Unitarity Triangle Angles

Time-dependent Transversity Analysis of \(B^0 \to J/\psi K^{*0}\)

Experiment	\(\boldsymbol{\sin(2\beta)\equiv\sin(2\phi_1)_{J/\psi K^*}}\)	\(\boldsymbol{\cos(2\beta)\equiv\cos(2\phi_1)_{J/\psi K^*}}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 88\,\mathrm{M}\)	\( -0.10 \pm 0.57 \pm 0.14 \)	\( 3.32 {}_{-0.96}^{+0.76} \pm 0.27 \)	\( -0.37 \) (stat)	PRD 71, 032005 (2005)
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 275\,\mathrm{M}\)	\( 0.24 \pm 0.31 \pm 0.05 \)	\( 0.56 \pm 0.79 \pm 0.11 \)	\( 0.22 \) (stat)	PRL 95 091601 (2005)
Average	\( 0.16 \pm 0.28 \) \( \chi^2 / \mathrm{dof} = 0.26 / 1 \) \( p = 0.61 \Rightarrow 0.5\sigma \)	\( 1.6 \pm 0.6 \) \( \chi^2 / \mathrm{dof} = 4.67 / 1 \) \( p = 0.03 \Rightarrow 2.2\sigma \)	uncorrelated averages	HFLAV
Figures	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output_0.json, output_1.json,

Time-dependent Analysis of \(B^0 \to D^{*} D^{*} K_S\)

Experiment	\(\boldsymbol{J_c / J_0}\)	\(\boldsymbol{(2 J_{s1}/ J_0)\sin(2\beta)\equiv(2 J_{s1}/ J_0)\sin(2\phi_1)}\)	\(\boldsymbol{(2 J_{s2}/ J_0)\cos(2\beta)\equiv(2 J_{s2}/ J_0)\cos(2\phi_1)}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 230\,\mathrm{M}\)	\( 0.76 \pm 0.18 \pm 0.07 \)	\( 0.10 \pm 0.24 \pm 0.06 \)	\( 0.38 \pm 0.24 \pm 0.05 \)	\( \) (stat)	PRD 74, 091101 (2006)
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 449\,\mathrm{M}\)	\( 0.60 {}_{-0.28}^{+0.25} \pm 0.08 \)	\( -0.17 \pm 0.42 \pm 0.09 \)	\( -0.23 {}_{-0.41}^{+0.43} \pm 0.13 \)	\( \) (stat)	PRD 76, 072004 (2007)
Average	\( 0.71 \pm 0.16 \) \( \chi^2 / \mathrm{dof} = 0.23 / 1 \) \( p = 0.63 \Rightarrow 0.5\sigma \)	\( 0.03 \pm 0.21 \) \( \chi^2 / \mathrm{dof} = 0.3 / 1 \) \( p = 0.58 \Rightarrow 0.5\sigma \)	\( 0.24 \pm 0.21 \) \( \chi^2 / \mathrm{dof} = 1.42 / 1 \) \( p = 0.23 \Rightarrow 1.2\sigma \)	uncorrelated averages	HFLAV
Figures	[pdf] [png]	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output_0.json, output_1.json, output_2.json,

Time-dependent CP Asymmetries in \(B^0 \to D_{CP} \pi^0\)

Experiment	\(\boldsymbol{\sin(2\beta) \equiv \sin(2\phi_1)}\)	\(\boldsymbol{C_{CP}}\)	Correlation	Reference
BaBar+Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 1243\,\mathrm{M}\)	\( 0.66 \pm 0.10 \pm 0.06 \)	\( -0.02 \pm 0.07 \pm 0.03 \)	\( -0.05 \) (stat)	PRL 115 (2015) 121604
Download input measurements: inputs.json.

Time-dependent CP Asymmetries in \(B^0 \to D^{(*)} h^0\), with Model-Dependent Dalitz Plot Analysis of Multibody D Decays

Experiment	\(\boldsymbol{\sin(2\beta) \equiv \sin(2\phi_1)}\)	\(\boldsymbol{\cos(2\beta) \equiv \cos(2\phi_1)}\)	Correlation	Reference
BaBar+Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 1240\,\mathrm{M}\)	\( 0.80 \pm 0.14 \pm 0.07 \)	\( 0.91 \pm 0.22 \pm 0.11 \)	\( 0.05 \) (stat)	PRL 121 (2018) 261801
Download input measurements: inputs.json.

Time-dependent CP Asymmetries in \(B^0 \to D^{(*)} h^0\), with Model-Independent Dalitz Plot Analysis of Multibody D Decays

Experiment	\(\boldsymbol{\sin(2\beta) \equiv \sin(2\phi_1)}\)	\(\boldsymbol{\cos(2\beta) \equiv \cos(2\phi_1)}\)	Correlation	Reference
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 772\,\mathrm{M}\)	\( 0.43 \pm 0.27 \pm 0.08 \)	\( 1.06 \pm 0.33 \pm 0.18 \)	\( -0.03 \) (stat)	PRD 94 (2016) 052004
Download input measurements: inputs.json.

Due to possible significant penguin pollution, both the cosine and sine coefficients of the Cabibbo-suppressed \(b \to c\bar{c} d\) decays are free parameters of the theory. Absence of penguin pollution would result in \(S_{c\bar{c}d} = -\eta_{CP} \sin(2\beta) \equiv -\eta_{CP}\sin(2\phi_1) \) and \( C_{c\bar{c}d} = 0 \) for the \( CP \) eigenstate final states ( \(\eta_{CP} = +1 \) for both \( J/\psi \pi^0 \) and \(D^+ D^- \) ).

At present we do not apply a rescaling of the results to a common updated set of input parameters.

Time-dependent \(C\!P\) asymmetries in \(B^0 \to J/\psi \pi^0\)

Experiment	\(\boldsymbol{S_{CP}(J/\psi \pi^0)}\)	\(\boldsymbol{C_{CP}(J/\psi \pi^0)}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 466\,\mathrm{M}\)	\( -1.23 \pm 0.21 \pm 0.04 \)	\( -0.20 \pm 0.19 \pm 0.03 \)	\( 0.20 \) (stat)	PRL 101 (2008) 021801
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 772\,\mathrm{M}\)	\( -0.59 \pm 0.19 \pm 0.03 \)	\( 0.150 \pm 0.140 \pm 0.035 \)	\( 0.01 \) (stat)	PRD 98 (2018) 112008(R)
Belle II \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 387\,\mathrm{M}\)	\( -0.88 \pm 0.17 \pm 0.03 \)	\( 0.13 \pm 0.12 \pm 0.03 \)	\( 0.08 \) (stat)	PRD 111 (2025) 012011
Average	\( -0.87 \pm 0.11 \)	\( 0.08 \pm 0.08 \)	\( 0.08 \) (stat)	HFLAV correlated average \( \chi^2 / \mathrm{dof} = 6.69 / 4 \) \( p = 0.15 \Rightarrow 1.4\sigma \)
Figures	[pdf] [png]	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output.json.

