timedependent amplitude analyses of the threebody decays B_{d} → K^{+}K^{−}K^{0} and B_{d} → π^{+}π^{−}K^{0} allow additional information to be extracted from the data. In particular, the cosine of the effective weak phase difference (cos(2β^{eff}) ≡ cos(2φ_{1}^{eff})) can be determined, as well as the sine term that is obtained from quasitwobody analysis. This information allows half of the degenerate solutions to be rejected. Furthermore, Dalitz plot analysis has enhanced sensitivity to direct CP violation.
A timedependent Dalitz plot analysis of B^{0} → K^{+}K^{−}K^{0} has been performed by BaBar (an update of their previous publication). As given above, parameters can be extracted in a form that allows a straightforward comparison/combination with those from timedependent CP asymmetries in quasitwobody b → qqbar s modes. In addition, the effective weak phase β^{eff} ≡ φ_{1}^{eff} is directly determined for two significant resonant contributions: φK^{0} and f_{0}K^{0}, the inclusive "high mass" (m_{K+K−} > 1.1 GeV/c^{2}) region, as well as the effective weak phase averaged over the Dalitz plot, with the CP properties of the individual components taken into account. In addition to the weak phase, BaBar also measure the timedependent direct CP violation parameter A_{CP} ( = C_{CP}). Belle have also performed a timedependent Dalitz plot analysis of B^{0} → K^{+}K^{−}K_{S}, using a model that includes contributions from f_{0}(980)K_{S}, φK_{S}, f_{X}(1500)K_{S}, χ_{c0}K_{S} and nonresonant terms (the f_{X}(1500)K_{S}, χ_{c0}K_{S} and nonresonant terms are constrained to share common CP parameters).
Experiment  K^{+}K^{−}K^{0} (whole DP)  Reference  

β^{eff}  A_{CP}  
BaBar
N(BB)=465M 
(25.3 ± 3.9 ± 0.9)°  0.03 ± 0.07 ± 0.02  arXiv:0808.0700 
Experiment  φK^{0}  f_{0}K^{0}  K^{+}K^{−}K^{0} (m_{K+K−} > 1.1 GeV/c^{2})  Correlation  Reference  

β^{eff}  A_{CP}  β^{eff}  A_{CP}  β^{eff}  A_{CP}  
BaBar
^{(*)}
N(BB)=465M 
(7.7 ± 7.7 ± 0.9)°  0.14 ± 0.19 ± 0.02  (8.5 ± 7.5 ± 1.8)°  0.01 ± 0.26 ± 0.07  (29.5 ± 4.5 ± 1.5)°  0.05 ± 0.09 ± 0.04  (stat)  arXiv:0808.0700 
Belle
^{(**)}
N(BB)=657M 
(32.2 ± 9.0 ± 2.6 ± 1.4)°  0.04 ± 0.20 ± 0.10 ± 0.02  (31.3 ± 9.0 ± 3.4 ± 4.0)°  −0.30 ± 0.29 ± 0.11 ± 0.09    (stat)  FPCP 2010 preliminary  
Average  (16.9 ± 6.0)°  0.11 ± 0.15  (16.9 ± 6.2)°  −0.07 ± 0.21  (29.5 ± 4.7)°  0.05 ± 0.10  (stat) 
HFAG correlated average
χ^{2} = 8.0/4 dof (CL=0.09 ⇒ 1.7σ) 

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φK^{0}(β^{eff}) vs
φK^{0}(A_{CP}):
eps.gz
png
f_{0}K^{0}(β^{eff}) vs f_{0}K^{0}(A_{CP}): eps.gz png φK^{0}(β^{eff}) vs f_{0}K^{0}(β^{eff}): eps.gz png φK^{0}(A_{CP}) vs f_{0}K^{0}(A_{CP}): eps.gz png 
^{(*)} The BaBar results on φK^{0} and f_{0}K^{0} suffer from an ambiguity in the solution. The results quoted here correspond to solution 1 presented in the paper.
^{(**)} The Belle results on φK^{0} and f_{0}K^{0} suffer from a fourfold ambiguity in the solution. The results quoted here correspond to solution 1 presented in the paper. The third source of uncertainty arises due to the composition of the Dalitz plot.
From the above results BaBar infer that the trigonometric reflection at π/2  &beta^{eff} is disfavoured at 4.8σ.
Timedependent Dalitz plot analyses of B^{0} → π^{+}π^{−}K_{S} have been performed by BaBar and Belle. As given above, parameters can be extracted in a form that allows a straightforward comparison/combination with those from timedependent CP asymmetries in quasitwobody b → qqbar s modes. In addition, the effective weak phase β^{eff} ≡ φ_{1}^{eff} is directly determined for two significant resonant contributions: f_{0}K_{S} and ρ^{0}K_{S} by both experiments. Both experiments find multiple solutions in the fits; in both cases we quote the results given as solution 1. BaBar also report parameters related to the intermediate states f_{2}(1270)K_{S}, f_{X}(1300)K_{S}, nonresonant π^{+}π^{−}K_{S} and χ_{c0}K_{S} (see b → ccbar s modes above). A number of additional parameters, for example relating to the Q2B modes K*^{+}π^{−}, are also extracted, but are not tabulated here.
The third error in the results given below is due to Dalitz model uncertainty.
Experiment  ρ^{0}K_{S}  f_{0}K_{S}  Correlation  Reference  

β^{eff} ≡ φ_{1}^{eff}  A_{CP}  β^{eff} ≡ φ_{1}^{eff}  A_{CP}  
BaBar
^{(*)}
N(BB)=383M 
(10.2 ± 8.9 ± 3.0 ± 1.9)°  0.05 ± 0.26 ± 0.10 ± 0.03  (36.0 ± 9.8 ± 2.1 ± 2.1)°  −0.08 ± 0.19 ± 0.03 ± 0.04  (stat)  PRD 80 (2009) 112001 
Belle
^{(*)}
N(BB)=657M 
(20.0 ^{+8.6} _{−8.5} ± 3.2 ± 3.5)°  0.03 ^{+0.23} _{−0.24} ± 0.11 ± 0.10  (12.7 ^{+6.9} _{−6.5} ± 2.8 ± 3.3)°  −0.06 ± 0.17 ± 0.07 ± 0.09  (stat)  PRD 79 (2009) 072004 
Average  (16.4 ± 6.8)°  0.06 ± 0.20  (20.6 ± 6.2)°  −0.07 ± 0.14  (stat) 
HFAG correlated average
χ^{2} = 4.1/4 dof (CL=0.39 ⇒ 0.9σ) 

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. 
Experiment  f_{2}K_{S}  f_{X}K_{S}  Nonresonant  χ_{c0}K_{S}  Correlation  Reference  

