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})  (1.527 ± 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) 
0.233 ± 0.010 ± 0.005 
BaBar: BABARCONF06/043
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)=348m  N(BB)=532m   
BaBar (hepex/0607107)
Belle (hepex/0608039) 
J/ψK_{S} (η_{CP}=1)  0.697 ± 0.041_{stat}  0.643 ± 0.038_{stat}  
ψ(2S)K_{S} (η_{CP}=1)  0.893 ± 0.119_{stat}    
χ_{c1}K_{S} (η_{CP}=1)  0.715 ± 0.174_{stat}    
η_{c}K_{S} (η_{CP}=1)  0.711 ± 0.229_{stat}    
J/ψK_{L} (η_{CP}=+1)  0.719 ± 0.080_{stat}  0.641 ± 0.057_{stat}  
J/ψK*^{0} (K*^{0} → K_{S}π^{0}) (η_{CP}= 12A_{⊥}^{2})  0.517 ± 0.283_{stat}    
All charmonium  0.710 ± 0.034 ± 0.019  0.642 ± 0.031 ± 0.017 
0.674 ± 0.026
(0.023_{statonly}) 
CL = 0.18 
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  0.675 ± 0.026 (0.023_{statonly})  CL = 0.36 
from which we obtain the following solutions for β ≡ φ_{1} (in [0, π])
β ≡ φ_{1} = (21.2 ± 1.0)°  or  β ≡ φ_{1} = (68.8 ± 1.0)° 
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)=348m  N(BB)=532m   
BaBar (hepex/0607107)
Belle (hepex/0608039) 
J/ψK_{S}    0.001 ± 0.028_{stat}  
ψ(2S)K_{S}      
χ_{c1}K_{S}      
η_{c}K_{S}      
J/ψK_{L}    −0.045 ± 0.033_{stat}  
J/ψK*^{0} (K*^{0} → K_{S}π^{0})      
All charmonium  0.070 ± 0.028 ± 0.018  −0.018 ± 0.021 ± 0.014 
0.012 ± 0.022
(0.017_{statonly}) 
CL = 0.02 
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.
BaBar divide 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. BaBar measures:
Experiment  J_{c}/J_{0}  (2J_{s1}/J_{0})sin(2β)  (2J_{s2}/J_{0})cos(2β)  Correlation  Reference 

BaBar
N(BB)=230m 
0.76 ± 0.18 ± 0.07  0.10 ± 0.24 ± 0.06  0.38 ± 0.24 ± 0.05    hepex/0608016 
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 the B_{s}–B_{s}bar mixing phase, φ_{s}. 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}.
The first measurement of φ_{s} from B_{s} → J/ψ φ has been performed by D0. Using an integrated luminosiy of 1 fb^{−1}, they perform an untagged, timedependent analysis from which they simultaneously measure the average B_{s} lifetime τ(B_{s}), ΔΓ, φ_{s}, the magnitude of the perpendicularly polarized component A_{⊥}, the difference in the fractions of the two CPeven components A_{0}^{2}  A_{}^{2}, and the strong phases associated with the two CPeven components δ_{0} and δ_{}. The results are given below.
The implicit convention above is that A_{⊥}^{2} + A_{0}^{2} + A_{}^{2} = 1, and the strong phases are measured relative to that of the A_{⊥} component (which is set to zero). The polarization components are defined at time t=0, ie. at the production (primary) vertex of the B_{s}. Note also that there is an ambiguity in the result for φ_{s}.
Experiment  τ(B_{s})  ΔΓ  φ_{s}  A_{⊥}  A_{0}^{2}  A_{}^{2}  δ_{}  δ_{0}  Correlation  Reference 