Note that the BaBar result is outside of the physical region, and the average is very close to the physical boundary. The interpretation of the average given above has to be done with the greatest care.

Time-dependent \(C\!P\) asymmetries in \(B^0 \to D^+ D^-\)

Experiment	\(\boldsymbol{S_{CP}(D^+ D^-)}\)	\(\boldsymbol{C_{CP}(D^+ D^-)}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 467\,\mathrm{M}\)	\( -0.63 \pm 0.36 \pm 0.05 \)	\( -0.07 \pm 0.23 \pm 0.03 \)	\( -0.01 \) (stat)	PRD 79, 032002 (2009)
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 772\,\mathrm{M}\)	\( -1.06 {}_{-0.14}^{+0.21} \pm 0.08 \)	\( -0.43 \pm 0.16 \pm 0.05 \)	\( -0.12 \) (stat)	PRD 85 (2012) 091106
LHCb \(\mathcal{L} = 9.0\,\mathrm{fb}^{-1}\)	\( -0.549 \pm 0.085 \pm 0.015 \)	\( (162 \pm 88 \pm 9) \times 10^{-3} \)	\( 0.47 \) (stat)	JHEP 01 (2025) 061
Average	\( -0.69 \pm 0.07 \)	\( -0.01 \pm 0.07 \)	\( 0.35 \) (stat)	HFLAV correlated average \( \chi^2 / \mathrm{dof} = 15.65 / 4 \) \( p = 0.00 \Rightarrow 2.9\sigma \)
Figures	[pdf] [png]	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output.json.

Note that the Belle result is outside of the physical region, and the average is very close to the physical boundary. The interpretation of the average given above has to be done with the greatest care.

Time-dependent \(C\!P\) asymmetries in \(B^0 \to J/\psi \rho^0\)

Experiment	\(\boldsymbol{S_{CP}(J/\psi \rho^0)}\)	\(\boldsymbol{C_{CP}(J/\psi \rho^0)}\)	Correlation	Reference
LHCb \(\mathcal{L} = 3.0\,\mathrm{fb}^{-1}\)	\( -0.66 {}_{-0.12}^{+0.13} \pm 0.06 \)	\( -0.063 \pm 0.056 \pm 0.016 \)	\( -0.01 \) (stat)	PLB 742 (2015) 38
Download input measurements: inputs.json.

Time-dependent \(C\!P\) asymmetries in \(B^0 \to D^{*+}D^{*-}\)

Experiment	\(\boldsymbol{S_{CP}(D^{*+} D^{*-})}\)	\(\boldsymbol{C_{CP}(D^{*+} D^{*-})}\)	\(\boldsymbol{R_{\perp}}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 467\,\mathrm{M}\)	\( -0.70 \pm 0.16 \pm 0.03 \)	\( 0.05 \pm 0.09 \pm 0.02 \)	\( 0.171 \pm 0.028 \pm 0.000 \)	\( \) (stat)	PRD 79, 032002 (2009)
BaBar part. rec. \(\mathcal{L} = 471.0\,\mathrm{fb}^{-1}\)	\( -0.49 \pm 0.18 \pm 0.08 \)	\( 0.15 \pm 0.09 \pm 0.04 \)	-	\( \) (stat)	PRD 86 (2012) 112006
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 772\,\mathrm{M}\)	\( -0.79 \pm 0.13 \pm 0.03 \)	\( -0.15 \pm 0.08 \pm 0.02 \)	\( 0.14 \pm 0.02 \pm 0.01 \)	\( \) (stat)	PRD 86 (2012) 071103(R)
Average	\( -0.71 \pm 0.09 \)	\( -0.01 \pm 0.05 \)	\( 0.151 \pm 0.017 \)	\( \) (stat)	HFLAV correlated average \( \chi^2 / \mathrm{dof} = 8.37 / 5 \) \( p = 0.14 \Rightarrow 1.5\sigma \)
Figures	[pdf] [png]	[pdf] [png]	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output.json.

Experiment	\(\boldsymbol{S_{+}(D^{*+}D^{*-})}\)	\(\boldsymbol{C_{+}(D^{*+}D^{*-})}\)	\(\boldsymbol{S_{-}(D^{*+}D^{*-})}\)	\(\boldsymbol{C_{-}(D^{*+}D^{*-})}\)	\(\boldsymbol{R_{\perp}(D^{*+}D^{*-})}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 467\,\mathrm{M}\)	\( -0.76 \pm 0.16 \pm 0.04 \)	\( 0.02 \pm 0.12 \pm 0.02 \)	\( -1.81 \pm 0.71 \pm 0.16 \)	\( 0.41 \pm 0.50 \pm 0.08 \)	\( (155 \pm 30 \pm 3) \times 10^{-3} \)	\( \) (stat)	PRD 79, 032002 (2009)
Download input measurements: inputs.json.

Time-dependent \(C\!P\) asymmetries in \(B^0 \to D^{*+}D^{-}\)

Experiment	\(\boldsymbol{S(D^{*\pm} D^{\mp})}\)	\(\boldsymbol{C(D^{*\pm} D^{\mp})}\)	\(\boldsymbol{\Delta S(D^{*\pm} D^{\mp})}\)	\(\boldsymbol{\Delta C(D^{*\pm} D^{\mp})}\)	\(\boldsymbol{A(D^{*\pm} D^{\mp})}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 467\,\mathrm{M}\)	\( -0.68 \pm 0.15 \pm 0.04 \)	\( 0.04 \pm 0.12 \pm 0.03 \)	\( 0.05 \pm 0.15 \pm 0.02 \)	\( 0.04 \pm 0.12 \pm 0.03 \)	\( 0.008 \pm 0.048 \pm 0.013 \)	\( \) (stat)	PRD 79, 032002 (2009)
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 772\,\mathrm{M}\)	\( -0.78 \pm 0.15 \pm 0.05 \)	\( -0.01 \pm 0.11 \pm 0.04 \)	\( -0.13 \pm 0.15 \pm 0.04 \)	\( 0.12 \pm 0.11 \pm 0.03 \)	\( 0.06 \pm 0.05 \pm 0.02 \)	\( \) (stat)	PRD 85 (2012) 091106
LHCb \(\mathcal{L} = 9.0\,\mathrm{fb}^{-1}\)	\( -0.861 \pm 0.077 \pm 0.019 \)	\( -0.059 \pm 0.092 \pm 0.020 \)	\( 0.019 \pm 0.075 \pm 0.012 \)	\( -0.031 \pm 0.092 \pm 0.016 \)	\( (8 \pm 14 \pm 6) \times 10^{-3} \)	\( \) (stat)	JHEP 03 (2020) 147
Average	\( -0.81 \pm 0.06 \)	\( -0.01 \pm 0.06 \)	\( 0.02 \pm 0.06 \)	\( 0.03 \pm 0.06 \)	\( 0.012 \pm 0.014 \)	\( \) (stat)	HFLAV correlated average \( \chi^2 / \mathrm{dof} = 4.36 / 10 \) \( p = 0.93 \Rightarrow 0.1\sigma \)
Figures	[pdf] [png]	[pdf] [png]	[pdf] [png]	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output.json.

