Time-dependent CP violation & Unitarity Triangle angles

Results on Time-Dependent CP Violation, and Measurements Related to the Angles of the Unitarity Triangle:
Winter 2007 (Moriond, Italy (etc.))

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

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| (from the BaBar results) for the following averages.

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	.

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 B_d → D*D*K_S

The decays B_d → D⁽*⁾D⁽*⁾K_S 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 B_d → D*D*K_S decays (with some theoretical input).

Analysis of the B_d → D*D*K_S decay has been performed by BaBar.

BaBar divide the Dalitz plot into two: m(D*⁺K_S)² > m(D*⁻K_S)² (η_y = +1) and m(D*⁺K_S)² < m(D*⁻K_S)² (η_y = -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

The parameters J₀, J_c, J_s1 and J_s2 are the integrals over the half-Dalitz plane m(D*⁺K_S)² < m(D*⁻K_S)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*K_S and B⁰-bar → D*D*K_S respectively. The parameter J_s2 (and hence J_s2/J₀) is predicted to be positive. BaBar measures:

From the above result and the assumption that J_s2>0, BaBar infer that cos(2β)>0 at the 94% confidence level.

Time-Dependent Analysis of B_s → J/ψ φ

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

The first measurement of φ_s from B_s → 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 B_s lifetime τ(B_s), ΔΓ, φ_s, 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 B_s. Note also that there is an ambiguity in the result for φ_s.

Interpretations:
D0 have combined the contour in the (φ_s, 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 φ_s = −0.70 ^+0.47_−0.39.

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, eg. 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. See e.g. Fleischer, NPB 659, 321 (2003).

Results of such an analysis are available from BaBar. The decays B_d → Dπ⁰, B_d → Dη, B_d → Dω, B_d → D*π⁰ and B_d → D*η are used. The daughter decay D* → Dπ⁰ is used. The CP-even D decay to K⁺K⁻ is used for all decay modes, with the CP-odd D decay to K_Sω also used in B_d → D⁽*⁾π⁰ and the additional CP-odd D decay to K_Sπ⁰ also used in B_d → Dω.

BaBar have performed separate fits for the cases where the intermediate D⁽*⁾ decays to CP-even and CP-odd final states, since these receive different contributions fom subleading amplitudes in the Standard Model. Since the effects of these corrections are expected to be negligible (~0.02) compared to the current experimental uncertainty, they have also performed a fit with all decays combined.

Time-Dependent CP Asymmetries in Colour Suppressed b → cu-bar d Transitions, with Multibody D Decays

Bondar, Gershon and Krokovny have shown that when multibody D decays, such as D → 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 both Belle and. BaBar. 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⁻. 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 = -η_CP × sin(2β^eff) ≡ -η_CP × sin(2φ₁^eff) 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(2β^eff) ≡ sin(2φ₁^eff) (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⁺K⁻K⁰ are also discussed in the next section). The BaBar results that appear in this table on K⁺K⁻K⁰ 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⁺K⁻K⁰. 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).

^(*) The BaBar results for φK⁰ are determined from their time-dependent Dalitz plot analysis of B⁰ → K⁺K⁻K⁰. The BaBar results for f₀K⁰ are a combination of results from the Dalitz plot analysis (sin(2β^eff) = 0.31 ± 0.32 ± 0.07, C_CP = −0.45 ± 0.28 ± 0.10), with those from the quasi-two-body analysis of B⁰ → f₀K_S, f₀ → π⁺π⁻ (sin(2β^eff) = 0.95 ^+0.23 _−0.32 ± 0.10, C_CP = −0.24 ± 0.31 ± 0.15, hep-ex/0408095). The BaBar results for K⁺K⁻K⁰ are taken from their previous quasi-two-body analysis (hep-ex/0507016).

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). [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 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.

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 B_d → K⁺K⁻K⁰

Time-dependent amplitude analysis of the three-body B_d → K⁺K⁻K⁰ decay allows additional information to be extracted from the data. In particular, the cosine of the effective weak phase difference (cos(2β^eff) ≡ cos(2φ₁^eff)) 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⁺K⁻K⁰ 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 β^eff ≡ φ₁^eff 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 A_CP ( = -C_CP).

