Time-dependent CP violation & Unitarity Triangle angles

Results on Time-Dependent CP Violation and Measurements Related to the Angles of the Unitarity Triangle: Winter conferences (Moriond etc.) 2024

Overview

$\beta \equiv {\phi }_1=\left({22.6}_{\mathrm{<mo>-</mo>0.4}}^{\mathrm{<mo>+</mo>0.5}}\right){}^{°}$	$\alpha \equiv {\phi }_2=\left({84.1}_{\mathrm{<mo>-</mo>3.8}}^{\mathrm{<mo>+</mo>4.5}}\right){}^{°}$	$\gamma \equiv {\phi }_3=\left({66.4}_{\mathrm{<mo>-</mo>3.0}}^{\mathrm{<mo>+</mo>2.8}}\right){}^{°}$	$\left(\stackrel{-}{\mathrm{\rho ,}},\stackrel{-}{\eta }\right)=\left(0.141\pm \mathrm{0.015,},0.359\pm 0.009\right)$
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Measurements related to the CKM angle β ≡ φ₁:

Measurements related to the CKM angle α ≡ φ₂:

Measurements related to the CKM angle γ ≡ φ₃:

Summary of the constraints on Unitarity Triangle Angles

Legend: if not stated otherwise,

• if two errors are given, the first is statistical and the second systematic
• if one error is given it represents the total error, where statistical and systematic uncertainties have been added in quadrature.
• in general: we do not rescale (inflate) the error of an average if inconsistencies between measurements occur.

We use Combos v3.20 (homepage, manual) for the rescaling of the experimental results to common sets of input parameters.

Time-dependent CP Asymmetries in b → cc-bar s Transitions

The experimental results have been rescaled to a common set of input parameters (see table below).

(*) We do not include an unpublished CDF preliminary result from 2007.

Additional note on commonly treated (correlated) systematic effects:

• systematic errors of 0.003 to sin(2β) ≡ sin(2φ₁) and 0.012 to cos(2β) ≡ cos(2φ₁) are applied for the uncertainty due to tag side CP asymmetries in doubly-Cabibbo-suppressed B decays (see PRL 91 (2003) 161801).
• a systematic uncertainty of 0.02 on ΔΓ_d/Γ_d (set to zero in the time-dependent CP fits) translates into an error of 0.001 on sin(2β) ≡ sin(2φ₁) and of 0.0009 on cos(2β) ≡ cos(2φ₁).

We obtain for sin(2β) ≡ sin(2φ₁) in the different decay modes:

^(*) The latest LHCb publication includes results for the J/ψK_S and ψ(2S)K_S final states obtained with LHC Run 2 data. The paper also includes averages with their previous results, based on the Run 1 sample, for J/ψK_S and J/ψK_S+ψ(2S)K_S combined. Since there is no combined LHCb Run 1+2 result for ψ(2S)K_S alone, we include the Run 1 and Run 2 results as separate inputs.
^(**) The BABAR result on χ_c0K_S comes from the time-dependent Dalitz plot analysis of B⁰ → 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.
^(****) Belle 2 have presented a first result with B⁰ → J/ψK_S, measuring sin(2β) ≡ sin(2φ₁) = 0.55 ± 0.21 ± 0.04, where the first uncertainty is statistical and the second is systematic. We do not include this result in our averages as it is not planned to be published. More recent preliminary results, as yet unsupported by publicly available documentation authored by the collaboration, are also not included.

Including earlier sin(2β) ≡ sin(2φ₁) measurements, as well as a measurement by Belle using Υ(5S) data and B-π tagging:

we find the slightly modified average:

from which we obtain the following solutions for β ≡ φ₁ (in [0, π])

Plots:

Averages of sin(2β) ≡ sin(2φ1) and C=−A from all experiments.			eps.gz png	eps.gz png
Same, but excluding less precise results to allow closer inspection of the detail.			eps.gz png	eps.gz png
Constraint on the ρ-bar-η-bar plane:			eps.gz png	eps.gz png

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 time-dependent CP analyses. We recompute C from |λ| (from the BaBar results) for the following averages.

^(†) The typical statistical correlation between S and C in LHCb measurements is 0.4–0.5, while those in e⁺e⁻ → Υ(4S) B factories are close to zero. These correlations are currently neglected in the averages.
^(*) The latest LHCb publication includes results for the J/ψK_S and ψ(2S)K_S final states obtained with LHC Run 2 data. The paper also includes averages with their previous results, based on the Run 1 sample, for J/ψK_S and J/ψK_S+ψ(2S)K_S combined. Since there is no combined LHCb Run 1+2 result for ψ(2S)K_S alone, we include both the Run 1 and Run 2 measurements as inputs.
^(**) The BABAR result on χ_c0K_S comes from the time-dependent Dalitz plot analysis of B⁰ → π⁺π⁻K_S. The third uncertainty is due to the Dalitz model.

Time-dependent Transversity Analysis of B⁰→ J/ψK*

The BaBar and Belle collaborations have performed measurements of sin(2β) & cos(2β) ≡ sin(2φ₁) & cos(2φ₁) in time-dependent transversity analyses of the pseudoscalar to vector-vector decay B⁰→ J/ψK*, where cos(2β) ≡ cos(2φ₁) enters as a factor in the interference between CP-even and CP-odd amplitudes. In principle, this analysis comes along with an ambiguity on the sign of cos(2β) ≡ cos(2φ₁) 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 P-wave phase relative to the slowly moving S-wave 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 s-quark 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(2φ1)J/ψK*	cos(2β) ≡ cos(2φ1)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 χ2 = 0.3/1 dof (CL = 0.61 → 0.5σ)	1.64 ± 0.62 χ2 = 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 non-Gaussian 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).

Time-dependent Analysis of B_d → D*D*K_S

The decays B_d → D⁽*⁾D⁽*⁾K_S are dominated by the b → cc-bar s transition, and are therefore sensitive to 2β ≡ 2φ₁. 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(2φ₁) 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 time-dependent asymmetry (defined in the usual way as the difference between the time-dependent distributions of B⁰-tagged and B⁰-bar-tagged events, divided by their sum) is given by

The parameters J₀, J_c, J_s1 and J_s2 are the integrals over the half-Dalitz plane m(D*⁺K_S)² < m(D*⁻K_S)² of the functions |a|² + |a-bar|², |a|² - |a-bar|², Re(a-bar a*) and Im(a-bar a*) respectively, where a and a-bar are the decay amplitudes of B⁰ → D*D*K_S and B⁰-bar → D*D*K_S respectively. The parameter J_s2 (and hence J_s2/J₀) 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	Jc/J0	(2Js1/J0)sin(2β) ≡ (2Js1/J0)sin(2φ1)	(2Js2/J0)cos(2β) ≡ (2Js2/J0)cos(2φ1)	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 χ2 = 0.2 (CL=0.63 ⇒ 0.5σ)	0.03 ± 0.21 χ2 = 0.3 (CL=0.59 ⇒ 0.5σ)	0.24 ± 0.22 χ2 = 1.4 (CL=0.23 ⇒ 1.2σ)	uncorrelated averages	HFLAV
Figures:	eps.gz png	eps.gz png	eps.gz png	.

