HFAG: Unitarity Triangle Parameters
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Results on Time-Dependent CP Violation, and Measurements Related to the Angles of the Unitarity Triangle:
Summer 2008 (ICHEP2008, USA & CKM2008, Italy)

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

§   Studies of b → cc-bar s Transitions
§   Studies of Colour Suppressed b → cu-bar d Transitions
§   Studies of b → cc-bar d Transitions
§   Studies of b → qq-bar s (penguin) Transitions
§   Studies of b → qq-bar d (penguin) Transitions
§   Studies of b → sγ Transitions
§   Studies of b → dγ Transitions
§   Studies of b → uu-bar d Transitions
§   Studies of Time-Dependent Interference Between b → cu-bar d and b-bar → u-bar cd-bar Transitions
§   Studies of Interference Between b → cu-bar s & b → uc-bar s Transitions

Legend: if not stated otherwise,

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


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).

Parameter Value Reference
τ(Bd) (1.530 ± 0.008) ps HFAG - Oscillations/Lifetime
Δmd (0.507 ± 0.004) ps−1 HFAG - Oscillations/Lifetime
|A|2
(CP-odd fraction in
B0→ J/ψK* CP sample)
0.233 ± 0.010 ± 0.005 BaBar: PRD 76 (2007) 031102
N(BB)=232m
0.195 ± 0.012 ± 0.008 Belle: PRL 95 (2005) 091601
N(BB)=275m
0.219 ± 0.009 Average
χ2 = 4.3/1 dof (CL=0.04 ⇒ 2.1σ)

Additional note on commonly treated (correlated) systematic effects:

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

Parameter: sin(2β) ≡ sin(2φ1)
Mode BaBar Belle Average Reference
Charmonium: N(BB)=465M N(BB)=535M - BaBar (arXiv:0808.1903)
Belle (PRL 98 (2007) 031802)
J/ψKSCP=-1) 0.660 ± 0.036 ± 0.012 0.643 ± 0.038stat
J/ψKLCP=+1) 0.698 ± 0.062 ± 0.031 0.641 ± 0.057stat
J/ψK0 0.670 ± 0.031 ± 0.013 0.642 ± 0.031 ± 0.017 0.657 ± 0.025
(0.022stat-only)
CL = 0.58
ψ(2S)KSCP=-1) 0.905 ± 0.101 ± 0.037 0.718 ± 0.090 ± 0.031
N(BB)=657M
0.802 ± 0.071
(0.067stat-only)
BaBar (arXiv:0808.1903)
Belle (PRD77 (2008) 091103(R))
χc1KSCP=-1) 0.619 ± 0.161 ± 0.040 - - BaBar (arXiv:0808.1903)
ηcKSCP=-1) 0.930 ± 0.160 ± 0.057 -
J/ψK*0 (K*0 → KSπ0) CP= 1-2|A|2) 0.608 ± 0.241 ± 0.088 -
All charmonium 0.691 ± 0.029 ± 0.014 0.650 ± 0.029 ± 0.018 0.671 ± 0.024
(0.021stat-only)
CL = 0.49

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

Parameter: sin(2β) ≡ sin(2φ1)
Experiment Value Reference
ALEPH 0.84 +0.82−1.04 ± 0.16 PL B492 (2000) 259-274
OPAL 3.2 +1.8−2.0 ± 0.5 EPJ C5 (1998) 379-388
CDF (full Run I) 0.79 +0.41−0.44(stat+syst) PRD 61 (2000) 072005

we find the only slightly modified average:

Parameter: sin(2β) ≡ sin(2φ1)
All charmonium 0.672 ± 0.024 (0.021stat-only) CL = 0.63

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

β ≡ φ1 = (21.1 ± 0.9)° or β ≡ φ1 = (68.9 ± 1.0)°

Plots:

Average of sin(2β) ≡ sin(2φ1) from all experiments.

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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

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Constraining the Unitarity Triangle (ρ, η):
The measurement of sin(2β) ≡ sin(2φ1) from charmonium modes can be compared in the ρ-bar-η-bar plane (ρ-bar, η-bar being the parameters in the exact (unitary) Wolfenstein parameterization of the CKM matrix) with the constraints from other experimental inputs.

Visit the CKMfitter and UTfit sites for results on global CKM fits using different fit techniques and input quantities.



The cosine coefficient:

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.

Parameter: C=−A (if not stated otherwise)
Mode BaBar Belle Average Reference
Charmonium N(BB)=465M N(BB)=535M - BaBar (arXiv:0808.1903)
Belle (PRL 98 (2007) 031802)
J/ψKS 0.029 ± 0.026 ± 0.017 0.001 ± 0.028stat
J/ψKL −0.030 ± 0.050 ± 0.027 −0.045 ± 0.033stat
J/ψK0 0.019 ± 0.023 ± 0.019 −0.018 ± 0.021 ± 0.014 −0.002 ± 0.020
(0.016stat-only)
CL = 0.28
ψ(2S)KS 0.092 ± 0.077 ± 0.021 −0.039 ± 0.069 ± 0.053
N(BB)=657M
0.033 ± 0.061
(0.051stat-only)
BaBar (arXiv:0808.1903)
Belle (PRD77 (2008) 091103(R))
χc1KS 0.133 ± 0.109 ± 0.026 - - BaBar (arXiv:0808.1903)
ηcKS 0.082 ± 0.125 ± 0.027 -
J/ψK*0 (K*0 → KSπ0) 0.028 ± 0.084 ± 0.054 -
All charmonium 0.026 ± 0.020 ± 0.016 −0.018 ± 0.025 (0.020stat-only) 0.005 ± 0.019
(0.014stat-only)
CL = 0.35


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
.

Interpretations:

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 Bd → D*D*KS

The decays Bd → D(*)D(*)KS 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 Bd → D*D*KS decays (with some theoretical input).