β^{eff} ≡ φ_{1}^{eff}  A_{CP}  β^{eff} ≡ φ_{1}^{eff}  A_{CP}  β^{eff} ≡ φ_{1}^{eff}  A_{CP}  β^{eff} ≡ φ_{1}^{eff}  A_{CP}  
BaBar
^{(*)}
N(BB)=383M 
(14.9 ± 17.9 ± 3.1 ± 5.2)°  −0.28 ^{+0.40} _{−0.35} ± 0.08 ± 0.07  (5.8 ± 15.2 ± 2.2 ± 2.3)°  −0.13 ^{+0.35} _{−0.33} ± 0.04 ± 0.09  (0.4 ± 8.8 ± 1.9 ± 3.8)°  −0.01 ± 0.25 ± 0.06 ± 0.05  (23.2 ± 22.4 ± 2.3 ± 4.2)°  0.29 ^{+0.44} _{−0.53} ± 0.03 ± 0.05  (stat)  PRD 80 (2009) 112001 
^{(*)} Both experiments suffer from ambiguities in the solutions. The results quoted here correspond to solution 1 presented in the papers.
Since parameters related to the B^{0} → f_{0}K_{S} decay are obtained in both B^{0} → K^{+}K^{−}K^{0} and B^{0} → π^{+}π^{−}K_{S}, we show compilations and naïve (uncorrelated) averages below.
Naïve (uncorrelated) averages for f_{0}K_{S} parameters 
eps.gz png 
eps.gz png 
. 
The final state in the decay B → φ K_{S} π^{0} is a mixture of CPeven and CPodd amplitudes. However, since only φ K*^{0} resonant states contribute (in particular, φ K*^{0}(892), φ K*^{0}_{0}(1430) and φ K*^{0}_{2}(1430) are seen), the composition can be determined from the analysis of B → φ K^{+} π^{−}, assuming only that the ratio of branching fractions B(K*^{0} → K_{S} π^{0})/B(K*^{0} → K^{+} π^{−}) is the same for each exited kaon state.
BaBar have performed a simultaneous analysis of B → φ K_{S} π^{0} and B → φ K^{+} π^{−} that is timedependent for the former mode and timeintegrated for the latter. Such an analysis allows, in principle, all parameters of the B → φ K*^{0} system to be determined, including mixinginduced CP violation effects. The latter is determined to be Δφ_{00} = 0.28 ± 0.42 ± 0.04, where Δφ_{00} is half the weak phase difference between B^{0} and B^{0}bar decays to φK*^{0}_{0}(1430). As presented above, this can also be presented in terms of the quasitwobody parameter sin(2β^{eff}_{00}) = sin(2β+2Δφ_{00}) = 0.97 ^{+0.03}_{−0.52}. The highly asymmetric uncertainty arises due to the conversion from the phase to the sine of the phase, and the proximity of the physical boundary.
Similar sin(2β^{eff}) parameters can be defined for each of the helicity amplitudes for both φ K*^{0}(892) and φ K*^{0}_{2}(1430). However, the relative phases between these decays are constrained due to the nature of the simultaneous analysis of B → φ K_{S} π^{0} and B → φ K^{+} π^{−}, and therefore these measurements are highly correlated. Instead of quoting all these results, BaBar provide an illustration of their measurements with the following differences:
where the first subscript indicates the helicity amplitude and the second indicates the spin of the kaon resonance. For the complete definitions of the Δδ and Δφ parameters, please refer to the BaBar paper.
Direct CP violation parameters for each of the contributing helicity amplitudes can also be measured. Again, these are determined from a simultaneous fit of B → φ K_{S} π^{0} and B → φ K^{+} π^{−}, with the precision being dominated by the statistics of the latter mode. The direct CP violation measurements are tabulated by HFAG  Rare Decays.
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_{CP} (J/ψ π^{0})  C_{CP} (J/ψ π^{0})  Correlation  Reference 

BaBar
N(BB)=466M 
−1.23 ± 0.21 ± 0.04  −0.20 ± 0.19 ± 0.03  0.20 (stat)  PRL 101 (2008) 021801 
Belle
N(BB)=535M 
−0.65 ± 0.21 ± 0.05  −0.08 ± 0.16 ± 0.05  −0.10 (stat)  PRD 77 (2008) 071101(R) 
Average  −0.93 ± 0.15  −0.10 ± 0.13  0.04 
HFAG correlated average
χ^{2} = 3.8/2 dof (CL=0.15 ⇒ 1.4σ) 

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We recall that we do NOT rescale (inflate) the errors due to measurement inconsistencies.
Experiment  S_{CP} (D^{+}D^{−})  C_{CP} (D^{+}D^{−})  Correlation  Reference 

BaBar
N(BB)=467M 
−0.65 ± 0.36 ± 0.05  −0.07 ± 0.23 ± 0.03  −0.01 (stat)  PRD 79, 032002 (2009) 
Belle
N(BB)=535M 
−1.13 ± 0.37 ± 0.09  −0.91 ± 0.23 ± 0.06  −0.04 (stat)  PRL 98, 221802 (2007) 
Average ^{(*)}  −0.89 ± 0.26  −0.48 ± 0.17  −0.02 
HFAG correlated average
χ^{2} = 7.3/2 dof (CL=0.025 ⇒ 2.2σ) 

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(*) Note that the Belle result is outside of the physical region, and the average is very close to the boundary. The interpretation of the average given above has to be done with the greatest care.
The vector particles in the pseudoscalar to vectorvector decay B_{d} → D*^{+}D*^{−} can have longitudinal and transverse relative polarization with different CP properties. The transversely polarized state (h_{⊥}) is CPodd, while the other two states in the transversity basis (h_{0} and h_{}) are CPeven. The CP parameters therefore have an important dependence on the fraction of the transversely polarized component R_{⊥}.
In the most recent results, Belle performs an initial fit to determine the transversely polarized fraction R_{⊥}, and then include effects due to its uncertainty together with other systematic errors. BaBar treat this variable as a free parameter in the fit and consequently this systematic is absorbed in the statistical error. We perform the average taking into account correlations of the CP parameters with each other as well as with R_{⊥}, though we are obliged to assume that the correlations of the Belle results with R_{⊥} are negligible.
Belle have performed a fit to the data assuming that the CP parameters for CPeven and CPodd transversity states are the same (up to a trivial change of sign for S_{CP}). BaBar have performed two fits to the data: in addition to a fit as above, an additional fit relaxes this assumption, so that differences between CPeven and CPodd parameters may be nontrivial. We use the first set of results to perform an average with Belle, and tabulate also the latter set of results.
Experiment  S_{CP} (D*^{+} D*^{−})  C_{CP} (D*^{+} D*^{−})  R_{⊥} (D*^{+} D*^{−})  Correlation  Reference 

