Charmless Three-body Decays of B Mesons

TL;DR

This paper investigates charmless three-body B meson decays using factorization and chiral perturbation theory, identifying dominant nonresonant backgrounds and analyzing CP asymmetries with results consistent with experimental data.

Contribution

It introduces a detailed framework combining factorization and chiral perturbation theory to analyze nonresonant contributions in B meson three-body decays, aligning predictions with experimental observations.

Findings

KKK modes are dominated by nonresonant background.

Predicted branching ratios agree with experimental data.

Direct CP asymmetry in B0→K+K−K0 is about -4%.

Abstract

Charmless 3-body decays of B mesons are studied in the framework of the factorization approach. The nonresonant contributions arising from $B\to P_1P_2$ transitions are evaluated using heavy meson chiral perturbation theory (HMChPT). The momentum dependence of nonresonant amplitudes is assumed to be in the exponential form $e^{-\alpha_{NR}} p_B\cdot(p_i+p_j)}$ so that the HMChPT results are recovered in the soft meson limit $p_i, p_j\to 0$. In addition, we have identified another large source of the nonresonant signal in the matrix elements of scalar densities, e.g. $<K\bar K|\bar ss|0>$, which can be constrained from the decay $\bar B^0\to K_SK_SK_S$ or $B^-\to K^-K_SK_S$. The intermediate vector meson contributions to 3-body decays are identified through the vector current, while the scalar meson resonances are mainly associated with the scalar density. Their effects are described in terms of the Breit-Wigner formalism. Our main results are: (i) All KKK modes are dominated by the nonresonant background. The predicted branching ratios of $K^+K^-K_{S(L)}$, $K^+K^-K^-$ and $K^-K_SK_S$ modes are consistent with the data within errors. (ii) Although the penguin-dominated $B^0\to K^+K^-K_{S}$ decay is subject to a potentially significant tree pollution, its effective $\sin 2\beta$ is very similar to that of the $K_SK_SK_S$ mode. However, direct CP asymmetry of the former, being of order -4%, is more prominent than the latter. (iii) For $B\to K\pi\pi$ decays, we found sizable nonresonant contributions in $K^-\pi^+\pi^-$ and $\bar K^0\pi^+\pi^-$ modes, in agreement with the Belle measurements but larger than the BaBar result.

Figures

Tables

Table 1. Table 1: Branching ratios of various charmless three-body decays of B 𝐵 B mesons. The fractions and the corresponding branching ratios of nonresonant (NR) components are included whenever available. The first and second entries are BaBar and Belle results, respectively.

Decay	BR(${10}^{-6}$)	BR${}_{\mathrm{NR}}{}^{}\left({10}^{-6}\right)$	NR fraction(%)	Ref.
$B^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^{-}$	$16.2\pm 1.2\pm 0.9$		$13.6\pm 6.1$	BaBarpipipi
	–	–	–
$B^{-}\to K^{-}{\pi }^{+}{\pi }^{-}$	$64.1\pm 2.4\pm 4.0$	$2.87\pm 0.65\pm {0.43}_{-0.25}^{+0.63}$	$4.5\pm 1.5$	BaBarKpipi
	$48.8\pm 1.1\pm 3.6$	$16.9\pm 1.3\pm {1.3}_{-0.9}^{+1.1}$	$34.0\pm {2.2}_{-1.8}^{+2.1}$	BelleKpipi
$B^{-}\to K^{+}{K}^{-}{K}^{-}$	$35.2\pm 0.9\pm 1.6$ 111When the intrinsic charm contribution is excluded, the charmless branching ratio will become $\left(33.5\pm 0.9\pm 1.6\right)\times {10}^{-6}$.	$50\pm 6\pm 4$	$141\pm 16\pm 9$	BaBarKpKpKm
	$32.1\pm 1.3\pm 2.4$ 222When the contribution from $B^{+}\to {\chi }_{c0}{K}^{+}$ is excluded, the charmless branching ratio will become $\left(30.6\pm 1.2\pm 2.3\right)\times {10}^{-6}$.	$24.0\pm 1.5\pm 1.5$ 333Belle found two solutions for the fractions and branching ratios. We follow Belle to use the large solution.	$74.8\pm 3.6$ 333Belle found two solutions for the fractions and branching ratios. We follow Belle to use the large solution.	BelleKpKpKm
$B^{-}\to K^{-}{K}_S{K}_S$	$10.7\pm 1.2\pm 1.0$			BaBarKmKsKs
	$13.4\pm 1.9\pm 1.5$			Belle2004
${\overline{B}}^0\to {\overline{K}}^0{\pi }^{+}{\pi }^{-}$	$43.0\pm 2.3\pm 2.3$			BaBarK0pippim
	$47.5\pm 2.4\pm 3.7$	$19.9\pm 2.5\pm {1.6}_{-1.2}^{+0.7}$	$41.9\pm 5.1\pm {0.6}_{-2.5}^{+1.4}$	BelleK0pipi
${\overline{B}}^0\to K^{-}{\pi }^{+}{\pi }^0$	$34.9\pm 2.1\pm 3.9$			BaBarKppimpi0
	${36.6}_{-4.3}^{+4.2}\pm 3.0$			BelleKppimpi0
${\overline{B}}^0\to K^{+}{K}^{-}{\overline{K}}^0$	$23.8\pm 2.0\pm 1.6$	$26.7\pm 4.6$	$112.0\pm 14.9$	BaBarKpKmK0
	$28.3\pm 3.3\pm 4.0$			Belle2004
${\overline{B}}^0\to K_S{K}_S{K}_S$	${6.9}_{-0.8}^{+0.9}\pm 0.6$			BaBarKsKsKs
	${4.2}_{-1.3}^{+1.6}\pm 0.8$			Belle2004

Table 2. Table 2: BaBar results for isobar amplitudes, phases, and fractions from the fit to the B 0 → K + ​ K − ​ K 0 → superscript 𝐵 0 superscript 𝐾 superscript 𝐾 superscript 𝐾 0 B^{0}\to K^{+}K^{-}K^{0} BaBarKpKmK0 . Three rows for non-resonant contribution correspond to coefficients of exponential functions in Eq. ( 6 ), while the fraction is given for the combined amplitude. For the nonresonant decay mode in K + ​ K − superscript 𝐾 superscript 𝐾 K^{+}K^{-} , the amplitude c 12 subscript 𝑐 12 c_{12} and the phase ϕ 12 subscript italic-ϕ 12 \phi_{12} in Eq. ( 6 ) are fixed to be one and zero, respectively. Errors are statistical only.

Decay		Amplitude $c_r$	Phase ${\varphi }_r$	Fraction (%)
$\varphi \left(1020\right)K^0$		$0.0085\pm 0.0010$	$-0.016\pm 0.234$	$12.5\pm 1.3$
$f_0\left(980\right)K^0$		$0.622\pm 0.046$	$-0.14\pm 0.14$	$40.2\pm 9.6$
$X_0\left(1550\right)K^0$		$0.114\pm 0.018$	$-0.47\pm 0.20$	$4.1\pm 1.3$
	${\left(K^{+}{K}^{-}\right)}_{\mathrm{NR}}{K}^0$	1 (fixed)	0 (fixed)
	${\left(K^{+}{K}^0\right)}_{\mathrm{NR}}{K}^{-}$	$0.33\pm 0.07$	$1.95\pm 0.27$	$112.0\pm 14.9$
	${\left(K^{-}{K}^0\right)}_{\mathrm{NR}}{K}^{+}$	$0.31\pm 0.08$	$-1.34\pm 0.37$
${\chi }_{c0}\left(1P\right)K^0$		$0.0306\pm 0.00649$	${}_{-2.33}{}^{0.81}\pm 0.54$	$3.0\pm 1.2$
$D^{+}{K}^{-}$		$1.11\pm 0.17$	–	$3.6\pm 1.5$
$D_s^{+}{K}^{-}$		$0.76\pm 0.14$	–	$1.8\pm 0.6$

Table 3. Table 3: Branching ratios (in units of 10 − 6 superscript 10 6 10^{-6} ) of resonant and nonresonant (NR) contributions to B ¯ 0 → K + ​ K − ​ K ¯ 0 → superscript ¯ 𝐵 0 superscript 𝐾 superscript 𝐾 superscript ¯ 𝐾 0 \overline{B}^{0}\to K^{+}K^{-}\overline{K}^{0} . Theoretical errors correspond to the uncertainties in (i) α NR subscript 𝛼 NR \alpha_{{}_{\rm NR}} , (ii) m s subscript 𝑚 𝑠 m_{s} , F 0 B ​ K subscript superscript 𝐹 𝐵 𝐾 0 F^{BK}_{0} and σ NR subscript 𝜎 NR \sigma_{{}_{\rm NR}} , and (iii) γ = ( 59 ± 7 ) ∘ 𝛾 superscript plus-or-minus 59 7 \gamma=(59\pm 7)^{\circ} . We do not have 1 / m b 1 subscript 𝑚 𝑏 1/m_{b} power corrections within this model. However, systematic errors due to model dependent assumptions may be sizable and are not included in the error estimates that we give. Experimental results are taken from Table 2 .

Decay mode	BaBar BaBarKpKmK0	Theory
$\varphi {\overline{K}}^0$	$2.98\pm 0.45$	${2.6}_{-0.0-0.4-0.0}^{+0.0+0.5+0.0}$
$f_0\left(980\right){\overline{K}}^0$	$9.57\pm 2.51$	${5.8}_{-0.0-0.5-0.0}^{+0.0+0.1+0.0}$
$X_0\left(1550\right)K^{-}$	$0.98\pm 0.33$	${0.93}_{-0.00-0.15-0.00}^{+0.00+0.16+0.00}$
NR	$26.7\pm 4.6$	${18.1}_{-0.7-3.8-0.2}^{+0.6+5.1+0.2}$
total	$23.8\pm 2.0\pm 1.6$	${19.8}_{-0.4-0.4-0.2}^{+0.4+0.5+0.1}$

Table 4. Table 4: Branching ratios for B ¯ → 0 K + K − K S , K S K S K S , K S K S K L \overline{B}{}^{0}\to K^{+}K^{-}K_{S},\,K_{S}K_{S}K_{S},\,K_{S}K_{S}K_{L} decays and the fraction of CP -even contribution to B ¯ 0 → K + ​ K − ​ K S → superscript ¯ 𝐵 0 superscript 𝐾 superscript 𝐾 subscript 𝐾 𝑆 \overline{B}^{0}\to K^{+}K^{-}K_{S} , f + subscript 𝑓 f_{+} . The branching ratio of CP -odd K + ​ K − ​ K S superscript 𝐾 superscript 𝐾 subscript 𝐾 𝑆 K^{+}K^{-}K_{S} with ϕ ​ K S italic-ϕ subscript 𝐾 𝑆 \phi K_{S} excluded is shown in parentheses. Results for ( K + ​ K − ​ K L ) C ​ P ± subscript superscript 𝐾 superscript 𝐾 subscript 𝐾 𝐿 limit-from 𝐶 𝑃 plus-or-minus (K^{+}K^{-}K_{L})_{CP\pm} are identical to those for ( K + ​ K − ​ K S ) C ​ P ∓ subscript superscript 𝐾 superscript 𝐾 subscript 𝐾 𝑆 limit-from 𝐶 𝑃 minus-or-plus (K^{+}K^{-}K_{S})_{CP\mp} . For theoretical errors, see Table 3 . Experimental results are taken from HFAG .

Final State	$ℬ{\left({10}^{-6}\right)}_{\mathrm{theory}}$	$ℬ{\left({10}^{-6}\right)}_{\mathrm{expt}}$
$K^{+}{K}^{-}{K}_S$	${9.89}_{-0.21-1.81-0.08}^{+0.19+2.28+0.07}$	$12.4\pm 1.2$
${\left(K^{+}{K}^{-}{K}_S\right)}_{CP+}$	${8.33}_{-0.12-1.49-0.06}^{+0.10+1.82+0.05}$
${\left(K^{+}{K}^{-}{K}_S\right)}_{CP-}$	${1.57}_{-0.10-0.32-0.02}^{+0.09+0.46+0.02}$
	$\left({0.14}_{-0.06-0.06-0.01}^{+0.06+0.14+0.01}\right)$
$K_S{K}_S{K}_S$	input	$6.2\pm 0.9$
$K_S{K}_S{K}_L$	${7.63}_{-0.01-1.19-0.03}^{+0.01+1.37+0.03}$
	$f_{+}^{\mathrm{theory}}$	$f_{+}^{\mathrm{expt}}$
$K^{+}{K}^{-}{K}_S$	${0.98}_{-0.01-0.02-0.00}^{+0.01+0.01+0.00}$	$0.91\pm 0.07$
	$f_{-}^{\mathrm{theory}}$
$K^{+}{K}^{-}{K}_L$	${0.98}_{-0.01-0.02-0.00}^{+0.01+0.01+0.00}$

Table 5. Table 5: Mixing-induced and direct CP  asymmetries sin ⁡ 2 ​ β eff 2 subscript 𝛽 eff \sin 2\beta_{\rm eff} (top) and A f subscript 𝐴 𝑓 A_{f} (in % percent \% , bottom), respectively, in B ¯ 0 → K + ​ K − ​ K S → superscript ¯ 𝐵 0 superscript 𝐾 superscript 𝐾 subscript 𝐾 𝑆 \overline{B}^{0}\to K^{+}K^{-}K_{S} and K S ​ K S ​ K S subscript 𝐾 𝑆 subscript 𝐾 𝑆 subscript 𝐾 𝑆 K_{S}K_{S}K_{S} decays. Experimental results for K + ​ K − ​ K S superscript 𝐾 superscript 𝐾 subscript 𝐾 𝑆 K^{+}K^{-}K_{S} and K + ​ K − ​ K L superscript 𝐾 superscript 𝐾 subscript 𝐾 𝐿 K^{+}K^{-}K_{L} modes are obtained from the data of B ¯ 0 → K + ​ K − ​ K ¯ 0 → superscript ¯ 𝐵 0 superscript 𝐾 superscript 𝐾 superscript ¯ 𝐾 0 \overline{B}^{0}\to K^{+}K^{-}\overline{K}^{0} . Results for ( K + ​ K − ​ K L ) C ​ P ± subscript superscript 𝐾 superscript 𝐾 subscript 𝐾 𝐿 limit-from 𝐶 𝑃 plus-or-minus (K^{+}K^{-}K_{L})_{CP\pm} are identical to those for ( K + ​ K − ​ K S ) C ​ P ∓ subscript superscript 𝐾 superscript 𝐾 subscript 𝐾 𝑆 limit-from 𝐶 𝑃 minus-or-plus (K^{+}K^{-}K_{S})_{CP\mp} . For theoretical errors, see Table 3 . Experimental results are taken from HFAG .