From the above result and the assumption that J_s2>0, BaBar infer that cos(2β)>0 at the 94% confidence level.

Time-Dependent Analysis of B_s → J/ψ φ (φ_s)

Decays of the B_s meson via the b → cc-bar 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 non-zero, 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(λ²).

The vector-vector final state J/ψ φ contains mixtures of polarization amplitudes: the CP-odd A_⊥, and the CP-even A₀ 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.

Measurements of φ_s, based on flavour-tagged 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⁻, B_s → J/ψ K⁺K⁻ in the high m(K⁺K⁻) region, 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⁻, B_s → φφ and K*⁰K*⁰ decays are reported below.

Time-Dependent CP Asymmetries in Colour Suppressed b → cu-bar d Transitions

B_d decays to final states such as Dπ⁰ are governed by the b → cu-bar d transitions. If one chooses a final state which is a CP eigenstate, eg. D_CPπ⁰, the usual time-dependence formulae are recovered, with the sine coefficient sensitive to sin(2β) ≡ sin(2φ₁). Since there is no penguin contribution to these decays, there is even less associated theoretical uncertainty than for b → cc-bar 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 CP-even and CP-odd 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(2φ₁) in these processes. All B_d → D⁽*⁾h⁰ decays are analysed together. The following CP-even final states are included: CP-even Dπ⁰ and Dη with D → K_Sπ⁰ and D → K_Sω, Dω with D → K_Sπ⁰, D*π⁰ and D*η with D* → Dπ⁰ and D → K⁺K⁻. The following CP-odd final states are included: Dπ⁰, Dη and Dω with D → K⁺K⁻, D*π⁰ and D*η with D* → Dπ⁰ and D → K_Sπ⁰.

Time-Dependent CP Asymmetries in Colour Suppressed b → cu-bar d Transitions, with Multibody D Decays

Bondar, Gershon and Krokovny have shown that when multibody D decays, such as D → K_Sπ⁺π⁻ are used, a time-dependent analysis of the Dalitz plot of the D decay allows a direct determination of the weak phase: β ≡ φ₁. Equivalently, both sin(2β) ≡ sin(2φ₁) and cos(2β) ≡ cos(2φ₁) can be measured. This information allows to resolve the ambiguity in the measurement of 2β ≡ 2φ₁ from sin(2β) ≡ sin(2φ₁) alone. Simlarly to the case of Dalitz plot analysis of B⁻ → DK⁻ with D → K_Sπ⁺π⁻, such an analysis can be carried out with either model-dependent or model-independent methods.

Time-Dependent CP Asymmetries in Colour Suppressed b → cu-bar d Transitions, with Model-Dependent Dalitz Plot Analysis of Multibody D Decays

A combined analysis of BaBar and Belle data has been done with the decays B_d → Dπ⁰, B_d → Dη, B_d → Dω, B_d → D*π⁰ and B_d → D*η are used. The daughter decays are D* → Dπ⁰, D → K_Sπ⁺π⁻, π⁰ → γγ, η → γγ and π⁺π⁻π⁰, and ω → π⁺π⁻π⁰. The D → K_Sπ⁺π⁻ decay model is obtained from a fit to D*⁺ → D⁰π⁺ 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.

Interpretations:
Treating β ≡ φ₁ as a free parameter in the fit, the result β ≡ φ₁ = (22.5 ± 4.4 ± 1.2 ± 0.6)° is obtained. This corresponds to an observation of CP violation (β ≡ φ₁ ≠ 0) at 5.1σ significance, and evidence for cos 2β ≡ cos 2φ₁ > 0 at 3.7σ. The ambiguous solution with the value of sin 2β ≡ sin 2φ₁ from b→cc-bar s transitions, but cos 2β ≡ cos 2φ₁ < 0, is ruled out at 7.3σ

Time-Dependent CP Asymmetries in Colour Suppressed b → cu-bar d Transitions, with Model-Independent Dalitz Plot Analysis of Multibody D Decays

A model-independent analysis has been performed by Belle. The decays B_d → Dπ⁰, B_d → Dη, B_d → Dη′, B_d → Dω, B_d → D*π⁰ and B_d → D*η are used. Note that due to the strong statistical and systematic correlations, model-dependent results and model-independent results from the same experiment cannot be combined.

Interpretations:
Belle disfavour the solution with the charmonium-kaon value of sin(2φ)₁ and a negative value for cos(2φ)₁, at 5.1 σ significance. The solution with the charmonium-kaon 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(2βeff) ≡ sin(2φ1eff) = −ηCPS from time-dependent b → cu-bar d analyses. The model-dependent results for Bd → D(*)h0, D → KSπ+π− are used. The results are compared to the values from the corresponding charmonium-kaon averages. The average value of sin(2β) ≡ sin(2φ1) in b → c ubar d transitions is sin(2βeff) ≡ sin(2φ1eff) = 0.71 ± 0.09 ( χ2 = 0.5/1 dof (CL=0.47 ⇒ 0.7σ) )

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Time-dependent CP Asymmetries in b → cc-bar d Transitions

Due to possible significant penguin pollution, both the cosine and the sine coefficients of the Cabibbo-suppressed b → cc-bar d decays are free parameters of the theory. Absence of penguin pollution would result in S_{cc-bar d} = − η_CP sin(2β) ≡ − η_CP sin(2φ₁) and C_{cc-bar d} = 0 for the CP eigenstate final states (η_CP = +1 for both J/ψπ⁰ and D⁺D⁻).

