Revisiting Charmless Hadronic B_{u,d} Decays in QCD Factorization

Hai-Yang Cheng; Chun-Khiang Chua

arXiv:0909.5229·hep-ph·November 20, 2014

Revisiting Charmless Hadronic B_{u,d} Decays in QCD Factorization

Hai-Yang Cheng, Chun-Khiang Chua

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TL;DR

This paper revisits charmless hadronic B meson decays within QCD factorization, exploring power corrections like penguin annihilation and soft effects to resolve CP puzzles and rate deficits, with predictions aligning well with experimental data.

Contribution

It introduces a detailed analysis of power correction effects in QCDF to address CP and rate puzzles in B decays, comparing with pQCD and SCET approaches.

Findings

01

Certain decay channels require penguin annihilation for correct CP violation.

02

Some decays need both power corrections to match measured CP asymmetries.

03

QCDF predictions agree well with experimental branching fractions and CP asymmetries.

Abstract

Within the framework of QCD factorization (QCDF), we consider two different types of power correction effects in order to resolve the CP puzzles and rate deficit problems with penguin-dominated two-body decays of B mesons and color-suppressed tree-dominated $π^{0} π^{0}$ and $ρ^{0} π^{0}$ modes: penguin annihilation and soft corrections to the color-suppressed tree amplitude. We emphasize that the electroweak penguin solution to the $B \to K π$ CP puzzle via New Physics is irrelevant for solving the CP and rate puzzles related to tree-dominated decays. While some channels e.g. $K^{-} π^{+}, K^{-} ρ^{0}, π^{+} π^{-}, ρ^{\pm} π^{\mp}$ need penguin annihilation to induce the correct magnitudes and signs for their CP violation, some other decays such as $B^{-} \to K^{-} π^{0}, π^{-} η, K^{-} η$ and $\overset{ˉ}{B}^{0} \to \overset{ˉ}{K}^{* 0} η, π^{0} π^{0}$ require the presence of both power corrections to account for the…

Figures1

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Tables19

Table 1. Table 1: Form factors for B → P → 𝐵 𝑃 B\to P transitions obtained in the QCD sum rules with B 2 g superscript subscript 𝐵 2 𝑔 B_{2}^{g} being the gluonic Gegenbauer moment.

$F_{0}^{B π} (0)$	$0.258 \pm 0.031$ Ball:PP	${0.26}_{- 0.03}^{+ 0.04}$ Melic:Bpi
$F_{0}^{B K} (0)$	$0.331 \pm 0.041$ Ball:PP	${0.36}_{- 0.04}^{+ 0.05}$ Melic:BK
$F_{0}^{B η} (0)$	$0.229 \pm 0.024 \pm 0.011$ Ball:Beta
$F_{0}^{B η^{'}} (0)$	$0.188 \pm 0.002 B_{2}^{g} \pm 0.019 \pm 0.009$ Ball:Beta

Table 2. Table 2: Input parameters. The values of the scale dependent quantities f V ⟂ ( μ ) subscript superscript 𝑓 perpendicular-to 𝑉 𝜇 f^{\perp}_{V}(\mu) and a 1 , 2 ⊥ , V ( μ ) subscript superscript 𝑎 bottom 𝑉 1 2 𝜇 a^{\bot,V}_{1,2}(\mu) are given for μ = 1 GeV 𝜇 1 GeV \mu=1\,\rm{GeV} . The values of Gegenbauer moments are taken from Ball2007 and Wolfenstein parameters from CKMfitter .

Light vector mesons
$V$	$f_{V} (MeV)$	$f_{V}^{⟂} (MeV)$	$a_{1}^{V}$	$a_{2}^{V}$	$a_{1}^{⊥, V}$	$a_{2}^{⊥, V}$
$ρ$	$216 \pm 3$	$165 \pm 9$	0	$0.15 \pm 0.07$	0	$0.14 \pm 0.06$
$ω$	$187 \pm 5$	$151 \pm 9$	0	$0.15 \pm 0.07$	0	$0.14 \pm 0.06$
$ϕ$	$215 \pm 5$	$186 \pm 9$	0	$0.18 \pm 0.08$	0	$0.14 \pm 0.07$
$K^{*}$	$220 \pm 5$	$185 \pm 10$	$0.03 \pm 0.02$	$0.11 \pm 0.09$	$0.04 \pm 0.03$	$0.10 \pm 0.08$
Light pseudoscalar mesons
$a_{1}^{π}$		$a_{2}^{π}$		$a_{1}^{K}$		$a_{2}^{K}$
0		$0.25 \pm 0.15$		$0.06 \pm 0.03$		$0.25 \pm 0.15$
$B$ mesons
$B$	$m_{B} (GeV$ )		$τ_{B} (ps)$	$f_{B} (MeV)$		$λ_{B} (MeV)$
$B_{u}$	$5.279$		$1.638$	$210 \pm 20$		$300 \pm 100$
$B_{d}$	$5.279$		$1.525$	$210 \pm 20$		$300 \pm 100$
$B_{s}$	$5.366$		$1.472$	$230 \pm 20$		$300 \pm 100$
Form factors at $q^{2} = 0$
$F_{0}^{B K} (0)$		$A_{0}^{B K^{*}} (0)$	$A_{1}^{B K^{*}} (0)$	$A_{2}^{B K^{*}} (0)$	$V_{0}^{B K^{*}} (0)$
$0.35 \pm 0.04$		$0.374 \pm 0.033$	$0.292 \pm 0.028$	$0.259 \pm 0.027$	$0.411 \pm 0.033$
$F_{0}^{B π} (0)$		$A_{0}^{B ρ} (0)$	$A_{1}^{B ρ} (0)$	$A_{2}^{B ρ} (0)$	$V_{0}^{B ρ} (0)$
$0.25 \pm 0.03$		$0.303 \pm 0.029$	$0.242 \pm 0.023$	$0.221 \pm 0.023$	$0.323 \pm 0.030$
$F_{0}^{B η_{q}} (0)$		$A_{0}^{B ω} (0)$	$A_{1}^{B ω} (0)$	$A_{2}^{B ω} (0)$	$V_{0}^{B ω} (0)$
$0.296 \pm 0.028$		$0.281 \pm 0.030$	$0.219 \pm 0.024$	$0.198 \pm 0.023$	$0.293 \pm 0.029$
Quark masses
$m_{b} (m_{b}) / GeV$		$m_{c} (m_{b}) / GeV$	$m_{c}^{pole} / m_{b}^{pole}$		$m_{s} (2.1 GeV) / GeV$
$4.2$		$0.91$	$0.3$		$0.095 \pm 0.020$
Wolfenstein parameters
$A$		$λ$	$\bar{ρ}$	$\bar{η}$	$γ$
$0.8116$		$0.2252$	$0.139$	$0.341$	${({67.8}_{- 3.9}^{+ 4.2})}^{\circ}$

Table 3. Table 3: The parameters ρ A subscript 𝜌 𝐴 \rho_{A} and ϕ A subscript italic-ϕ 𝐴 \phi_{A} for penguin annihilation. The fitted ρ A subscript 𝜌 𝐴 \rho_{A} and ϕ A subscript italic-ϕ 𝐴 \phi_{A} for B → V V → 𝐵 𝑉 𝑉 B\to VV decays are taken from ChengVV .

Mode	$ρ_{A}$	$ϕ_{A}$	Mode	$ρ_{A}$	$ϕ_{A}$
$B \to P P$	1.10	$- 50^{\circ}$	$B \to V P$	1.07	$- 70^{\circ}$
$B \to P V$	0.87	$- 30^{\circ}$	$B \to K ϕ$	0.70	$- 40^{\circ}$
$B \to K^{*} ρ$	0.78	$- 43^{\circ}$	$B \to K^{*} ϕ$	0.65	$- 53^{\circ}$

Table 4. Table 4: C P 𝐶 𝑃 CP -averaged branching fractions (in units of 10 − 6 superscript 10 6 10^{-6} ) and direct CP asymmetries (in %) of some selective B → P P → 𝐵 𝑃 𝑃 B\to PP decays obtained in QCD factorization for three distinct cases: (i) without any power corrections, (ii) with power corrections to penguin annihilation, and (iii) with power corrections to both penguin annihilation and color-suppressed tree amplitudes. The parameters ρ A subscript 𝜌 𝐴 \rho_{A} and ϕ A subscript italic-ϕ 𝐴 \phi_{A} are taken from Table 3 , ρ C = 1.3 subscript 𝜌 𝐶 1.3 \rho_{C}=1.3 and ϕ C = − 70 ∘ subscript italic-ϕ 𝐶 superscript 70 \phi_{C}=-70^{\circ} . The theoretical errors correspond to the uncertainties due to the variation of (i) Gegenbauer moments, decay constants, quark masses, form factors, the λ B subscript 𝜆 𝐵 \lambda_{B} parameter for the B 𝐵 B meson wave function, and (ii) ρ A , H subscript 𝜌 𝐴 𝐻 \rho_{A,H} , ϕ A , H subscript italic-ϕ 𝐴 𝐻 \phi_{A,H} , respectively.

Mode	W/o $ρ_{A, C}, ϕ_{A, C}$	With $ρ_{A}, ϕ_{A}$	With $ρ_{A, C}, ϕ_{A, C}$	Expt. HFAG
$ℬ ({\bar{B}}^{0} \to K^{-} π^{+})$	${13.1}_{- 3.5 - 0.7}^{+ 5.8 + 0.7}$	${19.3}_{- 4.8 - 6.2}^{+ 7.9 + 8.2}$	${19.3}_{- 4.8 - 6.2}^{+ 7.9 + 8.2}$	$19.4 \pm 0.6$
$ℬ ({\bar{B}}^{0} \to {\bar{K}}^{0} π^{0})$	${5.5}_{- 1.7 - 0.3}^{+ 2.8 + 0.3}$	${8.4}_{- 2.3 - 2.9}^{+ 3.8 + 3.8}$	${8.6}_{- 2.2 - 2.9}^{+ 3.8 + 3.8}$	$9.8 \pm 0.6$
$ℬ (B^{-} \to {\bar{K}}^{0} π^{-})$	${14.9}_{- 4.5 - 1.0}^{+ 6.9 + 0.9}$	${21.7}_{- 6.0 - 6.9}^{+ 9.2 + 9.0}$	${21.7}_{- 6.0 - 6.9}^{+ 9.2 + 9.0}$	$23.1 \pm 1.0$
$ℬ (B^{-} \to K^{-} π^{0})$	${9.1}_{- 2.3 + 0.5}^{+ 3.6 + 0.5}$	${12.6}_{- 3.0 - 3.7}^{+ 4.7 + 4.8}$	${12.5}_{- 3.0 - 3.8}^{+ 4.7 + 4.9}$	$12.9 \pm 0.6$
$ℬ (B^{-} \to K^{-} η)$	${1.6}_{- 0.7 - 0.4}^{+ 1.1 + 0.3}$	${2.4}_{- 1.1 - 1.0}^{+ 1.8 + 1.3}$	${2.4}_{- 1.1 - 1.0}^{+ 1.8 + 1.3}$	$2.36 \pm 0.27$
$ℬ ({\bar{B}}^{0} \to π^{+} π^{-})$	${6.2}_{- 0.6 - 0.4}^{+ 0.4 + 0.2}$	${7.0}_{- 0.7 - 0.7}^{+ 0.4 + 0.7}$	${7.0}_{- 0.7 - 0.7}^{+ 0.4 + 0.7}$	$5.16 \pm 0.22$
$ℬ ({\bar{B}}^{0} \to π^{0} π^{0})$	${0.42}_{- 0.11 - 0.08}^{+ 0.29 + 0.18}$	${0.52}_{- 0.10 - 0.10}^{+ 0.26 + 0.21}$	${1.1}_{- 0.4 - 0.3}^{+ 1.0 + 0.7}$	$1.55 \pm 0.19$ ¹¹1If an $S$ factor is included, the average will become $1.55 \pm 0.35$ .
$ℬ (B^{-} \to π^{-} π^{0})$	${4.9}_{- 0.5 - 0.3}^{+ 0.9 + 0.6}$	${4.9}_{- 0.5 - 0.3}^{+ 0.9 + 0.6}$	${5.9}_{- 1.1 - 1.1}^{+ 2.2 + 1.4}$	${5.59}_{- 0.40}^{+ 0.41}$
$ℬ (B^{-} \to π^{-} η)$	${4.4}_{- 0.3 - 0.2}^{+ 0.6 + 0.4}$	${4.5}_{- 0.3 - 0.3}^{+ 0.6 + 0.5}$	${5.0}_{- 0.6 - 0.7}^{+ 1.2 + 0.9}$	$4.1 \pm 0.3$
$A_{C P} ({\bar{B}}^{0} \to K^{-} π^{+})$	${4.0}_{- 0.7 - 1.1}^{+ 0.6 + 1.1}$	$- {7.4}_{- 1.5 - 4.8}^{+ 1.7 + 4.3}$	$- {7.4}_{- 1.5 - 4.8}^{+ 1.7 + 4.3}$	$- {9.8}_{- 1.1}^{+ 1.2}$
$A_{C P} ({\bar{B}}^{0} \to {\bar{K}}^{0} π^{0})$	$- {4.0}_{- 1.8 - 3.0}^{+ 1.2 + 3.5}$	${0.75}_{- 0.94 - 3.32}^{+ 1.88 + 2.56}$	$- {10.6}_{- 3.8 - 4.3}^{+ 2.7 + 5.6}$	$- 1 \pm 10$
$A_{C P} (B^{-} \to {\bar{K}}^{0} π^{-})$	${0.72}_{- 0.05 - 0.05}^{+ 0.06 + 0.05}$	${0.28}_{- 0.03 - 0.10}^{+ 0.03 + 0.09}$	${0.28}_{- 0.03 - 0.10}^{+ 0.03 + 0.09}$	$0.9 \pm 2.5$
$A_{C P} (B^{-} \to K^{-} π^{0})$	${7.3}_{- 1.2 - 2.7}^{+ 1.6 + 2.3}$	$- {5.5}_{- 1.8 - 4.6}^{+ 1.3 + 4.9}$	${4.9}_{- 2.1 - 5.4}^{+ 3.9 + 4.4}$	$5.0 \pm 2.5$
$A_{C P} (B^{-} \to K^{-} η)$	$- {22.1}_{- 16.7 - 7.3}^{+ 7.7 + 14.0}$	${12.7}_{- 5.0 - 15.0}^{+ 7.7 + 13.4}$	$- {11.0}_{- 21.6 - 10.1}^{+ 8.4 + 14.9}$	$- 37 \pm 9$
$A_{C P} ({\bar{B}}^{0} \to π^{+} π^{-})$	$- {6.2}_{- 0.5 - 1.8}^{+ 0.4 + 2.0}$	${17.0}_{- 1.2 - 8.7}^{+ 1.3 + 4.3}$	${17.0}_{- 1.2 - 8.7}^{+ 1.3 + 4.3}$	$38 \pm 6$
$A_{C P} ({\bar{B}}^{0} \to π^{0} π^{0})$	${33.4}_{- 10.6 - 37.7}^{+ 6.8 + 34.8}$	$- {26.9}_{- 6.0 - 37.5}^{+ 8.4 + 48.5}$	${57.2}_{- 20.8 - 34.6}^{+ 14.8 + 30.3}$	$43_{- 24}^{+ 25}$
$A_{C P} (B^{-} \to π^{-} π^{0})$	$- {0.06}_{- 0.01 - 0.02}^{+ 0.01 + 0.01}$	$- {0.06}_{- 0.01 - 0.02}^{+ 0.01 + 0.01}$	$- {0.11}_{- 0.01 - 0.03}^{+ 0.01 + 0.06}$	$6 \pm 5$
$A_{C P} (B^{-} \to π^{-} η)$	$- {11.4}_{- 1.0 - 2.7}^{+ 1.1 + 2.3}$	${11.4}_{- 0.9 - 9.1}^{+ 0.9 + 4.5}$	$- {5.0}_{- 3.4 - 10.3}^{+ 2.4 + 8.4}$	$- 13 \pm 7$

