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Results on Time-Dependent CP Measurements:
Summer 2005 (Lepton-Photon, Uppsala, Sweden and HEP 2005, Lisboa, Portugal).

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

Measurements related to the CKM angle β / φ1:

§   Time-Dependent CP Asymmetries in b → cc-bar s Transitions (sin2β/sin2φ1)
§   Time-Dependent CP Asymmetries in b → qq-bar s (penguin) Transitions (sin2βeff/sin2φ1,eff)
§   Time-Dependent Transversity Analysis of Bd → J/ψK* (cos2β/cos2φ1)
§   Time-Dependent CP Asymmetries in b → cc-bar d Transitions (eg. J/ψ π0, D(*)+D(*)−) (sin2βeff/sin2φ1,eff)
§   Time-Dependent CP Asymmetries in b → cu-bar d Transitions (eg. D π0) (2β/2φ1)
§   Time-Dependent Analysis of Bd → KSπ0γ

Measurements related to the CKM angle α / φ2:

§   Time-Dependent CP Asymmetries in Bd → π+π (sin2αeff/sin2φ2,eff)
§   Time-Dependent CP Asymmetries in Bd → (ρπ)0 (sin2αeff and α/sin2φ2,eff and φ2)
§   Time-Dependent CP Asymmetries in Bd → ρ+ρ (sin2αeff/sin2φ2,eff)
Measurements related to the CKM angle γ / φ3:

§   Time-Dependent CP Asymmetries in Bd → D+−π−+, etc. (sin(2β+γ)/sin(2φ13))
§   GLW Analyses of B → D(*)K(*) (γ/φ3)
§   ADS Analyses of B → D(*)K(*) and B → D(*)π (γ/φ3)
§   Dalitz Plot Analysis of B → D(*)K(*)− with D → KSπ+π, ... (γ/φ3)

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 Ref. / Comments
τ(Bd) (1.528 ± 0.009) ps HFAG - Oscillations/Lifetime
Δmd (0.509 ± 0.004) ps−1 HFAG - Oscillations/Lifetime
|A|2
(CP-odd fraction in
B0→ J/ψK* CP sample)
0.245 ± 0.015 ± 0.004
(note: acceptance-corrected central value; the uncorrected value is: 0.230)
BABAR: PRD 71 (2005) 032005
0.195 ± 0.012 ± 0.008 Belle: PRL 95 (2005) 091601
0.217 ± 0.010 Average

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 Ref. / Comments
Charmonium: N(BB)=227m N(BB)=386m - BABAR: PRL 94 (2005) 161803
BELLE-CONF-0569 (hep-ex/0507037)
J/ψKS, ψ(2S)KS, χc1KS, ηCKSCP=-1) 0.75 ± 0.04stat 0.668 ± 0.047stat (J/ψKS only)
J/ψKLCP=+1) 0.57 ± 0.09stat 0.619 ± 0.069stat
J/ψK*0 (K*0 → KSπ0) 0.96 ± 0.32stat -
All charmonium 0.722 ± 0.040 ± 0.023 0.652 ± 0.039 ± 0.020 0.685 ± 0.032
(0.028stat-only)
CL = 0.27

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

Parameter: sin(2β)/sin(2φ1)
Experiment Value Ref. / Comments
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 0.687 ± 0.032 (0.028stat-only) CL = 0.48

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

β/φ1 = (21.7 +1.3−1.2 or β/φ1 = (68.3 +1.2−1.3

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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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| for the following averages.

Parameter: C=−A (if not stated otherwise)
Mode BABAR Belle Average Ref. / Comments
Charmonium N(BB)=227m N(BB)=386m - BABAR: PRL 94 (2005) 161803
BELLE-CONF-0569 (hep-ex/0507037)
J/ψKS, ψ(2S)KS, χc1KS, ηCKS |λ| = 0.950 ± 0.031 ± 0.013
C = 0.051 ± 0.033 ± 0.014
C = 0.021 ± 0.034stat (J/ψKS only)
J/ψKL C = −0.049 ± 0.039stat
J/ψK*0 (K*0 → KSπ0) -
All charmonium −0.010 ± 0.026 ± 0.036 0.026 ± 0.041
(0.020stat-only)
CL = 0.31

Digression and plots:

Constraining CJ/ψ Ks from ACP(B+ → J/ψ K+) and ASL:

As suggested by Y. Nir, one can obtain a powerful SM constraint on |λ| = |q/p||A-bar/A| via the relations ASL = (1−|q/p|4)/(1+|q/p|4) and ACP(B+ → J/ψ K+) = (|A-bar/A|2−1)/(|A-bar/A|2+1), where ASL denotes the CP asymmetry in semileptonic B decays, and ACP(B+ → J/ψ K+) is the CP-violating charge asymmetry measured in B+ → J/ψ K+ decays. We take the ASL result from the HFAG oscillation group, ASL = −0.0026 ± 0.0067, and average the ACP(B+ → J/ψ K+) results from BABAR, Belle and CLEO, to find ACP(B+ → J/ψ K+) = −0.007 ± 0.019. These give |q/p| = 1.0013 ± 0.0034 and |A-bar/A| = 0.993 ± 0.018, and hence |λ|indirect = 0.994 ± 0.018, corresponding to C = 0.006 ± 0.018.

Discussion: the amplitude relation between neutral and charged B → J/ψ K decays has been found by Fleischer-Mannel to hold up to negligible corrections of the order O(λ3). However, the identification of |λ|, measured through the C coefficient in B0 → J/ψ K0, with |q/p||A-bar/A| assumes ΔΓBd=0. The systematic error on C from a width difference ΔΓBdBd~0.02 has been estimated by BABAR to be 0.0009.



Time-dependent CP Asymmetries in b → qq-bar s (penguin) Transitions

Parameter: sin(2βeff)/sin(2φ1eff) (sin(2β)/sin(2φ1)if β/φ1 dominant weak phase)
Mode BABAR
N(BB)=209-227m
Belle
N(BB)=386m
Average Ref. / Comments
φK0 0.50 ± 0.25 +0.07−0.04 0.44 ± 0.27 ± 0.05 0.47 ± 0.19
CL=0.87 (0.2σ)
BABAR: PRD 71 (2005) 091102(R)
BELLE-CONF-0569 (hep-ex/0507037)
η'K0 0.36 ± 0.13 ± 0.03 0.62 ± 0.12 ± 0.04 0.50 ± 0.09
CL=0.16 (1.4σ)
BABAR-CONF-05/014 (hep-ex/0507087)
BELLE-CONF-0569 (hep-ex/0507037)
f0KS 0.95 +0.23−0.32 ± 0.10 0.47 ± 0.36 ± 0.08 0.75 ± 0.24
CL=0.32 (1.0σ)
BABAR-CONF-04/019 (hep-ex/0408095)
BELLE-CONF-0569 (hep-ex/0507037)
π0KS 0.35 +0.30−0.33 ± 0.04 0.22 ± 0.47 ± 0.08 0.31 ± 0.26
CL=0.82 (0.2σ)
BABAR: PRD71 (2005) 111102
BELLE-CONF-0569 (hep-ex/0507037)
π0π0KS −0.84 ± 0.71 ± 0.08 - −0.84 ± 0.71 BABAR-CONF-05/020 (hep-ex/0508017)
ωKS 0.50 +0.34−0.38 ± 0.02 0.95 ± 0.53 +0.12−0.15 0.63 ± 0.30
CL=0.49 (0.7σ)
BABAR-CONF-05/001 (hep-ex/0503018)
BELLE-CONF-0569 (hep-ex/0507037)
K+KK0
(excluding φK0)
0.41 ± 0.18 ± 0.07 ± 0.11CP-even
(fCP-even= 0.89 ± 0.08 ± 0.06 [moments])
0.60 ± 0.18 ± 0.04 +0.19−0.12CP-even
(fCP-even= 0.93 ± 0.09 ± 0.05 [SU(2)])
0.51 ± 0.14 +0.11−0.08
CL=0.38 (0.9σ)
(rescaled to average fCP-even= 0.91 ± 0.07)
BABAR-CONF-05/002 (hep-ex/0507016)
BELLE-CONF-0569 (hep-ex/0507037)
KSKSKS 0.63 +0.28−0.32 ± 0.04 0.58 ± 0.36 ± 0.08 0.61 ± 0.23
CL=0.92 (0.1σ)
BABAR-CONF-05/012 (hep-ex/0507052)
BELLE-CONF-0569 (hep-ex/0507037)
Naïve b→s penguin average 0.50 ± 0.06 CL=0.79 (0.3σ)
Direct comparison of charmonium and s-penguin averages (see comments below): CL=0.0092 (2.6σ)