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

Experiment	SCP (J/ψ π0)	CCP (J/ψ π0)	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)=772M	−0.59 ± 0.19 ± 0.03	0.15 ± 0.14 +0.03 −0.04	0.01 (stat)	PR D98 (2018) 112008(R)
Average	−0.86 ± 0.14	0.04 ± 0.12	0.08	HFLAV correlated average χ2 = 6.4/2 dof (CL=0.04 ⇒ 2.0σ)
Figures:	eps.gz png	eps.gz png	eps.gz png

(*) 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	SCP (D+D−)	CCP (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 χ2 = 11/4 dof (CL=0.027 ⇒ 2.2σ)
Figures:	eps.gz png	eps.gz png	eps.gz png

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

The vector-vector final states J/ψρ⁰ and D*⁺D*⁻ are mixtures of CP-even and CP-odd; the longitudinally polarized component is CP-even. Note that in the general case of non-negligible penguin contributions, the penguin-tree 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 time-dependent amplitude analysis of B⁰ → J/ψπ⁺π^- decays, find a J/ψρ⁰ fit fraction of 65% and a longitudinal polarisation fraction of 57% (both consistent with results from a time-integrated 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.

The vector particles in the pseudoscalar to vector-vector decay B_d → D*⁺D*⁻ can have longitudinal and transverse relative polarization with different CP properties. The transversely polarized state (h_⊥) is CP-odd, while the other two states in the transversity basis (h₀ and h_||) are CP-even. 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 R₀ as free parameters in the fit. We do not include information on R₀ 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 CP-even and CP-odd 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 CP-even and CP-odd 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 CP-even and -odd components for the CP-odd dilution.

Experiment	SCP (D*+ D*−)	CCP (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 χ2 = 3.7/6 dof (CL=0.72 ⇒ 0.4σ)
Figures:	eps.gz png	eps.gz png	eps.gz png	eps.gz png

^(*) 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 non-CP 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+R²) and S₋ = 2 R sin(2β−δ)/(1+R²). [With alternative notation, S₊ = 2 R sin(2φ₁+δ)/(1+R²) and S₋ = 2 R sin(2φ₁−δ)/(1+R²)]. Here R is the ratio of the magnitudes of the amplitudes for B⁰ → D*⁺D⁻ and B⁰ → 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.

Compilation of results for (left) sin(2βeff) ≡ sin(2φ1eff) = −ηCPS and (right) C ≡ −A from time-dependent b → cc-bar d analyses with CP eigenstate final states. The results are compared to the values from the corresponding charmonium averages.	eps.gz png	eps.gz png
Same, but with separate CP-even and CP-odd results from D*+D*−	eps.gz png	eps.gz png
Same, but including results from D*+−D−+. (These measure the same quantity as other b → cc-bar 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.)	eps.gz png	eps.gz png
Same, but including a naïve b → c c-bar d average. Such an average assumes that penguin contributions to the b → c c-bar d decays are negligible. See the cautionary comments in the discussion on averaging the time-dependent CP violation parameters for b → qq-bar s transitions above. The results of the naïve average are sin(2βeff) ≡ sin(2φ1eff) = 0.73 ± 0.06 C ≡ −A = −0.04 ± 0.04 ( χ2 = 11/8 dof (CL=0.20 ⇒ 1.3σ) ) ( χ2 = 16/8 dof (CL=0.0045, ⇒ 2.0σ) )	eps.gz png	eps.gz png
2D comparisons of averages in the different b→c c-bar d modes.	eps png

Time-Dependent Asymmetries in B_s Decays Mediated by b → cc-bar d Transitions

Time-dependent CP asymmetries in B_s decays mediated by b → cc-bar 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 → cc-bar s transitions. Results in the b → cc-bar 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 → cc-bar s transitions, as described by Fleischer et al. (hep-ph/9903455, arXiv:1010.0089). If the penguin effect is large, it may also be possible to determine γ ≡ φ₃.

The parameters have been measured in B_s → J/ψ K_S⁰ decays by LHCb.

Time-dependent CP Asymmetries in b → qq-bar s (penguin) Transitions

Within the Standard Model, the b → s penguin transition carries approximately the same weak phase as the b → cc-bar s amplitude used above to obtain sin(2β) ≡ sin(2φ₁). When this single phase dominates the decay to a (quasi-)two-body CP eigenstate, the time-dependent CP violation parameters should therefore by given by S = −η_CP × sin(2β^eff) ≡ −η_CP × sin(2φ₁^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(2φ₁^eff) (b → qq-bar s) ∼ sin(2β) ≡ sin(2φ₁) (b → cc-bar s).

Various different final states have been used by BaBar and Belle to investigate time-dependent CP violation in hadronic b → s penguin transitions. These are summarised below. (Note that results from time-dependent Dalitz plot analyses of B⁰ → K⁺K⁻K⁰ and B⁰ → π⁺π⁻K_S are also discussed in the next section — results for φK⁰, ρ⁰K_S and f₀K_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 non-Gaussian errors. In that case, we perform uncorrelated averages (using the PDG prescription for asymmetric errors).

^(*) BaBar and Belle results for φK⁰, ρ⁰K_S and K⁺K⁻K⁰ (excluding φK⁰ and f₀K⁰) are determined from their time-dependent Dalitz plot analyses of B⁰ → K⁺K⁻K⁰ and B⁰ → 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 time-dependent Dalitz plot analysis of B⁰ → K⁺K⁻K_S, so we convert their results on φ₁.) The averages of the directly fitted parameters are more reliable than those of the Q2B parameters, therefore where possible we convert those results to give the averages quoted in the table above.
BaBar results for f₂K_S, f_XK_S and π⁺ π⁻ K_S nonresonant are determined from their time-dependent Dalitz plot analysis of B⁰ → π⁺π⁻K_S.

^(**) BaBar and Belle results for f₀K⁰ are combinations of results from the two Dalitz plot analyses: B⁰ → f₀K⁰ with f₀ → K⁺K⁻, and B⁰ → f₀K_S with f₀ → π⁺π⁻. Note that Q2B parameters extracted from Dalitz plot analyses are constrained to lie within the physical boundary (S_CP² + C_CP² < 1), and consequently the obtained errors can be highly non-Gaussian when the central value is close to the boundary. This is particularly evident in the BaBar results from B⁰ → f₀K_S with f₀ → π⁺π⁻. These results must be treated with extreme caution. As above, we convert the averages of the directly fitted parameters from the time-dependent Dalitz plot analyses back to the Q2B parameters given in the table above.

^(***) The BaBar results on φ K_S π⁰ come from a simultaneous angular analysis of B → φ K⁺ π⁻ and B → φ K_S π⁰, 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_CP² + C_CP² < 1), and consequently the obtained errors are highly non-Gaussian 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.

^(****) The results presented here are from a Belle II time-dependent analysis. They have additionally published results from a time-integrated analysis of B⁰ → π⁰K_S decays, which also provides sensitivity to A_CP =-C_CP. The time-integrated results are not included here.