Time-dependent Asymmetries in \(B^0_s\) Decays Mediated by \(b \to c \overline{c} d\) Transitions

Experiment	\(\boldsymbol{S_{CP}}\)	\(\boldsymbol{C_{CP}}\)	\(\boldsymbol{A_{\Delta \Gamma}}\)	Correlation	Reference
LHCb \(\mathcal{L} = 3.0\,\mathrm{fb}^{-1}\)	\( 0.49 {}_{-0.65}^{+0.77} \pm 0.06 \)	\( -0.28 \pm 0.41 \pm 0.08 \)	\( -0.08 \pm 0.40 \pm 0.08 \)	\( \) (stat)	JHEP 06 (2015) 131
Download input measurements: inputs.json.

Time-dependent \(C\!P\) asymmetries in \(B^0 \to \phi K^0\)

Experiment	\(\boldsymbol{\sin (2\beta^{\text{eff}}) \equiv \sin (2\phi_1^{\text{eff}})}\)	\(\boldsymbol{C_{CP}}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 470\,\mathrm{M}\)	\( 0.66 \pm 0.17 \pm 0.07 \)	\( 0.05 \pm 0.18 \pm 0.05 \)	\( \) (stat)	PRD 85 (2012) 112010
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 657\,\mathrm{M}\)	\( 0.90 {}_{-0.19}^{+0.09} \pm 0.00 \)	\( -0.04 \pm 0.20 \pm 0.10 \)	\( \) (stat)	PRD 82 (2010) 073011
Belle II \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 387\,\mathrm{M}\)	\( 0.54 \pm 0.26 \pm 0.07 \)	\( -0.31 \pm 0.20 \pm 0.05 \)	\( 0.01 \) (stat)	PRD 108 (2023) 072012
Average	\( 0.74 \pm 0.12 \) \( \chi^2 / \mathrm{dof} = 1.53 / 2 \) \( p = 0.47 \Rightarrow 0.7\sigma \)	\( -0.09 \pm 0.12 \) \( \chi^2 / \mathrm{dof} = 1.75 / 2 \) \( p = 0.42 \Rightarrow 0.8\sigma \)	uncorrelated averages	HFLAV
Figures	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output_0.json, output_1.json,

Time-dependent \(C\!P\) asymmetries in \(B^0 \to \eta^\prime K^0\)

Experiment	\(\boldsymbol{\sin (2\beta^{\text{eff}}) \equiv \sin (2\phi_1^{\text{eff}})}\)	\(\boldsymbol{C_{CP}}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 467\,\mathrm{M}\)	\( 0.57 \pm 0.08 \pm 0.02 \)	\( -0.08 \pm 0.06 \pm 0.02 \)	\( 0.03 \) (stat)	PRD 79 (2009) 052003
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 772\,\mathrm{M}\)	\( 0.68 \pm 0.07 \pm 0.03 \)	\( -0.03 \pm 0.05 \pm 0.03 \)	\( 0.03 \) (stat)	JHEP 10 (2014) 165
Belle II \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 387\,\mathrm{M}\)	\( 0.670 \pm 0.100 \pm 0.034 \)	\( -0.190 \pm 0.080 \pm 0.034 \)	\( 0.03 \) (stat)	PRD 110 (2024) 112002
Average	\( 0.64 \pm 0.05 \)	\( -0.08 \pm 0.04 \)	\( 0.02 \) (stat)	HFLAV correlated average \( \chi^2 / \mathrm{dof} = 3.42 / 4 \) \( p = 0.49 \Rightarrow 0.7\sigma \)
Figures	[pdf] [png]	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output.json.

Time-dependent \(C\!P\) asymmetries in \(B^0 \to K^0_S K^0_S K^0_S \)

Experiment	\(\boldsymbol{\sin (2\beta^{\text{eff}}) \equiv \sin (2\phi_1^{\text{eff}})}\)	\(\boldsymbol{C_{CP}}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 468\,\mathrm{M}\)	\( 0.94 {}_{-0.24}^{+0.21} \pm 0.06 \)	\( -0.17 \pm 0.18 \pm 0.04 \)	\( 0.16 \) (stat)	PRD 85 (2012) 054023
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 722\,\mathrm{M}\)	\( 0.71 \pm 0.23 \pm 0.05 \)	\( -0.12 \pm 0.16 \pm 0.05 \)	\( \) (stat)	PRD 103 (2021) 032003
Average	\( 0.83 \pm 0.17 \)	\( -0.15 \pm 0.12 \)	\( 0.07 \) (stat)	HFLAV correlated average \( \chi^2 / \mathrm{dof} = 0.55 / 2 \) \( p = 0.76 \Rightarrow 0.3\sigma \)
Figures	[pdf] [png]	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output.json.

Time-dependent \(C\!P\) asymmetries in \(B^0 \to \pi^0 K^0 \)

Experiment	\(\boldsymbol{\sin (2\beta^{\text{eff}}) \equiv \sin (2\phi_1^{\text{eff}})}\)	\(\boldsymbol{C_{CP}}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 467\,\mathrm{M}\)	\( 0.55 \pm 0.20 \pm 0.03 \)	\( 0.13 \pm 0.13 \pm 0.03 \)	\( 0.06 \) (stat)	PRD 79 (2009) 052003
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 657\,\mathrm{M}\)	\( 0.67 \pm 0.31 \pm 0.08 \)	\( -0.14 \pm 0.13 \pm 0.06 \)	\( -0.04 \) (stat)	PRD 81 (2010) 011101
Belle II \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 387\,\mathrm{M}\)	\( 0.75 {}_{-0.23}^{+0.20} \pm 0.04 \)	\( -0.04 {}_{-0.15}^{+0.14} \pm 0.05 \)	\( -0.02 \) (stat)	PRL 131 (2023) 111803
Average	\( 0.64 \pm 0.14 \)	\( -0.01 \pm 0.08 \)	\( 0.01 \) (stat)	HFLAV correlated average \( \chi^2 / \mathrm{dof} = 2.47 / 4 \) \( p = 0.65 \Rightarrow 0.5\sigma \)
Figures	[pdf] [png]	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output.json.

Time-dependent \(C\!P\) asymmetries in \(B^0 \to \rho^0 K^0 \)

Experiment	\(\boldsymbol{\sin (2\beta^{\text{eff}}) \equiv \sin (2\phi_1^{\text{eff}})}\)	\(\boldsymbol{C_{CP}}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 383\,\mathrm{M}\)	\( 0.35 {}_{-0.30}^{+0.27} \pm 0.12 \)	\( -0.05 \pm 0.26 \pm 0.10 \)	\( \) (stat)	PRD 80 (2009) 112001
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 657\,\mathrm{M}\)	\( 0.64 {}_{-0.25}^{+0.20} \pm 0.13 \)	\( -0.03 {}_{-0.23}^{+0.24} \pm 0.15 \)	\( \) (stat)	PRD 79 (2009) 072004
Average	\( 0.54 {}_{-0.21}^{+0.18} \)	\( -0.06 \pm 0.20 \)	-	HFLAV
Figures	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output.json.

