## Results on Time-Dependent CP Violation, and Measurements Related to the Angles of the Unitarity Triangle: ICHEP 2006 (Moscow, Russia)

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 Time-Dependent Dalitz Plot Asymmetries in Bd → D(*) h0, with D → KSπ+π− (sin2β & cos2β ≡ sin2φ1 & cos2&phi1) §   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 → uu-bar d Transitions §   Studies of Time-Dependent Interference Between b → cu-bar d and b-bar → u-bar cd-bar Transitions §   Studies of Interference Between b → cu-bar s & b → uc-bar s Transitions

Legend: if not stated otherwise,

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

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

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

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

Parameter Value Reference
τ(Bd) (1.527 ± 0.008) ps HFAG - Oscillations/Lifetime
Δmd (0.508 ± 0.004) ps−1 HFAG - Oscillations/Lifetime
|A|2
(CP-odd fraction in
B0→ J/ψK* CP sample)
0.233 ± 0.010 ± 0.005 BaBar: BABAR-CONF-06/043
N(BB)=232m
0.195 ± 0.012 ± 0.008 Belle: PRL 95 (2005) 091601
N(BB)=275m
0.219 ± 0.009 Average
χ2 = 4.3/1 dof (CL=0.04 ⇒ 2.1σ)

Additional note on commonly treated (correlated) systematic effects:

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

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

Parameter: sin(2β) ≡ sin(2φ1)
Mode BaBar Belle Average Reference
Charmonium: N(BB)=348m N(BB)=532m - BaBar (hep-ex/0607107)
Belle (hep-ex/0608039)
J/ψKSCP=-1) 0.697 ± 0.041stat 0.643 ± 0.038stat
ψ(2S)KSCP=-1) 0.893 ± 0.119stat -
χc1KSCP=-1) 0.715 ± 0.174stat -
ηcKSCP=-1) 0.711 ± 0.229stat -
J/ψKLCP=+1) 0.719 ± 0.080stat 0.641 ± 0.057stat
J/ψK*0 (K*0 → KSπ0) CP= 1-2|A|2) 0.517 ± 0.283stat -
All charmonium 0.710 ± 0.034 ± 0.019 0.642 ± 0.031 ± 0.017 0.674 ± 0.026
(0.023stat-only)
CL = 0.18

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

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

we find the only slightly modified average:

Parameter: sin(2β) ≡ sin(2φ1)
All charmonium 0.675 ± 0.026 (0.023stat-only) CL = 0.36

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

 β ≡ φ1 = (21.2 ± 1.0)° or β ≡ φ1 = (68.8 ± 1.0)°

Plots:

 Average of sin(2β) ≡ sin(2φ1) from all experiments. eps.gz png Averages of sin(2β) ≡ sin(2φ1) and C=-A from the B factories. eps.gz png eps.gz png Constraint on the ρ-bar-η-bar plane: eps.gz png eps.gz png

 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)=348m N(BB)=532m - BaBar (hep-ex/0607107)
Belle (hep-ex/0608039)
J/ψKS - 0.001 ± 0.028stat
ψ(2S)KS - -
χc1KS - -
ηcKS - -
J/ψKL - −0.045 ± 0.033stat
J/ψK*0 (K*0 → KSπ0) - -
All charmonium 0.070 ± 0.028 ± 0.018 −0.018 ± 0.021 ± 0.014 0.012 ± 0.022
(0.017stat-only)
CL = 0.02

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

The BaBar and Belle collaborations have performed measurements of sin(2β) & cos(2β) ≡ sin(2φ1) & cos(2φ1) in time-dependent transversity analyses of the pseudoscalar to vector-vector decay B0→ J/ψK*, where cos(2β) ≡ cos(2φ1) enters as a factor in the interference between CP-even and CP-odd amplitudes. In principle, this analysis comes along with an ambiguity on the sign of cos(2β) ≡ cos(2φ1) due to an incomplete determination of the strong phases occurring in the three transversity amplitudes. BaBar resolves this ambiguity by inserting the known variation of the rapidly moving P-wave phase relative to the slowly moving S-wave phase with the invariant mass of the Kπ system in the vicinity of the K*(892) resonance. The result is in agreement with the prediction obtained from s-quark helicity conservation. It corresponds to Solution II defined by Suzuki, which is the phase convention used for the averages given here.

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

Experiment sin(2β) ≡ sin(2φ1)J/ψK* cos(2β) ≡ cos(2φ1)J/ψK* Correlation Reference
BaBar
N(BB)=88M
−0.10 ± 0.57 ± 0.14 3.32 +0.76 −0.96 ± 0.27 −0.37 (stat) PRD 71, 032005 (2005)
Belle
N(BB)=275M
0.24 ± 0.31 ± 0.05 0.56 ± 0.79 ± 0.11
[using Solution II]
0.22 (stat) PRL 95 091601 (2005)
Average 0.16 ± 0.28
χ2 = 0.3/1 dof (CL = 0.61 → 0.5σ)
1.64 ± 0.62
χ2 = 4.7/1 dof (CL = 0.03 → 2.2σ)
uncorrelated averages HFAG
See remark below table
Figures:

eps.gz png

eps.gz png
.

Interpretations:

BaBar find a confidence level for cos(2β)>0 of 89%.
Note that due to the strong non-Gaussian character of the BaBar measurement, the interpretation of the average given above has to be done with the greatest care.
We perform uncorrelated averages (using the PDG prescription for asymmetric errors).

### Time-dependent Analysis of Bd → D*D*KS

The decays Bd → D(*)D(*)KS are dominated by the b → cc-bar s transition, and are therefore sensitive to 2β ≡ 2φ1. However, since the final state is not a CP eigenstate, extraction of the weak phases is difficult. Browder et al. have shown that terms sensitive to cos(2β) ≡ cos(2φ1) can be extracted from the analysis of Bd → D*D*KS decays (with some theoretical input).