Please note that

• Assuming that a single and vanishing weak phase dominates the decay amplitudes of the s-penguin modes, one has S = −η_CP×sin(2β) ≡ −η_CP×sin(2φ₁), where η_CP is the CP eigenvalue of the final state. Namely, η_CP = +1 for f₀K_S, f₂K_S, π⁰π⁰K_S, K_SK_SK_S, and η_CP = −1 for φK_S, η′K_S, π⁰K_S, ρK_S and ωK_S. We assume η_CP = +1 for f_XK_S and π⁺ π⁻ K_S nonresonant. Modes with K_L in the final state are related by η_CP (X K_S) = − η_CP (X K_L).
• Averaging over all s-penguin modes assumes that contributions with non-zero weak phases to the decay amplitudes can be neglected. Note that this assumption may be significantly violated due to doubly CKM-suppressed V_ub penguin amplitudes, and, in some cases, doubly CKM-suppressed and color-suppressed V_ub tree amplitudes that contribute to the decay amplitude. Recent theoretical analyses indicate that it is reasonable to expect that the modes φK⁰, η′K⁰ and K_SK_SK⁰ have theoretical uncertainties of the order of 0.05 or smaller, these can be significantly larger for the other modes (in particular for non-ss-bar-resonance modes: π⁰K_S, ρ⁰K_S, ωK_S and K⁺K⁻K⁰).
• Our naïve s-penguin average is, in fact, doubly naïve since it neglects both the theoretical uncertainty discussed above, and the fact that experimental systematic uncertainties are correlated between the measurements of individual modes. For these reasons, we do not advocate the use of these averages, and provide them only for academic interest. Use with extreme caution, if at all.
• The presence of a small O(λ²) weak phase in the decay of the s-penguin modes introduces a phase shift that corresponds to the weak phase 2β_s, which can be measured via time-dependent CP analysis of the decay B_s→ J/ψφ. Using the Wolfenstein parameterization, and neglecting higher order terms in ρ, η and λ, we can estimate this shift to be:
  S(s-penguin) = −η_CP×sin(2β)[cc-bar s]·(1 + Δ) ≡ −η_CP×sin(2φ₁)[cc-bar s]·(1 + Δ). Numerically, using the global CKM fit results for the Wolfenstein parameters, one finds: Δ = 3.3%, which corresponds to a shift of 2β^eff ≡ 2φ₁^eff of +2.1 degrees.

Compilation of results for −η×S ≈ sin(2βeff) ≡ sin(2φ1eff) and C from s-penguin decays.	eps png	eps png
Same, but without less precise results to allow closer inspection of the detail.	eps png	eps png
Comparisons of averages in the different b→q q-bar s modes	eps png	eps png
Same, but without f2KS, fXKS π0π0KS, π+ π− KS nonresonant and φ KS π0 to allow closer inspection of the detail.	eps png	eps png
2D comparisons of averages in the different b→q q-bar s modes.	eps png
Same, but without less precise results to allow closer inspection of the detail.	eps png

Time-dependent Dalitz plot analysis of B_d → K⁺K⁻K⁰ and B_d → π⁺π⁻K⁰

Time-dependent amplitude analyses of the three-body decays B_d → K⁺K⁻K⁰ and B_d → π⁺π⁻K⁰ allow additional information to be extracted from the data. In particular, the cosine of the effective weak phase difference (cos(2β^eff) ≡ cos(2φ₁^eff)) can be determined, as well as the sine term that is obtained from quasi-two-body analysis. This information allows half of the degenerate solutions to be rejected. Furthermore, Dalitz plot analysis has enhanced sensitivity to direct CP violation.

Time-dependent Dalitz plot analyses of B⁰ → 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 time-dependent CP asymmetries in quasi-two-body b → qq-bar s modes. In addition, the effective weak phase β^eff ≡ φ₁^eff is directly determined for two significant resonant contributions: φK⁰ and f₀K⁰ 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	φKS		f0KS		other K+K−KS		Correlation	Reference
	βeff ≡ φ1eff	ACP	βeff ≡ φ1eff	ACP	βeff ≡ φ1eff	ACP
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 χ2 = 1.8/6 dof (CL=0.93 ⇒ 0.1σ)
Figures:	eps.gz png	eps.gz png	eps.gz png	eps.gz png	eps.gz png	eps.gz png	.

^(*) 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 four-fold 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.

Time-dependent Dalitz plot analyses of B⁰ → 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 time-dependent CP asymmetries in quasi-two-body b → qq-bar s modes. In addition, the effective weak phase β^eff ≡ φ₁^eff is directly determined for two significant resonant contributions: f₀K_S and ρ⁰K_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 f₂(1270)K_S, f_X(1300)K_S, nonresonant π⁺π⁻K_S and χ_c0K_S (see b → cc-bar 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	ρ0KS		f0KS		Correlation	Reference
	βeff ≡ φ1eff	ACP	βeff ≡ φ1eff	ACP
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 χ2 = 4.1/4 dof (CL=0.39 ⇒ 0.9σ)
Figures:	eps.gz png	eps.gz png	eps.gz png	eps.gz png	.

^(*) 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 B⁰ → f₀K_S decay are obtained in both B⁰ → K⁺K⁻K⁰ and B⁰ → π⁺π⁻K_S, we show compilations and naïve (uncorrelated) averages below.

Figures: Naïve (uncorrelated) averages for f0KS parameters

eps.gz png

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.

Time-dependent Analysis of B → VV decays via b → qq-bar s (penguin) Transitions: B → φ K_S π⁰

The final state in the decay B → φ K_S π⁰ is a mixture of CP-even and CP-odd amplitudes. However, since only φ K*⁰ resonant states contribute (in particular, φ K*⁰(892), φ K*⁰₀(1430) and φ K*⁰₂(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 π⁰)/B(K*⁰ → K⁺ π⁻) is the same for each exited kaon state.

BaBar have performed a simultaneous analysis of B → φ K_S π⁰ and B → φ K⁺ π⁻ that is time-dependent for the former mode and time-integrated for the latter. Such an analysis allows, in principle, all parameters of the B → φ K*⁰ system to be determined, including mixing-induced CP violation effects. The latter is determined to be Δφ₀₀ = 0.28 ± 0.42 ± 0.04, where Δφ₀₀ is half the weak phase difference between B⁰ and B⁰-bar decays to φK*⁰₀(1430). As presented above, this can also be presented in terms of the quasi-two-body parameter sin(2β^eff₀₀) = sin(2β+2Δφ₀₀) = 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*⁰₂(1430). However, the relative phases between these decays are constrained due to the nature of the simultaneous analysis of B → φ K_S π⁰ 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 π⁰ 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.

