Time-dependent CP violation & Unitarity Triangle angles

Results on Time-Dependent CP Violation, and Measurements Related to the Angles of the Unitarity Triangle:
Winter 2009 (Moriond, Italy & FPCP, USA)

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

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).

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 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.

Including earlier sin(2β) ≡ sin(2φ₁) measurements using B_d → J/ψK_S decays:

we find the only slightly modified average:

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

Plots:

Average of sin(2β) ≡ sin(2φ1) from all experiments.			eps.gz png
Averages of sin(2β) ≡ sin(2φ1) and C=-A from the B factories.			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 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	HFAG See remark below table
Figures:	eps.gz png	eps.gz png	.

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	HFAG
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

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 from B_s → J/ψ φ have been performed by CDF and D0.

CDF have carried out a flavour-tagged, time-dependent analysis of B_s → J/ψ φ using 2.8 fb⁻¹ of data. They do not present a central value and its uncertainty, due to the highly non-Gaussian shape of the likelihood function. Instead, they present a confidence region in the φ_s-ΔΓ_s plane, from which they obtain φ_s ∈ [−0.56, −2.58] at the 68% confidence level. The consistency with the Standard Model expectation for (φ_s,ΔΓ_s) is 7% (corresponding to 1.8σ). These results supercede results from a previous flavour-tagged, time-dependent analysis using 1.35 fb⁻¹ of data, and an untagged analysis using 1.7 fb⁻¹.

D0 have performed a flavour-tagged, time-dependent analysis of B_s → J/ψ φ using 2.8 fb⁻¹ of data. They perform a fit in which the strong phase differences δ₀ and δ_|| (measured relative to δ_⊥; and denoted δ₂ and δ₁ in their analysis) are constrained to the equivalent values measured in B⁰ → J/ψ K*⁰ (see HFAG b → c), up to an error of π/5, allowing for SU(3) breaking effects. The results are given in the table below. They also obtain a 90% CL allowed interval φ_s ∈ [−1.20, +0.06]. These results supercede results from an untagged analysis using 1 fb⁻¹.

The implicit convention above is that |A_⊥|² + |A₀|² + |A_|||² = 1, and the strong phases are measured relative to that of the A_⊥ component (which is set to zero). The polarization components are defined at time t=0, ie. at the production (primary) vertex of the B_s. Note also that there is, in principle, an ambiguity in the result for φ_s (a four-fold ambiguity in the case of untagged analysis). However, the D0 result employs an assumption on the strong phases, that breaks the ambiguity, and the confidence region quoted in the CDF result includes both solutions.

Averaging of the above results is being carried out by the HFAG oscillation group. For the latest results, see here.

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, 321 (2003).

Results of such an analysis are available from BaBar. The decays B_d → Dπ⁰, B_d → Dη, B_d → Dω, B_d → D*π⁰ and B_d → D*η are used. The daughter decay D* → Dπ⁰ is used. The CP-even D decay to K⁺K⁻ is used for all decay modes, with the CP-odd D decay to K_Sω also used in B_d → D⁽*⁾π⁰ and the additional CP-odd D decay to K_Sπ⁰ also used in B_d → Dω.

BaBar have performed separate fits for the cases where the intermediate D⁽*⁾ decays to CP-even and CP-odd final states, since these receive different contributions fom subleading amplitudes in the Standard Model. Since the effects of these corrections are expected to be negligible (~0.02) compared to the current experimental uncertainty, they have also performed a fit with all decays combined.

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.

Results of such an analysis are available from both Belle and. BaBar. 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 and D → K_Sπ⁺π⁻. Note that BaBar quote uncertainties due to the D decay model separately from other systematic errors, while Belle do not.

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

Experiment	sin(2β) ≡ sin(2φ1)	cos(2β) ≡ cos(2φ1)	|λ|	Correlations	Reference
BaBar N(BB)=383M	0.29 ± 0.34 ± 0.03 ± 0.05	0.42 ± 0.49 ± 0.09 ± 0.13	1.01 ± 0.08 ± 0.02	(stat)	PRL 99, 231802 (2007)
Belle N(BB)=386M	0.78 ± 0.44 ± 0.22	1.87 +0.40 −0.53 +0.22 −0.32	-	-	PRL 97, 081801 (2006)
Average	0.45 ± 0.28 χ2 = 0.7 (CL=0.41 ⇒ 0.8σ)	1.01 ± 0.40 χ2 = 3.2 (CL=0.07 ⇒ 1.8σ)	-	uncorrelated averages	HFAG
Figures:	eps.gz png	eps.gz png	.