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

BaBar divide 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. BaBar measures:

Experiment Jc/J0 (2Js1/J0)sin(2β) (2Js2/J0)cos(2β) Correlation Reference
BaBar
N(BB)=230m
0.76 ± 0.18 ± 0.07 0.10 ± 0.24 ± 0.06 0.38 ± 0.24 ± 0.05 - hep-ex/0608016

Interpretations:

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

### Time-Dependent Analysis of Bs → J/ψ φ

Decays of the Bs meson via the b → cc-bar s transition probe the Bs–Bs-bar mixing phase, φs. 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.

The first measurement of φs from Bs → J/ψ φ has been performed by D0. Using an integrated luminosiy of 1 fb−1, they perform an untagged, time-dependent analysis from which they simultaneously measure the average Bs lifetime τ(Bs), ΔΓ, φs, the magnitude of the perpendicularly polarized component A, the difference in the fractions of the two CP-even components |A0|2 - |A|||2, and the strong phases associated with the two CP-even components δ0 and δ||. The results are given below.

The implicit convention above is that |A|2 + |A0|2 + |A|||2 = 1, and the strong phases are measured relative to that of the A component (which is set to zero). The polarization components are defined at time t=0, ie. at the production (primary) vertex of the Bs. Note also that there is an ambiguity in the result for φs.

Experiment τ(Bs) ΔΓ φs A |A0|2 - |A|||2 δ|| δ0 Correlation Reference
D0 1.49 ± 0.08 +0.01 −0.03 0.17 ± 0.09 ± 0.03 −0.79 ± 0.56 ± 0.01 0.46 ± 0.06 ± 0.01 0.37 ± 0.06 ± 0.01 3.30 ± 1.10 ± 0.00 0.70 ± 1.00 ± 0.00 (stat) hep-ex/0701012

Interpretations:
D0 have combined the contour in the (φs, ΔΓ) plane obtained above with a constraint obtained from the charge asymmetry in B–B-bar oscillations (see also HFAG - Oscillations), to obtain the result φs = −0.56 +0.44−0.41.

### 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.

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)=311m
0.45 ± 0.36 ± 0.05 ± 0.07 0.54 ± 0.54 ± 0.08 ± 0.18 0.975 +0.093−0.085 ± 0.012 ± 0.002 0.07 stat
between sin(2β) & cos(2β)
hep-ex/0607105
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 0.57 ± 0.30
χ2 = 0.3/1 dof (CL=0.59 ⇒ 0.5σ)
1.16 ± 0.42
χ2 = 2.5/1 dof (CL=0.12 ⇒ 1.6σ)
- 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 87% 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 the BaBar time-dependent Dalitz plot analysis of B0 → K+KK0 are also discussed in the next section). The BaBar results that appear in this table on K+KK0 come from previous analyses in which the final state is treated as a quasi-two-body system .

At present we do not apply a rescaling of the results to a common, updated set of input parameters. The exception is the CP-even fraction in the quasi-two-body analysis of B0 → K+KK0. We take correlations between S and C into account, 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)=347M
0.12 ± 0.31 ± 0.10 0.18 ± 0.20 ± 0.10 - hep-ex/0607112
Belle
N(BB)=535M
0.50 ± 0.21 ± 0.06 −0.07 ± 0.15 ± 0.05 0.05 (stat) hep-ex/0608039
Average 0.39 ± 0.18 0.01 ± 0.13 0.03 HFAG correlated average
χ2 = 1.8/2 dof (CL=0.41 ⇒ 0.8σ)
Figures:
eps.gz png eps.gz png eps.gz png
η′K0 BaBar
N(BB)=384M
0.58 ± 0.10 ± 0.03 −0.16 ± 0.07 ± 0.03 0.03 (stat) hep-ex/0609052
Belle
N(BB)=535M
0.64 ± 0.10 ± 0.04 0.01 ± 0.07 ± 0.05 0.09 (stat) hep-ex/0608039
Average 0.61 ± 0.07 −0.09 ± 0.06 0.04 HFAG correlated average
χ2 = 2.3/2 dof (CL=0.32 ⇒ 1.0σ)
Figures:
eps.gz png eps.gz png eps.gz png
KSKSKS BaBar
N(BB)=347M
0.66 ± 0.26 ± 0.08 −0.14 ± 0.22 ± 0.05 0.09 (stat) hep-ex/0607108
Belle
N(BB)=535M
0.30 ± 0.32 ± 0.08 −0.31 ± 0.20 ± 0.07 - hep-ex/0608039
Average 0.51 ± 0.21 −0.23 ± 0.15 0.04 HFAG correlated average
χ2 = 1.0/2 dof (CL=0.61 ⇒ 0.5σ)
Figures:
eps.gz png eps.gz png eps.gz png
π0KS BaBar
N(BB)=348M
0.33 ± 0.26 ± 0.04 0.20 ± 0.16 ± 0.03 −0.06 (stat) hep-ex/0607096
Belle
N(BB)=532M
0.33 ± 0.35 ± 0.08 0.05 ± 0.14 ± 0.05 −0.08 (stat) hep-ex/0609006
Average 0.33 ± 0.21 0.12 ± 0.11 −0.06 HFAG correlated average
χ2 = 0.5/2 dof (CL=0.79 ⇒ 0.3σ)
Figures:
eps.gz png eps.gz png eps.gz png
ρ0KS BaBar
N(BB)=227M
0.20 ± 0.52 ± 0.24 0.64 ± 0.41 ± 0.20 - PRL 98 (2007) 051803
ωKS BaBar
N(BB)=347M
0.62 +0.25 −0.30 ± 0.02 −0.43 +0.25 −0.23 ± 0.03 - hep-ex/0607101
Belle
N(BB)=532M
0.11 ± 0.46 ± 0.07 0.09 ± 0.29 ± 0.06 −0.04 (stat) hep-ex/0609006
Average 0.48 ± 0.24
χ2 = 0.9 (CL=0.35 ⇒ 0.9σ)
−0.21 ± 0.19
χ2 = 1.8 (CL=0.18 ⇒ 1.3σ)
uncorrelated averages HFAG
Figures:
eps.gz png eps.gz png .
f0K0 BaBar (*) 0.62 ± 0.23 −0.36 ± 0.23 - hep-ex/0607112, hep-ex/0408095 (*)
Belle
N(BB)=532M
0.18 ± 0.23 ± 0.11 0.15 ± 0.15 ± 0.07 −0.01 (stat) hep-ex/0609006
Average 0.42 ± 0.17 −0.02 ± 0.13 −0.00 HFAG correlated average
χ2 = 4.9/2 dof (CL=0.09 ⇒ 1.7σ)
Figures:
eps.gz png eps.gz png eps.gz png
π0π0KS BaBar
N(BB)=227M
−0.84 ± 0.71 ± 0.08 0.27 ± 0.52 ± 0.13 - hep-ex/0508017
K+KK0
(excluding φK0)
BaBar Q2B (*)
N(BB)=227M
0.41 ± 0.18 ± 0.07 ± 0.11CP-even
(fCP-even= 0.89 ± 0.08 ± 0.06 [moments])
0.23 ± 0.12 ± 0.07 - hep-ex/0507016
Belle
N(BB)=532M
0.68 ± 0.15 ± 0.03+0.21−0.13CP-even
(fCP-even= 0.93 ± 0.09 ± 0.05 [SU(2)])
0.09 ± 0.10 ± 0.05 −0.00 (stat) hep-ex/0609006
Average 0.58 ± 0.13 +0.12-0.09CP-even
(rescaled to average fCP-even= 0.91 ± 0.07)
χ2 = 1.6 (CL=0.21)
0.15 ± 0.09
χ2 = 0.6 (CL=0.43)
uncorrelated averages HFAG
Figures:
eps.gz png eps.gz png .
Naïve b→s penguin average 0.53 ± 0.05
χ2 = 13/15 dof (CL=0.59 ⇒ 0.5σ)
−0.01 ± 0.04
χ2 = 21/15 dof (CL=0.13 ⇒ 1.5σ)
uncorrelated averages HFAG
eps.gz png eps.gz png
Direct comparison of charmonium and s-penguin averages (see comments below): χ2 = 6.6 (CL=0.01 ⇒ 2.6σ)

