HFLAV: Timedependent CP violation & Unitarity Triangle angles













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})  (1.520 ± 0.004) ps  HFLAV  Oscillations/Lifetime 
Δm_{d}  (0.5064 ± 0.0019) ps^{−1}  HFLAV  Oscillations/Lifetime 
ΔΓ_{d}/Γ_{d}  −0.002 ± 0.010  HFLAV  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 

0.215 ± 0.032 ± 0.006 
CDF: PRL 94 (2005) 101803 (*)
∫Ldt=0.3 fb^{−1} 

0.183 ± 0.013 ± 0.025 
D0: PRL 102 (2009) 032001
∫Ldt=2.8 fb^{−1} 

0.201 ± 0.004 ± 0.008 
LHCb: PRD 88 (2013) 052002
∫Ldt=1.0 fb^{−1} 

0.209 ± 0.006 
Average
χ^{2} = 7.2/4 dof (CL=0.12 ⇒ 1.5σ) 
(*) We do not include an unpublished CDF preliminary result from 2007.
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  LHCb  Average  Reference 
N(BB)=465M  N(BB)=772M  ∫Ldt=3 fb^{−1}  
J/ψK_{S} (η_{CP}=1)  0.657 ± 0.036 ± 0.012  0.670 ± 0.029 ± 0.013 
0.731 ± 0.035 ± 0.020 0.83 ± 0.08 ± 0.01^{(*)} 
0.695 ± 0.019 (0.018_{statonly}) 
BaBar (PRD 79 (2009) 072009) Belle (PRL 108 (2012) 171802) LHCb (PRL 115 (2015) 031601, JHEP 11 (2017) 170) 
J/ψK_{L} (η_{CP}=+1)  0.694 ± 0.061 ± 0.031  0.642 ± 0.047 ± 0.021    0.663 ± 0.041 (0.037_{statonly}) 

J/ψK^{0}  0.666 ± 0.031 ± 0.013    0.75 ± 0.04  0.690 ± 0.018 (0.017_{statonly}) 

ψ(2S)K_{S} (η_{CP}=1)  0.897 ± 0.100 ± 0.036  0.738 ± 0.079 ± 0.036  0.84 ± 0.10 ± 0.01  0.817 ± 0.056 (0.053_{statonly}) 

ψ(nS)K^{0}      0.760 ± 0.034  0.701 ± 0.017 (0.016_{statonly}) 

χ_{c1}K_{S} (η_{CP}=1)  0.614 ± 0.160 ± 0.040  0.640 ± 0.117 ± 0.040    0.632 ± 0.099 (0.094_{statonly}) 

η_{c}K_{S} (η_{CP}=1)  0.925 ± 0.160 ± 0.057        BaBar (PRD 79 (2009) 072009) 
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  0.667 ± 0.023 ± 0.012  0.760 ± 0.034  0.698 ± 0.017 (0.016_{statonly}) 
CL = 0.09 
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}) 
0.667 ± 0.023 ± 0.012  0.760 ± 0.034  0.699 ± 0.017 (0.016_{statonly}) 
CL = 0.09 
^{(*)} The first LHCb result for J/ψK_{S} uses J/ψ decays to muons, while the second uses electrons. Their combination is given under J/ψK^{0}.
^{(**)} 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, as well as a measurement by Belle using Υ(5S) data and Bπ tagging:
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  
Belle (121/fb Υ(5S) data)  0.57 ± 0.58 ± 0.06  PRL 108 (2012) 171801 
we find the slightly modified average:
Parameter: sin(2β) ≡ sin(2φ_{1})  

All charmonium  0.699 ± 0.017 (0.016_{statonly})  CL = 0.21 
from which we obtain the following solutions for β ≡ φ_{1} (in [0, π])
β ≡ φ_{1} = (22.2 ± 0.7)°  or  β ≡ φ_{1} = (67.8 ± 0.7)° 
Plots:
Averages of sin(2β) ≡ sin(2φ_{1}) and C=−A from all experiments. 
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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  LHCb  Average  Reference 
N(BB)=465M  N(BB)=772M  ∫Ldt=3 fb^{−1}  
J/ψK_{S}  0.026 ± 0.025 ± 0.016  0.015 ± 0.021 ^{+0.023}_{−0.045} 
−0.038 ± 0.032 ± 0.005^{(†)} 0.12 ± 0.07 ± 0.02^{(*)} 
0.000 ± 0.020 (0.014_{statonly}) 
BaBar (PRD 79 (2009) 072009) Belle (PRL 108 (2012) 171802) LHCb (PRL 115 (2015) 031601, JHEP 11 (2017) 170) 
J/ψK_{L}  −0.033 ± 0.050 ± 0.027  −0.019 ± 0.026 ^{+0.041}_{−0.017}    −0.023 ± 0.030 (0.023_{statonly}) 

J/ψK^{0}  0.016 ± 0.023 ± 0.018    −0.014 ± 0.030  −0.007 ± 0.018 (0.012_{statonly}) 

ψ(2S)K_{S}  0.089 ± 0.076 ± 0.020  −0.104 ± 0.055 ^{+0.027}_{−0.047}  −0.05 ± 0.10 ± 0.01  −0.019 ± 0.048 (0.041_{statonly}) 

ψ(nS)K^{0}      −0.017 ± 0.029  −0.008 ± 0.017 (0.012_{statonly}) 

χ_{c1}K_{S}  0.129 ± 0.109 ± 0.025  0.017 ± 0.083 ^{+0.026}_{−0.046}    0.066 ± 0.074 (0.066_{statonly}) 

η_{c}K_{S}  0.080 ± 0.124 ± 0.029        BaBar (PRD 79 (2009) 072009) 
J/ψK*^{0} (K*^{0} → K_{S}π^{0})  0.025 ± 0.083 ± 0.054      
All charmonium  0.024 ± 0.020 ± 0.016  −0.006 ± 0.016 ± 0.012  −0.017 ± 0.029  −0.005 ± 0.015 (0.012_{statonly}) 
CL = 0.43 
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}) 
−0.006 ± 0.016 ± 0.012  −0.017 ± 0.029  −0.005 ± 0.015 (0.012_{statonly}) 
CL = 0.54 
^{(†)} The statistical correlation between the LHCb results for S and C from J/ψK_{S} with J/ψ decays to muons is 0.483. Similar correlations are found for the cases of J/ψK_{S} with J/ψ decays to electrons (0.46) and ψ(2S)K_{S} (0.48). These correlations are currently neglected in the averages.
^{(*)} The first LHCb result for J/ψK_{S} uses J/ψ decays to muons, while the second uses electrons. Their combination is given under J/ψK^{0}.
^{(**)} 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 
HFLAV
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  HFLAV 

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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}, based on flavourtagged analyses of B_{s} → J/ψ φ decays, have been performed by ATLAS, CMS, CDF, D0 and LHCb. LHCb have in addition performed measurements of φ_{s} from B_{s} → J/ψ π^{+}π^{−} and B_{s} → D_{s}^{+}D_{s}^{−}.
Averaging of the above results is being carried out by the HFLAV lifetimes and oscillation group. Measurements related to φ_{s} in B_{s} → K^{+}K^{−} and B_{s} → φφ decays are reported below.
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 (2003) 321. Effects of subleading amplitudes ∝ V_{ub} are expected to be negligible (~0.02) in the Standard Model, and to have opposite signs for the CPeven and CPodd final states.
A joint analysis of the BaBar and Belle data samples has been performed to obtain to obtain the best determination of sin(2β) ≡ sin(2φ_{1}) in these processes. All B_{d} → D^{(}*^{)}h^{0} decays are analysed together. The following CPeven final states are included: CPeven Dπ^{0} and Dη with D → K_{S}π^{0} and D → K_{S}ω, Dω with D → K_{S}π^{0}, D*π^{0} and D*η with D* → Dπ^{0} and D → K^{+}K^{−}. The following CPodd final states are included: Dπ^{0}, Dη and Dω with D → K^{+}K^{−}, D*π^{0} and D*η with D* → Dπ^{0} and D → K_{S}π^{0}.
Mode  Experiment  sin(2β) ≡ sin(2φ_{1})  C_{CP}  Correlation  Reference 

D^{(}*^{)} h^{0} 
BaBar+Belle
N(BB)=1243M 
0.66 ± 0.10 ± 0.06  −0.02 ± 0.07 ± 0.03  −0.05 (stat)  PRL 115 (2015) 121604 
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. Simlarly to the case of Dalitz plot analysis of B^{−} → DK^{−} with D → K_{S}π^{+}π^{−}, such an analysis can be carried out with either modeldependent or modelindependent methods.
A combined analysis of BaBar and Belle data has been done with 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}, D → K_{S}π^{+}π^{−}, π^{0} → γγ, η → γγ and π^{+}π^{−}π^{0}, and ω → π^{+}π^{−}π^{0}. The D → K_{S}π^{+}π^{−} decay model is obtained from a fit to D*^{+} → D^{0}π^{+} decays. The citation given in the table below corresponds to a Letter available as arXiv:1804.06152. A more detailed article on the same analysis is also available as arXiv:1804.06153. The results of this analysis supersede those from previous measurements by Belle and BaBar.
At present we do not apply a rescaling of the results to a common, updated set of input parameters. The third quoted uncertainty is due to Dalitz plot model dependence.
Experiment  sin(2β) ≡ sin(2φ_{1})  cos(2β) ≡ cos(2φ_{1})  Correlations  Reference 

BaBar+Belle
N(BB)=1240M 
0.80 ± 0.14 ± 0.06 ± 0.03  0.91 ± 0.22 ± 0.09 ± 0.07  0.05  arXiv:1804.06152 
Interpretations:
Treating β ≡ φ_{1} as a free parameter in the fit, the result β ≡ φ_{1} = (22.5 ± 4.4 ± 1.2 ± 0.6)° is obtained.
This corresponds to an observation of CP violation (β ≡ φ_{1} ≠ 0) at 5.1σ significance,
and evidence for cos 2β ≡ cos 2φ_{1} > 0 at 3.7σ.
The ambiguous solution with the value of sin 2β ≡ sin 2φ_{1} from b→ccbar s transitions, but cos 2β ≡ cos 2φ_{1} < 0, is ruled out at 7.3σ
A modelindependent analysis has been performed by Belle. The decays B_{d} → Dπ^{0}, B_{d} → Dη, B_{d} → Dη′, B_{d} → Dω, B_{d} → D*π^{0} and B_{d} → D*η are used. Note that due to the strong statistical and systematic correlations, modeldependent results and modelindependent results from the same experiment cannot be combined.
Experiment  sin 2β  cos 2β  Correlation  Reference 