Table 5. Table 5: C P 𝐶 𝑃 CP -averaged branching fractions (in units of 10 − 6 superscript 10 6 10^{-6} ) of B → P P → 𝐵 𝑃 𝑃 B\to PP decays obtained in various approaches. The pQCD results are taken from LiPP ; LuKK ; XiaoKeta ; Xiaoetaeta ; Xiaopieta . Note that there exist several pQCD calculations for B → K η ( ′ ) B\to K\eta^{(^{\prime})} Kou ; XiaoKeta ; ChenKeta ; LiKeta and here we cite the pQCD results with partial NLO corrections XiaoKeta . There are two solution sets with SCET predictions for decays involving η 𝜂 \eta and/or η ′ superscript 𝜂 ′ \eta^{\prime} Zupan .

Mode	QCDF (this work)	pQCD	SCET	Expt. HFAG
$B^{-} \to {\bar{K}}^{0} π^{-}$	${21.7}_{- 6.0 - 6.9}^{+ 9.2 + 9.0}$	${23.6}_{- 8.4}^{+ 14.5}$	$20.8 \pm 7.9 \pm 0.6 \pm 0.7$	$23.1 \pm 1.0$
$B^{-} \to K^{-} π^{0}$	${12.5}_{- 3.0 - 3.8}^{+ 4.7 + 4.9}$	${13.6}_{- 5.7}^{+ 10.3}$	$11.3 \pm 4.1 \pm 1.0 \pm 0.3$	$12.9 \pm 0.6$
${\bar{B}}^{0} \to K^{-} π^{+}$	${19.3}_{- 4.8 - 6.2}^{+ 7.9 + 8.2}$	${20.4}_{- 8.4}^{+ 16.1}$	$20.1 \pm 7.4 \pm 1.3 \pm 0.6$	$19.4 \pm 0.6$
${\bar{B}}^{0} \to {\bar{K}}^{0} π^{0}$	${8.6}_{- 2.2 - 2.9}^{+ 3.8 + 3.8}$	${8.7}_{- 3.4}^{+ 6.0}$	$9.4 \pm 3.6 \pm 0.2 \pm 0.3$	$9.8 \pm 0.6$
$B^{-} \to π^{-} π^{0}$	${5.9}_{- 1.1 - 1.1}^{+ 2.2 + 1.4}$	${4.0}_{- 1.9}^{+ 3.4}$	$5.2 \pm 1.6 \pm 2.1 \pm 0.6$	${5.59}_{- 0.40}^{+ 0.41}$
${\bar{B}}^{0} \to π^{+} π^{-}$	${7.0}_{- 0.7 - 0.7}^{+ 0.4 + 0.7}$	${6.5}_{- 3.8}^{+ 6.7}$	$5.4 \pm 1.3 \pm 1.4 \pm 0.4$	$5.16 \pm 0.22$
${\bar{B}}^{0} \to π^{0} π^{0}$	${1.1}_{- 0.4 - 0.3}^{+ 1.0 + 0.7}$	${0.29}_{- 0.20}^{+ 0.50}$	$0.84 \pm 0.29 \pm 0.30 \pm 0.19$	$1.55 \pm 0.19$
$B^{-} \to K^{-} K^{0}$	${1.8}_{- 0.5 - 0.5}^{+ 0.9 + 0.7}$	1.66	$1.1 \pm 0.4 \pm 1.4 \pm 0.03$	${1.36}_{- 0.27}^{+ 0.29}$
${\bar{B}}^{0} \to K^{+} K^{-}$	${0.10}_{- 0.02 - 0.03}^{+ 0.03 + 0.03}$	0.046	$-, -$	${0.15}_{- 0.10}^{+ 0.11}$
${\bar{B}}^{0} \to K^{0} {\bar{K}}^{0}$	${2.1}_{- 0.6 - 0.6}^{+ 1.0 + 0.8}$	1.75	$1.0 \pm 0.4 \pm 1.4 \pm 0.03$	${0.96}_{- 0.19}^{+ 0.21}$
$B^{-} \to K^{-} η$	${2.3}_{- 1.1 - 1.0}^{+ 1.8 + 1.3}$	${3.2}_{- 0.9 - 1.2 - 1.0}^{+ 1.2 + 2.7 + 1.1}$	$2.7 \pm 4.8 \pm 0.4 \pm 0.3$	$2.36 \pm 0.27$
			$2.3 \pm 4.5 \pm 0.4 \pm 0.3$
$B^{-} \to K^{-} η^{'}$	${78.4}_{- 26.8 - 19.5}^{+ 61.2 + 26.4}$	${51.0}_{- 8.2 - 6.2 - 3.5}^{+ 13.5 + 11.2 + 4.2}$	$69.5 \pm 27.0 \pm 4.4 \pm 7.7$	$71.1 \pm 2.6$
			$69.3 \pm 26.0 \pm 7.1 \pm 6.3$
${\bar{B}}^{0} \to {\bar{K}}^{0} η$	${1.6}_{- 0.9 - 0.8}^{+ 1.5 + 1.1}$	${2.1}_{- 0.6 - 1.0 - 0.9}^{+ 0.8 + 2.3 + 1.0}$	$2.4 \pm 4.4 \pm 0.2 \pm 0.3$	${1.12}_{- 0.28}^{+ 0.30}$
			$2.3 \pm 4.4 \pm 0.2 \pm 0.5$
${\bar{B}}^{0} \to {\bar{K}}^{0} η^{'}$	${74.2}_{- 24.9 - 18.4}^{+ 56.5 + 24.7}$	${50.3}_{- 8.2 - 6.2 - 2.7}^{+ 11.8 + 11.1 + 4.5}$	$63.2 \pm 24.7 \pm 4.2 \pm 8.1$	$66.1 \pm 3.1$
			$62.2 \pm 23.7 \pm 5.5 \pm 7.2$
$B^{-} \to π^{-} η$	${5.0}_{- 0.6 - 0.7}^{+ 1.2 + 0.9}$	${4.1}_{- 0.9 - 0.3 - 0.5}^{+ 1.3 + 0.4 + 0.6}$	$4.9 \pm 1.7 \pm 1.0 \pm 0.5$	$4.07 \pm 0.32$
			$5.0 \pm 1.7 \pm 1.2 \pm 0.4$
$B^{-} \to π^{-} η^{'}$	${3.8}_{- 0.6 - 0.6}^{+ 1.3 + 0.9}$	${2.4}_{- 0.5}^{+ 0.8} \pm 0.2 \pm 0.3$	$2.4 \pm 1.2 \pm 0.2 \pm 0.4$	${2.7}_{- 0.4}^{+ 0.5}$
			$2.8 \pm 1.2 \pm 0.3 \pm 0.3$
${\bar{B}}^{0} \to π^{0} η$	${0.36}_{- 0.02 - 0.10}^{+ 0.03 + 0.13}$	${0.23}_{- 0.03 - 0.03}^{+ 0.04 + 0.04} \pm 0.05$	$0.88 \pm 0.54 \pm 0.06 \pm 0.42$	$< 1.5$
			$0.68 \pm 0.46 \pm 0.03 \pm 0.41$
${\bar{B}}^{0} \to π^{0} η^{'}$	${0.42}_{- 0.09 - 0.12}^{+ 0.21 + 0.18}$	$0.19 \pm 0.02 \pm {0.03}_{- 0.05}^{+ 0.04}$	$2.3 \pm 0.8 \pm 0.3 \pm 2.7$	$1.2 \pm 0.4$
			$1.3 \pm 0.5 \pm 0.1 \pm 0.3$
${\bar{B}}^{0} \to η η$	${0.32}_{- 0.05 - 0.06}^{+ 0.13 + 0.07}$	${0.67}_{- 0.25}^{+ 0.32}$	$0.69 \pm 0.38 \pm 0.13 \pm 0.58$	$< 1.0$
			$1.0 \pm 0.4 \pm 0.3 \pm 1.4$
${\bar{B}}^{0} \to η η^{'}$	${0.36}_{- 0.10 - 0.08}^{+ 0.24 + 0.12}$	$0.18 \pm 0.11$	$1.0 \pm 0.5 \pm 0.1 \pm 1.5$	$< 1.2$
			$2.2 \pm 0.7 \pm 0.6 \pm 5.4$
${\bar{B}}^{0} \to η^{'} η^{'}$	${0.22}_{- 0.06 - 0.06}^{+ 0.14 + 0.08}$	${0.11}_{- 0.09}^{+ 0.12}$	$0.57 \pm 0.23 \pm 0.03 \pm 0.69$	$< 1.7$
			$1.2 \pm 0.4 \pm 0.3 \pm 3.7$

Table 6. Table 6: C P 𝐶 𝑃 CP -averaged branching fractions (in units of 10 − 6 superscript 10 6 10^{-6} ) of B → π η ( ′ ) B\to\pi\eta^{(^{\prime})} decays.

	$π^{-} η^{'}$	$π^{0} η^{'}$	$π^{-} η$	$π^{0} η$
BaBar	$3.5 \pm 0.6 \pm 0.2$ BaBar:Beta	$0.9 \pm 0.4 \pm 0.1$ BaBar:pi0eta	$4.00 \pm 0.40 \pm 0.24$ BaBar:Beta	$< 1.5$ BaBar:pi0eta
Belle	${1.8}_{- 0.6}^{+ 0.7} \pm 0.1$ Belle:pieta'	$2.8 \pm 1.0 \pm 0.3$ Belle:pieta'	$4.2 \pm 0.4 \pm 0.2$ Belle:Beta2	$< 2.5$ Belle:Beta1
Average	${2.7}_{- 0.4}^{+ 0.5}$	$1.2 \pm 0.4$	$4.1 \pm 0.3$	$< 1.5$

Table 7. Table 7: Same as Table 5 except for direct CP asymmetries (in %) of B → P P → 𝐵 𝑃 𝑃 B\to PP decays obtained in various approaches.