Final state	$\sin 2{\beta }_{\mathrm{eff}}$	Expt.
${\left(K^{+}{K}^{-}{K}_S\right)}_{\varphi K_S\mathrm{excluded}}$	${0.728}_{-0.002-0.001-0.020}^{+0.001+0.002+0.009}$	$0.73\pm 0.10$
${\left(K^{+}{K}^{-}{K}_S\right)}_{CP+}$	${0.732}_{-0.004-0.004-0.020}^{+0.003+0.006+0.009}$
${\left(K^{+}{K}^{-}{K}_L\right)}_{\varphi K_L\mathrm{excluded}}$	${0.728}_{-0.002-0.001-0.020}^{+0.001+0.002+0.009}$	$0.73\pm 0.10$
$K_S{K}_S{K}_S$	${0.719}_{-0.000-0.000-0.019}^{+0.000+0.000+0.008}$	$0.58\pm 0.20$
$K_S{K}_S{K}_L$	${0.718}_{-0.000-0.000-0.019}^{+0.000+0.000+0.008}$
	$A_f(\%)$	Expt.
${\left(K^{+}{K}^{-}{K}_S\right)}_{\varphi K_S\mathrm{excluded}}$	$-{4.63}_{-1.01-0.54-0.34}^{+1.35+0.53+0.40}$	$-7\pm 8$
${\left(K^{+}{K}^{-}{K}_S\right)}_{CP+}$	$-{4.86}_{-1.09-0.55-0.35}^{+1.43+0.52+0.42}$
${\left(K^{+}{K}^{-}{K}_L\right)}_{\varphi K_L\mathrm{excluded}}$	$-{4.63}_{-1.01-0.54-0.34}^{+1.35+0.53+0.40}$	$-7\pm 8$
$K_S{K}_S{K}_S$	${0.69}_{-0.01-0.01-0.06}^{+0.01+0.01+0.05}$	$14\pm 15$
$K_S{K}_S{K}_L$	${0.77}_{-0.01-0.03-0.07}^{+0.01+0.01+0.05}$

Table 6. Table 6: Branching ratios (in units of 10 − 6 superscript 10 6 10^{-6} ) of resonant and nonresonant (NR) contributions to B − → K + ​ K − ​ K − → superscript 𝐵 superscript 𝐾 superscript 𝐾 superscript 𝐾 B^{-}\to K^{+}K^{-}K^{-} . For theoretical errors, see Table 3 .

Decay mode	BaBar BaBarKpKpKm	Belle BelleKpKpKm	Theory
$\varphi K^{-}$	$4.14\pm 0.32\pm 0.33$	$4.72\pm 0.45\pm {0.35}_{-0.22}^{+0.39}$	${2.9}_{-0.0-0.5-0.0}^{+0.0+0.5+0.0}$
$f_0\left(980\right)K^{-}$	$6.5\pm 2.5\pm 1.6$		${7.0}_{-0.0-0.7-0.1}^{+0.0+0.4+0.1}$
$X_0\left(1550\right)K^{-}$	$43\pm 6\pm 3$		${1.1}_{-0.0-0.2-0.0}^{+0.0+0.2+0.0}$
$f_0\left(1710\right)K^{-}$	$1.7\pm 1.0\pm 0.3$
NR	$50\pm 6\pm 4$	$24.0\pm 1.5\pm {1.8}_{-5.7}^{+1.9}$	${25.3}_{-1.0-4.4-0.3}^{+0.9+4.8+0.3}$
Total	$35.2\pm 0.9\pm 1.6$	$32.1\pm 1.3\pm 2.4$	${25.5}_{-0.6-4.1-0.2}^{+0.5+4.4+0.2}$

Table 7. Table 7: Branching ratios (in units of 10 − 6 superscript 10 6 10^{-6} ) of resonant and nonresonant (NR) contributions to B − → K − ​ K S ​ K S → superscript 𝐵 superscript 𝐾 subscript 𝐾 𝑆 subscript 𝐾 𝑆 B^{-}\to K^{-}K_{S}K_{S} . For theoretical errors, see Table 3 .

Decay mode	$f_0\left(980\right)K^{-}$	$X_0\left(1550\right)K^{-}$	NR	total
Theory	${5.2}_{-0.0-0.5-0.1}^{+0.0+0.3+0.1}$	${0.92}_{-0.00-0.15-0.00}^{+0.00+0.16+0.00}$	${12.4}_{-0.3-2.0-0.1}^{+0.2+2.1+0.1}$	${12.2}_{-0.0-1.7-0.0}^{+0.0+1.5+0.0}$
Expt.				$11.5\pm 1.3$

Table 8. Table 8: Branching ratios (in units of 10 − 6 superscript 10 6 10^{-6} ) of resonant and nonresonant (NR) contributions to B − → K − ​ π + ​ π − → superscript 𝐵 superscript 𝐾 superscript 𝜋 superscript 𝜋 B^{-}\to K^{-}\pi^{+}\pi^{-} . For theoretical errors, see Table 3 .

Decay mode	BaBar BaBarKpipi	Belle BelleKpipi	Theory
${\overline{K}}^{\ast 0}{\pi }^{-}$	$9.04\pm 0.77\pm {0.53}_{-0.37}^{+0.21}$	$6.45\pm 0.43\pm {0.48}_{-0.35}^{+0.25}$	${3.0}_{-0.0-0.7-0.0}^{+0.0+0.8+0.0}$
${\overline{K}}_0^{\ast 0}\left(1430\right){\pi }^{-}$	$34.4\pm 1.7\pm {1.8}_{-1.4}^{+0.1}$	$32.0\pm 1.0\pm {2.4}_{-1.9}^{+1.1}$	${10.5}_{-0.0-2.7-0.1}^{+0.0+3.2+0.0}$
${\rho }^0{K}^{-}$	$5.08\pm 0.78\pm {0.39}_{-0.66}^{+0.22}$	$3.89\pm 0.47\pm {0.29}_{-0.29}^{+0.32}$	${1.3}_{-0.0-0.7-0.1}^{+0.0+1.9+0.1}$
$f_0\left(980\right)K^{-}$	$9.30\pm 0.98\pm {0.51}_{-0.72}^{+0.27}$	$8.78\pm 0.82\pm {0.65}_{-1.64}^{+0.55}$	${7.7}_{-0.0-0.8-0.1}^{+0.0+0.4+0.1}$
NR	$2.87\pm 0.65\pm {0.43}_{-0.25}^{+0.63}$	$16.9\pm 1.3\pm {1.3}_{-0.9}^{+1.1}$	${18.7}_{-0.6-6.3-0.2}^{+0.5+11.0+0.2}$
Total	$64.4\pm 2.5\pm 4.6$	$48.8\pm 1.1\pm 3.6$	${45.0}_{-0.4-10.5-0.1}^{+0.3+16.4+0.1}$

Table 9. Table 9: Same as Table 8 except for the decay B − → K ¯ 0 ​ π − ​ π 0 → superscript 𝐵 superscript ¯ 𝐾 0 superscript 𝜋 superscript 𝜋 0 B^{-}\to\overline{K}^{0}\pi^{-}\pi^{0} .

Decay mode	Theory	Decay mode	Theory
$K^{\ast -}{\pi }^0$	${1.5}_{-0.0-0.3-0.2}^{+0.0+0.3+0.2}$	${\overline{K}}^{\ast 0}{\pi }^{-}$	${1.5}_{-0.0-0.3-0.0}^{+0.0+0.4+0.0}$
$K_0^{\ast -}\left(1430\right){\pi }^0$	${5.5}_{-0.0-1.4-0.1}^{+0.0+1.6+0.1}$	${\overline{K}}_0^{\ast 0}\left(1430\right){\pi }^{-}$	${5.2}_{-0.0-1.4-0.0}^{+0.0+1.6+0.0}$
${\rho }^{-}{\overline{K}}^0$	${1.3}_{-0.0-0.9-0.0}^{+0.0+3.0+0.0}$	NR	${10.0}_{-0.2-3.7-0.0}^{+0.2+7.1+0.0}$
Total	${27.0}_{-0.2-8.8-0.2}^{+0.3+15.4+0.2}$

Table 10. Table 10: Same as Table 8 except for the decay B ¯ 0 → K ¯ 0 ​ π + ​ π − → superscript ¯ 𝐵 0 superscript ¯ 𝐾 0 superscript 𝜋 superscript 𝜋 \overline{B}^{0}\to\overline{K}^{0}\pi^{+}\pi^{-} .

Decay mode	Belle BelleK0pipi	Theory
$K^{\ast -}{\pi }^{+}$	$5.6\pm 0.7\pm {0.5}_{-0.3}^{+0.4}$	${2.1}_{-0.0-0.5-0.3}^{+0.0+0.5+0.3}$
$K_0^{\ast -}\left(1430\right){\pi }^{+}$	$30.8\pm 2.4\pm {2.4}_{-3.0}^{+0.8}$	${10.1}_{-0.0-2.5-0.2}^{+0.0+2.9+0.1}$
${\rho }^0{\overline{K}}^0$	$6.1\pm 1.0\pm {0.5}_{-1.1}^{+1.0}$	${2.0}_{-0.0-0.9-0.1}^{+0.0+1.9+0.1}$
$f_0\left(980\right){\overline{K}}^0$	$7.6\pm 1.7\pm {0.7}_{-0.7}^{+0.5}$	${7.7}_{-0.0-0.7-0.0}^{+0.0+0.4+0.0}$
NR	$19.9\pm 2.5\pm {1.6}_{-1.2}^{+0.7}$	${15.6}_{-0.1-4.9-0.0}^{+0.1+8.3+0.0}$
Total	$47.5\pm 2.4\pm 3.7$	${42.0}_{-0.2-10.8-0.0}^{+0.3+15.7+0.0}$

Table 11. Table 11: Branching ratios (in units of 10 − 6 superscript 10 6 10^{-6} ) of resonant and nonresonant (NR) contributions to B ¯ 0 → K − ​ π + ​ π 0 → superscript ¯ 𝐵 0 superscript 𝐾 superscript 𝜋 superscript 𝜋 0 \overline{B}^{0}\to K^{-}\pi^{+}\pi^{0} . Note that the branching ratios for K ∗ − ​ π + superscript 𝐾 absent superscript 𝜋 K^{*-}\pi^{+} and K ¯ ∗ 0 ​ π 0 superscript ¯ 𝐾 absent 0 superscript 𝜋 0 \overline{K}^{*0}\pi^{0} given in BaBarKppimpi0 and BelleKppimpi0 are their absolute ones. We have converted them into the product branching ratios, namely, ℬ ​ ( B → R ​ h ) × ℬ ​ ( R → h ​ h ) ℬ → 𝐵 𝑅 ℎ ℬ → 𝑅 ℎ ℎ {\cal B}(B\to Rh)\times{\cal B}(R\to hh) . For theoretical errors, see Table 3 .

Decay mode	BaBar BaBarKppimpi0	Belle BelleKppimpi0	Theory
$K^{\ast -}{\pi }^{+}$	$3.6\pm 0.8\pm 0.5$	${4.9}_{-1.5-0.3-0.3}^{+1.5+0.5+0.8}$	${1.0}_{-0.0-0.3-0.1}^{+0.0+0.3+0.1}$
${\overline{K}}^{\ast 0}{\pi }^0$	$2.0\pm 0.6\pm 0.3$		${1.0}_{-0.0-0.2-0.1}^{+0.0+0.3+0.2}$
$K_0^{\ast -}\left(1430\right){\pi }^{+}$	$11.2\pm 1.5\pm 3.5$	$5.1\pm {1.5}_{-0.7}^{+0.6}$	${5.0}_{-0.0-1.3-0.1}^{+0.0+1.5+0.1}$
${\overline{K}}_0^{\ast 0}\left(1430\right){\pi }^0$	$7.9\pm 1.5\pm 2.7$	${6.1}_{-1.5-0.6}^{+1.6+0.5}$	${4.2}_{-0.0-1.2-0.0}^{+0.0+1.4+0.0}$
${\rho }^{+}{K}^{-}$	$8.6\pm 1.4\pm 1.0$	${15.1}_{-3.3-1.5-2.1}^{+3.4+1.4+2.0}$	${2.5}_{-0.0-1.4-0.2}^{+0.0+3.6+0.2}$
NR			${9.6}_{-0.2-3.5-0.0}^{+0.3+6.6+0.0}$
Total	$34.9\pm 2.1\pm 3.9$	${36.6}_{-4.1}^{+4.2}\pm 3.0$	${28.9}_{-0.2-9.4-0.2}^{+0.2+16.1+0.2}$

Table 12. Table 12: Same as Table 8 except for the decay B ¯ 0 → K ¯ 0 ​ π 0 ​ π 0 → superscript ¯ 𝐵 0 superscript ¯ 𝐾 0 superscript 𝜋 0 superscript 𝜋 0 \overline{B}^{0}\to\overline{K}^{0}\pi^{0}\pi^{0} .

Decay mode	$f_0\left(980\right){\overline{K}}^0$	${\overline{K}}^{\ast 0}{\pi }^0$	${\overline{K}}_0^{\ast 0}\left(1430\right){\pi }^0$	NR	Total
Theory	${3.8}_{-0.0-0.4-0.0}^{+0.0+2.0+0.0}$	${0.55}_{-0.00-0.13-0.00}^{+0.00+0.16+0.00}$	${2.3}_{-0.0-0.6-0.0}^{+0.0+0.8+0.0}$	${5.3}_{-0.0-1.1-0.0}^{+0.0+1.8+0.0}$	${12.9}_{-0.0-3.0-0.1}^{+0.0+4.0+0.1}$

Table 13. Table 13: Branching ratios, mixing-induced and direct CP  asymmetries for B ¯ 0 → K S ​ π + ​ π − → superscript ¯ 𝐵 0 subscript 𝐾 𝑆 superscript 𝜋 superscript 𝜋 \overline{B}^{0}\to K_{S}\pi^{+}\pi^{-} decays. Results for ( K L ​ π ​ π ) C ​ P ± subscript subscript 𝐾 𝐿 𝜋 𝜋 limit-from 𝐶 𝑃 plus-or-minus (K_{L}\pi\pi)_{CP\pm} are identical to those for ( K S ​ π ​ π ) C ​ P ∓ subscript subscript 𝐾 𝑆 𝜋 𝜋 limit-from 𝐶 𝑃 minus-or-plus (K_{S}\pi\pi)_{CP\mp} . For theoretical errors, see Table 3 .