Time-Dependent CP Asymmetries in B_s decays via b → qq-bar s (penguin) Transitions (eg. B_s → K⁺K⁻)

The decay B_s → K⁺K⁻ involves a b → uu-bar s transition, and hence has both penguin and tree contributions. Both mixing-induced 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 U-spin, 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_CP² + C_CP² + A_ΔΓ² = 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.

(*) In JHEP 2103 (2021) 075 the results of the Run 1 analysis are updated to account for improved knowledge of Γ_s and ΔΓ_s. The central value of A^ΔΓ_CP changes to −0.97 ± 0.07, which is expected due to a shift in Γ_s with which it is strongly correlated. The combination of Run 1 and Run 2 results uses the updated values as inputs.

Time-Dependent CP Asymmetries in B_s → VV decays via b → qq-bar s (penguin) Transitions (eg. B_s → φφ, K*⁰K*⁰)

The decay B_s → φφ involves a b → ss-bar 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 vector-vector nature of the final state, angular analysis is needed to separate the CP-even and CP-odd 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 their combined LHCb Run 1 and Run 2 data sample, corresponding to ∫Ldt=9.0 fb⁻¹ (PRL 131 (2023) 171802), who measure φ_s(φφ) = −0.074 ± 0.069 and |λ| = 1.009 ± 0.030, where the statistical uncertainty is dominant. Results quoted with separated statistical and systematic uncertainties are available in the Run 1 and Run 2 publications.

The decay B_s → K*⁰K*⁰ involves a b → dd-bar 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 U-spin analysis with the B⁰ decay mode to the same final state can, however, be used to make clean SM tests (see, e.g. hep-ph/0703137 and arXiv:0705.0477).

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/c². In addition to the vector K*(892) resonance, contributions from Kπ S-wave and from the tensor K₂*(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).

Time-dependent CP Asymmetries in b → qq-bar d (penguin) Transitions

The b → qq-bar 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 ≡ φ₁) cancels that in the B⁰–B⁰-bar mixing, so that S = C = 0. However, even within the Standard Model, there may be non-negligible 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 ≡ φ₃, and hence test if any non-Standard Model contributions are present.

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

Experiment	SCP (KSKS)	CCP (KSKS)	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 χ2 = 2.5/2 dof (CL=0.29 ⇒ 1.1σ)
Figures:	eps.gz png	eps.gz png	eps.gz png

(*) 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.

Time-dependent Analysis of b → sγ Transitions

Time-dependent analyses of radiative b decays such as B⁰→ K_Sπ⁰γ, probe the polarization of the photon. In the SM, the photon helicity is dominantly left-handed for b → sγ, and right-handed for the conjugate process. As a consequence, B⁰ → K_Sπ⁰γ behaves like an effective flavor eigenstate, and mixing-induced 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(2φ₁) (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_Sπ⁰ can be performed, since the properties of the decay amplitudes are independent of the angular momentum of the K_Sπ⁰ system. However, if non-dipole operators contribute significantly to the amplitudes, then the Standard Model mixing-induced CP violation could be larger than the expectation given above, and the CPV parameters may vary slightly over the K_Sπ⁰γ Dalitz plot, for example as a function of the K_Sπ⁰ invariant mass.

An inclusive K_Sπ⁰γ analysis has been performed by Belle using the invariant mass range up to 1.8 GeV/c². Belle also gives results for the K*(892) region: 0.8 GeV/c² to 1.0 GeV/c². BaBar has measured the CP-violating asymmetries separately within and outside the K*(892) mass range: 0.8 GeV/c² to 1.0 GeV/c² is again used for K*(892)γ candidates, while events with invariant masses in the range 1.1 GeV/c² to 1.8 GeV/c² are used in the "K_Sπ⁰γ (not K*(892)γ)" analysis.

We quote two averages: one for K*(892) only, and the other one for the inclusive K_Sπ⁰γ 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_Sπ⁰γ final state, both BaBar and Belle have results using K_Sηγ and K_Sρ⁰γ (see footnotes below table), while Belle also has results using K_Sφγ.

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

Mode	Experiment	SCP (b → sγ)	CCP (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 χ2 = 1.9/2 dof (CL=0.40 ⇒ 0.9σ)
KSπ0γ (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 χ2 = 2.4/2 dof (CL=0.30 ⇒ 1.0σ)
KS 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)	PR D97 (2018) 092003
	Average	−0.49 ± 0.42	0.06 ± 0.29	−0.15	HFLAV correlated average χ2 = 2.9/2 dof (CL=0.24 ⇒ 1.2σ)
KS ρ0 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 χ2 = 1.9/2 dof (CL=0.38 ⇒ 0.9σ)
KS 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
Figures:		eps.gz png	eps.gz png	eps.gz png eps.gz png eps.gz png eps.gz png

(*) Due to the non-negligible width of the ρ⁰ meson, decays selected as B⁰ → K_Sρ⁰γ can include a significant contribution from K*⁺⁻π⁻⁺γ decays, which are flavour-specific and do not have the same oscillation phenomenology. Both BaBar and Belle measure S_eff for all B decay candidates with the ρ⁰ selection being 0.6 < m(π⁺π⁻) < 0.9 GeV/c², 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.

Time-dependent Analysis of B_s Decays Mediated by b → sγ Transitions

A similar analysis can be performed for radiative B_s decays to, for example, the φγ final state. As for other observables determined with self-conjugate final states produced in B_s decays the effective lifetime, or equivalently the A_ΔΓ parameter, also provides sensitivity. The LHCb collaboration has determined the relevant parameters in B_s → φγ decays.

Time-dependent Analysis of b → dγ Transitions

Similar to the b → sγ transitions discussed above, time-dependent analyses of radiative b decays such as B⁰→ ρ⁰γ 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 right-handed photons and carrying a new CP violating phase.

A time-dependent analysis of the B⁰→ ρ⁰γ 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.

Time-dependent CP Asymmetries in B_d→ π⁺π⁻

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 non-zero 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	SCP (π+π−)	CCP (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 Run 1 ∫Ldt=3.0 fb−1	−0.63 ± 0.05 ± 0.01	−0.34 ± 0.06 ± 0.01	0.45 (stat)	PR D98 (2018) 032004
LHCb Run 2 ∫Ldt=1.9 fb−1	−0.706 ± 0.042 ± 0.013	−0.311 ± 0.045 ± 0.015	0.394 (stat) 0.306 (syst)	JHEP 2103 (2021) 075
LHCb Average	−0.672 ± 0.034	−0.320 ± 0.038	0.405 (stat)	JHEP 2103 (2021) 075
Average	−0.666 ± 0.029	−0.311 ± 0.030	0.288	HFLAV correlated average χ2 = 0.8/4 dof (CL=0.94 ⇒ 0.1σ)
Figures:	eps.gz png	eps.gz png	eps.gz png

Interpretations:
The Gronau-London isospin analysis allows a constraint on α ≡ φ₂ to be extracted from the ππ 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 ππ decay modes (π⁺π⁻, π⁻π⁰, π⁰π⁰). (Constraints on α ≡ φ₂ can be obtained without information on S(π⁰π⁰), which has not yet been measured.) See our combination of results to obtain constraints on α ≡ φ₂.

Both Belle and BaBar give confidence level interpretations for α ≡ φ₂.
Belle exclude the range 9.5° < φ₂ < 81.6° at 68% confidence 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 ≡ φ₂ that a mirror solution at α → α + π ≡ φ₂ → φ₂ + π also exists.

For more details on the world average for α ≡ φ₂, calculated with different statistical treatments, refer to the CKMfitter and UTfit pages.

Time-dependent CP Asymmetries in B_d→ π⁺π⁻π⁰ (B_d→ ρ⁺⁻⁰π⁻⁺⁰)

Both BaBar and Belle have performed a full time-dependent Dalitz plot analyses of the decay B_d → (ρπ)⁰ → π⁺π⁻π⁰, which allows to simultaneously determine the complex decay amplitudes and the CP-violating weak phase α ≡ φ₂. 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 δ₊₋=arg[A⁻⁺A^+−*] and the UT angle α ≡ φ₂, 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.

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 "quasi-two-body" (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 time-independent. The time-dependent decay rate is given by

where Q_tag=+1(−1) when the tagging meson is a B⁰ (B⁰-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 quasi-two-body 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⁻⁺ quasi-two-body contributions to the π⁺π⁻π⁰ final state, there can also be a B_d→ ρ⁰π⁰ component. Both experiments have also extracted the quasi-two-body 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	ACP (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 χ2 = 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 χ2 = 1.5/2 dof (CL=0.47 ⇒ 0.7σ)
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Experiment	C (ρ0π0)	S (ρ0π0)	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 χ2 = 0.8/2 dof (CL=0.68 ⇒ 0.4σ)
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Interpretations:
The information given above can be used to extract α ≡ φ₂.
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 CP-violating charge asymmetries of all five ρπ decay modes (ρ⁺π⁻,ρ⁻π⁺, ρ⁰π⁰, ρ⁺π⁰, ρ⁰π⁺). With all information in the ρπ channels put together, Belle obtain the constraint 68° < φ₂ < 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 1-CL.

NB. It is implied in the above constraints on α ≡ φ₂ that a mirror solution at α → α + π ≡ φ₂ → φ₂ + π also exists.

For more details on the world average for α ≡ φ₂, calculated with different statistical treatments, refer to the CKMfitter and UTfit pages.

Time-dependent CP Asymmetries in B_d → ρρ (ρ⁺ρ⁻ and ρ⁰ρ⁰)

The vector particles in the pseudoscalar to vector-vector 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:

• BaBar measure f_long = 0.992 ± 0.024 ^+0.026_−0.013,
• Belle measure f_long = 0.988 ±0.012 ±0.023

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	SCP (ρ+ρ−)	CCP (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 χ2 = 0.03/2 dof (CL=0.99 ⇒ 0.02σ)
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Since the decay B⁰ → ρ⁰ρ⁰ 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 backgrounds can be overcome. BaBar measure the longitudinally polarized component to be

• f_long = 0.75 ^+0.11_−0.14 ± 0.04

It should be noted that the Belle results for the ρ⁰ρ⁰ polarisation are in some tension with those from BaBar. Belle determine

• f_long = 0.21 ^+0.18_−0.22 ± 0.13

There is also a measurement from LHCb of this quantity

• f_long = 0.745 ^+0.048_−0.058 ± 0.034

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).

Interpretations:
The Gronau-London isospin analysis allows a constraint on α ≡ φ₂ to be extracted from the ρρ 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 ρρ decay modes (ρ⁺ρ⁻, ρ⁻ρ⁰, ρ⁰ρ⁰). (A similar analysis could be done for each polarisation amplitude, but the others are found to not be statisticall significant.)

Constraints on α ≡ φ₂ have been set by both BaBar and Belle. The most recent values are

• BaBar: α = (92.4 ^+6.0_−6.5)°
• Belle: φ₂ = (93.7 ± 10.6)°

NB. It is implied in the above constraints on α ≡ φ₂ that a mirror solution at α → α + π ≡ φ₂ → φ₂ + π also exists.

For more details on the world average for α ≡ φ₂, calculated with different statistical treatments, refer to the CKMfitter and UTfit pages.

Time-dependent CP Asymmetries in B_d → a₁⁺⁻π⁻⁺

Both BaBar and Belle collaborations have performed Q2B analyses of the B_d → a₁⁺⁻π⁻⁺ decay, reconstructed in the final state π⁺π⁻π⁺π⁻.

Interpretations:
The parameter α_eff ≡ φ_2,eff, which reduces to α ≡ φ₂ 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 four-fold ambiguity in the above results.

For more details on the world average for α ≡ φ₂, calculated with different statistical treatments, refer to the CKMfitter and UTfit pages.

Combinations of Results on b → uu-bar d Processes to Obtain Constraints on α ≡ φ₂

World averages for α ≡ φ₂, 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, πππ⁰ and ρρ to obtain

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 → a₁⁺⁻π⁻⁺ does not significantly affect the average.

Independently from the constraints on α ≡ φ₂ 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 time-dependent CP violation parameters in B_d→ π⁺π⁻, π⁺π⁻π⁰ and ρ⁺ρ⁻ and ρ⁰ρ⁰. 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→ π⁰π⁰ mode, for which no time-dependent measurement is yet possible. The ratio of B⁺ to B⁰ lifetimes is taken from the HFLAV - Oscillations/Lifetime subgroup. Individual measurements are used as inputs, rather than the HFLAV averages, in order to facilitate cross-checks 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 ≡ φ₂ is

The fit has a χ² of 22.37 with 57 observables and 24 parameters. Using the χ² distribution, this corresponds to a p-value of 92% (or 0.1σ). A coverage check with pseudoexperiments gives a p-value of (91.9 ± 0.3)%.
An ambiguous solution also exists at α ≡ φ₂ ⇔ α + π ≡ φ₂ + 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 ≡ φ2, in terms of 1−CL, split by decay mode.