Mode	QCDF (this work)	pQCD	SCET	Expt. HFAG
$B^{-} \to {\bar{K}}^{0} π^{-}$	${0.28}_{- 0.03 - 0.10}^{+ 0.03 + 0.09}$	$0$	$< 5$	$0.9 \pm 2.5$
$B^{-} \to K^{-} π^{0}$	${4.9}_{- 2.1 - 5.4}^{+ 3.9 + 4.4}$	$- 1_{- 6}^{+ 3}$	$- 11 \pm 9 \pm 11 \pm 2$	$5.0 \pm 2.5$
${\bar{B}}^{0} \to K^{-} π^{+}$	$- {7.4}_{- 1.5 - 4.8}^{+ 1.7 + 4.3}$	$- 10_{- 8}^{+ 7}$	$- 6 \pm 5 \pm 6 \pm 2$	$- {9.8}_{- 1.1}^{+ 1.2}$
${\bar{B}}^{0} \to {\bar{K}}^{0} π^{0}$	$- {10.6}_{- 3.8 - 4.3}^{+ 2.7 + 5.6}$	$- 7_{- 4}^{+ 3}$	$5 \pm 4 \pm 4 \pm 1$	$- 1 \pm 10$
$B^{-} \to π^{-} π^{0}$	$- {0.11}_{- 0.01 - 0.03}^{+ 0.01 + 0.06}$	$0$	$< 4$	$6 \pm 5$
${\bar{B}}^{0} \to π^{+} π^{-}$	${17.0}_{- 1.2 - 8.7}^{+ 1.3 + 4.3}$	$18_{- 12}^{+ 20}$	$20 \pm 17 \pm 19 \pm 5$	$38 \pm 6$
${\bar{B}}^{0} \to π^{0} π^{0}$	${57.2}_{- 20.8 - 34.6}^{+ 14.8 + 30.3}$	$63_{- 34}^{+ 35}$	$- 58 \pm 39 \pm 39 \pm 13$	$43_{- 24}^{+ 25}$
$B^{-} \to K^{-} K^{0}$	$- {6.4}_{- 0.6 - 1.8}^{+ 0.8 + 1.8}$	11	$1.1 \pm 0.4 \pm 1.4 \pm 0.03$	$12_{- 18}^{+ 17}$
${\bar{B}}^{0} \to K^{+} K^{-}$	$0$	29
${\bar{B}}^{0} \to K^{0} {\bar{K}}^{0}$	$- {10.0}_{- 0.7 - 1.9}^{+ 0.7 + 1.0}$	0	$1.0 \pm 0.4 \pm 1.4 \pm 0.03$
$B^{-} \to K^{-} η$	$- {11.2}_{- 22.0 - 10.3}^{+ 8.5 + 15.2}$	$- {11.7}_{- 9.6 - 4.2 - 5.6}^{+ 6.8 + 3.9 + 2.9}$	$33 \pm 30 \pm 7 \pm 3$	$- 37 \pm 9$
			$- 33 \pm 39 \pm 10 \pm 4$
$B^{-} \to K^{-} η^{'}$	${0.52}_{- 0.53 - 0.90}^{+ 0.66 + 1.14}$	$- {6.2}_{- 1.1 - 1.0 - 1.0}^{+ 1.2 + 1.3 + 1.3}$	$- 10 \pm 6 \pm 7 \pm 5$	${1.3}_{- 1.7}^{+ 1.6}$
			$0.7 \pm 0.5 \pm 0.2 \pm 0.9$
${\bar{B}}^{0} \to {\bar{K}}^{0} η$	$- {21.4}_{- 22.9 - 11.3}^{+ 8.6 + 11.8}$	$- {12.7}_{- 4.1 - 1.5 - 6.7}^{+ 4.1 + 3.2 + 3.2}$	$21 \pm 20 \pm 4 \pm 3$
			$- 18 \pm 22 \pm 6 \pm 4$
${\bar{B}}^{0} \to {\bar{K}}^{0} η^{'}$	${3.0}_{- 0.5 - 0.8}^{+ 0.6 + 0.7}$	${2.3}_{- 0.4 - 0.6 - 0.1}^{+ 0.5 + 0.3 + 0.2}$	$11 \pm 6 \pm 12 \pm 2$	$5 \pm 5$
			$- 27 \pm 7 \pm 8 \pm 5$
$B^{-} \to π^{-} η$	$- {5.0}_{- 3.4 - 10.3}^{+ 2.4 + 8.4}$	$- 37_{- 6 - 4 - 1}^{+ 8 + 4 + 0}$	$5 \pm 19 \pm 21 \pm 5$	$- 13 \pm 7$
			$37 \pm 19 \pm 21 \pm 5$
$B^{-} \to π^{-} η^{'}$	${1.6}_{- 8.2 - 11.1}^{+ 5.0 + 9.4}$	$- 33_{- 4 - 6 - 2}^{+ 6 + 4 + 0}$	$21 \pm 12 \pm 10 \pm 14$	$6 \pm 15$
			$2 \pm 10 \pm 4 \pm 15$
${\bar{B}}^{0} \to π^{0} η$	$- {5.2}_{- 5.0 - 15.6}^{+ 2.8 + 24.6}$	$- 42_{- 12 - 2 - 3}^{+ 9 + 3 + 1}$	$3 \pm 10 \pm 12 \pm 5$
			$- 7 \pm 16 \pm 4 \pm 90$
${\bar{B}}^{0} \to π^{0} η^{'}$	$- {7.3}_{- 1.8 - 14.0}^{+ 1.0 + 17.6}$	$- 36_{- 9 - 1 - 3}^{+ 10 + 2 + 2}$	$- 24 \pm 10 \pm 19 \pm 24$
			$-, -$
${\bar{B}}^{0} \to η η$	$- {63.5}_{- 6.4 - 12.4}^{+ 10.4 + 9.8}$	$- 33_{- 2.8 - 3.8 - 0.0}^{+ 2.6 + 4.1 + 3.5}$	$- 9 \pm 24 \pm 21 \pm 4$
			$48 \pm 22 \pm 20 \pm 13$
${\bar{B}}^{0} \to η η^{'}$	$- {59.2}_{- 6.8 - 4.8}^{+ 7.2 + 3.8}$	${77.4}_{- 5.6 - 11.2 - 9.0}^{+ 0.0 + 6.9 + 8.0}$	$-, -$
			$70 \pm 13 \pm 20 \pm 4$
${\bar{B}}^{0} \to η^{'} η^{'}$	$- {44.9}_{- 3.1 - 9.2}^{+ 3.1 + 8.5}$	${23.7}_{- 6.9 - 16.9 - 8.5}^{+ 10.0 + 18.5 + 6.0}$	$-, -$
			$60 \pm 11 \pm 22 \pm 29$

Table 8. Table 8: Mixing-induced CP violation S f subscript 𝑆 𝑓 S_{f} in B → P P → 𝐵 𝑃 𝑃 B\to PP decays predicted in various approaches. The pQCD results are taken from LiPP ; XiaoKeta ; Xiaoetaeta ; Xiaopieta . For final states involving η 𝜂 \eta and/or η ′ superscript 𝜂 ′ \eta^{\prime} , there are two solutions with SCET predictions Zupan . The parameter η f = 1 subscript 𝜂 𝑓 1 \eta_{f}=1 except for K S ( π 0 , η , η ′ ) subscript 𝐾 𝑆 superscript 𝜋 0 𝜂 superscript 𝜂 ′ K_{S}(\pi^{0},\eta,\eta^{\prime}) modes where η f = − 1 subscript 𝜂 𝑓 1 \eta_{f}=-1 . Experimental results from BaBar (first entry) and Belle (second entry) are listed whenever available. The input values of sin ⁡ 2 β 2 𝛽 \sin 2\beta used at the time of theoretical calculations are displayed.

Mode	QCDF (this work)	pQCD	SCET	Expt. BaBar:etaK0 ; Belle:etaK0 ; Belle:piK0 ; BaBar:pipi ; Belle:pipi	Average
$\sin 2 β$	0.670	0.685	0.725
$η^{'} K_{S}$	${0.67}_{- 0.01 - 0.01}^{+ 0.01 + 0.01}$	${0.63}_{- 0.91}^{+ 0.50}$	$\begin{matrix} 0.706 \pm 0.008 \\ 0.715 \pm 0.010 \end{matrix}$	$\begin{matrix} 0.57 \pm 0.08 \pm 0.02 \\ 0.64 \pm 0.10 \pm 0.04 \end{matrix}$	$0.59 \pm 0.07$
$η K_{S}$	${0.79}_{- 0.06 - 0.06}^{+ 0.04 + 0.08}$	${0.62}_{- 0.92}^{+ 0.50}$	$\begin{matrix} 0.69 \pm 0.16 \\ 0.79 \pm 0.15 \end{matrix}$
$π^{0} K_{S}$	${0.79}_{- 0.04 - 0.04}^{+ 0.06 + 0.04}$	${0.74}_{- 0.03}^{+ 0.02}$	$0.80 \pm 0.03$	$\begin{matrix} 0.55 \pm 0.20 \pm 0.03 \\ 0.67 \pm 0.31 \pm 0.08 \end{matrix}$	$0.57 \pm 0.17$
$π^{+} π^{-}$	$- {0.69}_{- 0.10 - 0.09}^{+ 0.08 + 0.19}$	$- {0.42}_{- 0.56}^{+ 1.00}$	$- 0.86 \pm 0.10$	$\begin{matrix} - 0.68 \pm 0.10 \pm 0.03 \\ - 0.61 \pm 0.10 \pm 0.04 \end{matrix}$	$- 0.65 \pm 0.07$
$π^{0} η$	${0.08}_{- 0.12 - 0.23}^{+ 0.06 + 0.19}$	${0.067}_{- 0.010}^{+ 0.005}$	$\begin{matrix} - 0.90 \pm 0.24 \\ - 0.67 \pm 0.82 \end{matrix}$
$π^{0} η^{'}$	${0.16}_{- 0.07 - 0.14}^{+ 0.05 + 0.11}$	${0.067}_{- 0.011}^{+ 0.004}$	$\begin{matrix} - 0.96 \pm 0.12 \\ - 0.60 \pm 1.31 \end{matrix}$
$η η$	$- {0.77}_{- 0.05 - 0.06}^{+ 0.07 + 0.12}$	${0.535}_{- 0.004}^{+ 0.004}$	$\begin{matrix} - 0.98 \pm 0.11 \\ - 0.78 \pm 0.31 \end{matrix}$
$η η^{'}$	$- {0.76}_{- 0.05 - 0.03}^{+ 0.07 + 0.06}$	$- {0.131}_{- 0.050}^{+ 0.056}$	$\begin{matrix} - 0.82 \pm 0.77 \\ - 0.71 \pm 0.37 \end{matrix}$
$η^{'} η^{'}$	$- {0.85}_{- 0.02 - 0.06}^{+ 0.03 + 0.07}$	${0.93}_{- 0.12}^{+ 0.08}$	$\begin{matrix} - 0.59 \pm 1.10 \\ - 0.78 \pm 0.31 \end{matrix}$

Table 9. Table 9: Same as Table 8 except for Δ S f Δ subscript 𝑆 𝑓 \Delta S_{f} for penguin-dominated modes. The QCDF results obtained by Beneke BenekeS are displayed for comparison.

	QCDF (this work)
Mode	With $ρ_{C}, ϕ_{C}$	W/o $ρ_{C}$	QCDF (Beneke)	pQCD	SCET	Expt.	Average
$η^{'} K_{S}$	${0.00}_{- 0.01}^{+ 0.01}$	${0.01}_{- 0.01}^{+ 0.01}$	${0.01}_{- 0.01}^{+ 0.01}$	$- {0.06}_{- 0.91}^{+ 0.50}$	$\begin{matrix} - 0.02 \pm 0.01 \\ - 0.01 \pm 0.01 \end{matrix}$	$\begin{matrix} - 0.10 \pm 0.08 \\ - 0.03 \pm 0.11 \end{matrix}$	$- 0.08 \pm 0.07$
$η K_{S}$	${0.12}_{- 0.08}^{+ 0.09}$	${0.12}_{- 0.03}^{+ 0.04}$	${0.10}_{- 0.07}^{+ 0.11}$	$- {0.07}_{- 0.92}^{+ 0.50}$	$\begin{matrix} - 0.04 \pm 0.16 \\ 0.07 \pm 0.15 \end{matrix}$
$π^{0} K_{S}$	${0.12}_{- 0.06}^{+ 0.07}$	${0.09}_{- 0.06}^{+ 0.07}$	${0.07}_{- 0.04}^{+ 0.05}$	${0.06}_{- 0.03}^{+ 0.02}$	$0.08 \pm 0.03$	$\begin{matrix} - 0.12 \pm 0.20 \\ 0.00 \pm 0.32 \end{matrix}$	$- 0.10 \pm 0.17$

Table 10. Table 10: Same as Table 4 except for some selective B → V P → 𝐵 𝑉 𝑃 B\to VP decays with ρ C = 0.8 subscript 𝜌 𝐶 0.8 \rho_{C}=0.8 and ϕ C = − 80 ∘ subscript italic-ϕ 𝐶 superscript 80 \phi_{C}=-80^{\circ} .

