HFAG: CPV & Unitarity Triangle Parameters
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Results on Time-Dependent CP Violation, and Measurements Related to the Angles of the Unitarity Triangle:
Summer 2014 (FPCP 2014, France, ICHEP 2014, Spain, Beauty 2014, UK, CKM 2014, Austria, etc.)

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 Time-Dependent Interference Between b → cu-bar s and b-bar → u-bar cs-bar Transitions
§   Studies of Interference Between b → cu-bar s & b → uc-bar s Transitions (also covering interference between b → cu-bar d & b → uc-bar d 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) (1.519 ± 0.007) ps HFAG - Oscillations/Lifetime
Δmd (0.510 ± 0.004) ps−1 HFAG - Oscillations/Lifetime
ΔΓdd 0.015 ± 0.018 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
0.215 ± 0.032 ± 0.006 CDF: PRL 94 (2005) 101803 (*)
Ldt=0.3 fb−1
0.183 ± 0.013 ± 0.025 D0: PRL 102 (2009) 032001
Ldt=2.8 fb−1
0.201 ± 0.004 ± 0.008 LHCb: PRD 88 (2013) 052002
Ldt=1.0 fb−1
0.209 ± 0.006 Average
χ2 = 7.2/4 dof (CL=0.12 ⇒ 1.5σ)

(*) We do not include an unpublished CDF preliminary result from 2007.

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)=772M    
J/ψKSCP=-1) 0.657 ± 0.036 ± 0.012 0.670 ± 0.029 ± 0.013 0.665 ± 0.024
(0.023stat-only)
BaBar (PRD 79 (2009) 072009)
Belle (PRL 108 (2012) 171802)
J/ψKLCP=+1) 0.694 ± 0.061 ± 0.031 0.642 ± 0.047 ± 0.021 0.663 ± 0.041
(0.037stat-only)
J/ψK0 0.666 ± 0.031 ± 0.013 - 0.665 ± 0.022
(0.019stat-only)
ψ(2S)KSCP=-1) 0.897 ± 0.100 ± 0.036 0.738 ± 0.079 ± 0.036 0.807 ± 0.067
(0.062stat-only)
ψ(nS)K0 - - 0.676 ± 0.021
(0.018stat-only)
χc1KSCP=-1) 0.614 ± 0.160 ± 0.040 0.640 ± 0.117 ± 0.040 0.632 ± 0.099
(0.094stat-only)
ηcKSCP=-1) 0.925 ± 0.160 ± 0.057 - - BaBar (PRD 79 (2009) 072009)
J/ψK*0 (K*0 → KSπ0) CP= 1-2|A|2) 0.601 ± 0.239 ± 0.087 -
All charmonium 0.687 ± 0.028 ± 0.012 0.667 ± 0.023 ± 0.012 0.677 ± 0.020
(0.018stat-only)
CL = 0.57
χc0KSCP=+1) 0.69 ± 0.52 ± 0.04 ± 0.07 (*)
N(BB)=383M
- - BaBar (PRD 80 (2009) 112001)
J/ψKS, J/ψ → hadrons (ηCP=+1) 1.56 ± 0.42 ± 0.21 (**)
N(BB)=88M
- - BaBar (PRD 69 (2004) 052001)
All charmonium (incl. χc0KS etc.) 0.691 ± 0.031
(0.028stat-only)
0.667 ± 0.023 ± 0.012 0.679 ± 0.020
(0.018stat-only)
CL = 0.28

(*) The BABAR result on χc0KS comes from the time-dependent Dalitz plot analysis of B0 → π+πKS. The third uncertainty is due to the Dalitz model.

(**) BaBar (PRD 69 (2004) 052001) uses "hadronic and previously unused muonic decays of the J/ψ". We neglect a small possible correlation of this result with the main BaBar result that could be caused by reprocessing of the data.

Including earlier sin(2β) ≡ sin(2φ1) measurements, as well as a recent result from LHCb, using Bd → J/ψKS decays, and a measurement by Belle using Υ(5S) data and B-π tagging:

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
LHCb (1.0/fb) 0.73 ± 0.07 ± 0.04 PLB 721 (2013) 24
Belle (121/fb Υ(5S) data) 0.57 ± 0.58 ± 0.06 PRL 108 (2012) 171801

we find the only slightly modified average:

Parameter: sin(2β) ≡ sin(2φ1)
All charmonium 0.682 ± 0.019 (0.017stat-only) CL = 0.25

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

β ≡ φ1 = (21.5 +0.8−0.7 or β ≡ φ1 = (68.5 +0.7−0.8

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.

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eps.gz png
Constraint on the ρ-bar-η-bar plane:

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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)=772M    
J/ψKS 0.026 ± 0.025 ± 0.016 0.015 ± 0.021 +0.023−0.045 0.024 ± 0.026
(0.016stat-only)
BaBar (PRD 79 (2009) 072009)
Belle (PRL 108 (2012) 171802)
J/ψKL −0.033 ± 0.050 ± 0.027 −0.019 ± 0.026 +0.041−0.017 −0.023 ± 0.030
(0.023stat-only)
J/ψK0 0.016 ± 0.023 ± 0.018 - 0.006 ± 0.021
(0.013stat-only)
ψ(2S)KS 0.089 ± 0.076 ± 0.020 −0.104 ± 0.055 +0.027−0.047 −0.009 ± 0.055
(0.045stat-only)
ψ(nS)K0 - - 0.005 ± 0.020
(0.013stat-only)
χc1KS 0.129 ± 0.109 ± 0.025 0.017 ± 0.083 +0.026−0.046 0.066 ± 0.074
(0.066stat-only)
ηcKS 0.080 ± 0.124 ± 0.029 - - BaBar (PRD 79 (2009) 072009)
J/ψK*0 (K*0 → KSπ0) 0.025 ± 0.083 ± 0.054 -
All charmonium 0.024 ± 0.020 ± 0.016 −0.006 ± 0.016 ± 0.012 0.006 ± 0.017
(0.012stat-only)
CL = 0.29
χc0KSCP=+1) −0.29 +0.53−0.44 ± 0.03 ± 0.05 (*) - - BaBar (PRD 80 (2009) 112001)
All charmonium (incl. χc0KS) 0.023 ± 0.025
(0.020stat-only)
−0.006 ± 0.016 ± 0.012 0.005 ± 0.017
(0.012stat-only)
CL = 0.47

(*) The BABAR result on χc0KS comes from the time-dependent Dalitz plot analysis of B0 → π+πKS. The third uncertainty is due to the Dalitz model.

Including a recent result from LHCb, using Bd → J/ψKS decays:

Parameter: sin(2β) ≡ sin(2φ1)
Experiment Value Reference
LHCb (1.0/fb) 0.03 ± 0.09 ± 0.01 PLB 721 (2013) 24

The statistical correlation between the LHCb results for S and C is 0.42.

we find an average that is unchanged within the rounding:

Parameter: C(b → c c-bar s)
All charmonium 0.005 ± 0.017


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

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.

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 (φs)

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, based on flavour-tagged analyses of Bs → J/ψ φ decays, have been performed by ATLAS, CMS, CDF, D0 and LHCb. LHCb have in addition performed measurements of φs from Bs → J/ψ π+π and Bs → Ds+Ds.

Averaging of the above results is being carried out by the HFAG lifetimes and oscillation group. Measurements related to φs in Bs → K+K and Bs → φφ decays are reported below.