Final state	Branching ratio
${\left(K_S{\pi }^{+}{\pi }^{-}\right)}_{CP+}$	${13.52}_{-0.03-3.06-0.01}^{+0.02+4.03+0.01}$
${\left(K_S{\pi }^{+}{\pi }^{-}\right)}_{CP-}$	${7.45}_{-0.08-2.32-0.02}^{+0.10+3.79+0.02}$
$f_{+}$	${0.65}_{-0.00-0.04-0.00}^{+0.00+0.03+0.00}$
Final state	$\sin 2{\beta }_{\mathrm{eff}}$
${\left(K_S{\pi }^{+}{\pi }^{-}\right)}_{CP+}$	${0.693}_{-0.000-0.002-0.014}^{+0.000+0.003+0.003}$
${\left(K_S{\pi }^{+}{\pi }^{-}\right)}_{\mathrm{full}}$	${0.718}_{-0.001-0.007-0.018}^{+0.001+0.017+0.008}$
Final state	$A_f(\%)$
${\left(K_S{\pi }^{+}{\pi }^{-}\right)}_{CP+}$	${4.27}_{-0.00-0.12-0.35}^{+0.00+0.19+0.28}$
${\left(K_S{\pi }^{+}{\pi }^{-}\right)}_{\mathrm{full}}$	${4.94}_{-0.02-0.05-0.40}^{+0.03+0.03+0.32}$

Table 14. Table 14: Same as Table 8 except for the decay B − → K + ​ K − ​ π − → superscript 𝐵 superscript 𝐾 superscript 𝐾 superscript 𝜋 B^{-}\to K^{+}K^{-}\pi^{-} .

Decay mode	$f_0\left(980\right){\pi }^{-}$	$K^{\ast 0}{K}^{-}$	$K_0^{\ast 0}\left(1430\right)K^{-}$	NR	Total
Theory	${0.50}_{-0.00-0.04-0.02}^{+0.00+0.06+0.02}$	${0.23}_{-0.00-0.04-0.02}^{+0.00+0.04+0.02}$	${0.82}_{-0.00-0.16-0.08}^{+0.00+0.18+0.09}$	${1.8}_{-0.5-0.2-0.2}^{+0.5+0.4+0.2}$	${4.0}_{-0.6-0.5-0.3}^{+0.5+0.7+0.3}$
Expt.					BaBar2003
					Belle2004

Table 15. Table 15: Same as Table 8 except for B − → π + ​ π − ​ π − → superscript 𝐵 superscript 𝜋 superscript 𝜋 superscript 𝜋 B^{-}\to\pi^{+}\pi^{-}\pi^{-} . The nonresonant background is used as an input to fix the parameter α NR subscript 𝛼 NR \alpha_{{}_{\rm NR}} defined in Eq. ( 8 ).

Decay mode	BaBar BaBarpipipi	Theory
${\rho }^0{\pi }^{-}$	$8.8\pm 1.0\pm {0.6}_{-0.7}^{+0.1}$	${7.7}_{-0.0-1.6-0.2}^{+0.0+1.7+0.3}$
$f_0\left(980\right){\pi }^{-}$		${0.39}_{-0.00-0.01-0.02}^{+0.00+0.01+0.03}$
NR		input
Total	$16.2\pm 1.2\pm 0.9$	${12.0}_{-1.2-1.8-0.3}^{+1.1+2.0+0.4}$

Table 16. Table 16: Same as Table 8 except for the decay B ¯ 0 → π + ​ π − ​ π 0 → superscript ¯ 𝐵 0 superscript 𝜋 superscript 𝜋 superscript 𝜋 0 \overline{B}^{0}\to\pi^{+}\pi^{-}\pi^{0} .

Decay mode	${\rho }^{+}{\pi }^{-}$	${\rho }^{-}{\pi }^{+}$	${\rho }^0{\pi }^0$	$f_0\left(980\right){\pi }^0$	NR	Total
Theory	${8.5}_{-0.0-1.0-0.1}^{+0.0+1.1+0.2}$	${15.5}_{-0.0-3.5-0.3}^{+0.0+4.0+0.3}$	${1.0}_{-0.0-0.2-0.0}^{+0.0+0.3+0.0}$	${0.010}_{-0.000-0.002-0.000}^{+0.000+0.003+0.000}$	${0.05}_{-0.02-0.01-0.00}^{+0.02+0.01+0.00}$	${26.3}_{-0.0-5.0-0.2}^{+0.0+5.6+0.2}$

Table 17. Table 17: Direct CP  asymmetries (in %) for various charmless three-body B 𝐵 B decays. For theoretical errors, see Table 3 . Experimental results are taken from HFAG .

Final state	BaBar	Belle	Theory
$K^{+}{K}^{-}{K}^{-}$	$-2\pm 3\pm 2$		$-{10.4}_{-1.3-1.0-0.8}^{+1.7+0.9+0.9}$
$K^{-}{K}_S{K}_S$	$-4\pm 11\pm 2$		$-{3.9}_{-0.0-0.8-0.3}^{+0.0+0.6+0.3}$
$K^{+}{K}^{-}{\pi }^{-}$	$0\pm 10\pm 3$		${17.5}_{-3.8-3.4-0.2}^{+1.9+2.2+0.0}$
$K^{-}{\pi }^{+}{\pi }^{-}$	$-1.3\pm 3.7\pm 1.1$	$4.9\pm 2.6\pm 2.0$	$-{3.3}_{-0.5-0.4-0.2}^{+0.7+0.4+0.3}$
$K^{-}{\pi }^{+}{\pi }^0$		$7\pm 11\pm 1$	${6.3}_{-0.7-1.4-0.5}^{+0.6+1.4+0.5}$
${\overline{K}}^0{\pi }^{+}{\pi }^{-}$			${4.9}_{-0.0-0.1-0.4}^{+0.0+0.0+0.3}$
${\overline{K}}^0{\pi }^0{\pi }^0$	$-23\pm 52\pm 13$	$-17\pm 24\pm 6$	${0.28}_{-0.06-0.06-0.02}^{+0.09+0.07+0.02}$
${\overline{K}}^0{\pi }^{-}{\pi }^0$			${0.4}_{-0.0-0.4-0.0}^{+0.0+0.4+0.0}$
${\pi }^{+}{\pi }^{-}{\pi }^{-}$	$-1\pm 8\pm 3$		${4.4}_{-0.6-0.9-0.2}^{+0.8+1.2+0.0}$
${\pi }^{+}{\pi }^{-}{\pi }^0$			$-{3.0}_{-0.1-0.3-0.2}^{+0.1+0.2+0.3}$

Table 18. Table 18: Branching ratios of quasi-two-body decays B → V ​ P → 𝐵 𝑉 𝑃 B\to VP and B → S ​ P → 𝐵 𝑆 𝑃 B\to SP obtained from the studies of three-body decays based on the factorization approach. Unless specified, the experimental results are obtained from the 3-body Dalitz plot analyses given in previous Tables. Theoretical uncertainties have been added in quadrature. QCD factorization (QCDF) predictions taken from BN for V ​ P 𝑉 𝑃 VP modes and from CCY for S ​ P 𝑆 𝑃 SP channels are shown here for comparison.

Decay mode	BaBar	Belle	QCDF	This work
$\varphi K^0$	${8.4}_{-1.3}^{+1.5}\pm 0.5$ 111From the Dalitz plot analysis of $B^0\to K^{+}{K}^{-}{K}^0$ decay measured by BaBar (see Table 3), we obtain $ℬ\left(B^0\to \varphi K^0\right)=\left(6.2\pm 0.9\right)\times {10}^{-6}$. The experimental value of BaBar cited in the Table is obtained from a direct measurement of $B^0\to \varphi K^0$.	${9.0}_{-1.8}^{+2.2}\pm 0.7$ 222not determined directly from the Dalitz plot analysis of three-body decays.	${4.1}_{-0.4-1.6-1.9-3.0}^{+0.4+1.7+1.8+10.6}$	${5.3}_{-0.9}^{+1.0}$
$\varphi K^{-}$	$8.4\pm 0.7\pm 0.7$	$9.60\pm {0.92}_{-0.84}^{+1.05}$	${4.5}_{-0.4-1.7-2.1-3.3}^{+0.5+1.8+1.9+11.8}$	${5.9}_{-1.0}^{+1.1}$
${\overline{K}}^{\ast 0}{\pi }^{-}$	$13.5\pm {1.2}_{-0.9}^{+0.8}$	$9.8\pm {0.9}_{-1.2}^{+1.1}$	${3.6}_{-0.3-1.4-1.2-2.3}^{+0.4+1.5+1.2+7.7}$	${4.4}_{-1.0}^{+1.1}$
${\overline{K}}^{\ast 0}{\pi }^0$	$3.0\pm 0.9\pm 0.5$		${0.7}_{-0.1-0.4-0.3-0.5}^{+0.1+0.5+0.3+2.6}$	${1.5}_{-0.4}^{+0.5}$
$K^{\ast -}{\pi }^{+}$	$11.0\pm 1.5\pm 0.7$	$8.4\pm {1.1}_{-0.8}^{+0.9}$	${3.3}_{-1.2-1.2-0.8-1.6}^{+1.4+1.3+0.8+6.2}$	${3.1}_{-0.9}^{+0.9}$
$K^{\ast -}{\pi }^0$	$6.9\pm 2.0\pm 1.3$ 222not determined directly from the Dalitz plot analysis of three-body decays.		${3.3}_{-1.0-0.9-0.6-1.4}^{+1.1+1.0+0.6+4.4}$	${2.2}_{-0.5}^{+0.6}$
$K^{\ast 0}{K}^{-}$			${0.30}_{-0.09-0.10-0.09-0.19}^{+0.11+0.12+0.09+0.57}$	${0.35}_{-0.06}^{+0.06}$
${\rho }^0{K}^{-}$	$5.1\pm {0.8}_{-0.9}^{+0.6}$	$3.89\pm {0.47}_{-0.41}^{+0.43}$	${2.6}_{-0.9-1.4-0.6-1.2}^{+0.9+3.1+0.8+4.3}$	${1.3}_{-0.7}^{+1.9}$
${\rho }^0{\overline{K}}^0$	$4.9\pm 0.8\pm 0.9$	$6.1\pm 1.0\pm 1.1$	${4.6}_{-0.5-2.1-0.7-2.1}^{+0.5+4.0+0.7+6.1}$	${2.0}_{-0.9}^{+1.9}$
${\rho }^{+}{K}^{-}$	$8.6\pm 1.4\pm 1.0$	${15.1}_{-3.3-2.6}^{+3.4+2.4}$	${7.4}_{-1.9-3.6-1.1-3.5}^{+1.8+7.1+1.2+10.7}$	${2.5}_{-1.4}^{+3.6}$
${\rho }^{-}{\overline{K}}^0$	${8.0}_{-1.3}^{+1.4}\pm 0.5$ 222not determined directly from the Dalitz plot analysis of three-body decays.		${5.8}_{-0.6-3.3-1.3-3.2}^{+0.6+7.0+1.5+10.3}$	${1.3}_{-0.9}^{+3.0}$
${\rho }^0{\pi }^{-}$	$8.8\pm {1.0}_{-0.9}^{+0.6}$	${8.0}_{-2.0}^{+2.3}\pm 0.7$ 222not determined directly from the Dalitz plot analysis of three-body decays.	${11.9}_{-5.0-3.1-1.2-1.1}^{+6.3+3.6+2.5+1.3}$	${7.7}_{-1.6}^{+1.7}$
${\rho }^{-}{\pi }^{+}$			${21.2}_{-8.4-7.2-2.3-1.6}^{+10.3+8.7+1.3+2.0}$	${15.5}_{-3.5}^{+4.0}$
${\rho }^{+}{\pi }^{-}$			${15.4}_{-6.4-4.7-1.3-1.3}^{+8.0+5.5+0.7+1.9}$	${8.5}_{-1.0}^{+1.1}$
${\rho }^0{\pi }^0$	$1.4\pm 0.6\pm 0.3$	${3.1}_{-0.8-0.8}^{+0.9+0.6}$	${0.4}_{-0.2-0.1-0.3-0.3}^{+0.2+0.2+0.9+0.5}$	${1.0}_{-0.2}^{+0.3}$
$f_0\left(980\right)K^0;f_0\to {\pi }^{+}{\pi }^{-}$	$5.5\pm 0.7\pm 0.6$	$7.6\pm {1.7}_{-0.9}^{+0.8}$	${6.7}_{-0.1-1.5-1.1}^{+0.1+2.1+2.3}$ 333We have assumed $ℬ\left(f_0\left(980\right)\to {\pi }^{+}{\pi }^{-}\right)=0.50$ for the QCDF calculation.	${7.7}_{-0.7}^{+0.4}$
$f_0\left(980\right)K^{-};f_0\to {\pi }^{+}{\pi }^{-}$	$9.3\pm {1.0}_{-0.9}^{+0.6}$	$8.8\pm {0.8}_{-1.8}^{+0.9}$	${7.8}_{-0.2-1.6-1.2}^{+0.2+2.3+2.7}$ 333We have assumed $ℬ\left(f_0\left(980\right)\to {\pi }^{+}{\pi }^{-}\right)=0.50$ for the QCDF calculation.	${7.7}_{-0.8}^{+0.4}$
$f_0\left(980\right)K^0;f_0\to K^{+}{K}^{-}$	$5.3\pm 2.2$			${5.8}_{-0.5}^{+0.1}$
$f_0\left(980\right)K^{-};f_0\to K^{+}{K}^{-}$	$6.5\pm 2.5\pm 1.6$			${7.0}_{-0.7}^{+0.4}$
$f_0\left(980\right){\pi }^{-};f_0\to {\pi }^{+}{\pi }^{-}$			${0.5}_{-0.0-0.1-0.0}^{+0.0+0.2+0.1}$ 333We have assumed $ℬ\left(f_0\left(980\right)\to {\pi }^{+}{\pi }^{-}\right)=0.50$ for the QCDF calculation.	${0.39}_{-0.02}^{+0.03}$
$f_0\left(980\right){\pi }^{-};f_0\to K^{+}{K}^{-}$				${0.50}_{-0.04}^{+0.06}$
$f_0\left(980\right){\pi }^0;f_0\to {\pi }^{+}{\pi }^{-}$			${0.02}_{-0.01-0.00-0.01}^{+0.01+0.02+0.04}$ 333We have assumed $ℬ\left(f_0\left(980\right)\to {\pi }^{+}{\pi }^{-}\right)=0.50$ for the QCDF calculation.	${0.010}_{-0.002}^{+0.003}$
${\overline{K}}_0^{\ast 0}\left(1430\right){\pi }^{-}$	$36.6\pm 1.8\pm 4.7$	$51.6\pm {1.7}_{-7.4}^{+7.0}$	${11.0}_{-6.0-3.5-10.1}^{+10.3+7.5+49.9}$	${16.9}_{-4.4}^{+5.2}$
${\overline{K}}_0^{\ast 0}\left(1430\right){\pi }^0$	$12.7\pm 2.4\pm 4.4$	$9.8\pm 2.5\pm 0.9$	${6.4}_{-3.3-2.1-5.7}^{+5.4+2.2+26.1}$	${6.8}_{-1.9}^{+2.3}$
$K_0^{\ast -}\left(1430\right){\pi }^{+}$	$36.1\pm 4.8\pm 11.3$	$49.7\pm {3.8}_{-6.1}^{+4.0}$	${11.3}_{-5.8-3.7-9.9}^{+9.4+3.7+45.8}$	${16.2}_{-4.0}^{+4.7}$
$K_0^{\ast -}\left(1430\right){\pi }^0$			${5.3}_{-2.8-1.7-4.7}^{+4.7+1.6+22.3}$	${8.9}_{-2.2}^{+2.6}$
$K_0^{\ast 0}\left(1430\right)K^{-}$	222not determined directly from the Dalitz plot analysis of three-body decays.			${1.3}_{-0.3}^{+0.3}$

Table 19. Table 19: Form factors of B → π , K , K 0 ∗ ​ ( 1430 ) , ρ → 𝐵 𝜋 𝐾 superscript subscript 𝐾 0 1430 𝜌 B\to\pi,K,K_{0}^{*}(1430),\rho transitions obtained in the covariant light-front model CCH .