(*) The BaBar results for φK0 are determined from their time-dependent Dalitz plot analysis of B0 → K+KK0. The BaBar results for f0K0 are a combination of results from the Dalitz plot analysis (sin(2βeff) = 0.31 ± 0.32 ± 0.07, CCP = −0.45 ± 0.28 ± 0.10), with those from the quasi-two-body analysis of B0 → f0KS, f0 → π+π (sin(2βeff) = 0.95 +0.23 −0.32 ± 0.10, CCP = −0.24 ± 0.31 ± 0.15, hep-ex/0408095). The BaBar results for K+KK0 are taken from their previous quasi-two-body analysis (hep-ex/0507016).

• Assuming that a single and vanishing weak phase dominates the decay amplitudes of the s-penguin modes, one has S = −ηCP×sin(2β) ≡ −ηCP×sin(2φ1), where ηCP is the CP eigenvalue of the final state. Namely, ηCP = +1 for f0KS, π0π0KS, KSKSKS, and ηCP = −1 for φKS, η′KS, π0KS, ρKS and ωKS. Modes with KL in the final state are related by ηCP (X KS) = − ηCP (X KL).
• The mode K+KK0 may have both CP-even and CP-odd contributions. The CP-even fraction is experimentally derived from an isospin analysis (Belle) or a moments analysis (BABAR). [Also, see below for details of a time-dependent Dalitz plot analysis of this final state by BaBar.] We preaverage the CP-even fraction, in which there is an implicit assumption that the detection and reconstruction efficiencies have the same variation across the Dalitz plane in all analyses (and therefore the underlying CP-even fraction is indeed the same quantity).
• Averaging over all s-penguin modes assumes that contributions with non-zero weak phases to the decay amplitudes can be neglected. Note that this assumption may be significantly violated due to doubly CKM-suppressed Vub penguin amplitudes, and, in some cases, doubly CKM-suppressed and color-suppressed Vub tree amplitudes that contribute to the decay amplitude. Recent theoretical analyses indicate that it is reasonable to expect that the modes φK0, η′K0 and KSKSK0 have theoretical uncertainties of the order of 0.05 or smaller, these can be significantly larger for the other modes (in particular for non-ss-bar-resonance modes: π0KS, ρ0KS, ωKS and K+KK0).
• Our naïve s-penguin average is, in fact, doubly naïve since it neglects both the theoretical uncertainty discussed above, and the fact that experimental systematic uncertainties are correlated between the measurements of individual modes. For these reasons, we do not advocate the use of these averages, and provide them only for academic interest. Use with extreme caution, if at all.
• The presence of a small O(λ2) weak phase in the decay of the s-penguin modes introduces a phase shift that corresponds to the weak phase 2βs, which can be measured via time-dependent CP analysis of the decay Bs→ J/ψΦ. Using the Wolfenstein parameterization, and neglecting higher order terms in ρ, η and λ, we can estimate this shift to be:
S(s-penguin) = −ηCP×sin(2β)[cc-bar s]·(1 + Δ) ≡ −ηCP×sin(2φ1)[cc-bar s]·(1 + Δ). Numerically, using the global CKM fit results for the Wolfenstein parameters, one finds: Δ = 3.3%, which corresponds to a shift of 2βeff ≡ 2φ1eff of +2.1 degrees.

 Compilation of results for −η×S ≈ sin(2βeff) ≡ sin(2φ1eff) and C from s-penguin decays. eps png eps png Same, but without π0π0KS and ρ0KS, to allow closer inspection of the detail. eps png eps png Comparisons of averages in the different b→q q-bar s modes eps png eps png Same, but without π0π0KS and ρ0KS, to allow closer inspection of the detail. eps png eps png 2D comparisons of averages in the different b→q q-bar s modes. Taken from the PDG 2005 review on "CP Violation in Meson Decays" by D.Kirkby and Y.Nir. * This plot (and the averages) assume no correlations between the S and C measurements in each mode. An updated version of this plot is being prepared for ICHEP 2006. eps png

### Time-dependent Dalitz plot analysis of Bd → K+K−K0

Time-dependent amplitude analysis of the three-body Bd → K+KK0 decay allows additional information to be extracted from the data. In particular, the cosine of the effective weak phase difference (cos(2βeff) ≡ cos(2φ1eff)) can be determined, as well as the sine term that is obtained from quasi-two-body analysis. This information allows half of the degenerate solutions to be rejected. Furthermore, Dalitz plot analysis has enhanced sensitivity to direct CP violation.