Time-dependent \(C\!P\) asymmetries in \(B^0 \to \omega^0 K^0 \)

Experiment	\(\boldsymbol{\sin (2\beta^{\text{eff}}) \equiv \sin (2\phi_1^{\text{eff}})}\)	\(\boldsymbol{C_{CP}}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 467\,\mathrm{M}\)	\( 0.55 {}_{-0.29}^{+0.26} \pm 0.02 \)	\( -0.52 {}_{-0.20}^{+0.22} \pm 0.03 \)	\( 0.03 \) (stat)	PRD 79 (2009) 052003
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 772\,\mathrm{M}\)	\( 0.91 \pm 0.32 \pm 0.05 \)	\( 0.36 \pm 0.19 \pm 0.05 \)	\( -0.00 \) (stat)	PRD 90 (2014) 012002
Average	\( 0.71 \pm 0.21 \)	\( -0.04 \pm 0.14 \)	\( 0.01 \) (stat)	HFLAV correlated average \( \chi^2 / \mathrm{dof} = 9.92 / 2 \) \( p = 0.01 \Rightarrow 2.7\sigma \)
Figures	[pdf] [png]	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output.json.

Time-dependent \(C\!P\) asymmetries in \(B^0 \to f_0 K^0 \)

Experiment	\(\boldsymbol{\sin (2\beta^{\text{eff}}) \equiv \sin (2\phi_1^{\text{eff}})}\)	\(\boldsymbol{C_{CP}}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 470\,\mathrm{M}\)	\( 0.59 {}_{-0.18}^{+0.16} \pm 0.11 \)	\( 0.28 \pm 0.24 \pm 0.09 \)	\( \) (stat)	PRD 85 (2012) 112010
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 657\,\mathrm{M}\)	\( 0.89 {}_{-0.19}^{+0.10} \pm 0.08 \)	\( 0.30 \pm 0.29 \pm 0.14 \)	\( \) (stat)	PRD 82 (2010) 073011
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 383\,\mathrm{M}\)	\( 0.951 {}_{-0.159}^{+0.049} \pm 0.032 \)	\( 0.08 \pm 0.19 \pm 0.05 \)	\( \) (stat)	PRD 80 (2009) 112001
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 657\,\mathrm{M}\)	\( 0.43 {}_{-0.22}^{+0.20} \pm 0.14 \)	\( 0.06 \pm 0.17 \pm 0.11 \)	\( \) (stat)	PRD 79 (2009) 072004
Average	\( 0.69 {}_{-0.12}^{+0.10} \)	\( 0.14 \pm 0.12 \)	-	HFLAV
Figures	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output.json.

Time-dependent \(C\!P\) asymmetries in \(B^0 \to f_2 K^0 \)

Experiment	\(\boldsymbol{\sin (2\beta^{\text{eff}}) \equiv \sin (2\phi_1^{\text{eff}})}\)	\(\boldsymbol{C_{CP}}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 383\,\mathrm{M}\)	\( 0.50 {}_{-0.60}^{+0.41} \pm 0.18 \)	\( 0.28 {}_{-0.40}^{+0.35} \pm 0.11 \)	\( \) (stat)	PRD 80 (2009) 112001
Download input measurements: inputs.json.

Time-dependent \(C\!P\) asymmetries in \(B^0 \to f_X K^0 \)

Experiment	\(\boldsymbol{\sin (2\beta^{\text{eff}}) \equiv \sin (2\phi_1^{\text{eff}})}\)	\(\boldsymbol{C_{CP}}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 383\,\mathrm{M}\)	\( 0.20 {}_{-0.52}^{+0.47} \pm 0.11 \)	\( 0.13 {}_{-0.35}^{+0.33} \pm 0.10 \)	\( \) (stat)	PRD 80 (2009) 112001
Download input measurements: inputs.json.

Time-dependent \(C\!P\) asymmetries in \(B^0 \to \pi^0 \pi^0 K^0_S \)

Experiment	\(\boldsymbol{\sin (2\beta^{\text{eff}}) \equiv \sin (2\phi_1^{\text{eff}})}\)	\(\boldsymbol{C_{CP}}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 227\,\mathrm{M}\)	\( -0.72 \pm 0.71 \pm 0.08 \)	\( 0.23 \pm 0.52 \pm 0.13 \)	\( -0.02 \) (stat)	PRD 76 (2007) 071101
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 772\,\mathrm{M}\)	\( 0.92 {}_{-0.31}^{+0.27} \pm 0.11 \)	\( -0.28 \pm 0.21 \pm 0.04 \)	\( 0.00 \) (stat)	PRD 99 (2019) 011102
Average	\( 0.66 \pm 0.28 \)	\( -0.21 \pm 0.20 \)	\( -0.00 \) (stat)	HFLAV correlated average \( \chi^2 / \mathrm{dof} = 5.15 / 2 \) \( p = 0.08 \Rightarrow 1.8\sigma \)
Figures	[pdf] [png]	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output.json.

Time-dependent \(C\!P\) asymmetries in \(B^0 \to \phi K^0_S \pi^0 \)

Experiment	\(\boldsymbol{\sin (2\beta^{\text{eff}}) \equiv \sin (2\phi_1^{\text{eff}})}\)	\(\boldsymbol{C_{CP}}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 465\,\mathrm{M}\)	\( 0.97 {}_{-0.52}^{+0.03} \pm 0.00 \)	\( -0.20 \pm 0.14 \pm 0.06 \)	\( \) (stat)	PRD 78 (2008) 092008
Download input measurements: inputs.json.

Time-dependent \(C\!P\) asymmetries in \(B^0 \to \pi^+ \pi^- K^0_S\) nonresonant

Experiment	\(\boldsymbol{\sin (2\beta^{\text{eff}}) \equiv \sin (2\phi_1^{\text{eff}})}\)	\(\boldsymbol{C_{CP}}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 383\,\mathrm{M}\)	\( 0.01 {}_{-0.30}^{+0.30} \pm 0.15 \)	\( 0.01 \pm 0.25 \pm 0.08 \)	\( \) (stat)	PRD 80 (2009) 112001
Download input measurements: inputs.json.

Time-dependent \(C\!P\) asymmetries in \(B^0 \to K^+ K^- K^0_S\) (excluding \(\phi K^0\) and \(f_0 K^0\))

Experiment	\(\boldsymbol{\sin (2\beta^{\text{eff}}) \equiv \sin (2\phi_1^{\text{eff}})}\)	\(\boldsymbol{C_{CP}}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 470\,\mathrm{M}\)	\( 0.651 {}_{-0.121}^{+0.106} \pm 0.032 \)	\( 0.02 \pm 0.09 \pm 0.03 \)	\( \) (stat)	PRD 85 (2012) 112010
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 657\,\mathrm{M}\)	\( 0.76 {}_{-0.16}^{+0.12} \pm 0.07 \)	\( 0.14 \pm 0.11 \pm 0.09 \)	\( \) (stat)	PRD 82 (2010) 073011
Average	\( 0.68 {}_{-0.10}^{+0.09} \)	\( 0.06 \pm 0.08 \)	-	HFLAV
Figures	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output.json.