Interpretations:
Belle determine the sign of cos(2φ)₁ to be positive at 98.3% confidence level.
BaBar favour the solution of β with cos(2β)>0 at 86% confidence level.
Note that the Belle measurement has strongly non-Gaussian behaviour. The interpretation of the average given above has to be done with the greatest care.
We perform uncorrelated averages (using the PDG prescription for asymmetric errors).

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 and ρ⁰K_S are determined from their time-dependent Dalitz plot analyses of B⁰ → K⁺K⁻K⁰ and B⁰ → π⁺π⁻K_S, respectively. For the experimental results, we quote Q2B parameters that are given in the respective references, where possible. (Belle have not reported Q2B S parameters from their 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 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.
The BaBar results for K⁺K⁻K⁰ are for the inclusive "high mass" (m_K+K− > 1.1 GeV/c²) region in the time-dependent Dalitz plot analysis. Belle results for K⁺K⁻K⁰ are from a previous analysis in which the inclusive state (excluding φK⁰) was treated as a quasi-two-body mode.

^(**) 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.

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).
• The mode K⁺K⁻K⁰ may have both CP-even and CP-odd contributions. The CP-even fraction is experimentally derived from an isospin analysis (Belle) or a moments analysis (BABAR). [Also, see below for details of time-dependent Dalitz plot analyses of this final state by BaBar and Belle.] We preaverage the CP-even fraction, in which there is an implicit assumption that the detection and reconstruction efficiencies have the same variation across the Dalitz plane in all analyses (and therefore the underlying CP-even fraction is indeed the same quantity).
• 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 f2KS, fXKS π0π0KS, π+ π− KS nonresonant and φ KS π0 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

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.

A time-dependent Dalitz plot analysis of B⁰ → K⁺K⁻K⁰ has been performed by BaBar (an update of their previous publication). As given above, parameters can be extracted in a form that allows a straightforward comparison/combination with those from 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, the inclusive "high mass" (m_K+K− > 1.1 GeV/c²) region, as well as the effective weak phase averaged over the Dalitz plot, with the CP properties of the individual components taken into account. In addition to the weak phase, BaBar also measure the time-dependent direct CP violation parameter A_CP ( = -C_CP). Belle have also performed a time-dependent Dalitz plot analysis of B⁰ → K⁺K⁻K_S, using a model that includes contributions from f₀(980)K_S, φK_S, f_X(1500)K_S, χ_c0K_S and nonresonant terms (the f_X(1500)K_S, χ_c0K_S and nonresonant terms are constrained to share common CP parameters).

Experiment	φK0		f0K0		K+K−K0 (mK+K− > 1.1 GeV/c2)		Correlation	Reference
	βeff	ACP	βeff	ACP	βeff	ACP
BaBar (*) N(BB)=465M	(7.7 ± 7.7 ± 0.9)°	0.14 ± 0.19 ± 0.02	(8.5 ± 7.5 ± 1.8)°	0.01 ± 0.26 ± 0.07	(29.5 ± 4.5 ± 1.5)°	0.05 ± 0.09 ± 0.04	(stat)	arXiv:0808.0700
Belle (**) N(BB)=657M	(21.2 +9.8 −10.4 ± 2.0 ± 2.0)°	0.31 +0.21 −0.23 ± 0.04 ± 0.09	(28.2 +9.9 −9.8 ± 2.0 ± 2.0)°	−0.02 ± 0.34 ± 0.08 ± 0.09	-		(stat)	ICHEP 2008 preliminary
Average	(12.9 ± 5.6)°	0.23 ± 0.15	(16.3 ± 6.0)°	0.06 ± 0.19	(29.5 ± 4.7)°	0.05 ± 0.10	(stat)	HFAG correlated average χ2 = 2.9/4 dof (CL=0.58 ⇒ 0.6σ)
Figures:	eps.gz png	eps.gz png	eps.gz png	eps.gz png	φK0(βeff) vs φK0(ACP): eps.gz png f0K0(βeff) vs f0K0(ACP): eps.gz png φK0(βeff) vs f0K0(βeff): eps.gz png φK0(ACP) vs f0K0(ACP): eps.gz png

^(*) The BaBar results on φK⁰ and f₀K⁰ suffer from an ambiguity in the solution. The results quoted here correspond to solution 1 presented in the paper.