$F$	$F\left(0\right)$	$F\left(q_{\max }^2\right)$	$a$	$b$	$F$	$F\left(0\right)$	$F\left(q_{\max }^2\right)$	$a$	$b$
$F_1^{B\pi }$	$0.25$	$1.16$	1.73	0.95	$F_0^{B\pi }$	0.25	0.86	0.84	$0.10$
$F_1^{BK}$	$0.35$	$2.17$	1.58	0.68	$F_0^{BK}$	0.35	0.80	0.71	$0.04$
$F_1^{B{K}_0^{\ast }}$	$0.26$	$0.70$	1.52	0.64	$F_0^{B{K}_0^{\ast }}$	0.26	0.33	0.44	0.05
$V^{B\rho }$	$0.27$	$0.79$	1.84	1.28	$A_0^{B\rho }$	0.28	0.76	1.73	1.20
$A_1^{B\rho }$	0.22	0.53	0.95	0.21	$A_2^{B\rho }$	$0.20$	$0.57$	1.65	1.05
$V^{B{K}^{\ast }}$	$0.31$	$0.96$	1.79	1.18	$A_0^{B{K}^{\ast }}$	0.31	0.87	1.68	1.08
$A_1^{B{K}^{\ast }}$	0.26	0.58	0.93	0.19	$A_2^{B{K}^{\ast }}$	$0.24$	$0.70$	1.63	0.98

Equations

$$⟨\overline{K}\overline{K}K|{\mathcal{H}}_{\mathrm{eff}}|\overline{B}⟩=\frac{G_F}{\sqrt{2}}\sum _{p=u,c}{\lambda }_p^{(s)}⟨\overline{K}\overline{K}K|T_p|\overline{B}⟩,$$

$$⟨\overline{K}{}^0(p_1)K^{+}(p_2)|(\bar{u}b{)}_{V-A}|\overline{B}{}^0⟩$$

$$A_{\mathrm{current}-\mathrm{ind}}^{\mathrm{HMChPT}}$$

$${\omega }_{+}$$

$${\omega }_{-}$$

$$r$$

$$A_{Fig.1(c)}$$

$$A_{\mathrm{NR}}=(c_{12}{e}^{i{\varphi }_{12}}{e}^{-\alpha s_{12}^2}+c_{13}{e}^{i{\varphi }_{13}}{e}^{-\alpha s_{13}^2}+c_{23}{e}^{i{\varphi }_{23}}{e}^{-\alpha s_{23}^2})(1+b_{\mathrm{NR}}{e}^{i(\beta +{\delta }_{\mathrm{NR}})}),$$

$$A_{\mathrm{R}}=\sum _r{c}_r(1+b_r)f_r{e}^{i({\varphi }_r+{\delta }_r+\beta )},{\bar{A}}_{\mathrm{R}}=\sum _r{c}_r(1-b_r)f_r{e}^{i({\varphi }_r-{\delta }_r+\beta )}.$$

$$A_{\mathrm{current}-\mathrm{ind}}=A_{\mathrm{current}-\mathrm{ind}}^{\mathrm{HMChPT}}{e}^{-{\alpha }_{{}_{\mathrm{NR}}}{p}_B\cdot (p_1+p_2)}{e}^{i{\varphi }_{12}},$$

$${\alpha }_{{}_{\mathrm{NR}}}=0.103_{-0.011}^{+0.018}{\mathrm{GeV}}^{-2}.$$

$$⟨K^{+}(p_{K^{+}})K^{-}(p_{K^{-}})|\bar{q}{\gamma }_{\mu }q|0⟩$$

$$⟨K^0(p_{K^0}){\overline{K}}^0(p_{{\bar{K}}^0})|\bar{q}{\gamma }_{\mu }q|0⟩$$

$$F_{em}^{K^{+}{K}^{-}}=F_{\rho }+F_{\omega }+F_{\varphi }+F_{NR},F_{em}^{K^0{\bar{K}}^0}=-F_{\rho }+F_{\omega }+F_{\varphi }+F_{NR}^{\prime }.$$

$$F_u^{K^{+}{K}^{-}}$$

$$F_d^{K^{+}{K}^{-}}$$

$$F_s^{K^{+}{K}^{-}}$$

$$F_h(s_{23})=\frac{c_h}{m_h^2-s_{23}-i{m}_h{\Gamma }_h},F_{NR}^{(\prime )}(s_{23})=\left(\frac{x_1^{(\prime )}}{s_{23}}+\frac{x_2^{(\prime )}}{s_{23}^2}\right){\left[\ln \left(\frac{s_{23}}{{\tilde{\Lambda }}^2}\right)\right]}^{-1},$$

$$\begin{array}[]{lll}c_{\rho}=3c_{\omega}=c_{\phi}=0.363,&c_{\rho(1450)}=7.98\times 10^{-3},&c_{\rho(1700)}=1.71\times 10^{-3},\\ c_{\omega(1420)}=-7.64\times 10^{-2},&c_{\omega(1650)}=-0.116,&c_{\phi(1680)}=-2.0\times 10^{-2},\\ \end{array}$$

$$x_1=-3.26{\mathrm{GeV}}^2,x_2=5.02{\mathrm{GeV}}^4,x_1^{\prime }=0.47{\mathrm{GeV}}^2,x_2^{\prime }=0.$$

$$⟨\overline{K}{}^0(p_1)|(\bar{s}b{)}_{V-A}|\overline{B}{}^0⟩⟨K^{+}(p_2)K^{-}(p_3)|(\bar{q}q{)}_{V-A}|0⟩=(s_{12}-s_{13})F_1^{BK}(s_{23})F_q^{K^{+}{K}^{-}}(s_{23}).$$

$$⟨{\overline{K}}^0(p_1)|\bar{s}b|{\overline{B}}^0(p_B)⟩=\frac{m_B^2-m_K^2}{m_b-m_s}{F}_0^{BK}(s_{23}).$$

$$⟨K^{+}(p_2)K^{-}(p_3)|\bar{s}s|0⟩$$

$$f_s^{NR}$$

$$⟨K^{+}(p_2)K^{-}(p_3)|(\bar{d}b{)}_{{}_{V-A}}|{\overline{B}}^0{⟩}^R=\sum _i\frac{g^{{f_0}_i\to K^{+}{K}^{-}}}{m_{{f_0}_i}^2-s_{23}-i{m}_{{f_0}_i}{\Gamma }_{{f_0}_i}}⟨{f_0}_i|(\bar{d}b{)}_{{}_{V-A}}|{\overline{B}}^0⟩.$$

$$⟨{\overline{K}}^0(p_1)|(\bar{s}d{)}_{{}_{V-A}}|0⟩⟨K^{+}(p_2)K^{-}(p_3)|(\bar{d}b{)}_{{}_{V-A}}|{\overline{B}}^0{⟩}^R$$

$$⟨{\overline{K}}^0{K}^{+}{K}^{-}|T_p|{\overline{B}}^0{⟩}_{f_0}$$

$$\sigma =\frac{1}{\sqrt{2}}(u\bar{u}+d\bar{d}),f_0=s\bar{s},$$

$$|f_0(980)⟩=|s\bar{s}⟩\cos \theta +|n\bar{n}⟩\sin \theta ,|\sigma (600)⟩=-|s\bar{s}⟩\sin \theta +|n\bar{n}⟩\cos \theta ,$$

$$⟨\overline{K}{}^0(p_1)K^{+}(p_2)K^{-}(p_3)|(\bar{s}d{)}_{V-A}|0⟩⟨0|(\bar{d}b{)}_{V-A}|\overline{B}{}^0⟩\approx 0,$$

$$v=\frac{m_{K^{+}}^2}{m_u+m_s}=\frac{m_K^2-m_{\pi }^2}{m_s-m_d},$$

$${\sigma }_{{}_{\mathrm{NR}}}=e^{i\pi /4}\left(3.36_{-0.96}^{+1.12}\right)\mathrm{GeV}.$$

$$S_{KKK,CP\pm }$$

Full text

July, 2007

Charmless Three-body Decays of $B$ Mesons

Hai-Yang Cheng1, Chun-Khiang Chua2 and Amarjit Soni3

1 Institute of Physics, Academia Sinica

Taipei, Taiwan 115, Republic of China

2 Department of Physics, Chung Yuan Christian University

Chung-Li, Taiwan 320, Republic of China

3 Physics Department, Brookhaven National Laboratory

Upton, New York 11973

Abstract

An exploratory study of charmless 3-body decays of $B$ mesons is presented using a simple model based on the framework of the factorization approach. The nonresonant contributions arising from $B\to P_{1}P_{2}$ transitions are evaluated using heavy meson chiral perturbation theory (HMChPT). The momentum dependence of nonresonant amplitudes is assumed to be in the exponential form $e^{-\alpha_{{}_{\rm NR}}p_{B}\cdot(p_{i}+p_{j})}$ so that the HMChPT results are recovered in the soft meson limit $p_{i},~{}p_{j}\to 0$. In addition, we have identified another large source of the nonresonant signal in the matrix elements of scalar densities, e.g. $\langle K\overline{K}|\bar{s}s|0\rangle$, which can be constrained from the decay $\overline{B}^{0}\to K_{S}K_{S}K_{S}$ or $B^{-}\to K^{-}K_{S}K_{S}$. The intermediate vector meson contributions to 3-body decays are identified through the vector current, while the scalar meson resonances are mainly associated with the scalar density. Their effects are described in terms of the Breit-Wigner formalism. Our main results are: (i) All $KKK$ modes are dominated by the nonresonant background. The predicted branching ratios of $K^{+}K^{-}K_{S(L)}$, $K^{+}K^{-}K^{-}$ and $K^{-}K_{S}K_{S}$ modes are consistent with the data within errors. (ii) Although the penguin-dominated $B^{0}\to K^{+}K^{-}K_{S}$ decay is subject to a potentially significant tree pollution, its effective $\sin 2\beta$ is very similar to that of the $K_{S}K_{S}K_{S}$ mode. However, direct CP asymmetry of the former, being of order $-4\%$, is more prominent than the latter. (iii) For $B\to K\pi\pi$ decays, we found sizable nonresonant contributions in $K^{-}\pi^{+}\pi^{-}$ and $\overline{K}^{0}\pi^{+}\pi^{-}$ modes, in agreement with the Belle measurements but larger than the BaBar result. (iv) Time-dependent CP asymmetries in $K_{S}\pi^{0}\pi^{0}$, a purely $CP$-even state, and $K_{S}\pi^{+}\pi^{-}$, an admixture of $CP$-even and $CP$-odd components, are studied. (v) The $\pi^{+}\pi^{-}\pi^{0}$ mode is found to have a rate larger than $\pi^{+}\pi^{-}\pi^{-}$ even though the former involves a $\pi^{0}$ in the final state. They are both dominated by resonant $\rho$ contributions. (vi) We have computed the resonant contributions to 3-body decays and determined the rates for the quasi-two-body decays $B\to VP$ and $B\to SP$. The predicted $\rho\pi,~{}f_{0}(980)K$ and $f_{0}(980)\pi$ rates are in agreement with the data, while the calculated $\phi K,~{}K^{*}\pi,~{}\rho K$ and $K_{0}^{*}(1430)\pi$ are in general too small compared to experiment. (vii) Sizable direct CP asymmetry is found in $K^{+}K^{-}K^{-}$ and $K^{+}K^{-}\pi^{-}$ modes.

I Introduction

Three-body decays of heavy mesons are more complicated than the two-body case as they receive resonant and nonresonant contributions and involve 3-body matrix elements. The three-body meson decays are generally dominated by intermediate vector and scalar resonances, namely, they proceed via quasi-two-body decays containing a resonance state and a pseudoscalar meson. The analysis of these decays using the Dalitz plot technique enables one to study the properties of various resonances. The nonresonant background is usually believed to be a small fraction of the total 3-body decay rate. Experimentally, it is hard to measure the direct 3-body decays as the interference between nonresonant and quasi-two-body amplitudes makes it difficult to disentangle these two distinct contributions and extract the nonresonant one.

The Dalitz plot analysis of 3-body $B$ decays provides a nice methodology for extracting information on the unitarity triangle in the standard model (SM). For example, the Dalitz analysis combined with isospin symmetry allows one to extract the angle $\alpha$ from $B\to\rho\pi\to\pi\pi\pi$ Quinn . Recently, a method has been proposed in CPS for determining CKM parameters in 3-body decays $B\to K\pi\pi$ and $B_{s}\to K\pi\pi$. This method was extended further in GPSZ to $\Delta I=1,I(K^{*}\pi)=1/2$ amplitudes in the above decays and to $I=1$ amplitudes in $B_{s}\to K^{*}\overline{K}$ and $B_{s}\to\overline{K}^{*}K$.