Mode	W/o $ρ_{A, C}, ϕ_{A, C}$	With $ρ_{A}, ϕ_{A}$	With $ρ_{A, C}, ϕ_{A, C}$	Expt. HFAG
$ℬ ({\bar{B}}^{0} \to K^{-} ρ^{+})$	${6.5}_{- 2.6 - 0.4}^{+ 5.4 + 0.4}$	${8.6}_{- 2.8 - 4.5}^{+ 5.7 + 7.4}$	${8.6}_{- 2.8 - 4.5}^{+ 5.7 + 7.4}$	${8.6}_{- 1.1}^{+ 0.9}$
$ℬ ({\bar{B}}^{0} \to {\bar{K}}^{0} ρ^{0})$	${4.7}_{- 1.7 - 0.3}^{+ 3.3 + 0.3}$	${5.5}_{- 1.8 - 2.8}^{+ 3.5 + 4.3}$	${5.4}_{- 1.7 - 2.8}^{+ 3.3 + 4.3}$	$4.7 \pm 0.7$
$ℬ (B^{-} \to {\bar{K}}^{0} ρ^{-})$	${5.5}_{- 2.8 - 0.5}^{+ 6.1 + 0.7}$	${7.8}_{- 2.9 - 4.4}^{+ 6.3 + 7.3}$	${7.8}_{- 2.9 - 4.4}^{+ 6.3 + 7.3}$	${8.0}_{- 1.4}^{+ 1.5}$
$ℬ (B^{-} \to K^{-} ρ^{0})$	${1.9}_{- 1.0 - 0.2}^{+ 2.5 + 0.3}$	${3.3}_{- 1.1 - 1.7}^{+ 2.6 + 2.9}$	${3.5}_{- 1.2 - 1.8}^{+ 2.9 + 2.9}$	${3.81}_{- 0.46}^{+ 0.48}$
$ℬ ({\bar{B}}^{0} \to K^{* -} π^{+})$	${3.7}_{- 0.5 - 0.4}^{+ 0.5 + 0.4}$	${9.2}_{- 1.0 - 3.3}^{+ 1.0 + 3.7}$	${9.2}_{- 1.0 - 3.3}^{+ 1.0 + 3.7}$	${8.6}_{- 1.0}^{+ 0.9}$
$ℬ ({\bar{B}}^{0} \to {\bar{K}}^{* 0} π^{0})$	${1.1}_{- 0.2 - 0.2}^{+ 0.2 + 0.2}$	${3.5}_{- 0.5 - 1.5}^{+ 0.4 + 1.7}$	${3.5}_{- 0.4 - 1.4}^{+ 0.4 + 1.6}$	$2.4 \pm 0.7$
$ℬ (B^{-} \to {\bar{K}}^{* 0} π^{-})$	${4.0}_{- 0.9 - 0.6}^{+ 0.7 + 0.6}$	${10.4}_{- 1.5 - 3.9}^{+ 1.3 + 4.3}$	${10.4}_{- 1.5 - 3.9}^{+ 1.3 + 4.3}$	${9.9}_{- 0.9}^{+ 0.8}$
$ℬ (B^{-} \to K^{* -} π^{0})$	${3.2}_{- 0.4 - 0.3}^{+ 0.4 + 0.3}$	${6.8}_{- 0.7 - 2.2}^{+ 0.7 + 2.3}$	${6.7}_{- 0.7 - 2.2}^{+ 0.7 + 2.4}$	$6.9 \pm 2.3$
$ℬ ({\bar{B}}^{0} \to {\bar{K}}^{* 0} η)$	${11.0}_{- 3.5 - 1.0}^{+ 6.9 + 1.7}$	${15.4}_{- 4.0 - 7.1}^{+ 7.7 + 9.4}$	${15.6}_{- 4.1 - 7.1}^{+ 7.9 + 9.4}$	$15.9 \pm 1.0$
$ℬ ({\bar{B}}^{0} \to ω {\bar{K}}^{0})$	${2.9}_{- 1.6 - 0.4}^{+ 4.0 + 0.9}$	${3.9}_{- 1.6 - 2.2}^{+ 4.0 + 3.3}$	${4.1}_{- 1.7 - 2.2}^{+ 4.2 + 3.3}$	$5.0 \pm 0.6$
$ℬ ({\bar{B}}^{0} \to ρ^{0} π^{0})$	${0.76}_{- 0.37 - 0.31}^{+ 0.96 + 0.66}$	${0.58}_{- 0.32 - 0.22}^{+ 0.88 + 0.60}$	${1.3}_{- 0.6 - 0.6}^{+ 1.7 + 1.2}$	$2.0 \pm 0.5$ ¹¹1If an $S$ factor is included, the average will become $2.0 \pm 0.8$ .
$ℬ (B^{-} \to ρ^{-} π^{0})$	${11.6}_{- 0.9 - 0.5}^{+ 1.2 + 0.9}$	${11.8}_{- 0.9 - 0.6}^{+ 1.3 + 1.0}$	${11.8}_{- 1.1 - 1.4}^{+ 1.8 + 1.4}$	${10.9}_{- 1.5}^{+ 1.4}$
$ℬ (B^{-} \to ρ^{0} π^{-})$	${8.2}_{- 0.9 - 0.6}^{+ 1.8 + 1.2}$	${8.5}_{- 0.9 - 0.6}^{+ 1.8 + 1.2}$	${8.7}_{- 1.3 - 1.4}^{+ 2.7 + 1.7}$	${8.3}_{- 1.3}^{+ 1.2}$
$ℬ ({\bar{B}}^{0} \to ρ^{-} π^{+})$	${15.3}_{- 1.5 - 0.9}^{+ 1.0 + 0.5}$	${15.9}_{- 1.5 - 1.1}^{+ 1.1 + 0.9}$	${15.9}_{- 1.5 - 1.1}^{+ 1.1 + 0.9}$	$15.7 \pm 1.8$
$ℬ ({\bar{B}}^{0} \to ρ^{+} π^{-})$	${8.4}_{- 0.7 - 0.5}^{+ 0.4 + 0.3}$	${9.2}_{- 0.7 - 0.7}^{+ 0.4 + 0.5}$	${9.2}_{- 0.7 - 0.7}^{+ 0.4 + 0.5}$	$7.3 \pm 1.2$
$A_{C P} ({\bar{B}}^{0} \to K^{-} ρ^{+})$	$- {1.3}_{- 0.3 - 3.8}^{+ 0.7 + 3.8}$	${31.9}_{- 11.0 - 12.7}^{+ 11.5 + 19.6}$	${31.9}_{- 11.0 - 12.7}^{+ 11.5 + 19.6}$	$15 \pm 6$
$A_{C P} ({\bar{B}}^{0} \to {\bar{K}}^{0} ρ^{0})$	${6.8}_{- 1.2 - 4.9}^{+ 1.1 + 4.9}$	$- {5.0}_{- 6.4 - 4.5}^{+ 3.2 + 6.0}$	${8.7}_{- 1.2 - 6.8}^{+ 1.2 + 8.7}$	–
$A_{C P} (B^{-} \to {\bar{K}}^{0} ρ^{-})$	${0.24}_{- 0.15 - 0.07}^{+ 0.12 + 0.08}$	${0.27}_{- 0.27 - 0.17}^{+ 0.19 + 0.46}$	${0.27}_{- 0.27 - 0.17}^{+ 0.19 + 0.46}$	$- 12 \pm 17$
$A_{C P} (B^{-} \to K^{-} ρ^{0})$	$- {8.3}_{- 0.9 - 7.0}^{+ 3.5 + 7.0}$	${56.5}_{- 18.2 - 22.8}^{+ 16.1 + 30.0}$	${45.4}_{- 19.4 - 23.2}^{+ 17.8 + 31.4}$	$37 \pm 11$
$A_{C P} ({\bar{B}}^{0} \to K^{* -} π^{+})$	${15.6}_{- 0.7 - 4.7}^{+ 0.9 + 4.5}$	$- {12.1}_{- 0.5 - 16.0}^{+ 0.5 + 12.6}$	$- {12.1}_{- 0.5 - 16.0}^{+ 0.5 + 12.6}$	$- 23 \pm 8$
$A_{C P} ({\bar{B}}^{0} \to {\bar{K}}^{* 0} π^{0})$	$- {12.0}_{- 4.6 - 7.6}^{+ 2.4 + 11.3}$	$- {0.87}_{- 0.89 - 6.79}^{+ 1.71 + 6.04}$	$- {10.7}_{- 2.8 - 6.3}^{+ 1.8 + 9.1}$	$- 15 \pm 12$
$A_{C P} (B^{-} \to {\bar{K}}^{* 0} π^{-})$	${0.97}_{- 0.07 - 0.11}^{+ 0.11 + 0.12}$	${0.39}_{- 0.03 - 0.12}^{+ 0.04 + 0.10}$	${0.39}_{- 0.03 - 0.12}^{+ 0.04 + 0.10}$	$- 3.8 \pm 4.2$
$A_{C P} (B^{-} \to K^{* -} π^{0})$	${17.5}_{- 1.3 - 8.0}^{+ 2.0 + 6.3}$	$- {6.7}_{- 1.1 - 14.0}^{+ 0.7 + 11.8}$	${1.6}_{- 1.7 - 14.3}^{+ 3.1 + 11.1}$	$4 \pm 29$
$A_{C P} ({\bar{B}}^{0} \to {\bar{K}}^{* 0} η)$	${3.0}_{- 0.4 - 1.8}^{+ 0.4 + 1.9}$	${0.20}_{- 1.00 - 1.21}^{+ 0.51 + 2.00}$	${3.5}_{- 0.5 - 2.4}^{+ 0.4 + 2.7}$	$19 \pm 5$
$A_{C P} ({\bar{B}}^{0} \to ω {\bar{K}}^{0})$	$- {5.9}_{- 2.3 - 4.1}^{+ 1.9 + 3.4}$	${6.6}_{- 3.4 - 5.3}^{+ 4.7 + 6.0}$	$- {4.7}_{- 1.6 - 5.8}^{+ 1.8 + 5.5}$	$32 \pm 17$
$A_{C P} ({\bar{B}}^{0} \to ρ^{0} π^{0})$	$- {2.3}_{- 3.7 - 9.2}^{+ 2.4 + 9.9}$	${31.5}_{- 12.5 - 30.9}^{+ 13.3 + 21.5}$	${11.0}_{- 5.7 - 28.8}^{+ 5.0 + 23.5}$	$- 30 \pm 38$
$A_{C P} (B^{-} \to ρ^{-} π^{0})$	$- {5.4}_{- 0.3 - 2.1}^{+ 0.4 + 2.0}$	${16.3}_{- 1.2 - 10.5}^{+ 1.1 + 7.1}$	${9.7}_{- 3.1 - 10.3}^{+ 2.1 + 8.0}$	$2 \pm 11$
$A_{C P} (B^{-} \to ρ^{0} π^{-})$	${6.7}_{- 0.8 - 3.1}^{+ 0.5 + 3.5}$	$- {19.8}_{- 1.2 - 8.8}^{+ 1.7 + 12.6}$	$- {9.8}_{- 2.6 - 10.2}^{+ 3.4 + 11.4}$	$18_{- 17}^{+ 9}$
$A_{C P} ({\bar{B}}^{0} \to ρ^{-} π^{+})$	$- {3.5}_{- 0.2 - 0.9}^{+ 0.2 + 1.0}$	${4.4}_{- 0.3 - 6.8}^{+ 0.3 + 5.8}$	${4.4}_{- 0.3 - 6.8}^{+ 0.3 + 5.8}$	$11 \pm 6$
$A_{C P} ({\bar{B}}^{0} \to ρ^{+} π^{-})$	${0.6}_{- 0.1 - 2.2}^{+ 0.1 + 2.2}$	$- {22.7}_{- 1.1 - 4.4}^{+ 0.9 + 8.2}$	$- {22.7}_{- 1.1 - 4.4}^{+ 0.9 + 8.2}$	$- 18 \pm 12$

Table 11. Table 11: Branching fractions (in units of 10 − 6 superscript 10 6 10^{-6} ) of B → V P → 𝐵 𝑉 𝑃 B\to VP decays induced by the b → d → 𝑏 𝑑 b\to d ( Δ S = 0 Δ 𝑆 0 \Delta S=0 ) transition. We also cite the experimental data HFAG ; PDG and theoretical results given in pQCD Lupirho ; XiaoKKst ; Luphipi ; XiaoVeta and in SCET SCETVP .