Belle
N(BB)=772M 
0.43 ± 0.27 ± 0.08  1.06 ± 0.33 ^{+0.21} _{−0.15}  −0.03 (stat)  PR D94 (2016) 052004 
Interpretations:
Belle
disfavour the solution with the charmoniumkaon value of sin(2φ)_{1} and a negative value for cos(2φ)_{1}, at 5.1 σ significance.
The solution with the charmoniumkaon value of sin(2φ)_{1} and positive cos(2φ)_{1} is consistent with the data at the level of 1.3σ.
Compilation of results for
(left) sin(2β^{eff}) ≡ sin(2φ_{1}^{eff}) = −η_{CP}S
from timedependent b → cubar d analyses.
The modeldependent results for B_{d} → D^{(}*^{)}h^{0}, D → K_{S}π^{+}π^{−} are used.
The results are compared
to the values from the corresponding charmoniumkaon averages.
The average value of sin(2β) ≡ sin(2φ_{1}) in b → c ubar d transitions is

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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^{−}).
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 
HFLAV correlated average
χ^{2} = 3.8/2 dof (CL=0.15 ⇒ 1.4σ) 

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(*) Note that the BaBar 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.
Experiment  S_{CP} (D^{+}D^{−})  C_{CP} (D^{+}D^{−})  Correlation  Reference 

BaBar
N(BB)=467M 
−0.63 ± 0.36 ± 0.05  −0.07 ± 0.23 ± 0.03  −0.01 (stat)  PRD 79, 032002 (2009) 
Belle
N(BB)=772M 
−1.06 ^{+0.21} _{−0.14} ± 0.08  −0.43 ± 0.16 ± 0.05  −0.12 (stat)  PRD 85 (2012) 091106 
LHCb
∫Ldt=3 fb^{−1} 
−0.54 ^{+0.17} _{−0.16} ± 0.05  0.26 ^{+0.18} _{−0.17} ± 0.02  0.48 (stat)  PRL 117 (2016) 261801 
Average  −0.84 ± 0.12  −0.13 ± 0.10  0.18 
HFLAV correlated average
χ^{2} = 11/4 dof (CL=0.027 ⇒ 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 vectorvector final states J/ψρ^{0} and D*^{+}D*^{−} are mixtures 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^{+}).
Different techniques have been used to study these two VV modes. LHCb has performed a full timedependent amplitude analysis of B^{0} → J/ψπ^{+}π^{} decays, find a J/ψρ^{0} fit fraction of 65% and a longitudinal polarisation fraction of 57% (both consistent with results from a timeintegrated amplitude analysis). Fits are performed to obtain 2β^{eff} in the cases that all transversity amplitudes are assumed to have the same CP violation parameter, and allowing different parameters. The results in the former case are presented in terms of S_{CP} and C_{CP} below.
Experiment  S_{CP} (J/ψ ρ^{0})  C_{CP} (J/ψ ρ^{0})  Correlation  Reference 

LHCb
∫Ldt=3 fb^{−1} 
−0.66 ^{+0.13} _{−0.12} ^{+0.09} _{−0.03}  −0.06 ± 0.06 ^{+0.02} _{−0.01}  −0.01 (stat)  PLB 742 (2015) 38 
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. (In the most recent update Belle include R_{⊥} and also R_{0} as free parameters in the fit. We do not include information on R_{0} in the average for now.) BaBar treat R_{⊥} 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_{⊥}.
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. We also include the results of a separate analysis from BaBar based on partially reconstructed D*D* decays; in this analysis the BaBar measurement of R_{⊥} is used to correct the value of S fitted without separating CPeven and odd components for the CPodd dilution.
Experiment  S_{CP} (D*^{+} D*^{−})  C_{CP} (D*^{+} D*^{−})  R_{⊥} (D*^{+} D*^{−})  Correlation  Reference 

BaBar
N(BB)=467M 
−0.70 ± 0.16 ± 0.03  0.05 ± 0.09 ± 0.02  0.17 ± 0.03  (stat)  PRD 79, 032002 (2009) 
BaBar part. rec.
N(BB)=471M 
−0.49 ± 0.18 ± 0.07 ± 0.04  0.15 ± 0.09 ± 0.04    (stat)  PRD 86 (2012) 112006 
Belle
N(BB)=772M 
−0.79 ± 0.13 ± 0.03  −0.15 ± 0.08 ± 0.02  0.14 ± 0.02 ± 0.01  (stat)  PRD 86 (2012) 071103(R) 
Average  −0.71 ± 0.09  −0.01 ± 0.05  0.15 ± 0.02  (stat) 
HFLAV correlated average
χ^{2} = 3.7/6 dof (CL=0.72 ⇒ 0.4σ) 

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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.
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). An alternative notation, S = (S_{+} + S_{−})/2, Δ S = (S_{+} −S_{−})/2, C = (C_{+} + C_{−})/2, Δ C = (C_{+} −C_{−})/2, has been used in recent publications.
Experiment  S(D*^{+}D^{−})  C(D*^{+}D^{−})  ΔS(D*^{−}D^{+})  ΔC(D*^{−}D^{+})  A(D*^{+−}D^{−+})  Reference  

BaBar
N(BB)=467M 
−0.68 ± 0.15 ± 0.04  0.04 ± 0.12 ± 0.03  0.05 ± 0.15 ± 0.02  0.04 ± 0.12 ± 0.03  0.01 ± 0.05 ± 0.01  PRD 79, 032002 (2009)  
Belle
N(BB)=772M 
−0.78 ± 0.15 ± 0.05  −0.01 ± 0.11 ± 0.04  −0.13 ± 0.15 ± 0.04  0.12 ± 0.11 ± 0.03  0.06 ± 0.05 ± 0.02  PRD 85 (2012) 091106  
Average 
−0.73 ± 0.11
χ^{2} = 0.20 (CL=0.65 ⇒ 0.5σ) 
0.01 ± 0.09
χ^{2} = 0.1 (CL=0.77 ⇒ 0.3σ) 
−0.04 ± 0.11
χ^{2} = 0.7 (CL=0.41 ⇒ 0.8σ) 
0.08 ± 0.08
χ^{2} = 0.2 (CL=0.63 ⇒ 0.5σ) 
0.03 ± 0.04
χ^{2} = 0.5 (CL=0.48 ⇒ 0.7σ) 
HFLAV uncorrelated averages 


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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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Timedependent CP asymmetries in B_{s} decays mediated by b → ccbar d transitions provide a determination of φ_{s}^{eff} where possible effects from penguin amplitudes may cause a shift from the value of φ_{s} seen in b → ccbar s transitions. Results in the b → ccbar d case, with larger penguin effects, can be used together with flavour symmetries to derive limits on the possible size of penguin effects in the b → ccbar s transitions, as described by Fleischer et al. (hepph/9903455, arXiv:1010.0089). If the penguin effect is large, it may also be possible to determine γ ≡ φ_{3}.
The parameters have been measured in B_{s} → J/ψ K_{S}^{0} decays by LHCb.
Experiment  S_{CP}  C_{CP}  A_{ΔΓ}  Correlation  Reference 

LHCb
∫Ldt=3 fb^{−1} 
0.49 ^{+0.77} _{−0.65} ± 0.06  −0.28 ± 0.41 ± 0.08  −0.08 ± 0.40 ± 0.08  (stat)  JHEP 06 (2015) 131 
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)=470M 
0.66 ± 0.17 ± 0.07  0.05 ± 0.18 ± 0.05    PRD 85 (2012) 112010 
Belle
^{(*)}
N(BB)=657M 
0.90 ^{+0.09} _{−0.19}  −0.04 ± 0.20 ± 0.10 ± 0.02    PRD 82 (2010) 073011  
Average ^{(*)}  0.74 ^{+0.11} _{−0.13}  0.01 ± 0.14    HFLAV  

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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)=772M 
0.68 ± 0.07 ± 0.03  −0.03 ± 0.05 ± 0.03  0.03 (stat)  JHEP 1410 (2014) 165  
Average  0.63 ± 0.06  −0.05 ± 0.04  0.02 
HFLAV correlated average
χ^{2} = 1.3/2 dof (CL=0.53 ⇒ 0.6σ) 


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K_{S}K_{S}K_{S} 
BaBar
N(BB)=468M 
0.94 ^{+0.21} _{−0.24} ± 0.06  −0.17 ± 0.18 ± 0.04  0.16 (stat)  PRD 85 (2012) 054023 
Belle
N(BB)=535M 
0.30 ± 0.32 ± 0.08  −0.31 ± 0.20 ± 0.07    PRL 98 (2007) 031802  
Average  0.72 ± 0.19  −0.24 ± 0.14  0.09 
HFLAV 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)  PRD 81 (2010) 011101  
Average  0.57 ± 0.17  0.01 ± 0.10  0.02 
HFLAV 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    HFLAV  

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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)=772M 
0.91 ± 0.32 ± 0.05  0.36 ± 0.19 ± 0.05  −0.00 (stat)  PRD 90 (2014) 012002  
Average  0.71 ± 0.21  −0.04 ± 0.14  0.01 
HFLAV correlated average
χ^{2} = 9.9/2 dof (CL=0.007 ⇒ 2.7σ) 


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f_{0}K^{0}  BaBar ^{(**)}  0.74 ^{+0.12} _{−0.15}  0.15 ± 0.16    HFLAV ^{(**)} 
Belle ^{(**)}  0.63 ^{+0.16} _{−0.19}  0.13 ± 0.17    HFLAV ^{(**)}  
Average  0.69 ^{+0.10} _{−0.12}  0.14 ± 0.12    HFLAV  

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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} and f_{0}K^{0}) 
BaBar
^{(*)}
N(BB)=470M 
0.65 ± 0.12 ± 0.03  0.02 ± 0.09 ± 0.03    PRD 85 (2012) 112010 
Belle
^{(*)}
N(BB)=657M 
0.76 ^{+0.14} _{−0.18}  0.14 ± 0.11 ± 0.08 ± 0.03    PRD 82 (2010) 073011  
Average  0.68 ^{+0.09} _{−0.10} 
0.06 ± 0.08

  HFLAV  

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Naïve b→s penguin average 
0.655 ± 0.032
χ^{2} = 19/24 dof (CL=0.77 ⇒ 0.3σ) 
−0.006 ± 0.026
χ^{2} = 23/24 dof (CL=0.53 ⇒ 0.6σ) 
uncorrelated averages  HFLAV  
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Direct comparison of charmonium and spenguin averages (see comments below): χ^{2} = 0.5 (CL=0.47 ⇒ 0.7σ) 
^{(*)}
BaBar and Belle results for φK^{0},
ρ^{0}K_{S} and
K^{+}K^{−}K^{0} (excluding φK^{0} and f_{0}K^{0})
are determined from their
timedependent Dalitz plot analyses of
B^{0} → K^{+}K^{−}K^{0} and
B^{0} → π^{+}π^{−}K_{S}.
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}.
^{(**)} 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. 
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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. 