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)=470M
0.66 ± 0.17 ± 0.07 0.05 ± 0.18 ± 0.05 - PRD 85 (2012) 112010
Belle (*)
N(BB)=657M
0.90 +0.09 −0.19 −0.04 ± 0.20 ± 0.10 ± 0.02 - PRD 82 (2010) 073011
Average (*) 0.74 +0.11 −0.13 0.01 ± 0.14 - 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) PRD 79 (2009) 052003
Belle
N(BB)=772M
0.68 ± 0.07 ± 0.03 −0.03 ± 0.05 ± 0.03 0.03 (stat) arXiv:1408.5991
Average 0.63 ± 0.06 −0.05 ± 0.04 0.02 HFAG correlated average
χ2 = 1.3/2 dof (CL=0.53 ⇒ 0.6σ)
Figures:
eps.gz png eps.gz png eps.gz png
KSKSKS BaBar
N(BB)=468M
0.94 +0.21 −0.24 ± 0.06 −0.17 ± 0.18 ± 0.04 0.16 (stat) PRD 85 (2012) 054023
Belle
N(BB)=535M
0.30 ± 0.32 ± 0.08 −0.31 ± 0.20 ± 0.07 - PRL 98 (2007) 031802
Average 0.72 ± 0.19 −0.24 ± 0.14 0.09 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) PRD 79 (2009) 052003
Belle
N(BB)=657M
0.67 ± 0.31 ± 0.08 −0.14 ± 0.13 ± 0.06 −0.04 (stat) PRD 81 (2010) 011101
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
0.35 +0.26 −0.31 ± 0.06 ± 0.03 −0.05 ± 0.26 ± 0.10 ± 0.03 - PRD 80 (2009) 112001
Belle (*)
N(BB)=657M
0.64 +0.19 −0.25 ± 0.09 ± 0.10 −0.03 +0.24 −0.23 ± 0.11 ± 0.10 - PRD 79 (2009) 072004
Average (*) 0.54 +0.18 −0.21 −0.06 ± 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) PRD 79 (2009) 052003
Belle
N(BB)=772M
0.91 ± 0.32 ± 0.05 0.36 ± 0.19 ± 0.05 −0.00 (stat) PRD 90 (2014) 012002
Average 0.71 ± 0.21 −0.04 ± 0.14 0.01 HFAG correlated average
χ2 = 9.9/2 dof (CL=0.007 ⇒ 2.7σ)
Figures:
eps.gz png eps.gz png .
f0K0 BaBar (**) 0.74 +0.12 −0.15 0.15 ± 0.16 - HFAG (**)
Belle (**) 0.63 +0.16 −0.19 0.13 ± 0.17 - HFAG (**)
Average 0.69 +0.10 −0.12 0.14 ± 0.12 - HFAG
Figures:
eps.gz png eps.gz png .
f2KS BaBar (*)
N(BB)=383M
0.48 ± 0.52 ± 0.06 ± 0.10 0.28 +0.35 −0.40 ± 0.08 ± 0.07 0.01 (stat) PRD 80 (2009) 112001
fXKS BaBar (*)
N(BB)=383M
0.20 ± 0.52 ± 0.07 ± 0.07 0.13 +0.33 −0.35 ± 0.04 ± 0.09 0.29 (stat) PRD 80 (2009) 112001
π0π0KS (****) BaBar
N(BB)=227M
−0.72 ± 0.71 ± 0.08 0.23 ± 0.52 ± 0.13 −0.02 (stat) PRD 76 (2007) 071101
φ KS π0 BaBar (***)
N(BB)=465M
0.97 +0.03 −0.52 −0.20 ± 0.14 ± 0.06 - PRD 78 (2008) 092008
π+ π KS nonresonant BaBar (*)
N(BB)=383M
0.01 ± 0.31 ± 0.05 ± 0.09 0.01 ± 0.25 ± 0.06 ± 0.05 −0.11 (stat) PRD 80 (2009) 112001
K+KK0
(excluding φK0 and f0K0)
BaBar (*)
N(BB)=470M
0.65 ± 0.12 ± 0.03 0.02 ± 0.09 ± 0.03 - PRD 85 (2012) 112010
Belle (*)
N(BB)=657M
0.76 +0.14 −0.18 0.14 ± 0.11 ± 0.08 ± 0.03 - PRD 82 (2010) 073011
Average 0.68 +0.09 −0.10 0.06 ± 0.08
- HFAG
Figures:
eps.gz png eps.gz png .
Naïve b→s penguin average 0.655 ± 0.032
χ2 = 19/24 dof (CL=0.77 ⇒ 0.3σ)
−0.006 ± 0.026
χ2 = 23/24 dof (CL=0.53 ⇒ 0.6σ)
uncorrelated averages HFAG
eps.gz png eps.gz png
Direct comparison of charmonium and s-penguin averages (see comments below): χ2 = 0.5 (CL=0.47 ⇒ 0.7σ)

(*) BaBar and Belle results for φK0, ρ0KS and K+KK0 (excluding φK0 and f0K0) are determined from their time-dependent Dalitz plot analyses of B0 → K+KK0 and B0 → π+πKS. 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.
BaBar results for f2KS, fXKS and π+ π KS nonresonant are determined from their time-dependent Dalitz plot analysis of B0 → π+πKS.

(**) 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.

(****) We do not include a preliminary result from Belle on π0π0KS that remains unpublished after more than two years.

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 f2KS, fXKS π0π0KS, π+ π KS nonresonant 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 f2KS, fXKS π0π0KS, π+ π KS nonresonant 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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Same, but without f2KS, fXKS π0π0KS, π+ π KS nonresonant and φ KS π0 to allow closer inspection of the detail.
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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.

Time-dependent Dalitz plot analyses of B0 → K+KKS have been performed by BaBar and Belle. As given above, parameters can be extracted in a form that allows a straightforward comparison/combination with those from time-dependent CP asymmetries in quasi-two-body b → qq-bar s modes. In addition, the effective weak phase βeff ≡ φ1eff is directly determined for two significant resonant contributions: φK0 and f0K0 and for the rest of the charmless contributions to the Dalitz plot combined, with the CP properties of the individual components taken into account.

Experiment φKS f0KS other K+KKS Correlation Reference
βeff ≡ φ1eff ACP βeff ≡ φ1eff ACP βeff ≡ φ1eff ACP
BaBar (*)
N(BB)=470M
(21 ± 6 ± 2)° −0.05 ± 0.18 ± 0.05 (18 ± 6 ± 4)° −0.28 ± 0.24 ± 0.09 (20.3 ± 4.3 ± 1.2)° −0.02 ± 0.09 ± 0.03 (stat) PRD 85 (2012) 112010
Belle (**)
N(BB)=657M
(32.2 ± 9.0 ± 2.6 ± 1.4)° 0.04 ± 0.20 ± 0.10 ± 0.02 (31.3 ± 9.0 ± 3.4 ± 4.0)° −0.30 ± 0.29 ± 0.11 ± 0.09 (24.9 ± 6.4 ± 2.1 ± 2.5)° −0.14 ± 0.11 ± 0.08 ± 0.03 (stat) PRD 82 (2010) 073011
Average (24 ± 5)° −0.01 ± 0.14 (22 ± 6)° −0.29 ± 0.20 (21.6 ± 3.7)° −0.06 ± 0.08 (stat) HFAG correlated average
χ2 = 1.8/6 dof (CL=0.93 ⇒ 0.1σ)
Figures:

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(*) The BaBar fit to K+KKS finds a global minimum and four other local minima with -2 ln L values within 9 units of the global minimum (the closest is separated by 3.9 units). Values of the CP violation parameters are only quoted for the global minimum.

(**) The Belle fit to K+KKS results in a four-fold ambiguity in the solution, all with consistent values of the CP parameters. The results quoted here correspond to solution 1 presented in the paper (which is preferred based on comparisons of the relative branching fractions of the scalar resonances to π+π and K+K to external measurements). The third source of uncertainty arises due to the composition of the Dalitz plot.