Nonresonant 3-body decays of charmed mesons have been measured in several channels and the nonresonant signal in charm decays are found to be less than 10% PDG . In the past few years, some of the charmless $B$ to 3-body decay modes have also been measured at $B$ factories and studied using the Dalitz plot analysis. The measured fractions and the corresponding branching ratios of nonresonant components for some of 3-body $B$ decay modes are listed in Table 1. We see that the nonresonant 3-body decays could play an essential role in $B$ decays. It is now well established that the $B\to KKK$ modes are dominated by the nonresonant background. For example, the nonresonant fraction is about 90% in $\overline{B}^{0}\to K^{+}K^{-}\overline{K}^{0}$ decay. While this is a surprise in view of the rather small nonresonant contributions in 3-body charm decays, it is not entirely unexpected because the energy release scale in weak $B$ decays is of order 5 GeV, whereas the major resonances lie in the energy region of 0.77 to 1.6 GeV. Consequently, it is likely that 3-body $B$ decays may receive sizable nonresonant contributions. At any rate, it is important to understand and identify the underlying mechanism for nonresonant decays.

The direct nonresonant three-body decays of mesons in general receive two distinct contributions: one from the point-like weak transition and the other from the pole diagrams that involve three-point or four-point strong vertices. For $D$ decays, attempts of applying the effective $SU(4)\times SU(4)$ chiral Lagrangian to describe the $DP\to DP$ and $PP\to PP$ scattering at energies $\sim m_{D}$ have been made by several authors Singer ; KP ; Cheng86 ; CC90 ; Botella to calculate the nonresonant $D$ decays, though in principle it is not justified to employ the SU(4) chiral symmetry. As shown in CC90 ; Botella , the predictions of the nonresonant decay rates in chiral perturbation theory are in general too small when compared with experiment. With the advent of heavy quark symmetry and its combination with chiral symmetry Yan ; Wise ; Burdman , the nonresonant $D$ decays can be studied reliably at least in the kinematic region where the final pseuodscalar mesons are soft. Some of the direct 3-body $D$ decays were studied based on this approach Zhang ; Ivanov .

For the case of $B$ mesons, consider the three-body $B$ decay $B\to P_{1}P_{2}P_{3}$. Under the factorization hypothesis, one of the nonresonant contributions arises from the transitions $B\to P_{1}P_{2}$. The nonresonant background in charmless three-body $B$ decays due to the transition $B\to P_{1}P_{2}$ has been studied extensively Deshpande ; Fajfer1 ; Fajfer2 ; Deandrea1 ; Deandrea ; Fajfer3 based on heavy meson chiral perturbation theory (HMChPT) Yan ; Wise ; Burdman . However, the predicted decay rates are, in general, unexpectedly large. For example, the branching ratio of the nonresonant decay $B^{-}\to\pi^{+}\pi^{-}\pi^{-}$ is predicted to be of order $10^{-5}$ in Deshpande ; Fajfer1 , which is too large compared to the limit $4.6\times 10^{-6}$ set by BaBar BaBarpipipi . Therefore, it is important to reexamine and clarify the existing calculations.

The issue has to do with the applicability of HMChPT. In order to apply this approach, two of the final-state pseudoscalars in $B\to P_{1}P_{2}$ transition have to be soft. The momentum of the soft pseudoscalar should be smaller than the chiral symmetry breaking scale of order 1 GeV. For 3-body charmless $B$ decays, the available phase space where chiral perturbation theory is applicable is only a small fraction of the whole Dalitz plot. Therefore, it is not justified to apply chiral and heavy quark symmetries to a certain kinematic region and then generalize it to the region beyond its validity. In this work we shall assume the momentum dependence of nonresonant amplitudes in the exponential form $e^{-\alpha_{{}_{\rm NR}}p_{B}\cdot(p_{i}+p_{j})}$ so that the HMChPT results are recovered in the soft meson limit $p_{i},~{}p_{j}\to 0$. We shall see that the parameter $\alpha_{{}_{\rm NR}}$ can be fixed from the tree-dominated decay $B^{-}\to\pi^{+}\pi^{-}\pi^{-}$.

However, the nonresonant background in $B\to P_{1}P_{2}$ transition does not suffice to account for the experimental observation that the penguin-dominated decay $B\to KKK$ is dominated by the nonresonant contributions. This implies that the two-body matrix element e.g. $\langle K\overline{K}|\bar{s}s|0\rangle$ induced by the scalar density should have a large nonresonant component. In the absence of first-principles calculation, we will use the $\overline{B}^{0}\to K_{S}K_{S}K_{S}$ mode in conjunction with the mass spectrum in $\overline{B}^{0}\to K^{+}K^{-}\overline{K}^{0}$ to fix the nonresonant contribution to $\langle K\overline{K}|\bar{s}s|0\rangle$.

In this work, we shall study the charmless 3-body decays of $B$ mesons using the factorization approach. Besides the nonresonant background as discussed above, we will also study resonant contributions to 3-body decays. Vector meson and scalar resonances contribute to the two-body matrix elements $\langle P_{1}P_{2}|V_{\mu}|0\rangle$ and $\langle P_{1}P_{2}|S|0\rangle$, respectively. They can also contribute to the three-body matrix element $\langle P_{1}P_{2}|V_{\mu}-A_{\mu}|B\rangle$. Resonant effects are described in terms of the usual Breit-Wigner formalism. In this manner we are able to figure out the relevant resonances which contribute to the 3-body decays of interest and compute the rates of $B\to VP$ and $B\to SP$. In conjunction with the nonresonant contribution, we are ready to calculate the total rates for three-body decays.

It should be stressed from the outset that in this work we take the factorization approximation as a working hypothesis rather than a first-principles starting point. If we start with theories such as QCD factorization BBNS , or pQCD Li or soft-collinear effective theory SCET , then we can take power corrections seriously and make an estimation. Since factorization has not been proved for three-body B decays, we shall work in the phenomenological factorization model rather than in the established theories such as QCDF. That is, we start with the simple idea of factorization and see if it works for three-body decays, in the hope that it will provide a useful zeroth step for others to try to improve.

The penguin-induced three-body decays $B^{0}\to K^{+}K^{-}K_{S}$ and $K_{S}K_{S}K_{S}$ deserve special attention as the current measurements of the deviation of $\sin 2\beta_{\rm eff}$ in $KKK$ modes from $\sin 2\beta_{J/\psi K_{S}}$ may indicate New Physics in $b\to s$ penguin-induced modes. It is of great importance to examine and estimate how much of the deviation of $\sin 2\beta_{\rm eff}$ is allowed in the SM. Owing to the presence of color-allowed tree contributions in $B^{0}\to K^{+}K^{-}K_{S}$, this mode is subject to a potentially significant tree pollution and the deviation of the mixing-induced CP asymmetry from that measured in $B\to J/\psi K_{S}$ could be as large as ${\cal O}(0.10)$. Since the tree amplitude is tied to the nonresonant background, it is very important to understand the nonresonant contributions in order to have a reliable estimate of $\sin 2\beta_{\rm eff}$ in $KKK$ modes.

The layout of the present paper is as follows. In Sec. II we shall apply the factorization approach to study $B^{0}\to K^{+}K^{-}K_{S}$ and $K_{S}K_{S}K_{S}$ decays and discuss resonant and nonresonant contributions. In order to set up the framework for calculations we will discuss $B\to KKK$ modes in most details. We then turn to $K\pi\pi$ modes in Sec. III. The tree-dominated modes $KK\pi$ in Sec. IV, and $\pi\pi\pi$ in Sec. V. In Sec. VI, we determine the rates for $B\to VP$ and $B\to SP$ and compare our results with the approach of QCD factorization. Sec. VII contains our conclusions. The factorizable amplitudes of various $B\to P_{1}P_{2}P_{3}$ decays are summarized in Appendix A. The relevant input parameters such as decay constants, form factors, etc. are collected in Appendix B.

II $B\to KKK$ decays

For 3-body $B$ decays, the $b\to sq\bar{q}$ penguin transitions contribute to the final states with odd number of kaons, namely, $KKK$ and $K\pi\pi$, while $b\to uq\bar{q}$ tree and $b\to dq\bar{q}$ penguin transitions contribute to final states with even number of kaons, e.g. $KK\pi$ and $\pi\pi\pi$. We shall first discuss the $b\to s$ penguin dominated 3-body decays in details and then turn to $b\to u$ tree dominated modes. For $B\to KKK$ modes, we shall first consider the neutral $B$ decays as they involve mixing-induced CP asymmetries.

II.1 $\overline{B}^{0}\to KKK$ decays

We consider the decay $\overline{B}^{0}\to K^{+}K^{-}\overline{K}^{0}$ as an illustration. Under the factorization approach, the $\overline{B}{}^{0}\to K^{+}K^{-}\overline{K}{}^{0}$ decay amplitude consists of three distinct factorizable terms: (i) the current-induced process with a meson emission, $\langle\overline{B}^{0}\to K^{+}\overline{K}^{0}\rangle\times\langle 0\to K^{-}\rangle$, (ii) the transition process, $\langle\overline{B}^{0}\to\overline{K}^{0}\rangle\times\langle 0\to K^{+}K^{-}\rangle$, and (iii) the annihilation process $\langle\overline{B}^{0}\to 0\rangle\times\langle 0\to K^{+}K^{-}\overline{K}^{0}\rangle$, where $\langle A\to B\rangle$ denotes a $A\to B$ transition matrix element. In the factorization approach, the matrix element of the $\overline{B}\to\overline{K}\,\overline{K}K$ decay amplitude is given by

[TABLE]

where $\lambda_{p}^{(s)}\equiv V_{pb}V^{*}_{ps}$ and the explicit expression of $T_{p}$ in terms of four-quark operators is given in Eq. (66). The factorizable $\overline{B}^{0}\to K^{+}K^{-}\overline{K}^{0}$ decay amplitude is given in Eq. (68). Note that the OZI suppressed matrix element $\langle K^{+}K^{-}|(\bar{d}d)_{V-A}|0\rangle$ is included in the factorizable amplitude since it could be enhanced through the long-distance pole contributions via the intermediate vector mesons such as $\rho^{0}$ and $\omega$. Likewise, the OZI-suppressed matrix elements $\langle K^{+}K^{-}|(\bar{d}b)_{{}_{V-A}}|\overline{B}^{0}\rangle$ and $\langle K^{+}K^{-}|\bar{d}(1-\gamma_{5})b|\overline{B}^{0}\rangle$ are included as they receive contributions from the scalar resonances like $f_{0}(980)$.

For the current-induced process, the two-meson transition matrix element $\langle\overline{K}{}^{0}K^{+}|(\bar{u}b)_{V-A}|\overline{B}{}^{0}\rangle$ has the general expression LLW

[TABLE]

where $(\bar{q}_{1}q_{2})_{{}_{V-A}}\equiv\bar{q}_{1}\gamma_{\mu}(1-\gamma_{5})q_{2}$. This leads to

[TABLE]

where $s_{ij}\equiv(p_{i}+p_{j})^{2}$. To compute the form factors $r$, $\omega_{\pm}$ and $h$, one needs to consider not only the point-like contact diagram, Fig. 1(a), but also various pole diagrams depicted in Fig. 1. In principle, one can apply HMChPT to evaluate the form factors $r,~{}\omega_{+}$ and $\omega_{-}$ LLW . However, this will lead to too large decay rates in disagreement with experiment Cheng:2002qu . The heavy meson chiral Lagrangian given in Yan ; Wise ; Burdman is needed to compute the strong $B^{*}BP$, $B^{*}B^{*}P$ and $BBPP$ vertices. The results for the form factors are LLW ; Fajfer1

[TABLE]

where $f_{\pi}=132$ MeV, $g$ is a heavy-flavor independent strong coupling which can be extracted from the CLEO measurement of the $D^{*+}$ decay width, $|g|=0.59\pm 0.01\pm 0.07$ CLEOg . We shall follow Yan to fix its sign to be negative. The point-like diagram Fig. 1(a) characterized by the term $f_{B}/(2f_{\pi}^{2})$ contributes to the form factors $\omega_{+}$ and $r$, while Figs. 1(b) and 1(d) contribute to $r$ and Fig. 1(c) contributes to all the form factors.

A direct calculation indicates that the branching ratio of $\overline{B}^{0}\to K^{+}K^{-}\overline{K}^{0}$ arising from the current-induced process alone is already at the level of $77\times 10^{-6}$ which exceeds the measured total branching ratio of $25\times 10^{-6}$ (see Table 1). The issue has to do with the applicability of HMChPT. In order to apply this approach, two of the final-state pseudoscalars ($K^{+}$ and $\overline{K}^{0}$ in this example) have to be soft. The momentum of the soft pseudoscalar should be smaller than the chiral symmetry breaking scale $\Lambda_{\chi}$ of order $0.83-1.0$ GeV. For 3-body charmless $B$ decays, the available phase space where chiral perturbation theory is applicable is only a small fraction of the whole Dalitz plot. Therefore, it is not justified to apply chiral and heavy quark symmetries to a certain kinematic region and then generalize it to the region beyond its validity. If the soft meson result is assumed to be the same in the whole Dalitz plot, the decay rate will be greatly overestimated.

In CCSKKK ; Cheng:2002qu we have tried to circumvent the aforementioned problem by applying HMChPT only to the strong vertex and use the form factors to describe the weak vertex. Moreover, we introduced a form factor to take care of the off-shell effect. For example, Fig. 1(c) can be evaluated by considering the strong interaction $\overline{B}{}^{0}\to\overline{K}{}^{0}\overline{B}_{s}^{*}$ followed by the weak transition $\overline{B}_{s}^{*}\to K^{+}$ and the result is Cheng:2002qu

[TABLE]

where $F^{B_{s}K}_{0,1}$ are the $B_{s}\to K$ weak transition from factors in the standard convention BSW and we have introduced a form factor $F(s_{23},m_{B_{s}^{*}})$ to take into account the off-shell effect of the $B_{s}^{*}$ pole CCSKKK . It is parameterized as $F(s_{23},m_{B_{s}^{*}})=(\Lambda^{2}-m_{B_{s}^{*}}^{2})/(\Lambda^{2}-s_{23})$ with the cut-off parameter $\Lambda$ chosen to be $\Lambda=m_{B^{*}_{s}}+\Lambda_{\rm QCD}$. Needless to say, this parametrization of the form factor is somewhat arbitrary. Moreover, the nonresonant contribution thus calculated is too small compared to experiment.