Time-dependent Dalitz plot analysis of \(B^0\to K^+ K^- K^0_S\)

Experiment	\(\boldsymbol{\beta^{\text{eff}}\equiv \phi_1^{\text{eff}} [^\circ] \text{ (} \phi K^0_S \text{)}}\)	\(\boldsymbol{A_{CP} \text{ (} \phi K^0_S \text{)}}\)	\(\boldsymbol{\beta^{\text{eff}}\equiv \phi_1^{\text{eff}} [^\circ] \text{ (} f_0 K^0_S \text{)}}\)	\(\boldsymbol{A_{CP} \text{ (} f_0 K^0_S \text{)}}\)	\(\boldsymbol{\beta^{\text{eff}}\equiv \phi_1^{\text{eff}} [^\circ] \text{ (other} K^+ K^- K^0_S \text{)}}\)	\(\boldsymbol{A_{CP} \text{ (other} K^+ K^- K^0_S \text{)}}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 470\,\mathrm{M}\)	\( 21 \pm 6 \pm 2 \)	\( -0.05 \pm 0.18 \pm 0.05 \)	\( 18 \pm 6 \pm 4 \)	\( -0.28 \pm 0.24 \pm 0.09 \)	\( 20.3 \pm 4.3 \pm 1.2 \)	\( -0.02 \pm 0.09 \pm 0.03 \)	\( \) (stat)	PRD 85 (2012) 112010
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 657\,\mathrm{M}\)	\( 32.2 \pm 9.0 \pm 3.0 \)	\( 0.04 \pm 0.20 \pm 0.10 \)	\( 31 \pm 9 \pm 5 \)	\( -0.30 \pm 0.29 \pm 0.14 \)	\( 24.9 \pm 6.4 \pm 3.3 \)	\( -0.14 \pm 0.11 \pm 0.09 \)	\( \) (stat)	PRD 82 (2010) 073011
Average	\( 24 \pm 5 \)	\( -0.01 \pm 0.14 \)	\( 22 \pm 6 \)	\( -0.29 \pm 0.20 \)	\( 22 \pm 4 \)	\( -0.06 \pm 0.08 \)	\( \) (stat)	HFLAV correlated average \( \chi^2 / \mathrm{dof} = 1.85 / 6 \) \( p = 0.93 \Rightarrow 0.1\sigma \)
Figures	[pdf] [png]	[pdf] [png]	[pdf] [png]	[pdf] [png]	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output.json.

Time-dependent Dalitz plot analysis of \(B^0 \to \pi^+\pi^- K^0_S\)

Experiment	\(\boldsymbol{\beta^{\text{eff}}\equiv \phi_1^{\text{eff}} [^\circ] \text{ (} \rho^0 K^0_S \text{)}}\)	\(\boldsymbol{A_{CP} \text{ (} \rho^0 K^0_S \text{)}}\)	\(\boldsymbol{\beta^{\text{eff}}\equiv \phi_1^{\text{eff}} [^\circ] \text{ (} f_0 K^0_S \text{)}}\)	\(\boldsymbol{A_{CP} \text{ (} f_0 K^0_S \text{)}}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 383\,\mathrm{M}\)	\( 10 \pm 9 \pm 4 \)	\( 0.05 \pm 0.26 \pm 0.10 \)	\( 36.0 \pm 9.8 \pm 3.0 \)	\( -0.08 \pm 0.19 \pm 0.05 \)	\( \) (stat)	PRD 80 (2009) 112001
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 657\,\mathrm{M}\)	\( 20 {}_{-9}^{+9} \pm 5 \)	\( 0.03 {}_{-0.24}^{+0.23} \pm 0.15 \)	\( 13 {}_{-7}^{+7} \pm 4 \)	\( -0.06 \pm 0.17 \pm 0.11 \)	\( \) (stat)	PRD 79 (2009) 072004
Average	\( 16 \pm 7 \)	\( 0.06 \pm 0.20 \)	\( 21 \pm 6 \)	\( -0.07 \pm 0.14 \)	\( \) (stat)	HFLAV correlated average \( \chi^2 / \mathrm{dof} = 4.14 / 4 \) \( p = 0.39 \Rightarrow 0.9\sigma \)
Figures	[pdf] [png]	[pdf] [png]	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output.json.

Experiment	\(\boldsymbol{\beta^{\text{eff}}\equiv{\phi_1^{\text{eff}}} [^\circ] \text{ (} f_2 K^0_S \text{)}}\)	\(\boldsymbol{A_{CP} \text{ (} f_2 K^0_S \text{)}}\)	\(\boldsymbol{\beta^{\text{eff}}\equiv{\phi_1^{\text{eff}}} [^\circ] \text{ (} f_X K^0_S \text{)}}\)	\(\boldsymbol{A_{CP} \text{ (} f_X K^0_S \text{)}}\)	\(\boldsymbol{\beta^{\text{eff}}\equiv{\phi_1^{\text{eff}}} [^\circ] \text{ (Nonresonant)}}\)	\(\boldsymbol{A_{CP} \text{ (Nonresonant)}}\)	\(\boldsymbol{\beta^{\text{eff}}\equiv{\phi_1^{\text{eff}}} [^\circ] \text{ (} \chi_{c0} K^0_S \text{)}}\)	\(\boldsymbol{A_{CP} \text{ (} \chi_{c0} K^0_S \text{)}}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 383\,\mathrm{M}\)	\( 15 \pm 18 \pm 6 \)	\( -0.28 {}_{-0.35}^{+0.40} \pm 0.11 \)	\( 5.8 \pm 15.2 \pm 3.2 \)	\( -0.13 {}_{-0.33}^{+0.35} \pm 0.10 \)	\( 0 \pm 9 \pm 4 \)	\( -0.01 \pm 0.25 \pm 0.08 \)	\( 23 \pm 22 \pm 5 \)	\( 0.29 {}_{-0.53}^{+0.44} \pm 0.06 \)	\( \) (stat)	PRD 80 (2009) 112001
Download input measurements: inputs.json.

Time-dependent \(C\!P\) asymmetries in \(B^0 \to K^0_S K^0_S\)

Experiment	\(\boldsymbol{S_{CP} (K^0_S K^0_S)}\)	\(\boldsymbol{C_{CP} (K^0_S K^0_S)}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 350\,\mathrm{M}\)	\( -1.28 {}_{-0.73}^{+0.80} \pm 0.14 \)	\( -0.40 \pm 0.41 \pm 0.06 \)	\( -0.32 \) (stat)	PRL 97 (2006) 171805
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 657\,\mathrm{M}\)	\( -0.38 {}_{-0.77}^{+0.69} \pm 0.09 \)	\( 0.38 \pm 0.38 \pm 0.05 \)	\( 0.48 \) (stat)	PRL 100 (2008) 121601
Average	\( -1.1 \pm 0.5 \)	\( -0.06 \pm 0.26 \)	\( 0.14 \) (stat)	HFLAV correlated average \( \chi^2 / \mathrm{dof} = 2.5 / 2 \) \( p = 0.29 \Rightarrow 1.1\sigma \)
Figures	[pdf] [png]	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output.json.