A time-dependent Dalitz plot analysis of B0 → K+KK0 has been performed by BaBar. At present, the extracted parameters are not 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. Rather, the effective weak phase βeff ≡ φ1eff is directly determined for two significant resonant contributions: φK0 and f0K0, as well as the effective weak phase averaged over the Dalitz plot, with the CP properties of the individual components taken into account. In addition to the weak phase, BaBar also measure the time-dependent direct CP violation parameter ACP ( = -CCP).

Experiment φK0 f0K0 K+KK0 Reference
βeff ACP βeff ACP βeff ACP
BaBar
N(BB)=347m
0.06 ± 0.16 ± 0.05 −0.18 ± 0.20 ± 0.10 0.18 ± 0.19 ± 0.04 0.45 ± 0.28 ± 0.10 0.361 ± 0.079 ± 0.037 −0.034 ± 0.079 ± 0.025 hep-ex/0607112

Interpretations:

From the above results BaBar infer that the trigonometric reflection at π/2 - &betaeff is disfavoured at 4.6σ.

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

Due to possible significant penguin pollution, both the cosine and the sine coefficients of the Cabibbo-suppressed b → cc-bar d decays are free parameters of the theory. Absence of penguin pollution would result in Scc-bar d = − ηCP sin(2β) ≡ − ηCP sin(2φ1) and Ccc-bar d = 0 for the CP eigenstate final states (ηCP = +1 for both J/ψπ0 and D+D). For the non-CP eigenstates D*+−D−+, absence of penguin pollution (ie. no direct CP violation) gives A = 0, C+ = −C (but is not necessarily zero), S+ = 2 R sin(2β+δ)/(1+R2) and S = 2 R sin(2β−δ)/(1+R2). [With alternative notation, S+ = 2 R sin(2φ1+δ)/(1+R2) and S = 2 R sin(2φ1−δ)/(1+R2)]. Here R is the ratio of the magnitudes of the amplitudes for B0 → D*+D and B0 → D*D+, while δ is the strong phase between them. If there is no CP violation of any kind, then S+ = −S (but is not necessarily zero). The vector-vector final state D*+D* is a mixture of CP-even and CP-odd; the longitudinally polarized component is CP-even. Note that in the general case of non-negligible penguin contributions, the penguin-tree ratio and strong phase differences do not have to be the same for each helicity amplitude (likewise, they do not have to be the same for D*+D and D*D+).

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

Experiment SCP (J/ψ π0) CCP (J/ψ π0) Correlation Reference
BaBar
N(BB)=232M
−0.68 ± 0.30 ± 0.04 −0.21 ± 0.26 ± 0.06 0.08 (stat) PRD 74 011101 (2006)
Belle
N(BB)=152M
−0.72 ± 0.42 ± 0.09 0.01 ± 0.29 ± 0.03 −0.12 (stat) PRL 93, 261801 (2004)
Average −0.68 ± 0.25 −0.11 ± 0.20 0.00 HFAG correlated average
χ2 = 0.3/2 dof (CL=0.86 ⇒ 0.2σ)
Figures:

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Experiment SCP (D+D) CCP (D+D) Correlation Reference
BaBar
N(BB)=232M
−0.29 ± 0.63 ± 0.06 0.11 ± 0.35 ± 0.06 - PRL 95 (2005) 131802
Belle
N(BB)=535M
−1.12 ± 0.37 ± 0.09 −0.92 ± 0.23 ± 0.05 0.04 CKM2006 preliminary
Average −0.89 ± 0.33 −0.60 ± 0.20 0.03 HFAG correlated average
χ2 = 7.1/2 dof (CL=0.029 ⇒ 2.2σ)
Figures:

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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 experiments obtain the fraction of the transversely polarized component:

• BaBar measure RT = 0.125 ± 0.044 ± 0.007
• Belle measure RT = 0.19 ± 0.08 ± 0.01

We convert Im(λ) = S/(1 + C) and |λ|2 = (1 − C)/(1 + C), taking into account correlations. Note that Belle quote uncertainties due to the transversely polarized fraction RT separately from other systematic errors, while BaBar do not.

Experiment SCP (D*+ D*) CCP (D*+ D*) Correlation Reference
BaBar
N(BB)=227M
−0.75 ± 0.25 ± 0.03 0.06 ± 0.17 ± 0.03 0.04 (stat) PRL 95 (2005) 151804
Belle
N(BB)=152M
−0.75 ± 0.56 ± 0.10 ± 0.06 0.26 ± 0.26 ± 0.05 ± 0.01 - PLB 618 (2005) 34
Average −0.75 ± 0.23 0.12 ± 0.14 0.03 HFAG correlated average
χ2 = 0.4/2 dof (CL=0.82⇒ 0.2σ)
Figures:

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(*) Note that we have not pre-averaged the CP-odd fractions (and then accordingly rescaled the average sine coefficient). Since both data samples are independent, the results are (approximately) invariant under such a treatment, compared to the direct average that is performed here.