From the above results BaBar infer that the trigonometric reflection at π/2 - &beta^eff 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 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	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:	eps.gz png	eps.gz png	eps.gz png

Experiment	SCP (D+D−)	CCP (D+D−)	Correlation	Reference
BaBar N(BB)=364M	−0.54 ± 0.34 ± 0.06	0.11 ± 0.22 ± 0.07	−0.17 (stat)	arXiv:0705.1190
Belle N(BB)=535M	−1.13 ± 0.37 ± 0.09	−0.91 ± 0.23 ± 0.06	−0.04 (stat)	hep-ex/0702031
Average (*)	−0.75 ± 0.26	−0.37 ± 0.17	−0.10	HFAG correlated average χ2 = 12/2 dof (CL=0.003 ⇒ 3.0σ)
Figures:	eps.gz png	eps.gz png	eps.gz png

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

The vector particles in the pseudoscalar to vector-vector decay B_d → 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 R_T = 0.125 ± 0.044 ± 0.007
• Belle measure R_T = 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 R_T 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:	eps.gz png	eps.gz png	eps.gz 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.

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.

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	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:		eps.gz png	eps.gz png	eps.gz png eps.gz 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	SCP (π+π−)	CCP (π+π−)	Correlation	Reference
BaBar N(BB)=383M	−0.60 ± 0.11 ± 0.03	−0.21 ± 0.09 ± 0.02	−0.07 (stat)	hep-ex/0703016
Belle N(BB)=532M	−0.61 ± 0.10 ± 0.04	−0.55 ± 0.08 ± 0.05	−0.15 (stat)	hep-ex/0608035
Average	−0.61 ± 0.08	−0.38 ± 0.07	−0.09	HFAG correlated average χ2 = 6.7/2 dof (CL=0.034 ⇒ 2.1σ)
Figures:	eps.gz png	eps.gz png	eps.gz png

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

Belle exclude the range 9° < φ₂ < 81° at the 95.4% 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 B_d→ π⁺π⁻π⁰ (B_d→ ρ⁺⁻⁰π⁻⁺⁰)

Both BaBar and Belle have performed a full time-dependent Dalitz plot analyses 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), 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.

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

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.

We average the quasi-two-body parameters provided by the experiments, which should be equivalent to determining average values directly from the averaged bilinear coefficients.

As shown by Charles it can be convenient to transform the experimentally motivated CP parameters A_CP(ρπ) and C_ρπ into the physically motivated choices
A⁺⁻(ρπ) = (|κ⁺⁻|²−1)/(|κ⁺⁻|²+1) = −(A_CP(ρπ)+C_ρπ+A_CP(ρπ)ΔC_ρπ)/(1+ΔC_ρπ + A_CP(ρπ)C_ρπ),
A⁻⁺(ρπ) = (|κ⁻⁺|²−1)/(|κ⁻⁺|²+1) = (−A_CP(ρπ)+C_ρπ+A_CP(ρπ)ΔC_ρπ)/(−1+ΔC_ρπ + A_CP(ρπ)C_ρπ),
where κ⁺⁻ = (q/p)Abar⁻⁺/A⁺⁻ and κ⁻⁺ = (q/p)Abar⁺⁻/A⁻⁺. With this definition A⁻⁺(ρπ) (A⁺⁻(ρπ)) describes CP violation in B_d 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 B_d→ ρ⁺⁻π⁻⁺ quasi-two-body contributions to the π⁺π⁻π⁰ final state, there can also be a B_d→ ρ⁰π⁰ component. Both experiments have also extracted the quasi-two-body parameters associated with this intermediate state.

Note again that at present we do not apply a rescaling of the results to a common, updated set of input parameters. Correlations due to possible common systematics are neglected in the following averages.

Experiment	ACP (ρ+−π−+)	C (ρ+−π−+)	S (ρ+−π−+)	ΔC (ρ+−π−+)	ΔS (ρ+−π−+)	Correlations	Reference
BaBar N(BB)=375M	−0.14 ± 0.05 ± 0.02	0.15 ± 0.09 ± 0.05	−0.03 ± 0.11 ± 0.04	0.39 ± 0.09 ± 0.09	−0.01 ± 0.14 ± 0.06	(stat)	hep-ex/0703008
Belle N(BB)=449M	−0.12 ± 0.05 ± 0.04	−0.13 ± 0.09 ± 0.05	0.06 ± 0.13 ± 0.05	0.36 ± 0.10 ± 0.05	−0.08 ± 0.13 ± 0.05	(stat)	hep-ex/0701015
Average	−0.13 ± 0.04	0.01 ± 0.07	0.01 ± 0.09	0.37 ± 0.08	−0.04 ± 0.10	(stat)	HFAG correlated average χ2 = 4.2/5 dof (CL=0.52 ⇒ 0.6σ)
Figures:	eps.gz png	eps.gz png	eps.gz png	eps.gz png	eps.gz png	.