BaBar
N(BB)=467M 
−0.71 ± 0.16 ± 0.03  0.05 ± 0.09 ± 0.02  0.17 ± 0.03  (stat)  PRD 79, 032002 (2009) 
Belle
N(BB)=657M 
−0.96 ± 0.25 ^{+0.12} _{−0.16}  −0.15 ± 0.13 ± 0.04  0.12 ± 0.04 ± 0.02  (stat)  PRD 80 (2009) 111104 
Average  −0.77 ± 0.14  −0.02 ± 0.08  0.16 ± 0.02  (stat) 
HFAG correlated average
χ^{2} = 2.9/3 dof (CL=0.41 ⇒ 0.8σ) 

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Experiment  S_{+} (D*^{+} D*^{−})  C_{+} (D*^{+} D*^{−})  S_{−} (D*^{+} D*^{−})  C_{−} (D*^{+} D*^{−})  R_{⊥} (D*^{+} D*^{−})  Correlation  Reference 

BaBar
N(BB)=467M 
−0.76 ± 0.16 ± 0.04  0.02 ± 0.12 ± 0.02  −1.81 ± 0.71 ± 0.16  0.41 ± 0.50 ± 0.08  0.15 ± 0.03  (stat)  PRD 79, 032002 (2009) 
^{(*)} Note that the BaBar values of R_{⊥} in these tables are not corrected for efficiency; the efficiency corrected value is R_{⊥} = 0.158 ± 0.028 ± 0.006. (Belle measure R_{T} = 0.125 ± 0.043 ± 0.023.)
Experiment  S_{+−}(D*^{+}D^{−})  C_{+−}(D*^{+}D^{−})  S_{−+}(D*^{−}D^{+})  C_{−+}(D*^{−}D^{+})  A(D*^{+−}D^{−+})  Correlation  Reference 

BaBar
N(BB)=467M 
−0.63 ± 0.21 ± 0.03  0.08 ± 0.17 ± 0.04  −0.74 ± 0.23 ± 0.05  0.00 ± 0.17 ± 0.03  0.01 ± 0.05 ± 0.01  (stat)  PRD 79, 032002 (2009) 
Belle
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 
−0.61 ± 0.19
χ^{2} = 0.03 (CL=0.86 ⇒ 0.2σ) 
−0.09 ± 0.14
χ^{2} = 2.5 (CL=0.12 ⇒ 1.6σ) 
−0.79 ± 0.21
χ^{2} = 0.2 (CL=0.66 ⇒ 0.4σ) 
0.07 ± 0.14
χ^{2} = 0.6 (CL=0.46 ⇒ 0.7σ) 
0.02 ± 0.04
χ^{2} = 0.4 (CL=0.54 ⇒ 0.6σ) 
uncorrelated averages  HFAG 

eps.gz png  eps.gz png  eps.gz png  eps.gz png  eps.gz png  . 
Compilation of results for (left) sin(2β^{eff}) ≡ sin(2φ_{1}^{eff}) = −η_{CP}S and (right) C ≡ −A from timedependent b → ccbar d analyses with CP eigenstate final states. The results are compared to the values from the corresponding charmonium averages. 
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Same, but with separate CPeven and CPodd results from D*^{+}D*^{−} 



Same, but including results from
D*^{+−}D^{−+}.
(These measure the same quantity as other b → ccbar d modes when the strong phase difference between the two decay amplitudes vanishes. This is in addition to the usual assumption of negligible penguin contributions.) 



Same, but including a naïve b → c cbar d average.
Such an average assumes that penguin contributions to the
b → c cbar d decays are negligible.
See the cautionary comments in the discussion on averaging
the timedependent CP violation parameters for
b → qqbar s transitions above.
The results of the naïve average are

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2D comparisons of averages in the different b→c cbar d modes. 
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The b → qqbar d penguin transitions are suppressed in the Standard Model, leading to small numbers of events available in these final states. If the top quark dominates in the loop, the phase in the decay amplitude (β ≡ φ_{1}) cancels that in the B^{0}–B^{0}bar mixing, so that S = C = 0. However, even within the Standard Model, there may be nonnegligible contributions with u or c quarks in the penguin loop (or from rescattering, etc.) so that different values of S and C are possible. In this case, these can be used to obtain constraints on γ ≡ φ_{3}, and hence test if any nonStandard Model contributions are present.
At present we do not apply a rescaling of the results to a common, updated set of input parameters.
Experiment  S_{CP} (K_{S}K_{S})  C_{CP} (K_{S}K_{S})  Correlation  Reference 

BaBar
N(BB)=350M 
−1.28 ^{+0.80} _{−0.73} ^{+0.11} _{−0.16}  −0.40 ± 0.41 ± 0.06  −0.32 (stat)  PRL 97 (2006) 171805 
Belle
N(BB)=657M 
−0.38 ^{+0.69} _{−0.77} ± 0.09  0.38 ± 0.38 ± 0.05  0.48 (stat)  PRL 100 (2008) 121601 
Average  −1.08 ± 0.49  −0.06 ± 0.26  0.14 
HFAG correlated average
χ^{2} = 2.5/2 dof (CL=0.29 ⇒ 1.1σ) 

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(*) Note that the BaBar result is outside of the physical region, as is the average. The interpretation of the results given above has to be done with the greatest care.
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.
In addition to results with the K_{S}π^{0}γ final state, BABAR has results using K_{S}ηγ, while Belle has results using K_{S}ρ^{0}γ.
At present we do not apply a rescaling of the results to a common, updated set of input parameters.
Mode  Experiment  S_{CP} (b → sγ)  C_{CP} (b → sγ)  Correlation  Reference  

K*(892)γ 
BaBar
N(BB)=467M 
−0.03 ± 0.29 ± 0.03  −0.14 ± 0.16 ± 0.03  0.05 (stat)  PRD 78 (2008) 071102  
Belle
N(BB)=535M 
−0.32 ^{+0.36} _{−0.33} ± 0.05  0.20 ± 0.24 ± 0.05  0.08 (stat)  PRD 74 (2006) 111104  
Average  −0.16 ± 0.22  −0.04 ± 0.14  0.06 
HFAG correlated average
χ^{2} = 1.9/2 dof (CL=0.40 ⇒ 0.9σ) 

K_{S}π^{0}γ
(incl. K*γ) 
BaBar
N(BB)=467M 
−0.17 ± 0.26 ± 0.03  −0.19 ± 0.14 ± 0.03  0.04 (stat)  PRD 78 (2008) 071102  
Belle
N(BB)=535M 
−0.10 ± 0.31 ± 0.07  0.20 ± 0.20 ± 0.06  0.08 (stat)  PRD 74 (2006) 111104(R)  
Average  −0.15 ± 0.20  −0.07 ± 0.12  0.05 
HFAG correlated average
χ^{2} = 2.4/2 dof (CL=0.30 ⇒ 1.0σ) 