Experiment S+−(D*+D) C+−(D*+D) S−+(D*D+) C−+(D*D+) A(D*+−D−+) Correlations Reference
BABAR'05
N(BB)=232m
−0.54 ± 0.35 ± 0.07 0.09 ± 0.25 ± 0.06 −0.29 ± 0.33 ± 0.07 0.17 ± 0.24 ± 0.04 - PRL 95 (2005) 131802
Belle'04
N(BB)=152m
(combined fully and
partially rec. B decays)
−0.55 ± 0.39 ± 0.12 −0.37 ± 0.22 ± 0.06 −0.96 ± 0.43 ± 0.12 0.23 ± 0.25 ± 0.06 0.07 ± 0.08 ± 0.04 - PRL 93 (2004) 201802
Average −0.54 ± 0.27
CL=0.99 (0.0σ)
−0.16 ± 0.17
CL=0.18 (1.3σ)
−0.53 ± 0.27
CL=0.23 (1.2σ)
0.20 ± 0.18
CL=0.87 (0.2σ)
0.07 ± 0.09 uncorrelated averages

 Compilation of results for (left) sin(2βeff) ≡ sin(2φ1eff) = −ηCPS and (right) C from time-dependent b → cc-bar d analyses with CP eigenstate final states. The results are compared to the values from the corresponding charmonium averages. eps.gz png eps.gz png

### 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

### 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.

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)=232M
−0.21 ± 0.40 ± 0.05 −0.40 ± 0.23 ± 0.04 0.07 (stat) PRD 72 (2005) 051103
Belle
N(BB)=532M
−0.32 +0.36 −0.33 ± 0.05 0.20 ± 0.24 ± 0.05 0.08 (stat) PRD 74 (2006) 111104
Average −0.28 ± 0.26 −0.11 ± 0.17 0.07 HFAG correlated average
χ2 = 3.2/2 dof (CL=0.20 ⇒ 1.3σ)
KSπ0γ
(incl. K*γ)
BaBar
N(BB)=232M
−0.06 ± 0.37 −0.48 ± 0.22 0.05 (stat) PRD 72 (2005) 051103
Belle
N(BB)=532M
−0.10 ± 0.31 ± 0.07 0.20 ± 0.20 ± 0.06 0.08 (stat) PRD 74 (2006) 111104
Average −0.09 ± 0.24 −0.12 ± 0.15 0.06 HFAG correlated average
χ2 = 5.1/2 dof (CL=0.08 ⇒ 1.8σ)
Figures:

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### Time-dependent CP Asymmetries in Bd→ π+π−

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

Experiment SCP+π) CCP+π) Correlation Reference
BaBar
N(BB)=350M
−0.53 ± 0.14 ± 0.02 −0.16 ± 0.11 ± 0.03 −0.08 (stat) hep-ex/0607106
Belle
N(BB)=532M
−0.61 ± 0.10 ± 0.04 −0.55 ± 0.08 ± 0.05 −0.15 (stat) hep-ex/0608035
Average −0.59 ± 0.09 −0.39 ± 0.07 −0.10 HFAG correlated average
χ2 = 7.4/2 dof (CL=0.024 ⇒ 2.3σ)
Figures:

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Interpretations:
The Gronau-London isospin analysis allows a constraint on α ≡ φ2 to be extracted from the ππ system even in the presence of non-negligible penguin contributions. The analysis involves the SU(2) partners of the Bd→ π+π decay. The relevant branching ratios (given in units of 10−6) and CP-violating charge asymmetries are taken from HFAG - Rare Decays.

 BR(B0 → π+π−) = 5.2 ± 0.2 - BR(B+ → π+π0) = 5.7 ± 0.4 ACP(B+ → π+π0) = 0.04 ± 0.05 BR(B0 → π0π0) = 1.3 ± 0.2 ACP(B0 → π0π0) = 0.36 +0.33−0.31

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

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+π−π0 (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. eps.gz png eps.gz png

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 is convenient to transform the experimentally motivated CP parameters ACP(ρπ) and Cρπ into the physically motivated ones
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. Belle 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.

Experiment ACP+−π−+) C (ρ+−π−+) S (ρ+−π−+) ΔC (ρ+−π−+) ΔS (ρ+−π−+) Correlations Reference
BaBar
N(BB)=347M
−0.14 ± 0.04 ± 0.01 0.15 ± 0.09 ± 0.04 0.01 ± 0.12 ± 0.03 0.38 ± 0.09 ± 0.02 0.06 ± 0.13 ± 0.03 (total) hep-ex/0608002
Belle
N(BB)=447M
−0.12 ± 0.05 ± 0.03 −0.13 ± 0.09 ± 0.06 0.06 ± 0.13 ± 0.07 0.35 ± 0.10 ± 0.06 −0.12 ± 0.14 ± 0.07 (total) hep-ex/0609003
Average −0.13 ± 0.03 0.03 ± 0.07 0.03 ± 0.09 0.36 ± 0.07 −0.02 ± 0.10 (total) HFAG correlated average
χ2 = 4.7/5 dof (CL=0.45 ⇒ 0.8σ)
Figures:

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.
Experiment A−++−π−+) A+−+−π−+) Correlation Reference
BaBar
N(BB)=347M
−0.38 +0.15 −0.16 ± 0.07 0.03 ± 0.07 ± 0.03 0.62 hep-ex/0608002
Belle
N(BB)=447M
0.08 ± 0.17 ± 0.12 0.22 ± 0.08 ± 0.05 0.53 hep-ex/0609003
Average −0.19 ± 0.13 0.11 ± 0.06 0.46 HFAG correlated average
χ2 = 3.8/2 dof (CL=0.15 ⇒ 1.4σ)
Figures:

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Experiment C (ρ0π0) S (ρ0π0) Correlation Reference
Belle
N(BB)=447M
0.45 ± 0.35 ± 0.32 0.15 ± 0.57 ± 0.43 0.07 hep-ex/0609003

Interpretations:
The information given above can be used to extract α ≡ φ2.
From the measured form factor bilinears, BaBar extract a confidence level interpretation for α, and constraint 75° < α < 152° at 68% confidence level.
Belle has performed a similar analysis. In addition, Belle has also included information from the SU(2) partners of B → ρπ, which can be used to constrain α ≡ φ2 via an isospin pentagon relation. The isospin analysis uses as input the branching fractions and CP-violating charge asymmetries of all five ρπ decay modes (ρ+ππ+, ρ0π0, ρ+π0, ρ0π+). The relevant information is taken from HFAG - Rare Decays and is tabulated below. [Branching fractions are given in units of 10−6.] Those values that are not included in the table below can be obtained from the bilinear form-factors, eg. a measurement BR(B0 → ρ0π0)/ BR(B0 → ρ+−π−+) = 0.133 ± 0.022 ± 0.023, is extracted. With all information in the ρπ channels put together, Belle obtain the tighter constraint φ2 = (83+12−23)° (where the errors correspond to 1σ ie. 68.3% confidence level).