^(**) The Belle results on φK⁰ and f₀K⁰ suffer from a four-fold ambiguity in the solution. The results quoted here correspond to solution 1 presented in the paper. The third source of uncertainty arises due to the composition of the Dalitz plot.

From the above results BaBar infer that the trigonometric reflection at π/2 - &beta^eff is disfavoured at 4.8σ.

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)	arXiv:0905.3615
Belle (*) N(BB)=657M	(20.0 +8.6 −8.5 ± 3.2 ± 3.5)°	0.03 +0.23 −0.24 ± 0.11 ± 0.10	(12.7 +6.9 −6.5 ± 2.8 ± 3.3)°	−0.06 ± 0.17 ± 0.07 ± 0.09	(stat)	PRD 79 (2009) 072004
Average	(16.4 ± 6.8)°	0.06 ± 0.20	(20.6 ± 6.2)°	−0.07 ± 0.14	(stat)	HFAG correlated average 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

eps.gz png

.

Time-dependent analysis of B → VV decays: 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 HFAG - Rare Decays.

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⁻). 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). The vector-vector final state D*⁺D*⁻ is a mixture 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⁺).

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)=535M	−0.65 ± 0.21 ± 0.05	−0.08 ± 0.16 ± 0.05	−0.10 (stat)	PRD 77 (2008) 071101(R)
Average	−0.93 ± 0.15	−0.10 ± 0.13	0.04	HFAG correlated average χ2 = 3.8/2 dof (CL=0.15 ⇒ 1.4σ)
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We recall that we do NOT rescale (inflate) the errors due to measurement inconsistencies.

Experiment	SCP (D+D−)	CCP (D+D−)	Correlation	Reference
BaBar N(BB)=467M	−0.65 ± 0.36 ± 0.05	−0.07 ± 0.23 ± 0.03	−0.01 (stat)	PRD 79, 032002 (2009)
Belle N(BB)=535M	−1.13 ± 0.37 ± 0.09	−0.91 ± 0.23 ± 0.06	−0.04 (stat)	PRL 98, 221802 (2007)
Average (*)	−0.89 ± 0.26	−0.48 ± 0.17	−0.02	HFAG correlated average χ2 = 7.3/2 dof (CL=0.025 ⇒ 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 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. BaBar treat this variable as a free parameter in the fit and consequently this systematic is absorbed in the statistical error. We perform the average taking into account correlations of the CP parameters with each other as well as with R_⊥, though we are obliged to assume that the correlations of the Belle results with R_⊥ are negligible.

Belle have performed a fit to the data assuming that the CP parameters for 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.

Experiment	SCP (D*+ D*−)	CCP (D*+ D*−)	R⊥ (D*+ D*−)	Correlation	Reference
BaBar N(BB)=467M	−0.71 ± 0.16 ± 0.03	0.05 ± 0.09 ± 0.02	0.17 ± 0.03	(stat)	PRD 79, 032002 (2009)
Belle N(BB)=657M	−0.96 ± 0.25 +0.12 −0.16	−0.15 ± 0.13 ± 0.04	0.12 ± 0.04 ± 0.02	(stat)	arXiv:0901.4057
Average	−0.77 ± 0.14	−0.02 ± 0.08	0.16 ± 0.02	(stat)	HFAG correlated average χ2 = 2.9/3 dof (CL=0.41 ⇒ 0.8σ)
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. (Belle measure R_T = 0.125 ± 0.043 ± 0.023.)

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) = 0.85 ± 0.10 C ≡ −A = −0.10 ± 0.06 ( χ2 = 5.7/5 dof (CL=0.33 ⇒ 1.0σ) ) ( χ2 = 15/5 dof (CL=0.01, ⇒ 2.5σ) )	eps.gz png	eps.gz png
2D comparisons of averages in the different b→c c-bar d modes.	eps png

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	HFAG 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, BABAR has results using K_Sηγ, while Belle has results using K_Sρ⁰γ.

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	HFAG 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	HFAG 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
KS ρ0 γ	Belle N(BB)=657M	0.11 ± 0.33 +0.05 −0.09	−0.05 ± 0.18 ± 0.06	0.04 (stat)	PRL 101 (2008) 251601
Figures:		eps.gz png	eps.gz png	eps.gz png eps.gz png

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⁻

Please note that at present we do not apply a rescaling of the results to a common, updated set of input parameters. Correlation due to common systematics are neglected in the following averages. We recall that we do NOT rescale (inflate) the errors due to measurement inconsistencies.

Experiment	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)	arXiv:0807.4226
Belle N(BB)=535M	−0.61 ± 0.10 ± 0.04	−0.55 ± 0.08 ± 0.05	−0.15 (stat)	PRL 98 (2007) 211801
Average	−0.65 ± 0.07	−0.38 ± 0.06	−0.08	HFAG correlated average χ2 = 5.8/2 dof (CL=0.055 ⇒ 1.9σ)
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 analysis involves the SU(2) partners of the B_d→ π⁺π⁻ decay. The relevant branching ratios (given in units of 10⁻⁶) and CP-violating charge asymmetries are taken from HFAG - Rare Decays.

Belle exclude the range 9° < φ₂ < 81° at the 95.4% confidence level.
BaBar give a confidence level interpretation for α and exclude the range 23° < α < 67° at the 90% confidence level.

NB. It is implied in the above constraints on α ≡ 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.

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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.

As shown by Charles it can be convenient to transform the experimentally motivated CP parameters A_CP(ρπ) and C_ρπ into the physically motivated choices
A⁺⁻(ρπ) = (|κ⁺⁻|²−1)/(|κ⁺⁻|²+1) = −(A_CP(ρπ)+C_ρπ+A_CP(ρπ)ΔC_ρπ)/(1+ΔC_ρπ + A_CP(ρπ)C_ρπ),
A⁻⁺(ρπ) = (|κ⁻⁺|²−1)/(|κ⁻⁺|²+1) = (−A_CP(ρπ)+C_ρπ+A_CP(ρπ)ΔC_ρπ)/(−1+ΔC_ρπ + A_CP(ρπ)C_ρπ),
where κ⁺⁻ = (q/p)Abar⁻⁺/A⁺⁻ and κ⁻⁺ = (q/p)Abar⁺⁻/A⁻⁺. With this definition A⁻⁺(ρπ) (A⁺⁻(ρπ)) describes CP violation in B_d decays where the ρ is emitted (not emitted) by the spectator interaction. Both experiments obtain values for A⁺⁻ and A⁻⁺, which we average. Again, this procedure should be equivalent to extracting these values directly from the previous results.

In addition to the B_d→ ρ⁺⁻π⁻⁺ 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 arXiv:0710.4974 [hep-ex].

Experiment	ACP (ρ+−π−+)	C (ρ+−π−+)	S (ρ+−π−+)	ΔC (ρ+−π−+)	ΔS (ρ+−π−+)	Correlations	Reference
BaBar N(BB)=375M	−0.14 ± 0.05 ± 0.02	0.15 ± 0.09 ± 0.05	−0.03 ± 0.11 ± 0.04	0.39 ± 0.09 ± 0.09	−0.01 ± 0.14 ± 0.06	(stat)	PRD 76 (2007) 012004
Belle N(BB)=449M	−0.12 ± 0.05 ± 0.04	−0.13 ± 0.09 ± 0.05	0.06 ± 0.13 ± 0.05	0.36 ± 0.10 ± 0.05	−0.08 ± 0.13 ± 0.05	(stat)	PRL 98 (2007) 221602
Average	−0.13 ± 0.04	0.01 ± 0.07	0.01 ± 0.09	0.37 ± 0.08	−0.04 ± 0.10	(stat)	HFAG correlated average χ2 = 4.2/5 dof (CL=0.52 ⇒ 0.6σ)
Figures:	eps.gz png	eps.gz png	eps.gz png	eps.gz png	eps.gz png	.