Analysis of the Bd → D*D*KS decay has been performed by BaBar. and Belle.

The analyses proceed by dividing the Dalitz plot into two: m(D*+KS)2 > m(D*KS)2y = +1) and m(D*+KS)2 < m(D*KS)2y = -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

A(Δt) = ηy (Jc/J0) cos(ΔmdΔt) − [ (2Js1/J0)sin(2β) + ηy (2Js2/J0)cos(2β) ] sin(ΔmdΔt)

The parameters J0, Jc, Js1 and Js2 are the integrals over the half-Dalitz plane m(D*+KS)2 < m(D*KS)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*KS and B0-bar → D*D*KS respectively. The parameter Js2 (and hence Js2/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
.

Interpretations:

From the above result and the assumption that Js2>0, BaBar infer that cos(2β)>0 at the 94% confidence level.



Time-Dependent Analysis of Bs → J/ψ φ

Decays of the Bs meson via the b → cc-bar s transition probe φs, a CP violating phase related to Bs–Bs-bar mixing. An important difference with respect to the Bd–Bd-bar system, is that the value of ΔΓ is predicted to significantly non-zero, allowing information on φs to be extracted without tagging the flavour of the decaying B meson. Within the Standard Model, φs 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 φs, and their interference provides additional sensitivity. The sensitivity to φs depends strongly on ΔΓ, and less strongly on the perpendicularly polarized fraction, |A|2.

In this discussion we make the approximation φs ≈ −2βs where φs ≡ arg[ − M12 / Γ12 ] and 2βs ≡ 2 arg[ − VtsVtb* / VcsVcb* ]. This is a reasonable approximation since, although the equality does not hold in the Standard Model, both are much smaller than the current experimental resolution, whereas new physics contributions add a phase φNP to φs and subtract the same phase from 2βs, so that the approximation remains valid.

Measurements of φs from Bs → J/ψ φ have been performed by CDF and D0.

CDF have carried out a flavour-tagged, time-dependent analysis of Bs → 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 φs-ΔΓs plane, from which they obtain φs ∈ [−0.56, −2.58] at the 68% confidence level. The consistency with the Standard Model expectation for (φs,ΔΓs) 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 Bs → 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 φs ∈ [−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 Bs. Note also that there is, in principle, an ambiguity in the result for φs (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.

Experiment τ(Bs) ΔΓ φs A |A0|2 - |A|||2 Correlation Reference
D0 1.52 ± 0.06 ± 0.01 0.19 ± 0.07 +0.02 −0.01 −0.57 +0.24 −0.30 +0.07 −0.02 0.41 ± 0.04 +0.01 −0.02 0.34 ± 0.05 ± 0.03 (stat) arXiv:0802.2255


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

Bd 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. DCPπ0, 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 Bd → J/ψ KS. See e.g. Fleischer, NPB 659, 321 (2003).

Results of such an analysis are available from BaBar. The decays Bd → Dπ0, Bd → Dη, Bd → Dω, Bd → D*π0 and Bd → 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 KSω also used in Bd → D(*)π0 and the additional CP-odd D decay to KSπ0 also used in Bd → 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.

Mode Experiment −sin(2β) ≡ −sin(2φ1) CCP Correlation Reference
D(*)CP+ h0 BaBar
N(BB)=383M
−0.65 ± 0.26 ± 0.06 −0.33 ± 0.19 ± 0.04 0.04 (stat) PRL 99, 081801 (2007)
D(*)CP− h0 −0.46 ± 0.46 ± 0.13 −0.03 ± 0.28 ± 0.07 −0.14 (stat)
D(*) h0 −0.56 ± 0.23 ± 0.05 −0.23 ± 0.16 ± 0.04 −0.02 (stat)


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 → KSπ+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 Bd → Dπ0, Bd → Dη, Bd → Dω, Bd → D*π0 and Bd → D*η are used. The daughter decays are D* → Dπ0 and D → KSπ+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 = -ηCP × sin(2βeff) ≡ -ηCP × sin(2φ1eff) 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(2βeff) ≡ sin(2φ1eff) (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+KK0 and B0 → π+πKS are also discussed in the next section — results for φK0, ρ0KS and f0KS 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).