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. 
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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. 

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.
Timedependent Dalitz plot analyses of B^{0} → K^{+}K^{−}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: φK^{0} and f_{0}K^{0} and for the rest of the charmless contributions to the Dalitz plot combined, with the CP properties of the individual components taken into account.
Experiment  φK_{S}  f_{0}K_{S}  other K^{+}K^{−}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}  
BaBar
^{(*)}
N(BB)=470M 
(21 ± 6 ± 2)°  −0.05 ± 0.18 ± 0.05  (18 ± 6 ± 4)°  −0.28 ± 0.24 ± 0.09  (20.3 ± 4.3 ± 1.2)°  −0.02 ± 0.09 ± 0.03  (stat)  PRD 85 (2012) 112010 
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  (24.9 ± 6.4 ± 2.1 ± 2.5)°  −0.14 ± 0.11 ± 0.08 ± 0.03  (stat)  PRD 82 (2010) 073011 
Average  (24 ± 5)°  −0.01 ± 0.14  (22 ± 6)°  −0.29 ± 0.20  (21.6 ± 3.7)°  −0.06 ± 0.08  (stat) 
HFLAV correlated average
χ^{2} = 1.8/6 dof (CL=0.93 ⇒ 0.1σ) 

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. 
^{(*)} The BaBar fit to K^{+}K^{−}K_{S} finds a global minimum and four other local minima with 2 ln L values within 9 units of the global minimum (the closest is separated by 3.9 units). Values of the CP violation parameters are only quoted for the global minimum.
^{(**)} The Belle fit to K^{+}K^{−}K_{S} results in a fourfold ambiguity in the solution, all with consistent values of the CP parameters. The results quoted here correspond to solution 1 presented in the paper (which is preferred based on comparisons of the relative branching fractions of the scalar resonances to π^{+}π^{−} and K^{+}K^{−} to external measurements). 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  β^{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) 
HFLAV 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 
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. 
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 HFLAV  Rare Decays.
The decay B_{s} → K^{+}K^{−} involves a b → uubar s transition, and hence has both penguin and tree contributions. Both mixinginduced and direct CP violation effects may arise, and additional input is needed to disentangle the contributions and determine γ and β_{s}^{eff}. For example, the observables in B_{d} → π^{+}π^{−} can be related using Uspin, as proposed by Fleischer.
The observables are S_{CP}, C_{CP}, and A_{ΔΓ}. The alternative notations A_{mix} = S_{CP} and A_{dir} = −C_{CP} are sometimes found in the literature. All three observables can be treated as free parameters, but they are physically constrained to satisfy S_{CP}^{2} + C_{CP}^{2} + A_{ΔΓ}^{2} = 1. Note that the untagged decay distribution, from which an "effective lifetime" can be measured, retains sensitivity to A_{ΔΓ}. Averages of effective lifetimes are performed by the HFLAV lifetimes and oscillation group.
The observables have been measured by LHCb, who treat S_{CP}, C_{CP}, and A_{ΔΓ} as independent parameters.
Experiment  S_{CP}  C_{CP}  A^{ΔΓ}_{CP}  Correlation  Reference 

LHCb
∫Ldt=3.0 fb^{−1} 
0.18 ± 0.06 ± 0.02  0.20 ± 0.06 ± 0.02  −0.79 ± 0.07 ± 0.10  (stat)  LHCbPAPER2018006 
The decay B_{s} → φφ involves a b → ssbar s transition, and hence is a "pure penguin" mode. Since the mixing phase and the decay phase are expected to cancel in the Standard Model, the prediction for the phase from the interference of mixing and decay is predicted to be φ_{s}(φφ) = 0 with low uncertainty. Due to the vectorvector nature of the final state, angular analysis is needed to separate the CPeven and CPodd contributions. Such an analysis also makes it possible to fit directly for φ_{s}(φφ).
A constraint on φ_{s}(φφ) has been obtained by LHCb using ∫Ldt=5.0 fb^{−1} (LHCbCONF2018001), who measure φ_{s}(φφ) = −0.06 ± 0.13 (stat) ± 0.03 (syst) rad.
The decay B_{s} → K*^{0}K*^{0} involves a b → ddbar s transition, and similarly is a "pure penguin" mode. In this case the possibility of SM contributions with a different weak phase to the penguin amplitude makes the theoretical prediction less straightforward. A Uspin analysis with the B^{0} decay mode to the same final state can, however, be used to make clean SM tests (see hepph/0703137).
A significant complication arises due to the width of the K*(892)^{0} meson. This has been addressed by LHCb, through a full analysis of the B_{s} → (K^{+}π^{−})(K^{−}π^{+}) decay, using a Kπ mass window from 750 to 1600 MeV/c^{2}. In addition to the vector K*(892) resonance, contributions from Kπ Swave and from the tensor K_{2}*(1430) state are included. Assuming all amplitudes have contibutions only from the same weak phase, a value of φ_{s}((K^{+}π^{−})(K^{−}π^{+})) = −0.10 ± 0.13 (stat) ± 0.14 (syst) rad is measured, from a data sample of ∫Ldt=3.0 fb^{−1} (JHEP 1803 (2018) 140).
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 
HFLAV 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, both BaBar and Belle have results using K_{S}ηγ and K_{S}ρ^{0}γ (see footnotes below table), while Belle also has results using K_{S}φγ.
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 
HFLAV 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 
HFLAV 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  
Belle
N(BB)=772M 
−1.32 ± 0.77 ± 0.36  0.48 ± 0.41 ± 0.07  −0.15 (stat)  arXiv:1803.07774  
Average  −0.49 ± 0.42  0.06 ± 0.29  −0.15 
HFLAV correlated average
χ^{2} = 2.9/2 dof (CL=0.24 ⇒ 1.2σ) 

K_{S} ρ^{0} γ (*) 
BaBar
N(BB)=471M 
−0.18 ± 0.32 ^{+0.06} _{−0.05}  −0.39 ± 0.20 ^{+0.03} _{−0.02}  −0.09 (stat)  PRD 93 (2016) 052013  
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  
Average(*)  −0.06 ± 0.23  −0.22 ± 0.14  −0.02 
HFLAV correlated average
χ^{2} = 1.9/2 dof (CL=0.38 ⇒ 0.9σ) 

K_{S} φ γ 
Belle
N(BB)=772M 
0.74 ^{+0.72} _{−1.05} ^{+0.10} _{−0.24}  −0.35 ± 0.58 ^{+0.10} _{−0.23}    PRD 84 (2011) 071101  

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(*) Due to the nonnegligible width of the ρ^{0} meson, decays selected as B^{0} → K_{S}ρ^{0}γ can include a significant contribution from K*^{+−}π^{−+}γ decays, which are flavourspecific and do not have the same oscillation phenomenology. Both BaBar and Belle measure S_{eff} for all B decay candidates with the ρ^{0} selection being 0.6 < m(π^{+}π^{−}) < 0.9 GeV/c^{2}, obtaining 0.14 ± 0.25 ±0.03 (BaBar) and 0.09 ± 0.27 ^{+0.04}_{−0.07} (Belle). These values are then corrected for a "dilution factor", that is evaluated with different methods in the two experiments: BaBar obtain −0.78 ^{+0.19}_{−0.17} while Belle obtain 0.83 ^{+0.19}_{−0.03}. Until the discrepancy between these values is understood, the average of the results should be treated with extreme caution.
A similar analysis can be performed for radiative B_{s} decays to, for example, the φγ final state. As for other observables determined with selfconjugate final states produced in B_{s} decays, the effective lifetime also provides sensitivity, and can be determined without tagging the initial flavour of the decaying meson. The LHCb collaboration have determined the associated parameter A_{ΔΓ}(φγ) = −0.98 ^{+0.46}_{−0.52}^{+0.23}_{−0.20}.
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 
The observables have been measured by BaBar, Belle & LHCb. Note that at the B factories the observables are in principle uncorrelated (due to the fact that the time variable, Δt, has the range −∞ < Δt < +∞ – small correlations can be induced e.g.by backgrounds), at hadron colliders a nonzero correlation is expected (the time variable t takes the range 0 < t < +∞). 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.
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)  PRD 87 (2013) 052009 
Belle
N(BB)=772M 
−0.64 ± 0.08 ± 0.03  −0.33 ± 0.06 ± 0.03  −0.10 (stat)  PRD 88 (2013) 092003 
LHCb
∫Ldt=3.0 fb^{−1} 
−0.63 ± 0.05 ± 0.01  −0.34 ± 0.06 ± 0.01  0.45 (stat)  LHCbPAPER2018006 
Average  −0.63 ± 0.04  −0.32 ± 0.04  0.21 
HFLAV correlated average
χ^{2} = 1.1/4 dof (CL=0.90 ⇒ 0.13σ) 