 BR(B0 → ρ+−π−+) = 24.0 ± 2.5 - BR(B+ → ρ+π0) = 10.8 +1.4−1.5 ACP(B+ → ρ+π0) = 0.02 ± 0.11 BR(B+ → π+ρ0) = 8.7 +1.0−1.1 ACP(B+ → π+ρ0) = −0.07 +0.12−0.13

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

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

### Time-dependent CP Asymmetries in Bd → ρ+ρ−

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:

• BABAR measure flong = 0.977 ± 0.024 +0.015−0.013,
• Belle measure flong = 0.941 +0.034−0.040 ±0.030

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

Experiment SCP+ρ) CCP+ρ) Correlation Reference
BaBar
N(BB)=350M
−0.19 ± 0.21 +0.05 −0.07 −0.07 ± 0.15 ± 0.06 −0.06 (stat) hep-ex/0607098
Belle
N(BB)=535M
0.19 ± 0.30 ± 0.07 −0.16 ± 0.21 ± 0.07 0.10 (stat) hep-ex/0702009
Average −0.06 ± 0.18 −0.11 ± 0.13 −0.00 HFAG correlated average
χ2 = 1.2/2 dof (CL=0.56 ⇒ 0.6σ)
Figures:

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Interpretations:
The Gronau-London isospin analysis allows a constraint on α ≡ φ2 to be extracted from the ρρ system even in the presence of non-negligible penguin contributions. The analysis involves the SU(2) partners of the Bd→ ρ+ρ decay. The relevant branching ratios (given in units of 10−6) and CP-violating charge asymmetries are taken from HFAG - Rare Decays.

 BR(B0 → ρ+ρ−) = 23.1 +3.2−3.3 - BR(B+ → ρ+ρ0) = 18.2 ± 3.0 ACP(B+ → ρ+ρ0) = −0.08 ± 0.13 BR(B0 → ρ0ρ0) = 1.2 ± 0.5 ACP(B0 → ρ0ρ0) NOT MEASURED YET

Belle obtain φ2 = (88 ± 17)° or 59° < φ2 < 117° at 90% confidence level.
BaBar find α ∈ [74, 117]° at 68% confidence level.

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

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

### Time-dependent CP Asymmetries in Bd → 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 - hep-ex/0612050

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

BaBar obtain αeff = (78.6 ± 7.3)°

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

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

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

Neutral B meson decays such as Bd → D+−π−+, Bd → D*+−π−+ and Bd → D+−ρ−+ provide sensitivity to γ ≡ φ3 because of the interference between the Cabibbo-favoured amplitude (e.g. B0 → Dπ+) with the doubly Cabibbo-suppressed amplitude (e.g. B0 → D+π). The relative weak phase between these two amplitudes is −γ ≡ −φ3 and, when combined with the BdBd-bar mixing phase, the total phase difference is −(2β+γ) ≡ −(2φ13).

The size of the CP violating effect in each mode depends on the ratio of magnitudes of the suppressed and favoured amplitudes, e.g., r = |A(B0 → D+π)/A(B0 → Dπ+)|. Each of the ratios r, rD*π and r is expected to be about 0.02, and can be obtained experimentally from the corresponding suppressed charged B decays, (e.g., B+ → D+π0) using isospin, or from self-tagging decays with strangeness (e.g., B0 → Ds+π), using SU(3). In the latter case, the theoretical uncertainties are hard to quantify. The smallness of the r values makes direct extractions from, e.g., the D+−π−+ system very difficult.

Both BABAR and Belle exploit partial reconstructions of D*+−π−+ to increase the available statistics. Both experiments also reconstruct D+−π−+ and D*+−π−+ fully, and BABAR includes the mode D+−ρ−+. Additional states with similar quark content are also possible, but for vector-vector final states an angular analysis is required, while states containing higher resonances may suffer from uncertainties due to nonresonant or other contributions.

BABAR and Belle use different observables:

• BABAR defines the time-dependent PDF by
f+−(η, Δt) = e−|Δt|/τ/4τ × [ 1 −+ Sζ sin(ΔmΔt) −+ η Cζ cos(ΔmΔt) ]
where the first (second) sign is used when the tagging meson is a B0 (B0-bar). [Note here that a tagging B0 (B0-bar) corresponds to −Sζ (+Sζ).] The parameters η and ζ take the values +1 and + (−1 and −) if the final state is, e.g., Dπ+ (D+π). The parameters actually measured by BABAR are denoted a and c. In the fit, the substitutions Cζ = 1 and Sζ = a −+ η bi − η ci are made. [Note that S+ = a − c and S = a + c, so a = (S+ + S)/2, c = (S − S+)/2.] The subscript i denotes the tagging category. These are motivated by the possibility of CP violation on the tag side, which is absent for semileptonic B decays (mostly lepton tags). The parameter a is independent of tag side CPV. These parameters are the same for the partially and fully reconstructed B decay analyses from BABAR.
• The parameters used by Belle in the analysis using partially reconstructed B decays, are similar to the S parameters defined above. However, in the Belle definition, a tagging B0 corresponds to a + sign in front of the sine coefficient; furthermore the correspondance between the super/subscript and the final state is opposite, so that S+− (BABAR) = − S−+ (Belle). In this analysis, only lepton tags are used, so there is no effect from tag-side CPV. In the Belle analysis using fully reconstructed B decays, this effect is measured and taken into account using D*+−l−+ν decays; in neither Belle analysis are the a, b and c parameters used. In the latter case, the measured parameters are 2RD(*)π sin( 2φ13 +− δD(*)π ); the definition is such that S+− (Belle) = −2RD*πsin( 2φ13 +− δD*π ). However, the definition includes an angular momentum factor (−1)L, and so for the results in the Dπ system, there is an additional factor of −1 in the conversion.