Mode Experiment sin(2βeff) ≡ sin(2φ1eff) CCP Correlation Reference
φK0 BaBar (*)
N(BB)=465M
0.26 ± 0.26 ± 0.03 −0.14 ± 0.19 ± 0.02 - arXiv:0808.0700
Belle (*)
N(BB)=657M
0.67 +0.22 −0.32 −0.31 +0.21 −0.23 ± 0.04 ± 0.09 - ICHEP 2008 preliminary
Average (*) 0.44 +0.17 −0.18 −0.23 ± 0.15 - HFAG
Figures:
eps.gz png eps.gz png .
η′K0 BaBar
N(BB)=467M
0.57 ± 0.08 ± 0.02 −0.08 ± 0.06 ± 0.02 0.03 (stat) arXiv:0809.1174
Belle
N(BB)=535M
0.64 ± 0.10 ± 0.04 0.01 ± 0.07 ± 0.05 0.09 (stat) PRL 98 (2007) 031802
Average 0.59 ± 0.07 −0.05 ± 0.05 0.04 HFAG correlated average
χ2 = 0.9/2 dof (CL=0.63 ⇒ 0.5σ)
Figures:
eps.gz png eps.gz png eps.gz png
KSKSKS BaBar
N(BB)=465M
0.90 +0.18 −0.20 +0.03 −0.04 −0.16 ± 0.17 ± 0.03 0.10 (stat) CKM2008 preliminary
Belle
N(BB)=535M
0.30 ± 0.32 ± 0.08 −0.31 ± 0.20 ± 0.07 - PRL 98 (2007) 031802
Average 0.74 ± 0.17 −0.23 ± 0.13 0.06 HFAG correlated average
χ2 = 2.7/2 dof (CL=0.26 ⇒ 1.1σ)
Figures:
eps.gz png eps.gz png eps.gz png
π0K0 BaBar
N(BB)=467M
0.55 ± 0.20 ± 0.03 0.13 ± 0.13 ± 0.03 0.06 (stat) arXiv:0809.1174
Belle
N(BB)=657M
0.67 ± 0.31 ± 0.08 −0.14 ± 0.13 ± 0.06 −0.04 (stat) arXiv:0809.4366
Average 0.57 ± 0.17 0.01 ± 0.10 0.02 HFAG correlated average
χ2 = 2.0/2 dof (CL=0.37 ⇒ 0.9σ)
Figures:
eps.gz png eps.gz png eps.gz png
ρ0KS BaBar (*)
N(BB)=383M
0.61 +0.22 −0.24 ± 0.09 ± 0.08 0.02 ± 0.27 ± 0.08 ± 0.06 - arXiv:0708.2097
Belle (*)
N(BB)=657M
0.64 +0.19 −0.25 ± 0.09 ± 0.10 −0.03 +0.24 −0.23 ± 0.11 ± 0.11 - arXiv:0811.3665
Average (*) 0.63 +0.17 −0.21 −0.01 ± 0.20 - HFAG
Figures:
eps.gz png eps.gz png .
ωKS BaBar
N(BB)=467M
0.55 +0.26 −0.29 ± 0.02 −0.52 +0.22 −0.20 ± 0.03 0.03 (stat) arXiv:0809.1174
Belle
N(BB)=535M
0.11 ± 0.46 ± 0.07 0.09 ± 0.29 ± 0.06 −0.04 (stat) PRD 76 (2007) 091103(R)
Average 0.45 ± 0.24 −0.32 ± 0.17 0.01 HFAG correlated average
χ2 = 3.4/2 dof (CL=0.18 ⇒ 1.3σ)
Figures:
eps.gz png eps.gz png .
f0K0 BaBar (**) 0.64 +0.15 −0.18 0.16 ± 0.19 - arXiv:0808.0700,
arXiv:0708.2097 (**)
Belle (**) 0.60 +0.16 −0.19 0.05 ± 0.18 - CKM2008 preliminary
Average 0.62 +0.11 −0.13 0.10 ± 0.13 - HFAG
Figures:
eps.gz png eps.gz png .
π0π0KS BaBar
N(BB)=227M
−0.72 ± 0.71 ± 0.08 0.23 ± 0.52 ± 0.13 −0.02 (stat) PRD 76 (2007) 071101
Belle
N(BB)=657M
−0.43 ± 0.49 ± 0.09 0.17 ± 0.24 ± 0.06 0.09 (stat) arXiv:0708.1845
Average −0.52 ± 0.41 0.18 ± 0.22 0.06 HFAG correlated average
χ2 = 0.1/2 dof (CL=0.94 ⇒ 0.1σ)
Figures:
eps.gz png eps.gz png eps.gz png
φ KS π0 BaBar (***)
N(BB)=465M
0.97 +0.03 −0.52 −0.20 ± 0.14 ± 0.06 - arXiv:0808.3586
K+KK0
(excluding φK0)
BaBar (*)
N(BB)=465M
0.86 ± 0.08 ± 0.03 −0.05 ± 0.09 ± 0.04 - arXiv:0808.0700
Belle
N(BB)=535M
0.68 ± 0.15 ± 0.03+0.21 −0.13CP-even
(fCP-even= 0.93 ± 0.09 ± 0.05 [SU(2)])
0.09 ± 0.10 ± 0.05 −0.00 (stat) PRD 76 (2007) 091103(R)
Average 0.82 ± 0.07
χ2 = 1.0 (CL=0.31 ⇒ 1.0σ)
0.01 ± 0.07
χ2 = 0.9 (CL=0.35 ⇒ 0.9σ)
uncorrelated averages HFAG
Figures:
eps.gz png eps.gz png .
Naïve b→s penguin average 0.64 ± 0.04
χ2 = 24/21 dof (CL=0.27 ⇒ 1.1σ)
−0.05 ± 0.03
χ2 = 19/21 dof (CL=0.58 ⇒ 0.6σ)
uncorrelated averages HFAG
eps.gz png eps.gz png
Direct comparison of charmonium and s-penguin averages (see comments below): χ2 = 0.51 (CL=0.47 ⇒ 0.7σ)

(*) BaBar and Belle results for φK0 and ρ0KS are determined from their time-dependent Dalitz plot analyses of B0 → K+KK0 and B0 → π+πKS, 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+KKS, 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.
The BaBar results for K+KK0 are for the inclusive "high mass" (mK+K− > 1.1 GeV/c2) region in the time-dependent Dalitz plot analysis. Belle results for K+KK0 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 → f0KS with f0 → π+π. Note that Q2B parameters extracted from Dalitz plot analyses are constrained to lie within the physical boundary (SCP2 + CCP2 < 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 → f0KS 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 KS π0 come from a simultaneous angular analysis of B → φ K+ π and B → φ KS 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 (SCP2 + CCP2 < 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(2βeff) in this result includes both statistical and systematic uncertainties.

Please note that



Compilation of results for −η×S ≈ sin(2βeff) ≡ sin(2φ1eff) and C from s-penguin decays.

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Same, but without π0π0KS and φ KS π0 to allow closer inspection of the detail.
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Comparisons of averages in the different b→q q-bar s modes

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Same, but without π0π0KS and φ KS π0 to allow closer inspection of the detail.
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2D comparisons of averages in the different b→q q-bar s modes.