Interpretations:

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


Time-dependent Dalitz plot analyses of B0 → π+πKS have been performed by BaBar and Belle. As given above, parameters can be extracted in a form that allows a straightforward comparison/combination with those from time-dependent CP asymmetries in quasi-two-body b → qq-bar s modes. In addition, the effective weak phase βeff ≡ φ1eff is directly determined for two significant resonant contributions: f0KS and ρ0KS by both experiments. Both experiments find multiple solutions in the fits; in both cases we quote the results given as solution 1. BaBar also report parameters related to the intermediate states f2(1270)KS, fX(1300)KS, nonresonant π+πKS and χc0KS (see b → cc-bar s modes above). A number of additional parameters, for example relating to the Q2B modes K*+π, are also extracted, but are not tabulated here.

The third error in the results given below is due to Dalitz model uncertainty.

Experiment ρ0KS f0KS Correlation Reference
βeff ≡ φ1eff ACP βeff ≡ φ1eff ACP
BaBar (*)
N(BB)=383M
(10.2 ± 8.9 ± 3.0 ± 1.9)° 0.05 ± 0.26 ± 0.10 ± 0.03 (36.0 ± 9.8 ± 2.1 ± 2.1)° −0.08 ± 0.19 ± 0.03 ± 0.04 (stat) PRD 80 (2009) 112001
Belle (*)
N(BB)=657M
(20.0 +8.6 −8.5 ± 3.2 ± 3.5)° 0.03 +0.23 −0.24 ± 0.11 ± 0.10 (12.7 +6.9 −6.5 ± 2.8 ± 3.3)° −0.06 ± 0.17 ± 0.07 ± 0.09 (stat) PRD 79 (2009) 072004
Average (16.4 ± 6.8)° 0.06 ± 0.20 (20.6 ± 6.2)° −0.07 ± 0.14 (stat) HFAG correlated average
χ2 = 4.1/4 dof (CL=0.39 ⇒ 0.9σ)
Figures:

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Experiment f2KS fXKS Nonresonant χc0KS Correlation Reference
βeff ≡ φ1eff ACP βeff ≡ φ1eff ACP βeff ≡ φ1eff ACP βeff ≡ φ1eff ACP
BaBar (*)
N(BB)=383M
(14.9 ± 17.9 ± 3.1 ± 5.2)° −0.28 +0.40 −0.35 ± 0.08 ± 0.07 (5.8 ± 15.2 ± 2.2 ± 2.3)° −0.13 +0.35 −0.33 ± 0.04 ± 0.09 (0.4 ± 8.8 ± 1.9 ± 3.8)° −0.01 ± 0.25 ± 0.06 ± 0.05 (23.2 ± 22.4 ± 2.3 ± 4.2)° 0.29 +0.44 −0.53 ± 0.03 ± 0.05 (stat) PRD 80 (2009) 112001

(*) Both experiments suffer from ambiguities in the solutions. The results quoted here correspond to solution 1 presented in the papers.


Since parameters related to the B0 → 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 via b → qq-bar s (penguin) Transitions: 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 Bs decays via b → qq-bar s (penguin) Transitions (eg. Bs → K+K)

The decay Bs → K+K involves a b → uu-bar s transition, and hence has both penguin and tree contributions. Both mixing-induced and direct CP violation effects may arise, and additional input is needed to disentangle the contributions and determine γ and βseff. For example, the observables in Bd → π+π can be related using U-spin, as proposed by Fleischer.

The observables are SCP, CCP, and AΔΓ. The alternative notations Amix = SCP and Adir = −CCP are sometimes found in the literature. All three observables can be treated as free parameters, but they are physically constrained to satisfy SCP2 + CCP2 + AΔΓ2 = 1. Note that the untagged decay distribution, from which an "effective lifetime" can be measured, retains sensitivity to AΔΓ. Averages of effective lifetimes are performed by the HFAG lifetimes and oscillation group.

The observables have been measured by LHCb, who impose the constraint mentioned above to eliminate AΔΓ.

Experiment S C Correlation Reference
LHCb
Ldt=1.0 fb−1
0.30 ± 0.12 ± 0.04 0.14 ± 0.11 ± 0.03 0.02 (stat) JHEP 1310 (2013) 183


Time-Dependent CP Asymmetries in Bs → VV decays via b → qq-bar s (penguin) Transitions (eg. Bs → φφ)

The decay Bs → φφ involves a b → ss-bar s transition, and hence is a "pure penguin" mode. Since the mixing phase and the decay phase are expected to cancel in the Standard Model, the prediction for the phase from the interference of mixing and decay is predicted to be φs(φφ) = 0 with low uncertainty. Due to the vector-vector nature of the final state, angular analysis is needed to separate the CP-even and CP-odd contributions. Such an analysis also makes it possible to fit directly for φs(φφ).

A constraint on φs(φφ) has been obtained by LHCb using ∫Ldt=3.0 fb−1 (arXiv:1407.2222 [hep-ex]), who measure φs(φφ) = −0.17 ± 0.15 (stat) ± 0.03 (syst) rad.



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). An alternative notation, S = (S+ + S)/2, Δ S = (S+ −S)/2, C = (C+ + C)/2, Δ C = (C+ −C)/2, has been used in recent publications.

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



Experiment SCP (D+D) CCP (D+D) Correlation Reference
BaBar
N(BB)=467M
−0.63 ± 0.36 ± 0.05 −0.07 ± 0.23 ± 0.03 −0.01 (stat) PRD 79, 032002 (2009)
Belle
N(BB)=772M
−1.06 +0.21 −0.14 ± 0.08 −0.43 ± 0.16 ± 0.05 −0.12 (stat) PRD 85 (2012) 091106
Average −0.98 ± 0.17 −0.31 ± 0.14 −0.08 HFAG correlated average
χ2 = 2.8/2 dof (CL=0.25 ⇒ 1.2σ)
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. (In the most recent update Belle include R and also R0 as free parameters in the fit. We do not include information on R0 in the average for now.) BaBar treat R as a free parameter in the fit and consequently this systematic is absorbed in the statistical error. We perform the average taking into account correlations of the CP parameters with each other as well as with R.

Belle have performed a fit to the data assuming that the CP parameters for CP-even and CP-odd transversity states are the same (up to a trivial change of sign for 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. We also include the results of a separate analysis from BaBar based on partially reconstructed D*D* decays; in this analysis the BaBar measurement of R is used to correct the value of S fitted without separating CP-even and -odd components for the CP-odd dilution.

Experiment SCP (D*+ D*) CCP (D*+ D*) R (D*+ D*) Correlation Reference
BaBar
N(BB)=467M
−0.70 ± 0.16 ± 0.03 0.05 ± 0.09 ± 0.02 0.17 ± 0.03 (stat) PRD 79, 032002 (2009)
BaBar part. rec.
N(BB)=471M
−0.49 ± 0.18 ± 0.07 ± 0.04 0.15 ± 0.09 ± 0.04 - (stat) PRD 86 (2012) 112006
Belle
N(BB)=772M
−0.79 ± 0.13 ± 0.03 −0.15 ± 0.08 ± 0.02 0.14 ± 0.02 ± 0.01 (stat) PRD 86 (2012) 071103(R)
Average −0.71 ± 0.09 −0.01 ± 0.05 0.15 ± 0.02 (stat) HFAG correlated average
χ2 = 3.7/6 dof (CL=0.72 ⇒ 0.4σ)
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) PRD 79, 032002 (2009)

(*) 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.