K_{S} η γ 
BaBar
N(BB)=465M 
−0.18 ^{+0.49} _{−0.46} ± 0.12  −0.32 ^{+0.40} _{−0.39} ± 0.07  −0.17 (stat)  PRD 79 (2009) 011102  
K_{S} ρ^{0} γ 
Belle
N(BB)=657M 
0.11 ± 0.33 ^{+0.05} _{−0.09}  −0.05 ± 0.18 ± 0.06  0.04 (stat)  PRL 101 (2008) 251601  

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Similar to the b → sγ transitions discussed above, timedependent analyses of radiative b decays such as B^{0}→ ρ^{0}γ probe the polarization of the photon emitted in radiative b → dγ decays. However, since the CP violating phase from the b → d decay amplitude cancels that from the B_{d}–B_{d}bar mixing (to an approximation that is exact in the limit of top quark dominance in the loops), the asymmetry is suppressed even further in the Standard Model. An observable signal would be a sign of a new physics amplitude emitting righthanded photons and carrying a new CP violating phase.
A timedependent analysis of the B^{0}→ ρ^{0}γ channel has been carried out by Belle.
At present we do not apply a rescaling of the results to a common, updated set of input parameters.
Experiment  S_{CP} (b → dγ)  C_{CP} (b → dγ)  Correlation  Reference 

Belle
N(BB)=657M 
−0.83 ± 0.65 ± 0.18  0.44 ± 0.49 ± 0.14  −0.08 (stat)  PRL 100 (2008) 021602 
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_{CP} (π^{+}π^{−})  C_{CP} (π^{+}π^{−})  Correlation  Reference 

BaBar
N(BB)=467M 
−0.68 ± 0.10 ± 0.03  −0.25 ± 0.08 ± 0.02  −0.06 (stat)  arXiv:0807.4226 
Belle
N(BB)=535M 
−0.61 ± 0.10 ± 0.04  −0.55 ± 0.08 ± 0.05  −0.15 (stat)  PRL 98 (2007) 211801 
Average  −0.65 ± 0.07  −0.38 ± 0.06  −0.08 
HFAG correlated average
χ^{2} = 5.8/2 dof (CL=0.055 ⇒ 1.9σ) 

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Interpretations:
The
GronauLondon
isospin analysis allows a constraint on α ≡ φ_{2}
to be extracted from the ππ system even in the presence of nonnegligible
penguin contributions.
The analysis involves the SU(2) partners of
the B_{d}→ π^{+}π^{−} decay.
The relevant branching ratios (given in units of 10^{−6})
and CPviolating charge asymmetries are taken from
HFAG  Rare Decays.
BR(B^{0} → π^{+}π^{−}) = 5.16 ± 0.22   
BR(B^{+} → π^{+}π^{0}) = 5.59 ^{+0.41}_{0.41}  A_{CP}(B^{+} → π^{+}π^{0}) = 0.06 ± 0.05 
BR(B^{0} → π^{0}π^{0}) = 1.55 ± 0.19  A_{CP}(B^{0} → π^{0}π^{0}) = 0.43 ^{+0.25}_{−0.24} 
Belle
exclude the range 9° < φ_{2} < 81° at the 95.4% confidence level.
BaBar
give a confidence level interpretation for α
and exclude the range 23° < α < 67° at the 90% confidence level.
NB. It is implied in the above constraints on α ≡ φ_{2} that a mirror solution at α → α + π ≡ φ_{2} → φ_{2} + π also exists.
For more details on the world average for α ≡ φ_{2}, calculated with different statistical treatments, refer to the CKMfitter and UTfit pages.
Both BaBar and Belle have performed a full timedependent Dalitz plot analyses 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), implemented as suggested by Quinn and Silva. The experiments determine 27 coefficients of the form factor bilinears from the fit to data. Physics parameters, such as the quasitwobody parameters, and the phases δ_{+−}=arg[A^{−+}A^{+−*}] and the UT angle α ≡ φ_{2}, are determined from subsequent fits to the bilinear coefficients.
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.
[The table of averages of the form factor bilinears is suppressed here for the benefit of the nonspecialist. Those interested in the details can find them here.] 
Compilation of averages of the form factor bilinears. 
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From the bilinear coefficients given above, both experiments extract "quasitwobody" (Q2B) parameters. Considering only the charged ρ bands in the Dalitz plot, 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
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.
We average the quasitwobody parameters provided by the experiments, which should be equivalent to determining average values directly from the averaged bilinear coefficients.
As shown by
Charles
it can be convenient to transform the experimentally motivated
CP parameters A_{CP}(ρπ) and C_{ρπ}
into the physically motivated choices
A^{+−}(ρπ) =
(κ^{+−}^{2}−1)/(κ^{+−}^{2}+1) =
−(A_{CP}(ρπ)+C_{ρπ}+A_{CP}(ρπ)ΔC_{ρπ})/(1+ΔC_{ρπ} + A_{CP}(ρπ)C_{ρπ}),
A^{−+}(ρπ) =
(κ^{−+}^{2}−1)/(κ^{−+}^{2}+1) =
(−A_{CP}(ρπ)+C_{ρπ}+A_{CP}(ρπ)ΔC_{ρπ})/(−1+ΔC_{ρπ} + A_{CP}(ρπ)C_{ρπ}),
where
κ^{+−} = (q/p)Abar^{−+}/A^{+−} and
κ^{−+} = (q/p)Abar^{+−}/A^{−+}.
With this definition A^{−+}(ρπ) (A^{+−}(ρπ))
describes CP violation in B_{d} decays
where the ρ is emitted (not emitted) by the spectator interaction.
Both experiments obtain values for A^{+−} and A^{−+},
which we average.
Again, this procedure should be equivalent to extracting these values
directly from the previous results.
In addition to the B_{d}→ ρ^{+−}π^{−+} quasitwobody contributions to the π^{+}π^{−}π^{0} final state, there can also be a B_{d}→ ρ^{0}π^{0} component. Both experiments have also extracted the quasitwobody parameters associated with this intermediate state.
Note again that at present we do not apply a rescaling of the results to a common, updated set of input parameters. Correlations due to possible common systematics are neglected in the following averages.
The citation given for Belle in the tables below corresponds to a short article published in PRL. A more detailed article on the same analysis is also available as arXiv:0710.4974 [hepex].
Experiment  A_{CP} (ρ^{+−}π^{−+})  C (ρ^{+−}π^{−+})  S (ρ^{+−}π^{−+})  ΔC (ρ^{+−}π^{−+})  ΔS (ρ^{+−}π^{−+})  Correlations  Reference 

BaBar
N(BB)=375M 
−0.14 ± 0.05 ± 0.02  0.15 ± 0.09 ± 0.05  −0.03 ± 0.11 ± 0.04  0.39 ± 0.09 ± 0.09  −0.01 ± 0.14 ± 0.06  (stat)  PRD 76 (2007) 012004 
Belle
N(BB)=449M 
−0.12 ± 0.05 ± 0.04  −0.13 ± 0.09 ± 0.05  0.06 ± 0.13 ± 0.05  0.36 ± 0.10 ± 0.05  −0.08 ± 0.13 ± 0.05  (stat)  PRL 98 (2007) 221602 
Average  −0.13 ± 0.04  0.01 ± 0.07  0.01 ± 0.09  0.37 ± 0.08  −0.04 ± 0.10  (stat) 
HFAG correlated average
χ^{2} = 4.2/5 dof (CL=0.52 ⇒ 0.6σ) 