D0  1.49 ± 0.08 ^{+0.01} _{−0.03}  0.17 ± 0.09 ± 0.03  −0.79 ± 0.56 ± 0.01  0.46 ± 0.06 ± 0.01  0.37 ± 0.06 ± 0.01  3.30 ± 1.10 ± 0.00  0.70 ± 1.00 ± 0.00  (stat)  hepex/0701012 
Interpretations:
D0
have combined the contour in the (φ_{s}, ΔΓ) plane obtained above
with a constraint obtained from the
charge asymmetry
in B–Bbar oscillations
(see also HFAG  Oscillations),
to obtain the result
φ_{s} = −0.56 ^{+0.44}_{−0.41}.
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}.
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)=311m 
0.45 ± 0.36 ± 0.05 ± 0.07  0.54 ± 0.54 ± 0.08 ± 0.18  0.975 ^{+0.093}_{−0.085} ± 0.012 ± 0.002 
0.07 stat
between sin(2β) & cos(2β) 
hepex/0607105 
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  0.57 ± 0.30
χ^{2} = 0.3/1 dof (CL=0.59 ⇒ 0.5σ) 
1.16 ± 0.42
χ^{2} = 2.5/1 dof (CL=0.12 ⇒ 1.6σ) 
  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 87% 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 the BaBar timedependent Dalitz plot analysis of B^{0} → K^{+}K^{−}K^{0} are also discussed in the next section). The BaBar results that appear in this table on K^{+}K^{−}K^{0} come from previous analyses in which the final state is treated as a quasitwobody system .
At present we do not apply a rescaling of the results to a common, updated set of input parameters. The exception is the CPeven fraction in the quasitwobody analysis of B^{0} → K^{+}K^{−}K^{0}. We take correlations between S and C into account, 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)=347M 
0.12 ± 0.31 ± 0.10  0.18 ± 0.20 ± 0.10    hepex/0607112 
Belle
N(BB)=535M 
0.50 ± 0.21 ± 0.06  −0.07 ± 0.15 ± 0.05  0.05 (stat)  hepex/0608039  
Average  0.39 ± 0.18  0.01 ± 0.13  0.03 
HFAG correlated average
χ^{2} = 1.8/2 dof (CL=0.41 ⇒ 0.8σ) 


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η′K^{0} 
BaBar
N(BB)=384M 
0.58 ± 0.10 ± 0.03  −0.16 ± 0.07 ± 0.03  0.03 (stat)  hepex/0609052 
Belle
N(BB)=535M 
0.64 ± 0.10 ± 0.04  0.01 ± 0.07 ± 0.05  0.09 (stat)  hepex/0608039  
Average  0.61 ± 0.07  −0.09 ± 0.06  0.04 
HFAG correlated average
χ^{2} = 2.3/2 dof (CL=0.32 ⇒ 1.0σ) 


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K_{S}K_{S}K_{S} 
BaBar
N(BB)=347M 
0.66 ± 0.26 ± 0.08  −0.14 ± 0.22 ± 0.05  0.09 (stat)  hepex/0607108 
Belle
N(BB)=535M 
0.30 ± 0.32 ± 0.08  −0.31 ± 0.20 ± 0.07    hepex/0608039  
Average  0.51 ± 0.21  −0.23 ± 0.15  0.04 
HFAG correlated average
χ^{2} = 1.0/2 dof (CL=0.61 ⇒ 0.5σ) 


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π^{0}K_{S} 
BaBar
N(BB)=348M 
0.33 ± 0.26 ± 0.04  0.20 ± 0.16 ± 0.03  −0.06 (stat)  hepex/0607096 
Belle
N(BB)=532M 
0.33 ± 0.35 ± 0.08  0.05 ± 0.14 ± 0.05  −0.08 (stat)  hepex/0609006  
Average  0.33 ± 0.21  0.12 ± 0.11  −0.06 
HFAG correlated average
χ^{2} = 0.5/2 dof (CL=0.79 ⇒ 0.3σ) 


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ρ^{0}K_{S} 
BaBar
N(BB)=227M 
0.20 ± 0.52 ± 0.24  0.64 ± 0.41 ± 0.20    PRL 98 (2007) 051803 
ωK_{S} 
BaBar
N(BB)=347M 
0.62 ^{+0.25} _{−0.30} ± 0.02  −0.43 ^{+0.25} _{−0.23} ± 0.03    hepex/0607101 
Belle
N(BB)=532M 
0.11 ± 0.46 ± 0.07  0.09 ± 0.29 ± 0.06  −0.04 (stat)  hepex/0609006  
Average 
0.48 ± 0.24
χ^{2} = 0.9 (CL=0.35 ⇒ 0.9σ) 
−0.21 ± 0.19
χ^{2} = 1.8 (CL=0.18 ⇒ 1.3σ) 
uncorrelated averages  HFAG  

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f_{0}K^{0}  BaBar ^{(*)}  0.62 ± 0.23  −0.36 ± 0.23    hepex/0607112, hepex/0408095 ^{(*)} 
Belle
N(BB)=532M 
0.18 ± 0.23 ± 0.11  0.15 ± 0.15 ± 0.07  −0.01 (stat)  hepex/0609006  
Average  0.42 ± 0.17  −0.02 ± 0.13  −0.00 
HFAG correlated average
χ^{2} = 4.9/2 dof (CL=0.09 ⇒ 1.7σ) 