Mode	QCDF (this work)	pQCD	SCET 1	SCET 2	Expt.
$B^{-} \to ρ^{-} π^{0}$	${11.8}_{- 1.1 - 1.4}^{+ 1.8 + 1.4}$	$6 \sim 9$	${8.9}_{- 0.1 - 1.0}^{+ 0.3 + 1.0}$	${11.4}_{- 0.6 - 0.9}^{+ 0.6 + 1.1}$	${10.9}_{- 1.5}^{+ 1.4}$
$B^{-} \to ρ^{0} π^{-}$	${8.7}_{- 1.3 - 1.4}^{+ 2.7 + 1.7}$	$5 \sim 6$	${10.7}_{- 0.7 - 0.9}^{+ 0.7 + 1.0}$	${7.9}_{- 0.1 - 0.8}^{+ 0.2 + 0.8}$	${8.3}_{- 1.3}^{+ 1.2}$
${\bar{B}}^{0} \to ρ^{\pm} π^{\mp}$	${25.1}_{- 2.2 - 1.8}^{+ 1.5 + 1.4}$	$18 \sim 45$	${13.4}_{- 0.5 - 1.2}^{+ 0.6 + 1.2}$	${16.8}_{- 0.5 - 1.5}^{+ 0.5 + 1.6}$	$23.0 \pm 2.3$
${\bar{B}}^{0} \to ρ^{+} π^{-}$	${9.2}_{- 0.7 - 0.7}^{+ 0.4 + 0.5}$		${5.9}_{- 0.5 - 0.5}^{+ 0.5 + 0.5}$	${6.6}_{- 0.1 - 0.7}^{+ 0.2 + 0.7}$	$7.3 \pm 1.2$
${\bar{B}}^{0} \to ρ^{-} π^{+}$	${15.9}_{- 1.5 - 1.1}^{+ 1.1 + 0.9}$		${7.5}_{- 0.1 - 0.8}^{+ 0.3 + 0.8}$	${10.2}_{- 0.5 - 0.9}^{+ 0.4 + 0.9}$	$15.7 \pm 1.8$
${\bar{B}}^{0} \to ρ^{0} π^{0}$	${1.3}_{- 0.6 - 0.6}^{+ 1.7 + 1.2}$	$0.07 \sim 0.11$	${2.5}_{- 0.1 - 0.2}^{+ 0.2 + 0.2}$	${1.5}_{- 0.1 - 0.1}^{+ 0.1 + 0.1}$	$2.0 \pm 0.5$
$B^{-} \to ω π^{-}$	${6.7}_{- 1.0 - 1.1}^{+ 2.1 + 1.3}$	$4 \sim 8$	${6.7}_{- 0.3 - 0.6}^{+ 0.4 + 0.7}$	${8.5}_{- 0.3 - 0.8}^{+ 0.3 + 0.8}$	$6.9 \pm 0.5$
${\bar{B}}^{0} \to ω π^{0}$	${0.01}_{- 0.00 - 0.01}^{+ 0.02 + 0.04}$	$0.10 \sim 0.28$	${0.0003}_{- 0.0000 - 0.0000}^{+ 0.0299 + 0.0000}$	${0.015}_{- 0.000 - 0.002}^{+ 0.024 + 0.002}$	$< 0.5$
$B^{-} \to K^{* 0} K^{-}$	${0.80}_{- 0.17 - 0.28}^{+ 0.20 + 0.31}$	${0.32}_{- 0.07}^{+ 0.12}$	${0.49}_{- 0.20 - 0.08}^{+ 0.26 + 0.09}$	${0.51}_{- 0.16 - 0.06}^{+ 0.18 + 0.07}$	$0.68 \pm 0.19$ ¹¹1from the preliminary Belle measurement Belle:KstK .
$B^{-} \to K^{* -} K^{0}$	${0.46}_{- 0.17 - 0.26}^{+ 0.37 + 0.42}$	${0.21}_{- 0.13}^{+ 0.14}$	${0.54}_{- 0.21 - 0.08}^{+ 0.26 + 0.10}$	${0.51}_{- 0.17 - 0.07}^{+ 0.21 + 0.08}$
${\bar{B}}^{0} \to K^{* +} K^{-}$	${0.08}_{- 0.01 - 0.02}^{+ 0.01 + 0.02}$	${0.083}_{- 0.067}^{+ 0.072}$
${\bar{B}}^{0} \to K^{* -} K^{+}$	${0.07}_{- 0.01 - 0.03}^{+ 0.01 + 0.04}$	${0.017}_{- 0.011}^{+ 0.027}$
${\bar{B}}^{0} \to K^{* 0} {\bar{K}}^{0}$	${0.70}_{- 0.15 - 0.25}^{+ 0.18 + 0.28}$	${0.24}_{- 0.06}^{+ 0.07}$	${0.45}_{- 0.19 - 0.07}^{+ 0.24 + 0.09}$	${0.47}_{- 0.14 - 0.05}^{+ 0.17 + 0.06}$
${\bar{B}}^{0} \to {\bar{K}}^{* 0} K^{0}$	${0.47}_{- 0.17 - 0.27}^{+ 0.36 + 0.43}$	${0.49}_{- 0.09}^{+ 0.15}$	${0.51}_{- 0.20 - 0.08}^{+ 0.24 + 0.09}$	${0.48}_{- 0.16 - 0.06}^{+ 0.20 + 0.07}$	$< 1.9$
$B^{-} \to ϕ π^{-}$	$\approx 0.043$ ²²2due to the $ω - ϕ$ mixing effect.	${0.032}_{- 0.014}^{+ 0.012}$	$\approx 0.003$	$\approx 0.003$	$< 0.24$
${\bar{B}}^{0} \to ϕ π^{0}$	${0.01}_{- 0.01 - 0.01}^{+ 0.03 + 0.02}$	${0.0068}_{- 0.0008}^{+ 0.0010}$	$\approx 0.001$	$\approx 0.001$	$< 0.28$
$B^{-} \to ρ^{-} η$	${8.3}_{- 0.6 - 0.9}^{+ 1.0 + 0.9}$	${6.7}_{- 1.9}^{+ 2.6}$	${3.9}_{- 1.7 - 0.4}^{+ 2.0 + 0.4}$	${3.3}_{- 1.6 - 0.3}^{+ 1.9 + 0.3}$	$6.9 \pm 1.0$
$B^{-} \to ρ^{-} η^{'}$	${5.6}_{- 0.5 - 0.7}^{+ 0.9 + 0.8}$	${4.6}_{- 1.4}^{+ 1.6}$	${0.37}_{- 0.22 - 0.07}^{+ 2.46 + 0.08}$	${0.44}_{- 0.20 - 0.05}^{+ 3.18 + 0.06}$	${9.1}_{- 2.8}^{+ 3.7}$
${\bar{B}}^{0} \to ρ^{0} η$	${0.10}_{- 0.01 - 0.03}^{+ 0.02 + 0.04}$	${0.13}_{- 0.06}^{+ 0.13}$	${0.04}_{- 0.01 - 0.00}^{+ 0.20 + 0.00}$	${0.14}_{- 0.13 - 0.01}^{+ 0.33 + 0.01}$	$< 1.5$
${\bar{B}}^{0} \to ρ^{0} η^{'}$	${0.09}_{- 0.04 - 0.03}^{+ 0.10 + 0.07}$	${0.10}_{- 0.05}^{+ 0.05}$	${0.43}_{- 0.12 - 0.05}^{+ 2.51 + 0.05}$	${1.0}_{- 0.9 - 0.1}^{+ 3.5 + 0.1}$	$< 1.3$
${\bar{B}}^{0} \to ω η$	${0.85}_{- 0.26 - 0.24}^{+ 0.65 + 0.40}$	${0.71}_{- 0.28}^{+ 0.37}$	${0.91}_{- 0.49 - 0.09}^{+ 0.66 + 0.09}$	${1.4}_{- 0.6 - 0.1}^{+ 0.8 + 0.1}$	${0.94}_{- 0.31}^{+ 0.36}$
${\bar{B}}^{0} \to ω η^{'}$	${0.59}_{- 0.20 - 0.18}^{+ 0.50 + 0.33}$	${0.55}_{- 0.26}^{+ 0.31}$	${0.18}_{- 0.10 - 0.03}^{+ 1.31 + 0.04}$	${3.1}_{- 2.6 - 0.3}^{+ 4.9 + 0.3}$	${1.01}_{- 0.39}^{+ 0.47}$
${\bar{B}}^{0} \to ϕ η$	$\approx 0.005$ ²²2due to the $ω - ϕ$ mixing effect.	${0.011}_{- 0.009}^{+ 0.062}$	$\approx 0.0004$	$\approx 0.0008$	$< 0.5$
${\bar{B}}^{0} \to ϕ η^{'}$	$\approx 0.004$	${0.017}_{- 0.010}^{+ 0.161}$	$\approx 0.0001$	$\approx 0.0007$	$< 0.5$

Table 12. Table 12: Branching fractions (in units of 10 − 6 superscript 10 6 10^{-6} ) of B → V P → 𝐵 𝑉 𝑃 B\to VP decays induced by the b → s → 𝑏 𝑠 b\to s ( Δ S = 1 Δ 𝑆 1 \Delta S=1 ) transition. We also cite the average of the experimental data HFAG ; PDG and theoretical results given in pQCD LiVP ; ChenKeta and in SCET SCETVP .

Mode	QCDF (this work)	pQCD	SCET 1	SCET 2	Expt.
$B^{-} \to K^{* -} π^{0}$	${6.7}_{- 0.7 - 2.2}^{+ 0.7 + 2.4}$	${4.3}_{- 2.2}^{+ 5.0}$	${4.2}_{- 1.7 - 0.7}^{+ 2.2 + 0.8}$	${6.5}_{- 1.7 - 0.7}^{+ 1.9 + 0.7}$	$6.9 \pm 2.3$
$B^{-} \to {\bar{K}}^{* 0} π^{-}$	${10.4}_{- 1.5 - 3.9}^{+ 1.3 + 4.3}$	${6.0}_{- 1.5}^{+ 2.8}$	${8.5}_{- 3.6 - 1.4}^{+ 4.7 + 1.7}$	${9.9}_{- 3.0 - 1.1}^{+ 3.5 + 1.3}$	${9.9}_{- 0.9}^{+ 0.8}$
${\bar{B}}^{0} \to {\bar{K}}^{* 0} π^{0}$	${3.5}_{- 0.4 - 1.4}^{+ 0.4 + 1.6}$	${2.0}_{- 0.6}^{+ 1.2}$	${4.6}_{- 1.8 - 0.7}^{+ 2.3 + 0.9}$	${3.7}_{- 1.2 - 0.5}^{+ 1.4 + 0.5}$	$2.4 \pm 0.7$
${\bar{B}}^{0} \to K^{* -} π^{+}$	${9.2}_{- 1.0 - 3.3}^{+ 1.0 + 3.7}$	${6.0}_{- 2.6}^{+ 6.8}$	${8.4}_{- 3.4 - 1.3}^{+ 4.4 + 1.6}$	${9.5}_{- 2.8 - 1.1}^{+ 3.2 + 1.2}$	${8.6}_{- 1.0}^{+ 0.9}$
$B^{-} \to ρ^{0} K^{-}$	${3.5}_{- 1.2 - 1.8}^{+ 2.9 + 2.9}$	${5.1}_{- 2.8}^{+ 4.1}$	${6.7}_{- 2.2 - 0.9}^{+ 2.7 + 1.0}$	${4.6}_{- 1.5 - 0.6}^{+ 1.8 + 0.7}$	${3.81}_{- 0.46}^{+ 0.48}$
$B^{-} \to ρ^{-} {\bar{K}}^{0}$	${7.8}_{- 2.9 - 4.4}^{+ 6.3 + 7.3}$	${8.7}_{- 4.4}^{+ 6.8}$	${9.3}_{- 3.7 - 1.4}^{+ 4.7 + 1.7}$	${10.1}_{- 3.3 - 1.3}^{+ 4.0 + 1.5}$	${8.0}_{- 1.4}^{+ 1.5}$
${\bar{B}}^{0} \to ρ^{0} {\bar{K}}^{0}$	${5.4}_{- 1.7 - 2.8}^{+ 3.4 + 4.3}$	${4.8}_{- 2.3}^{+ 4.3}$	${3.5}_{- 1.5 - 0.6}^{+ 2.0 + 0.7}$	${5.8}_{- 1.8 - 0.7}^{+ 2.1 + 0.8}$	$4.7 \pm 0.7$
${\bar{B}}^{0} \to ρ^{+} K^{-}$	${8.6}_{- 2.8 - 4.5}^{+ 5.7 + 7.4}$	${8.8}_{- 4.5}^{+ 6.8}$	${9.8}_{- 3.7 - 1.4}^{+ 4.6 + 1.7}$	${10.2}_{- 3.2 - 1.2}^{+ 3.8 + 1.5}$	${8.6}_{- 1.1}^{+ 0.9}$
$B^{-} \to ω K^{-}$	${4.8}_{- 1.9 - 2.3}^{+ 4.4 + 3.5}$	${10.6}_{- 5.8}^{+ 10.4}$	${5.1}_{- 1.9 - 0.8}^{+ 2.4 + 0.9}$	${5.9}_{- 1.7 - 0.7}^{+ 2.1 + 0.8}$	$6.7 \pm 0.5$
${\bar{B}}^{0} \to ω {\bar{K}}^{0}$	${4.1}_{- 1.7 - 2.2}^{+ 4.2 + 3.3}$	${9.8}_{- 4.9}^{+ 8.6}$	${4.1}_{- 1.7 - 0.7}^{+ 2.1 + 0.8}$	${4.9}_{- 1.6 - 0.6}^{+ 1.9 + 0.7}$	$5.0 \pm 0.6$
$B^{-} \to ϕ K^{-}$	${8.8}_{- 2.7 - 3.6}^{+ 2.8 + 4.7}$	${7.8}_{- 1.8}^{+ 5.9}$	${9.7}_{- 3.9 - 1.5}^{+ 4.9 + 1.8}$	${8.6}_{- 2.7 - 1.0}^{+ 3.2 + 1.2}$	$8.30 \pm 0.65$
${\bar{B}}^{0} \to ϕ {\bar{K}}^{0}$	${8.1}_{- 2.5 - 3.3}^{+ 2.6 + 4.4}$	${7.3}_{- 1.6}^{+ 5.4}$	${9.1}_{- 3.6 - 1.4}^{+ 4.6 + 1.7}$	${8.0}_{- 2.5 - 1.0}^{+ 3.0 + 1.1}$	${8.3}_{- 1.0}^{+ 1.2}$
$B^{-} \to K^{* -} η$	${15.7}_{- 4.3 - 7.1}^{+ 8.5 + 9.4}$	${22.13}_{- 0.27}^{+ 0.26}$	${17.9}_{- 5.4 - 2.9}^{+ 5.5 + 3.5}$	${18.6}_{- 4.8 - 2.2}^{+ 4.5 + 2.5}$	$19.3 \pm 1.6$
$B^{-} \to K^{* -} η^{'}$	${1.7}_{- 0.4 - 1.6}^{+ 2.7 + 4.1}$	$6.38 \pm 0.26$	${4.5}_{- 3.9 - 0.8}^{+ 6.6 + 0.9}$	${4.8}_{- 3.7 - 0.6}^{+ 5.3 + 0.8}$	${4.9}_{- 1.9}^{+ 2.1}$ ¹¹1This is from the BaBar data BaBar:Ksteta' . Belle obtained an upper limit $2.9 \times 10^{- 6}$ Belle:Ksteta' .
${\bar{B}}^{0} \to {\bar{K}}^{* 0} η$	${15.6}_{- 4.1 - 7.1}^{+ 7.9 + 9.4}$	${22.31}_{- 0.29}^{+ 0.28}$	${16.6}_{- 5.0 - 2.7}^{+ 5.1 + 3.2}$	${16.5}_{- 4.3 - 2.0}^{+ 4.1 + 2.3}$	$15.9 \pm 1.0$
${\bar{B}}^{0} \to {\bar{K}}^{* 0} η^{'}$	${1.5}_{- 0.4 - 1.7}^{+ 2.4 + 3.9}$	${3.35}_{- 0.27}^{+ 0.29}$	${4.1}_{- 3.6 - 0.7}^{+ 6.2 + 0.9}$	${4.0}_{- 3.4 - 0.6}^{+ 4.7 + 0.7}$	$3.8 \pm 1.2$ ²²2This is from the BaBar data BaBar:Ksteta' . Belle obtained an upper limit $2.6 \times 10^{- 6}$ Belle:Ksteta' .

Table 13. Table 13: Same as Table 11 except for direct CP asymmetries involving b → d → 𝑏 𝑑 b\to d ( Δ S = 0 Δ 𝑆 0 \Delta S=0 ) transitions.