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. 
Experiment  A_{−+} (ρ^{+−}π^{−+})  A_{+−} (ρ^{+−}π^{−+})  Correlation  Reference 

BaBar
N(BB)=375M 
−0.37 ^{+0.16} _{−0.10} ± 0.09  0.03 ± 0.07 ± 0.04  0.62  PRD 76 (2007) 012004 
Belle
N(BB)=449M 
0.08 ± 0.16 ± 0.11  0.21 ± 0.08 ± 0.04  0.47  PRL 98 (2007) 221602 
Average  −0.18 ± 0.12  0.11 ± 0.06  0.40 
HFAG correlated average
χ^{2} = 4.0/2 dof (CL=0.14 ⇒ 1.5σ) 

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Experiment  C (ρ^{0}π^{0})  S (ρ^{0}π^{0})  Correlation  Reference 

BaBar
N(BB)=375M 
−0.10 ± 0.40 ± 0.53  0.04 ± 0.44 ± 0.18  0.35  PRD 76 (2007) 012004 
Belle
N(BB)=449M 
0.49 ± 0.36 ± 0.28  0.17 ± 0.57 ± 0.35  0.08  PRL 98 (2007) 221602 
Average  0.30 ± 0.38  0.12 ± 0.38  0.12 
HFAG correlated average
χ^{2} = 0.5/2 dof (CL=0.76 ⇒ 0.3σ) 

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Interpretations:
The information given above can be used to extract α ≡ φ_{2}.
From the measured form factor bilinears,
BaBar
extract a confidence level interpretation for α,
and constrain α = (87^{+45}_{−13})° at 68% confidence level.
Belle
has performed a similar analysis.
In addition, Belle
has also included information from the SU(2) partners of B → ρπ,
which can be used to constrain α ≡ φ_{2} via an
isospin pentagon
relation.
The isospin analysis uses as input the branching fractions and
CPviolating charge asymmetries
of all five ρπ decay modes
(ρ^{+}π^{−},ρ^{−}π^{+},
ρ^{0}π^{0}, ρ^{+}π^{0},
ρ^{0}π^{+}).
The relevant information is taken from
HFAG  Rare Decays and is tabulated below.
[Branching fractions are given in units of 10^{−6}.]
Those values that are not included in the table below
can be obtained from the bilinear formfactors,
eg. a measurement
BR(B^{0} → ρ^{0}π^{0})/
BR(B^{0} → ρ^{+−}π^{−+}) = 0.133 ± 0.022 ± 0.023,
is extracted.
With all information in the ρπ channels put together,
Belle
obtain the tighter constraint
68° < φ_{2} < 95° at 68% confidence level,
for the solution consistent with the Standard Model.
BR(B^{0} → ρ^{+−}π^{−+}) = 23.0 ± 2.3   
BR(B^{+} → ρ^{+}π^{0}) = 10.9 ^{+1.4}_{−1.5}  A_{CP}(B^{+} → ρ^{+}π^{0}) = 0.02 ± 0.11 
BR(B^{+} → π^{+}ρ^{0}) = 8.3 ^{+1.2}_{−1.3}  A_{CP}(B^{+} → π^{+}ρ^{0}) = 0.18 ^{+0.09}_{−0.17} 
NB. It is implied in the above constraints on α ≡ φ_{2} that a mirror solution at α → α + π ≡ φ_{2} → φ_{2} + π also exists.
For more details on the world average for α ≡ φ_{2}, calculated with different statistical treatments, refer to the CKMfitter and UTfit pages.
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 decay is found to be dominated by the longitudinally polarized component:
At present we do not apply a rescaling of the results to a
common, updated set of input parameters.
The CP parameters measured are those for the longitudinally polarized component
(ie. S_{ρρ,long}, C_{ρρ,long}).
Experiment  S_{CP} (ρ^{+}ρ^{−})  C_{CP} (ρ^{+}ρ^{−})  Correlation  Reference 

BaBar
N(BB)=387M 
−0.17 ± 0.20 ^{+0.05} _{−0.06}  0.01 ± 0.15 ± 0.06  −0.04 (stat)  PRD 76 (2007) 052007 
Belle
N(BB)=535M 
0.19 ± 0.30 ± 0.07  −0.16 ± 0.21 ± 0.07  0.10 (stat)  PRD 76 (2007) 011104 
Average  −0.05 ± 0.17  −0.06 ± 0.13  0.01 
HFAG correlated average
χ^{2} = 1.4/2 dof (CL=0.50 ⇒ 0.7σ) 

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Since the decay B_{d} → ρ^{0}ρ^{0} results in an all charged particle final state, its timedependent CP violation parameters can be determined experimentally, if difficulties related to the small branching fraction and large backgrounds can be overcome. BaBar have performed the first such analysis. They measure the longitudinally polarized component to be
At present we do not apply a rescaling of the results to a
common, updated set of input parameters.
The CP parameters measured are those for the longitudinally polarized component
(ie. S_{ρρ,long}, C_{ρρ,long}).
Experiment  S_{CP} (ρ^{0}ρ^{0})  C_{CP} (ρ^{0}ρ^{0})  Correlation  Reference 

BaBar
N(BB)=465M 
0.3 ± 0.7 ± 0.2  0.2 ± 0.8 ± 0.3  0.0 (stat)  PRD 78 (2008) 071104(R) 
Interpretations:
The
GronauLondon
isospin analysis allows a constraint on α ≡ φ_{2}
to be extracted from the ρρ system even in the presence of nonnegligible
penguin contributions.
The analysis involves the SU(2) partners of
the B_{d}→ ρ^{+}ρ^{−} decay.
The relevant branching ratios (given in units of 10^{−6})
and CPviolating charge asymmetries are taken from
HFAG  Rare Decays.
BR(B^{0} → ρ^{+}ρ^{−}) = 24.2 ^{+3.1}_{−3.2}  — 
BR(B^{+} → ρ^{+}ρ^{0}) = 24.0 ^{+1.9}_{−2.0}  A_{CP}(B^{+} → ρ^{+}ρ^{0}) = −0.051 ± 0.054 
BR(B^{0} → ρ^{0}ρ^{0}) = 0.73 ^{+0.27}_{−0.28}  — 
Belle
obtain φ_{2} = (88 ± 17)° or
59° < φ_{2} < 117° at 90% confidence level.
Including an upper limit for the
B^{0} → ρ^{0} ρ^{0} decay,
and world average values for the other parameters, this becomes
φ_{2} = (91.7 ± 14.9)°.
BaBar
find α ∈ [73, 117]° at 68% confidence level.
In the BaBar analysis of
B_{d} → ρ^{0}ρ^{0},
a constraint of α  α_{eff} < 15.7° (17.6°)
is obtained at the 68% (90%) CL.
The solution at α  α_{eff} = +11.3° is preferred.
In the BaBar analysis of
B^{+} → ρ^{+}ρ^{0},
a constraint of α = (92.4 ^{+6.0}_{−6.5})°
is obtained.
NB. It is implied in the above constraints on α ≡ φ_{2} that a mirror solution at α → α + π ≡ φ_{2} → φ_{2} + π also exists.
For more details on the world average for α ≡ φ_{2}, calculated with different statistical treatments, refer to the CKMfitter and UTfit pages.
The BaBar collaboration have performed a Q2B analysis of the B_{d} → a_{1}^{+−}π^{−+} decay, reconstructed in the final state π^{+}π^{−}π^{+}π^{−}.
Experiment  A_{CP} (a_{1}^{+−}π^{−+})  C (a_{1}^{+−}π^{−+})  S (a_{1}^{+−}π^{−+})  ΔC (a_{1}^{+−}π^{−+})  ΔS (a_{1}^{+−}π^{−+})  Correlations  Reference 