Experiment	A−+ (ρ+−π−+)	A+− (ρ+−π−+)	Correlation	Reference
BaBar N(BB)=375M	−0.37 +0.16 −0.10 ± 0.09	0.03 ± 0.07 ± 0.04	0.62	PRD 76 (2007) 012004
Belle N(BB)=449M	0.08 ± 0.16 ± 0.11	0.21 ± 0.08 ± 0.04	0.47	PRL 98 (2007) 221602
Average	−0.18 ± 0.12	0.11 ± 0.06	0.40	HFAG correlated average χ2 = 4.0/2 dof (CL=0.14 ⇒ 1.5σ)
Figures:	eps.gz png	eps.gz png	eps.gz png

Experiment	C (ρ0π0)	S (ρ0π0)	Correlation	Reference
BaBar N(BB)=375M	−0.10 ± 0.40 ± 0.53	0.04 ± 0.44 ± 0.18	0.35	PRD 76 (2007) 012004
Belle N(BB)=449M	0.49 ± 0.36 ± 0.28	0.17 ± 0.57 ± 0.35	0.08	PRL 98 (2007) 221602
Average	0.30 ± 0.38	0.12 ± 0.38	0.12	HFAG correlated average χ2 = 0.5/2 dof (CL=0.76 ⇒ 0.3σ)
Figures:	eps.gz png	eps.gz png	eps.gz png

Interpretations:
The information given above can be used to extract α ≡ φ₂.
From the measured form factor bilinears, BaBar extract a confidence level interpretation for α, and constrain α = (87⁺⁴⁵₋₁₃)° at 68% confidence level.
Belle has performed a similar analysis. In addition, Belle has also included information from the SU(2) partners of B → ρπ, which can be used to constrain α ≡ φ₂ 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⁺). The relevant information is taken from HFAG - Rare Decays and is tabulated below. [Branching fractions are given in units of 10⁻⁶.] Those values that are not included in the table below can be obtained from the bilinear form-factors, eg. a measurement BR(B⁰ → ρ⁰π⁰)/ BR(B⁰ → ρ⁺⁻π⁻⁺) = 0.133 ± 0.022 ± 0.023, is extracted. With all information in the ρπ channels put together, Belle obtain the tighter constraint 68° < φ₂ < 95° at 68% confidence level, for 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 → 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 = 0.941 ^+0.034_−0.040 ±0.030

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)=535M	0.19 ± 0.30 ± 0.07	−0.16 ± 0.21 ± 0.07	0.10 (stat)	PRD 76 (2007) 011104
Average	−0.05 ± 0.17	−0.06 ± 0.13	0.01	HFAG correlated average χ2 = 1.4/2 dof (CL=0.50 ⇒ 0.7σ)
Figures:	eps.gz png	eps.gz png	eps.gz png

Since the decay B_d → 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 have performed the first such analysis. They measure the longitudinally polarized component to be

• f_long = 0.75 ^+0.11_−0.14 ± 0.04

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 analysis involves the SU(2) partners of the B_d→ ρ⁺ρ⁻ decay. The relevant branching ratios (given in units of 10⁻⁶) and CP-violating charge asymmetries are taken from HFAG - Rare Decays.

Belle obtain φ₂ = (88 ± 17)° or 59° < φ₂ < 117° at 90% confidence level. Including an upper limit for the B⁰ → ρ⁰ ρ⁰ decay, and world average values for the other parameters, this becomes φ₂ = (91.7 ± 14.9)°.
BaBar find α ∈ [73, 117]° at 68% confidence level.
In the BaBar analysis of B_d → ρ⁰ρ⁰, a constraint of |α - α_eff| < 15.7° (17.6°) is obtained at the 68% (90%) CL. The solution at α - α_eff = +11.3° is preferred.
In the BaBar analysis of B⁺ → ρ⁺ρ⁰, a constraint of α = (92.4 ^+6.0_−6.5)° is obtained.

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⁻⁺

The BaBar collaboration have performed a Q2B analysis 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)°

NB. There is a four-fold ambiguity in the above result.

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 → 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-bar 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⁺)|. 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.

Both BABAR and Belle exploit partial reconstructions of D*⁺⁻π⁻⁺ to increase the available statistics. Both experiments also reconstruct D⁺⁻π⁻⁺ and D*⁺⁻π⁻⁺ fully, and BABAR includes the mode D⁺⁻ρ⁻⁺. Additional states with similar quark content are also possible, but for 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 and Belle 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-bar). [Note here that a tagging B⁰ (B⁰-bar) 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. These parameters are the same for the partially and fully reconstructed B decay analyses from BABAR.