Experiment	A−+ (ρ+−π−+)	A+− (ρ+−π−+)	Correlation	Reference
BaBar N(BB)=375M	−0.37 +0.16 −0.10 ± 0.09	0.03 ± 0.07 ± 0.04	0.62	hep-ex/0703008
Belle N(BB)=449M	0.08 ± 0.16 ± 0.11	0.21 ± 0.08 ± 0.04	0.47	hep-ex/0701015
Average	−0.18 ± 0.12	0.11 ± 0.06	0.40	HFAG correlated average χ2 = 4.0/2 dof (CL=0.14 ⇒ 1.5σ)
Figures:	eps.gz png	eps.gz png	eps.gz png

Experiment	C (ρ0π0)	S (ρ0π0)	Correlation	Reference
BaBar N(BB)=375M	−0.10 ± 0.40 ± 0.53	0.04 ± 0.44 ± 0.18	0.35	hep-ex/0703008
Belle N(BB)=449M	0.49 ± 0.36 ± 0.28	0.17 ± 0.57 ± 0.35	0.08	hep-ex/0701015
Average	0.30 ± 0.38	0.12 ± 0.38	0.12	HFAG correlated average χ2 = 0.5/2 dof (CL=0.76 ⇒ 0.3σ)
Figures:	eps.gz png	eps.gz png	eps.gz png

Interpretations:
The information given above can be used to extract α ≡ φ₂.
From the measured form factor bilinears, BaBar extract a confidence level interpretation for α, and constrain α = (87⁺⁴⁵₋₁₃)° at 68% confidence level.
Belle has performed a similar analysis. In addition, Belle has also included information from the SU(2) partners of B → ρπ, which can be used to constrain α ≡ φ₂ via an isospin pentagon relation. The isospin analysis uses as input the branching fractions and CP-violating charge asymmetries of all five ρπ decay modes (ρ⁺π⁻,ρ⁻π⁺, ρ⁰π⁰, ρ⁺π⁰, ρ⁰π⁺). The relevant information is taken from HFAG - Rare Decays and is tabulated below. [Branching fractions are given in units of 10⁻⁶.] Those values that are not included in the table below can be obtained from the bilinear form-factors, eg. a measurement BR(B⁰ → ρ⁰π⁰)/ BR(B⁰ → ρ⁺⁻π⁻⁺) = 0.133 ± 0.022 ± 0.023, is extracted. With all information in the ρπ channels put together, Belle obtain the tighter constraint 68° < φ₂ < 95° at 68% confidence level, for the solution consistent with the Standard Model.

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 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 decay is found to be dominated by the longitudinally polarized component:

• BABAR measure f_long = 0.977 ± 0.024 ^+0.015_−0.013,
• Belle measure 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.
The CP parameters measured are those for the longitudinally polarized component (ie. S_ρρ,long, C_ρρ,long).

Experiment	SCP (ρ+ρ−)	CCP (ρ+ρ−)	Correlation	Reference
BaBar N(BB)=347M	−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:	eps.gz png	eps.gz png	eps.gz png

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

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 B_d → a₁⁺⁻π⁻⁺

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

Interpretations:
The parameter α_eff ≡ φ_2,eff, which reduces to α ≡ φ₂ in the limit of no penguin contributions, can be extracted from the above results.

BaBar obtain α_eff = (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 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_ADS and R_ADS, where (for example for the B⁻ → DK⁻, D → K⁺π⁻ mode)

(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⁺π⁻π⁰.

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_ADS (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.

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:

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, 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. 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⁻ → DK_Sπ⁻ 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*⁻ → K_Sπ⁻ is the same in both experiments (so that κ, r_s and δ_s 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 assigned to the averages given in these plots do not include model errors.

eps.gz png

eps.gz png

eps.gz png

eps.gz png

Digression:

Dalitz Plot Analysis of B⁻ → D K⁻ with D → π⁺π⁻π⁰

BaBar have performed a similar Dalitz plot analysis using the decay D → π⁺π⁻π⁰. In this case the measured yields of B⁻ → DK⁻ and B⁺ → DK⁺ events are found to make a significant contribution to the sensitivity to CP violation and this information is included into the fit. Consequently, an alternative set of fit parameters is used in order to avoid significant biasing and nonlinear correlations. The result is parameterized in terms of polar coordinates:

where the constant x₀ = 0.850 depends on the amplitude structure of the D → π⁺π⁻π⁰ decay, and z_± = r_B e^{i( δ_B ± γ )} ≡ r_B e^{i( δ_B ± φ₃ )}. This choice of variables is motivated by the fact that the yields of B^± decays are proportional to 1 + ρ_±² - x₀². The uncertainty due to the D decay model is included in the systematic error.