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Time-dependent Dalitz plot analysis of Bd → K+KK0 and Bd → π+πK0

Time-dependent amplitude analyses of the three-body decays Bd → K+KK0 and Bd → π+πK0 allow additional information to be extracted from the data. In particular, the cosine of the effective weak phase difference (cos(2βeff) ≡ cos(2φ1eff)) 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+KK0 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 βeff ≡ φ1eff is directly determined for two significant resonant contributions: φK0 and f0K0, the inclusive "high mass" (mK+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 ACP ( = -CCP). Belle have also performed a time-dependent Dalitz plot analysis of B0 → K+KKS, using a model that includes contributions from f0(980)KS, φKS, fX(1500)KS, χc0KS and nonresonant terms (the fX(1500)KS, χc0KS and nonresonant terms are constrained to share common CP parameters).

Experiment K+KK0 (whole DP) Reference
βeff ACP
BaBar
N(BB)=465M
(25.3 ± 3.9 ± 0.9)° 0.03 ± 0.07 ± 0.02 arXiv:0808.0700
Experiment φK0 f0K0 K+KK0 (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:

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φK0eff) vs φK0(ACP): eps.gz png
f0K0eff) vs f0K0(ACP): eps.gz png
φK0eff) vs f0K0eff): 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.


Interpretations:

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

A time-dependent Dalitz plot analysis of B0 → π+πKS 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 βeff ≡ φ1eff is directly determined for two significant resonant contributions: f0KS and ρ0KS. A number of additional parameters, for example relating to the Q2B modes K*+π, are also extracted, but are not tabulated here. A similar analysis has been carried out by Belle.

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
(19 +10−9 ± 3 ± 3)° −0.02 ± 0.27 ± 0.08 ± 0.06 (44 +11−10 ± 3 ± 4)° −0.35 ± 0.27 ± 0.07 ± 0.04 - arXiv:0708.2097
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) arXiv:0811.3665
Average 19.5 ± 7.1
χ2 = 0.005 (CL=0.94 ⇒ 0.1σ)
0.01 ± 0.20
χ2 = 0.02 (CL=0.90 ⇒ 0.1σ)
23.6 ± 6.6
χ2 = 5.1 (CL=0.024 ⇒ 2.3σ)
−0.16 ± 0.17
χ2 = 0.7 (CL=0.41 ⇒ 0.8σ)
- HFAG
Figures:

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.

Since parameters related to the B0 → f0KS decay are obtained in both B0 → K+KK0 and B0 → π+πKS, we show compilations and naïve (uncorrelated) averages below.

Figures:
Naïve (uncorrelated) averages for f0KS parameters

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.


Time-dependent analysis of B → VV decays: B → φ KS π0

The final state in the decay B → φ KS 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 → KS π0)/B(K*0 → K+ π) is the same for each exited kaon state.

BaBar have performed a simultaneous analysis of B → φ KS 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(2βeff00) = 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(2βeff) 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 KS 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:

sin(2β − 2Δδ01) − sin(2β) = −0.42+0.26−0.34
sin(2β − 2Δφ||1) − sin(2β) = −0.32+0.22−0.30
sin(2β − 2Δφ⊥1) − sin(2β) = −0.30+0.23−0.32
sin(2β − 2Δφ⊥1) − sin(2β − 2Δφ||1) = 0.02 ± 0.23
sin(2β − 2Δδ02) − sin(2β) = −0.10+0.18−0.29

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 KS 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 Scc-bar d = − ηCP sin(2β) ≡ − ηCP sin(2φ1) and Ccc-bar d = 0 for the CP eigenstate final states (ηCP = +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)
Figures:

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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) arXiv:0808.1866
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:

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

The vector particles in the pseudoscalar to vector-vector decay Bd → 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 SCP). 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) arXiv:0808.1866
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:

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Experiment S+ (D*+ D*) C+ (D*+ D*) S (D*+ D*) C (D*+ D*) R (D*+ D*) Correlation Reference
BaBar
N(BB)=467M
−0.76 ± 0.16 ± 0.04 0.02 ± 0.12 ± 0.02 −1.81 ± 0.71 ± 0.16 0.41 ± 0.50 ± 0.08 0.15 ± 0.03 (stat) arXiv:0808.1866

(*) 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 RT = 0.125 ± 0.043 ± 0.023.)

Experiment S+−(D*+D) C+−(D*+D) S−+(D*D+) C−+(D*D+) A(D*+−D−+) Correlation Reference
BaBar
N(BB)=467M
−0.63 ± 0.21 ± 0.03 0.08 ± 0.17 ± 0.04 −0.74 ± 0.23 ± 0.05 0.00 ± 0.17 ± 0.03 0.01 ± 0.05 ± 0.01 (stat) arXiv:0808.1866
Belle
N(BB)=152M
(combined fully and partially rec. B decays)
−0.55 ± 0.39 ± 0.12 −0.37 ± 0.22 ± 0.06 −0.96 ± 0.43 ± 0.12 0.23 ± 0.25 ± 0.06 0.07 ± 0.08 ± 0.04 - PRL 93 (2004) 201802
Average −0.61 ± 0.19
χ2 = 0.03 (CL=0.86 ⇒ 0.2σ)
−0.09 ± 0.14
χ2 = 2.5 (CL=0.12 ⇒ 1.6σ)
−0.79 ± 0.21
χ2 = 0.2 (CL=0.66 ⇒ 0.4σ)
0.07 ± 0.14
χ2 = 0.6 (CL=0.46 ⇒ 0.7σ)
0.02 ± 0.04
χ2 = 0.4 (CL=0.54 ⇒ 0.6σ)
uncorrelated averages HFAG
Figures:
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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.

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Same, but with separate CP-even and CP-odd results from D*+D*
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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.)
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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) )

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2D comparisons of averages in the different b→c c-bar d modes.

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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:

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(*) Note that the BaBar result is outside of the physical region, as is the average. The interpretation of the results given above has to be done with the greatest care.



Time-dependent Analysis of b → sγ Transitions

Time-dependent analyses of radiative b decays such as B0→ KSπ0γ, 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 → KSπ0γ behaves like an effective flavor eigenstate, and mixing-induced CP violation is expected to be small - a simple estimation gives: S ~ −2(ms/mb)sin(2β) ≡ −2(ms/mb)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 KSπ0 can be performed, since the properties of the decay amplitudes are independent of the angular momentum of the KSπ0 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 KSπ0γ Dalitz plot, for example as a function of the KSπ0 invariant mass.

An inclusive KSπ0γ 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 "KSπ0γ (not K*(892)γ)" analysis.

We quote two averages: one for K*(892) only, and the other one for the inclusive KSπ0γ 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 KSπ0γ final state, BABAR has results using KSηγ, while Belle has results using KSρ0γ.

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) arXiv:0807.3103
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) arXiv:0807.3103
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) arXiv:0805.1317
KS ρ0 γ Belle
N(BB)=657M
0.11 ± 0.33 +0.05 −0.09 −0.05 ± 0.18 ± 0.06 0.04 (stat) arXiv:0806.1980
Figures:

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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 Bd–Bd-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.

Experiment SCP (b → dγ) CCP (b → dγ) Correlation Reference
Belle
N(BB)=657M
−0.83 ± 0.65 ± 0.18 0.44 ± 0.49 ± 0.14 −0.08 (stat) PRL 100 (2008) 021602



Time-dependent CP Asymmetries in Bd→ 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) CCP+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:

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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 Bd→ π+π decay. The relevant branching ratios (given in units of 10−6) and CP-violating charge asymmetries are taken from HFAG - Rare Decays.

BR(B0 → π+π) = 5.16 ± 0.22 -
BR(B+ → π+π0) = 5.59 +0.410.41 ACP(B+ → π+π0) = 0.06 ± 0.05
BR(B0 → π0π0) = 1.55 ± 0.19 ACP(B0 → π0π0) = 0.43 +0.25−0.24

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 Bd→ π+ππ0 (Bd→ ρ+−0π−+0)

Both BaBar and Belle have performed a full time-dependent Dalitz plot analyses of the decay Bd → (ρπ)0 → π+ππ0, 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.

[The table of averages of the form factor bilinears is suppressed here for the benefit of the nonspecialist. Those interested in the details can find them here.]
Compilation of averages of the form factor bilinears.

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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 ACP(ρπ) − is time-independent. The time-dependent decay rate is given by

Γ( B → ρ+−π−+ (Δt) ) = (1 +− ACP(ρπ)) e−|Δt|/τ/8τ × [1 + Qtag(Sρπ+−ΔSρπ)sin(ΔmΔt) − Qtag(Cρπ+−ΔCρπ)cos(ΔmΔt)],

where Qtag=+1(−1) when the tagging meson is a B0 (B0-bar). CP symmetry is violated if either one of the following conditions is true: ACP(ρπ)≠0, Cρπ≠0 or Sρπ≠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 Bd 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 ACP(ρπ) and Cρπ into the physically motivated choices
A+−(ρπ) = (|κ+−|2−1)/(|κ+−|2+1) = −(ACP(ρπ)+Cρπ+ACP(ρπ)ΔCρπ)/(1+ΔCρπ + ACP(ρπ)Cρπ),
A−+(ρπ) = (|κ−+|2−1)/(|κ−+|2+1) = (−ACP(ρπ)+Cρπ+ACP(ρπ)ΔCρπ)/(−1+ΔCρπ + ACP(ρπ)Cρπ),
where κ+− = (q/p)Abar−+/A+− and κ−+ = (q/p)Abar+−/A−+. With this definition A−+(ρπ) (A+−(ρπ)) describes CP violation in Bd 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 Bd→ ρ+−π−+ quasi-two-body contributions to the π+ππ0 final state, there can also be a Bd→ 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+−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:

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.
Experiment A−++−π−+) A+−+−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:

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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:

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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+ππ+, 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.

BR(B0 → ρ+−π−+) = 23.0 ± 2.3 -
BR(B+ → ρ+π0) = 10.9 +1.4−1.5 ACP(B+ → ρ+π0) = 0.02 ± 0.11
BR(B+ → π+ρ0) = 8.7 +1.0−1.1 ACP(B+ → π+ρ0) = −0.07 +0.12−0.13

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 Bd → ρρ (ρ+ρ and ρ0ρ0)

The vector particles in the pseudoscalar to vector-vector decay Bd → ρ+ρ can have longitudinal and transverse relative polarization with different CP properties. The decay is found to be dominated by the longitudinally polarized component:

At present we do not apply a rescaling of the results to a common, updated set of input parameters.
The CP parameters measured are those for the longitudinally polarized component (ie. Sρρ,long, Cρρ,long).

Experiment SCP+ρ) CCP+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:

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Since the decay Bd → 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

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. Sρρ,long, Cρρ,long).

Experiment SCP0ρ0) CCP0ρ0) Correlation Reference
BaBar
N(BB)=465M
0.3 ± 0.7 ± 0.2 0.2 ± 0.8 ± 0.3 0.0 (stat) arXiv:0807.4977

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 Bd→ ρ+ρ decay. The relevant branching ratios (given in units of 10−6) and CP-violating charge asymmetries are taken from HFAG - Rare Decays.

BR(B0 → ρ+ρ) = 24.2 +3.1−3.2
BR(B+ → ρ+ρ0) = 18.2 ± 3.0 ACP(B+ → ρ+ρ0) = −0.08 ± 0.13
BR(B0 → ρ0ρ0) = 0.68 ± 0.27

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 Bd → ρ0ρ0, a constraint of |α - αeff| < 15.7° (17.6°) is obtained at the 68% (90%) CL. The solution at α - αeff = +11.3° is preferred.

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 Bd → a1+−π−+

The BaBar collaboration have performed a Q2B analysis of the Bd → a1+−π−+ decay, reconstructed in the final state π+ππ+π.

