Results on Time-Dependent CP Measurements:
Winter 2006 (Moriond, Italy (etc.) & FPCP, Canada)

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:

Measurements related to the CKM angle α ≡ φ₂:

Legend: if not stated otherwise,

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

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

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

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

Additional note on commonly treated (correlated) systematic effects:

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

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

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

we find the only slightly modified average:

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

Plots:

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

Historically the experiments determined |λ| for the charmonium modes; more recently the parameters C = −A = (1−|λ|²)/(1+|λ|²) are being used, as they are in all other time-dependent CP analyses. We recompute C from |λ| for the following averages.

Digression and plots:

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

Please note that

• Assuming that a single and vanishing weak phase dominates the decay amplitudes of the s-penguin modes, one has S = −η_CP×sin(2β) ≡ −η_CP×sin(2φ₁), where η_CP is the CP eigenvalue of the final state. Namely, η_CP = +1 for f₀K_S, π⁰π⁰K_S, K_SK_SK_S, and η_CP = −1 for φK_S, η′K_S, π⁰K_S, ρK_S and ωK_S. Modes with K_L in the final state are related by η_CP (X K_S) = − η_CP (X K_L).
• The mode K⁺K⁻K⁰ may have both CP-even and CP-odd contributions. The CP-even fraction is experimentally derived from an isospin analysis (Belle) or a moments analysis (BABAR). We preaverage the CP-even fraction, in which there is an implicit assumption that the detection and reconstruction efficiencies have the same variation across the Dalitz plane in all analyses (and therefore the underlying CP-even fraction is indeed the same quantity).
• Averaging over all s-penguin modes assumes that contributions with non-zero weak phases to the decay amplitudes can be neglected. Note that this assumption may be significantly violated due to doubly CKM-suppressed V_ub penguin amplitudes, and, in some cases, doubly CKM-suppressed and color-suppressed V_ub tree amplitudes that contribute to the decay amplitude. Recent theoretical analyses indicate that it is reasonable to expect that the modes φK⁰, η′K⁰ and K_SK_SK⁰ have theoretical uncertainties of the order of 0.05 or smaller, these can be significantly larger for the other modes (in particular for non-ss-bar-resonance modes: π⁰K_S, ρ⁰K_S, ωK_S and K⁺K⁻K⁰).
• Our naïve s-penguin average is, in fact, doubly naïve since it neglects both the theoretical uncertainty discussed above, and the fact that experimental systematic uncertainties are correlated between the measurements of individual modes. For these reasons, we do not advocate the use of these averages, and provide them only for academic interest. Use with extreme caution, if at all.
• The presence of a small O(λ²) weak phase in the decay of the s-penguin modes introduces a phase shift that corresponds to the weak phase 2β_s, which can be measured via time-dependent CP analysis of the decay B_s→ J/ψΦ. Using the Wolfenstein parameterization, and neglecting higher order terms in ρ, η and λ, we can estimate this shift to be:
  S(s-penguin) = −η_CP×sin(2β)[cc-bar s]·(1 + Δ) ≡ −η_CP×sin(2φ₁)[cc-bar s]·(1 + Δ). Numerically, using the global CKM fit results for the Wolfenstein parameters, one finds: Δ = 3.3%, which corresponds to a shift of 2β^eff ≡ 2φ₁^eff of +2.1 degrees.

The cosine coefficient:

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
Naïve s-penguin averages	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.	eps png

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.

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

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

Experiment	SJ/ψπ0	CJ/ψπ0 = −AJ/ψπ0	Correlation	Ref. / Comments
BABAR'06 N(BB)=232m	−0.68 ± 0.30 ± 0.04	−0.21 ± 0.26 ± 0.06	0.08	PRD 74 (2006) 011101
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:	eps png	eps png

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:	eps png	eps png

⁽*⁾ 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.

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 png

eps png

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

B_d decays to final states such as Dπ⁰ are governed by the b → cu-bar d transitions. If one chooses a final state which is a CP eigenstate, ie. D_CPπ⁰, the usual time-dependence formulae are recovered, with the sine coefficient sensitive to sin(2β) ≡ sin(2φ₁). Since there is no penguin contribution to these decays, there is even less associated theoretical uncertainty than for b → cc-bar s decays like B_d → J/ψ K_S.