BaBar
N(BB)=384M 
−0.07 ± 0.07 ± 0.02  −0.10 ± 0.15 ± 0.09  0.37 ± 0.21 ± 0.07  0.26 ± 0.15 ± 0.07  −0.14 ± 0.21 ± 0.06  (stat)  PRL 98 (2007) 181803 
Experiment  A_{−+} (a_{1}^{+−}π^{−+})  A_{+−} (a_{1}^{+−}π^{−+})  Correlation  Reference 

BaBar
N(BB)=384M 
0.07 ± 0.21 ± 0.15  0.15 ± 0.15 ± 0.07  0.63  PRL 98 (2007) 181803 
Interpretations:
The parameter α_{eff} ≡ φ_{2,eff},
which reduces to α ≡ φ_{2} in the limit of
no penguin contributions, can be extracted from the above results.
BaBar obtain α_{eff} = (78.6 ± 7.3)°
NB. There is a fourfold ambiguity in the above result.
For more details on the world average for α ≡ φ_{2}, calculated with different statistical treatments, refer to the CKMfitter and UTfit pages.
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  Reference  

partially reconstructed N(BB)=232m 
fully reconstructed N(BB)=232m 
partially reconstructed N(BB)=657m 
fully reconstructed N(BB)=386m 

a_{D*π}  −0.034 ± 0.014 ± 0.009  −0.040 ± 0.023 ± 0.010  −0.047 ± 0.014 ± 0.012  −0.039 ± 0.020 ± 0.013  −0.040 ± 0.010
CL=0.96 (0.0σ) 
BaBar: PRD 71 (2005) 112003 (partially reco.) BaBar: PRD 73 (2006) 111101 (fully reco.) Belle: arXiv:0809.3203 [hepex] (partially reco.) Belle: PRD 73 (2006) 092003 (fully reco.) 
c_{D*π}  −0.019 ± 0.022 ± 0.013
(lepton tags only) 
0.049 ± 0.042 ± 0.015
(lepton tags only) 
−0.009 ± 0.014 ± 0.012  −0.011 ± 0.020 ± 0.013  −0.007 ± 0.012 CL=0.61 (0.5σ) 

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. 
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Averages of the D*π results. 
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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. 
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Timedependent analyses of transitions such as B_{d} → D^{+−}K_{S}π^{−+} can be used to probe sin(2β+γ) ≡ sin(2φ_{1}+φ_{3}) in a similar way to that discussed above. Since the final state contains three particles, a Dalitz plot analysis is necessary to maximise the sensitivity. BaBar have carried out such an analysis. They obtain 2β+γ = (83 ± 53 ± 20)° (with an ambiguity 2β+γ → 2β+γ+π) assuming the ratio of the b → u and b → c amplitude to be constant across the Dalitz plot at 0.3.
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, Belle and CDF 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+−} = 2 [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.
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−}  Reference 

D_{CP}K^{−} 
BaBar
N(BB)=382M 
0.27 ± 0.09 ± 0.04  −0.09 ± 0.09 ± 0.02  1.06 ± 0.10 ± 0.05  1.03 ± 0.10 ± 0.05  PRD 77, 111102 (2008) 
Belle
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)  
CDF  0.39 ± 0.17 ± 0.04    1.30 ± 0.24 ± 0.12    PRD 81, 031105(R) (2010)  
Average 
0.24 ± 0.07
χ^{2} = 2.3 (CL=0.32 ⇒ 1.0σ) 
−0.10 ± 0.08
χ^{2} = 0.03 (CL=0.86 ⇒ 0.2σ) 
1.10 ± 0.09
χ^{2} = 0.7 (CL=0.70 ⇒ 0.4σ) 
1.06 ± 0.10
χ^{2} = 0.4 (CL=0.54 ⇒ 0.6σ) 
HFAG  

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D*_{CP}K^{−} 
BaBar
N(BB)=383M 
−0.11 ± 0.09 ± 0.01  0.06 ± 0.10 ± 0.02  1.31 ± 0.13 ± 0.03  1.09 ± 0.12 ± 0.04  PRD 78, 092002 (2008) 
Belle
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 
−0.12 ± 0.08
χ^{2} = 0.14 (CL=0.71 ⇒ 0.4σ) 
0.07 ± 0.10
χ^{2} = 0.05 (CL=0.83 ⇒ 0.2σ) 
1.33 ± 0.12
χ^{2} = 0.12 (CL=0.73 ⇒ 0.4σ) 
1.10 ± 0.12
χ^{2} = 0.03 (CL=0.87 ⇒ 0.2σ) 
HFAG  

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D_{CP}K*^{−} 
BaBar
N(BB)=379M 
0.09 ± 0.13 ± 0.06  −0.23 ± 0.21 ± 0.07  2.17 ± 0.35 ± 0.09  1.03 ± 0.27 ± 0.13  PRD 80 (2009) 092001 
Belle  NO RESULTS AVAILABLE (*)    
Average  0.09 ± 0.14  −0.23 ± 0.22  2.17 ± 0.36  1.03 ± 0.30  HFAG 
Compilation of the above results. 
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CP+ only 
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CP only 
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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. In addition, the Cartesian coordinates x_{±} (discussed below in the context of analysis of B→DK with multibody D decay) can be extracted from the observables measured in GLW analysis. The relations are
There is no direct sensitivity to y_{+−}, but indirect bounds can be obtained using