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π^{0}π^{0}K_{S} 
BaBar
N(BB)=227M 
−0.84 ± 0.71 ± 0.08  0.27 ± 0.52 ± 0.13    hepex/0508017 
K^{+}K^{−}K^{0}
(excluding φK^{0}) 
BaBar Q2B
^{(*)}
N(BB)=227M 
0.41 ± 0.18 ± 0.07 ± 0.11_{CPeven}
(f_{CPeven}= 0.89 ± 0.08 ± 0.06 [moments]) 
0.23 ± 0.12 ± 0.07    hepex/0507016 
Belle
N(BB)=532M 
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)  hepex/0609006  
Average 
0.58 ± 0.13 ^{+0.12}_{0.09}_{CPeven}
(rescaled to average f_{CPeven}= 0.91 ± 0.07) χ^{2} = 1.6 (CL=0.21) 
0.15 ± 0.09
χ^{2} = 0.6 (CL=0.43) 
uncorrelated averages  HFAG  

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Naïve b→s penguin average 
0.53 ± 0.05
χ^{2} = 13/15 dof (CL=0.59 ⇒ 0.5σ) 
−0.01 ± 0.04
χ^{2} = 21/15 dof (CL=0.13 ⇒ 1.5σ) 
uncorrelated averages  HFAG  
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Direct comparison of charmonium and spenguin averages (see comments below): χ^{2} = 6.6 (CL=0.01 ⇒ 2.6σ) 
^{(*)} The BaBar results for φK^{0} are determined from their timedependent Dalitz plot analysis of B^{0} → K^{+}K^{−}K^{0}. The BaBar results for f_{0}K^{0} are a combination of results from the Dalitz plot analysis (sin(2β^{eff}) = 0.31 ± 0.32 ± 0.07, C_{CP} = −0.45 ± 0.28 ± 0.10), with those from the quasitwobody analysis of B^{0} → f_{0}K_{S}, f_{0} → π^{+}π^{−} (sin(2β^{eff}) = 0.95 ^{+0.23} _{−0.32} ± 0.10, C_{CP} = −0.24 ± 0.31 ± 0.15, hepex/0408095). The BaBar results for K^{+}K^{−}K^{0} are taken from their previous quasitwobody analysis (hepex/0507016).
Please note that
Compilation of results for −η×S ≈ sin(2β^{eff}) ≡ sin(2φ_{1}^{eff}) and C from spenguin decays. 
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Same, but without π^{0}π^{0}K_{S} and ρ^{0}K_{S}, 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 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. An updated version of this plot is being prepared for ICHEP 2006. 
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Timedependent amplitude analysis of the threebody B_{d} → K^{+}K^{−}K^{0} decay allows 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. At present, the extracted parameters are not in a form that allows a straightforward comparison/combination with those from timedependent CP asymmetries in quasitwobody b → qqbar s modes. Rather, the effective weak phase β^{eff} ≡ φ_{1}^{eff} is directly determined for two significant resonant contributions: φK^{0} and f_{0}K^{0}, 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}).
Experiment  φK^{0}  f_{0}K^{0}  K^{+}K^{−}K^{0}  Reference  

β^{eff}  A_{CP}  β^{eff}  A_{CP}  β^{eff}  A_{CP}  
BaBar
N(BB)=347m 
0.06 ± 0.16 ± 0.05  −0.18 ± 0.20 ± 0.10  0.18 ± 0.19 ± 0.04  0.45 ± 0.28 ± 0.10  0.361 ± 0.079 ± 0.037  −0.034 ± 0.079 ± 0.025  hepex/0607112 
From the above results BaBar infer that the trigonometric reflection at π/2  &beta^{eff} is disfavoured at 4.6σ.
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)=232M 
−0.68 ± 0.30 ± 0.04  −0.21 ± 0.26 ± 0.06  0.08 (stat)  PRD 74 011101 (2006) 
Belle
N(BB)=152M 
−0.72 ± 0.42 ± 0.09  0.01 ± 0.29 ± 0.03  −0.12 (stat)  PRL 93, 261801 (2004) 
Average  −0.68 ± 0.25  −0.11 ± 0.20  0.00  HFAG correlated average χ^{2} = 0.3/2 dof (CL=0.86 ⇒ 0.2σ) 

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Experiment  S_{CP} (D^{+}D^{−})  C_{CP} (D^{+}D^{−})  Correlation  Reference 

BaBar
N(BB)=232M 
−0.29 ± 0.63 ± 0.06  0.11 ± 0.35 ± 0.06    PRL 95 (2005) 131802 
Belle
N(BB)=535M 
−1.12 ± 0.37 ± 0.09  −0.92 ± 0.23 ± 0.05  0.04  CKM2006 preliminary 
Average  −0.89 ± 0.33  −0.60 ± 0.20  0.03 
HFAG correlated average
χ^{2} = 7.1/2 dof (CL=0.029 ⇒ 2.2σ) 

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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 experiments obtain the fraction of the transversely polarized component:
We convert Im(λ) = S/(1 + C) and λ^{2} = (1 − C)/(1 + C), taking into account correlations. Note that Belle quote uncertainties due to the transversely polarized fraction R_{T} separately from other systematic errors, while BaBar do not.
Experiment  S_{CP} (D*^{+} D*^{−})  C_{CP} (D*^{+} D*^{−})  Correlation  Reference 

BaBar
N(BB)=227M 
−0.75 ± 0.25 ± 0.03  0.06 ± 0.17 ± 0.03  0.04 (stat)  PRL 95 (2005) 151804 
Belle
N(BB)=152M 
−0.75 ± 0.56 ± 0.10 ± 0.06  0.26 ± 0.26 ± 0.05 ± 0.01    PLB 618 (2005) 34 
Average  −0.75 ± 0.23  0.12 ± 0.14  0.03  HFAG correlated average χ^{2} = 0.4/2 dof (CL=0.82⇒ 0.2σ) 

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^{(}*^{)} 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.
Experiment  S_{+−}(D*^{+}D^{−})  C_{+−}(D*^{+}D^{−})  S_{−+}(D*^{−}D^{+})  C_{−+}(D*^{−}D^{+})  A(D*^{+−}D^{−+})  Correlations  Reference 

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. 
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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 
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_{CP} (b → sγ)  C_{CP} (b → sγ)  Correlation  Reference  

K*(892)γ 
BaBar
N(BB)=232M 
−0.21 ± 0.40 ± 0.05  −0.40 ± 0.23 ± 0.04  0.07 (stat)  PRD 72 (2005) 051103  
Belle
N(BB)=532M 
−0.32 ^{+0.36} _{−0.33} ± 0.05  0.20 ± 0.24 ± 0.05  0.08 (stat)  PRD 74 (2006) 111104  
Average  −0.28 ± 0.26  −0.11 ± 0.17  0.07 
HFAG correlated average
χ^{2} = 3.2/2 dof (CL=0.20 ⇒ 1.3σ) 

K_{S}π^{0}γ
(incl. K*γ) 
BaBar
N(BB)=232M 
−0.06 ± 0.37  −0.48 ± 0.22  0.05 (stat)  PRD 72 (2005) 051103  
Belle
N(BB)=532M 
−0.10 ± 0.31 ± 0.07  0.20 ± 0.20 ± 0.06  0.08 (stat)  PRD 74 (2006) 111104  
Average  −0.09 ± 0.24  −0.12 ± 0.15  0.06 
HFAG correlated average
χ^{2} = 5.1/2 dof (CL=0.08 ⇒ 1.8σ) 


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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)=350M 
−0.53 ± 0.14 ± 0.02  −0.16 ± 0.11 ± 0.03  −0.08 (stat)  hepex/0607106 
Belle
N(BB)=532M 
−0.61 ± 0.10 ± 0.04  −0.55 ± 0.08 ± 0.05  −0.15 (stat)  hepex/0608035 
Average  −0.59 ± 0.09  −0.39 ± 0.07  −0.10  HFAG correlated average χ^{2} = 7.4/2 dof (CL=0.024 ⇒ 2.3σ) 