Experiment ACP (a1+−π−+) C (a1+−π−+) S (a1+−π−+) ΔC (a1+−π−+) ΔS (a1+−π−+) Correlations Reference
BaBar
N(BB)=384M
−0.07 ± 0.07 ± 0.02 −0.10 ± 0.15 ± 0.09 0.37 ± 0.21 ± 0.07 0.26 ± 0.15 ± 0.07 −0.14 ± 0.21 ± 0.06 (stat) PRL 98 (2007) 181803
Experiment A−+ (a1+−π−+) A+− (a1+−π−+) Correlation Reference
BaBar
N(BB)=384M
0.07 ± 0.21 ± 0.15 0.15 ± 0.15 ± 0.07 0.63 PRL 98 (2007) 181803

Interpretations:
The parameter αeff ≡ φ2,eff, which reduces to α ≡ φ2 in the limit of no penguin contributions, can be extracted from the above results.

BaBar obtain αeff = (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 Bd → D+−π−+, Bd → D*+−π−+ and Bd → D+−ρ−+ provide sensitivity to γ ≡ φ3 because of the interference between the Cabibbo-favoured amplitude (e.g. B0 → Dπ+) 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 BdBd-bar mixing phase, the total phase difference is −(2β+γ) ≡ −(2φ13).

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 = |A(B0 → D+π)/A(B0 → Dπ+)|. Each of the ratios r, rD*π and r 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 → Ds+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:

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

Belle D*π (partial reconstruction): aπ* = − (S+ + S)/2
cπ* = − (S+ − S)/2
Belle D*π (full reconstruction): aπ* = + ( 2 RD*π sin( 2φ13 + δD*π ) + 2 RD*π sin( 2φ13 − δD*π ) )/2
cπ* = + ( 2 RD*π sin( 2φ13 + δD*π ) − 2 RD*π sin( 2φ13 − δD*π ) )/2
Belle Dπ (full reconstruction): aπ = − ( 2 R sin( 2φ13 + δ ) + 2 R sin( 2φ13 − δ ) )/2
cπ = − ( 2 R sin( 2φ13 + δ ) − 2 R sin( 2φ13 − δ ) )/2

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

Observable BABAR Belle Average Reference
partially
reconstructed
N(BB)=232m
fully
reconstructed
N(BB)=232m
partially
reconstructed
N(BB)=657m
fully
reconstructed
N(BB)=386m
aD*π −0.034 ± 0.014 ± 0.009 −0.040 ± 0.023 ± 0.010 −0.047 ± 0.014 ± 0.012 −0.039 ± 0.020 ± 0.013 −0.040 ± 0.010
CL=0.96 (0.0σ)
BaBar: PRD 71 (2005) 112003
(partially reco.)

BaBar: PRD 73 (2006) 111101
(fully reco.)


Belle: arXiv:0809.3203 [hep-ex]
(partially reco.)

Belle: PRD 73, 092003 (2006)
(fully reco.)
cD*π −0.019 ± 0.022 ± 0.013
(lepton tags only)
0.049 ± 0.042 ± 0.015
(lepton tags only)
−0.009 ± 0.014 ± 0.012 −0.011 ± 0.020 ± 0.013 −0.007 ± 0.012
CL=0.61 (0.5σ)
a - −0.010 ± 0.023 ± 0.007 - −0.050 ± 0.021 ± 0.012 −0.030 ± 0.017
CL=0.24 (1.2σ)
c - −0.033 ± 0.042 ± 0.012
(lepton tags only)
- −0.019 ± 0.021 ± 0.012 −0.022 ± 0.021
CL=0.78 (0.3σ)
a - −0.024 ± 0.031 ± 0.009 - - −0.024 ± 0.033
c - −0.098 ± 0.055 ± 0.018
(lepton tags only)
- - −0.098 ± 0.058

Compilation of the above results.

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Averages of the D*π results.

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Digression:

Constraining 2β+γ ≡ 2φ13:
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 ≡ 2φ13), of which two are different for each decay mode. Therefore, there is not enough information to solve directly for 2β+γ ≡ 2φ13. However, for each choice of R and 2β+γ ≡ 2φ13, 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 ≡ 2φ13. 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.


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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 ACP+− and RCP+−, where

ACP+− = [Γ(B → D(*)CP+−K(*)) − Γ(B+ → D(*)CP+−K(*)+)] / Sum ,
RCP+− = [Γ(B → D(*)CP+−K(*)) + Γ(B+ → D(*)CP+−K(*)+)] / [Γ(B → D(*)0 K(*)) + Γ(B+ → D(*)0-bar K(*)+)].

Experimentally, it is convenient to measure RCP+− 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.