Time-dependent \(CP\) asymmetries in \(B^0 \to \pi^+ \pi^-\)

The Gronau-London isospin analysis allows a constraint on \(\alpha \equiv \phi_2\) to be extracted from the \(\pi\pi\) system even in the presence of non-negligible penguin contributions. The isospin analysis uses as input the branching fractions and \(CP\)-violating charge asymmetries of all three \(\pi\pi\) decays modes ( \(\pi^+ \pi^-\), \(pi^\pm \pi^0\), \(\pi^0 \pi^0\) ). Constraints on \(\alpha\equiv\phi_2\) can be obtained without information on \(S(\pi^0\pi^0)\), which has not yet been measured. See our combination of results to obtain constraints on \(\alpha \equiv \phi_2\).

The observables for \( B^0 \to \pi^+ \pi^-\) have been measured by BaBar, Belle and LHCb. Note that at the \(B\) factories the observables are in principle uncorrelated (due to the fact that the time variable, \(\Delta t\), has the range \(-\infty \lt \Delta t \lt +\infty\), although small correlations can be induced by backgrounds), at hadron colliders a non-zero correlation is expected (the time variable takes the range \( 0 \lt t \lt + \infty\)). Please note that at present we do not apply a rescaling of the results to a common, updated set of input parameters. Correlations due to common systematics are neglected in the following averages.

Both Belle and BaBar give confidence level interpretations for \(\alpha \equiv \phi_2\). Belle exclude the range \( 9.5^\circ \lt \phi_2 \lt 81.6^\circ\) at the 68% confidence level. BaBar state that the true value lies in the range \( 71^\circ \lt \alpha \lt 109^\circ \) at the 68% confidence level (considering the solution consistent with the Standard Model). NB. It is implied in these constraints on \(\alpha \equiv \phi_2\) that a mirror solution at \( \alpha \to \alpha + \pi = \phi_2 \to \phi_2 + \pi \) also exists.

Experiment	\(\boldsymbol{S_{CP}(\pi^+ \pi^-)}\)	\(\boldsymbol{C_{CP}(\pi^+ \pi^-)}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 467\,\mathrm{M}\)	\( -0.68 \pm 0.10 \pm 0.03 \)	\( -0.25 \pm 0.08 \pm 0.02 \)	\( -0.06 \) (stat)	PRD 87 (2013) 052009
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 772\,\mathrm{M}\)	\( -0.64 \pm 0.08 \pm 0.03 \)	\( -0.33 \pm 0.06 \pm 0.03 \)	\( -0.10 \) (stat)	PRD 88 (2013) 092003
LHCb \(\mathcal{L} = 4.9\,\mathrm{fb}^{-1}\)	\( -0.672 \pm 0.034 \)	\( -0.32 \pm 0.04 \)	\( 0.41 \) (stat)	JHEP 03 (2021) 075
Average	\( -0.666 \pm 0.029 \)	\( -0.311 \pm 0.030 \)	\( 0.29 \) (stat)	HFLAV correlated average \( \chi^2 / \mathrm{dof} = 0.78 / 4 \) \( p = 0.94 \Rightarrow 0.1\sigma \)
Figures	[pdf] [png]	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output.json.

Time-dependent Dalitz plot analysis of \(B^0 \to \pi^+ \pi^-\pi^0\) (\(B^0 \to \rho^{\pm 0} \pi^{\mp 0}\))

Both BaBar and Belle have performed a full time-dependent Dalitz plot analysis of the decay \(B^0 \to (\rho\pi)^0 \to \pi^+ \pi^- \pi^0\), which allows to simultaneously determine the complex decay amplitudes and the \(CP\)-violating weak phase \(\alpha\equiv\phi_2\). 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 quasi-two-body parameters, and the phases \(\delta_\pm = \arg[A^\mp A^{\pm*}]\) and the UT angle \(\alpha\equiv\phi_2\), 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.

NB this is the place to add links to the averages of the form factor bilinears.

From the bilinear coefficients given above, both experiments extract quasi-two-body (Q2B) parameters. Considering only the charged \(\rho\) bands in the Dalitz plot, the Q2B analysis involves 5 different parameters, one of which - the charge asymmetry \(A_{CP}(\rho\pi)\) - is time-independent. The time-dependent decay rate is given by \[ \Gamma( B^0 \to \rho^\pm \pi^mp)(\Delta t) = ( 1 \pm A_{CP}(\rho^\pm\pi^\mp)) e^{-|\Delta t|/\tau} / 8\tau \times [ 1 + Q_{\rm{tag}}( S(\rho^\pm\pi^\mp) \pm \Delta S(\rho^\pm\pi^\mp))\sin(\Delta m\Delta t) - Q_{\rm{tag}}( C(\rho^\pm\pi^\mp) \pm \Delta C(\rho^\pm\pi^\mp))\cos(\Delta m \Delta t) ], \] where \(Q_{\rm{tag}} = +1(-1)\) when the tagging meson is a \(B^0\) (\(\bar{B}^0\)). \(CP\) symmetry is violated if either one of the foloowing conditions is true: \(A_{CP}(\rho^\pm\pi^\mp) \neq 0\), \(C(\rho^\pm\pi^\mp)\neq 0\) or \(S(\rho^\pm\pi^\mp)\neq 0\). 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 \(B^0\) mixing.

We average the Q2B 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, \(A_{CP}(\rho^\pm\pi^\mp)\) and \(C(\rho^\pm\pi^\mp)\), into the physically motivated choices \[ \begin{align} A^{\pm}(\rho\pi) &= (|\kappa^\pm|^2 - 1) / ( |\kappa^\pm|^2 + 1) = -( A_{CP}(\rho^\pm\pi^\mp) + C(\rho^\pm\pi^\mp) + A_{CP}(\rho^\pm\pi^\mp)\Delta C(\rho^\pm\pi^\mp) ) / ( 1 + \Delta C(\rho^\pm\pi^\mp) + A_{CP} (\rho^\pm\pi^\mp)C(\rho^\pm\pi^\mp) ), \\ A^{\mp}(\rho\pi) &= (|\kappa^\mp|^2 - 1) / ( |\kappa^\mp|^2 + 1) = ( -A_{CP}(\rho^\pm\pi^\mp) + C(\rho^\pm\pi^\mp) + A_{CP}(\rho^\pm\pi^\mp)\Delta C(\rho^\pm\pi^\mp) ) / (-1 + \Delta C(\rho^\pm\pi^\mp) + A_{CP} (\rho^\pm\pi^\mp)C(\rho^\pm\pi^\mp) ), \end{align} \] where \(k^\pm = (q/p)\bar{A}^\mp / A^\pm\) and \(\kappa^\mp = (q/p)\bar{A}^\pm / A^\mp\). With this definition \(A^\mp(\rho\pi) \) ( \(A^\pm(\rho\pi)\) ) describes \(CP\) violation in \(B^0\) decays where the \(\rho\) is emitted (not emitted) by the spectator interaction. Both experiments obtain values for \(A^\pm\) and \(A^\mp\), which we average. Again, this procedure should be equivalent to extracting these values directly from the previous results.

In addition to the \(B^0 \to \rho^\pm \pi^\mp\) Q2B contributions to the \(\pi^+\pi^-\pi^0\) final state, there can also be a \(B^0 \to \rho^0 \pi^0\) component. Both experiments have also extracted the Q2B 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 give 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 PRL 77 (2008) 072001.

The information given below can be used to extract \(\alpha\equiv\phi_2\). A confidence level interpretation for \(\alpha\) can be obtained by scanning over the measured form factor bilinears. In addition, information from the $B \to (\rho\pi)$ SU(2) partners can be included via an isospin pentagon relation. The isospin analysis uses as input the branching fractions and \(CP\)-violating charge asymmetries of all give \(\rho \pi\) decay modes (\(\rho^+\pi^-\), \(\rho^-\pi^+\), \(\rho^0\pi^0\), \(\rho^+\pi^0\), \(\rho^0 \pi^+\)). With all information in the \(\rho\pi\) channels put together, Belle obtain the constraint \(68^\circ \lt \phi_2 \lt 95^\circ\) at the 68% confidence level, for the solution consistent with the Standard Model. BaBar present a scan, but not an interval, for \(\alpha\), since their studies indicate that the scan is not statistically robust and cannot be interpreted as \(1-CL\). NB. It is implied in these constraints on \(\alpha \equiv \phi_2\) that a mirror solution at \( \alpha \to \alpha + \pi = \phi_2 \to \phi_2 + \pi \) also exists.

Experiment	\(\boldsymbol{A_{CP}(\rho^{\pm} \pi^{\mp})}\)	\(\boldsymbol{C(\rho^{\pm} \pi^{\mp})}\)	\(\boldsymbol{S_{CP}(\rho^{\pm} \pi^{\mp})}\)	\(\boldsymbol{\Delta C(\rho^{\pm} \pi^{\mp})}\)	\(\boldsymbol{\Delta S(\rho^{\pm} \pi^{\mp})}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 471\,\mathrm{M}\)	\( -0.100 \pm 0.029 \pm 0.021 \)	\( 0.02 \pm 0.06 \pm 0.04 \)	\( 0.053 \pm 0.081 \pm 0.034 \)	\( 0.23 \pm 0.06 \pm 0.05 \)	\( 0.05 \pm 0.08 \pm 0.04 \)	\( \) (stat)	PRD 88 (2013) 012003
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 449\,\mathrm{M}\)	\( -0.12 \pm 0.05 \pm 0.04 \)	\( -0.13 \pm 0.09 \pm 0.05 \)	\( 0.06 \pm 0.13 \pm 0.05 \)	\( 0.36 \pm 0.10 \pm 0.05 \)	\( -0.08 \pm 0.13 \pm 0.05 \)	\( \) (stat)	PRL 98 (2007) 221602
Average	\( -0.107 \pm 0.031 \)	\( -0.03 \pm 0.06 \)	\( 0.06 \pm 0.07 \)	\( 0.27 \pm 0.06 \)	\( 0.01 \pm 0.08 \)	\( \) (stat)	HFLAV correlated average \( \chi^2 / \mathrm{dof} = 3.76 / 5 \) \( p = 0.58 \Rightarrow 0.5\sigma \)
Figures	[pdf] [png]	[pdf] [png]	[pdf] [png]	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output.json.

Experiment	\(\boldsymbol{A_{\mp}(\rho^{\pm} \pi^{\mp})}\)	\(\boldsymbol{A_{\pm}(\rho^{\pm} \pi^{\mp})}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 471\,\mathrm{M}\)	\( -0.12 \pm 0.08 \pm 0.04 \)	\( 0.09 {}_{-0.06}^{+0.05} \pm 0.04 \)	\( 0.55 \) (stat)	PRD 88 (2013) 012003
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 449\,\mathrm{M}\)	\( 0.08 \pm 0.16 \pm 0.11 \)	\( 0.21 \pm 0.08 \pm 0.04 \)	\( 0.47 \) (stat)	PRL 98 (2007) 221602
Average	\( -0.07 \pm 0.08 \)	\( 0.13 \pm 0.05 \)	\( 0.52 \) (stat)	HFLAV correlated average \( \chi^2 / \mathrm{dof} = 1.37 / 2 \) \( p = 0.50 \Rightarrow 0.7\sigma \)
Figures	[pdf] [png]	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output.json.

Experiment	\(\boldsymbol{C(\rho^{0} \pi^{0})}\)	\(\boldsymbol{S(\rho^{0} \pi^{0})}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 471\,\mathrm{M}\)	\( 0.19 \pm 0.23 \pm 0.15 \)	\( -0.37 \pm 0.34 \pm 0.20 \)	\( 0.00 \) (stat)	PRD 88 (2013) 012003
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 449\,\mathrm{M}\)	\( 0.49 \pm 0.36 \pm 0.28 \)	\( 0.17 \pm 0.57 \pm 0.35 \)	\( 0.08 \) (stat)	PRL 98 (2007) 221602
Average	\( 0.26 \pm 0.23 \)	\( -0.24 \pm 0.34 \)	\( 0.02 \) (stat)	HFLAV correlated average \( \chi^2 / \mathrm{dof} = 0.76 / 2 \) \( p = 0.68 \Rightarrow 0.4\sigma \)
Figures	[pdf] [png]	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output.json.

Time-dependent \(CP\) asymmetries in \(B^0 \to \rho^+ \rho^-\)

The Gronau-London isospin analysis allows a constraint on \(\alpha \equiv \phi_2\) to be extracted from the \(\rho\rho\) system even in the presence of non-negligible penguin contributions. The isospin analysis uses as input the branching fractions and \(CP\)-violating charge asymmetries of the longitudinal components of all three \(\rho\rho\) decays modes ( \(\rho^+ \rho^-\), \(\rho^\pm \rho^0\), \(\rho^0 \rho^0\) ). A similar analysis could be done for each polarisation amplitude, but the other are found to not be statistically significant. Constraints on \(\alpha\equiv\phi_2\) can be obtained without information on \(S(\pi^0\pi^0)\), which has not yet been measured. See our combination of results to obtain constraints on \(\alpha \equiv \phi_2\).

Constraints on \(\alpha\equiv\phi_2\) have been set by both BaBar and Belle. The most recent values are

• BaBar: \( \alpha = (92.4^{+6.0}_{-6.5})^\circ \)
• Belle: \( \phi_2 = (93.7 \pm 10.6)^\circ \)

NB. It is implied in these constraints on \(\alpha \equiv \phi_2\) that a mirror solution at \( \alpha \to \alpha + \pi = \phi_2 \to \phi_2 + \pi \) also exists.