Please note that

The cosine coefficient:

Parameter: C=−A (if not stated otherwise)
Mode BABAR
N(BB)=209-227m
Belle
N(BB)=386m
Average Ref. / Comments
φK0 0.00 ± 0.23 ± 0.05 −0.14 ± 0.17 ± 0.07 −0.09 ± 0.14
CL=0.64 (0.5σ)
BABAR: PRD 71 (2005) 091102(R)
BELLE-CONF-0569 (hep-ex/0507037)
η'K0 −0.16 ± 0.09 ± 0.02 0.04 ± 0.08 ± 0.06 −0.07 ± 0.07
CL=0.14 (1.5σ)
BABAR-CONF-05/014 (hep-ex/0507087)
BELLE-CONF-0569 (hep-ex/0507037)
f0KS −0.24 ± 0.31 ± 0.15 0.23 ± 0.23 ± 0.13 0.06 ± 0.21
CL=0.28 (1.1σ)
BABAR-CONF-04/019 (hep-ex/0408095)
BELLE-CONF-0569 (hep-ex/0507037)
π0KS 0.06 ± 0.18 ± 0.03 −0.11 ± 0.18 ± 0.08 −0.02 ± 0.13
CL=0.53 (0.6σ)
BABAR: PRD71 (2005) 111102
BELLE-CONF-0569 (hep-ex/0507037)
π0π0KS 0.27 ± 0.52 ± 0.13 - 0.27 ± 0.54 BABAR-CONF-05/020 (hep-ex/0508017)
ωKS −0.56 +0.29−0.27 ± 0.03 −0.19 ± 0.39 ± 0.13 −0.44 ± 0.23
CL=0.46 (0.7σ)
BABAR-CONF-05/001 (hep-ex/0503018)
BELLE-CONF-0569 (hep-ex/0507037)
K+KK0
(excluding φK0)
0.23 ± 0.13 0.06 ± 0.11 ± 0.07 0.15 ± 0.09
CL=0.36 (0.9σ)
BABAR-CONF-05/002 (hep-ex/0507016)
BELLE-CONF-0569 (hep-ex/0507037)
KSKSKS −0.10 ± 0.25 ± 0.05 −0.50 ± 0.23 ± 0.06 −0.31 ± 0.17
CL=0.25 (1.1σ)
BABAR-CONF-05/012 (hep-ex/0507052)
BELLE-CONF-0569 (hep-ex/0507037)
Naïve b→s penguin average −0.04 ± 0.04 CL=0.30 (1.0σ)


Compilation of results for −η×S ≈ sin(2β)/sin(2φ1) and C from charmonium and s-penguin decays.

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

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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.

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Time-dependent Transversity Analysis of B0→ J/ψK* (cos(2β)/cos(2φ1))

The BABAR and Belle collaborations have performed measurements of (cos(2β)/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.

Experiment sin(2β)/sin(2φ1)J/ψK* cos(2β)/cos(2φ1)J/ψK* Correlation Ref. / Comments
BABAR'04
N(BB)=88m
−0.10 ± 0.57 ± 0.14 3.32 +0.76−0.96 ± 0.27 −0.37 PRD 71 (2005) 032005
Belle'04
N(BB)=275m
0.24 ± 0.31 ± 0.05 0.56 ± 0.79 ± 0.11
[using Solution II]
+0.22 PRL 95 (2005) 091601
Average IN PREPARATION
(CL = → xσ)
IN PREPARATION
(CL = → xσ)
? See remark below table

Note that due to the strong non-Gaussian character of the BABAR measurement (although the result is far positive, the confidence level for cos(2β)>0 is only 89%), the interpretation of the average given above has to be done with the greatest care.


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)/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 SJ/ψπ0 CJ/ψπ0 = −AJ/ψπ0 Correlation Ref. / Comments
BABAR'05
N(BB)=232m
−0.68 ± 0.30 ± 0.04 −0.21 ± 0.26 ± 0.09 0.08 BABAR-CONF-05/016 (hep-ex/0507074)
Belle'04
N(BB)=152m
−0.72 ± 0.42 ± 0.09 0.01 ± 0.29 ± 0.03 −0.12 PRL 93 (2004) 261801
Average −0.69 ± 0.25
CL=0.94 (0.1σ)
−0.11 ± 0.20
CL=0.58 (0.6σ)
uncorrelated averages
Figures:

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Experiment SD+D− CD+D− Correlation Ref. / Comments
BABAR'05
N(BB)=232m
−0.29 ± 0.63 ± 0.06 0.11 ± 0.35 ± 0.06 - PRL 95 (2005) 131802
Belle NOT YET AVAILABLE NOT YET AVAILABLE - -
Average −0.29 ± 0.63 0.11 ± 0.36 - -

We convert Im(λ) = S/(1 + C) and |λ|2 = (1 − C)/(1 + C), taking into account correlations:

Experiment SD*+D*− CD*+D*− Correlation Ref. / Comments
BABAR'05
N(BB)=227m
−0.75 ± 0.25 ± 0.03 0.06 ± 0.17 ± 0.03 Table PRL 95 (2005) 151804
RT = 0.125 ± 0.044 ± 0.007
Belle'04
N(BB)=152m
−0.75 ± 0.56 ± 0.10 ± 0.06[RT] 0.26 ± 0.26 ± 0.05 ± 0.01[RT] - PLB 618 (2005) 34
RT = 0.19 ± 0.08 ± 0.01
Average(*) −0.75 ± 0.23
CL=1.00 (0.0σ)
0.12 ± 0.14
CL=0.52 (0.6σ)
uncorrelated averages
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. Also: due to the lack of correlations coefficients, we have performed an uncorrelated average here.

Experiment S+−(D*+D) C+−(D*+D) S−+(D*D+) C−+(D*D+) A(D*+−D−+) Ref. / Comments
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.

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Time-Dependent CP Asymmetries in 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, ie. 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. This information allows to resolve the ambiguity in the measurement of 2β/2φ1 from sin(2β)/sin(2φ1).

Results of such an analysis are available from Belle. 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.

Experiment β/φ1 Reference / comments
Belle'05
N(BB)=386m
16 ± 21 ± 12 BELLE-CONF-0546 (hep-ex/0507065)

Note that direct measurements of phases often suffer from unusual and non-Gaussian behaviour. The statistical uncertainty of the Belle measurement is obtained from a toy Monte Carlo study.


Time-dependent Analysis of B0→ KSπ0γ

Time-dependent analysis of B0→ KSπ0γ, probes 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)/sin(2φ1) (with an assumption that the Standard Model dipole operator is dominant). Corrections to the above 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 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 SKsπ0γ CKsπ0γ = −AKsπ0γ Correlation Ref. / Comments
K*(892)γ BABAR'05
N(BB)=232m
−0.21 ± 0.40 ± 0.05 −0.40 ± 0.23 ± 0.04 −0.064 PRD 72 (2005) 051103
Belle'05
N(BB)=386m
0.01 ± 0.52 ± 0.11 −0.11 ± 0.33 ± 0.09 0.002 BELLE-CONF-0570 (hep-ex/0507059)
Average −0.13 ± 0.32
CL=0.74 (0.3σ)
−0.31 ± 0.19
CL=0.48 (0.7σ)
- uncorrelated average
KSπ0γ
(incl. K*γ)
BABAR'05
N(BB)=232m
−0.06 ± 0.37 −0.48 ± 0.22 −0.066 PRD 72 (2005) 051103
Belle'05
N(BB)=386m
0.08 ± 0.41 ± 0.10 −0.12 ± 0.27 ± 0.10 0.004 BELLE-CONF-0570 (hep-ex/0507059)
Average 0.00 ± 0.28
CL=0.80 (0.3σ)
−0.35 ± 0.17
CL=0.32 (1.0σ)
- uncorrelated average
Figures:

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K*γ only: eps png
KSπ0 only: eps png

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K*γ only: eps png
KSπ0 only: eps png



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 Sππ Cππ = −Aππ Correlation Ref. / Comments
BABAR'04
N(BB)=227m
−0.30 ± 0.17 ± 0.03 −0.09 ± 0.15 ± 0.04 −0.016 PRL 95 (2005) 151803
Belle'05
N(BB)=275m
−0.67 ± 0.16 ± 0.06 −0.56 ± 0.12 ± 0.06 −0.09 PRL 95 (2005) 101801
Average −0.50 ± 0.12 −0.37 ± 0.10 −0.056 χ2 = 7.9 (CL=0.019 ⇒ 2.3σ)
Figures:

eps gif gif (high res)

eps gif gif (high res)

Digression and plots:

(The following numerical exercises involve the SU(2) and SU(3) partners of the Bd→ π+π decay. The relevant branching ratios and CP-violating charge asymmetries are taken from HFAG - Rare Decays (Moriond 2005) averages.)

Constraining α: using as input the measured Cππ and Sππ coefficients together with the present (HFAG) ππ branching fractions and CP asymmetries (including the direct CP-asymmetry measurement for B0→ π0π0), one can perform the Gronau-London isospin analysis. The plot on the right hand side uses the statistical interpretation of the CKMfitter analysis (Rfit) (see the UTfit pages for a Bayesian interpretation). Here, it does not include the Fiertz treatment of electroweak penguins for ππ. Including it would lead to a shift in α of approximately −2 deg. All other SU(2)-breaking effects are also neglected.
eps gif gif (high res)
The Penguin-to-tree ratio: using as input the measured Cππ and Sππ coefficients together with the Wolfenstein parameters ρ and η from the Global CKM fit using standard constraints, one can infer module and phase of the complex penguin to tree (P/T) ratio in Bd→ π+π decays within the Standard Model. Note that the definition of P/T is convention-dependent (see, e.g., GroRos02). One can choose to eliminate the charm quark in the penguin loop using CKM unitarity, so that the amplitudes are parameterized as follows:
A(Bd→ π+π)  =  Ru e i γ T + Rt e−i β P,
A(Bd-bar→ π+π)  =  Ru e−i γ T + Rt e i β P.
Plots for confidence level representations of the P/T phase versus its module can be found on the corresponding CKMfitter and UTfit pages.


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

The "Quasi-two-body" (Q2B) CP(t) analysis of Bd→ ρ+−π−+ decays (performed by Belle) assumes a narrow width approximation for the ρ meson. The interference regions in the π+ππ0 Dalitz plot are removed by kinematic cuts. Dilution of the CP results due to residual interference effects is not accounted for in the systematic errors. 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. Note that the BABAR analysis uses a full Dalitz plot approach and hence avoids the systematic effects due to the Q2B approximation.

At present we do not apply a rescaling of the results due to the fit dependence on the Bd lifetime and oscillation frequency.

Experiment ACP(ρπ) Sρπ Cρπ ΔSρπ ΔCρπ Correlations Ref. / Comments
BABAR'04
N(BB)=213m
−0.088 ± 0.049 ± 0.013 −0.10 ± 0.14 ± 0.04 0.34 ± 0.11 ± 0.05 0.22 ± 0.15 ± 0.03 0.15 ± 0.11 ± 0.03 Table BABAR-CONF-04/038, hep-ex/0408099
Belle'04
N(BB)=152m
−0.16 ± 0.10 ± 0.02 −0.28 ± 0.23 +0.10−0.08 0.25 ± 0.17 +0.02−0.06 −0.30 ± 0.24 ± 0.09 0.38 ± 0.18 +0.02−0.04 Table PRL 94 (2005) 121801
Average −0.102 ± 0.045 −0.13 ± 0.13 0.31 ± 0.10 0.09 ± 0.13 0.22 ± 0.10 Table  
Significance of CPV in the decay: Δχ2 = χ2(Acp=C=0) − χ2 = 14.5 (CL = 0.00070, that is: 3.4σ)

Digression and plots:

CP violation in the decay: 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.

Taking into account experimental correlations, one finds
A+−(ρπ) = −0.15 ± 0.09,
A−+(ρπ) = −0.47 +0.13−0.14.
The two quantities have a linear correlation coefficient of +59%. See right hand plot for a confidence level representation in the A+−(ρπ) versus A−+(ρπ) plane.


eps gif gif (high res)

Flavor-charge specific branching fractions:
the charge and flavor asymmetry parameters ACP(ρπ), Cρπ and ΔCρπ can be used to derive flavor-charge specific rates from the HFAG branching fraction BR(Bd→ ρ+−π−+) = (24.0 ± 2.5)×10−6.

For individual B flavors and ρ charges, we define:
BR( Bf → ρ+-π−+ ) =
0.5 × ( 1 +- ACP(ρπ)) × ( 1 + f (Cρπ +- ΔCρπ) ) × BR( B → ρπ ),
where Bf = Bd for f=1 and Bf = Bd-bar for f=−1.
BR( B → ρπ ) indicates the total BF for all ρπ modes combined.
One finds
BR( Bd → ρ+π ) = (16.5 +2.7−2.5)×10−6,
BR( Bd → ρπ+ ) = (14.4 +2.4−2.2)×10−6,
BR( Bd-bar → ρ+π ) = (5.1 +1.9−1.7)×10−6,
BR( Bd-bar → ρπ+ ) = (12.0 +2.2−2.0)×10−6.

For flavor-averaged inclusive branching fractions:
BRB→ ρπ(+−) = 0.5(BRB→ρ+π + BRB-bar→ ρπ+),
BRB→ ρπ(−+) = 0.5(BRB→ρπ+ + BRB-bar→ ρ+π),
where the individual charge-flavor branching fractions are defined on the left. The total (flavor-averaged) ρπ branching fraction is the sum BRB→ ρπ(+−) + BRB→ ρπ(−+). One finds

BRB→ ρπ(+−) = (13.9 +2.2−2.1)×10−6,
BRB→ ρπ(−+) = (10.1 +2.1−1.9)×10−6,
with an anti-correlation of −28% among the two branching fractions.

The BABAR Collaboration has performed a full time-dependent Dalitz plot analysis of the decay Bd → (ρπ)0 → π+ππ0, which allows to simultaneously determine the complex decay amplitudes and the CP-violating weak phase α. The analysis follows the idea of Snyder and Quinn (1993). BABAR uses a model that consists of charged and neutral ρ(770) resonances and their radial excitations ρ(1450) and ρ(1700). No non-resonant contributions are found. BABAR determines 16 coefficients of the form factor bilinears from the fit to data. The unknown amplitude parameters, among which are the phases δ+−=arg[A−+A+−*] and the UT angle α, are determined from a subsequent fit to the 16 bilinear coefficients.

Experiment α/φ2 (°) δ+− (°) Ref. / Comments
BABAR'04
N(BB)=213m
113 +27−17 ± 6 −67 +28−31 ± 7 BABAR-CONF-04/038, hep-ex/0408099
Belle not yet available
Confidence levels for α (left hand plot) and δ+− (right hand plot) as found by BABAR
eps gif gif (high res)

eps gif gif (high res)

Note that Dalitz plot phases are non-Gaussian quantities in general. Only marginal constraints are obtained beyond 2σ.


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 BABAR Collaboration determines the fraction of longitudinally polarized events with an angular analysis to be flong = 0.978 ± 0.014+0.021−0.029, so that a per-event transversity analysis can be avoided and only the longitudinal CP parameters are determined. The Belle collaboration determine the same quantity to be flong = 0.951 +0.033−0.039 +0.029−0.031 At present we do not apply a rescaling of the results to a common, updated set of input parameters.

Experiment Sρρ,long Cρρ,long Correlation Ref. / Comments
BABAR'05
N(BB)=232m
−0.33 ± 0.24 +0.08−0.14
flong = 0.978 ± 0.014+0.021−0.029
−0.03 ± 0.18 ± 0.09 −0.042 PRL 95 (2005) 041805
Belle'05
N(BB)=275m
0.09 ± 0.42 ± 0.08
flong = 0.951 +0.033−0.039 +0.029−0.031
0.00 ± 0.30 +0.09−0.10 0.06 BELLE-CONF-0545 (hep-ex/0507039)
Average −0.21 ± 0.22 −0.03 ± 0.17 0.01  

Digression and plots:

(The following numerical exercises involve the SU(2) partners of the Bd → ρ+ρ decay. The relevant branching ratios, CP-violating charge asymmetries and fractions of longitudinal polarization are taken from HFAG - Rare Decays averages. Tests of the isospin relations show that within the present experimental uncertainties the branching fraction for ρ+ρ00ρ0) is expected to decrease (rise) in the future.)