Mode	QCDF (this work)	pQCD	SCET 1	SCET 2	Expt.
$B^{-} \to ρ^{-} π^{0}$	${9.7}_{- 3.1 - 10.3}^{+ 2.1 + 8.0}$	$0 \sim 20$	${15.5}_{- 18.9 - 1.4}^{+ 16.9 + 1.6}$	${12.3}_{- 10.0 - 1.1}^{+ 9.4 + 0.9}$	$2 \pm 11$
$B^{-} \to ρ^{0} π^{-}$	$- {9.8}_{- 2.6 - 10.2}^{+ 3.4 + 11.4}$	$- 20 \sim 0$	$- {10.8}_{- 12.7 - 0.7}^{+ 13.1 + 0.9}$	$- {19.2}_{- 13.4 - 1.9}^{+ 15.5 + 1.7}$	$18_{- 17}^{+ 9}$
${\bar{B}}^{0} \to ρ^{+} π^{-}$	$- {22.7}_{- 1.1 - 4.4}^{+ 0.9 + 8.2}$		$- {9.9}_{- 16.7 - 0.7}^{+ 17.2 + 0.9}$	$- {12.4}_{- 15.3 - 1.2}^{+ 17.6 + 1.1}$	$- 18 \pm 12$
${\bar{B}}^{0} \to ρ^{-} π^{+}$	${4.4}_{- 0.3 - 6.8}^{+ 0.3 + 5.8}$		${11.8}_{- 20.0 - 1.1}^{+ 17.5 + 1.2}$	${10.8}_{- 10.2 - 1.0}^{+ 9.4 + 0.9}$	$11 \pm 6$
${\bar{B}}^{0} \to ρ^{0} π^{0}$	${11.0}_{- 5.7 - 28.8}^{+ 5.0 + 23.5}$	$- 75 \sim 0$	$- {0.6}_{- 21.9 - 0.1}^{+ 21.4 + 0.1}$	$- {3.5}_{- 20.3 - 0.3}^{+ 21.4 + 0.3}$	$- 30 \pm 38$
$B^{-} \to ω π^{-}$	$- {13.2}_{- 2.1 - 10.7}^{+ 3.2 + 12.0}$	$\sim 0$	${0.5}_{- 19.6 - 0.0}^{+ 19.1 + 0.1}$	${2.3}_{- 13.2 - 0.2}^{+ 13.4 + 0.2}$	$- 4 \pm 6$
${\bar{B}}^{0} \to ω π^{0}$	$- {17.0}_{- 22.8 - 82.3}^{+ 55.4 + 98.6}$	$- 20 \sim 75$	$- {9.4}_{- 0.0 - 0.9}^{+ 24.0 + 1.1}$	${39.5}_{- 185.5 - 3.1}^{+ 79.1 + 3.4}$
$B^{-} \to K^{* 0} K^{-}$	$- {8.9}_{- 1.1 - 2.4}^{+ 1.1 + 2.8}$	$- {6.9}_{- 5.3 - 0.3 - 6.5 - 6.0}^{+ 5.6 + 1.0 + 9.2 + 4.0}$	$- {3.6}_{- 5.3 - 0.4}^{+ 6.1 + 0.4}$	$- {4.4}_{- 4.1 - 0.2}^{+ 4.1 + 0.2}$
$B^{-} \to K^{* -} K^{0}$	$- {7.8}_{- 4.1 - 10.0}^{+ 5.9 + 4.1}$	${6.5}_{- 7.3 - 1.4 - 7.7 - 3.9}^{+ 7.9 + 1.1 + 9.1 + 2.1}$	$- {1.5}_{- 2.3 - 0.1}^{+ 2.6 + 0.1}$	$- {1.2}_{- 1.7 - 0.1}^{+ 1.7 + 0.1}$
${\bar{B}}^{0} \to K^{* +} K^{-}$	$- {4.7}_{- 0.2 - 2.7}^{+ 0.1 + 4.7}$
${\bar{B}}^{0} \to K^{* -} K^{+}$	${5.5}_{- 0.2 - 5.5}^{+ 0.2 + 7.0}$
${\bar{B}}^{0} \to K^{* 0} {\bar{K}}^{0}$	$- {13.5}_{- 1.7 - 2.3}^{+ 1.6 + 1.4}$		$- {3.6}_{- 5.3 - 0.4}^{+ 6.1 + 0.4}$	$- {4.4}_{- 4.1 - 0.2}^{+ 4.1 + 0.2}$
${\bar{B}}^{0} \to {\bar{K}}^{* 0} K^{0}$	$- {3.5}_{- 1.7 + 2.0}^{+ 1.3 + 0.7}$		$- {1.5}_{- 2.3 - 0.1}^{+ 2.6 + 0.1}$	$- {1.2}_{- 1.7 - 0.1}^{+ 1.7 + 0.1}$
$B^{-} \to ϕ π^{-}$	0	$- {8.0}_{- 1.0 - 0.1}^{+ 0.9 + 1.5}$
${\bar{B}}^{0} \to ϕ π^{0}$	0	$- {6.3}_{- 0.5 - 2.5}^{+ 0.7 + 2.5}$
$B^{-} \to ρ^{-} η$	$- {8.5}_{- 0.4 - 5.3}^{+ 0.4 + 6.5}$	${1.9}_{- 0.0 - 0.3 - 0.0 - 0.5}^{+ 0.1 + 0.2 + 0.1 + 0.6}$	$- {6.6}_{- 21.3 - 0.7}^{+ 21.5 + 0.6}$	$- {9.1}_{- 15.8 - 0.8}^{+ 16.7 + 0.9}$	$11 \pm 11$
$B^{-} \to ρ^{-} η^{'}$	${1.4}_{- 2.2 - 11.7}^{+ 0.8 + 14.0}$	$- {25.0}_{- 0.3 - 1.6 - 0.7 - 1.8}^{+ 0.4 + 4.1 + 0.8 + 2.1}$	$- {19.8}_{- 37.5 - 3.1}^{+ 66.5 + 2.8}$	$- {21.7}_{- 24.3 - 1.7}^{+ 135.9 + 2.1}$	$4 \pm 28$
${\bar{B}}^{0} \to ρ^{0} η$	${86.2}_{- 5.8 - 21.4}^{+ 3.7 + 10.4}$	$- {89.6}_{- 0.9 - 3.9 - 0.1 - 9.0}^{+ 1.9 + 13.7 + 0.7 + 4.6}$	$- {46.7}_{- 74.3 - 3.7}^{+ 170.4 + 2.9}$	${33.3}_{- 62.4 - 2.8}^{+ 66.9 + 3.1}$
${\bar{B}}^{0} \to ρ^{0} η^{'}$	${53.5}_{- 7.9 - 57.6}^{+ 4.5 + 39.5}$	$- {75.7}_{- 4.8 - 7.0 - 4.0 - 9.9}^{+ 5.6 + 13.1 + 6.3 + 12.9}$	$- {51.7}_{- 42.9 - 3.9}^{+ 103.3 + 3.4}$	${52.2}_{- 80.6 - 4.1}^{+ 19.9 + 4.4}$
${\bar{B}}^{0} \to ω η$	$- {44.7}_{- 9.9 - 11.6}^{+ 13.1 + 17.7}$	${33.5}_{- 1.4 - 4.6 - 6.8 - 4.4}^{+ 1.0 + 0.8 + 5.9 + 3.9}$	$- {9.4}_{- 30.2 - 1.0}^{+ 30.7 + 0.9}$	$- {9.6}_{- 16.8 - 0.9}^{+ 17.8 + 0.9}$
${\bar{B}}^{0} \to ω η^{'}$	$- {41.4}_{- 2.4 - 14.4}^{+ 2.5 + 19.5}$	${16.0}_{- 0.9 - 3.9 - 3.2 - 2.0}^{+ 0.1 + 3.3 + 2.2 + 1.7}$	$- {43.0}_{- 38.8 - 5.1}^{+ 87.5 + 4.8}$	$- {27.2}_{- 29.7 - 2.2}^{+ 18.1 + 2.4}$
${\bar{B}}^{0} \to ϕ η$	0	0
${\bar{B}}^{0} \to ϕ η^{'}$	0	0

Table 14. Table 14: Same as Table 12 except for direct CP asymmetries (in % percent \% ) involving Δ S = 1 Δ 𝑆 1 \Delta S=1 processes.

Mode	QCDF (this work)	pQCD	SCET 1	SCET 2	Expt.
$B^{-} \to K^{* -} π^{0}$	${1.6}_{- 1.7 - 14.4}^{+ 3.1 + 11.1}$	$- 32_{- 28}^{+ 21}$	$- {17.8}_{- 24.6 - 2.0}^{+ 30.3 + 2.2}$	$- {12.9}_{- 12.2 - 0.8}^{+ 12.0 + 0.8}$	$4 \pm 29$
$B^{-} \to {\bar{K}}^{* 0} π^{-}$	${0.4}_{- 1.6 - 3.9}^{+ 1.3 + 4.3}$	$- 1_{- 0}^{+ 1}$	$0$	$0$	$- 3.8 \pm 4.2$
${\bar{B}}^{0} \to {\bar{K}}^{* 0} π^{0}$	$- {10.8}_{- 2.8 - 6.3}^{+ 1.8 + 9.1}$	$- 11_{- 5}^{+ 7}$	${5.0}_{- 8.4 - 0.5}^{+ 7.5 + 0.5}$	${5.4}_{- 5.1 - 0.5}^{+ 4.8 + 0.4}$	$- 15 \pm 12$
${\bar{B}}^{0} \to K^{* -} π^{+}$	$- {12.1}_{- 0.5 - 16.0}^{+ 0.5 + 12.6}$	$- 60_{- 19}^{+ 32}$	$- {11.2}_{- 16.2 - 1.3}^{+ 19.0 + 1.3}$	$- {12.2}_{- 11.3 - 0.8}^{+ 11.4 + 0.8}$	$- 23 \pm 8$
$B^{-} \to ρ^{0} K^{-}$	${45.4}_{- 19.4 - 23.2}^{+ 17.8 + 31.4}$	$71_{- 35}^{+ 25}$	${9.2}_{- 16.1 - 0.7}^{+ 15.2 + 0.7}$	${16.0}_{- 22.4 - 1.6}^{+ 20.5 + 1.3}$	$37 \pm 11$
$B^{-} \to ρ^{-} {\bar{K}}^{0}$	${0.3}_{- 0.3 - 0.2}^{+ 0.2 + 0.5}$	$1 \pm 1$	$0$	$0$	$- 12 \pm 17$
${\bar{B}}^{0} \to ρ^{0} {\bar{K}}^{0}$	${8.7}_{- 1.2 - 6.8}^{+ 1.2 + 8.7}$	$7_{- 5}^{+ 8}$	$- {6.6}_{- 9.7 - 0.9}^{+ 11.6 + 0.8}$	$- {3.5}_{- 4.8 - 0.2}^{+ 4.8 + 0.3}$	$6 \pm 20$
${\bar{B}}^{0} \to ρ^{+} K^{-}$	${31.9}_{- 11.0 - 12.7}^{+ 11.5 + 19.6}$	$64_{- 30}^{+ 24}$	${7.1}_{- 12.4 - 0.7}^{+ 11.2 + 0.7}$	${9.6}_{- 13.5 - 0.9}^{+ 13.0 + 0.7}$	$15 \pm 6$
$B^{-} \to ω K^{-}$	${22.1}_{- 12.8 - 13.0}^{+ 13.7 + 14.0}$	$32_{- 17}^{+ 15}$	${11.6}_{- 20.4 - 1.1}^{+ 18.2 + 1.1}$	${12.3}_{- 17.3 - 1.1}^{+ 16.6 + 0.8}$	$2 \pm 5$
${\bar{B}}^{0} \to ω {\bar{K}}^{0}$	$- {4.7}_{- 1.6 - 5.8}^{+ 1.8 + 5.5}$	$- 3_{- 4}^{+ 2}$	${5.2}_{- 9.2 - 0.6}^{+ 8.0 + 0.6}$	${3.8}_{- 5.4 - 0.3}^{+ 5.2 + 0.3}$	$32 \pm 17$ ¹¹1Note that the measurements of $52_{- 20}^{+ 22} \pm 3$ by BaBar BaBar:omegaK and $- 9 \pm 29 \pm 6$ by Belle Belle:omegaK are of opposite sign.
$B^{-} \to ϕ K^{-}$	${0.6}_{- 0.1 - 0.1}^{+ 0.1 + 0.1}$	$1_{- 1}^{+ 0}$	$0$	$0$	$- 1 \pm 6$
${\bar{B}}^{0} \to ϕ {\bar{K}}^{0}$	${0.9}_{- 0.1 - 0.1}^{+ 0.2 + 0.2}$	$3_{- 2}^{+ 1}$	$0$	$0$	$23 \pm 15$
$B^{-} \to K^{* -} η$	$- {9.7}_{- 3.7 - 7.1}^{+ 3.9 + 6.2}$	$- {24.57}_{- 0.27}^{+ 0.72}$	$- {2.6}_{- 5.5 - 0.3}^{+ 5.4 + 0.3}$	$- {1.9}_{- 3.6 - 0.1}^{+ 3.4 + 0.1}$	$2 \pm 6$
$B^{-} \to K^{* -} η^{'}$	${65.5}_{- 39.5 - 50.2}^{+ 10.1 + 34.2}$	${4.60}_{- 1.32}^{+ 1.16}$	${2.7}_{- 19.5 - 0.3}^{+ 27.4 + 0.4}$	${2.6}_{- 32.9 - 0.2}^{+ 26.7 + 0.2}$	$- 30_{- 33}^{+ 37}$
${\bar{B}}^{0} \to {\bar{K}}^{* 0} η$	${3.5}_{- 0.5 - 2.4}^{+ 0.4 + 2.7}$	$0.57 \pm 0.011$	$- {1.1}_{- 2.4 - 0.1}^{+ 2.3 + 0.1}$	$- {0.7}_{- 1.3 - 0.0}^{+ 1.2 + 0.1}$	$19 \pm 5$
${\bar{B}}^{0} \to {\bar{K}}^{* 0} η^{'}$	${6.8}_{- 9.2 - 50.2}^{+ 10.7 + 33.2}$	$- 1.30 \pm 0.08$	${9.6}_{- 11.0 - 1.2}^{+ 8.9 + 1.3}$	${9.9}_{- 4.3 - 0.9}^{+ 6.2 + 0.9}$	$8 \pm 25$

Table 15. Table 15: Mixing-induced CP violation S f subscript 𝑆 𝑓 S_{f} in B ¯ → V P → ¯ 𝐵 𝑉 𝑃 \bar{B}\to VP decays predicted in various approaches. The pQCD results are taken from LiVP ; XiaoVeta . There are two solutions with SCET predictions SCETVP . The parameter η f = 1 subscript 𝜂 𝑓 1 \eta_{f}=1 except for ( ϕ , ρ , ω ) K S italic-ϕ 𝜌 𝜔 subscript 𝐾 𝑆 (\phi,\rho,\omega)K_{S} modes where η f = − 1 subscript 𝜂 𝑓 1 \eta_{f}=-1 . Experimental results from BaBar (first entry) and Belle (second entry) are listed whenever available. The input values of sin ⁡ 2 β 2 𝛽 \sin 2\beta used at the time of theoretical calculations which are needed for the calculation of Δ S f Δ subscript 𝑆 𝑓 \Delta S_{f} are displayed.