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

Mode Experiment ACP+ ACP− RCP+ RCP− Reference
DCPK BaBar
N(BB)=382M
0.27 ± 0.09 ± 0.04 −0.09 ± 0.09 ± 0.02 1.06 ± 0.10 ± 0.05 1.03 ± 0.10 ± 0.05 PRD 77, 111102 (2008)
Belle
N(BB)=275M
0.06 ± 0.14 ± 0.05 −0.12 ± 0.14 ± 0.05 1.13 ± 0.16 ± 0.08 1.17 ± 0.14 ± 0.14 PRD 73, 051106 (2006)
CDF 0.39 ± 0.17 ± 0.04 - 1.30 ± 0.24 ± 0.12 - arXiv:0809.4809
Average 0.24 ± 0.07
χ2 = 2.3 (CL=0.32 ⇒ 1.0σ)
−0.10 ± 0.08
χ2 = 0.03 (CL=0.86 ⇒ 0.2σ)
1.10 ± 0.09
χ2 = 0.7 (CL=0.70 ⇒ 0.4σ)
1.06 ± 0.10
χ2 = 0.4 (CL=0.54 ⇒ 0.6σ)
HFAG
Figures:
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D*CPK BaBar
N(BB)=383M
−0.11 ± 0.09 ± 0.01 0.06 ± 0.10 ± 0.02 1.31 ± 0.13 ± 0.03 1.09 ± 0.12 ± 0.04 PRD 78, 092002 (2008)
Belle
N(BB)=275M
−0.20 ± 0.22 ± 0.04 0.13 ± 0.30 ± 0.08 1.41 ± 0.25 ± 0.06 1.15 ± 0.31 ± 0.12 PRD 73, 051106 (2006)
Average −0.12 ± 0.08
χ2 = 0.14 (CL=0.71 ⇒ 0.4σ)
0.07 ± 0.10
χ2 = 0.05 (CL=0.83 ⇒ 0.2σ)
1.33 ± 0.12
χ2 = 0.12 (CL=0.73 ⇒ 0.4σ)
1.10 ± 0.12
χ2 = 0.03 (CL=0.87 ⇒ 0.2σ)
HFAG
Figures:
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DCPK* BaBar
N(BB)=379M
0.09 ± 0.13 ± 0.05 −0.23 ± 0.21 ± 0.07 2.17 ± 0.35 ± 0.09 1.03 ± 0.27 ± 0.13 CKM2008 preliminary
Belle NO RESULTS AVAILABLE (*) -
Average 0.09 ± 0.14 −0.23 ± 0.22 2.17 ± 0.36 1.03 ± 0.30 HFAG
(*) We do not include a preliminary result from Belle on DCPK* (BELLE-CONF-0316) which is more than two years old.
Compilation of the above results.

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CP+ only

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CP- only

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Digression:

Constraining γ ≡ φ3: The rate ratios and asymmetries of the GLW method can be expressed in terms of amplitude ratios and strong phase differences, as well as the weak phase difference γ ≡ φ3. For the GLW observables, one has:
RCP+− = 1 + rB2 +− 2rBcos(δB)cos(γ) ≡ 1 + rB2 +− 2rBcos(δB)cos(φ3),
ACP+− = +− 2rBsin(δB)sin(γ) / RCP+− ≡ +− 2rBsin(δB)sin(φ3) / RCP+−,

where rB = |A(b→u)/A(b→c)| and δB = arg[A(b→u)/A(b→c)]. Only the weak phase difference γ ≡ φ3 is universal, while the other parameters depend on the decay process.

In addition, the Cartesian coordinates x± (discussed below in the context of analysis of B→DK with multibody D decay) can be extracted from the observables measured in GLW analysis. The relations are

x+− = (RCP+ (1−+ACP+) − RCP− (1−+ACP−))/4

There is no direct sensitivity to y+−, but indirect bounds can be obtained using

rB2 = x+−2 + y+−2 = (RCP+ + RCP−)/2
Plots upcoming.


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 AADS and RADS, where (for example for the B → DK, D → K+π mode)

AADS = [Γ(B → [K+π]DK) − Γ(B+ → [Kπ+]DK+)] / [Γ(B → [K+π]DK) + Γ(B+ → [Kπ+]DK+)] ,
RADS = [Γ(B → [K+π]DK) + Γ(B+ → [Kπ+]DK+)] / [Γ(B → [Kπ+]DK) + Γ(B+ → [K+π]DK+)] .

(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+ππ0.

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

Mode Experiment RADS AADS Reference
DK
D→Kπ
BaBar
N(BB)=232M
0.013 +0.011 −0.009 - PRD 72 (2005) 032004
Belle
N(BB)=657M
0.008 ± 0.006 +0.002 −0.003 −0.13 +0.97 −0.88 ± 0.26 PRD 78 (2008) 071901
Average 0.010 ± 0.005
χ2 = 0.19 (CL=0.67 ⇒ 0.4σ)
−0.13 +1.00 −0.92 HFAG
D*K
D* → Dπ0
D→Kπ
BaBar
N(BB)=232M
−0.002 +0.010 −0.006 - PRD 72 (2005) 032004
Average −0.002 +0.010 −0.006 - HFAG
D*K
D* → Dγ
D→Kπ
BaBar
N(BB)=232M
0.011 +0.018 −0.013 - PRD 72 (2005) 032004
Average 0.011 +0.018 −0.013 - HFAG
DK*
D→Kπ
BaBar
N(BB)=379M
0.066 ± 0.029 ± 0.008 −0.34 ± 0.45 ± 0.16 CKM2008 preliminary
Average 0.066 ± 0.031 −0.34 ± 0.48 HFAG
DK
D→Kππ0
BaBar
N(BB)=226M
0.012 ± 0.012 ± 0.009 - PRD 76 (2007) 111101
Average 0.012 ± 0.015 - HFAG
Compilation of the above results.

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Digression:

Constraining γ ≡ φ3: As for the GLW method, the rate ratios and asymmetries of the ADS method can be expressed in terms of amplitude ratios and strong phase differences, as well as the weak phase difference γ ≡ φ3. For the ADS observables, one has:

RADS = rB2 + rD2 + 2 rB rD cos(δBD) cos(γ) ≡ rB2 + rD2 + 2 rB rD cos(δBD) cos(φ3),
AADS = 2 rB rD sin(δBD) sin(γ) / RADS ≡ 2 rB rD sin(δBD) sin(φ3) / RADS,

where rB = |A(b→u)/A(b→c)| and δB = arg[A(b→u)/A(b→c)] as before. rD and δD are the corresponding amplitude ratio and strong phase difference of the D meson decay amplitudes. The value of rD2 is obtained from the ratio of the suppressed-to-allowed branching fractions BR(D0 → K+π) = (1.43 ± 0.04)×10−4 and BR(D0 → Kπ+) = (3.80 ± 0.07)×10−2 [PDG 2006], respectively. With this it is found rD = 0.0613 ± 0.0010.

The strong phase, δB, is different, in general, for decays to D and D* mesons. Bondar and Gershon have pointed out that there is an effective strong phase shift of π between the cases that D* is reconstructed in the Dπ0 and Dγ final states, which in principle allows γ ≡ φ3 to be measured using the ADS technique with B+− → D* K+− alone.