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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^{−}) = 5.2 ± 0.2   
BR(B^{+} → π^{+}π^{0}) = 5.7 ± 0.4  A_{CP}(B^{+} → π^{+}π^{0}) = 0.04 ± 0.05 
BR(B^{0} → π^{0}π^{0}) = 1.3 ± 0.2  A_{CP}(B^{0} → π^{0}π^{0}) = 0.36 ^{+0.33}_{−0.31} 
Belle
exclude the range 9° < φ_{2} < 81° at the 95.4% confidence level.
BaBar
give a confidence level interpretation for α.
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 is convenient to transform the experimentally motivated
CP parameters A_{CP}(ρπ) and C_{ρπ}
into the physically motivated ones
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. Belle 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.
Experiment  A_{CP} (ρ^{+−}π^{−+})  C (ρ^{+−}π^{−+})  S (ρ^{+−}π^{−+})  ΔC (ρ^{+−}π^{−+})  ΔS (ρ^{+−}π^{−+})  Correlations  Reference 

BaBar
N(BB)=347M 
−0.14 ± 0.04 ± 0.01  0.15 ± 0.09 ± 0.04  0.01 ± 0.12 ± 0.03  0.38 ± 0.09 ± 0.02  0.06 ± 0.13 ± 0.03  (total)  hepex/0608002 
Belle
N(BB)=447M 
−0.12 ± 0.05 ± 0.03  −0.13 ± 0.09 ± 0.06  0.06 ± 0.13 ± 0.07  0.35 ± 0.10 ± 0.06  −0.12 ± 0.14 ± 0.07  (total)  hepex/0609003 
Average  −0.13 ± 0.03  0.03 ± 0.07  0.03 ± 0.09  0.36 ± 0.07  −0.02 ± 0.10  (total) 
HFAG correlated average
χ^{2} = 4.7/5 dof (CL=0.45 ⇒ 0.8σ) 

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

BaBar
N(BB)=347M 
−0.38 ^{+0.15} _{−0.16} ± 0.07  0.03 ± 0.07 ± 0.03  0.62  hepex/0608002 
Belle
N(BB)=447M 
0.08 ± 0.17 ± 0.12  0.22 ± 0.08 ± 0.05  0.53  hepex/0609003 
Average  −0.19 ± 0.13  0.11 ± 0.06  0.46 
HFAG correlated average
χ^{2} = 3.8/2 dof (CL=0.15 ⇒ 1.4σ) 

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

Belle
N(BB)=447M 
0.45 ± 0.35 ± 0.32  0.15 ± 0.57 ± 0.43  0.07  hepex/0609003 
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 constraint 75° < α < 152° 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
φ_{2} = (83^{+12}_{−23})°
(where the errors correspond to 1σ ie. 68.3% confidence level).
BR(B^{0} → ρ^{+−}π^{−+}) = 24.0 ± 2.5   
BR(B^{+} → ρ^{+}π^{0}) = 10.8 ^{+1.4}_{−1.5}  A_{CP}(B^{+} → ρ^{+}π^{0}) = 0.02 ± 0.11 
BR(B^{+} → π^{+}ρ^{0}) = 8.7 ^{+1.0}_{−1.1}  A_{CP}(B^{+} → π^{+}ρ^{0}) = −0.07 ^{+0.12}_{−0.13} 
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)=350M 
−0.19 ± 0.21 ^{+0.05} _{−0.07}  −0.07 ± 0.15 ± 0.06  −0.06 (stat)  hepex/0607098 
Belle
N(BB)=535M 
0.19 ± 0.30 ± 0.07  −0.16 ± 0.21 ± 0.07  0.10 (stat)  hepex/0702009 
Average  −0.06 ± 0.18  −0.11 ± 0.13  −0.00 
HFAG correlated average
χ^{2} = 1.2/2 dof (CL=0.56 ⇒ 0.6σ) 

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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^{−}) = 23.1 ^{+3.2}_{−3.3}   
BR(B^{+} → ρ^{+}ρ^{0}) = 18.2 ± 3.0  A_{CP}(B^{+} → ρ^{+}ρ^{0}) = −0.08 ± 0.13 
BR(B^{0} → ρ^{0}ρ^{0}) = 1.2 ± 0.5  A_{CP}(B^{0} → ρ^{0}ρ^{0}) NOT MEASURED YET 
Belle
obtain φ_{2} = (88 ± 17)° or
59° < φ_{2} < 117° at 90% confidence level.
BaBar
find α ∈ [74, 117]° at 68% 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.
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    hepex/0612050 
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)=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.) BaBar: PRD 73 (2006) 111101 (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. 
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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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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−}  Reference 