Experiment S(D*+D) C(D*+D) ΔS(D*D+) ΔC(D*D+) A(D*+−D−+) Reference
BaBar
N(BB)=467M
−0.68 ± 0.15 ± 0.04 0.04 ± 0.12 ± 0.03 0.05 ± 0.15 ± 0.02 0.04 ± 0.12 ± 0.03 0.01 ± 0.05 ± 0.01 PRD 79, 032002 (2009)
Belle
N(BB)=772M
−0.78 ± 0.15 ± 0.05 −0.01 ± 0.11 ± 0.04 −0.13 ± 0.15 ± 0.04 0.12 ± 0.11 ± 0.03 0.06 ± 0.05 ± 0.02 PRD 85 (2012) 091106
Average −0.73 ± 0.11
χ2 = 0.20 (CL=0.65 ⇒ 0.5σ)
0.01 ± 0.09
χ2 = 0.1 (CL=0.77 ⇒ 0.3σ)
−0.04 ± 0.11
χ2 = 0.7 (CL=0.41 ⇒ 0.8σ)
0.08 ± 0.08
χ2 = 0.2 (CL=0.63 ⇒ 0.5σ)
0.03 ± 0.04
χ2 = 0.5 (CL=0.48 ⇒ 0.7σ)
HFAG
uncorrelated averages
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) = 0.79 ± 0.07 C ≡ −A = −0.05 ± 0.05
( χ2 = 8.9/6 dof (CL=0.18 ⇒ 1.3σ) ) ( χ2 = 13/6 dof (CL=0.05, ⇒ 2.0σ) )

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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, both BaBar and Belle have results using KSηγ and KSρ0γ (see footnote below table), while Belle also has results using KSφγ.

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

Mode Experiment SCP (b → sγ) CCP (b → sγ) Correlation Reference
K*(892)γ BaBar
N(BB)=467M
−0.03 ± 0.29 ± 0.03 −0.14 ± 0.16 ± 0.03 0.05 (stat) PRD 78 (2008) 071102
Belle
N(BB)=535M
−0.32 +0.36 −0.33 ± 0.05 0.20 ± 0.24 ± 0.05 0.08 (stat) PRD 74 (2006) 111104
Average −0.16 ± 0.22 −0.04 ± 0.14 0.06 HFAG correlated average
χ2 = 1.9/2 dof (CL=0.40 ⇒ 0.9σ)
KSπ0γ
(incl. K*γ)
BaBar
N(BB)=467M
−0.17 ± 0.26 ± 0.03 −0.19 ± 0.14 ± 0.03 0.04 (stat) PRD 78 (2008) 071102
Belle
N(BB)=535M
−0.10 ± 0.31 ± 0.07 0.20 ± 0.20 ± 0.06 0.08 (stat) PRD 74 (2006) 111104(R)
Average −0.15 ± 0.20 −0.07 ± 0.12 0.05 HFAG correlated average
χ2 = 2.4/2 dof (CL=0.30 ⇒ 1.0σ)
KS η γ BaBar
N(BB)=465M
−0.18 +0.49 −0.46 ± 0.12 −0.32 +0.40 −0.39 ± 0.07 −0.17 (stat) PRD 79 (2009) 011102
Belle
N(BB)=772M
−1.32 ± 0.77 ± 0.36 0.48 ± 0.41 ± 0.07 −0.14 (stat) LLWI 2014 preliminary
Average −0.49 ± 0.42 0.06 ± 0.29 −0.15 HFAG correlated average
χ2 = 2.9/2 dof (CL=0.24 ⇒ 1.2σ)
KS ρ0 γ * BaBar
N(BB)=471M
0.25 ± 0.46 +0.08 −0.06 −0.39 ± 0.20 ± 0.05 −0.09 (stat) BEACH 2014 preliminary
Belle
N(BB)=657M
0.11 ± 0.33 +0.05 −0.09 −0.05 ± 0.18 ± 0.06 0.04 (stat) PRL 101 (2008) 251601
Average 0.14 ± 0.27 −0.20 ± 0.14 −0.01 HFAG correlated average
χ2 = 1.5/2 dof (CL=0.47 ⇒ 0.7σ)
KS φ γ Belle
N(BB)=772M
0.74 +0.72 −1.05 +0.10 −0.24 −0.35 ± 0.58 +0.10 −0.23 - PRD 84 (2011) 071101
Figures:

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* Due to the non-negligible width of the ρ0 meson, decays selected as B0 → KSρ0γ can include a significant contribution from K*+−π−+γ decays, which are flavour-specific and do not have the same oscillation phenomenology. Both BaBar and Belle measure Seff for all B decay candidates with the ρ0 selection being 0.6 < m(π+π) < 0.9 GeV/c2, obtaining 0.14 ± 0.25 +0.04−0.03 (BaBar) and 0.09 ± 0.27 +0.04−0.07 (Belle). These values are then corrected for a "dilution factor", that is evaluated with different methods in the two experiments: BaBar obtain 0.549 +0.096−0.094 while Belle obtain 0.83 +0.19−0.03. Until the discrepancy between these values is understood, the average of the results should be treated with caution.



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

The observables have been measured by BaBar, Belle & LHCb. Note that at the B factories the observables are in principle uncorrelated (due to the fact that the time variable, Δt, has the range −∞ < Δt < +∞ – small correlations can be induced e.g.by backgrounds), at hadron colliders a non-zero correlation is expected (the time variable t takes the range 0 < t < +∞). Please note that at present we do not apply a rescaling of the results to a common, updated set of input parameters. Correlation due to common systematics are neglected in the following averages.

Experiment SCP+π) CCP+π) Correlation Reference
BaBar
N(BB)=467M
−0.68 ± 0.10 ± 0.03 −0.25 ± 0.08 ± 0.02 −0.06 (stat) PRD 87 (2013) 052009
Belle
N(BB)=772M
−0.64 ± 0.08 ± 0.03 −0.33 ± 0.06 ± 0.03 −0.10 (stat) PRD 88 (2013) 092003
LHCb
Ldt=1.0 fb−1
−0.71 ± 0.13 ± 0.02 −0.38 ± 0.15 ± 0.02 0.38 (stat) JHEP 1310 (2013) 183
Average −0.66 ± 0.06 −0.31 ± 0.05 0.00 HFAG correlated average
χ2 = 0.9/4 dof (CL=0.92 ⇒ 0.1σ)
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 isospin analysis uses as input the branching fractions and CP-violating charge asymmetries of all three ππ decay modes (π+π, ππ0, π0π0). (Constraints on α ≡ φ2 can be obtained without information on S(π0π0), which has not yet been measured.)

Both Belle and BaBar give confidence level interpretations for α ≡ φ2.
Belle exclude the range 23.8° < φ2 < 66.8° at 1σ level.
BaBar state that the true value lies in the range 71° < α < 109° at the 68% confidence level (considering the solution consistent with the Standard Model).

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

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



Time-dependent CP Asymmetries in 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.
The data in these plots is taken from (BaBar) PRD 88 (2013) 012003 and (Belle) PRL 98 (2007) 221602.

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From the bilinear coefficients given above, both experiments extract "quasi-two-body" (Q2B) parameters. Considering only the charged ρ bands in the Dalitz plot, the Q2B analysis involves 5 different parameters, one of which − the charge asymmetry 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 PRD 77 (2008) 072001.

Experiment ACP+−π−+) C (ρ+−π−+) S (ρ+−π−+) ΔC (ρ+−π−+) ΔS (ρ+−π−+) Correlations Reference
BaBar
N(BB)=471M
−0.10 ± 0.03 ± 0.02 0.02 ± 0.06 ± 0.04 0.05 ± 0.08 ± 0.03 0.23 ± 0.06 ± 0.05 0.05 ± 0.08 ± 0.04 (stat) PRD 88 (2013) 012003
Belle
N(BB)=449M
−0.12 ± 0.05 ± 0.04 −0.13 ± 0.09 ± 0.05 0.06 ± 0.13 ± 0.05 0.36 ± 0.10 ± 0.05 −0.08 ± 0.13 ± 0.05 (stat) PRL 98 (2007) 221602
Average −0.11 ± 0.03 −0.03 ± 0.06 0.06 ± 0.07 0.27 ± 0.06 0.01 ± 0.08 (stat) HFAG correlated average
χ2 = 3.5/5 dof (CL=0.63 ⇒ 0.5σ)
Figures:

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.
Experiment A−++−π−+) A+−+−π−+) Correlation Reference
BaBar
N(BB)=471M
−0.12 ± 0.08 +0.04 −0.05 0.09 +0.05 −0.06 ± 0.04 0.55 PRD 88 (2013) 012003
Belle
N(BB)=449M
0.08 ± 0.16 ± 0.11 0.21 ± 0.08 ± 0.04 0.47 PRL 98 (2007) 221602
Average −0.08 ± 0.08 0.13 ± 0.05 0.37 HFAG correlated average
χ2 = 1.5/2 dof (CL=0.47 ⇒ 0.7σ)
Figures:

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Experiment C (ρ0π0) S (ρ0π0) Correlation Reference
BaBar
N(BB)=471M
0.19 ± 0.23 ± 0.15 −0.37 ± 0.34 ± 0.20 0.00 PRD 88 (2013) 012003
Belle
N(BB)=449M
0.49 ± 0.36 ± 0.28 0.17 ± 0.57 ± 0.35 0.08 PRL 98 (2007) 221602
Average 0.27 ± 0.24 −0.23 ± 0.34 0.02 HFAG correlated average
χ2 = 0.8/2 dof (CL=0.68 ⇒ 0.4σ)
Figures:

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Interpretations:
The information given above can be used to extract α ≡ φ2.
A confidence level interpretation for α can be obtained by scanning over the measured form factor bilinears. In addition, information from the B → ρπ SU(2) partners can be included via an isospin pentagon relation. The isospin analysis uses as input the branching fractions and CP-violating charge asymmetries of all five ρπ decay modes (ρ+ππ+, ρ0π0, ρ+π0, ρ0π+). With all information in the ρπ channels put together, Belle obtain the constraint 68° < φ2 < 95° at 68% confidence level, for the solution consistent with the Standard Model. BaBar present a scan, but not an interval, for α, since their studies indicate that the scan is not statistically robust and cannot be interpreted as 1-CL.

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

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



Time-dependent CP Asymmetries in 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 B0 → ρ0ρ0 results in an all charged particle final state, its time-dependent CP violation parameters can be determined experimentally, if difficulties related to the small branching fraction and large backgrounds can be overcome. BaBar measure the longitudinally polarized component to be

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

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) PRD 78 (2008) 071104(R)

Interpretations:
The Gronau-London isospin analysis allows a constraint on α ≡ φ2 to be extracted from the ρρ system even in the presence of non-negligible penguin contributions. The isospin analysis uses as input the branching fractions and CP-violating charge asymmetries of the longitudinal components of all three ρρ decay modes (ρ+ρ, ρρ0, ρ0ρ0). (A similar analysis could be done for each polarisation amplitude, but the others are found to not be statisticall significant.)

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

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
Belle
N(BB)=772M
−0.06 ± 0.05 ± 0.07 −0.01 ± 0.11 ± 0.09 −0.51 ± 0.14 ± 0.08 0.54 ± 0.11 ± 0.07 −0.09 ± 0.14 ± 0.06 (stat) PRD 86 (2012) 092012
Average −0.06 ± 0.06 −0.05 ± 0.11 −0.20 ± 0.13 0.43 ± 0.10 −0.10 ± 0.12 (stat) HFAG correlated average
χ2 = 12/5 dof (CL=0.03 ⇒ 2.1σ)
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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 (stat) PRL 98 (2007) 181803
Belle
N(BB)=772M
−0.04 ± 0.26 ± 0.19 0.07 ± 0.08 ± 0.10 0.61 (stat) PRD 86 (2012) 092012
Average 0.02 ± 0.20 0.10 ± 0.10 0.38 HFAG correlated average
χ2 = 0.2/2 dof (CL=0.92 ⇒ 0.1σ)
Figures:
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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)°. Belle obtain φ2,eff = (107.3 ± 6.6 (stat) ± 4.8 (syst))°.

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

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



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

World averages for α ≡ φ2, combining all available measurements and calculated with different statistical treatments, have been calculated by the CKMfitter and UTfit groups. In addition, the BaBar and Belle collaborations have combined their results on B → ππ, πππ0 and ρρ to obtain

α ≡ φ2 = (88 ± 5)°

The above solution is that consistent with the Standard Model (an ambiguous solution shifted by 180° exists). The strongest constraint currently comes from the B → ρρ system. The inclusion of results from Bd → a1+−π−+ does not significantly affect the average.




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 −0.046 ± 0.013 ± 0.015 −0.039 ± 0.020 ± 0.013 −0.039 ± 0.010
CL=0.97 (0.03σ)
BaBar: PRD 71 (2005) 112003
(partially reco.)

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


Belle: PRD 84 (2011) 021101(R)
(partially reco.)

Belle: PRD 73 (2006) 092003
(fully reco.)
cD*π −0.019 ± 0.022 ± 0.013
(lepton tags only)
0.049 ± 0.042 ± 0.015
(lepton tags only)
−0.015 ± 0.013 ± 0.015 −0.011 ± 0.020 ± 0.013 −0.010 ± 0.013
CL=0.59 (0.6σ)
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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Time-Dependent CP Asymmetries in b → cu-bar s Transitions (eg. Bd → D+−KSπ−+, etc.) (sin(2β+γ) ≡ sin(2φ13))

Time-dependent analyses of transitions such as Bd → D+−KSπ−+ can be used to probe sin(2β+γ) ≡ sin(2φ13) in a similar way to that discussed above. Since the final state contains three particles, a Dalitz plot analysis is necessary to maximise the sensitivity. BaBar have carried out such an analysis. They obtain 2β+γ = (83 ± 53 ± 20)° (with an ambiguity 2β+γ → 2β+γ+π) assuming the ratio of the b → u and b → c amplitude to be constant across the Dalitz plot at 0.3.



Time-dependent CP Asymmetries in B0s decays via b → cu-bar s Transitions (eg. Analyses of B0s → D−+sK+−, etc.) (sin(γ−2βs) ≡ sin(φ3−2βs+))

LHCb have measured the time-dependent CP violation parameters in B0s → D−+sK+− decays.

Experiment C AΔΓ AΔΓbar S Sbar Correlation Reference
LHCb
Ldt=1 fb−1
0.53 ± 0.25 ± 0.04 0.37 ± 0.42 ± 0.20 0.20 ± 0.41 ± 0.20 −1.09 ± 0.33 ± 0.08 −0.36 ± 0.34 ± 0.08 (stat) (syst) arXiv:1407.6127

Digression:

Constraining γ
From these results, and a constraint on 2βs from independent LHCb measurements, LHCb determine γ = (115 +28−43) °, δDsK = (3 +19−20) ° and rDsK = 0.53 +0.17−0.16.



GLW Analyses of B → D(*)K(*)

A theoretically clean measurement of the angle γ ≡ φ3 can be obtained from the rate and asymmetry measurements of B → D(*)CPK(*)− decays, where the D(*) meson decays to CP even (D(*)CP+) and CP odd (D(*)CP−) eigenstates. The method benefits from the interference between the dominant b→cu-bar s transitions with the corresponding doubly CKM-suppressed b→uc-bar s transition. It was proposed by Gronau, Wyler and Gronau, London (GLW).

BaBar, Belle, CDF and LHCb use consistent definitions for ACP+− and RCP+−, where

ACP+− = [Γ(B → D(*)CP+−K(*)) − Γ(B+ → D(*)CP+−K(*)+)] / Sum ,
RCP+− = 2 [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)=467M
0.25 ± 0.06 ± 0.02 −0.09 ± 0.07 ± 0.02 1.18 ± 0.09 ± 0.05 1.07 ± 0.08 ± 0.04 PRD 82 (2010) 072004
Belle
N(BB)=772M
0.29 ± 0.06 ± 0.02 −0.12 ± 0.06 ± 0.01 1.03 ± 0.07 ± 0.03 1.13 ± 0.09 ± 0.05 LP 2011 preliminary
CDF
Ldt=1 fb−1
0.39 ± 0.17 ± 0.04 - 1.30 ± 0.24 ± 0.12 - PRD 81 (2010) 031105(R)
LHCb
Ldt=1 fb−1
0.14 ± 0.03 ± 0.01 - 1.01 ± 0.04 ± 0.01 - PLB 712 (2012) 203
Average 0.19 ± 0.03
χ2 = 6.5/3 dof (CL=0.09 ⇒ 1.7σ)
−0.11 ± 0.05
χ2 = 0.10 (CL=0.75 ⇒ 0.3σ)
1.03 ± 0.03
χ2 = 3.5/3 dof (CL=0.33 ⇒ 1.0σ)
1.10 ± 0.07
χ2 = 0.2 (CL=0.66 ⇒ 0.4σ)
HFAG
Figures:
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Click here for averages excluding KSφ from the CP-odd channels.
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)=772M
−0.14 ± 0.10 ± 0.01 0.22 ± 0.11 ± 0.01 1.19 ± 0.13 ± 0.03 1.03 ± 0.13 ± 0.03 CKM2012 preliminary
Average −0.12 ± 0.07
χ2 = 0.05 (CL=0.82 ⇒ 0.2σ)
0.13 ± 0.07
χ2 = 1.1 (CL=0.29 ⇒ 1.1σ)
1.25 ± 0.09
χ2 = 0.40 (CL=0.52 ⇒ 0.6σ)
1.06 ± 0.09
χ2 = 0.11 (CL=0.74 ⇒ 0.3σ)
HFAG
Figures:
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DCPK* BaBar
N(BB)=379M
0.09 ± 0.13 ± 0.06 −0.23 ± 0.21 ± 0.07 2.17 ± 0.35 ± 0.09 1.03 ± 0.27 ± 0.13 PRD 80 (2009) 092001
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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eps.gz png
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. The experiments 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+)] .

Digression:

Recently it has been noted that the observables (R+, R) may be more suitable for use than (RADS, AADS) since the former are better behaved (they are statistically independent observables, while the uncertainty on AADS depends on the central value of RADS). The definitions are

R+ = Γ(B+ → [Kπ+]DK+) / Γ(B+ → [K+π]DK+)       R = Γ(B → [K+π]DK) / Γ(B → [Kπ+]DK)

They are related to (RADS, AADS) by

RADS = (R+ + R)/2       AADS = (R − R+) / (R + R+)

We may switch to using this set of variables at a later time, but presently the majority of experimental results are presented in the (RADS, AADS) format.

(Some of) these observables have been measured so far for the D(*)K(*)− modes. BaBar, Belle, CDF and LHCb have presented results for B → DK while BaBar and Belle have also presented results using B → D*K, with both D* → Dπ0 and D* → Dγ. BaBar have also presented results on 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 AADS RADS Reference
DK
D→Kπ
BaBar
N(BB)=467M
−0.86 ± 0.47 +0.12 −0.16 0.011 ± 0.006 ± 0.002 PRD 82 (2010) 072006
Belle
N(BB)=772M
−0.39 +0.26 −0.28 +0.04 −0.03 0.0163 +0.0044 −0.0041 +0.0007 −0.0013 PRL 106 (2011) 231803
CDF
Ldt=7 fb−1
−0.82 ± 0.44 ± 0.09 0.0220 ± 0.0086 ± 0.0026 PRD 84 (2011) 091504
LHCb
Ldt=1 fb−1
−0.52 ± 0.15 ± 0.02 0.0152 ± 0.0020 ± 0.0004 PLB 712 (2012) 203
Average −0.54 ± 0.12
χ2 = 1.1/3 dof (CL=0.77 ⇒ 0.3σ)
0.0153 ± 0.0017
χ2 = 1.1/3 dof (CL=0.78 ⇒ 0.3σ)
HFAG
Figures:
eps.gz png eps.gz png .
D*K
D* → Dπ0
D→Kπ
BaBar
N(BB)=467M
0.77 ± 0.35 ± 0.12 0.018 ± 0.009 ± 0.004 PRD 82 (2010) 072006
Belle
N(BB)=772M
0.4 +1.1 −0.7 +0.2 −0.1 0.010 +0.008 −0.007 +0.001 −0.002 LP 2011 preliminary
Average 0.72 ± 0.34
χ2 = 0.1 (CL=0.71 ⇒ 0.4σ)
0.013 ± 0.006
χ2 = 0.4 (CL=0.52 ⇒ 0.6σ)
HFAG
Figures:
eps.gz png eps.gz png .
D*K
D* → Dγ
D→Kπ
BaBar
N(BB)=467M
0.36 ± 0.94 +0.25 −0.41 0.013 ± 0.014 ± 0.008 PRD 82 (2010) 072006
Belle
N(BB)=772M
−0.51 +0.33 −0.29 ± 0.08 0.036 +0.014 −0.012 ± 0.002 LP 2011 preliminary
Average −0.43 ± 0.31
χ2 = 0.7 (CL=0.42 ⇒ 0.8σ)
0.027 ± 0.010
χ2 = 1.3 (CL=0.26 ⇒ 1.1σ)
HFAG
Figures:
eps.gz png eps.gz png .
DK*
D→Kπ
BaBar
N(BB)=379M
−0.34 ± 0.43 ± 0.16 0.066 ± 0.031 ± 0.010 PRD 80 (2009) 092001
DK
D→Kππ0
BaBar
N(BB)=474M
- 0.0091 +0.0082 −0.0076 +0.0014 −0.0037 PRD 84 (2011) 012002
Belle
N(BB)=772M
0.41 ± 0.30 ± 0.05 0.0198 ± 0.0062 ± 0.0024 PRD 88 (2013) 091104(R)
Average - 0.0156 ± 0.0052
χ2 = 1.0 (CL=0.32 ⇒ 1.0σ)
HFAG
Figures:
. eps.gz png .
DK
D→K3π
LHCb
Ldt=1 fb−1
−0.42 ± 0.22 0.0124 ± 0.0027 PLB 723 (2013) 44
Compilation of the above results.

eps.gz png

eps.gz png

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,
R± = rB2 + rD2 + 2 rB rD cos(δB±γ) = rB2 + rD2 + 2 rB rD cos(δB±φ3)

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, ensuring that mixing effects are corrected for. The value of δD can be determined directly using quantum correlated D mesons produced in ψ(3770) decay, as has been done by CLEO. The most precise values of both quantities come from the averages performed by HFAG to obtain the mixing parameters in the charm system.

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. Some of the observables have been measured by BaBar, Belle, CDF and LHCb in the various D(*)π modes.

Mode Experiment AADS RADS Reference

D→Kπ
BaBar
N(BB)=467M
0.03 ± 0.17 ± 0.04 0.0033 ± 0.0006 ± 0.0004 PRD 82 (2010) 072006
Belle
N(BB)=772M
−0.04 ± 0.11 +0.02 −0.01 0.00328 +0.00038 −0.00036 +0.00012 −0.00018 PRL 106 (2011) 231803
CDF
Ldt=7 fb−1
0.13 ± 0.25 ± 0.02 0.0028 ± 0.0007 ± 0.0004 PRD 84 (2011) 091504
LHCb
Ldt=1 fb−1
0.143 ± 0.062 ± 0.011 0.00410 ± 0.00025 ± 0.00005 PLB 712 (2012) 203
Average 0.09 ± 0.05
χ2 = 2.2/3 dof (CL=0.53 ⇒ 0.6σ)
0.00375 ± 0.00020
χ2 = 5.1/3 dof (CL=0.17 ⇒ 1.4σ)
HFAG
Figures:
eps.gz png eps.gz png .
D*π
D* → Dπ0
D→Kπ
BaBar
N(BB)=467M
−0.09 ± 0.27 ± 0.05 0.0032 ± 0.0009 ± 0.0008 PRD 82 (2010) 072006
Belle
N(BB)=772M
−0.07 ± 0.23 ± 0.05 0.0040 +0.0010 −0.0009 ± 0.0003 LP 2011 preliminary
Average −0.08 ± 0.18
χ2 = 0.003 (CL=0.96 ⇒ 0.1σ)
0.0037 ± 0.0008
χ2 = 0.27 (CL=0.61 ⇒ 0.5σ)
HFAG
Figures:
eps.gz png eps.gz png .
D*π
D* → Dγ
D→Kπ
BaBar
N(BB)=467M
−0.65 ± 0.55 ± 0.22 0.0027 ± 0.0014 ± 0.0022 PRD 82 (2010) 072006
Belle
N(BB)=772M
−0.10 +0.26 −0.25 ± 0.02 0.0041 +0.0011 −0.0010 ± 0.0001 LP 2011 preliminary
Average −0.19 ± 0.23
χ2 = 0.73 (CL=0.39 ⇒ 0.9σ)
0.0039 ± 0.0010
χ2 = 0.25 (CL=0.62 ⇒ 0.5σ)
HFAG
Figures:
eps.gz png eps.gz png .