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_{ADS} and R_{ADS}, where (for example for the B^{−} → DK^{−}, D → K^{+}π^{−} mode)
A_{ADS} = [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_{ADS} = [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. Belle have presented results for B^{−} → DK^{−} while BaBar have also presented results using B^{−} → D*K^{−}, with both D* → Dπ^{0} and D* → Dγ, and B^{−} → DK*^{−}. For all the above the D → K^{+}π^{−} mode is used. In addition, BaBar have presented results using B^{−} → DK^{−} with D → K^{+}π^{−}π^{0}.
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_{ADS}  R_{ADS}  Reference 

DK^{−}
D→Kπ 
BaBar
N(BB)=426M 
−0.70 ± 0.35 ^{+0.09} _{−0.14}  0.014 ± 0.005 ± 0.003  EPS 2009 preliminary 
Belle
N(BB)=657M 
−0.13 ^{+0.97} _{−0.88} ± 0.26  0.008 ± 0.006 ^{+0.002} _{−0.003}  PRD 78 (2008) 071901  
Average 
−0.62 ± 0.34
χ^{2} = 0.32 (CL=0.57 ⇒ 0.6σ) 
0.011 ± 0.004
χ^{2} = 0.39 (CL=0.53 ⇒ 0.6σ) 
HFAG  
D*K^{−}
D* → Dπ^{0} D→Kπ 
BaBar
N(BB)=426M 
0.77 ± 0.35 ± 0.12  0.018 ± 0.009 ± 0.004  EPS 2009 preliminary 
Average  0.77 ± 0.37  0.018 ± 0.010  HFAG  
D*K^{−}
D* → Dγ D→Kπ 
BaBar
N(BB)=426M 
0.36 ± 0.94 ^{+0.25} _{−0.41}  0.013 ± 0.014 ± 0.007  EPS 2009 preliminary 
Average  0.36 ^{+0.97} _{−1.03}  0.013 ± 0.016  HFAG  
DK*^{−}
D→Kπ 
BaBar
N(BB)=379M 
−0.34 ± 0.43 ± 0.16  0.066 ± 0.031 ± 0.010  PRD 80 (2009) 092001 
Average  −0.34 ± 0.46  0.066 ± 0.033  HFAG  
DK^{−}
D→Kππ^{0} 
BaBar
N(BB)=226M 
  0.012 ± 0.012 ± 0.009  PRD 76 (2007) 111101 
Average    0.012 ± 0.015  HFAG 
Compilation of the above results. 
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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^{−}) = (1.43 ± 0.04)×10^{−4} and BR(D^{0} → K^{−}π^{+}) = (3.80 ± 0.07)×10^{−2} [PDG 2006], respectively. With this it is found r_{D} = 0.0613 ± 0.0010. The strong phase, δ_{B}, is different, in general, for decays to 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. The situation for D→Kππ^{0} is slightly more complicated since the hadronic parameters can vary across the phase space (Dalitz plane). Effective hadronic parameters can be used, and eventually a Dalitz analysis (either binned or unbinned) may be possible to extract the maximum information from the decay. 
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_{ADS} (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_{ADS}  R_{ADS}  Reference 

Dπ^{−}
D→Kπ 
BaBar
N(BB)=426M 
  0.0033 ± 0.0006 ± 0.0003  EPS 2009 preliminary 
Belle
N(BB)=657M 
−0.023 ± 0.218 ± 0.071  0.0034 ^{+0.0006} _{−0.0005} ^{+0.0001} _{−0.0002}  PRD 78 (2008) 071901  
Average  −0.02 ± 0.23 
0.0034 ± 0.0004
χ^{2} = 0.013 (CL=0.91 ⇒ 0.1σ) 
HFAG  
D*π^{−}
D* → Dπ^{0} D→Kπ 
BaBar
N(BB)=426M 
  0.0032 ± 0.0009 ± 0.0009  EPS 2009 preliminary 
Average    0.0032 ± 0.0012  HFAG  
D*π^{−}
D* → Dγ D→Kπ 
BaBar
N(BB)=426M 
  0.0027 ± 0.0014 ± 0.0022  EPS 2009 preliminary 
Average    0.0027 ± 0.0026  HFAG 
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*^{−}. Both BaBar and Belle use both D* decays to Dπ^{0} and Dγ, taking the effective strong phase shift into account. Both experiments use the decay D → K_{S}π^{+}π^{−}; BaBar also use D → K_{S}K^{+}K^{−} (though not for B^{−} → DK*^{−}).
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 using use an alternative parametrization [replacing r_{B} and δ_{B} with κr_{s} and δ_{s}, respectively] suggested by Gronau.
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} ) 
Note that (x_{+},y_{+}) are determined from B^{+} decays, while (x_{−},y_{−}) are determined from B^{−} decays.
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. For details of correlations in the model uncertainty assigned by Belle, (See Appendix of Ref.) The Belle results also include an additional source of uncertainty due to background from B^{−} → DK_{S}π^{−} other than B^{−} → DK*^{−}, which we have not included here.
Averages are performed using the following procedure.
Mode  Experiment  x+  y+  x  y  Correlation  Reference 

DK^{−} 
BaBar
N(BB)=468M 
−0.103 ± 0.037 ± 0.006 ± 0.007  −0.021 ± 0.048 ± 0.004 ± 0.009  0.060 ± 0.039 ± 0.007 ± 0.006  0.062 ± 0.045 ± 0.004 ± 0.006  (stat) (syst) (model)  arXiv:1005.1096 
Belle
N(BB)=657M 
−0.107 ± 0.043 ± 0.011 ± 0.055  −0.067 ± 0.059 ± 0.018 ± 0.063  0.105 ± 0.047 ± 0.011 ± 0.064  0.177 ± 0.060 ± 0.018 ± 0.054  (stat) (model)  PRD 81 (2010) 112002  
Average No model error 
−0.104 ± 0.029  −0.038 ± 0.038  0.085 ± 0.030  0.105 ± 0.036  (stat+syst) 
HFAG correlated average
χ^{2} = 3.6/4 dof (CL=0.47 ⇒ 0.7σ) 

NB. The contours in these plots do not include model errors. 
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D*K^{−} 
BaBar
N(BB)=468M 
0.147 ± 0.053 ± 0.017 ± 0.003  −0.032 ± 0.077 ± 0.008 ± 0.006  −0.104 ± 0.051 ± 0.019 ± 0.002  −0.052 ± 0.063 ± 0.009 ± 0.007  (stat) (syst) (model)  arXiv:1005.1096 
Belle
^{(*)}
N(BB)=657M 
0.083 ± 0.092 ± 0.081  0.157 ± 0.109 ± 0.063  −0.036 ± 0.127 ± 0.090  −0.249 ± 0.118 ± 0.049  (stat) (model)  PRD 81 (2010) 112002  
Average No model error 
0.130 ± 0.048  0.031 ± 0.063  −0.090 ± 0.050  −0.099 ± 0.056  (stat+syst) 
HFAG correlated average
χ^{2} = 5.0/4 dof (CL=0.29 ⇒ 1.1σ) 