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. 
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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. 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_{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 preliminary 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'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  PRD 72 (2005) 071104 
Average  −0.22 ± 0.63  0.046 ± 0.032  
DK^{−}
D→Kππ^{0} 
BaBar'06
N(BB)=226m 
  0.012 ± 0.012 ± 0.009  hepex/0607065 
Average    0.012 ± 0.015 
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 
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.
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, which is based on a set of reasonable, though imperfect, assumptions.
Mode  Experiment  x+  y+  x  y  Correlation  Reference 

DK^{−}
D→K_{S}π^{+}π^{−} 
BaBar
N(BB)=347M 
−0.072 ± 0.056 ± 0.014 ± 0.029  −0.033 ± 0.066 ± 0.007 ± 0.018  0.041 ± 0.059 ± 0.018 ± 0.011  0.056 ± 0.071 ± 0.007 ± 0.023  (stat)  hepex/0607104 
Belle
N(BB)=386M 
−0.135 ^{+0.069} _{−0.070} ± 0.017 ± 0.051  −0.085 ^{+0.090} _{−0.086} ± 0.009 ± 0.066  0.025 ^{+0.072} _{−0.080} ± 0.013 ± 0.068  0.170 ^{+0.093} _{−0.117} ± 0.016 ± 0.049  (stat) (model)  PRD 73, 112009 (2006)  
Average No model error 
−0.097 ± 0.045  −0.051 ± 0.053  0.045 ± 0.047  0.093 ± 0.058  (stat+syst) 
HFAG correlated average
χ^{2} = 1.5/4 dof (CL=0.83 ⇒ 0.2σ) 

NB. The contours in these plots do not include model errors. 
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D*K^{−}
D*→Dπ^{0} & D*→Dγ D→K_{S}π^{+}π^{−} 
BaBar
N(BB)=347M 
0.084 ± 0.088 ± 0.015 ± 0.018  0.096 ± 0.111 ± 0.032 ± 0.017  −0.106 ± 0.091 ± 0.020 ± 0.009  −0.019 ± 0.096 ± 0.022 ± 0.016  (stat)  hepex/0607104 
Belle
N(BB)=386M 
0.032 ^{+0.120} _{−0.116} ± 0.004 ± 0.049  0.008 ^{+0.137} _{−0.136} ± 0.011 ± 0.074  −0.128 ^{+0.167} _{−0.146} ± 0.023 ± 0.071  −0.339 ^{+0.172} _{−0.158} ± 0.027 ± 0.053  (stat) (model)  PRD 73, 112009 (2006)  
Average No model error 
0.067 ± 0.071  0.061 ± 0.088  −0.110 ± 0.080  −0.101 ± 0.085  (stat+syst)  HFAG correlated average χ^{2} = 3.2/4 dof (CL=0.52 ⇒ 0.6σ) 

NB. The contours in these plots do not include model errors. 
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DK^{*−}
K^{*−} → K_{S}π^{−} D→K_{S}π^{+}π^{−} 
BaBar
N(BB)=227M 
−0.070 ± 0.230 ± 0.130 ± 0.030  −0.010 ± 0.320 ± 0.180 ± 0.050  −0.200 ± 0.200 ± 0.110 ± 0.030  0.260 ± 0.300 ± 0.160 ± 0.030    hepex/0507101 
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 
−0.094 ± 0.144  −0.007 ± 0.146  −0.480 ± 0.173  −0.056 ± 0.253  (stat+syst)  HFAG correlated average χ^{2} = 4.6/4 dof (CL=0.33 ⇒ 1.0σ) 

NB. The contours in these plots do not include model errors. 
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NB. The uncertainities assigned to the averages given in these plots do not include model errors. 
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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 = (92 ± 41 ± 11 ± 12)° (from DK^{−} & D*K^{−}) 
Belle obtain
φ_{3} = (53 ^{+15}_{−18} ± 3 ± 9)° (from DK^{−},D*K^{−} & DK*^{−}) 

The experiments also obtain values for the hadronic parameters  
r_{B} (DK^{−}) < 0.140 (1σ)  δ _{B} (DK^{−}) = (118 ± 63 ± 19 ± 36)°  r_{B} (DK^{−}) = 0.16 ± 0.05 ± 0.01 ± 0.05  δ _{B} (DK^{−}) = (146 ^{+19}_{−20} ± 3 ± 23)° 
0.017 < r_{B} (D*K^{−}) < 0.203  δ _{B} (D*K^{−}) = (298 ± 59 ± 18 ± 10)°  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 )° 
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. 