• The parameters used by Belle in the analysis using partially reconstructed B decays, 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). In this analysis, only lepton tags are used, so there is no effect from tag-side CPV. In the Belle analysis using fully reconstructed B decays, this effect is measured and taken into account using D*⁺⁻l⁻⁺ν decays; in neither Belle analysis are the a, b and c parameters used. In the latter case, the measured parameters are 2R_D⁽*⁾π sin( 2φ₁+φ₃ +− δ_D⁽*⁾π ); the definition is such that S⁺⁻ (Belle) = −2R_D*πsin( 2φ₁+φ₃ +− δ_D*π ). However, the definition includes an angular momentum factor (−1)^L, and so for the results in the Dπ system, there is an additional factor of −1 in the conversion.

Here we convert the Belle results to express them in terms of a and c. Explicitly, the conversion reads:

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

Compilation of the above results.	eps.gz png	eps.gz png
Averages of the D*π results.	eps.gz png	eps.gz png

Digression:

Constraining 2β+γ ≡ 2φ1+φ3: For each of Dπ, D*π and Dρ, there are two measurements (a and c, or S+ and S−) which depend on three unknowns (R, δ and 2β+γ ≡ 2φ1+φ3), of which two are different for each decay mode. Therefore, there is not enough information to solve directly for 2β+γ ≡ 2φ1+φ3. However, for each choice of R and 2β+γ ≡ 2φ1+φ3, one can find the value of δ that allows a and c to be closest to their measured values, and calculate the distance in terms of numbers of standard deviations. (We currently neglect experimental correlations in this analysis.) These values of N(σ)min can then be plotted as a function of R and 2β+γ ≡ 2φ1+φ3. These plots are given for the Dπ and D*π modes; the uncertainties in the Dρ mode are currently too large to give any meaningful constraint. The constraints can be tightened if one is willing to use theoretical input on the values of R and/or δ. One popular choice is the use of SU(3) symmetry to obtain R by relating the suppressed decay mode to B decays involving Ds mesons. For more information, visit the CKMfitter and UTfit sites.	eps.gz png	eps.gz png CL: eps.gz png
	eps.gz png	eps.gz png CL: eps.gz png

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

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.

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 and CDF use consistent definitions for A_CP+− and R_CP+−, where

Experimentally, it is convenient to measure R_CP+− using double ratios, in which similar ratios for B⁻ → D⁽*⁾ π⁽*⁾⁻ decays are used for normalization.

These observables have been measured so far for three D⁽*⁾K⁽*⁾⁻ modes.

• BABAR, Belle and CDF all 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 and K_Sω, Belle additionally include K_Sφ (BABAR include this mode in their Dalitz analysis of D → K_SK^*K⁻).
• In B⁻ → D*_CPK⁻ decays, Belle use the same D decay modes as they use for B⁻ → D_CPK⁻ (listed above), and use only the D* → Dπ⁰ decay, which gives CP(D*) = CP(D). BABAR use the same CP even D decays as above, and the CP odd D decays to K_Sπ⁰, K_Sω and K_Sφ. They also use both D* → Dπ⁰ and D* → Dγ decays, the latter giving the opposite CP: CP(D*) = −CP(D).
• 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.

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

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:

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. BABAR and Belle use consistent definitions for A_ADS and R_ADS, where (for example for the B⁻ → DK⁻, D → K⁺π⁻ mode)

(Some of) these observables have been measured so far for the D⁽*⁾K⁻ modes. Belle have presented results for B⁻ → DK⁻ while BaBar have also presented results using B⁻ → D*K⁻, with both D* → Dπ⁰ and D* → Dγ, and B⁻ → DK*⁻. For all the above the D → K⁺π⁻ mode is used. In addition, BaBar have presented results using B⁻ → DK⁻ with D → K⁺π⁻π⁰.

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

Compilation of the above results.

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Digression:

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. The observables have been measured by Belle in the Dπ⁻ mode.

Dalitz Plot Analysis of B⁻ → D⁽*⁾ K⁽*⁾⁻ with D → K_Sπ⁺π⁻, ...