The Dalitz plot analysis of $\overline{B}^{0}\to K^{+}K^{-}\overline{K}^{0}$ has been recently performed by BaBar BaBarKpKmK0 . In the BaBar analysis, a phenomenological parametrization of the non-resonant amplitudes is described by

[TABLE]

and resonant terms are described by

[TABLE]

The BaBar results for isobar amplitudes, phases and fractions from the fit to the $B^{0}\to K^{+}K^{-}K^{0}$ are summarized in Table 2. It is evident that this decay is dominated by the nonresonant background. For our purpose, we will parametrize the current-induced nonresonant amplitude Eq. (3) as

[TABLE]

so that the HMChPT results are recovered in the chiral limit $p_{1},~{}p_{2}\to 0$. That is, the nonresonant amplitude in the soft meson region is described by HMChPT, but its energy dependence beyond the chiral limit is governed by the exponential term $e^{-\alpha_{{}_{\rm NR}}p_{B}\cdot(p_{1}+p_{2})}$. In what follows, we shall use the tree-dominated $B^{-}\to\pi^{+}\pi^{-}\pi^{-}$ decay data to fix $\alpha_{{}_{\rm NR}}$, which turns out to be

[TABLE]

This is very close to the naive expectation of $\alpha_{{}_{\rm NR}}\sim{\cal O}(1/(2m_{B}\Lambda_{\chi}))$ based on the dimensional argument. The phase $\phi_{12}$ of the nonresonant amplitude in the $(K^{+}\overline{K}^{0})$ system will be set to zero for simplicity.

For the transition amplitude, we need to evaluate the 2-kaon creation matrix element which can be expressed in terms of time-like kaon current form factors as

[TABLE]

The weak vector form factors $F^{K^{+}K^{-}}_{q}$ and $F^{K^{0}\bar{K}^{0}}_{q}$ can be related to the kaon electromagnetic (e.m.) form factors $F^{K^{+}K^{-}}_{em}$ and $F^{K^{0}\bar{K}^{0}}_{em}$ for the charged and neutral kaons, respectively. Phenomenologically, the e.m. form factors receive resonant and nonresonant contributions and can be expressed by

[TABLE]

It follows from Eqs. (II.1) and (11) that

[TABLE]

where use of isospin symmetry has been made.

The resonant and nonresonant terms in Eq. (11) can be parametrized as

[TABLE]

with $\tilde{\Lambda}\approx 0.3$ GeV. The expression for the nonresonant form factor is motivated by the asymptotic constraint from pQCD, namely, $F(t)\to(1/t)[\ln(t/\tilde{\Lambda}^{2})]^{-1}$ in the large $t$ limit Brodsky . The unknown parameters $c_{h}$, $x_{i}$ and $x^{\prime}_{i}$ are fitted from the kaon e.m. data, giving the best fit values (in units of GeV2 for $c_{h}$)  DKK :

[TABLE]

and

[TABLE]

Note that the form factors $F_{\rho,\omega,\phi}$ in Eqs. (11) and (12) include the contributions from the vector mesons $\rho(770),\,\rho(1450),\,\rho(1700)$, $\omega(782),\,\omega(1420),\,\omega(1650),$ $\phi(1020)$ and $\phi(1680)$. It is interesting to note that (i) the fitted values of $c_{V}$ are very close to the vector meson dominance expression $g_{{}_{V\gamma}}g_{VKK}$ for $V=\rho,\omega,\phi$ DM2 ; PDG , where $g_{{}_{V\gamma}}$ is the e.m. coupling of the vector meson defined by $\langle V|j_{em}|0\rangle=g_{V\gamma}\varepsilon^{*}_{V}$ and $g_{VKK}$ is the $V\to KK$ strong coupling with $-g_{\phi K^{+}K^{-}}\simeq g_{\rho K^{+}K^{-}}/\sqrt{2}=g_{\omega K^{+}K^{-}}/\sqrt{2}\simeq 3.03$, and (ii) the vector-meson pole contributions alone yield $F^{K^{+}K^{-}}_{u,s}(0)\approx 1,-1$ and $F^{K^{+}K^{-}}_{d}(0)\approx 0$ as the charged kaon does not contain the valence $d$ quark. The matrix element for the current-induced decay process then has the expression

[TABLE]

We also need to specify the 2-body matrix elements $\langle K^{+}K^{-}|\bar{s}s|0\rangle\langle\overline{K}^{0}|\bar{s}b|\overline{B}^{0}\rangle$ induced from the scalar densities. The use of the equation of motion leads to

[TABLE]

The matrix element $\langle K^{+}K^{-}|\bar{s}s|0\rangle$ receives resonant and non-resonant contributions:

[TABLE]

where ${f_{0}}_{i}$ denote the generic $f_{0}$-type scalar mesons, ${f_{0}}_{i}=f_{0}(980),f_{0}(1370),f_{0}(1500),X_{0}(1550),\cdots$, the scalar decay constant $\bar{f}_{{f_{0}}_{i}}^{s}$ is defined by $\langle{f_{0}}_{i}|\bar{s}s|0\rangle=m_{{f_{0}}_{i}}\bar{f}^{s}_{{f_{0}}_{i}}$ [see Eq. (81)], $g^{{f_{0}}_{i}\to K^{+}K^{-}}$ is the ${f_{0}}_{i}\to K^{+}K^{-}$ strong coupling, and the nonresonant terms are related to those in $F_{s}^{K^{+}K^{-}}$ through the equation of motion. The presence of the nonresonant $\sigma_{{}_{\rm NR}}$ term will be explained shortly. The main scalar meson pole contributions are those that have dominant $s\bar{s}$ content and large coupling to $K\overline{K}$. We consider the scalar mesons $f_{0}(980)$ and $X_{0}(1550)$ (denoted as $f_{X}(1500)$ by Belle) which are supposed to have the largest couplings with the $K\overline{K}$ pair. Note that the nature of the broad state $X_{0}(1550)$ observed by BaBar and Belle, for example, what is its relation with $f_{0}(1500)$, is not clear. To proceed with the numerical calculations, we shall use $g^{f_{0}(980)\to K^{+}K^{-}}=4.3$ GeV,111This is different from the coupling $g^{f_{0}(980)\to K^{+}K^{-}}=1.5$ GeV originally employed in CCSKKK . The coupling $g^{f_{0}(980)\to\pi^{+}\pi^{-}}\sim 1.33$ GeV can be fixed from a recent Belle measurement of $\Gamma(f_{0}(980)\to\pi^{+}\pi^{-})$ [see Eq. (III)]. Using the BES result $(g^{f_{0}(980)\to KK}/g^{f_{0}(980)\to\pi\pi})^{2}=4.21\pm 0.25\pm 0.21$ BES , one can deduce that $g^{f_{0}(980)\to KK}=2.7\pm 0.6$ GeV. In this work, we found that a slightly large coupling $g^{f_{0}(980)\to KK}$ will give better numerical results. $g^{X_{0}(1550)\to K^{+}K^{-}}=1.4$ GeV, $\Gamma_{f_{0}(980)}=80$ MeV, $\Gamma_{X_{0}(1550)}=0.257$ GeV BaBarKpKpKm , $\bar{f}_{f_{0}(980)}(\mu=m_{b}/2)\simeq 0.46$ GeV Cheng:2005ye and $\bar{f}_{f_{0}(1530)}\simeq 0.30$ GeV. The sign of the resonant terms is fixed by $f_{s}^{K^{+}K^{-}}(0)=v$ from a chiral perturbation theory calculation (see, for example, Cheng:1988va ). It should be stressed that although the nonresonant contributions to $f_{s}^{KK}$ and $F_{s}^{KK}$ are related through the equation of motion, the resonant ones are different and not related a priori. As stressed in CCSKKK , to apply the equation of motion, the form factors should be away from the resonant region. In the presence of the resonances, we thus need to introduce a nonresonant term characterized by the parameter $\sigma_{\rm NR}$ in Eq. (II.1) which will be specified later. The parameter $\alpha$ appearing in the same equation should be close to the value of $\alpha_{{}_{\rm NR}}$ given in Eq. (9). We will use the experimental measurement $\alpha=(0.14\pm 0.02)\,{\rm GeV}^{-2}$ BaBarKpKmK0 .

As noticed before, the matrix elements $\langle K^{+}K^{-}|(\bar{d}b)_{{}_{V-A}}|\overline{B}^{0}\rangle$ and $\langle K^{+}K^{-}|\bar{d}(1-\gamma_{5})b|\overline{B}^{0}\rangle$ are included in Eq. (68) as they receive intermediate scalar pole contributions. More explicitly,

[TABLE]

Hence,

[TABLE]

The superscript $u$ of the form factor $F_{0}^{B{f_{0}}_{i}^{u}}$ reminds us that it is the $u\bar{u}$ quark content that gets involved in the $B$ to ${f_{0}}_{i}$ form factor transition. In short, the relevant $f_{0}(980)$ pole contributions to $\overline{B}^{0}\to K^{+}K^{-}\overline{K}^{0}$ are

[TABLE]

where we have employed Eq. (II.1) and applied equations of motion to the matrix elements $\langle\overline{K}^{0}|\bar{s}\gamma_{5}d|0\rangle\langle K^{+}K^{-}|\bar{d}\gamma_{5}b|\overline{B}^{0}\rangle$. Comparing this equation with Eq. (A6) of CCY , we see that the expression inside $\{\cdots\}$ is identical to that of $\overline{B}^{0}\to f_{0}(980)\overline{K}^{0}$, as it should be.

We digress for a moment to discuss the wave function of the $f_{0}(980)$. What is the quark structure of the light scalar mesons below or near 1 GeV has been quite controversial. In this work we shall consider the conventional $q\bar{q}$ assignment for the $f_{0}(980)$. In the naive quark model, the flavor wave functions of the $f_{0}(980)$ and $\sigma(600)$ read

[TABLE]

where the ideal mixing for $f_{0}$ and $\sigma$ has been assumed. In this picture, $f_{0}(980)$ is purely an $s\bar{s}$ state. However, there also exist some experimental evidences indicating that $f_{0}(980)$ is not purely an $s\bar{s}$ state. First, the observation of $\Gamma(J/\psi\to f_{0}\omega)\approx{1\over 2}\Gamma(J/\psi\to f_{0}\phi)$ PDG clearly indicates the existence of the non-strange and strange quark content in $f_{0}(980)$. Second, the fact that $f_{0}(980)$ and $a_{0}(980)$ have similar widths and that the $f_{0}$ width is dominated by $\pi\pi$ also suggests the composition of $u\bar{u}$ and $d\bar{d}$ pairs in $f_{0}(980)$; that is, $f_{0}(980)\to\pi\pi$ should not be OZI suppressed relative to $a_{0}(980)\to\pi\eta$. Therefore, isoscalars $\sigma(600)$ and $f_{0}$ must have a mixing

[TABLE]

with $n\bar{n}\equiv(\bar{u}u+\bar{d}d)/\sqrt{2}$. Experimental implications for the $f_{0}\!-\!\sigma$ mixing angle have been discussed in detail in ChengDSP . It is found that $\theta$ lies in the ranges of $25^{\circ}<\theta<40^{\circ}$ and $-40^{\circ}<\theta<-15^{\circ}$ (or $140^{\circ}<\theta<165^{\circ}$). Note that the phenomenological analysis of the radiative decays $\phi\to f_{0}(980)\gamma$ and $f_{0}(980)\to\gamma\gamma$ favors a solution of the $\theta$ to be negative (or in the second quadrant). In this work, we shall use $\theta=-25^{\circ}$.

Finally, the matrix elements involving 3-kaon creation are given by Cheng:2002qu

[TABLE]

where

[TABLE]

characterizes the quark-order parameter $\langle\bar{q}q\rangle$ which spontaneously breaks the chiral symmetry. Both relations in Eq. (II.1) are originally derived in the chiral limit Cheng:2002qu and hence the quark masses appearing in Eq. (25) are referred to the scale $\sim$ 1 GeV . The first relation reflects helicity suppression which is expected to be even more effective for energetic kaons. For the second relation, we introduce the form factor $F^{KKK}$ to extrapolate the chiral result to the physical region. Following Cheng:2002qu we shall take $F^{KKK}(q^{2})=1/[1-(q^{2}/\Lambda^{2}_{\chi})]$ with $\Lambda_{\chi}=0.83$ GeV being a chiral symmetry breaking scale.

To proceed with the numerical calculations, we need to specify the input parameters. The relevant CKM matrix elements, decay constants, form factors, the effective Wilson coefficients $a_{i}^{p}$ and the running quark masses are collected in Appendix B. As for the parameter $\sigma_{{}_{\rm NR}}$ in Eq. (II.1), in principle we can set its phase $\phi_{\sigma}$ to zero and use the measured $K_{S}K_{S}K_{S}$ rate, namely, ${\cal B}(\overline{B}^{0}\to K_{S}K_{S}K_{S})=(6.2\pm 0.9)\times 10^{-6}$ HFAG , to fix the parameter $\sigma_{{}_{\rm NR}}$ and then use the data obtained from the Dalitz plot analysis to determine the strong phases $\phi_{r}$ for resonant amplitudes. However, in doing so one needs the data of invariant mass spectra. In the absence of such information, instead we will treat $\phi_{\sigma}$ as a free parameter and do not assign any other strong phases to the resonant amplitudes except for those arising from the Breit-Wigner formalism. It turns out that if $\phi_{\sigma}$ is small, the $K^{+}K^{-}$ mass spectrum in $\overline{B}^{0}\to K^{+}K^{-}K_{S}$ will have a prominent hump at the invariant mass $m_{K^{+}K^{-}}=3$ GeV, which is not seen experimentally (see Fig. 2(c)). We found that $\phi_{\sigma}\approx\pi/4$ will yield $K^{+}K^{-}$ mass spectrum consistent with the data

[TABLE]

Note that the phase of $\sigma_{{}_{\rm NR}}$ is consistent with the BaBar measurement shown in Table 2, namely, $\phi_{\sigma}^{\rm BaBar}=1.19\pm 0.37$.

The calculated branching ratios of resonant and nonresonant contributions to $\overline{B}^{0}\to K^{+}K^{-}\overline{K}^{0}$ are summarized in Table 3. The theoretical errors shown there are from the uncertainties in (i) the parameter $\alpha_{\rm NR}$ which governs the momentum dependence of the nonresonant amplitude, (ii) the strange quark mass $m_{s}$, the form factor $F^{BK}_{0}$ and the nonresonant parameter $\sigma_{{}_{\rm NR}}$, and (iii) the unitarity angle $\gamma$.

In QCD calculations based on a heavy quark expansion, one faces uncertainties arising from power corrections such as annihilation and hard-scattering contributions. For example, in QCD factorization, there are large theoretical uncertainties related to the modelling of power corrections corresponding to weak annihilation effects and the chirally-enhanced power corrections to hard spectator scattering. Even for two-body $B$ decays, power corrections are of order (10-20)% for tree-dominated modes, but they are usually bigger than the central values for penguin-dominated decays. Needless to say, $1/m_{b}$ power corrections for three-body decays may well be larger. However, as stressed in Introduction, in this exploratory work we use the phenomenological factorization model rather than in the established theories based on a heavy quark expansion. Consequently, uncertainties due to power corrections, at this stage, are not included in our calculations, by assumption. In view of such shortcomings we must emphasize that the additional errors due to such model dependent assumptions may be sizable.

From Table 3 we see that the predicted rates for resonant and nonresonant components are consistent with experiment within errors. The nonresonant contribution arises dominantly from the transition process (88%) via the scalar-density-induced vacuum to $K\bar{K}$ transition, namely, $\langle K^{+}K^{-}|\bar{s}s|0\rangle$, and slightly from the current-induced process (3%). Therefore, it is natural to conjecture that nonresonant decays could also play a prominent role in other penguin dominated 3-body $B$ decays.

The $K^{+}K^{-}K_{S}$ mode is an admixture of $CP$-even and $CP$-odd components. By excluding the major $CP$-odd contribution from $\phi K_{S}$, the 3-body $K^{+}K^{-}K_{S}$ final state is primarily $CP$-even. The $K^{+}K^{-}$ mass spectra of the $\overline{B}{}^{0}\to K^{+}K^{-}K_{S}$ decay from $CP$-even and $CP$-odd contributions are shown in Fig. 2. For the $CP$-even spectrum, there are peaks at the threshold and $m_{K^{+}K^{-}}=1.5$ GeV region. The threshold enhancement arises from the $f_{0}(980)K_{S}$ and the nonresonant $f_{S}^{K^{+}K^{-}}$ contributions [see Eq. (II.1)]. 222In our previous work CCSKKK we have argued that the spectrum should have a peak at the large $m_{K^{+}K^{-}}$ end. This is because we have introduced an additional nonresonant contribution to the $\omega_{-}$ parameter parametrized as $\omega^{NR}_{-}=\kappa\,\frac{2p_{B}\cdot p_{2}}{s^{2}_{12}}$ and employed the $B^{-}\to D^{0}K^{0}K^{-}$ data and applied isospin symmetry to the $\overline{B}\to K\overline{K}$ matrix elements to determine the unknown parameter $\kappa$. Since this nonresonant term favors a small $m_{K^{+}K_{S}}$ region, a peak of the spectrum at large $m_{K^{+}K^{-}}$ is thus expected. However, such a bump is not seen experimentally BaBarKpKmK0 . In this work we will no longer consider this term. For the $CP$-odd spectrum, the peak on the lower end corresponds to the $\phi K_{S}$ contribution, which is also shown in the insert. The $b\to u$ transition is governed by the current-induced process $\langle\overline{B}^{0}\to K^{+}\overline{K}^{0}\rangle\times\langle 0\to K^{-}\rangle$ [see Eq. (68)]. From Eq. (8) it is clear that the $b\to u$ amplitude prefers a small invariant mass of $K^{+}$ and $\overline{K}^{0}$ and hence a large invariant mass of $K^{+}$ and $K^{-}$. In contrast, the $b\to c$ amplitude prefers a small $s_{23}$. Consequently, their interference is largely suppressed. The full $K^{+}K^{-}K_{S}$ spectrum, which is the sum of the $CP$-even and the $CP$-odd parts, has been measured by BaBar [Fig. 2(c)]. It clearly shows the phenomenon of threshold enhancement and the scalar resonances $X_{0}(1550)$ and $\chi_{c0}$.

The decay $\overline{B}^{0}\to K_{S}K_{S}K_{S}$ is a pure penguin-induced mode [cf. Eq. (71)] and it receives intermediate pole contributions only from the iso-singlet scalar mesons such as $f_{0}(980)$. Just like other $KKK$ modes, this decay is governed by the nonresonant background dominated by the $\sigma_{{}_{\rm NR}}$ term defined in Eq. (II.1). Hence, this mode is ideal for determining the unknown parameter $\sigma_{{}_{\rm NR}}$ which is given in Eq. (26). Time-dependent CP violation in neutral 3-body decay modes with fixed CP parity was first discussed by Gershon and Hazumi Gershon .

Results for the decay rates and CP asymmetries in $\overline{B}{}^{0}\to K^{+}K^{-}K_{S(L)},\,K_{S}K_{S}K_{S(L)}$ are displayed in Table 4 and Table 5, respectively. (For the decay amplitudes of $\overline{B}{}^{0}\to K_{S}K_{S}K_{S(L)}$, see CCSKKK for details.) The mixing-induced CP violations are defined by

[TABLE]

where $A$ is the decay amplitude of $\overline{B}^{0}\to K^{+}K^{-}K_{S(L)}$ or $K_{S}K_{S}K_{S(L)}$ and $\bar{A}$ is the conjugated $B^{0}$ decay amplitude, and $f_{+}$ is the CP even fraction defined by

[TABLE]

Generally, it is more convenient to define an effective $\sin 2\beta$ via $S_{f}\equiv-\eta_{f}\sin 2\beta_{\rm eff}$ with $\eta_{f}=2f_{+}-1$ for $K^{+}K^{-}K_{S}$. The predicted value of $f_{+}$ is consistent with the data but it is on the higher end of the experimental measurement because the $CP$-odd contributions from the vector mesons $\rho,\omega,\cdots,$ are OZI suppressed and the $CP$-odd nonresonant contribution is constrained by the $\pi^{+}\pi^{-}\pi^{-}$ rate.

The deviation of the mixing-induced CP asymmetry in $B^{0}\to K^{+}K^{-}K_{S}$ and $K_{S}K_{S}K_{S}$ from that measured in $B\to\phi_{c\bar{c}}K_{S}$, i.e. $\sin 2\beta_{\phi_{c\bar{c}}K_{S}}=0.681\pm 0.025$ HFAG , namely, $\Delta\sin 2\beta_{\rm eff}\equiv\sin 2\beta_{\rm eff}-\sin 2\beta_{\phi_{c\bar{c}}K_{S}}$, is calculated from Table 5 to be

[TABLE]

The corresponding experimental values are $0.049\pm 0.10$ and $-0.101\pm 0.20$, respectively. Due to the presence of color-allowed tree contributions in $\overline{B}^{0}\to K^{+}K^{-}K_{S}$, it is naively expected that this penguin-dominated mode is subject to a potentially significant tree pollution and hence $\Delta\sin 2\beta_{\rm eff}$ can be as large as ${\cal O}(10\%)$. However, our calculation indicates the deviation of the mixing-induced CP asymmetry in $\overline{B}^{0}\to K^{+}K^{-}K_{S}$ from that measured in $\overline{B}^{0}\to\phi_{c\bar{c}}K_{S}$ is very similar to that of the $K_{S}K_{S}K_{S}$ mode as the tree pollution effect in the former is somewhat washed out. Nevertheless, direct $CP$ asymmetry of the former, being of order $-4\%$, is more prominent than the latter.333 In our previous work CCSKKK , $\Delta\sin 2\beta_{\rm eff}$ is found to be

$\displaystyle\Delta\sin 2\beta_{K^{+}K^{-}K_{S}}=0.06^{+0.09}_{-0.04}\,,\qquad\Delta\sin 2\beta_{K_{S}K_{S}K_{S}}=0.06^{+0.03}_{-0.04}\,,$

for $\sin 2\beta_{J/\psi K_{S}}=0.687\pm 0.032$, while direct CP asymmetry is less than 1% in both modes. Note that due to an oversight the experimental error bars were not included in our previous paper for the theoretical calculation of $\Delta\sin 2\beta_{\rm eff}$.

II.2 $B^{-}\to KKK$ decays

The $B^{-}\to K^{+}K^{-}K^{-}$ decay amplitude has a similar expression as Eq. (68) except that one also needs to add the contributions from the interchange $s_{23}\to s_{12}$ and put a factor of 1/2 in the decay rate to account for the identical particle effect.

Branching ratios of resonant and nonresonant contributions to $B^{-}\to K^{+}K^{-}K^{-}$ are shown in Table 6. It is clear that the predicted rates of resonant and nonresonant components are consistent with the data except for the broad scalar resonance $X_{0}(1550)$. Both BaBar and Belle have seen a large fraction from $X_{0}(1550)$, $(121\pm 19\pm 6)\%$ by BaBar BaBarKpKpKm and $(63.4\pm 6.9)\%$ by Belle BelleKpKpKm , 444Belle BelleKpKpKm actually found two solutions for the fraction of $X_{0}(1550)K^{-}$: $(63.4\pm 6.9)\%$ and $(8.21\pm 1.94)\%$. The first solution is preferred by Belle. while our prediction is similar to that in $\overline{B}^{0}\to K^{+}K^{-}\overline{K}^{0}$. It is not clear why there is a huge disparity between $B^{-}\to K^{+}K^{-}K^{-}$ and $\overline{B}^{0}\to K^{+}K^{-}\overline{K}^{0}$ as far as the $X_{0}(1550)$ contribution is concerned. Obviously, a refined measurement of the $X_{0}(1550)$ contribution to the $K^{+}K^{-}K^{-}$ mode is urgently needed in order to clarify this issue. Our result for the nonresonant contribution is in good agreement with Belle, but disagrees with BaBar. Notice that Belle did not see the scalar resonance $f_{0}(980)$ as Belle employed the E791 result E791 for $g^{f_{0}\to K\bar{K}}$ which is smaller than $g^{f_{0}\to\pi\pi}$. In contrast to E791, the ratio $g^{f_{0}\to K\bar{K}}/g^{f_{0}\to\pi\pi}$ is measured to be larger than 4 in the existing $e^{+}e^{-}$ experiments BES ; epem

We next turn to the decay $B^{-}\to K^{-}K_{S}K_{S}$. Following Gronau3body , let us consider the symmetric state of $K^{0}\overline{K}^{0}$

[TABLE]

Hence,

[TABLE]

The factorizable amplitude of $B^{-}\to K^{-}K^{0}\overline{K}^{0}$ is given by Eq. (72). Just as other $KKK$ modes, this decay is also expected to be dominated by the nonresonant contribution (see Table 7). The calculated total rate is in good agreement with experiment. Just as the pure penguin mode $K_{S}K_{S}K_{S}$, the decay $B^{-}\to K^{-}K_{S}K_{S}$ also can be used to constrain the nonresonant parameter $\sigma_{{}_{\rm NR}}$.

As pointed out in Gronau3body , isospin symmetry implies the relation

[TABLE]

This leads to

[TABLE]

Experimentally, this relation is well satisfied: LHS=$(23.0\pm 2.6)\times 10^{-6}$ and RHS=$(22.1\pm 2.1)\times 10^{-6}$. Hence, the isospin relation Eq. (32) is well respected.

III $B\to K\pi\pi$ decays

In this section we shall consider five $B\to K\pi\pi$ decays, namely, $B^{-}\to K^{-}\pi^{+}\pi^{-}$, $\overline{K}^{0}\pi^{-}\pi^{0}$, $\overline{B}^{0}\to K^{-}\pi^{+}\pi^{0}$, $\overline{K}^{0}\pi^{+}\pi^{-}$ and $\overline{K}^{0}\pi^{0}\pi^{0}$. They are dominated by $b\to s$ penguin transition and consist of three decay processes: (i) the current-induced process, $\langle B\to\pi\pi\rangle\times\langle 0\to K\rangle$, (ii) the transition processes, $\langle B\to\pi\rangle\times\langle 0\to\pi K\rangle$, and $\langle B\to K\rangle\times\langle 0\to\pi\pi\rangle$, and (iii) the annihilation process $\langle B\to 0\rangle\times\langle 0\to K\pi\pi\rangle$.

The factorizable amplitudes for $B^{-}\to K^{-}\pi^{+}\pi^{-}$, $\overline{K}^{0}\pi^{-}\pi^{0}$, $\overline{B}^{0}\to K^{-}\pi^{+}\pi^{0}$, $\overline{K}^{0}\pi^{+}\pi^{-}$ and $\overline{K}^{0}\pi^{0}\pi^{0}$ are given in Eqs. (74-78), respectively. All five channels have the three-body matrix element $\langle\pi\pi|(\bar{q}b)_{V-A}|B\rangle$ which has the similar expression as Eqs. (3) and (II.1) except that the pole $B_{s}^{*}$ is replaced by $B^{*}$ and the kaon is replaced by the pion. However, there are additional resonant contributions to this three-body matrix element due to the intermediate vector $\rho$ and scalar $f_{0}$ mesons

[TABLE]

where $\rho_{i}$ denote generic $\rho$-type vector mesons, e.g. $\rho=\rho(770),\rho(1450),\rho(1700),\cdots$. Applying Eqs. (81) and (B) we are led to

[TABLE]

Likewise, the 3-body matrix element $\langle K^{-}\pi^{+}|(\bar{s}b)_{{}_{V-A}}|\overline{B}^{0}\rangle$ appearing in $\overline{B}^{0}\to K^{-}\pi^{+}\pi^{0}$ also receives the following resonant contributions

[TABLE]

with $K_{i}^{*}=K^{*}(892),K^{*}(1410),K^{*}(1680),\cdots$.

For the two-body matrix elements $\langle\pi^{+}K^{-}|(\bar{s}d)_{V-A}|0\rangle$, $\langle\pi^{+}\pi^{-}|(\bar{u}u)_{V-A}|0\rangle$ and $\langle\pi^{+}\pi^{-}|\bar{s}s|0\rangle$, we note that

[TABLE]

where we have taken into account the sign flip arising from interchanging the operators $s\leftrightarrow d$. Hence,

[TABLE]

However, the form factor $F_{1}$ also receives resonant contributions

[TABLE]

with

[TABLE]

where ${K^{*}_{0}}_{i}=K^{*}_{0}(1430),\cdots$. Hence, the resonant contributions to the form factor $F_{1}^{K\pi}$ are

[TABLE]

In principle, the weak vector form factor $F^{\pi^{+}\pi^{-}}$ defined by

[TABLE]

can be related to the time-like pion electromagnetic form factors. However, unlike the kaon case, the time-like e.m. form factors of the pions are not well measured enough allowing us to determine the resonant and nonresonant parts. Therefore, we shall only consider the resonant part which has the expression

[TABLE]

Following Eq. (II.1), the relevant matrix elements of scalar densities read

[TABLE]

and

[TABLE]

Note that for the scalar meson, the decay constants $f_{S}$ and $\bar{f}_{S}$ are defined in Eq. (81) and they are related via Eq. (82). The nonresonant contribution $\langle\pi^{+}(p_{2})\pi^{-}(p_{3})|\bar{s}s|0\rangle^{NR}$ vanishes under the OZI rule, while under SU(3) symmetry555The matrix elements of scalar densities can be generally decomposed into $D$-, $F$- and $S$(singlet)-type components. Assuming that the singlet component is OZI suppressed, SU(3) symmetry leads to, for example, the relation $\langle K\pi|\bar{s}q|0\rangle^{NR}=\langle K\bar{K}|\bar{s}s|0\rangle^{NR}$.

[TABLE]

with the expression of $f_{s}^{NR}$ given in Eq. (II.1).

It is known that in the narrow width approximation, the 3-body decay rate obeys the factorization relation

[TABLE]

with $R$ being a resonance. This means that the amplitudes $A(B\to RP\to P_{1}P_{2}P)$ and $A(B\to RP)$ should have the same expressions apart from some factors. Hence, using the known results for quasi-two-body decay amplitude $A(B\to RP)$, one can have a cross check on the three-body decay amplitude of $B\to RP\to P_{1}P_{2}P$. For example, from Eq. (76) we obtain the factorizable amplitude $A(\overline{B}^{0}\to K_{0}^{*0}(1430)\pi^{0};K^{*0}_{0}(1430)\to K^{-}\pi^{+})$ as

[TABLE]

where

[TABLE]

The expression inside $\{\cdots\}$ is indeed the amplitude of $\overline{B}^{0}\to K_{0}^{*0}(1430)\pi^{0}$ given in Eq. (A6) of CCY .

The strong coupling constants such as $g^{\rho\to\pi^{+}\pi^{-}}$ and $g^{f_{0}(980)\to\pi^{+}\pi^{-}}$ are determined from the measured partial widths through the relations

[TABLE]

for scalar and vector mesons, respectively, where $p_{c}$ is the c.m. momentum. The numerical results are

[TABLE]

In determining the coupling of $f_{0}\to\pi^{+}\pi^{-}$, we have used the partial width

[TABLE]

measured by Belle Bellef0 . The momentum dependence of the weak form factor $F^{K\pi}(q^{2})$ is parametrized as

[TABLE]

where $\Lambda_{\chi}\approx 830$ MeV is the chiral-symmetry breaking scale Cheng88 and $\Gamma_{R}$ is the width of the relevant resonance, which is taken to be 200 MeV Cheng:2002qu .

The results of the calculation are summarized in Tables 8-12. We see that except for $f_{0}(980)K$, the predicted rates for $K^{*}\pi$, $K^{*}_{0}(1430)\pi$ and $\rho K$ are smaller than the data. Indeed, the predictions based on QCD factorization for these decays are also generally smaller than experiment by a factor of 2$\sim$5. This will be discussed in more details in Sec. VI.

While Belle has found a sizable fraction of order $(35\sim 40)\%$ for the nonresonant signal in $K^{-}\pi^{+}\pi^{-}$ and $\overline{K}^{0}\pi^{+}\pi^{-}$ modes (see Table 1), BaBar reported a small fraction of order 4.5% in $K^{-}\pi^{+}\pi^{-}$. The huge disparity between BaBar and Belle is ascribed to the different parameterizations adopted by both groups. BaBar BaBarKpipi used the LASS parametrization to describe the $K\pi$ $S$-wave and the nonresonant component by a single amplitude suggested by the LASS collaboration to describe the scalar amplitude in elastic $K\pi$ scattering. As commented in BelleKpipi , while this approach is experimentally motivated, the use of the LASS parametrization is limited to the elastic region of M(K\pi){\ \lower-1.2pt\vbox{\hbox{\hbox to0.0pt{<\hss}\lower 5.0pt\vbox{\hbox{\sim} }}}\ }2.0 GeV, and an additional amplitude is still required for a satisfactory description of the data. In our calculations we have taken into account the nonresonant contributions to the two-body matrix elements of scalar densities, $\langle K\pi|\bar{s}q|0\rangle$. Recall that a large nonresonant contribution from $\langle K\overline{K}|\bar{s}s|0\rangle$ is needed in order to explain the observed decay rates of $B^{0}\to K_{S}K_{S}K_{S}$ and $B^{-}\to K^{-}K_{S}K_{S}$. From Tables 8-12 we see that our predicted nonresonant rates are in agreement with the Belle measurements. The reason why the nonresonant fraction is as large as 90% in $KKK$ decays, but becomes only $(35\sim 40)\%$ in $K\pi\pi$ channels (see Table 1) can be explained as follows. Under SU(3) flavor symmetry, we have the relation $\langle K\pi|\bar{s}q|0\rangle^{NR}=\langle K\bar{K}|\bar{s}s|0\rangle^{NR}$. Hence, the nonresonant rates in the $K^{-}\pi^{+}\pi^{-}$ and $\overline{K}^{0}\pi^{+}\pi^{-}$ modes should be similar to that in $K^{+}K^{-}\overline{K}^{0}$ or $K^{+}K^{-}K^{-}$. Since the $KKK$ channel receives resonant contributions only from $\phi$ and $f_{0_{i}}$ mesons, while $K^{*}_{i},K^{*}_{0i},\rho_{i},f_{0i}$ resonances contribute to $K\pi\pi$ modes, this explains why the nonresonant fraction is of order 90% in the former and becomes of order 40% in the latter. Note that the predicted nonresonant contribution in the $K^{-}\pi^{+}\pi^{0}$ mode is larger than the BaBar’s upper bound and barely consistent with the Belle limit. It is conceivable that the SU(3) breaking effect in $\langle K\pi|\bar{s}q|0\rangle^{NR}$ may lead to a result consistent with the Belle limit.

It is interesting to notice that, based on a simple fragmentation model and SU(3) symmetry, Gronau and Rosner Gronau3body found the relations

[TABLE]

Again, a large nonresonant background in $K^{-}\pi^{+}\pi^{-}$ and $\overline{K}^{0}\pi^{+}\pi^{-}$ is favored by this model.

Although the $\overline{B}^{0}\to K_{S}\pi^{0}\pi^{0}$ rate has not been measured, its time-dependent CP asymmetries have been studied by BaBar BaBarK0pi0pi0 with the results

[TABLE]

Note that this mode is a CP-even eigenstate. We found that its branching ratio is not so small, of order $6\times 10^{-6}$, in spite of the presence of two neutral pions in the final state (see Table 12). Theoretically, we obtain

[TABLE]

Finally, we consider the mode $K_{S}\pi^{+}\pi^{-}$ which is an admixture of CP-even and CP-odd components. Results for the decay rates and CP asymmetries are displayed in Table 13. We see that the effective $\sin 2\beta$ is of order 0.718 and direct CP asymmetry of order 4.9% for $K_{S}\pi^{+}\pi^{-}$.

IV $B\to KK\pi$ decays

We now turn to the three-body decay modes dominated by $b\to u$ tree and $b\to d$ penguin transitions, namely, $KK\pi$ and $\pi\pi\pi$. We first consider the decay $B^{-}\to K^{+}K^{-}\pi^{-}$ whose factorizable amplitude is given by Eq. (73). Note that we have included the matrix element $\langle K^{+}K^{-}|\bar{d}d|0\rangle$. Although its nonresonant contribution vanishes as $K^{+}$ and $K^{-}$ do not contain the valence $d$ or $\bar{d}$ quark, this matrix element does receive a contribution from the scalar $f_{0}$ pole

[TABLE]

where $\langle f_{0}|\bar{d}d|0\rangle=m_{f_{0}}\bar{f}_{f_{0}}^{d}$. In the 2-quark model for $f_{0}(980)$, $\bar{f}_{f_{0}(980)}^{d}=\bar{f}_{f_{0}(980)}\sin\theta/\sqrt{2}$. Also note that the matrix element $\langle K^{-}(p_{3})|(\bar{s}b)_{V-A}|B^{-}\rangle\langle\pi^{-}(p_{1})K^{+}(p_{2})|(\bar{d}s)_{V-A}|0\rangle$ has a similar expression as Eq. (38) except for a sign difference

[TABLE]

As in Eq. (41), the form factor $F_{1}^{K\pi}$ receives a resonant contribution for the $K^{*}$ pole.

The nonresonant and various resonant contributions to $B^{-}\to K^{+}K^{-}\pi^{-}$ are shown in Table 16. The predicted total rate is consistent with upper limits set by BaBar and Belle.

V $B\to\pi\pi\pi$ decays

The factorizable amplitudes of the tree-dominated decay $B^{-}\to\pi^{+}\pi^{-}\pi^{-}$ and $\overline{B}^{0}\to\pi^{+}\pi^{-}\pi^{0}$ are given by Eqs. (79) and (A), respectively. We see that the former is dominated by the $\rho^{0}$ pole, while the latter receives $\rho^{\pm}$ and $\rho^{0}$ contributions. As a consequence, the $\pi^{+}\pi^{-}\pi^{0}$ mode has a rate larger than $\pi^{+}\pi^{-}\pi^{-}$ even though the former involves a $\pi^{0}$ in the final state.

The $\pi^{+}\pi^{-}\pi^{-}$ mode receives nonresonant contributions mostly from the $b\to u$ transition as the nonresonant contribution in the matrix element $\langle\pi^{+}\pi^{-}|\bar{d}d|0\rangle$ is suppressed by the smallness of penguin Wilson coefficients $a_{6}$ and $a_{8}$. Therefore, the measurement of the nonresonant contribution in this decay can be used to constrain the nonresonant parameter $\alpha_{{}_{\rm NR}}$ in Eq. (8)

VI Direct $CP$ asymmetries

Direct CP asymmetries for various charmless three-body $B$ decays are collected in Table 17. Mixing-induced and direct CP asymmetries in $B^{0}\to K^{+}K^{-}K_{S,L}$ and $K_{S}K_{S}K_{S,L}$ decays are already shown in Table 5. It appears that direct CP violation is sizable in $K^{+}K^{-}K^{-}$ and $K^{+}K^{-}\pi^{-}$ modes.

The major uncertainty with direct CP violation comes from the strong phases which are needed to induce partial rate CP asymmetries. In this work, the strong phases arise from the effective Wilson coefficients $a_{i}^{p}$ listed in (A) and from the Breit-Wigner formalism for resonances. Since direct CP violation in charmless two-body $B$ decays can be significantly affected by final-state rescattering CCSfsi , it is natural to extend the study of final-state rescattering effects to the case of three-body $B$ decays. We will leave this to a future investigation.

VII Two-body $B\to VP$ and $B\to SP$ decays

Thus far we have considered the branching ratio products ${\cal B}(B\to Rh_{1}){\cal B}(R\to h_{2}h_{3})$ with the resonance $R$ being a vector meson or a scalar meson. Using the experimental information on ${\cal B}(R\to h_{2}h_{3})$ PDG

[TABLE]

one can extract the branching ratios of $B\to VP$ and $B\to SP$. The results are summarized in Table 18.

Two remarks about the experimental branching ratios are in order: (i) The BaBar results for the branching ratios of $\overline{B}^{0}\to K^{*-}\pi^{+},~{}\overline{K}^{*0}\pi^{0},~{}K^{*-}_{0}(1430)\pi^{+}$ are inferred from the three-body decays $\overline{B}^{0}\to\overline{K}^{0}\pi^{+}\pi^{-}$ (see Table 11) and Belle results are taken from $\overline{B}^{0}\to K^{-}\pi^{+}\pi^{0}$ (see Table 10). (ii) Branching ratios of $\overline{B}^{0}\to\phi\overline{K}^{0}$ shown in Table 18 are not inferred from the Dalitz plot analysis of $B\to KKK$ decays.

For comparison, the predictions of the QCD factorization approach for $B\to VP$ BN and $B\to SP$ CCY are also exhibited in Table 18. In order to compare theory with experiment for $B\to f_{0}(980)K\to\pi^{+}\pi^{-}K$, we need an input for ${\cal B}(f_{0}(980)\to\pi^{+}\pi^{-})$. To do this, we shall use the BES measurement BES

[TABLE]

Assuming that the dominance of the $f_{0}(980)$ width by $\pi\pi$ and $K\overline{K}$ and applying isospin relation, we obtain

[TABLE]

At first sight, it appears that the ratio defined by

[TABLE]

is not consistent with the value of $0.69\pm 0.32$ inferred from the BaBar data (see Tables 6 and 8)

[TABLE]

where we have applied the narrow width approximation Eq. (47).

The above-mentioned discrepancy can be resolved by noting that the factorization relation Eq. (47) for the resonant three-body decay is applicable only when the two-body decays $B\to RP$ and $R\to P_{1}P_{2}$ are kinematically allowed and the resonance is narrow, the so-called narrow width approximation. However, as the decay $f_{0}(980)\to K^{+}K^{-}$ is kinematically barely or even not allowed, the off resonance peak effect of the intermediate resonant state will become important. Therefore, it is necessary to take into account the finite width effect of the $f_{0}(980)$ which has a width of order 40-100 MeV PDG . In short, one cannot determine the ratio $R$ by applying the narrow width approximation to the three-body decays. That is, one should employ the decays $B\to K\pi\pi$ rather than $B\to KKK$ to extract the experimental branching ratio for $B\to f_{0}(980)K$ provided ${\cal B}(f_{0}(980)\to\pi\pi)$ is available.

We now compare the present work for $B\to VP$ and $B\to SP$ with the approach of QCD factorization BBNS ; CCY . In this work, our calculation of 3-body $B$ decays is similar to the simple generalized factorization approach Ali ; CCTY by assuming a set of universal and process independent effective Wilson coefficients $a_{i}^{p}$ with $p=u,c$ in Eq. (A). In QCDF, the calculation of $a_{i}^{p}$ is rather sophisticated. They are basically the Wilson coefficients in conjunction with short-distance nonfactorizable corrections such as vertex corrections and hard spectator interactions. In general, they have the expressions BBNS ; BN

[TABLE]

where $i=1,\cdots,10$, the upper (lower) signs apply when $i$ is odd (even), $c_{i}$ are the Wilson coefficients, $C_{F}=(N_{c}^{2}-1)/(2N_{c})$ with $N_{c}=3$, $M_{2}$ is the emitted meson and $M_{1}$ shares the same spectator quark with the $B$ meson. The quantities $V_{i}(M_{2})$ account for vertex corrections, $H_{i}(M_{1}M_{2})$ for hard spectator interactions with a hard gluon exchange between the emitted meson and the spectator quark of the $B$ meson and $P_{i}(M_{2})$ for penguin contractions. Hence, the effective Wilson coefficients $a_{i}^{p}(M_{1}M_{2})$ depend on the nature of $M_{1}$ and $M_{2}$; that is, they are process dependent. Moreover, they depend on the order of the argument, namely, $a_{i}^{p}(M_{2}M_{1})\neq a_{i}^{p}(M_{1}M_{2})$ in general. In the above equation, $N_{i}(M_{2})$