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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 isospin analysis uses as input the branching fractions and
CPviolating charge asymmetries of all three ππ decay modes
(π^{+}π^{−},
π^{−}π^{0},
π^{0}π^{0}).
(Constraints on α ≡ φ_{2} can be obtained without information on S(π^{0}π^{0}), which has not yet been measured.)
Both Belle and
BaBar
give confidence level interpretations for α ≡ φ_{2}.
Belle
exclude the range 23.8° < φ_{2} < 66.8° at 1σ level.
BaBar
state that the true value lies in the range 71° < α < 109° at the 68% confidence level (considering the solution consistent with the Standard Model).
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.
The data in these plots is taken from (BaBar) PRD 88 (2013) 012003 and (Belle) PRL 98 (2007) 221602. 
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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.
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 PRD 77 (2008) 072001.
Experiment  A_{CP} (ρ^{+−}π^{−+})  C (ρ^{+−}π^{−+})  S (ρ^{+−}π^{−+})  ΔC (ρ^{+−}π^{−+})  ΔS (ρ^{+−}π^{−+})  Correlations  Reference 

BaBar
N(BB)=471M 
−0.10 ± 0.03 ± 0.02  0.02 ± 0.06 ± 0.04  0.05 ± 0.08 ± 0.03  0.23 ± 0.06 ± 0.05  0.05 ± 0.08 ± 0.04  (stat)  PRD 88 (2013) 012003 
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  −0.11 ± 0.03  −0.03 ± 0.06  0.06 ± 0.07  0.27 ± 0.06  0.01 ± 0.08  (stat) 
HFLAV correlated average
χ^{2} = 3.5/5 dof (CL=0.63 ⇒ 0.5σ) 

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

BaBar
N(BB)=471M 
−0.12 ± 0.08 ^{+0.04} _{−0.05}  0.09 ^{+0.05} _{−0.06} ± 0.04  0.55  PRD 88 (2013) 012003 
Belle
N(BB)=449M 
0.08 ± 0.16 ± 0.11  0.21 ± 0.08 ± 0.04  0.47  PRL 98 (2007) 221602 
Average  −0.08 ± 0.08  0.13 ± 0.05  0.37 
HFLAV correlated average
χ^{2} = 1.5/2 dof (CL=0.47 ⇒ 0.7σ) 

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

BaBar
N(BB)=471M 
0.19 ± 0.23 ± 0.15  −0.37 ± 0.34 ± 0.20  0.00  PRD 88 (2013) 012003 
Belle
N(BB)=449M 
0.49 ± 0.36 ± 0.28  0.17 ± 0.57 ± 0.35  0.08  PRL 98 (2007) 221602 
Average  0.27 ± 0.24  −0.23 ± 0.34  0.02 
HFLAV correlated average
χ^{2} = 0.8/2 dof (CL=0.68 ⇒ 0.4σ) 

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Interpretations:
The information given above can be used to extract α ≡ φ_{2}.
A confidence level interpretation for α can be obtained by scanning over the measured form factor bilinears.
In addition, information from the B → ρπ SU(2) partners can be included 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^{+}).
With all information in the ρπ channels put together,
Belle
obtain the constraint 68° < φ_{2} < 95° at 68% confidence level,
for the solution consistent with the Standard Model.
BaBar
present a scan, but not an interval, for α, since their studies indicate that the scan is not statistically robust and cannot be interpreted as 1CL.
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)=772M 
−0.13 ± 0.15 ± 0.05  0.00 ± 0.10 ± 0.06  −0.02 (stat)  PRD 93 (2016) 032010 
Average  −0.14 ± 0.13  0.00 ± 0.09  −0.02 
HFLAV correlated average
χ^{2} = 0.03/2 dof (CL=0.99 ⇒ 0.02σ) 

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Since the decay B^{0} → 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 measure the longitudinally polarized component to be
It should be noted that the Belle results for the ρ^{0}ρ^{0} polarisation are in some tension with those from BaBar. Belle determine
There is also a measurement from LHCb of this quantity
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 isospin analysis uses as input the branching fractions and
CPviolating charge asymmetries of the longitudinal components of all three ρρ decay modes
(ρ^{+}ρ^{−},
ρ^{−}ρ^{0},
ρ^{0}ρ^{0}).
(A similar analysis could be done for each polarisation amplitude, but the others are found to not be statisticall significant.)
Constraints on α ≡ φ_{2} have been set by both BaBar and Belle. The most recent values are
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 
Belle
N(BB)=772M 
−0.06 ± 0.05 ± 0.07  −0.01 ± 0.11 ± 0.09  −0.51 ± 0.14 ± 0.08  0.54 ± 0.11 ± 0.07  −0.09 ± 0.14 ± 0.06  (stat)  PRD 86 (2012) 092012 
Average  −0.06 ± 0.06  −0.05 ± 0.11  −0.20 ± 0.13  0.43 ± 0.10  −0.10 ± 0.12  (stat) 
HFLAV correlated average
χ^{2} = 12/5 dof (CL=0.03 ⇒ 2.1σ) 

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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 (stat)  PRL 98 (2007) 181803 
Belle
N(BB)=772M 
−0.04 ± 0.26 ± 0.19  0.07 ± 0.08 ± 0.10  0.61 (stat)  PRD 86 (2012) 092012 
Average  0.02 ± 0.20  0.10 ± 0.10  0.38 
HFLAV correlated average
χ^{2} = 0.2/2 dof (CL=0.92 ⇒ 0.1σ) 

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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)°. Belle obtain φ_{2,eff} = (107.3 ± 6.6 (stat) ± 4.8 (syst))°.
NB. There is a fourfold ambiguity in the above results.
For more details on the world average for α ≡ φ_{2}, calculated with different statistical treatments, refer to the CKMfitter and UTfit pages.
World averages for α ≡ φ_{2}, combining all available measurements and calculated with different statistical treatments, have been calculated by the CKMfitter and UTfit groups. In addition, the BaBar and Belle collaborations have combined their results on B → ππ, πππ^{0} and ρρ to obtain
α ≡ φ_{2} = (88 ± 5)° 
The above solution is that consistent with the Standard Model (an ambiguous solution shifted by 180° exists). The strongest constraint currently comes from the B → ρρ system. The inclusion of results from B_{d} → a_{1}^{+−}π^{−+} does not significantly affect the average.
Independently from the constraints on α ≡ φ_{2} obtained by the experiments or other groups, the results summarised above are statistically combined to produce world average constraints. The combination is performed with the GammaCombo framework and follows a frequentist procedure, similar to that described in detail in arXiv:1705.02981.
The input measurements used in the combination are those listed above for timedependent CP violation parameters in B_{d}→ π^{+}π^{−}, π^{+}π^{−}π^{0} and ρ^{+}ρ^{−} and ρ^{0}ρ^{0}. Additional inputs for the branching fractions and (for ρρ) polarisation fractions, for these modes and for their isospin partners are taken from the HFLAV  Rare Decays subgroup. The CP asymmetry parameters for the isospin partner B^{+} decay modes are also taken from the HFLAV  Rare Decays subgroup, as is the corresponding parameter for B_{d}→ π^{0}π^{0} mode, for which no timedependent measurement is possible. The ratio of B^{+} to B^{0} lifetimes is taken from the HFLAV  Oscillations/Lifetime subgroup. Individual measurements are used as inputs, rather than the HFLAV averages, in order to facilitate crosschecks and to ensure the most appropriate treatment of correlations. A combination based on the HFLAV averages for each of the quantities measured by experiments gives consistent results.
The obtained world average for the Unitarity Triangle angle α ≡ φ_{2} is
Parameter: α ≡ φ_{2} from all b → uubar d modes  

α ≡ φ_{2}  (84.9 ^{+5.1}_{−4.5})° 
An ambiguous solution also exists at α ≡ φ_{2} ⇔ α + π ≡ φ_{2} + π. The secondary minimum close to zero is disfavoured by physical arguments not included in the isospin analysis, as discussed in arXiv:1705.02981.
Figures:
World average of α ≡ φ_{2}, in terms of 1−CL, split by decay mode. 
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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  −0.046 ± 0.013 ± 0.015  −0.039 ± 0.020 ± 0.013  −0.039 ± 0.010
CL=0.97 (0.03σ) 
BaBar: PRD 71 (2005) 112003 (partially reco.) BaBar: PRD 73 (2006) 111101 (fully reco.) Belle: PRD 84 (2011) 021101(R) (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) 
−0.015 ± 0.013 ± 0.015  −0.011 ± 0.020 ± 0.013  −0.010 ± 0.013 CL=0.59 (0.6σ) 

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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eps.gz png CL: eps.gz png 
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eps.gz png CL: eps.gz png 
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.
LHCb have measured the timedependent CP violation parameters in B^{0}_{s} → D^{−+}_{s}K^{+−} decays.
Digression:
Constraining γ

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, CDF and LHCb 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 often convenient to measure R_{CP+−} using double ratios, in which similar ratios for B^{−} → D^{(}*^{)} π^{(}*^{)}^{−} decays are used for normalization. These double ratios are equivalent to R_{CP+−} in the limit of vanishing b → u transition in the normalisation channel.
These observables have been measured so far for three D^{(}*^{)}K^{(}*^{)−} modes.
For the DK*^{−} mode some care is needed due to other possible contributions to the B^{−} → DK_{S}π^{−} final state. This can be handled in the relations between the observables and γ ≡φ_{3} through the introduction of a coherence factor, as discussed below. It is assumed that the selection of the K* region of the DK_{S}π^{−} Dalitz plot is the same for all experiments, so that the combination of results is valid.
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−}  Correlations  Reference 