NB. The contours in these plots do not include model errors. 
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DK^{*−} 
BaBar
N(BB)=468M 
−0.151 ± 0.083 ± 0.029 ± 0.006  0.045 ± 0.106 ± 0.036 ± 0.008  0.075 ± 0.096 ± 0.029 ± 0.007  0.127 ± 0.095 ± 0.027 ± 0.006  (stat) (syst) (model)  arXiv:1005.1096 
Belle
N(BB)=386M 
−0.105 ^{+0.177} _{−0.167} ± 0.006 ± 0.088  −0.004 ^{+0.164} _{−0.156} ± 0.013 ± 0.095  −0.784 ^{+0.249} _{−0.295} ± 0.029 ± 0.097  −0.281 ^{+0.440} _{−0.335} ± 0.046 ± 0.086  (stat) (model)  PRD 73, 112009 (2006)  
Average No model error 
−0.152 ± 0.077  0.024 ± 0.091  −0.043 ± 0.094  0.091 ± 0.096  (stat+syst) 
HFAG correlated average
χ^{2} = 13/4 dof (CL=0.011 ⇒ 2.5σ) 

NB. The contours in these plots do not include model errors. 
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Compilation of (x_{±},y_{±}) measurements from B → D^{(}*^{)}K^{(}*^{)} decays with D → K_{S}π^{+}π^{−} and D → K_{S}K^{+}K^{−}. NB. The uncertainities in these plots do not include model errors. 
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Compilation of x_{+} and x_{−} measurements including results from Dalitz and GLW analyses. NB. The uncertainities in these plots do not include model errors. 
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. 
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. 
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 = (68 ^{+15}_{−14} ± 4 ± 3)° (from DK^{−}, D*K^{−} & DK*^{−}) 
Belle obtain
φ_{3} = (78 ^{+11}_{−12} ± 4 ± 9)° (from DK^{−} & D*K^{−}) 

The experiments also obtain values for the hadronic parameters  
r_{B} (DK^{−}) = 0.096 ± 0.029 ± 0.005 ± 0.004  δ_{B} (DK^{−}) = (119 ^{+19}_{−20} ± 3 ± 3)°  r_{B} (DK^{−}) = 0.160 ^{+0.040}_{−0.038} ± 0.011^{+0.05}_{−0.010}  δ_{B} (DK^{−}) = (138 ^{+13}_{−16} ± 4 ± 23)° 
r_{B} (D*K^{−}) = 0.133 ^{+0.042}_{−0.039} ± 0.014 ± 0.003  δ_{B} (D*K^{−}) = (−82 ± 21 ± 5 ± 3)°  r_{B} (D*K^{−}) = 0.196 ^{+0.072}_{−0.069} ± 0.012 ^{+0.062}_{−0.012}  δ_{B} (D*K^{−}) = (342 ^{+19}_{−21} ± 3 ± 23)° 
κr_{s} = 0.149 ^{+0.066}_{−0.062} ± 0.026 ± 0.006  δ_{s} = (111 ± 32 ± 11 ± 3)°  ( r_{B} (DK*^{−}) = 0.56 ^{+0.22}_{−0.16} ± 0.04 ± 0.08 *)  (δ _{B} (DK*^{−}) = (243^{+20}_{−23} ± 3 ± 50 )° *) 
For attempts to extract γ ≡ φ_{3}
from the combined BaBar and Belle results,
visit the
CKMfitter
and
UTfit sites.
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. 
BaBar have performed a similar Dalitz plot analysis using the decay D → π^{+}π^{−}π^{0}. In this case the measured yields of B^{−} → DK^{−} and B^{+} → DK^{+} events are found to make a significant contribution to the sensitivity to CP violation and this information is included into the fit. Consequently, an alternative set of fit parameters is used in order to avoid significant biasing and nonlinear correlations. The result is parameterized in terms of polar coordinates:
ρ_{±} ≡  z_{±}  x_{0}   θ_{±} ≡ tan^{− 1} (Im(z_{±}) / (Re(z_{±})  x_{0})) 
where the constant x_{0} = 0.850 depends on the amplitude structure of the D → π^{+}π^{−}π^{0} decay, and z_{±} = r_{B} e^{i( δB ± γ )} ≡ r_{B} e^{i( δB ± φ3 )}. This choice of variables is motivated by the fact that the yields of B^{±} decays are proportional to 1 + ρ_{±}^{2}  x_{0}^{2}. The uncertainty due to the D decay model is included in the systematic error.
Mode  Experiment  ρ^{+}  θ^{+}  ρ^{−}  θ^{−}  Reference 

DK^{−}
D→ π^{+}π^{−}π^{0} 
BaBar
N(BB)=324M 
0.75 ± 0.11 ± 0.04  (147 ± 23 ± 1)°  0.72 ± 0.11 ± 0.04  (173 ± 42 ± 2)°  PRL 99 (2007) 251801 
Average  0.75 ± 0.12  (147 ± 23)°  0.72 ± 0.12  (173 ± 42)° 
Digression:
Constraining γ ≡ φ_{3}:
The measurements of ρ_{+,−} and θ_{+,−} can be used to place bounds on γ ≡ φ_{3} and the hadronic parameters. BaBar use a frequentist technique to obtain −30° < γ < 76°, 0.06 < r_{B} (DK^{−}) < 0.78 and −27° < δ _{B} (DK^{−}) < 78° at the 68% confidence level. 
BaBar have presented results on B^{0} → DK*^{0} with D → K^{−}π^{+}, D → K^{−}π^{+} π^{0} and D → K^{−}π^{+} π^{+} π^{−}. The following 95% CL limits are set:
BaBar
N(BB)=465M 
R_{ADS}(Kπ) < 0.244  R_{ADS}(Kππ^{0}) < 0.181  R_{ADS}(Kπππ) < 0.391  PRD 80 (2009) 031102 
(See above for a definition of the parameters).
Combining the results and using additional input from CLEOc (here and here) a limit on the ratio between the b→u and b→c amplitudes of r_{s} ∈ [0.07,0.41] at 95% CL limits is set. 
BaBar have performed a similar Dalitz plot analysis to that described above using neutral B decays. In order to avoid complications due to B^{0}–B^{0}bar oscillations (see here), the decay to the selftagging final state DK*^{0}, with K*^{0} → K^{+}π^{−}, is used. Effects due to the natural width of the K*^{0} are handled using the parametrization suggested by Gronau.
Constraining γ ≡ φ_{3}:
BaBar extract the threedimensional likelihood for the parameters (γ, δ_{S}, r_{S}) and, combining with a separately measured PDF for r_{S} (using a Bayesian technique), obtain bounds on each of the three parameters. 
γ = (162 ± 56)°, δ_{S} = (62 ± 57)° r_{S} < 0.55 at 95% probability 
Note that there is an ambiguity in the solutions for γ and δ_{S} (γ, δ_{S} → γ+π, δ_{S}+π). 