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.

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From the bilinear coefficients given above, both experiments extract "quasi-two-body" (Q2B) parameters. Considering only the charged ρ bands in the Dalitz plot, the Q2B analysis involves 5 different parameters, one of which − the charge asymmetry 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 is convenient to transform the experimentally motivated CP parameters A_CP(ρπ) and C_ρπ into the physically motivated ones
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. Belle have also extracted the quasi-two-body parameters associated with this intermediate state.

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

Experiment	ACP (ρ+−π−+)	C (ρ+−π−+)	S (ρ+−π−+)	ΔC (ρ+−π−+)	ΔS (ρ+−π−+)	Correlations	Reference
BaBar N(BB)=347M	−0.14 ± 0.04 ± 0.01	0.15 ± 0.09 ± 0.04	0.01 ± 0.12 ± 0.03	0.38 ± 0.09 ± 0.02	0.06 ± 0.13 ± 0.03	(total)	hep-ex/0608002
Belle N(BB)=447M	−0.12 ± 0.05 ± 0.03	−0.13 ± 0.09 ± 0.06	0.06 ± 0.13 ± 0.07	0.35 ± 0.10 ± 0.06	−0.12 ± 0.14 ± 0.07	(total)	hep-ex/0609003
Average	−0.13 ± 0.03	0.03 ± 0.07	0.03 ± 0.09	0.36 ± 0.07	−0.02 ± 0.10	(total)	HFAG correlated average χ2 = 4.7/5 dof (CL=0.45 ⇒ 0.8σ)
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Experiment	A−+ (ρ+−π−+)	A+− (ρ+−π−+)	Correlation	Reference
BaBar N(BB)=347M	−0.38 +0.15 −0.16 ± 0.07	0.03 ± 0.07 ± 0.03	0.62	hep-ex/0608002
Belle N(BB)=447M	0.08 ± 0.17 ± 0.12	0.22 ± 0.08 ± 0.05	0.53	hep-ex/0609003
Average	−0.19 ± 0.13	0.11 ± 0.06	0.46	HFAG correlated average χ2 = 3.8/2 dof (CL=0.15 ⇒ 1.4σ)
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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 75° < α < 152° at 68% confidence level.
Belle has performed a similar analysis. In addition, Belle has also included information from the SU(2) partners of B → ρπ, which can be used to constrain α ≡ φ₂ via an isospin pentagon relation. The relevant branching ratios (given in units of 10⁻⁶) and CP-violating charge asymmetries are taken from HFAG - Rare Decays and are tabulated below. With this information, and a measurement BR(B⁰ → ρ⁰π⁰)/ BR(B⁰ → ρ⁺⁻π⁻⁺) = 0.133 ± 0.022 ± 0.023, Belle obtain the tighter constraint φ₂ = (83⁺¹²₋₂₃)° (where the errors correspond to 1σ ie. 68.3% 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.