D → Kππ0
Belle
N(BB)=772M
0.16 ± 0.27 +0.03 −0.04 0.00189 ± 0.00054 +0.00022 −0.00025 PRD 88 (2013) 091104(R)

D → K3π
LHCb
Ldt=1 fb−1
0.13 ± 0.10 0.0037 ± 0.0004 PLB 723 (2013) 44
Compilation of the above results.

eps.gz png

eps.gz png


Analysis of B → D(*) K(*)− with D → KSh+h

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 analysis can be performed either with the assumption of a Dalitz plot model for the D meson decay (model-dependent) or without such an assumption (model-independent). In the latter case, it is necessary to bin the Dalitz plot and use measurements of quantities related to the average strong phase difference between the amplitudes D and D-bar decays in each bin. Such measurements can be obtained using ψ(3770) → DD-bar decays.

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, experiments 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



Model-Dependent Dalitz Plot Analysis of B → D(*) K(*)− with D → KSπ+π, ...

Results are available from both Belle and BaBar using B → D K, B → D*K and B → DK*. Both BaBar and Belle use both D* decays to Dπ0 and Dγ, taking 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*). Results are also available from LHCb using B → D K with D → KSπ+π.

For the DK* mode, both BaBar and Belle 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 using use an alternative parametrization [replacing rB and δB with κrs and δs, respectively] suggested by Gronau. [BaBar do not obtain constraints on rB and δB in this decay.]

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 the Appendix of Ref.) The Belle results using B → DK* 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)=468M
−0.103 ± 0.037 ± 0.006 ± 0.007 −0.021 ± 0.048 ± 0.004 ± 0.009 0.060 ± 0.039 ± 0.007 ± 0.006 0.062 ± 0.045 ± 0.004 ± 0.006 (stat) (syst) (model) PRL 105 (2010) 121801
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) PRD 81 (2010) 112002
LHCb
Ldt=1 fb−1
−0.084 ± 0.045 ± 0.009 ± 0.005 −0.032 ± 0.048 +0.010 −0.009 ± 0.008 0.027 ± 0.044 +0.010 −0.008 ± 0.001 0.013 ± 0.048 +0.009 −0.007 ± 0.003 (stat) (model) NPB 888 (2014) 169
Average −0.098 ± 0.024 −0.036 ± 0.030 0.070 ± 0.025 0.075 ± 0.029 (stat) HFAG correlated average
χ2 = 7.2/8 dof (CL=0.52 ⇒ 0.7σ)
Figures:
NB. The contours in these plots
do not include model errors.

eps.gz png

eps.gz png

eps.gz png
D*K BaBar
N(BB)=468M
0.147 ± 0.053 ± 0.017 ± 0.003 −0.032 ± 0.077 ± 0.008 ± 0.006 −0.104 ± 0.051 ± 0.019 ± 0.002 −0.052 ± 0.063 ± 0.009 ± 0.007 (stat) (syst) (model) PRL 105 (2010) 121801
Belle (*)
N(BB)=657M
0.083 ± 0.092 ± 0.081 0.157 ± 0.109 ± 0.063 −0.036 ± 0.127 ± 0.090 −0.249 ± 0.118 ± 0.049 (stat) (model) PRD 81 (2010) 112002
Average
No model error
0.130 ± 0.048 0.031 ± 0.063 −0.090 ± 0.050 −0.099 ± 0.056 (stat+syst) HFAG correlated average
χ2 = 5.0/4 dof (CL=0.29 ⇒ 1.1σ)
Figures:
NB. The contours in these plots
do not include model errors.

eps.gz png

eps.gz png

eps.gz png
DK*− BaBar
N(BB)=468M
−0.151 ± 0.083 ± 0.029 ± 0.006 0.045 ± 0.106 ± 0.036 ± 0.008 0.075 ± 0.096 ± 0.029 ± 0.007 0.127 ± 0.095 ± 0.027 ± 0.006 (stat) (syst) (model) PRL 105 (2010) 121801
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
−0.152 ± 0.077 0.024 ± 0.091 −0.043 ± 0.094 0.091 ± 0.096 (stat+syst) HFAG correlated average
χ2 = 13/4 dof (CL=0.011 ⇒ 2.5σ)
Figures:
NB. The contours in these plots
do not include model errors.

eps.gz png

eps.gz png

eps.gz png
(*) The Belle results for D*K are our average of their results on D*K with D*→Dπ0 and D*K with D*→Dγ. The average is performed using the statistical correlations provided, and neglecting all systematic correlations; model uncertainties are not included. The first uncertainty on the quoted results is combined statistical and systematic, the second is the model error (taken from the Belle results on D*K with D*→Dπ0).
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.

eps.gz png

eps.gz png

eps.gz png

eps.gz png

Digression:

r
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. All experiments have done so using frequentist techniques. Note that the uncertainty on γ ≡ φ3 is approximately inversely proportional to the central value of rB.
BaBar obtain
γ = (68 +15−14 ± 4 ± 3)°
(from DK, D*K & DK*)
Belle obtain
φ3 = (78 +11−12 ± 4 ± 9)°
(from DK & D*K)
LHCb obtain
γ = (84 +49−42
(from DK using ∫Ldt=1 fb−1; a more precise result using ∫Ldt=3 fb−1 and the model-independent method is reported below)
The experiments also obtain values for the hadronic parameters
rB (DK) = 0.096 ± 0.029 ± 0.005 ± 0.004 δB (DK) = (119 +19−20 ± 3 ± 3)° rB (DK) = 0.160 +0.040−0.038 ± 0.011+0.05−0.010 δB (DK) = (138 +13−16 ± 4 ± 23)° rB (DK) = 0.06 ± 0.04 δB (DK) = (115 +41−51
rB (D*K) = 0.133 +0.042−0.039 ± 0.014 ± 0.003 δB (D*K) = (−82 ± 21 ± 5 ± 3)° rB (D*K) = 0.196 +0.072−0.069 ± 0.012 +0.062−0.012 δB (D*K) = (342 +19−21 ± 3 ± 23)° . .
κrs = 0.149 +0.066−0.062 ± 0.026 ± 0.006 δs = (111 ± 32 ± 11 ± 3)° ( rB (DK*) = 0.56 +0.22−0.16 ± 0.04 ± 0.08 *) B (DK*) = (243+20−23 ± 3 ± 50 )° *) . .
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.


Model-Independent Dalitz Plot Analysis of B → D(*) K(*)− with D → KSπ+π, ...

A model-independent approach to the analysis of B → D(*) K with multibody D decays was proposed by Giri, Grossman, Soffer and Zupan, and further developed by Bondar and Poluektov (see also here). The method relies on information on the average strong phase difference between D0 and D0-bar decays in bins of Dalitz plot position that can be obtained from quantum-correlated Ψ(3770) → D0D0-bar events. This information is measured in the form of parameters ci and si that are the amplitude weighted averages of the cosine and sine of the strong phase difference in a Dalitz plot bin labelled by i, respectively. These quantities have been obtained for D → KSπ+π (and D → KSK+K) by CLEOc (see also here). [Preliminary results from BESIII are also available.]