Constraining α:
Using as input the measured Cρρ,long and Sρρ,long coefficients together with the average (HFAG) ρρ branching fractions and longitudinal polarization fractions (including the limit on ρ0ρ0, for which the polarization is unknown), one can perform the Gronau-London isospin analysis (electroweak penguins can be taken into account, while other SU(2)-breaking effects are usually neglected). Plots for confidence level representations of the P/T phase versus its module can be found on the corresponding CKMfitter and UTfit pages.
The Penguin-to-tree ratio:
Using as input the measured Cρρ,long and Sρρ,long coefficients together with the Wolfenstein parameters ρ and η using standard constraints, one can infer module and phase of the complex penguin to tree (P/T) ratio as done in the ππ case. Plots for confidence level representations of the P/T phase versus its module can be found on the corresponding CKMfitter and UTfit pages.

Combined α constraint from b → uu-bar d transitions:
Averaging the confidence level curves from the ππ and ρρ isospin analyses as well as the ρπ Dalitz plot analysis, leads to the combined constraint:
α = (99 +12 −9[1σ] +22−16[2σ]) °,
where the first errors given are at one and the second at two standard deviations, respectively. The isospin analyses are performed following the statistical interpretation of the CKMfitter analysis (Rfit) (see the UTfit pages for a Bayesian interpretation). Here, it does not include the Fiertz treatment of electroweak penguins for ππ and ρρ, which leads to a shift in α of approximately −2 deg. All other SU(2)-breaking effects are also neglected.

eps gif



Time-dependent CP Asymmetries in Bd → D+−π−+, Bd → D*+−π−+ and Bd → D+−ρ−+

The decays 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 Ref. / Comments
partially
reconstructed
N(BB)=232m
fully
reconstructed
N(BB)=232m
partially
reconstructed
N(BB)=152m
fully
reconstructed
N(BB)=152m
aπ* −0.034 ± 0.014 ± 0.009 −0.043 ± 0.023 ± 0.010 −0.030 ± 0.028 ± 0.018 0.060 ± 0.040 ± 0.019 −0.028 ± 0.012
CL=0.22 (1.2σ)
BABAR: PRD 71 (2005) 112003
(partially reco.)

BABAR-CONF-05/015 (hep-ex/0507075)
(fully reco.)


Belle: PLB 624 (2005) 11
(partially reco.)

Belle: PRL 93 (2004) 031802
Erratum-ibid. 93 (2004) 059901
(fully reco.)
cπ* −0.019 ± 0.022 ± 0.013
(lepton tags only)
0.047 ± 0.042 ± 0.015
(lepton tags only)
−0.005 ± 0.028 ± 0.018 0.049 ± 0.040 ± 0.019 0.004 ± 0.017
CL=0.42 (0.8σ)
aπ - −0.013 ± 0.022 ± 0.007 - −0.062 ± 0.037 ± 0.018 −0.025 ± 0.020
CL=0.30 (1.0σ)
cπ - −0.043 ± 0.042 ± 0.011
(lepton tags only)
- −0.025 ± 0.037 ± 0.018 −0.034 ± 0.030
CL=0.76 (0.3σ)
aρ - −0.024 ± 0.031 ± 0.010 - - −0.024 ± 0.033
cρ - −0.098 ± 0.055 ± 0.019
(lepton tags only)
- - −0.098 ± 0.058

Compilation of the above results.

eps.gz png

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Averages of the D*π results.

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eps.gz png

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.


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 for DK and 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− Ref. / Comments
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 BABAR-PUB-05/051 (submitted to PRD(R))
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 hep-ex/0601032 (submitted to PRD(R))
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 hep-ex/0601032 (submitted to PRD(R))
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)
Belle'03
N(BB)=96m
−0.02 ± 0.33 ± 0.07 0.19 ± 0.50 ± 0.04 - - Belle-CONF-0316 (hep-ex/0307074)
Average −0.06 ± 0.18 −0.08 ± 0.32 1.96 ± 0.41 0.65 ± 0.27  
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),
ACP+− = +− 2rBsin(δB)sin(γ) / 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 A and R, where

A = [Γ(B → [K+π]D(*)K(*)) − Γ(B+ → [Kπ+]D(*)K(*)+)] / [Γ(B → [K+π]D(*)K(*)) + Γ(B+ → [Kπ+]D(*)K(*)+)] ,
R = [Γ(B → [K+π]D(*)K(*)) + Γ(B+ → [Kπ+]D(*)K(*)+)] / [Γ(B → [Kπ+]D(*)K(*)) + Γ(B+ → [K+π]D(*)K(*)+)] .