D_{CP}K^{−} 
BaBar
N(BB)=467M 
0.25 ± 0.06 ± 0.02  −0.09 ± 0.07 ± 0.02  1.18 ± 0.09 ± 0.05  1.07 ± 0.08 ± 0.04    PRD 82 (2010) 072004 
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 (2006) 051106  
CDF
∫Ldt=1 fb^{−1} 
0.39 ± 0.17 ± 0.04    1.30 ± 0.24 ± 0.12      PRD 81 (2010) 031105(R)  
LHCb KK
∫Ldt=5 fb^{−1} 
0.126 ± 0.014 ± 0.002    0.988 ± 0.015 ± 0.011      PLB 777 (2018) 16  
LHCb pipi
∫Ldt=5 fb^{−1} 
0.115 ± 0.025 ± 0.007    0.992 ± 0.027 ± 0.015      PLB 777 (2018) 16  
Average 
0.129 ± 0.012
χ^{2} = 6.4/4 dof (CL=0.17 ⇒ 1.4σ) 
−0.10 ± 0.07
χ^{2} = 0.03 (CL=0.86 ⇒ 0.2σ) 
0.996 ± 0.016
! χ^{2} = 5.2/4 dof (CL=0.26 ⇒ 1.1σ) 
1.09 ± 0.08
χ^{2} = 0.2 (CL=0.65 ⇒ 0.5σ) 
HFLAV uncorrelated averages  

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Click here for averages excluding K_{S}φ from the CPodd channels.  
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 (2006) 051106  
LHCb
∫Ldt=5 fb^{−1} 
−0.151 ± 0.033 ± 0.011  0.276 ± 0.094 ± 0.047  1.138 ± 0.029 ± 0.016  0.902 ± 0.087 ± 0.112  (stat) (syst)  PLB 777 (2018) 16  
Average  −0.142 ± 0.032  0.15 ± 0.07  1.140 ± 0.031  1.03 ± 0.09  (stat) 
HFLAV correlated average
χ^{2} = 5.8/8 dof (CL=0.67 ⇒ 0.4σ) 


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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 (*)    
LHCb KK
∫Ldt=4.8 fb^{−1} 
0.06 ± 0.07 ± 0.01    1.22 ± 0.09 ± 0.01      JHEP 11 (2017) 156  
LHCb pipi
∫Ldt=4.8 fb^{−1} 
0.15 ± 0.13 ± 0.02    1.08 ± 0.14 ± 0.03      JHEP 11 (2017) 156  
Average 
0.08 ± 0.06
χ^{2} = 0.37 (CL=0.83 ⇒ 0.21σ) 
−0.23 ± 0.22 
1.22 ± 0.07
χ^{2} = 7.9 (CL=0.02 ⇒ 2.3σ) 
1.03 ± 0.30  HFLAV uncorrelated avearges  

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D_{CP}K^{−}π^{+}π^{−} 
LHCb KK
∫Ldt=3 fb^{−1} 
−0.045 ± 0.064 ± 0.011    1.043 ± 0.069 ± 0.034      PRD 92 (2015) 112005 
LHCb ππ
∫Ldt=3 fb^{−1} 
−0.054 ± 0.101 ± 0.011    1.035 ± 0.108 ± 0.038     
(*) We do not include preliminary results from Belle on D_{CP}K*^{−} (BELLECONF0316), on D_{CP}K^{−} (LP 2011 preliminary) or on D*_{CP}K^{−} (CKM 2012 preliminary) which are all unpublished after more than two years.
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

If a multibody D decay can be shown to be dominated by one CPeigenstate, it can be used in a "GLWlike" (sometimes called "quasiGLW") analysis. The same observables R_{CP}, A_{CP} as for the GLW case are measured, but an additional factor of (2F_{+}1), where F_{+} is the fractional CPeven content, enters the expressions relating these observables to γ ≡ φ_{3}. The F_{+} factors have been measured using CLEOc data (see also here) to be
The GLWlike observables for D→π^{+}π^{−}π^{0} and K^{+}K^{−}π^{0} have been measured by LHCb. The A_{GLW} observable for D→π^{+}π^{−}π^{0} was measured in an earlier analysis (discussed more below) by BaBar. The GLWlike observables for D→π^{+}π^{−}π^{+}π^{−} have been measured by LHCb.
Mode  Experiment  A_{GLW}  R_{GLW}  Reference 

D_{πππ0}K^{−} 
LHCb
∫Ldt=3 fb^{−1} 
0.05 ± 0.09 ± 0.01  0.98 ± 0.11 ± 0.05  PR D91 (2015) 112014 
BaBar
N(BB)=324M 
−0.02 ± 0.15 ± 0.03    PRL 99 (2007) 251801  
Average 
0.03 ± 0.08
χ^{2} = 0.2 (CL=0.68 ⇒ 0.4σ) 
0.98 ± 0.12  HFLAV  

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D_{KKπ0}K^{−} 
LHCb
∫Ldt=3 fb^{−1} 
0.30 ± 0.20 ± 0.02  0.95 ± 0.22 ± 0.04  PR D91 (2015) 112014 
D_{ππππ}K^{−} 
LHCb
∫Ldt=3 fb^{−1} 
0.10 ± 0.03 ± 0.02  0.97 ± 0.04 ± 0.02  PL B760 (2016) 117 
D_{ππππ}K*^{−} 
LHCb
∫Ldt=4.8 fb^{−1} 
0.02 ± 0.11 ± 0.01  1.08 ± 0.13 ± 0.03  JHEP 1711 (2017) 156 
Compilation of the above results together with those for CPeven modes. 
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In the limit that a multibody decay is a pure CPeigenstate, there is no benefit to analysing the distribution of the decays across the phase space (i.e. of performing analyses such as those discussed below), but otherwise additional sensitivity can be obtained. BaBar have performed an analysis of the D decay Dalitz plot in the decay chain B^{−} → D K^{−}, D → π^{+}π^{−}π^{0}. Since a significant contribution to the sensitivity to CP violation comes from the A_{GLW} observable, this information is included into the fit via a choice of fit parameters that avoids 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 (x_{0} ≡ 2F_{+}  1, where F_{+} is the fractional CPeven content defined above), 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. 
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. The experiments 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^{+})] . 
Digression:
It has been noted that the observables (R^{+}, R^{−}) may be more suitable for use than (R_{ADS}, A_{ADS}) since the former are better behaved (they are statistically independent observables, while the uncertainty on A_{ADS} depends on the central value of R_{ADS}). The definitions are 
R^{+} = Γ(B^{+} → [K^{−}π^{+}]_{D}K^{+}) / Γ(B^{+} → [K^{+}π^{−}]_{D}K^{+}) R^{−} = Γ(B^{−} → [K^{+}π^{−}]_{D}K^{−}) / Γ(B^{−} → [K^{−}π^{+}]_{D}K^{−}) 
They are related to (R_{ADS}, A_{ADS}) by 
R_{ADS} = (R^{+} + R^{−})/2 A_{ADS} = (R^{−} − R^{+}) / (R^{−} + R^{+}) 
We may switch to using this set of variables at a later time, but presently the majority of experimental results are presented in the (R_{ADS}, A_{ADS}) format. 
(Some of) these observables have been measured so far for the D^{(}*^{)}K^{(}*^{)−} modes. BaBar, Belle, CDF and LHCb have presented results for B^{−} → DK^{−} while BaBar and Belle have also presented results using B^{−} → D*K^{−}, with both D* → Dπ^{0} and D* → Dγ. BaBar have also presented results on B^{−} → DK*^{−}. For all the above the D → K^{+}π^{−} mode is used. In addition, BaBar and LHCb have presented results using B^{−} → DK^{−} with D → K^{+}π^{−}π^{0} and LHCb have presented results using B^{−} → DK^{−} with D → K^{+}π^{−}&pi^{+}π^{−}.
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)=467M 
−0.86 ± 0.47 ^{+0.12} _{−0.16}  0.011 ± 0.006 ± 0.002  PRD 82 (2010) 072006  
Belle
N(BB)=772M 
−0.39 ^{+0.26} _{−0.28} ^{+0.04} _{−0.03}  0.0163 ^{+0.0044} _{−0.0041} ^{+0.0007} _{−0.0013}  PRL 106 (2011) 231803  
CDF
∫Ldt=7 fb^{−1} 
−0.82 ± 0.44 ± 0.09  0.0220 ± 0.0086 ± 0.0026  PRD 84 (2011) 091504  
LHCb
∫Ldt=3 fb^{−1} 
−0.403 ± 0.056 ± 0.011  0.0188 ± 0.0011 ± 0.0010  PL B760 (2016) 117  
Average 
−0.41 ± 0.06
χ^{2} = 1.7/3 dof (CL=0.64 ⇒ 0.5σ) 
0.0183 ± 0.0014
χ^{2} = 1.8/3 dof (CL=0.61 ⇒ 0.5σ) 
HFLAV  

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DK^{−}
D→Kππ^{0} 
BaBar
N(BB)=474M 
  0.0091 ^{+0.0082} _{−0.0076} ^{+0.0014} _{−0.0037}  PRD 84 (2011) 012002  
Belle
N(BB)=772M 
0.41 ± 0.30 ± 0.05  0.0198 ± 0.0062 ± 0.0024  PRD 88 (2013) 091104(R)  
LHCb
∫Ldt=3 fb^{−1} 
−0.20 ± 0.27 ± 0.03  0.0140 ± 0.0047 ± 0.0019  PR D91 (2015) 112014  
Average 
0.07 ± 0.20
χ^{2} = 2.2 (CL=0.13 ⇒ 1.5σ) 
0.0148 ± 0.0036
χ^{2} = 1.1/2 dof (CL=0.59 ⇒ 0.5σ) 
HFLAV  