Decay	QCDF (this work)	pQCD	SCET	Expt. BaBar:phiK0 ; Belle:etaK0 ; BaBar:rhoK0 ; Belle:rhoK0 ; BaBar:etaK0 ; Belle:omegaK0 ; BaBar:rhopi ; Belle:rhopi	Average
$\sin 2 β$	0.670	0.687	0.687
$ϕ K_{S}$	${0.692}_{- 0.000 - 0.002}^{+ 0.003 + 0.002}$	$0.71 \pm 0.01$	$\begin{matrix} 0.69 \\ 0.69 \end{matrix}$	$\begin{matrix} 0.26 \pm 0.26 \pm 0.03 \\ {0.67}_{- 0.32}^{+ 0.22} \end{matrix}$	${0.44}_{- 0.18}^{+ 0.17}$
$ω K_{S}$	${0.84}_{- 0.05 - 0.06}^{+ 0.05 + 0.04}$	${0.84}_{- 0.07}^{+ 0.03}$	$\begin{matrix} {0.51}_{- 0.06 - 0.02}^{+ 0.05 + 0.02} \\ {0.80}_{- 0.02 - 0.01}^{+ 0.02 + 0.01} \end{matrix}$	$\begin{matrix} {0.55}_{- 0.29}^{+ 0.26} \pm 0.02 \\ 0.11 \pm 0.46 \pm 0.07 \end{matrix}$	$0.45 \pm 0.24$
$ρ^{0} K_{S}$	${0.50}_{- 0.14 - 0.12}^{+ 0.07 + 0.06}$	${0.50}_{- 0.06}^{+ 0.10}$	$\begin{matrix} {0.85}_{- 0.05 - 0.01}^{+ 0.04 + 0.01} \\ {0.56}_{- 0.03 - 0.01}^{+ 0.02 + 0.01} \end{matrix}$	$\begin{matrix} {0.35}_{- 0.31}^{+ 0.26} \pm 0.06 \pm 0.03 \\ {0.64}_{- 0.25}^{+ 0.19} \pm 0.09 \pm 0.10 \end{matrix}$	${0.54}_{- 0.21}^{+ 0.18}$
$ρ^{0} π^{0}$	$- {0.24}_{- 0.14 - 0.22}^{+ 0.15 + 0.20}$		$\begin{matrix} - {0.11}_{- 0.14 - 0.15}^{+ 0.14 + 0.10} \\ - {0.19}_{- 0.14 - 0.15}^{+ 0.14 + 0.10} \end{matrix}$	$\begin{matrix} 0.04 \pm 0.44 \pm 0.18 \\ 0.17 \pm 0.57 \pm 0.35 \end{matrix}$	$0.12 \pm 0.38$
$ω π^{0}$	${0.78}_{- 0.20 - 1.39}^{+ 0.14 + 0.20}$		$\begin{matrix} - {0.87}_{- 0.00 - 0.01}^{+ 0.44 + 0.02} \\ {0.72}_{- 1.54 - 0.11}^{+ 0.36 + 0.07} \end{matrix}$
$ρ^{0} η$	${0.51}_{- 0.07 - 0.32}^{+ 0.08 + 0.19}$	${0.23}_{- 0.37}^{+ 0.30}$	$\begin{matrix} {0.86}_{- 2.03 - 0.07}^{+ 0.15 + 0.03} \\ {0.29}_{- 0.44 - 0.15}^{+ 0.36 + 0.09} \end{matrix}$
$ρ^{0} η^{'}$	${0.80}_{- 0.09 - 0.43}^{+ 0.04 + 0.24}$	$- {0.49}_{- 0.20}^{+ 0.25}$	$\begin{matrix} {0.79}_{- 1.73 - 0.09}^{+ 0.20 + 0.05} \\ {0.38}_{- 1.24 - 0.14}^{+ 0.22 + 0.09} \end{matrix}$
$ω η$	$- {0.16}_{- 0.13 - 0.16}^{+ 0.13 + 0.17}$	${0.39}_{- 0.66}^{+ 0.51}$	$\begin{matrix} {0.12}_{- 0.20 - 0.17}^{+ 0.19 + 0.10} \\ - {0.16}_{- 0.15 - 0.15}^{+ 0.14 + 0.10} \end{matrix}$
$ω η^{'}$	$- {0.28}_{- 0.13 - 0.13}^{+ 0.14 + 0.16}$	${0.77}_{- 0.53}^{+ 0.22}$	$\begin{matrix} {0.23}_{- 1.10 - 0.10}^{+ 0.59 + 0.10} \\ - {0.27}_{- 0.33 - 0.14}^{+ 0.17 + 0.09} \end{matrix}$

Table 16. Table 16: Same as Table 15 except for Δ S f Δ subscript 𝑆 𝑓 \Delta S_{f} for penguin-dominated modes. The QCDF results obtained by Beneke BenekeS are included for comparison.

	QCDF (this work)
Decay	With $ρ_{C}, ϕ_{C}$	W/o $ρ_{C}$	QCDF (Beneke)	pQCD	SCET	Expt.	Average
$ϕ K_{S}$	${0.022}_{- 0.002}^{+ 0.004}$	${0.022}_{- 0.002}^{+ 0.004}$	${0.02}_{- 0.01}^{+ 0.01}$	$0.02 \pm 0.01$	$\begin{matrix} \sim 0 \\ \sim 0 \end{matrix}$	$\begin{matrix} - 0.43 \pm 0.26 \\ {0.02}_{- 0.32}^{+ 0.22} \end{matrix}$	$- {0.25}_{- 0.18}^{+ 0.17}$
$ω K_{S}$	${0.17}_{- 0.08}^{+ 0.06}$	${0.13}_{- 0.04}^{+ 0.06}$	${0.13}_{- 0.08}^{+ 0.08}$	${0.15}_{- 0.07}^{+ 0.03}$	$\begin{matrix} - {0.18}_{- 0.06}^{+ 0.05} \\ {0.11}_{- 0.02}^{+ 0.02} \end{matrix}$	$\begin{matrix} - 0.43 \pm 0.26 \\ {0.02}_{- 0.32}^{+ 0.22} \end{matrix}$	$- {0.14}_{- 0.29}^{+ 0.26}$
$ρ^{0} K_{S}$	$- {0.17}_{- 0.18}^{+ 0.09}$	$- {0.11}_{- 0.11}^{+ 0.07}$	$- {0.08}_{- 0.12}^{+ 0.08}$	$- {0.19}_{- 0.06}^{+ 0.10}$	$\begin{matrix} {0.16}_{- 0.05}^{+ 0.04} \\ - {0.13}_{- 0.03}^{+ 0.02} \end{matrix}$	$\begin{matrix} - {0.34}_{- 0.31}^{+ 0.27} \\ - {0.05}_{- 0.28}^{+ 0.23} \end{matrix}$	$- 0.10 \pm 0.17$

Table 17. Table 17: Various CP -violating parameters in the decays B ¯ 0 → ρ ± π ∓ → superscript ¯ 𝐵 0 superscript 𝜌 plus-or-minus superscript 𝜋 minus-or-plus \bar{B}^{0}\to\rho^{\pm}\pi^{\mp} . SCET results are quoted from SCETVP . Experimental results are taken from BaBar:rho0pi0 ; Belle:rho0pi0 and the world average from HFAG .

Parameter	QCDF (this work)	SCET 1	SCET 2	Expt.
$𝒜_{ρ π}$	$- {0.11}_{- 0.00 - 0.05}^{+ 0.00 + 0.07}$	$- {0.12}_{- 0.05 - 0.03}^{+ 0.04 + 0.04}$	$- {0.21}_{- 0.02 - 0.03}^{+ 0.03 + 0.02}$	$- 0.13 \pm 0.04$
$C$	${0.09}_{- 0.00 - 0.07}^{+ 0.00 + 0.05}$	$- {0.01}_{- 0.12 - 0.00}^{+ 0.13 + 0.00}$	${0.01}_{- 0.10 - 0.00}^{+ 0.09 + 0.00}$	$0.01 \pm 0.07$
$S$	$- {0.04}_{- 0.01 - 0.09}^{+ 0.01 + 0.10}$	$- {0.11}_{- 0.08 - 0.13}^{+ 0.07 + 0.08}$	$- {0.01}_{- 0.07 - 0.14}^{+ 0.06 + 0.08}$	$0.01 \pm 0.09$
$Δ C$	${0.26}_{- 0.02 - 0.02}^{+ 0.02 + 0.02}$	${0.11}_{- 0.13 - 0.01}^{+ 0.12 + 0.01}$	${0.12}_{- 0.10 - 0.01}^{+ 0.09 + 0.01}$	$0.37 \pm 0.08$
$Δ S$	$- {0.02}_{- 0.00 - 0.02}^{+ 0.00 + 0.03}$	$- {0.47}_{- 0.06 - 0.04}^{+ 0.08 + 0.05}$	${0.43}_{- 0.07 - 0.03}^{+ 0.05 + 0.03}$	$- 0.04 \pm 0.10$

Table 18. Table 18: C P 𝐶 𝑃 CP -averaged branching fractions (in units of 10 − 6 superscript 10 6 10^{-6} ) and polarization fractions for B ¯ → V V → ¯ 𝐵 𝑉 𝑉 \bar{B}\to VV decays. For QCDF, the annihilation parameters are specified to be ρ A = 0.78 subscript 𝜌 𝐴 0.78 \rho_{A}=0.78 and ϕ A = − 43 ∘ subscript italic-ϕ 𝐴 superscript 43 \phi_{A}=-43^{\circ} for K ∗ ρ , K ∗ K ¯ ∗ superscript 𝐾 𝜌 superscript 𝐾 superscript ¯ 𝐾 K^{*}\rho,K^{*}\bar{K}^{*} and ρ A = 0.65 subscript 𝜌 𝐴 0.65 \rho_{A}=0.65 and ϕ A = − 53 ∘ subscript italic-ϕ 𝐴 superscript 53 \phi_{A}=-53^{\circ} for K ∗ ϕ superscript 𝐾 italic-ϕ K^{*}\phi and K ∗ ω superscript 𝐾 𝜔 K^{*}\omega by default. The world averages of experimental results are taken from HFAG . The pQCD results are taken from Li:rhorho ; Lu:rhoomega ; Lu:KstV ; Lu:KstKst . There are two distinct pQCD predictions for the branching fractions and longitudinal polarization fractions of B → K ∗ ( ρ , ϕ , ω ) → 𝐵 superscript 𝐾 𝜌 italic-ϕ 𝜔 B\to K^{*}(\rho,\phi,\omega) decays, depending on the type of wave functions. Numbers in parentheses are for asymptotic wave functions. Estimates of uncertainties are not available in many of pQCD predictions.