Model-independent determinations of γ ≡ φ3 has been performed by Belle and LHCb. Both Belle and LHCb have used B → D K with D → KSπ+π. The LHCb results also include D → KSK+K -- we do not attempt to separate the contribution from this mode in our combination (although LHCb results using D → KSπ+π only are also available).

The variables (x±, y±), defined above are determined from the data. Note that due to the strong statistical and systematic correlations with the model-dependent results given above, these results cannot be combined.

The results below have three sets of errors, which are statistical, systematic, and uncertainty coming from the knowledge of ci and si respectively. To perform the average, we remove the last uncertainty, which should be 100% correlated between the measurements. Since the size of the uncertainty from ci and si is found to depend on the size of the B → DK data sample, we assign the LHCb uncertainties (which are mostly the smaller of the Belle and LHCb values) to the averaged result. This procedure should be conservative.

Mode Experiment x+ y+ x- y- Correlation Reference
DK
D→KSπ+π
Belle
N(BB)=772M
−0.110 ± 0.043 ± 0.014 ± 0.007 −0.050 +0.052 −0.055 ± 0.011 ± 0.007 0.095 ± 0.045 ± 0.014 ± 0.010 0.137 +0.053 −0.057 ± 0.015 ± 0.023 (stat) PRD 85 (2012) 112014
LHCb
Ldt=3 fb−1
−0.077 ± 0.024 ± 0.010 ± 0.004 −0.022 ± 0.025 ± 0.004 ± 0.010 0.025 ± 0.025 ± 0.010 ± 0.005 0.075 ± 0.029 ± 0.005 ± 0.014 (stat) arXiv:1408.2748
Average −0.085 ± 0.023 ± 0.004 −0.027 ± 0.023 ± 0.010 0.044 ± 0.023 ± 0.005 0.090 ± 0.026 ± 0.014 (stat) HFAG correlated average
χ2 = 4.1/4 dof (CL=0.39 ⇒ 0.9σ)
Figures:
NB. The contours in these plots
do not include model errors.

eps.gz png

eps.gz png

eps.gz png
.

Digression:

Constraining γ ≡ φ3:
As above, the measurements of x+,− and y+,− can be used to place bounds on γ ≡ φ3. Belle and LHCb have done so using a frequentist technique.
Belle obtain
φ3 = (77.3 +15.1−14.9 ± 4.1 ± 4.3)°
LHCb obtain
γ = (62 +15−14
rB (DK) = 0.145 ± 0.030 ± 0.010 ± 0.011 δB (DK) = (129.9 ± +15.0 ± 3.8 ± 4.7)° rB (DK) = 0.080 +0.019−0.021 δB (DK) = (134 +14−15
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.


Model-Independent Analysis of B → D(*) K(*)− with D → KSK+−π−+, ...

Decays of D mesons to KSK+−π−+ can be used in a similar approach to that discussed above to determine γ ≡ φ3. Since these decays are less abundant, the event samples available to date have not been sufficient for a fine binning of the Dalitz plots, but the analysis can be performed using only an overall coherence factor and related strong phase difference for the decay. These quantities have been determined by CLEO both for the full Dalitz plots and in a restricted region ±100 MeV/c2 around the peak of the K*(892)± resonance.

LHCb have reported results of an analysis of B → D K and B → D π decays with D → KSK+−π−+. The decays with different final states of the D meson are distinguished by the charge of the kaon from the decay of the D meson relative to the charge of the B meson, and are labelled "same sign" (SS) and "opposite sign" (OS). Six observables potentially sensitive to γ ≡ φ3 are measured: two ratios of rates for DK and Dπ decays (one each for SS and OS) and four asymmetries (for DK & Dπ, SS & OS). This is done both for the full Dalitz plot and for the K*(892)±-dominated region (with the same boundaries as used by CLEO). Note that there is a significant overlap of events between the two samples. The results do not yet have sufficient precision to set significant constraints on γ ≡ φ3.

Mode Experiment RSS ROS ASS,DK AOS,DK ASS,Dπ AOS,Dπ Correlation Reference
DK
D→KSK+−π−+
LHCb
Ldt=3 fb−1
0.092 ± 0.009 ± 0.004 0.066 ± 0.009 ± 0.002 0.040 ± 0.091 ± 0.018 0.233 ± 0.129 ± 0.024 −0.025 ± 0.024 ± 0.010 −0.052 ± 0.029 ± 0.017 (stat) PLB 733 (2014) 36
DK
D→K*(892)+−K−+
LHCb
Ldt=3 fb−1
0.084 ± 0.011 ± 0.003 0.056 ± 0.013 ± 0.002 0.026 ± 0.109 ± 0.029 0.336 ± 0.208 ± 0.026 −0.012 ± 0.028 ± 0.010 −0.054 ± 0.043 ± 0.017 (stat) PLB 733 (2014) 36


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.


GLW and ADS Analyses of B0 → DK*0

LHCb have presented results on B0 → DK*0 with D → K+K and D → π+π

Experiment ACP+ RCP+ Reference
LHCb (K+K)
Ldt=3 fb−1
−0.20 ± 0.15 ± 0.02 1.05 +0.17 −0.15 ± 0.04 arXiv:1407.8136
LHCb (π+π)
Ldt=3 fb−1
−0.09 ± 0.22 ± 0.02 1.21 +0.28 −0.25 ± 0.05 arXiv:1407.8136

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

BaBar
N(BB)=465M
RADS(Kπ) < 0.244 RADS(Kππ0) < 0.181 RADS(Kπππ) < 0.391 PRD 80 (2009) 031102
Belle
N(BB)=772M
RADS(Kπ) < 0.16 - - PRD 86 (2012) 011101
LHCb have presented results on B0 → DK*0 with D → Kπ+, and measured the parameters R+ and R.
Experiment R+ R Reference
LHCb
Ldt=3 fb−1
0.06 ± 0.03 ± 0.01 0.06 ± 0.03 ± 0.01 arXiv:1407.8136

(See above for a definition of the parameters).

Digression:

Combining the results and using additional input from CLEOc (here and here), BaBar set a limit on the ratio between the b→u and b→c amplitudes of rB(DK*0) ∈ [0.07,0.41] at 95% CL.
Belle set at limit of rB(DK*0) < 0.4 at 95% CL.
LHCb take input from HFAG charm and obtain rB(DK*0) = 0.240 +0.055−0.048 (different from zero with 2.7σ significance).


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).


GLW Analyses of B+ → DK+ π+π

LHCb have presented results on B+ → DK+ π+π with D → K+K and π+π

Experiment ACP+ RCP+ Reference
LHCb
Ldt=1 fb−1
−0.14 ± 0.10 ± 0.01 0.95 ± 0.11 ± 0.02 LHCb-CONF-2012-021


Combinations of Results on B → DK Processes to Obtain Constraints on γ ≡ φ3

BaBar, Belle and LHCb have all presented constraints on γ ≡ φ3 from combinations of their results on B+ → DK+ and related processes.

BaBar obtain γ = (69 +17−16
rB = 0.092 +0.013−0.012
δB = (105 +16−17
PRD 87 (2013) 052015
Belle obtain φ3 = (68 +15−14
rB = 0.112 +0.014−0.015
δB = (116 +18−21
CKM2012 preliminary
LHCb obtain γ = (73 +910
rB = 0.0914 +0.0083−0.0088
δB = (127 +10−12
LHCb-CONF-2014-004

For attempts to extract γ ≡ φ3 from all available experimental results, visit the CKMfitter and UTfit sites.


This page is maintained by Tim Gershon.


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