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DK^{−}
D→K3π 
LHCb
∫Ldt=3 fb^{−1} 
−0.313 ± 0.102 ± 0.038  0.0140 ± 0.0015 ± 0.0006  PL B760 (2016) 117  
D*K^{−}
D* → Dπ^{0} D→Kπ 
BaBar
N(BB)=467M 
0.77 ± 0.35 ± 0.12  0.018 ± 0.009 ± 0.004  PRD 82 (2010) 072006  
Belle  NO RESULTS AVAILABLE (*)    
D*K^{−}
D* → Dγ D→Kπ 
BaBar
N(BB)=467M 
0.36 ± 0.94 ^{+0.25} _{−0.41}  0.013 ± 0.014 ± 0.008  PRD 82 (2010) 072006  
Belle  NO RESULTS AVAILABLE (*)    
DK*^{−}
D→Kπ 
BaBar
N(BB)=379M 
−0.34 ± 0.43 ± 0.16  0.066 ± 0.031 ± 0.010  PRD 80 (2009) 092001  
LHCb
∫Ldt=4.8 fb^{−1} 
−0.81 ± 0.17 ± 0.04  0.011 ± 0.004 ± 0.001  JHEP 1711 (2017) 156  
Average 
−0.75 ± 0.16
χ^{2} = 0.92 (CL=0.34 ⇒ 1.0σ) 
0.012 ± 0.004
χ^{2} = 2.8 (CL=0.09 ⇒ 1.7σ) 
HFLAV  

eps.gz png  eps.gz png  .  
DK^{−}π^{+}π^{−}
D→Kπ 
LHCb
∫Ldt=3 fb^{−1} 
−0.32 ^{+0.27} _{−0.34}  0.0082 ^{+0.0038} _{−0.0030}  PRD 92 (2015) 112005 
(*) We do not include preliminary results from Belle on B^{−} → D*K^{−}, D* → Dπ^{0} or Dγ, D→Kπ (LP 2011 preliminary) which are unpublished after more than two years.
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, ensuring that mixing effects are corrected for. The value of δ_{D} can be determined directly using quantum correlated D mesons produced in ψ(3770) decay, as has been done by CLEO. The most precise values of both quantities come from the averages performed by HFLAV to obtain the mixing parameters in the charm system. 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. In the case of D decay to a multibody D final state, such K^{+}π^{−}π^{0} or K^{+}π^{−}&pi^{+}π^{−}, the relations between the observables and γ ≡ φ_{3} are modified by the introduction of a coherence factor. These parameters have been determined in arXiv:1602.07430, combining results from CLEOc and LHCb data. The modification to the formalism is similar to that discussed for "GLWlike" analyses above, but since the possibility to use multibody decays was considered in the original ADS papers, it is not usually referred to as "ADSlike" (or "quasiADS"). Eventually it may be possible to extract the maximum information from the decay by exploiting the Dalitz plot distribution in either a binned or unbinned approach. 
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}. Some of the observables have been measured by BaBar, Belle, CDF and LHCb in the various D^{(}*^{)}π^{−} modes.
Mode  Experiment  A_{ADS}  R_{ADS}  Reference  

Dπ^{−}
D→Kπ 
BaBar
N(BB)=467M 
0.03 ± 0.17 ± 0.04  0.0033 ± 0.0006 ± 0.0004  PRD 82 (2010) 072006  
Belle
N(BB)=772M 
−0.04 ± 0.11 ^{+0.02} _{−0.01}  0.00328 ^{+0.00038} _{−0.00036} ^{+0.00012} _{−0.00018}  PRL 106 (2011) 231803  
CDF
∫Ldt=7 fb^{−1} 
0.13 ± 0.25 ± 0.02  0.0028 ± 0.0007 ± 0.0004  PRD 84 (2011) 091504  
LHCb
∫Ldt=3 fb^{−1} 
0.100 ± 0.031 ± 0.009  0.00360 ± 0.00012 ± 0.00009  PL B760 (2016) 117  
Average 
0.088 ± 0.030
χ^{2} = 1.6/3 dof (CL=0.66 ⇒ 0.4σ) 
0.00353 ± 0.00014
χ^{2} = 1.5/3 dof (CL=0.68 ⇒ 0.4σ) 
HFLAV  

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Dπ^{−}
D → Kππ^{0} 
Belle
N(BB)=772M 
0.16 ± 0.27 ^{+0.03} _{−0.04}  0.00189 ± 0.00054 ^{+0.00022} _{−0.00025}  PRD 88 (2013) 091104(R)  
LHCb
∫Ldt=3 fb^{−1} 
0.44 ± 0.19 ± 0.01  0.00235 ± 0.00049 ± 0.00004  PR D91 (2015) 112014  
Average 
0.35 ± 0.16
χ^{2} = 0.71 (CL=0.40 ⇒ 0.8σ) 
0.00216 ± 0.00038
χ^{2} = 0.4 (CL=0.55 ⇒ 0.6σ) 
HFLAV  

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Dπ^{−}
D → K3π 
LHCb
∫Ldt=3 fb^{−1} 
0.023 ± 0.048 ± 0.005  0.00377 ± 0.00018 ± 0.00006  PL B760 (2016) 117  
D*π^{−}
D* → Dπ^{0} D→Kπ 
BaBar
N(BB)=467M 
−0.09 ± 0.27 ± 0.05  0.0032 ± 0.0009 ± 0.0008  PRD 82 (2010) 072006  
Belle  NO RESULTS AVAILABLE (*)    
D*π^{−}
D* → Dγ D→Kπ 
BaBar
N(BB)=467M 
−0.65 ± 0.55 ± 0.22  0.0027 ± 0.0014 ± 0.0022  PRD 82 (2010) 072006  
Belle  NO RESULTS AVAILABLE (*)    
D3π^{−}
D→Kπ 
LHCb
∫Ldt=3 fb^{−1} 
−0.003 ± 0.090  0.00427 ± 0.00043  PRD 92 (2015) 112005 
(*) We do not include preliminary results from Belle on B^{−} → D*π^{−}, D* → Dπ^{0} or Dγ, D→Kπ (LP 2011 preliminary) which are unpublished after more than two years.
Compilation of the above results. 
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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 (GGSZ) and the Belle Collaboration.
The analysis can be performed either with the assumption of a Dalitz plot model for the D meson decay (modeldependent) or without such an assumption (modelindependent). In the latter case, it is necessary to bin the Dalitz plot and use measurements of quantities related to the average strong phase difference between the amplitudes D and Dbar decays in each bin. Such measurements can be obtained using ψ(3770) → DDbar decays. Note that due to the strong statistical and systematic correlations, modeldependent results and modelindependent results from the same experiment cannot be combined.
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, experiments 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}
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*^{−}). Results are also available from LHCb using B^{−} → D K^{−} with D → K_{S}π^{+}π^{−}.
For the DK*^{−} mode, both BaBar and Belle 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. [BaBar do not obtain constraints on r_{B} and δ_{B} in this decay.]
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 the Appendix of their paper. The Belle results using B^{−} → DK*^{−} 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)  PRL 105 (2010) 121801 
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  
LHCb
∫Ldt=1 fb^{−1} 
−0.084 ± 0.045 ± 0.009 ± 0.005  −0.032 ± 0.048 ^{+0.010} _{−0.009} ± 0.008  0.027 ± 0.044 ^{+0.010} _{−0.008} ± 0.001  0.013 ± 0.048 ^{+0.009} _{−0.007} ± 0.003  (stat) (model)  NPB 888 (2014) 169  
Average  −0.098 ± 0.024  −0.036 ± 0.030  0.070 ± 0.025  0.075 ± 0.029  (stat) 
HFLAV correlated average
χ^{2} = 7.2/8 dof (CL=0.52 ⇒ 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)  PRL 105 (2010) 121801 
Belle
^{(*)}
N(BB)=657M 
0.100 ± 0.074 ± 0.081  0.155 ± 0.101 ± 0.063  −0.023 ± 0.112 ± 0.090  −0.252 ± 0.112 ± 0.049  (stat) (model)  PRD 81 (2010) 112002  
Average No model error 
0.132 ± 0.044  0.037 ± 0.061  −0.081 ± 0.049  −0.107 ± 0.055  (stat+syst) 
HFLAV correlated average
χ^{2} = 5.8/4 dof (CL=0.22 ⇒ 1.2σ) 

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)  PRL 105 (2010) 121801 
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) 
HFLAV 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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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}. All experiments have done so using frequentist techniques. Note that the uncertainty on γ ≡ φ_{3} is approximately inversely proportional to the central value of r_{B}. 

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^{−}) 
LHCb obtain
γ = (84 ^{+49}_{−42})° (from DK^{−} using ∫Ldt=1 fb^{−1}; a more precise result using ∫Ldt=3 fb^{−1} and the modelindependent method is reported below) 

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} (DK^{−}) = 0.06 ± 0.04  δ_{B} (DK^{−}) = (115 ^{+41}_{−51})° 
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 )° *)  .  . 
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. 
A modelindependent approach to the analysis of B^{−} → D^{(}*^{)} K^{−} with multibody D decays was proposed by Giri, Grossman, Soffer and Zupan, and further developed by Bondar and Poluektov (see also here). The method relies on information on the average strong phase difference between D^{0} and D^{0}bar decays in bins of Dalitz plot position that can be obtained from quantumcorrelated Ψ(3770) → D^{0}D^{0}bar events. This information is measured in the form of parameters c_{i} and s_{i} that are the amplitude weighted averages of the cosine and sine of the strong phase difference in a Dalitz plot bin labelled by i, respectively. These quantities have been obtained for D → K_{S}π^{+}π^{−} (and D → K_{S}K^{+}K^{−}) by CLEOc (see also here). [Preliminary results from BESIII are also available.]
Modelindependent determinations of γ ≡ φ_{3} has been performed by Belle and LHCb. Both Belle and LHCb have used B^{−} → D K^{−} with D → K_{S}π^{+}π^{−}. The LHCb results also include D → K_{S}K^{+}K^{−}  we do not attempt to separate the contribution from this mode in our combination (although LHCb results using D → K_{S}π^{+}π^{−} only are also available).
The variables (x_{±}, y_{±}), defined above are determined from the data.
The results below have three sets of errors, which are statistical, systematic, and uncertainty coming from the knowledge of c_{i} and s_{i} respectively. To perform the average, we remove the last uncertainty, which should be 100% correlated between the measurements. Since the size of the uncertainty from c_{i} and s_{i} is found to depend on the size of the B → DK data sample, we assign the LHCb uncertainties (which are mostly the smaller of the Belle and LHCb values) to the averaged result. This procedure should be conservative.
Mode  Experiment  x+  y+  x  y  Correlation  Reference  

DK^{−}
D→K_{S}π^{+}π^{−} 
Belle
N(BB)=772M 
−0.110 ± 0.043 ± 0.014 ± 0.007  −0.050 ^{+0.052} _{−0.055} ± 0.011 ± 0.007  0.095 ± 0.045 ± 0.014 ± 0.010  0.137 ^{+0.053} _{−0.057} ± 0.015 ± 0.023  (stat)  PRD 85 (2012) 112014  
LHCb
∫Ldt=3 fb^{−1} 
−0.077 ± 0.024 ± 0.010 ± 0.004  −0.022 ± 0.025 ± 0.004 ± 0.010  0.025 ± 0.025 ± 0.010 ± 0.005  0.075 ± 0.029 ± 0.005 ± 0.014  (stat)  JHEP 1410 (2014) 097  
Average (*)  −0.085 ± 0.023 ± 0.004  −0.027 ± 0.023 ± 0.010  0.044 ± 0.023 ± 0.005  0.090 ± 0.026 ± 0.014  (stat) 
HFLAV correlated average
χ^{2} = 4.1/4 dof (CL=0.39 ⇒ 0.9σ) 


eps.gz png 
eps.gz png 
eps.gz png 
. 
Digression:
Constraining γ ≡ φ_{3}:
As above, the measurements of x_{+,−} and y_{+,−} can be used to place bounds on γ ≡ φ_{3}. Belle and LHCb have done so using a frequentist technique. 