	$ℬ$			$f_{L}$
Decay	QCDF	pQCD	Expt.	QCDF	pQCD	Expt.
$B^{-} \to ρ^{-} ρ^{0}$	${20.0}_{- 1.9 - 0.9}^{+ 4.0 + 2.0}$	${16.0}_{- 8.1}^{+ 15.0}$ ¹¹1There exist several pQCD calculations for $ρ ρ$ modes Lu:rhoomega ; ChenVV ; Li:rhorho . Here we cite the NLO results from Li:rhorho .	${24.0}_{- 2.0}^{+ 1.9}$	${0.96}_{- 0.01 - 0.02}^{+ 0.01 + 0.02}$		$0.950 \pm 0.016$
${\bar{B}}^{0} \to ρ^{+} ρ^{-}$	${25.5}_{- 2.6 - 1.5}^{+ 1.5 + 2.4}$	${25.3}_{- 13.8}^{+ 25.3}$ ¹¹1There exist several pQCD calculations for $ρ ρ$ modes Lu:rhoomega ; ChenVV ; Li:rhorho . Here we cite the NLO results from Li:rhorho .	${24.2}_{- 3.2}^{+ 3.1}$	${0.92}_{- 0.02 - 0.02}^{+ 0.01 + 0.01}$		${0.978}_{- 0.022}^{+ 0.025}$
${\bar{B}}^{0} \to ρ^{0} ρ^{0}$	${0.9}_{- 0.4 - 0.2}^{+ 1.5 + 1.1}$	${0.92}_{- 0.56}^{+ 1.10}$ ¹¹1There exist several pQCD calculations for $ρ ρ$ modes Lu:rhoomega ; ChenVV ; Li:rhorho . Here we cite the NLO results from Li:rhorho .	${0.73}_{- 0.28}^{+ 0.27}$	${0.92}_{- 0.04 - 0.37}^{+ 0.03 + 0.06}$	0.78	${0.75}_{- 0.15}^{+ 0.12}$
$B^{-} \to ρ^{-} ω$	${16.9}_{- 1.6 - 0.9}^{+ 3.2 + 1.7}$	$19 \pm 2 \pm 1$	$15.9 \pm 2.1$	${0.96}_{- 0.01 - 0.03}^{+ 0.01 + 0.02}$	0.97	$0.90 \pm 0.06$
${\bar{B}}^{0} \to ρ^{0} ω$	${0.08}_{- 0.02 - 0.00}^{+ 0.02 + 0.36}$	$1.9 \pm 0.2 \pm 0.2$	$< 1.5$	${0.52}_{- 0.25 - 0.36}^{+ 0.11 + 0.50}$	0.87
${\bar{B}}^{0} \to ω ω$	${0.7}_{- 0.3 - 0.2}^{+ 0.9 + 0.7}$	$1.2 \pm 0.2 \pm 0.2$	$< 4.0$	${0.94}_{- 0.01 - 0.20}^{+ 0.01 + 0.04}$	0.82
$B^{-} \to K^{* 0} K^{* -}$	${0.6}_{- 0.1 - 0.3}^{+ 0.1 + 0.3}$	${0.48}_{- 0.08}^{+ 0.12}$	$1.2 \pm 0.5$	${0.45}_{- 0.04 - 0.38}^{+ 0.02 + 0.55}$	0.82	${0.75}_{- 0.26}^{+ 0.16}$
${\bar{B}}^{0} \to K^{* -} K^{* +}$	${0.1}_{- 0.0 - 0.1}^{+ 0.0 + 0.1}$	${0.064}_{- 0.010}^{+ 0.005}$	$< 2.0$	$\approx 1$	0.99
${\bar{B}}^{0} \to K^{* 0} {\bar{K}}^{* 0}$	${0.6}_{- 0.1 - 0.3}^{+ 0.1 + 0.2}$	${0.35}_{- 0.07}^{+ 0.13}$	${1.28}_{- 0.32}^{+ 0.37}$ ²²2This is from the BaBar data BaBar:KstKst . The Belle’s new measurement yields $(0.3 \pm 0.3 \pm 0.1) \times 10^{- 6}$ Belle:KstK .	${0.52}_{- 0.07 - 0.48}^{+ 0.04 + 0.48}$	0.78	${0.80}_{- 0.13}^{+ 0.12}$
$B^{-} \to {\bar{K}}^{* 0} ρ^{-}$ ³³3This mode is employed as an input for extracting the parameters $ρ_{A}$ and $ϕ_{A}$ for $B \to K^{*} ρ$ decays.	${9.2}_{- 1.1 - 5.4}^{+ 1.2 + 3.6}$	$17 (13)$	$9.2 \pm 1.5$	${0.48}_{- 0.04 - 0.40}^{+ 0.03 + 0.52}$	$0.82 (0.76)$	$0.48 \pm 0.08$
$B^{-} \to K^{* -} ρ^{0}$	${5.5}_{- 0.5 - 2.5}^{+ 0.6 + 1.3}$	9.0 (6.4)	$< 6.1$	${0.67}_{- 0.03 - 0.48}^{+ 0.02 + 0.31}$	0.85 (0.78)	${0.96}_{- 0.16}^{+ 0.06}$ ⁴⁴4A recent BaBar measurement gives $f_{L} (K^{* -} ρ^{0}) = 0.9 \pm 0.2$ BaBar:KVrhonew , but it has only $2.5 σ$ significance.
${\bar{B}}^{0} \to K^{* -} ρ^{+}$	${8.9}_{- 1.0 - 5.5}^{+ 1.1 + 4.8}$	13 (9.8)	$< 12$	${0.53}_{- 0.03 - 0.32}^{+ 0.02 + 0.45}$	0.78 (0.71)
${\bar{B}}^{0} \to {\bar{K}}^{* 0} ρ^{0}$	${4.6}_{- 0.5 - 3.5}^{+ 0.6 + 3.5}$	5.9 (4.7)	$3.4 \pm 1.0$	${0.39}_{- 0.00 - 0.31}^{+ 0.00 + 0.60}$	0.74 (0.68)	$0.57 \pm 0.12$
$B^{-} \to K^{* -} ϕ$ ⁵⁵5This mode is employed as an input for extracting the parameters $ρ_{A}$ and $ϕ_{A}$ for $B \to K^{*} ϕ$ decays.	${10.0}_{- 1.3 - 6.1}^{+ 1.4 + 12.3}$	⁶⁶6See footnote 6 in Sec.VI.B.	$10.0 \pm 1.1$	${0.49}_{- 0.07 - 0.42}^{+ 0.04 + 0.51}$	⁶⁶6See footnote 6 in Sec.VI.B.	$0.50 \pm 0.05$
${\bar{B}}^{0} \to {\bar{K}}^{* 0} ϕ$	${9.5}_{- 1.2 - 5.9}^{+ 1.3 + 11.9}$	⁶⁶6See footnote 6 in Sec.VI.B.	$9.8 \pm 0.7$	${0.50}_{- 0.06 - 0.43}^{+ 0.04 + 0.51}$	⁶⁶6See footnote 6 in Sec.VI.B.	$0.480 \pm 0.030$
$B^{-} \to K^{* -} ω$	${3.0}_{- 0.3 - 1.5}^{+ 0.4 + 2.5}$	7.9 (5.5)	$< 7.4$	${0.67}_{- 0.04 - 0.39}^{+ 0.03 + 0.32}$	$0.81 (0.73)$	$0.41 \pm 0.19$
${\bar{B}}^{0} \to {\bar{K}}^{* 0} ω$	${2.5}_{- 0.4 - 1.5}^{+ 0.4 + 2.5}$	9.6 (6.6)	$2.0 \pm 0.5$	${0.58}_{- 0.10 - 0.14}^{+ 0.07 + 0.43}$	$0.82 (0.74)$	$0.70 \pm 0.13$

Table 19. Table 19: Direct CP asymmetries (in %) of B ¯ → V V → ¯ 𝐵 𝑉 𝑉 \bar{B}\to VV decays. The pQCD results are taken from Lu:rhoomega for ρ ρ , ρ ω , K ∗ K ¯ ∗ 𝜌 𝜌 𝜌 𝜔 superscript 𝐾 superscript ¯ 𝐾 \rho\rho,\rho\omega,K^{*}\bar{K}^{*} . CP asymmetries of K ∗ ρ superscript 𝐾 𝜌 K^{*}\rho and K ∗ ω superscript 𝐾 𝜔 K^{*}\omega in the pQCD approach are shown in Fig. 4 of Lu:KstV as a function of γ 𝛾 \gamma and only the signs of A C P ( K ∗ ρ ) subscript 𝐴 𝐶 𝑃 superscript 𝐾 𝜌 A_{CP}(K^{*}\rho) and A C P ( K ∗ ω ) subscript 𝐴 𝐶 𝑃 superscript 𝐾 𝜔 A_{CP}(K^{*}\omega) are displayed here. Note that the definition of A C P subscript 𝐴 𝐶 𝑃 A_{CP} in Lu:KstV has a sign opposite to the usual convention.

Decay	QCDF (this work)	pQCD	Expt. HFAG
$B^{-} \to ρ^{-} ρ^{0}$	$0.06$	0	$- 5.1 \pm 5.4$
${\bar{B}}^{0} \to ρ^{+} ρ^{-}$	$- 4_{- 0 - 3}^{+ 0 + 3}$	$- 7$	$6 \pm 13$
${\bar{B}}^{0} \to ρ^{0} ρ^{0}$	$30_{- 16 - 26}^{+ 17 + 14}$	80
$B^{-} \to ρ^{-} ω$	$- 8_{- 1 - 4}^{+ 1 + 3}$	$- 23 \pm 7$	$- 20 \pm 9$
${\bar{B}}^{0} \to ρ^{0} ω$	$3_{- 6 - 76}^{+ 2 + 51}$
${\bar{B}}^{0} \to ω ω$	$- 30_{- 14 - 18}^{+ 15 + 16}$
$B^{-} \to K^{* 0} K^{* -}$	$16_{- 3 - 34}^{+ 1 + 17}$	$- 15$
${\bar{B}}^{0} \to K^{* -} K^{* +}$	$0$	$- 65$
${\bar{B}}^{0} \to K^{* 0} {\bar{K}}^{* 0}$	$- 14_{- 1 - 2}^{+ 1 + 6}$	0
$B^{-} \to {\bar{K}}^{* 0} ρ^{-}$	$- {0.3}_{- 0 - 0}^{+ 0 + 2}$	$+$	$- 1 \pm 16$
$B^{-} \to K^{* -} ρ^{0}$	$43_{- 3 - 28}^{+ 6 + 12}$	$+$	$20_{- 29}^{+ 32}$
${\bar{B}}^{0} \to K^{* -} ρ^{+}$	$32_{- 3 - 14}^{+ 1 + 2}$	$+$
${\bar{B}}^{0} \to {\bar{K}}^{* 0} ρ^{0}$	$- 15_{- 8 - 14}^{+ 4 + 16}$	$-$	$9 \pm 19$
$B^{-} \to K^{* -} ϕ$	$0.05$		$- 1 \pm 8$
${\bar{B}}^{0} \to {\bar{K}}^{* 0} ϕ$	${0.8}_{- 0 - 0.5}^{+ 0 + 0.4}$		$1 \pm 5$
$B^{-} \to K^{* -} ω$	$56_{- 4 - 43}^{+ 3 + 4}$	$+$	$29 \pm 35$
${\bar{B}}^{0} \to {\bar{K}}^{* 0} ω$	$23_{- 5 - 18}^{+ 9 + 5}$	$+$	$45 \pm 25$

Equations229

P

P

A (\overset{ˉ}{B}^{0} \to K^{-} π^{+})

A (\overset{ˉ}{B}^{0} \to K^{-} π^{+})

A (\overset{ˉ}{B}^{0} \to \overset{ˉ}{K}^{0} π^{0})

A (B^{-} \to \overset{ˉ}{K}^{0} π^{-})

A (B^{-} \to K^{-} π^{0})

C = [λ_{u} a_{2}^{u}]_{SD} + [λ_{u} a_{2}^{u}]_{LD} + FSIs + \dots .

C = [λ_{u} a_{2}^{u}]_{SD} + [λ_{u} a_{2}^{u}]_{LD} + FSIs + \dots .

⟨ M_{1} M_{2} ∣ H_{eff} ∣ \overline{B} ⟩ = \frac{G _{F}}{2} p = u, c \sum λ_{p}^{(q)} ⟨ M_{1} M_{2} ∣ T_{A}^{h, p} + T_{B}^{h, p} ∣ \overline{B} ⟩,

⟨ M_{1} M_{2} ∣ H_{eff} ∣ \overline{B} ⟩ = \frac{G _{F}}{2} p = u, c \sum λ_{p}^{(q)} ⟨ M_{1} M_{2} ∣ T_{A}^{h, p} + T_{B}^{h, p} ∣ \overline{B} ⟩,

T_{A}^{h}

T_{A}^{h}

X^{(\overset{ˉ}{B} P_{1}, P_{2})}

X^{(\overset{ˉ}{B} P_{1}, P_{2})}

X^{(\overset{ˉ}{B} P, V)}

X^{(\overset{ˉ}{B} V, P)}

X_{h}^{(\overset{ˉ}{B} V_{1}, V_{2})}

X_{0}^{(\overline{B} V_{1}, V_{2})}

X_{0}^{(\overline{B} V_{1}, V_{2})}

X_{\pm}^{(\overline{B} V_{1}, V_{2})}

\displaystyle a_{i}^{p,h}(M_{1}M_{2})=\left(c_{i}+{c_{i\pm 1}\over N_{c}}\right)N_{i}^{h}(M_{2})+{c_{i\pm 1}\over N_{c}}\,{C_{F}\alpha_{s}\over 4\pi}\Big{[}V_{i}^{h}(M_{2})+{4\pi^{2}\over N_{c}}H_{i}^{h}(M_{1}M_{2})\Big{]}+P_{i}^{h,p}(M_{2}),

\displaystyle a_{i}^{p,h}(M_{1}M_{2})=\left(c_{i}+{c_{i\pm 1}\over N_{c}}\right)N_{i}^{h}(M_{2})+{c_{i\pm 1}\over N_{c}}\,{C_{F}\alpha_{s}\over 4\pi}\Big{[}V_{i}^{h}(M_{2})+{4\pi^{2}\over N_{c}}H_{i}^{h}(M_{1}M_{2})\Big{]}+P_{i}^{h,p}(M_{2}),

\displaystyle N_{i}^{h}(M_{2})=\begin{cases}0,&$i=6,8$,\cr 1,&{\rm else}.\cr\end{cases}

\displaystyle N_{i}^{h}(M_{2})=\begin{cases}0,&$i=6,8$,\cr 1,&{\rm else}.\cr\end{cases}

\frac{G _{F}}{2} p = u, c \sum λ_{p} ⟨ M_{1} M_{2} ∣ T_{B}^{h, p} ∣ \overline{B}^{0} ⟩

\frac{G _{F}}{2} p = u, c \sum λ_{p} ⟨ M_{1} M_{2} ∣ T_{B}^{h, p} ∣ \overline{B}^{0} ⟩

b_{1}

b_{1}

b_{2}

b_{3, EW}

b_{4, EW}

α_{1}^{h} (M_{1} M_{2})

α_{1}^{h} (M_{1} M_{2})

α_{2}^{h} (M_{1} M_{2})

α_{3}^{h, p} (M_{1} M_{2})

α_{4}^{h, p} (M_{1} M_{2})

α_{3, EW}^{h, p} (M_{1} M_{2})

α_{4, EW}^{h, p} (M_{1} M_{2})

β_{i}^{p} (M_{1} M_{2}) = \frac{i f _{B} f _{M_{1}} f _{M_{2}}}{X ^{(\overline{B} M_{1}, M_{2})}} b_{i}^{p} .

β_{i}^{p} (M_{1} M_{2}) = \frac{i f _{B} f _{M_{1}} f _{M_{2}}}{X ^{(\overline{B} M_{1}, M_{2})}} b_{i}^{p} .

r_{χ}^{P} (μ) = \frac{2 m _{P}^{2}}{m _{b} ( μ ) ( m _{2} + m _{1} ) ( μ )}, r_{χ}^{V} (μ) = \frac{2 m _{V}}{m _{b} ( μ )} \frac{f _{V}^{⊥} ( μ )}{f _{V}} .

r_{χ}^{P} (μ) = \frac{2 m _{P}^{2}}{m _{b} ( μ ) ( m _{2} + m _{1} ) ( μ )}, r_{χ}^{V} (μ) = \frac{2 m _{V}}{m _{b} ( μ )} \frac{f _{V}^{⊥} ( μ )}{f _{V}} .

F^{B η} = F^{B η_{q}} cos θ,