Bondar, Gershon and Krokovny have shown that when multibody D decays, such as D → K_Sπ⁺π⁻ are used, a time-dependent analysis of the Dalitz plot of the D decay allows a direct determination of the weak phase: β ≡ φ₁. Equivalently, both sin(2β) ≡ sin(2φ₁) and cos(2β) ≡ cos(2φ₁) can be measured. This information allows to resolve the ambiguity in the measurement of 2β ≡ 2φ₁ from sin(2β) ≡ sin(2φ₁) alone.

Results of such an analysis are available from Belle. The decays B_d → Dπ⁰, B_d → Dη, B_d → Dω, B_d → D*π⁰ and B_d → D*η are used. The daughter decays are D* → Dπ⁰ and D → K_Sπ⁺π⁻.

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

Note that direct measurements of phases often suffer from unusual and non-Gaussian behaviour. Belle determine the sign of cos(2φ)₁ to be positive at 98.3% confidence level.

Time-dependent Analysis of b → sγ Transitions

Time-dependent analyses of radiative b decays such as B⁰→ K_Sπ⁰γ, 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 → K_Sπ⁰γ behaves like an effective flavor eigenstate, and mixing-induced CP violation is expected to be small - a simple estimation gives: S ~ −2(m_s/m_b)sin(2β) ≡ −2(m_s/m_b)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 K_Sπ⁰ can be performed, since the properties of the decay amplitudes are independent of the angular momentum of the K_Sπ⁰ 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 K_Sπ⁰γ Dalitz plot, for example as a function of the K_Sπ⁰ invariant mass.

An inclusive K_Sπ⁰γ 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 "K_Sπ⁰γ (not K*(892)γ)" analysis.

We quote two averages: one for K*(892) only, and the other one for the inclusive K_Sπ⁰γ 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:		eps png K*γ only: eps png KSπ0 only: eps png	eps png K*γ only: eps png KSπ0 only: eps png

Time-dependent CP Asymmetries in B_d→ 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 B_d→ π⁺π⁻ decay. The relevant branching ratios and CP-violating charge asymmetries are taken from HFAG - Rare Decays.

Constraining α ≡ φ2: 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 ≡ φ2 of approximately −2 deg. All other SU(2)-breaking effects are also neglected.

eps gif gif (high res)

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

The "Quasi-two-body" (Q2B) CP(t) analysis of B_d→ ρ⁺⁻π⁻⁺ decays (performed by Belle) assumes a narrow width approximation for the ρ meson. The interference regions in the π⁺π⁻π⁰ 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 A_CP(ρπ) − is time-independent. The time-dependent decay rate is given by
Γ( B → ρ⁺⁻π⁻⁺ (Δt) ) = (1 +− A_CP(ρπ)) e^−|Δt|/τ/8τ × [1 + Q_tag(S_ρπ+−ΔS_ρπ)sin(ΔmΔt) − Q_tag(C_ρπ+−ΔC_ρπ)cos(ΔmΔt)],
where Q_tag=+1(−1) when the tagging meson is a B⁰ (B⁰-bar). CP symmetry is violated if either one of the following conditions is true: A_CP(ρπ)≠0, C_ρπ≠0 or S_ρπ≠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 B_d 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 B_d lifetime and oscillation frequency.

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)

Flavour-charge specific branching fractions: the charge and flavour asymmetry parameters ACP(ρπ), Cρπ and ΔCρπ can be used to derive flavour-charge specific rates from the HFAG branching fraction BR(Bd→ ρ+−π−+) = (24.0 ± 2.5)×10−6. For individual B flavours 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 → ρ−π+ ) = (14.4 +2.4−2.2)×10−6, BR( Bd-bar → ρ+π− ) = (5.1 +1.9−1.7)×10−6, BR( Bd-bar → ρ−π+ ) = (12.0 +2.2−2.0)×10−6. For flavour-averaged inclusive branching fractions: BRB→ ρπ(+−) = 0.5(BRB→ρ+π− + BRB-bar→ ρ−π+), BRB→ ρπ(−+) = 0.5(BRB→ρ−π+ + BRB-bar→ ρ+π−), where the individual charge-flavour branching fractions are defined on the left. The total (flavour-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 B_d → (ρπ)⁰ → π⁺π⁻π⁰, 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). 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 ≡ φ₂, 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 ≡ φ2 (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 B_d → ρ⁺ρ⁻

The vector particles in the pseudoscalar to vector-vector decay B_d → 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 f_long = 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 f_long = 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.