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Time-dependent CP Asymmetries in b → cu-bar d Transitions

Neutral B meson decays such as B_d → D⁺⁻π⁻⁺, B_d → D*⁺⁻π⁻⁺ and B_d → D⁺⁻ρ⁻⁺ provide sensitivity to γ ≡ φ₃ because of the interference between the Cabibbo-favoured amplitude (e.g. B⁰ → D⁻π⁺) with the doubly Cabibbo-suppressed amplitude (e.g. B⁰ → D⁺π⁻). The relative weak phase between these two amplitudes is −γ ≡ −φ₃ and, when combined with the B_dB_d mixing phase, the total phase difference is −(2β+γ) ≡ −(2φ₁+φ₃).

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(B⁰ → D⁺π⁻)/A(B⁰ → D⁻π⁺)|, as well as the strong phase between the amplitudes, usually denoted e.g. δ_Dπ. 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⁺π⁰) using isospin, or from self-tagging decays with strangeness (e.g., B⁰ → 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. There are different conventions in use for the definition of the strong phase difference, which can make a comparison of results in terms of this parameter non-trivial.

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⁻⁺. LHCb have also performed the analysis with fully reconstructed D⁺⁻π⁻⁺ decays. Additional states with similar quark content are also possible, but for vector-vector final states an angular analysis is required, while states containing higher resonances may suffer from uncertainties due to nonresonant or other contributions.

BaBar, Belle and LHCb use different observables:

• BaBar defines the time-dependent PDF by
  f⁺⁻(η, Δt) = e^−|Δt|/τ/4τ × [ 1 −+ S_ζ sin(ΔmΔt) −+ η C_ζ cos(ΔmΔt) ]
  where the first (second) sign is used when the tagging meson is a B⁰ (B⁰). [Note here that a tagging B⁰ (B⁰) corresponds to −S_ζ (+S_ζ).] The parameters η and ζ take the values +1 and + (−1 and −) if the final state is, e.g., D⁻π⁺ (D⁺π⁻). The parameters actually measured by BaBar are denoted a and c. In the fit, the substitutions C_ζ = 1 and S_ζ = a −+ η b_i − η c_i are made. [Note that S₊ = a − c and S₋ = a + c, so a = (S₊ + S₋)/2, c = (S₋ − S₊)/2.] The subscript i denotes the tagging category. These are motivated by the possibility of CP violation on the tag side, which is absent for semileptonic B decays (mostly lepton tags). The parameter a is independent of tag side CPV.

• The parameters used by Belle are similar to the S parameters defined above. However, in the Belle definition, a tagging B⁰ corresponds to a + sign in front of the sine coefficient; furthermore the correspondance between the super/subscript and the final state is opposite, so that S₊₋ (BaBar) = − S⁻⁺ (Belle).

• LHCb uses essentially the same convention as Belle, but with the notation (S_f, S_f) = (S⁻, S⁺)

Here we convert the Belle and LHCb results to express them in terms of a and c. Correlations are taken into account in the LHCb case, where significant correlations are reported. Explicitly, the conversion reads: a = −(S⁺ + S⁻)/2, c = −(S⁺ − S⁻)/2

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	c	Correlation	Reference
D−+π+−	BaBar N(BB)=232M	−0.010 ± 0.023 ± 0.007	−0.033 ± 0.042 ± 0.012	-	PRD 73 (2006) 111101
	Belle N(BB)=386M	−0.050 ± 0.021 ± 0.012	0.019 ± 0.021 ± 0.012	-	PRD 73 (2006) 092003
	LHCb ∫Ldt=3 fb−1	−0.048 ± 0.018 ± 0.005	0.010 ± 0.009 ± 0.008	0.00 (stat) −0.46 (syst)	JHEP 1806 (2018) 084
	Average	−0.038 ± 0.013	0.009 ± 0.010	−0.05	HFLAV correlated average χ2 = 3.0/4 dof (CL=0.56 ⇒ 0.6σ)
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Mode	Experiment	a	c	Reference
D*−+π+− Full reconstruction	BaBar N(BB)=232M	−0.040 ± 0.023 ± 0.010	0.049 ± 0.042 ± 0.015	PRD 73 (2006) 111101
	Belle N(BB)=386M	−0.039 ± 0.020 ± 0.013	−0.011 ± 0.020 ± 0.013	PRD 73 (2006) 092003
D*−+π+− Partial reconstruction	BaBar N(BB)=232M	−0.034 ± 0.014 ± 0.009	−0.019 ± 0.022 ± 0.013	PRD 71 (2005) 112003
	Belle N(BB)=657M	−0.046 ± 0.013 ± 0.015	−0.015 ± 0.013 ± 0.015	PRD 84 (2011) 021101(R)
D*−+π+− All	Average	−0.039 ± 0.010 χ2 = 0.2/3 dof (CL=0.97 ⇒ 0.03σ)	−0.010 ± 0.013 χ2 = 1.9/3 dof (CL=0.0.59 ⇒ 0.6σ)	HFLAV
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D−+ρ+−	BaBar N(BB)=232M	−0.024 ± 0.031 ± 0.009	−0.098 ± 0.055 ± 0.018	PRD 73 (2006) 111101

Compilation of the above results.

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Time-Dependent CP Asymmetries in b → cu-bar s Transitions (eg. B_d → D⁺⁻K_Sπ⁻⁺, etc.) (sin(2β+γ) ≡ sin(2φ₁+φ₃))

Time-dependent analyses of transitions such as B_d → D⁺⁻K_Sπ⁻⁺ can be used to probe sin(2β+γ) ≡ sin(2φ₁+φ₃) 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.

Time-dependent CP Asymmetries in B⁰_s decays via b → cu-bar s Transitions (eg. Analyses of B⁰_s → D⁻⁺_sK⁺⁻, etc.) (sin(γ−2β_s) ≡ sin(φ₃−2β_s+))

LHCb have measured the time-dependent CP violation parameters in B⁰_s → D⁻⁺_sK⁺⁻ decays. Moreover, LHCb have also studied time-dependent CP violation in B⁰_s → D⁻⁺_sK⁺⁻π⁺π⁻ decays. In the latter analysis, both model-dependent and -independent analysis are done; results for the latter are quoted. These are given in terms of the quasi-two-body parameters (C_f, A^ΔΓ_f, A^ΔΓ_f, S_f, S_f), but in this case it should be understood that these are effective parameters, integrated over the phase space of the B⁰_s → D⁻⁺_sK⁺⁻π⁺π⁻ decay.

Digression:

GLW Analyses of B⁻ → D⁽*⁾K⁽*⁾⁻

A theoretically clean measurement of the angle γ ≡ φ₃ can be obtained from the rate and asymmetry measurements of B⁻ → D⁽*⁾_CPK⁽*⁾⁻ 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→cu-bar s transitions with the corresponding doubly CKM-suppressed b→uc-bar 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

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.

• BaBar, Belle, CDF and LHCb all generally use the CP even D decays to K⁺K⁻ and π⁺π⁻ in B⁻ → D_CPK⁻ decays, while BaBar and Belle also study the CP odd D decays to K_Sπ⁰, K_Sω and K_Sφ. In the Belle + Belle II analysis, only D → K⁺K⁻ and D → K_Sπ⁰ decays are used. (BaBar also provide results excluding K_Sφ due to the statistical overlap with events used in their Dalitz plot analysis of D → K_SK^*K⁻ – click here for averages using these results).
• In B⁻ → D*_CPK⁻ decays, BaBar and Belle use the same D decay modes as they use for B⁻ → D_CPK⁻ (listed above). Belle use only the D* → Dπ⁰ decay, which gives CP(D*) = CP(D). BaBar use both D* → Dπ⁰ and D* → Dγ decays, the latter giving the opposite CP: CP(D*) = −CP(D). LHCb use only the CP even D decays to K⁺K⁻ and π⁺π⁻, but by virtue of reconstructing both D* → Dπ⁰ and D* → Dγ decays with a partial reconstruction technique, both CP-even and -odd D* final states are accessed. The partial reconstruction technique provides statistical separation between the D* → Dπ⁰ and D* → Dγ decays (and certain backgrounds), and the observables are therefore correlated. By constrast, correlations are negligible in the case of full reconstruction.
• LHCb have also reported results for corresponding observables in B⁻ → D_CPπ⁻ and D*_CPπ⁻ decays. Since these are expected to have little sensitivity to γ ≡ φ₃ we do not tabulate those results here.
• In B⁻ → D_CPK*⁻ decays, BaBar use CP even D decays to K⁺K⁻ and π⁺π⁻, and CP odd D decays to K_Sπ⁰, K_Sω and K_Sφ. LHCb use only the CP even D decays to K⁺K⁻ and π⁺π⁻. Both experiments reconstruct the K*⁻ → K_Sπ⁻ decay
• In B⁻ → D_CPK⁻π⁺π⁻ decays, LHCb use CP even D decays to K⁺K⁻ and π⁺π⁻. The results are presented separately for the two modes to allow for possibly different CP violation effects in the D decays.

For the DK*⁻ mode some care is needed due to other possible contributions to the B⁻ → DK_Sπ⁻ final state. This can be handled in the relations between the observables and γ ≡φ₃ through the introduction of a coherence factor, as discussed below. It is assumed that the selection of the K* region of the DK_Sπ⁻ 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.

(*) We do not include preliminary results from Belle on D_CPK*⁻ (BELLE-CONF-0316), on D_CPK⁻ (LP 2011 preliminary) or on D*_CPK⁻ (CKM 2012 preliminary) which are all unpublished after more than two years.

Compilation of the above results.	eps.gz png	eps.gz png
CP+ only	eps.gz png	eps.gz png
CP- only	eps.gz png	eps.gz png

Digression:

GLW-like Analyses of B⁻ → D⁽*⁾K⁽*⁾⁻

If a multibody D decay can be shown to be dominated by one CP-eigenstate, it can be used in a "GLW-like" (sometimes called "quasi-GLW") 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 CP-even content, enters the expressions relating these observables to γ ≡ φ₃. The F₊ factors have been measured, with first measurements using CLEOc data (see also here) and more recent and precise results from BESIII (also here)

• F₊(π⁺π⁻π⁰) = 0.973 ± 0.017
• F₊(K⁺K⁻π⁰) = 0.732 ± 0.055
• F₊(π⁺π⁻π⁺π⁻) = 0.735 ± 0.015 ± 0.005
• F₊(K⁺K⁻π⁺π⁻) = 0.730 ± 0.037 ± 0.021

The GLW-like observables for D→π⁺π⁻π⁰ and K⁺K⁻π⁰ have been measured by LHCb. The A_GLW observable for D→π⁺π⁻π⁰ was measured in an earlier analysis (discussed more below) by BaBar. The GLW-like observables for D→π⁺π⁻π⁺π⁻ have been measured by LHCb.

(*) Results from a binned analysis of B⁻→D_KKππK⁻ based on the same data sample are reported below.

Compilation of the above results together with those for CP-even modes.

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Dalitz Plot Analysis of B⁻ → D K⁻ with D → π⁺π⁻π⁰

In the limit that a multibody decay is a pure CP-eigenstate, 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⁻π⁰. 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:

where the constant x₀ = 0.850 depends on the amplitude structure of the D → π⁺π⁻π⁰ decay (x₀ ≡ 2F₊ - 1, where F₊ is the fractional CP-even content defined above), and z_± = r_B e^{i( δ_B ± γ )} ≡ r_B e^{i( δ_B ± φ₃ )}. This choice of variables is motivated by the fact that the yields of B^± decays are proportional to 1 + ρ_±² - x₀². The uncertainty due to the D decay model is included in the systematic error.

Digression:

ADS Analyses of B⁻ → D⁽*⁾K⁽*⁾⁻ and B⁻ → D⁽*⁾π⁻

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 CKM-suppressed D decay interferes with the suppressed (b→u) B decay followed by the CKM-favored 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)

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(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* → Dπ⁰ and D* → Dγ. LHCb have also presented results for B⁻ → D*K⁻ using a partial reconstruction technique that provides statistical separation between the D* → Dπ⁰ and D* → Dγ decays (and certain backgrounds); the corresponding observables are therefore correlated. 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⁻π⁰ and LHCb have presented results using B⁻ → DK⁻ with D → K⁺π⁻π⁺π⁻.

The primary results of the LHCb analysis of B⁻ → D⁽*⁾K⁻ with D → K⁺π⁻ are the (R⁻, R⁺) observables mentioned above, for which separate statistical and systematic uncertainties are reported. We take the results in terms of (R_ADS, A_ADS), for which only total uncertainties are given, in order to be able to average with results from other experiments.

At present we do not rescale the results to a common set of input parameters. Also, common systematic errors are not considered.

(*) We do not include preliminary results from Belle on B⁻ → D*K⁻, D* → Dπ⁰ or Dγ, D→Kπ (LP 2011 preliminary) which are unpublished after more than two years.
(**) LHCb have also performed a binned analysis of B → DK, D → K3π, with the same data sample. The unbinned and binned measurements are not independent and therefore should not be combined. The binned analysis has more sensitivity to γ ≡ φ₃.

Compilation of the above results.

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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_Br_D / (r_B²+r_D²). 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 ≡ φ₃. Some of the observables have been measured by BaBar, Belle, CDF and LHCb in the various D⁽*⁾π⁻ modes.