Belle obtain
φ_{3} = (77.3 ^{+15.1}_{−14.9} ± 4.1 ± 4.3)° 
LHCb obtain
γ = (62 ^{+15}_{−14})° 

r_{B} (DK^{−}) = 0.145 ± 0.030 ± 0.010 ± 0.011  δ_{B} (DK^{−}) = (129.9 ± +15.0 ± 3.8 ± 4.7)°  r_{B} (DK^{−}) = 0.080 ^{+0.019}_{−0.021}  δ_{B} (DK^{−}) = (134 ^{+14}_{−15} )° 
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. 
Decays of D mesons to K_{S}K^{+−}π^{−+} can be used in a similar approach to that discussed above to determine γ ≡ φ_{3}. Since these decays are less abundant, the event samples available to date have not been sufficient for a fine binning of the Dalitz plots, but the analysis can be performed using only an overall coherence factor and related strong phase difference for the decay. These quantities have been determined by CLEO both for the full Dalitz plots and in a restricted region ±100 MeV/c^{2} around the peak of the K*(892)^{±} resonance.
LHCb have reported results of an analysis of B^{−} → D K^{−} and B^{−} → D π^{−} decays with D → K_{S}K^{+−}π^{−+}. The decays with different final states of the D meson are distinguished by the charge of the kaon from the decay of the D meson relative to the charge of the B meson, and are labelled "same sign" (SS) and "opposite sign" (OS). Six observables potentially sensitive to γ ≡ φ_{3} are measured: two ratios of rates for DK and Dπ decays (one each for SS and OS) and four asymmetries (for DK & Dπ, SS & OS). This is done both for the full Dalitz plot and for the K*(892)^{±}dominated region (with the same boundaries as used by CLEO). Note that there is a significant overlap of events between the two samples. The results do not yet have sufficient precision to set significant constraints on γ ≡ φ_{3}.
Mode  Experiment  R_{SS}  R_{OS}  A_{SS,DK}  A_{OS,DK}  A_{SS,Dπ}  A_{OS,Dπ}  Correlation  Reference 

DK^{−}
D→K_{S}K^{+−}π^{−+} 
LHCb
∫Ldt=3 fb^{−1} 
0.092 ± 0.009 ± 0.004  0.066 ± 0.009 ± 0.002  0.040 ± 0.091 ± 0.018  0.233 ± 0.129 ± 0.024  −0.025 ± 0.024 ± 0.010  −0.052 ± 0.029 ± 0.017  (stat)  PLB 733 (2014) 36 
DK^{−}
D→K*(892)^{+−}K^{−+} 
LHCb
∫Ldt=3 fb^{−1} 
0.084 ± 0.011 ± 0.003  0.056 ± 0.013 ± 0.002  0.026 ± 0.109 ± 0.029  0.336 ± 0.208 ± 0.026  −0.012 ± 0.028 ± 0.010  −0.054 ± 0.043 ± 0.017  (stat)  PLB 733 (2014) 36 
LHCb have presented results on B^{0} → DK*^{0} with D → K^{+}K^{−} and D → π^{+}π^{−}. The sensitivity to γ ≡ φ_{3} arises in the same way as discussed above, except that the magnitude of the ratio of amplitudes r_{B} is expected to be larger, ≈ 0.3, because both amplitudes are coloursuppressed. Effects due to the natural width of the K*^{0} are handled in a quasitwobody approximation using the parametrization suggested by Gronau, whereby an additional parameter &kappa, referred to as a coherence factor, is introduced. In addition, the hadronic factors r_{B} and δ_{B} are replaced by effective parameters r_{B} and δ_{B}. These parameters depend on the K*^{0} selection window used; all analyses make very similar requirements (the emerging consensus is to use m(K^{+}π^{−}) within 50 MeV/c^{2} of the K*(892)^{0} pole mass and helicity angle satisfying cos(&theta_{K*0})>0.4).
Experiment  A_{CP+}  R_{CP+}  Reference 

LHCb KK
∫Ldt=3 fb^{−1} 
−0.20 ± 0.15 ± 0.02  1.05 ^{+0.17} _{−0.15} ± 0.04  PRD 90 (2014) 112002 
LHCb pipi
∫Ldt=3 fb^{−1} 
−0.09 ± 0.22 ± 0.02  1.21 ^{+0.28} _{−0.25} ± 0.05  PRD 90 (2014) 112002 
BaBar have presented results on B^{0} → DK*^{0} with D → K^{−}π^{+}, D → K^{−}π^{+} π^{0} and D → K^{−}π^{+} π^{+} π^{−}. Belle have presented results with the D → K^{−}π^{+} mode. 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 
Belle
N(BB)=772M 
R_{ADS}(Kπ) < 0.16      PRD 86 (2012) 011101 
Experiment  R_{+}  R_{−}  Reference 

LHCb
∫Ldt=3 fb^{−1} 
0.06 ± 0.03 ± 0.01  0.06 ± 0.03 ± 0.01  PRD 90 (2014) 112002 
(See above for a definition of the parameters).
Digression:
Combining the results and using additional input from CLEOc
(here and
here),
BaBar set a limit on the ratio between the b→u and b→c amplitudes of r_{B}(DK*^{0}) ∈ [0.07,0.41] at 95% CL.
Belle set at limit of r_{B}(DK*^{0}) < 0.4 at 95% CL. LHCb take input from HFLAV charm and obtain r_{B}(DK*^{0}) = 0.240 ^{+0.055}_{−0.048} (different from zero with 2.7σ significance). 
As pointed out by Gershon (see also Gershon and Williams), a Dalitz plot analysis of B^{0} → DK^{+}π^{−} decays provides more sensitivity to γ than the quasitwobody DK*^{0} approach. The analysis provides direct sensitivity to the hadronic parameters r_{B} and δ_{B} associated with the B^{0} → DK*^{0} decay amplitudes, rather than effective hadronic parameters averaged over the K*^{0} selection window as in the quasitwobody case. The results of the analysis can be presented in terms of Cartesian parameters, as done for the GGSZ analysis, or alternatively directly in terms of γ, r_{B} and δ_{B}.
Such an analysis has been performed by LHCb. A simultaneous fit is performed to the B^{0} → DK^{+}π^{−} Dalitz plots with the neutral D meson reconstructed in the K^{+}π^{−}, K^{+}K^{−} and π^{+}π^{−} final states. The reported results are for the parameters associated with the B^{0} → DK*(892)^{0} decay.
Experiment  x+  y+  x−  y−  Correlation  Reference 

LHCb
∫Ldt=3 fb^{−1} 
0.04 ± 0.16 ± 0.11  −0.47 ± 0.28 ± 0.22  −0.02 ± 0.13 ± 0.14  −0.35 ± 0.26 ± 0.41  (stat) (syst)  PR D93 (2016) 112018 
Note that, since the measurements use overlapping data samples, these results cannot be combined with the LHCb results for GLW observables in B^{0} → DK*^{0} decays reported above.
Digression:
LHCb use these results to obtain confidence levels for γ, r_{B}(DK*^{0}) and δ_{B}(DK*^{0}). In addition, results are reported for the hadronic parameters needed to relate these results to quasitwobody measurements of B^{0} → DK*^{0} decays, where a selection window of m(K^{+}π^{−}) within 50 MeV/c^{2} of the pole mass and helicity angle satisfying cos(&theta_{K*0})>0.4 is assumed. These parameters are the coherence factor κ, the ratio of quasitwobody and amplitude level r_{B} values, R_{B} = r_{B}/r_{B}, and the difference between quasitwobody and amplitude level δ_{B} values, Δδ_{B} = δ_{B}−δ_{B}.  
LHCb obtain κ = 0.958 ^{+0.005}_{−0.010} ^{+0.002}_{−0.045}, R_{B} = 1.02 ^{+0.03}_{−0.01} ±0.06, Δδ_{B} = 0.02 ^{+0.03}_{−0.02} ±0.11 
As discussed above for B^{−} → DK^{−}, Dalitz plot analysis of a threebody D decay allows simultaneous determination of the weak phase difference γ ≡ φ_{3}, the strong phase difference δ_{B} and the ratio of amplitudes r_{B}. 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 accommodated through the coherence factor κ, and the effective hadronic factors r_{B} and δ_{B}, as discussed above. Results obtained with both modeldependent and modelindependent approaches are available. Note that due to the strong statistical and systematic correlations, modeldependent results and modelindependent results from the same experiment cannot be combined.
BaBar have performed a similar modeldependent Dalitz plot analysis to that described above using neutral B decays, presenting results in terms of γ ≡ φ_{3} only. LHCb have done a similar analysis, using B^{0} → DK*^{0} with D → K_{S}π^{+}π^{−} decays, and presented results in terms of the Cartesian parameters. The third uncertainty in the results is due to model dependence.
Experiment  x−  y−  x+  y+  Correlation  Reference 