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

 Belle D*π (partial reconstruction): aπ* = − (S+ + S−)/2 cπ* = − (S+ − S−)/2 Belle D*π (full reconstruction): aπ* = + ( 2 RD*π sin( 2φ1+φ3 + δD*π ) + 2 RD*π sin( 2φ1+φ3 − δD*π ) )/2 cπ* = + ( 2 RD*π sin( 2φ1+φ3 + δD*π ) − 2 RD*π sin( 2φ1+φ3 − δD*π ) )/2 Belle Dπ (full reconstruction): aπ = − ( 2 RDπ sin( 2φ1+φ3 + δDπ ) + 2 RDπ sin( 2φ1+φ3 − δDπ ) )/2 cπ = − ( 2 RDπ sin( 2φ1+φ3 + δDπ ) − 2 RDπ sin( 2φ1+φ3 − δDπ ) )/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)=386m
fully
reconstructed
N(BB)=386m
aD*π −0.034 ± 0.014 ± 0.009 −0.040 ± 0.023 ± 0.010 −0.041 ± 0.019 ± 0.017 −0.039 ± 0.020 ± 0.013 −0.037 ± 0.011
CL=0.96 (0.0σ)
BaBar: PRD 71 (2005) 112003
(partially reco.)

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

Belle: PRD 73, 092003 (2006)
- COMBINED: −0.040 ± 0.014 ± 0.011
cD*π −0.019 ± 0.022 ± 0.013
(lepton tags only)
0.049 ± 0.042 ± 0.015
(lepton tags only)
−0.007 ± 0.019 ± 0.017 −0.011 ± 0.020 ± 0.013 −0.006 ± 0.014
CL=0.41 (0.8σ)
- COMBINED: −0.009 ± 0.014 ± 0.011
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. eps.gz png eps.gz png Averages of the D*π results. eps.gz png eps.gz png

Digression:

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

### 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 and Belle use consistent definitions for ACP+− and RCP+−, where

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

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

These observables have been measured so far for three D(*)K(*)− modes. Both Belle and BABAR use the CP even D decays to K+K and π+π in all three modes; both experiments also use only the D* → Dπ0 decay, which gives CP(D*) = CP(D). For CP-odd D decay modes, Belle use KSπ0, KSφ and KSω in all three analyses, and also use KSη in DK and D*K analyses. BABAR use KSπ0, KSφ and KSω for DK analysis.

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'05
N(BB)=232m
0.35 ± 0.13 ± 0.04 −0.06 ± 0.13 ± 0.04 0.90 ± 0.12 ± 0.04 0.86 ± 0.10 ± 0.05 PRD 73, 051105 (2006)
Belle'06
N(BB)=275m
0.06 ± 0.14 ± 0.05 −0.12 ± 0.14 ± 0.05 1.13 ± 0.16 ± 0.08 1.17 ± 0.14 ± 0.14 PRD 73, 051106 (2006)
Average 0.22 ± 0.10 −0.09 ± 0.10 0.98 ± 0.10 0.94 ± 0.10
D*CPK BABAR'04
N(BB)=123m
−0.10 ± 0.23 +0.03−0.04 - 1.06 ± 0.26 +0.10−0.09 - PRD 71 (2005) 031102
Belle'06
N(BB)=275m
−0.20 ± 0.22 ± 0.04 0.13 ± 0.30 ± 0.08 1.41 ± 0.25 ± 0.06 1.15 ± 0.31 ± 0.12 PRD 73, 051106 (2006)
Average −0.15 ± 0.16 0.13 ± 0.31 1.25 ± 0.19 1.15 ± 0.33
DCPK* BABAR'05
N(BB)=232m
−0.08 ± 0.19 ± 0.08 −0.26 ± 0.40 ± 0.12 1.96 ± 0.40 ± 0.11 0.65 ± 0.26 ± 0.08 PRD 72 (2005) 071103(R)
Average −0.08 ± 0.21 −0.26 ± 0.42 1.96 ± 0.41 0.65 ± 0.27
(*) 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. eps.gz png eps.gz png CP+ only eps.gz png eps.gz png CP- only eps.gz png eps.gz png

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.

Plots upcoming.

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

A modification of the GLW idea has been suggested by Atwood, Dunietz and Soni, where B → DK with D → K+π (or similar) and the charge conjugate decays are used. Here, the favoured (b→c) B decay followed by the doubly CKM-suppressed D decay interferes with the suppressed (b→u) B decay followed by the CKM-favored D decay. The relative similarity of the combined decay amplitudes enhances the possible CP asymmetry. BABAR and Belle use consistent definitions for AADS and RADS, where (for example for the B → DK, D → K+π mode)

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

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

DK
D→Kπ
BaBar'05
N(BB)=232m
- 0.013 +0.011−0.009 PRD 72 (2005) 032004
Belle'05
N(BB)=386m
- 0.000 ± 0.008 ± 0.001 BELLE-CONF-0552 (hep-ex/0508048)
Average - 0.006 ± 0.006
D*K
D* → Dπ0
D→Kπ
BaBar'05
N(BB)=232m
- −0.002+0.010−0.006 PRD 72 (2005) 032004
Average - −0.002+0.010−0.006
D*K
D* → Dγ
D→Kπ
BaBar'05
N(BB)=232m
- 0.011+0.018−0.013 PRD 72 (2005) 032004
Average - 0.011+0.018−0.013
DK*
D→Kπ
BaBar'05
N(BB)=232m
−0.22 ± 0.61 ± 0.17 0.046 ± 0.031 ± 0.008 PRD 72 (2005) 071104
Average −0.22 ± 0.63 0.046 ± 0.032
DK
D→Kππ0
BaBar'06
N(BB)=226m
- 0.012 ± 0.012 ± 0.009 hep-ex/0607065
Average - 0.012 ± 0.015
 Compilation of the above results. 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(δB+δD) cos(γ) ≡ rB2 + rD2 + 2 rB rD cos(δB+δD) cos(φ3), AADS = 2 rB rD sin(δB+δD) sin(γ) / RADS ≡ 2 rB rD sin(δB+δD) sin(φ3) / RADS,

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

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

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

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

D→Kπ
Belle'05
N(BB)=386m
0.10 ± 0.22 ± 0.06 0.0035 +0.0008−0.0007 ± 0.0003 BELLE-CONF-0552 (hep-ex/0508048)
Average 0.10 ± 0.23 0.0035 +0.0009−0.0008

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

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

Results are available from both Belle and BaBar using B → D K, B → D*K and B → DK*. Belle use the D* decay to Dπ0 only, while BaBar also use Dγ, and take the effective strong phase shift into account. In all cases the decay D → KSπ+π is used.