The vector particles in the pseudoscalar to vector-vector decay \(B^0 \to \rho^+ \rho^-\) can have longitundinal and transverse relative polarisation with different \(CP\) properties. The decay is found to be dominated by the longitundinally polarised component:

• BaBar measure \( f_L = 0.992 \pm 0.024 ^{+0.026}_{-0.013} \)
• Belle measure \( f_L = 0.988 \pm 0.012 \pm 0.023 \)
• Belle II measure \( f_L = 0.921 ^{+0.024}_{-0.025}{}^{+0.017}_{-0.015} \)

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 longitundinally polarised component (i.e. \(S_{\rho\rho, \rm{long}}\), \(C_{\rho\rho, \rm{long}}\) ).

Experiment	\(\boldsymbol{S_{CP}(\rho^{+} \rho^{-})}\)	\(\boldsymbol{C_{CP}(\rho^{+} \rho^{-})}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 387\,\mathrm{M}\)	\( -0.17 \pm 0.20 \pm 0.06 \)	\( 0.01 \pm 0.15 \pm 0.06 \)	\( -0.04 \) (stat)	PRD 76 (2007) 052007
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 772\,\mathrm{M}\)	\( -0.13 \pm 0.15 \pm 0.05 \)	\( 0.00 \pm 0.10 \pm 0.06 \)	\( -0.02 \) (stat)	PRD 93 (2016) 032010
Belle 2 \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 387\,\mathrm{M}\)	\( -0.26 \pm 0.19 \pm 0.08 \)	\( -0.02 \pm 0.12 \pm 0.06 \)	\( -0.06 \) (stat)	PRD 111 (2025) 092001
Average	\( -0.18 \pm 0.11 \)	\( -0.01 \pm 0.08 \)	\( -0.03 \) (stat)	HFLAV correlated average \( \chi^2 / \mathrm{dof} = 0.28 / 4 \) \( p = 0.99 \Rightarrow 0.0\sigma \)
Figures	[pdf] [png]	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output.json.

Time-dependent \(CP\) asymmetries in \(B^0 \to \rho^0 \rho^0\)

Since the decay \(B^0 \to rho^0\rho^0\) results in an all charged particle final state, its time-dependent \(CP\) violation parameters can be determined experimentally, if difficulties related to the small branching fraction and large brackgrounds can be overcome. The longitundinally polarised component is determined to be

• BaBar measure \( f_L = 0.75^{+0.11}_{-0.14} \pm 0.04 \)
• Belle measure \( f_L = 0.21^{+0.18}_{-0.22} \pm 0.13 \)
• LHCb measure \( f_L = 0.745^{+0.048}+{-0.058} \pm 0.034 \)

It should be noted that the Belle results for the \(\rho^0\rho^0\) polarisation are in some tension with those from BaBar and LHCb.

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 longitundinally polarised component (i.e. \(S_{\rho\rho, \rm{long}}\), \(C_{\rho\rho, \rm{long}}\) ).

Experiment	\(\boldsymbol{S_{CP}(\rho^0 \rho^0)}\)	\(\boldsymbol{C_{CP}(\rho^0 \rho^0)}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 465\,\mathrm{M}\)	\( 0.3 \pm 0.7 \pm 0.2 \)	\( 0.2 \pm 0.8 \pm 0.3 \)	\( -0.04 \) (stat)	PRD 78 (2008) 071104
Download input measurements: inputs.json.

Time-dependent \(CP\) asymmetries in \(B^0 \to a_1^{\pm} \pi^{\mp}\)

Both BaBar and Belle have performed Q2B analyses of the \(B^0 \to a_1^\pm \pi^\mp\) decay, reconstructed in the final state \(\pi^+\pi^-\pi^+\pi^-\).

The parameter \( \alpha_{\rm eff} \equiv \phi_{2, \rm{eff}} \), which reduces to \( \alpha \equiv \phi_2 \) in the limit of no penguin contributions, can be extracted from these results.

• BaBar obtain \( \alpha_{\rm eff} = (78.6 \pm 7.3)^\circ \)
• Belle obtain \( \phi_{2, \rm{eff}} = (107.3 \pm 6.6 \pm 4.8)^\circ

NB. There is a four-fold ambiguity in the results below.

Experiment	\(\boldsymbol{A_{CP}(a_{1}^{\pm} \pi^{\mp})}\)	\(\boldsymbol{C(a_{1}^{\pm} \pi^{\mp})}\)	\(\boldsymbol{S(a_{1}^{\pm} \pi^{\mp})}\)	\(\boldsymbol{\Delta S (a_{1}^{\pm} \pi^{\mp})}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 384\,\mathrm{M}\)	\( -0.07 \pm 0.07 \pm 0.02 \)	\( -0.10 \pm 0.15 \pm 0.09 \)	\( 0.37 \pm 0.21 \pm 0.07 \)	\( -0.14 \pm 0.21 \pm 0.06 \)	\( \) (stat)	PRL 98 (2007) 181803
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 772\,\mathrm{M}\)	\( -0.06 \pm 0.05 \pm 0.07 \)	\( -0.01 \pm 0.11 \pm 0.09 \)	\( -0.51 \pm 0.14 \pm 0.08 \)	\( -0.09 \pm 0.14 \pm 0.06 \)	\( \) (stat)	PRD 86 (2012) 092012
Average	\( -0.06 \pm 0.06 \)	\( -0.05 \pm 0.11 \)	\( -0.20 \pm 0.13 \)	\( -0.10 \pm 0.12 \)	\( \) (stat)	HFLAV correlated average \( \chi^2 / \mathrm{dof} = 12.24 / 5 \) \( p = 0.03 \Rightarrow 2.1\sigma \)
Figures	[pdf] [png]	[pdf] [png]	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output.json.

Experiment	\(\boldsymbol{A_{\mp}(a_{1}^{\pm} \pi^{\mp})}\)	\(\boldsymbol{A_{\pm}(a_{1}^{\pm} \pi^{\mp})}\)	Correlation	Reference
BaBar \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 384\,\mathrm{M}\)	\( 0.07 \pm 0.21 \pm 0.15 \)	\( 0.15 \pm 0.15 \pm 0.07 \)	\( 0.63 \) (stat)	PRL 98 (2007) 181803
Belle \(N(B\smash{\bar{\phantom{B}}}\kern-14mu B) = 772\,\mathrm{M}\)	\( -0.04 \pm 0.26 \pm 0.19 \)	\( 0.07 \pm 0.08 \pm 0.10 \)	\( 0.61 \) (stat)	PRD 86 (2012) 092012
Average	\( 0.02 \pm 0.20 \)	\( 0.1 \pm 0.1 \)	\( 0.38 \) (stat)	HFLAV correlated average \( \chi^2 / \mathrm{dof} = 0.16 / 2 \) \( p = 0.92 \Rightarrow 0.1\sigma \)
Figures	[pdf] [png]	[pdf] [png]	[pdf] [png]
Download input measurements: inputs.json. Download output average: output.json.