Results on Time-Dependent CP Violation, and Measurements Related to the Angles of the Unitarity Triangle:
Winter 2008 (Moriond, Italy (etc.) and FPCP2008, Taiwan).

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:

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.

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 the first flavour-tagged, time-dependent analysis of B_s → J/ψ φ using 1.35 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 ∈ [−0.32, −2.82] at the 68% confidence level. The consistency with the Standard Model expectation for (φ_s,ΔΓ_s) is 15%. These results supercede results from 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.

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

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). The third error in the results for ρ⁰K_S 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).

^(*) The BaBar results for φK⁰ are determined from their time-dependent Dalitz plot analysis of B⁰ → K⁺K⁻K⁰. The BaBar results for ρ⁰K_S 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.

^(**) The BaBar results for f₀K⁰ are a combination of results from the two Dalitz plot analyses: sin(2β^eff) = 0.25 ± 0.26 ± 0.10, C_CP = −0.41 ± 0.23 ± 0.07 from B⁰ → f₀K⁰ with f₀ → K⁺K⁻, and sin(2β^eff) = 0.94 ^+0.02_−0.07 ^+0.03_−0.05 ± 0.02, C_CP = 0.35 ± 0.27 ± 0.07 ± 0.04 from 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 are highly non-Gaussian when the central value is close to the boundary. This is particularly evident in the results from B⁰ → f₀K_S with f₀ → π⁺π⁻. These results must be treated with extreme caution.

^(***) Due to the highly non-Gaussian errors of the result from B⁰ → f₀K_S with f₀ → π⁺π⁻, and the fact that this result has a significant effect on the χ² of the naïve b→s penguin average (see comments below), we also calculate this average excluding the outlying point. In this case, the BaBar result on f₀K⁰ is that from B⁰ → f₀K⁰ with f₀ → K⁺K⁻.

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, π⁰π⁰K_S, K_SK_SK_S, and η_CP = −1 for φK_S, η′K_S, π⁰K_S, ρK_S and ωK_S. 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 a time-dependent Dalitz plot analysis of this final state by BaBar.] 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 excluding the outlying measurement of B0 → f0KS with f0 → π+π− (see comments above).	eps png	eps png
Same, but without π0π0KS and ρ0KS, 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 excluding the outlying measurement of B0 → f0KS with f0 → π+π− (see comments above).	eps png	eps png
Same, but without π0π0KS and ρ0KS, 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. 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).

From the above results BaBar infer that the trigonometric reflection at π/2 - &beta^eff is disfavoured at 4.5σ.

A time-dependent Dalitz plot analysis of B⁰ → π⁺π⁻K_S has been performed by BaBar. 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. 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.

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)	arXiv:0804.0896
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)
Figures:	eps.gz png	eps.gz png	eps.gz png

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)=364M	−0.54 ± 0.34 ± 0.06	0.11 ± 0.22 ± 0.07	−0.17 (stat)	PRL 99, 071801 (2007)
Belle N(BB)=535M	−1.13 ± 0.37 ± 0.09	−0.91 ± 0.23 ± 0.06	−0.04 (stat)	PRL 98, 221802 (2007)
Average (*)	−0.75 ± 0.26	−0.37 ± 0.17	−0.10	HFAG correlated average χ2 = 12/2 dof (CL=0.003 ⇒ 3.0σ)
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.