LHCb
∫Ldt=3 fb^{−1} 
−0.15 ± 0.14 ± 0.03 ± 0.01  0.25 ± 0.15 ± 0.06 ± 0.01  0.05 ± 0.24 ± 0.04 ± 0.01  −0.65 ^{+0.24} _{−0.23} ± 0.08 ± 0.01  (stat)  JHEP 1608 (2016) 137 
Digression:
Constraining γ ≡ φ_{3}:
BaBar extract the threedimensional likelihood for the parameters (γ, δ_{B}(DK*^{0}), r_{B}(DK*^{0})) and, combining with a separately measured PDF for r_{B}(DK*^{0}) (using a Bayesian technique), obtain bounds on each of the three parameters. LHCb convert their results on the Cartesian parameters into bounds on γ, δ_{B}(DK*^{0}) and r_{B}(DK*^{0}) using similar techniques to those discussed above. 
BaBar obtain γ = (162 ± 56)°, δ_{B}(DK*^{0}) = (62 ± 57)°, r_{B}(DK*^{0}) < 0.55 at 95% probability 
LHCb obtain γ = (80 ^{+21}_{−22})°, δ_{B}(DK*^{0}) = (197 ^{+24}_{−20})°, r_{B}(DK*^{0}) = 0.39 ± 0.13 
Note that there is an ambiguity in the solutions for γ and δ_{B} (γ, δ_{B} → γ+π, δ_{B}+π). 
Belle and LHCb have performed similar modelindependent Dalitz plot analyses to those described above, but using B^{0} → DK*^{0} decays. Both experiments used D → K_{S}π^{+}π^{−} decays, and LHCb also included their D → K_{S}K^{+}K^{−} sample. Results are presented in terms of the socalled Cartesian parameters. Belle give separately the uncertainties due to imperfect knowledge of the c_{i} and s_{i} parameters obtained by CLEOc while LHCb constrain these parameters within uncertainties in the fit to data, and hence the effect is absorbed in their statistical uncertainties.
Experiment  x+  y+  x−  y−  Correlation  Reference 

Belle
N(BB)=772M 
0.1 ^{+0.7} _{−0.4} ^{+0.0} _{−0.1} ± 0.1  0.3 ^{+0.5} _{−0.8} ^{+0.0} _{−0.1} ± 0.1  0.4 ^{+1.0} _{−0.6} ^{+0.0} _{−0.1} ± 0.0  −0.6 ^{+0.8} _{−1.0} ^{+0.1} _{−0.0} ± 0.1  (stat)  PTEP 2016 043C01 
LHCb
∫Ldt=3 fb^{−1} 
0.05 ± 0.35 ± 0.02  −0.81 ± 0.28 ± 0.06  −0.31 ± 0.20 ± 0.04  0.31 ± 0.21 ± 0.05  (stat)  JHEP 1606 (2016) 131 
Average  0.10 ± 0.30  −0.63 ± 0.26  −0.27 ± 0.20  0.27 ± 0.21  (stat) 
HFLAV correlated average
χ^{2} = 4.2/4 dof (CL=0.38 ⇒ 0.9σ) 
(*) Note that the Belle results have significantly nonGaussian and asymmetric uncertainties, and therefore the averages and plots should be interpreted carefully.
Digression:
Constraining γ ≡ φ_{3}:
From these results, Belle set the constraint r_{B}(DK*^{0}) < 0.87 at 68% confidence level. LHCb obtain constraints on γ = (71 ± 20)°, r_{B}(DK*^{0}) and δ_{B}(DK*^{0}) using a frequentist technique. 

γ = (71 ± 20)°, δ_{B}(DK*^{0}) = (204 ^{+21}_{−20})°, r_{B}(DK*^{0}) = 0.56 ± 0.17  
Note that there is an ambiguity in the solutions for γ and δ_{B} (γ, δ_{B} → γ+π, δ_{B}+π). 
BaBar and LHCb have presented constraints on γ ≡ φ_{3} from combinations of their results on B^{+} → DK^{+} and related processes.
(*) Belle presented in 2012 a preliminary combination of their results, but this remains unpublished after more than two years and therefore we do not list it.
BaBar obtain γ = (69 ^{+17}_{−16})° r_{B}(DK^{+}) = 0.092 ^{+0.013}_{−0.012} δ_{B}(DK^{+}) = (105 ^{+16}_{−17})° PRD 87 (2013) 052015 . . . . . . . . 
LHCb obtain γ = (76.8 ^{+5.1}_{−5.7})° r_{B}(DK^{+}) = 0.1043 ±0.0054 δ_{B}(DK^{+}) = (139.9 ^{+4.5}_{−5.2})° r_{B}(D*K^{+}) = 0.21 ^{+0.10}_{−0.09} δ_{B}(D*K^{+}) = (335 ^{+10}_{−19})° r_{B}(DK*^{+}) = 0.052 ^{+0.022}_{−0.021} δ_{B}(DK*^{+}) = (72 ^{+43}_{−32})° r_{B}(DK*^{0}) = 0.225 ^{+0.045}_{−0.048} δ_{B}(DK*^{0}) = (187 ^{+21}_{−20})° LHCbCONF2017004 
Independently from the constraints on γ ≡ φ_{3} obtained by the experiments, the results summarised above are statistically combined to produce world average constraints on γ ≡ φ_{3} and the hadronic parameters involved. The combination is performed with the GammaCombo framework and follows a frequentist procedure, similar to those used by the experiments.
The input measurements used in the combination are those listed above. Individual measurements are used as inputs, rather than the HFLAV averages, in order to facilitate crosschecks and to ensure the most appropriate treatment of correlations. A combination based on the HFLAV averages for each of the quantities measured by experiments gives consistent results.
In certain measurements, the experiments have explored different approaches to analysing the same data sample, and therefore choices of which results in include are needed. Concerning results of GGSZ analyses of B^{−} → D^{(}*^{)} K^{(}*^{)−} decays, the modeldependent results from the BaBar and Belle experiments are used, whilst the modelindependent results are used for LHCb. This choice is made in order to maintain consistency of the approach across experiments whilst maximising the size of the samples used to obtain inputs for the combination. For GGSZ analysis of B^{0} → DK*^{0} the modelindependent result from LHCb is used for consistency. The LHCb results from the GLW analysis of B^{0} → DK*^{0} are not used because of the overlap with the GLWDalitz analysis which is used instead. Results from timedependent analyses of B_{d} → D^{+−}π^{−+}, B_{d} → D*^{+−}π^{−+} and B_{d} → D^{+−}ρ^{−+} are not used as there are insufficient constraints on the associated hadronic parameters. Similarly, results from B_{d} → D^{+−}K_{S}π^{−+} are not used.
Auxiliary inputs are used in the combination in order to constrain the D system parameters and subsequently improve the determination of γ ≡ φ_{3}. These include the ratio of suppressed to favoured decay amplitudes and the strong phase difference for D → K^{+−}π^{−+} decays, taken from the HFLAV Charm Physics subgroup global fits. The amplitude ratios, strong phase differences and coherence factors of D→K^{+−}π^{−+}π^{0}, D→K^{+−}π^{−+}π^{+}π^{−} and D→K_{S}K^{+−}π^{+−} decays are taken from CLEOc and LHCb measurements. The fraction of CPeven content for quasiGLW D→π^{+}π^{−}π^{+}π^{−}, D→K^{+}K^{−}π^{0} and D→π^{+}π^{−}π^{0} decays are taken from CLEOc measurements. Constraints required to relate the hadronic parameters of the B^{0} → DK*^{0} GLWDalitz analysis to the effective hadronic parameters of the quasitwobody approaches are taken from LHCb measurements. Finally, the value of −2β_{s} is taken from the HFLAV Lifetimes and Oscillations subgroup; this is required to obtain sensitivity to γ ≡ φ_{3} from the timedependent analysis of B^{0}_{s} → D^{−+}_{s}K^{+−} decays decays.
The following reasonable, although imperfect, assumptions are made when performing the averages.
The obtained world averages for the Unitarity Triangle angle γ ≡ φ_{3} and related hadronic parameters are
Parameter: γ ≡ φ_{3} from all B → DK and similar b → cubar s & b → ucbar s modes  

γ ≡ φ_{3}  (73.5 ^{+4.2}_{−5.1})°  
r_{B}(DK^{+}) = 0.103 ± 0.005  δ_{B}(DK^{+}) = (136.9 ^{+4.6}_{−5.2})°  
r_{B}(D*K^{+}) = 0.142 ^{+0.019}_{−0.020}  δ_{B}(D*K^{+}) = (321 ^{+8}_{−9})°  
r_{B}(DK*^{+}) = 0.075 ^{+0.017}_{−0.018}  δ_{B}(DK*^{+}) = (106 ^{+18}_{−25})°  
r_{B}(DK*^{0}) = 0.223 ^{+0.042}_{−0.046}  δ_{B}(DK*^{0}) = (193 ^{+27}_{−21})° 
An ambiguous solution also exists at γ ≡ φ_{3} ⇔ γ + π ≡ φ_{3} + π (all strong phases δ are also simultaneously shifted by π).
Figures:
World average of γ ≡ φ_{3}, in terms of 1−CL, split by decay mode. 
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World average of γ ≡ φ_{3}, in terms of 1−CL, split by analysis method. 
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World averages for the hadronic parameters r_{B} in the different decay modes, in terms of 1−CL. 
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Contributions to the combination from different input measurements, shown in the plane of the relevant r_{B} parameter vs. γ ≡ φ_{3}. Contours show the twodimensional 68% and 95% CL regions. 
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The plots below show the constraints, in the (ρ, η) plane on
Whether or not the three allowed areas overlap at a common point in the plane is a consistency test of the Standard Model. Similar plots including additional constraints, for example on the lengths of the sides of the Unitarity Triangle, can be found at the CKMfitter and UTfit pages.
Figures:
World average constraints on the angles of the Unitarity Triangle, in the (ρ, η) plane 
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