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

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

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

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

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

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

Averages are performed using the following procedure, which is based on a set of reasonable, though imperfect, assumptions.

• It is assumed that both experiments use the same D decay model. Therefore, we do not rescale the results to a common model.
• It is further assumed that the model uncertainty is 100% correlated between experiments, and therefore this source of error is not used in the averaging procedure.
• We include in the average the effect of correlations within each experiments set of measurements.
• At present it is unclear how to assign an average model uncertainty. We have not attempted to do so. Our average includes only statistical and systematic error. An unknown amount of model uncertainty should be added to the final error.
• We follow the suggestion of Gronau in making the DK* averages. Explicitly, we assume that the selection of K* → KSπ is the same in both experiments (so that κ, rs and δs are the same), and drop the additional source of model uncertainty assigned by Belle due to possible nonresonant decays.
• We do not consider common systematic errors, other than the D decay model.
Mode Experiment x+ y+ x- y- Correlation Reference
DK
D→KSπ+π
BaBar
N(BB)=347M
−0.072 ± 0.056 ± 0.014 ± 0.029 −0.033 ± 0.066 ± 0.007 ± 0.018 0.041 ± 0.059 ± 0.018 ± 0.011 0.056 ± 0.071 ± 0.007 ± 0.023 (stat) hep-ex/0607104
Belle
N(BB)=386M
−0.135 +0.069 −0.070 ± 0.017 ± 0.051 −0.085 +0.090 −0.086 ± 0.009 ± 0.066 0.025 +0.072 −0.080 ± 0.013 ± 0.068 0.170 +0.093 −0.117 ± 0.016 ± 0.049 (stat) (model) PRD 73, 112009 (2006)
Average
No model error
−0.097 ± 0.045 −0.051 ± 0.053 0.045 ± 0.047 0.093 ± 0.058 (stat+syst) HFAG correlated average
χ2 = 1.5/4 dof (CL=0.83 ⇒ 0.2σ)
Figures:
NB. The contours in these plots
do not include model errors.

eps.gz png

eps.gz png

eps.gz png
D*K
D*→Dπ0 & D*→Dγ
D→KSπ+π
BaBar
N(BB)=347M
0.084 ± 0.088 ± 0.015 ± 0.018 0.096 ± 0.111 ± 0.032 ± 0.017 −0.106 ± 0.091 ± 0.020 ± 0.009 −0.019 ± 0.096 ± 0.022 ± 0.016 (stat) hep-ex/0607104
Belle
N(BB)=386M
0.032 +0.120 −0.116 ± 0.004 ± 0.049 0.008 +0.137 −0.136 ± 0.011 ± 0.074 −0.128 +0.167 −0.146 ± 0.023 ± 0.071 −0.339 +0.172 −0.158 ± 0.027 ± 0.053 (stat) (model) PRD 73, 112009 (2006)
Average
No model error
0.067 ± 0.071 0.061 ± 0.088 −0.110 ± 0.080 −0.101 ± 0.085 (stat+syst) HFAG correlated average
χ2 = 3.2/4 dof (CL=0.52 ⇒ 0.6σ)
Figures:
NB. The contours in these plots
do not include model errors.

eps.gz png

eps.gz png

eps.gz png
DK*−
K*− → KSπ
D→KSπ+π
BaBar
N(BB)=227M
−0.070 ± 0.230 ± 0.130 ± 0.030 −0.010 ± 0.320 ± 0.180 ± 0.050 −0.200 ± 0.200 ± 0.110 ± 0.030 0.260 ± 0.300 ± 0.160 ± 0.030 - hep-ex/0507101
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
−0.094 ± 0.144 −0.007 ± 0.146 −0.480 ± 0.173 −0.056 ± 0.253 (stat+syst) HFAG correlated average
χ2 = 4.6/4 dof (CL=0.33 ⇒ 1.0σ)
Figures:
NB. The contours in these plots
do not include model errors.

eps.gz png

eps.gz png

eps.gz png
 Figures: NB. The uncertainities assignedto the averages given in these plotsdo not include model errors. eps.gz png eps.gz png eps.gz png eps.gz png

Digression:

 Constraining γ ≡ φ3: The measurements of x+,− and y+,− in the various D(*)K(*) decay modes can be used to place bounds on γ ≡ φ3. Both experiments have done so using frequentist techniques. BaBar obtain γ = (92 ± 41 ± 11 ± 12)° (from DK− & D*K−) Belle obtain φ3 = (53 +15−18 ± 3 ± 9)° (from DK−,D*K− & DK*−) The experiments also obtain values for the hadronic parameters rB (DK−) < 0.140 (1σ) δ B (DK−) = (118 ± 63 ± 19 ± 36)° rB (DK−) = 0.16 ± 0.05 ± 0.01 ± 0.05 δ B (DK−) = (146 +19−20 ± 3 ± 23)° 0.017 < rB (D*K−) < 0.203 δ B (D*K−) = (298 ± 59 ± 18 ± 10)° rB (D*K−) = 0.18 +0.11−0.10 ± 0.01 ± 0.05 δ B (D*K−) = (302 +34−35 ± 6 ± 23)° . . rB (DK*−) = 0.56 +0.22−0.16 ± 0.04 ± 0.08 δ B (DK*−) = (243+20−23 ± 3 ± 50 )° For attempts to extract γ ≡ φ3 from the combined BaBar and Belle results, visit the CKMfitter and UTfit sites. Note that the above results suffer an ambiguity: γ → γ + π ≡ φ3 → φ3 + π, δ → δ + π. We quote the result which is consistent with the Standard Model fit.

This page is maintained by Gianluca Cavoto, Tim Gershon and Karim Trabelsi,