Another method to extract γ ≡ φ₃ from the interference between B⁻ → D⁽*⁾⁰ K⁻ and B⁻ → D⁽*⁾⁰-bar K⁻ uses multibody D decays. A Dalitz plot analysis allows simultaneous determination of the weak phase difference γ ≡ φ₃, the strong phase difference δ_B and the ratio of amplitudes r_B. This idea was proposed by Giri, Grossman, Soffer and Zupan and the Belle Collaboration. The assumption of a D decay model results in an additional model uncertainty.

Results are available from both Belle and BaBar using B⁻ → D K⁻, B⁻ → D*K⁻ and B⁻ → DK*⁻. Belle use the D* decay to Dπ⁰ only, while BaBar also use Dγ, and take the effective strong phase shift into account. Both experiments use the decay D → K_Sπ⁺π⁻; BaBar also use D → K_SK⁺K⁻ (though not for B⁻ → DK*⁻).

For the DK*⁻ mode, both collaborations use K*⁻ → K_Sπ⁻; in this case some care is needed due to other possible contributions to the B⁻ → DK_Sπ⁻ final state. Belle assign an additional (model) uncertainty, while BaBar use an alternative parametrization [replacing r_B and δ_B with κr_s and δ_s, respectively] suggested by Gronau.

If the values of γ ≡ φ₃, δ_B and r_B are obtained by directly fitting the data, the extracted value of r_B is biased (since it is positive definite by nature). Since the error on γ ≡ φ₃ depends on the value of r_B some statistical treatment is necessary to correctly estimate the uncertainty. To obviate this effect, both experiments now use a different set of variables in the fits:

Note that (x₊,y₊) are determined from B⁺ decays, while (x₋,y₋) are determined from B⁻ decays.

These parameters have the advantage of having (approximately) Gaussian distributions, and of having small statistical correlations. Some statistical treatment is necessary to convert these measurements into constraints on the underlying physical parameters γ ≡ φ₃, δ_B and r_B [BaBar do not obtain constraints on r_B and δ_B for the B⁻ → DK*⁻ decay due to the reparametrization described above]. Both experiments use frequentist procedures, though there are differences in the details.

The results below have three sets of errors, which are statistical, systematic, and model related uncertainties respectively. For details of correlations in the model uncertainty assigned by Belle, (See Appendix of Ref.) The Belle results also include an additional source of uncertainty due to background from B⁻ → DK_Sπ⁻ other than B⁻ → DK*⁻, which we have not included here.

Averages are performed using the following procedure.

• It is assumed that both experiments use the same D decay model. Therefore, we do not rescale the results to a common model.
• It is further assumed that the model uncertainty is 100% correlated between experiments, and therefore this source of error is not used in the averaging procedure.
• Note that while the above two assumptions may be reasonable in the case that both experiments are using the same decay mode (as was the case until recently with both using only D → K_Sπ⁺π⁻), now that BaBar include also D → K_SK⁺K⁻; this is certainly invalid. However, separate results for the two D decay modes are not available, making a better treatment difficult at present.
• We include in the average the effect of correlations within each experiments set of measurements.
• At present it is unclear how to assign an average model uncertainty. We have not attempted to do so. Our average includes only statistical and systematic error. An unknown amount of model uncertainty should be added to the final error.
• We follow the suggestion of Gronau in making the DK* averages. Explicitly, we assume that the selection of K*⁻ → K_Sπ⁻ is the same in both experiments (so that κ, r_s and δ_s are the same), and drop the additional source of model uncertainty assigned by Belle due to possible nonresonant decays.
• We do not consider common systematic errors, other than the D decay model.