The situation for D→Kππ0 is slightly more complicated since the hadronic parameters can vary across the phase space (Dalitz plane). Effective hadronic parameters can be used, and eventually a Dalitz analysis (either binned or unbinned) may be possible to extract the maximum information from the decay.

As can be seen from the expressions above, the maximum size of the asymmetry, for given values of rB and rD is given by: AADS (max) = 2rBrD / (rB2+rD2). Thus, sizeable asymmetries may be found also for B → D(*)π decays, despite the expected smallness (~0.01) of rB for this case, providing sensitivity to γ ≡ φ3. The observables have been measured by Belle in the Dπ mode.

Mode Experiment RADS AADS Reference

D→Kπ
Belle
N(BB)=657M
0.0034 +0.0006 −0.0005 +0.0001 −0.0002 −0.023 ± 0.218 ± 0.071 PRD 78 (2008) 071901
Average 0.0034 ± 0.0006 −0.023 ± 0.229  


Dalitz Plot Analysis of B → D(*) K(*)− with D → KSπ+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 δB and the ratio of amplitudes rB. 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 → KSπ+π; BaBar also use D → KSK+K (though not for B → DK*).

For the DK* mode, both collaborations use K* → KSπ; in this case some care is needed due to other possible contributions to the B → DKSπ final state. Belle assign an additional (model) uncertainty, while BaBar use an alternative parametrization [replacing rB and δB with κrs and δs, respectively] suggested by Gronau.

If the values of γ ≡ φ3, δB and rB are obtained by directly fitting the data, the extracted value of rB is biased (since it is positive definite by nature). Since the error on γ ≡ φ3 depends on the value of rB 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:

x+ = rB cos( δB+γ ) ≡ rB cos( δB3 ) y+ = rB sin( δB+γ ) ≡ rB sin( δB3 )
x = rB cos( δB−γ ) ≡ rB cos( δB−φ3 ) y = rB sin( δB−γ ) ≡ rB sin( δB−φ3 )

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, δB and rB [BaBar do not obtain constraints on rB and δB 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 → DKSπ other than B → DK*, which we have not included here.

Averages are performed using the following procedure.

Mode Experiment x+ y+ x- y- Correlation Reference
DK BaBar
N(BB)=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.

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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.

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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.

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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.

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.
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.

Digression:

Constraining γ ≡ φ3:
The measurements of x+,− and y+,− in the various D(*)K(*) decay modes can be used to place bounds on γ ≡ φ3. Both experiments have done so using frequentist techniques.
BaBar obtain
γ = (76 ± 22 ± 5 ± 5)°
(from DK, D*K & DK*)
Belle obtain
φ3 = (76 +12−13 ± 4 ± 9)°
(from DK & D*K)
The experiments also obtain values for the hadronic parameters
rB (DK) = 0.086 ± 0.035 δB (DK) = (109 +28−31 rB (DK) = 0.16 ± 0.04 ± 0.01 ± 0.05 δB (DK) = (136 +14−16 ± 4 ± 23)°
rB (D*K) = 0.135 ± 0.051 δB (D*K) = (297 +30−28 rB (D*K) = 0.21 ± 0.08 ± 0.02 ± 0.05 δB (D*K) = (343 +20−22 ± 4 ± 23)°
κrs = 0.163 +0.088−0.105 δs = (104 +43−41 ( rB (DK*) = 0.56 +0.22−0.16 ± 0.04 ± 0.08 *) B (DK*) = (243+20−23 ± 3 ± 50 )° *)
For attempts to extract γ ≡ φ3 from the combined BaBar and Belle results, visit the CKMfitter and UTfit sites.
Note that the above results suffer an ambiguity: γ → γ + π ≡ φ3 → φ3 + π, δ → δ + π. We quote the result which is consistent with the Standard Model fit.


Dalitz Plot Analysis of B → D K with D → π+ππ0

BaBar have performed a similar Dalitz plot analysis using the decay D → π+ππ0. 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:

ρ± ≡ | z± - x0 | θ± ≡ tan− 1 (Im(z±) / (Re(z±) - x0))

where the constant x0 = 0.850 depends on the amplitude structure of the D → π+ππ0 decay, and z± = rB ei( δB ± γ ) ≡ rB ei( δB ± 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.

Mode Experiment ρ+ θ+ ρ θ Reference
DK
D→ π+ππ0
BaBar
N(BB)=324M
0.75 ± 0.11 ± 0.04 (147 ± 23 ± 1)° 0.72 ± 0.11 ± 0.04 (173 ± 42 ± 2)° PRL 99 (2007) 251801
Average 0.75 ± 0.12 (147 ± 23)° 0.72 ± 0.12 (173 ± 42)°  

Digression:

Constraining γ ≡ φ3:
The measurements of ρ+,− and θ+,− can be used to place bounds on γ ≡ φ3 and the hadronic parameters.
BaBar use a frequentist technique to obtain −30° < γ < 76°, 0.06 < rB (DK) < 0.78 and −27° < δ B (DK) < 78° at the 68% confidence level.


ADS Analyses of B0 → DK*0

BaBar have presented preliminary results on B0 → DK*0 with D → Kπ+, D → Kπ+ π0 and D → Kπ+ π+ π.
Details will be added here once public documentation is available.



Dalitz Plot Analysis of B0 → DK*0 with D → KSπ+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.

Constraining γ ≡ φ3:
BaBar extract the three-dimensional likelihood for the parameters (γ, δS, rS) and, combining with a separately measured PDF for rS (using a Bayesian technique), obtain bounds on each of the three parameters.
γ = (162 ± 56)°,   δS = (62 ± 57)°   rS < 0.55 at 95% probability
Note that there is an ambiguity in the solutions for γ and δS (γ, δS → γ+π, δS+π).




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