(Some of) these observables have been measured so far for the 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 A R Ref. / Comments
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 BABAR-PUB-05/039 (hep-ex/0508001)
Average −0.22 ± 0.63 0.046 ± 0.032  
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:

R = rB2 + rD2 + 2rBrDcos(δBD)cos(γ),
A = 2rBrDsin(δBD)sin(γ) / R,

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. We obtain rD2 from the ratio of the suppressed-to-allowed branching fractions BR(D0 → K+π) = (1.38 ± 0.11)×10−4 and BR(D0 → Kπ+) = (3.80 ± 0.09)×10−2 [PDG 2004], respectively. With this we find rD = 0.0603 ± 0.0025. The strong phase is different, in general, for 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.

Plots upcoming.

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

Mode Experiment A R Ref. / Comments

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 Belle and BABAR using B → D K and B → D*K. 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. Both collaborations also have results using B → DK*, using K*− → KSπ; in this case some care is needed due to other possible contributions to the B → DKSπ final state. Belle assign an additional 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. At present, the two experiments use different procedures: Belle fit the data for γ/φ3, δB and rB and then use a frequentist treatment to correct for bias. BABAR instead use a different set of variables in their fit:

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

These parameters have the advantage of having (approximately) Gaussian distributions, and of having small statistical correlations. BABAR use a frequentist treatment to convert these measurements into constraints on the underlying physical parameters γ/φ3, δB and rB [constraints on rB and δB for the B → DK* decay are not directly obtained due to the reparametrization described above].

At present, we make no attempt to average the results.

Experiment Mode γ/φ3 (°) δB (°) rB Ref. / Comments
Belle'04
N(BB)=275m
DK
D→KSπ+π
64 ± 19 ± 13 ± 11 157 ± 19 ± 11 ± 21 0.21 ± 0.08 ± 0.03 ± 0.04 Belle-CONF-0476 (hep-ex/0411049)
D*K
D*→Dπ0
D→KSπ+π
75 ± 57 ± 11 ± 11 321 ± 57 ± 11 ± 21 0.12 +0.16−0.11 ± 0.02 ± 0.04
Combined 68 +14−15 ± 13 ± 11 - -
Belle'05
N(BB)=275m
DK*−
K*− → KSπ
D→KSπ+π
112 ± 35 ± 9 ± 11 ± 8(*) 353 ± 35 ± 8 ± 21 ± 49(*) 0.25 ± 0.18 ± 0.09 ± 0.04 ± 0.08(*) BELLE-CONF-0502 (hep-ex/0504013)

(*) The final error is due to the possible contribution from nonresonant B → D KSπ.

Experiment Mode x+ y+ x y Ref. / Comments
BABAR'05
N(BB)=227m
DK
D→KSπ+π
−0.13 ± 0.07 ± 0.03 ± 0.03 0.02 ± 0.08 ± 0.02 ± 0.02 0.08 ± 0.07 ± 0.03 ± 0.02 0.06 ± 0.09 ± 0.04 ± 0.04 PRL 95, 121802 (2005);
BABAR-CONF-05/018 (hep-ex/0507101)
D*K
D*→Dπ0 & D*→Dγ
D→KSπ+π
0.14 ± 0.09 ± 0.03 ± 0.03 0.01 ± 0.12 ± 0.04 ± 0.06 −0.13 ± 0.09 ± 0.03 ± 0.02 −0.14 ± 0.11 ± 0.02 ± 0.03
DK*−
K*− → KSπ
D→KSπ+π
−0.07 ± 0.23 ± 0.13 ± 0.03 −0.01 ± 0.32 ± 0.18 ± 0.05 −0.20 ± 0.20 ± 0.11 ± 0.03 0.26 ± 0.30 ± 0.16 ± 0.03
Combined γ = (67 ± 28 ± 13 ± 11)°
rB(DK) = 0.118 ± 0.079 ± 0.034 +0.036−0.034 δB(DK) = (104 ± 45 +17-21 +16-24
rB(D*K) = 0.169 ± 0.096 +0.030−0.028 +0.029−0.026 δB(D*K) = (296 ± 41 +14−12 ± 15)°

At present, we make no attempt to average the results.





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