Mode	Experiment	x+	y+	x-	y-	Correlation	Reference
DK−	BaBar N(BB)=383M	−0.067 ± 0.043 ± 0.014 ± 0.011	−0.015 ± 0.055 ± 0.006 ± 0.008	0.090 ± 0.043 ± 0.015 ± 0.011	0.053 ± 0.056 ± 0.007 ± 0.015	(stat) (syst) (model)	PRD 78 (2008) 034023
	Belle N(BB)=657M	−0.107 ± 0.043 ± 0.011 ± 0.055	−0.067 ± 0.059 ± 0.018 ± 0.063	0.105 ± 0.047 ± 0.011 ± 0.064	0.177 ± 0.060 ± 0.018 ± 0.054	(stat) (model)	arXiv:0803.3375
	Average No model error	−0.087 ± 0.032	−0.037 ± 0.041	0.104 ± 0.033	0.111 ± 0.042	(stat+syst)	HFAG correlated average χ2 = 3.1/4 dof (CL=0.54 ⇒ 0.6σ)
Figures: NB. The contours in these plots do not include model errors.		eps.gz png		eps.gz png		eps.gz png
D*K−	BaBar N(BB)=383M	0.137 ± 0.068 ± 0.014 ± 0.005	0.080 ± 0.102 ± 0.010 ± 0.012	−0.111 ± 0.069 ± 0.014 ± 0.004	−0.051 ± 0.080 ± 0.009 ± 0.010	(stat) (syst) (model)	PRD 78 (2008) 034023
	Belle N(BB)=657M	0.133 ± 0.083 ± 0.018 ± 0.081	0.130 ± 0.120 ± 0.022 ± 0.063	0.024 ± 0.140 ± 0.018 ± 0.090	−0.243 ± 0.137 ± 0.022 ± 0.049	(stat) (model)	arXiv:0803.3375
	Average No model error	0.136 ± 0.054	0.100 ± 0.078	−0.061 ± 0.061	−0.122 ± 0.067	(stat+syst)	HFAG correlated average χ2 = 3.2/4 dof (CL=0.53 ⇒ 0.6σ)
Figures: NB. The contours in these plots do not include model errors.		eps.gz png		eps.gz png		eps.gz png
DK*−	BaBar N(BB)=383M	−0.113 ± 0.107 ± 0.028 ± 0.018	0.125 ± 0.139 ± 0.051 ± 0.010	0.115 ± 0.138 ± 0.039 ± 0.014	0.226 ± 0.142 ± 0.058 ± 0.011	(stat) (syst) (model)	PRD 78 (2008) 034023
	Belle N(BB)=386M	−0.105 +0.177 −0.167 ± 0.006 ± 0.088	−0.004 +0.164 −0.156 ± 0.013 ± 0.095	−0.784 +0.249 −0.295 ± 0.029 ± 0.097	−0.281 +0.440 −0.335 ± 0.046 ± 0.086	(stat) (model)	PRD 73, 112009 (2006)
	Average No model error	−0.117 ± 0.092	0.067 ± 0.108	−0.097 ± 0.127	0.161 ± 0.143	(stat+syst)	HFAG correlated average χ2 = 14/4 dof (CL=0.008 ⇒ 2.7σ)
Figures: NB. The contours in these plots do not include model errors.		eps.gz png		eps.gz png		eps.gz png

Figures: Compilation of (x±,y±) measurements from B → D(*)K(*) decays with D → KSπ+π− and D → KSK+K−. NB. The uncertainities in these plots do not include model errors.

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Figures: Compilation of x+ and x− measurements including results from Dalitz and GLW analyses. NB. The uncertainities in these plots do not include model errors.	eps.gz png	eps.gz png	eps.gz png	eps.gz png
	.	eps.gz png	eps.gz png	.

Digression:

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

BaBar have performed a similar Dalitz plot analysis using the decay D → π⁺π⁻π⁰. In this case the measured yields of B⁻ → DK⁻ and B⁺ → DK⁺ events are found to make a significant contribution to the sensitivity to CP violation and this information is included into the fit. Consequently, an alternative set of fit parameters is used in order to avoid significant biasing and nonlinear correlations. The result is parameterized in terms of polar coordinates:

where the constant x₀ = 0.850 depends on the amplitude structure of the D → π⁺π⁻π⁰ decay, 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 B⁰ → DK*⁰

BaBar have presented results on B⁰ → DK*⁰ with D → K⁻π⁺, D → K⁻π⁺ π⁰ and D → K⁻π⁺ π⁺ π⁻. The following 95% CL limits are set:

(See above for a definition of the parameters).

Dalitz Plot Analysis of B⁰ → DK*⁰ with D → K_Sπ⁺π⁻

BaBar have performed a similar Dalitz plot analysis to that described above using neutral B decays. In order to avoid complications due to B⁰–B⁰-bar oscillations (see here), the decay to the self-tagging final state DK*⁰, with K*⁰ → K⁺π⁻, is used. Effects due to the natural width of the K*⁰ are handled using the parametrization suggested by Gronau.