Digression and plots:

The following numerical exercises involve the SU(2) partners of the B_d → ρ⁺ρ⁻ decay. The relevant branching ratios, CP-violating charge asymmetries and fractions of longitudinal polarization are taken from HFAG - Rare Decays averages.

Combined α ≡ φ2 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 ≡ φ2 = (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 ≡ φ2 of approximately −2 deg. All other SU(2)-breaking effects are also neglected.

eps gif

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

Neutral B meson decays such as B_d → D⁺⁻π⁻⁺, B_d → D*⁺⁻π⁻⁺ and B_d → D⁺⁻ρ⁻⁺ provide sensitivity to γ ≡ φ₃ because of the interference between the Cabibbo-favoured amplitude (e.g. B⁰ → D⁻π⁺) 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 B_dB_d-bar mixing phase, the total phase difference is −(2β+γ) ≡ −(2φ₁+φ₃).

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_Dπ = |A(B⁰ → D⁺π⁻)/A(B⁰ → D⁻π⁺)|. Each of the ratios r_Dπ, r_D*π and r_Dρ 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 → D_s⁺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 −+ S_ζ sin(ΔmΔt) −+ η C_ζ 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 −S_ζ (+S_ζ).] The parameters η and ζ take the values +1 and + (−1 and −) if the final state is, e.g., D⁻π⁺ (D⁺π⁻). The parameters actually measured by BABAR are denoted a and c. In the fit, the substitutions C_ζ = 1 and S_ζ = a −+ η b_i − η c_i 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 2R_D⁽*⁾π sin( 2φ₁+φ₃ +− δ_D⁽*⁾π ); the definition is such that S⁺⁻ (Belle) = −2R_D*πsin( 2φ₁+φ₃ +− δ_D*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:

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

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 A_CP+− and R_CP+−, where

Experimentally, it is convenient to measure R_CP+− 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 K_Sπ⁰, K_Sφ and K_Sω in all three analyses, and also use K_Sη in DK⁻ and D*K⁻ analyses. BABAR use K_Sπ⁰, K_Sφ and K_Sω 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.

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:

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_Kπ and R_Kπ, where

(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.

Compilation of the above results.

eps.gz png

Digression:

As can be seen from the expressions above, the maximum size of the asymmetry, for given values of r_B and r_D is given by: A_Kπ (max) = 2r_Br_D / (r_B²+r_D²). Thus, sizeable asymmetries may be found also for B⁻ → D⁽*⁾π⁻ decays, despite the expected smallness (~0.01) of r_B for this case, providing sensitivity to γ ≡ φ₃. The observables have been measured by Belle in the Dπ⁻ mode.

Dalitz Plot Analysis of B⁻ → D⁽*⁾ K⁽*⁾⁻ with D → K_Sπ⁺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 δ_B and the ratio of amplitudes r_B. 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 → K_Sπ⁺π⁻ is used.

For the DK*⁻ mode, both collaborations use K^*− → K_Sπ⁻; in this case some care is needed due to other possible contributions to the B⁻ → DK_Sπ⁻ final state. Belle assign an additional (model) uncertainty, while BABAR use an alternative parametrization [replacing r_B and δ_B with κr_s and δ_s, respectively] suggested by Gronau. Due to these different treatments, the averages for this mode should be treated with care.

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

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, δ_B and r_B [BaBar do not obtain constraints on r_B and δ_B 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. Note that the Belle results for (x₊, y₊, x₋, y₋) do not include model uncertainty. (See Appendix of Ref.) Since the model uncertainty should be 100% correlated between experiments, we average the values of (x₊, y₊, x₋, y₋) without including this source of error, and then add it back on to the averaged values.

Digression: