Quasi-two-body decays B → Kρ → Kππ in perturbative QCD approach Wen-Fei Wang1,2∗ and Hsiang-nan Li1† 1
arXiv:1609.04614v1 [hep-ph] 15 Sep 2016
2
Institute of Physics, Academia Sinica, Taipei, Taiwan 115, Republic of China and
Institute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006, China (Dated: September 16, 2016)
We analyze the quasi-two-body decays B → Kρ → Kππ in the perturbative QCD (PQCD) approach, in which final-state interactions between the pions in the resonant regions associated with the P -wave states ρ(770) and ρ′ (1450) are factorized into two-pion distribution amplitudes. Adopting experimental inputs for the time-like pion form factors involved in two-pion distribution amplitudes, we calculate branching ratios and direct CP asymmetries of the B → Kρ(770), Kρ′ (1450) → Kππ modes. It is shown that agreement of theoretical results with data can be achieved, through which Gegenbauer moments of the P -wave two-pion distribution amplitudes are determined. The consistency between the three-body and two-body analyses of the B → Kρ(770) → Kππ decays supports the PQCD factorization framework for exclusive hadronic B meson decays. PACS numbers: 13.20.He, 13.25.Hw, 13.30.Eg
I.
INTRODUCTION
Strong dynamics contained in three-body hadronic B meson decays is much more complicated than in two-body cases, because of entangled nonresonant and resonant contributions, and significant final-state interactions [1]. Nonresonant contributions may not be negligible in these decays, as indicated by the observations made in Refs [2–7]. Quasi-two-body channels through intermediate scalar, vector and tensor resonances, which produce hadron pairs with final-state interactions, usually dominate total branching fractions. An amplitude for a three-body hadronic B meson decay, as a coherent sum of nonresonant and resonant contributions, leads to nonuniform distributions of events described by differential branching fractions [2–16] and of direct CP asymmetries [17–20] in a Dalitz plot [21]. Dalitzplot analyses of abundant three-body hadronic B meson decays from different collaborations (BaBar [2–4, 11–16], Belle [5, 6, 8–10] and LHCb [17–19]) have revealed valuable information on involved strong and weak dynamics. On the theoretical side, substantial progress on three-body hadronic B meson decays by means of symmetry principles and factorization theorems has been made, although rigorous justification of these approaches is not yet available. Isospin, U-spin and flavor SU(3) symmetries were adopted in [22–31], and the role of the CP T invariance in three-body B meson decays was discussed in Refs [32, 33]. The QCD factorization [34, 35] has been widely applied to studies of three-body charmless hadronic B meson decays [36–48], including, for instance, detailed investigation on factorization properties of the B + → π + π + π − mode in various regions of phase space [49]. The perturbative QCD (PQCD) approach based on the kT factorization theorem [50, 51] has been employed in Refs. [52–56], where strong dynamics between two final-state hadrons in resonant regions are factorized into a new nonperturbative input, the two-hadron distribution amplitudes. An advantage of the PQCD factorization approach is that both nonresonant and resonant contributions can be accommodated into this new input. A model that combines the heavy quark effective theory and the chiral Lagrangian was proposed in Ref. [57] to compute nonresonant decay amplitudes. The B meson transition to a meson pair has been analyzed in the heavy-mass and large-energy limits [58], and in the light-cone sum rules [59] also in terms of two-meson distribution amplitudes. Nonresonant contributions to the above transition were evaluated in the heavy meson chiral perturbation theory [60] in Refs. [43, 44, 46]. In this Letter we will focus on resonant contributions to three-body hadronic B meson decays in the PQCD approach, extending our previous work on S-wave resonances to P -wave ones. We have determined the Gegenbauer moments of 0 the S-wave two-pion distribution amplitudes by fitting our formalism to the B(s) → J/ψπ + π − and Bs → π + π − ℓ+ ℓ− data. Here we will consider the quasi-two-body decays B → Kρ → Kππ, which receive contributions mainly from the ρ(770) and ρ′ (1450) intermediate states. These resonant contributions are parametrized into the time-like pion form factors involved in the two-pion distribution amplitudes, for which there exist experimental inputs from the e+ e−
∗ Electronic † Electronic
address:
[email protected] address:
[email protected]
2 annihilation. It will be demonstrated that agreement of theoretical results with data can be achieved by choosing appropriate Gegenbauer moments of the P -wave two-pion distribution amplitudes. On one hand, the consistency between the three-body and two-body analyses of the quasi-two-body modes B → Kρ(770) → Kππ to be verified below supports the PQCD factorization for exclusive hadronic B meson decays. On the other hand, with both the S-wave and P -wave distribution amplitudes being ready, we can proceed to predictions for branching ratios and direct CP asymmetries of three-body hadronic B meson decays in various localized regions of two-pion phase space. The rest of this Letter is organized as follows. The PQCD framework for three-body hadronic B meson decays is reviewed in Sec. II, where the P -wave two-pion distribution amplitudes up to twist 3 are parametrized. Numerical results for branching ratios and direct CP asymmetries of the various B → Kρ → Kππ modes are presented and compared with those from the two-body analysis in Sec. III. The straightforward extension of the present formalism to other P -wave resonant contributions is highlighted. Section IV contains the Conclusion. The factorization formulas for the relevant three-body decay amplitudes are collected in the Appendix. II.
FRAMEWORK
In the rest frame of the B meson, we write the B meson momentum pB and the light spectator quark momentum kB as mB mB (1) pB = √ (1, 1, 0T ), kB = 0, √ xB , kBT , 2 2 in the light-cone coordinates, with mB being the B meson mass and xB the momentum fraction. For the B → Kρ → Kππ decays, we define the resonant state momentum p (in the plus z direction) and the associated spectator quark momentum k, and the kaon momentum p3 (in the minus z direction) and the associated non-strange quark momentum k3 as mB mB mB mB p = √ (1, η, 0T ), k = √ z, 0, kT , p3 = √ (0, 1 − η, 0T ), k3 = 0, √ (1 − η)x3 , k3T , (2) 2 2 2 2 p with the variable η = w2 /m2B , w = p2 being the invariant mass of the resonant state, and the momentum fractions z and x3 . The momenta p1 and p2 for the two pions from the resonant state have the components [54] mB p+ 1 = ζ √ , 2
mB p− 1 = (1 − ζ)η √ , 2
mB p+ 2 = (1 − ζ) √ , 2
mB p− 2 = ζη √ , 2
in which the momentum fraction ζ of the first pion runs between 0 and 1. In Ref. [55] we have introduced the distribution amplitudes for the pion pair [61–63] Z 1 dy − −izp+ y− + ¯ − )γν T ψ(0)|0i, φIvν (z, ζ, w2 ) = √ e hπ (p1 )π − (p2 )|ψ(y 2π 2 2Nc Z p+ dy − −izp+ y− + 1 ¯ − )T ψ(0)|0i, e hπ (p1 )π − (p2 )|ψ(y φIs (z, ζ, w2 ) = √ 2π 2 2Nc w ⊥ Z 1 p+ f2π dy − −izp+ y− + ¯ − )iσµν nµ T ψ(0)|0i, φItν (z, ζ, w2 ) = √ e hπ (p1 )π − (p2 )|ψ(y − 2 2π 2 2Nc w
(3)
(4) (5) (6)
where Nc is the number of colors, n− = (0, 1, 0T ) is a dimensionless vector, T = τ 3 /2 is chosen for the isovector ⊥ I = 1 state, ψ represents the u-d quark doublet, and f2π is a normalization constant. For I = 1, the P -wave is the I=1 I=1 I=1 leading partial wave, to which φvν=− and φtν=⊥ contribute at twist 2, and φI=1 , and φI=1 tν=+ contribute at vν=⊥ , φs 2 twist 3. With w being a variable, the above two-pion distribution amplitudes contain both nonresonant and resonant contributions from the pion pair. The P -wave two-pion distribution amplitudes are organized into p/1 p/2 − p/2 p/1 I=1 1 I=1 2 I=1 2 2 √ p / φ (z, ζ, w ) + wφ (z, ζ, w ) + φI=1 = φ (z, ζ, w ) , (7) vν=− s ππ w(2ζ − 1) tν=+ 2Nc
3
FIG. 1: Typical Feynman diagrams for the quasi-two-body decays B → Kρ → Kππ, in which the symbol ⊗ stands for the weak vertex, × denotes possible attachments of hard gluons, and the green rectangle represents intermediate states.
whose components are parametrized as h i 3Fπ (w2 ) 3/2 √ z(1 − z) 1 + a02 C2 (1 − 2z) P1 (2ζ − 1) , 2Nc 3F (w2 ) s (1 − 2z) 1 + as2 1 − 10z + 10z 2 P1 (2ζ − 1) , φI=1 (z, ζ, w2 ) ≡ φs (z, ζ, w2 ) = √ s 2 2Nc h i 2 3F t (w ) 3/2 2 t 2 √ (1 − 2z)2 1 + at2 C2 (1 − 2z) P1 (2ζ − 1) , φI=1 tν=+ (z, ζ, w ) ≡ φ (z, ζ, w ) = 2 2Nc
2 0 2 φI=1 vν=− (z, ζ, w ) ≡ φ (z, ζ, w ) =
(8) (9) (10)
3/2 with the Gegenbauer polynomial C2 (t) = 3 5t2 − 1 /2 and the Legendre polynomial P1 (2ζ − 1) = 2ζ − 1. In principle, the time-like form factors associated with the second Gegenbauer moments a0,t,s can differ from Fπ,s,t (w2 ) 2 associated with the leading ones. Here we assume that they are the same, which can then be factored out and serve as the normalization of the two-pion distribution amplitudes. The moments a0,t,s will be regarded as free parameters and 2 determined in this work. Up to the second Gegenbauer terms, the Legendre polynomial P3 (2ζ − 1) also contributes. However, more unknown form factors will be introduced, and currently available data are not sufficient for their extraction. The time-like pion form factor Fπ (w2 ) has attracted considerable theoretical effort [64–78] and been measured with high precision by the CMD-2 [79–81], KLOE [82–85], BaBar [86, 87], BESIII [88], ALEPH [89, 90], CLEO [91, 92], and Belle [93] Collaborations. The ρ meson dominance model for Fπ has been established in Ref. [94]. Guaranteed by the Watson theorem [95], strong interactions between the ρ meson and the pion pair, including elastic rescattering of the two pions, can be factorized into Fπ . In experimental investigations of three-body hadronic B meson decays, the ρ resonant contribution is usually parametrized as the Gounaris-Sakurai (GS) model [96] based on the BreitWigner (BW) function [97]. Taking into account the ρ-ω interference and excited-state contributions, we write Fπ as a coherent sum [86] X −1 1 + cω BWω (w2 , mω , Γω ) X Fπ (w2 ) = GSρ (w2 , mρ , Γρ ) , (11) + ci GSi (w2 , mi , Γi ) 1 + ci 1 + cω
with i = ρ′ (1450), ρ′′ (1700) and ρ′′′ (2254). The explicit expressions of the auxiliary functions GS and BW in Eq. (11) are referred to Refs. [86, 96]. The inputs for the masses m and widths Γ of ρ′ , ρ′′ , and ρ′′′ , and for the complex parameters c can be found in Ref. [86]. We have cω = 0 for a charged ρ meson, because of no interference between it and a ω meson. Note that the Gegenbauer moments a0,t,s are the same for all the resonant states ρ, ρ′ , ρ′′ ,... in the 2 above parametrization of the two-pion distribution amplitudes. The quasi-two-body decays B → Kρ → Kππ can be also analyzed in an alternative approach based on two-body decays: the quark pair q q¯ from a hard decay kernel forms the ρ meson, followed by its BW propagator, and then by the ρ → ππ transition with the strength gρππ . The equivalence between the framework with the ρ meson propagator and the present one with the two-pion distribution amplitudes hints the relation, Fπρ (w2 ) ≈
gρππ wfρ , Dρ (w2 )
(12)
where Fπρ represents the ρ component of Eq. (11), fρ is the ρ meson decay constant, and Dρ is the denominator of the BW function for the ρ resonance. We have the similar relations for the ρ components in the other two form factors, ρ Fs,t (w2 ) ≈ gρππ wfρT /Dρ (w2 ), in which the decay constant fρT normalizes the twist-3 ρ meson distribution amplitudes. Due to the dominance of the ρ resonant contributions to the time-like form factors [86], it is legitimate to postulate the approximation Fs,t (w2 ) ≈ (fρT /fρ )Fπ (w2 ).
4 The amplitude A for the quasi-two-body decays B → Kρ → Kππ in the PQCD approach is, according to Fig. 1, given by [52, 53] A = φB ⊗ H ⊗ φK ⊗ φI=1 ππ ,
(13)
where the hard kernel H contains only one hard gluon exchange at leading order in the strong coupling αs as in the two-body formalism, the symbol ⊗ means convolutions in parton momenta, and the B meson (kaon, two-pion) distribution amplitude φB (φK , φI=1 ππ ) absorbs nonperturbative dynamics in the decay processes. We then have their differential branching fractions [98] −→ dB |− p→ π ||pK | = τB |A|2 , dw 32π 3 m3B
(14)
− → τB being the B meson mean lifetime. The magnitudes of the pion and kaon momenta, |− p→ π | and |pK |, are written, in the center-of-mass frame of the pion pair, as rh i 1p 2 2 − →| = 1 2 , p→ | = w − 4m | p (m2B − m2K ) − 2 (m2B + m2K ) w2 + w4 /w2 , (15) |− π K π 2 2
with the pion mass mπ and the kaon mass mK . The B → Kρ → Kππ decay amplitudes A are collected in the Appendix, which are similar to those in Ref. [99] for the two-body B meson decay into a pseudoscalar meson and a vector meson. III.
RESULTS
For the numerical study, we adopt the inputs (in units of GeV) [98] (f =4)
ΛMS
= 0.250,
mπ± = 0.140,
mB ±,0 = 5.280,
mK ± = 0.494,
mK 0 = 0.498,
mπ0 = 0.135,
mρ = 0.775,
Γρ = 0.149,
(16)
the mean lifetimes τB 0 = 1.519 × 10−12 s and τB ± = 1.638 × 10−12 s, and the Wolfenstein parameters from Ref. [98]. The decay constant fρ has been extracted from the τ ± → ρ± ντ decay rate for the charged ρ± meson and from ρ0 → e+ e− for the neutral ρ0 meson. In this work we take their arithmetic average value fρ = (0.216 ± 0.003) GeV [100, 101]. The decay constant fρT has been computed in lattice QCD [102–105], for which we choose fρT = 0.184 GeV [102]. The ratio fρT /fρ then determines the ratios Fs,t /Fπ postulated in the previous section. The B meson and kaon distribution amplitudes are the same as widely adopted in the PQCD approach [54, 106–108]. TABLE I: PQCD results for the CP averaged branching ratios and the direct CP asymmetries of the B → Kρ → Kππ decays. The corresponding data are quoted from Particle Data Group [98]. Mode B + → K + ρ0 → K + π + π − B + → K 0 ρ+ → K 0 π + π 0 B 0 → K + ρ− → K + π − π 0 B 0 → K 0 ρ0 → K 0 π + π −
B (10−6 ) ACP B (10−6 ) ACP B (10−6 ) ACP B (10−6 ) ACP
Results +0.44 s 0 +0.29 K +0.39 t +0.39 (a (ω ) 3.42+0.78 B −0.39 2 )−0.38 (m0 )−0.32 (a2 )−0.28 (a2 ) −0.55 t K 0 s 0.43+0.04 −0.05 (ωB ) ± 0.06(a2 ) ± 0.03(m0 ) ± 0.03(a2 ) ± 0.01(a2 ) +1.65 +1.92 s 0 +0.53 K +0.60 t +0.88 7.43−1.31 (ωB )−1.42 (a2 )−0.91 (m0 )−0.62 (a2 )−0.47 (a2 ) +0.04 0 s t K +0.01 0.15+0.02 −0.01 (ωB )−0.05 (a2 ) ± 0.01(m0 )−0.00 (a2 ) ± 0.00(a2 ) +1.71 +0.58 t +0.78 K +0.67 0 +0.39 s 6.51−1.12 (ωB )−0.61 (a2 )−0.77 (m0 )−0.64 (a2 )−0.47 (a2 ) +0.09 K 0 s t +0.03 0.31+0.00 −0.01 (ωB )−0.08 (a2 )−0.02 (m0 ) ± 0.01(a2 ) ± 0.02(a2 ) +0.73 t +0.52 K +0.28 0 +0.26 s 3.76+1.09 −0.74 (ωB )−0.60 (a2 )−0.47 (m0 )−0.25 (a2 )−0.23 (a2 ) +0.00 +0.01 0 t K +0.00 0.06−0.02 (ωB )−0.01 (a2 ) ± 0.00(m0 )−0.01 (a2 ) ± 0.00(as2 )
Data [98] 3.7 ± 0.5 0.37 ± 0.10 8.0 ± 1.5 − 0.12 ± 0.17 7.0 ± 0.9 0.20 ± 0.11 4.7 ± 0.6 −
We first single out the ρ(770) component of the time-like pion form factor in Eq. (11). The fit to the data in Table I determines the Gegenbauer moments a02 = 0.25, as2 = 0.75, and at2 = −0.60, which differ from those in the distribution amplitudes for a longitudinally polarized ρ meson [109, 110]. The resultant CP averaged branching ratios (B) and direct CP asymmetries (ACP ) for the B + → K + ρ0 → K + π + π − , B + → K 0 ρ+ → K 0 π + π 0 , B 0 → K + ρ− → K + π − π 0 and B 0 → K 0 ρ0 → K 0 π + π − modes are presented in Table I. The theoretical uncertainties come from the variations
5
2.5
1.0
BB+
] ]
B B
0.8
)
2.0
K-[ K+[
1.5
+
0
0
K [ +
K [ 0
K [
] ] ] ]
CP
0
+
K [
1.0
0.4
0.5
0.2
d
/dw (10
B
0.6
-5
GeV
-1
B
+
0.0
0.0 0.5
1.0
1.5
2.0
0.5
1.0
w (GeV)
w (GeV)
(a)
(b)
1.5
2.0
FIG. 2: (a) differential branching ratios for the B ± → K ± ρ0 → K ± π + π − decays, and (b) differential distributions of ACP in w for the B → Kρ → Kππ decays.
of the shape parameter of the B meson distribution amplitude ωB = 0.40 ± 0.04 GeV, at2 = −0.60 ± 0.20, the chiral 0 s scale associate with the kaon mK 0 = 1.6 ± 0.1 GeV, a2 = 0.25 ± 0.10, and a2 = 0.75 ± 0.25. The uncertainties from τB ± , τB 0 , the Gegenbauer moments of the kaon distribution amplitudes, and the Wolfenstein parameters in [98] are small and have been neglected. It is observed that the uncertainties of ACP are much smaller than those of B, and that the consistency between our results and the data is satisfactory. Examining the distributions of these branching ratios in the pion-pair invariant mass w, we find that the main portion of the branching ratios lies in the region around the pole mass of the ρ resonance as expected: the differential branching ratios of the B ± → K ± ρ0 → K ± π + π − decays in Fig. 2(a) exhibit peaks at the ρ meson mass. The central values of B are 1.78 × 10−6 and 2.46 × 10−6 for the B + → K + ρ0 → K + π + π − decay in the ranges of w, [mρ − 0.5Γρ , mρ + 0.5Γρ ] and [mρ − Γρ , mρ + Γρ ], respectively, which amount to 52% and 72% of B = 3.42 × 10−6 in Table I. The branching fraction 3.27 × 10−6 is accumulated in the range [2mπ , 1.5 GeV] for this mode. Figure 2(b) displays the differential distributions of ACP for the four B → Kρ → Kππ modes, in which a falloff of ACP with w is seen for B + → K + ρ0 → K + π + π − , B + → K 0 ρ+ → K 0 π + π 0 , and B 0 → K + ρ− → K + π − π 0 . It implies that the direct CP asymmetries in the above three quasi-two-body decays, if calculated as the two-body decays B → Kρ with the ρ resonance mass being fixed to mρ , may be overestimated. The ascent of the differential distribution of ACP with w for B 0 → K 0 ρ0 → K 0 π + π − implies that its direct CP asymmetry, if calculated in the two-body formalism, may be underestimated . To verify the above observation, we treat the B → Kρ → Kππ modes as the two-body decays B → Kρ in the PQCD approach [99] by imposing the replacement η → rρ2 for the momenta in Eqs. (2) and (3), with the mass ratio rρ = mρ /mB . Employing the same Gegenbauer moments a0,t,s for the ρ meson distribution amplitudes, we obtain 2 ( +0.40 t +0.42 −6 K +0.47 0 +0.25 s , B = (3.52+0.67 −0.45 (ωB )−0.34 (a2 )−0.38 (m0 )−0.43 (a2 )−0.24 (a2 )) × 10 (17) B + → K + ρ0 +0.02 +0.09 t +0.00 0 s ACP = 0.55−0.04 (ωB )−0.08 (a2 ) ± 0.03(mK ) (a ) ± 0.01(a ) , 0 −0.01 2 2 ( +1.69 t +1.04 −6 K +0.84 0 +0.43 s , B = (7.66+1.79 + 0 + −1.19 (ωB )−1.44 (a2 )−0.95 (m0 )−0.73 (a2 )−0.41 (a2 )) × 10 (18) B →K ρ +0.03 t K 0 ACP = 0.22 ± 0.03(ωB )−0.05 (a2 ) ± 0.01(m0 ) ± 0.00(a2 ) ± 0.00(as2 ) , ( +0.67 t +0.86 K +0.91 0 +0.42 s −6 B = (6.92+1.58 , 0 + − −1.04 (ωB )−0.53 (a2 )−0.81 (m0 )−0.80 (a2 )−0.40 (a2 )) × 10 B →K ρ (19) +0.01 +0.01 +0.00 +0.13 t +0.03 0 s ) . ) (a ACP = 0.34−0.01 (ωB )−0.12 (a2 )−0.02 (mK ) (a 0 −0.02 2 −0.02 2 ( +0.70 t +0.55 −6 s K +0.40 0 , B = (4.01+1.07 0 0 0 −0.71 (ωB )−0.63 (a2 )−0.50 (m0 )−0.35 (a2 ) ± 0.19(a2 )) × 10 (20) B →K ρ K +0.00 0 s t ACP = 0.04 ± 0.01(ωB ) ± 0.00(a2 ) ± 0.00(m0 )−0.01 (a2 ) ± 0.00(a2 ) . The comparison of Table I with Eqs. (17)-(20) confirms that the branching ratios of the four quasi-two-body modes in the three-body and two-body frameworks are close to each other. The tiny distinction between them suggests that the PQCD approach is a consistent theory for exclusive hadronic B meson decays. The total ACP for the decays B + → K + ρ0 → K + π + π − , B + → K 0 ρ+ → K 0 π + π 0 , and B 0 → K + ρ− → K + π − π 0 in Table I, compared with the corresponding values in Eqs. (17)-(19), have been, as explained above, moderated by the finite width of the ρ resonance appearing in the time-like form factor Fπ . Because ACP in Table I agree better with the data, it may be more appropriate to treat B → Kρ as three-body decays.
6 TABLE II: PQCD predictions for the CP averaged branching ratios and the direct CP asymmetries of the B → Kρ′ → Kππ decays. Mode B + → K + ρ′0 → K + π + π −
B (10−7 ) ACP B (10−7 ) ACP B (10−7 ) ACP B (10−7 ) ACP
B + → K 0 ρ′+ → K 0 π + π 0 B 0 → K + ρ′− → K + π − π 0 B 0 → K 0 ρ′0 → K 0 π + π −
Results +0.81 t +0.59 s +0.40 K +0.13 0 4.32+1.17 −0.99 (ωB )−0.79 (a2 )−0.64 (a2 )−0.46 (m0 )−0.17 (a2 ) +0.06 K s +0.02 t +0.01 0.32−0.04 (ωB ) ± 0.03(a2 )−0.02 (a2 )−0.01 (m0 ) ± 0.01(a02 ) +3.14 t +1.26 s +1.13 K +0.42 0 10.37+3.72 −2.36 (ωB )−2.71 (a2 )−1.03 (a2 )−0.92 (m0 )−0.37 (a2 ) +0.03 +0.02 0 0.12 ± 0.02(ωB )−0.01 (at2 )−0.02 (as2 ) ± 0.01(mK 0 ) ± 0.01(a2 ) +1.32 +2.37 0 K +0.26 s +0.86 t +1.17 7.61−1.90 (ωB )−1.03 (a2 )−0.88 (a2 )−0.75 (m0 )−0.22 (a2 ) t +0.00 s K 0 0.27+0.02 −0.01 (ωB ) ± 0.06(a2 )−0.01 (a2 ) ± 0.02(m0 ) ± 0.01(a2 ) +1.11 +1.82 0 K +0.14 t s +0.48 4.84−1.32 (ωB )−1.05 (a2 ) ± 0.50(a2 )−0.46 (m0 )−0.16 (a2 ) +0.02 t s K 0 0.08+0.00 −0.01 (ωB )−0.00 (a2 ) ± 0.01(a2 ) ± 0.01(m0 ) ± 0.01(a2 )
10
K+[ K+[
] ]
1
/dw
(10
-5
GeV
-1
)
B+ B+
d
0.1
0.01 0.5
1.0
1.5
2.0
2.5
w (GeV)
FIG. 3: CP averaged differential branching ratios for the decays B + → K + ρ0 → K + π + π − and B + → K + ρ′0 → K + π + π − .
The parametrization of the time-like pion form factor in Eq. (11) also allows to single out the the ρ′ (1450) component. Adopting the two-pion distribution amplitudes in Eqs. (8)-(10), we derive the CP averaged branching ratios and the direct CP asymmetries for the decays B + → K + ρ′0 → K + π + π − , B + → K 0 ρ′+ → K 0 π + π 0 , B 0 → K + ρ′− → K + π − π 0 , and B 0 → K 0 ρ′0 → K 0 π + π − listed in Table II, whose errors have the same sources as in Table I. We compare the differential branching ratios for the B + → K + ρ0 → K + π + π − and B + → K + ρ′0 → K + π + π − decays in Fig. 3, whose difference is mainly governed by the corresponding BW functions. All these predictions can be confronted with data in the future. To extract the branching ratios for the two-body decays B → Kρ′ from the quasi-two-body ones B → Kρ′ → Kππ, we need the branching fraction for ρ′ → ππ, which is inferred from the ratio of the widths, Γππ /Γρ′ . The width Γππ for ρ′ → ππ was evaluated in the Nambu-Jona-Lasinio quark model, and found to be 22 MeV [111], consistent with 17 ∼ 25 MeV obtained from the e+ e− annihilation data [112]. Taking Γρ′ = 0.311 ± 0.062 GeV [112], we get the branching fraction B(ρ′ → ππ) = 4.56% ∼ 10.0%. The ρ′ → ππ branching fraction can be also estimated from the relation [113] Γρ′ →ππ =
2 3 p→ gρ2′ ππ |− π (mρ′ )| . 2 6π mρ′
(21) ′
The coupling gρ′ ππ is read off the ρ′ component of the time-like form factor Fπ in Eq. (11) according to Fπρ (w2 ) ≈ gρ′ ππ wfρ′ /Dρ′ (w2 ) at w = mρ′ , which is similar to Eq. (12) for the ρ component. We adopt the decay constant fρ′ = 0.185+0.030 −0.035 GeV resulting from the data Γρ′ →e+ e− = 1.6 ∼ 3.4 keV [112], which agrees with fρ′ = (0.182 ± 0.005) GeV from the perturbative analysis in the large-Nc limit [114], fρ′ = (0.186 ± 0.014) GeV from the double-pole QCD sum rules [115], and fρ′ = 0.128 GeV from the relativistic constituent quark model [116]. Equation (21) then yields ′ B(ρ′ → ππ) = 10.04+5.23 −2.61 %, compatible with B(ρ → ππ) = 4.56% ∼ 10.0% from the width ratio Γππ /Γρ′ .
7 With B(ρ′ → ππ) = 10.04%, we extract the B → Kρ′ branching ratios from Table II (in units of 10−6 ), +0.80 t +0.59 s +0.40 K +0.13 0 B(B + → K + ρ′0 ) = 4.30+1.16 −0.99 (ωB )−0.79 (a2 )−0.64 (a2 )−0.45 (m0 )−0.17 (a2 ) ,
(22)
B(B 0 → K + ρ′− ) =
(24)
+
0 ′+
B(B → K ρ ) = 0
0 ′0
B(B → K ρ ) =
+3.13 t +1.26 s +1.13 K +0.41 0 10.33+3.71 −2.35 (ωB )−2.70 (a2 )−1.03 (a2 )−0.92 (m0 )−0.37 (a2 ) , +1.31 t +1.16 s +0.86 K +0.26 0 7.57+2.36 −1.89 (ωB )−1.03 (a2 )−0.87 (a2 )−0.74 (m0 )−0.22 (a2 ) , +1.11 t s +0.47 K +0.14 0 4.82+1.82 −1.31 (ωB )−1.04 (a2 ) ± 0.50(a2 )−0.46 (m0 )−0.16 (a2 ) .
(23) (25)
Note that the data B(B 0 → K + ρ′− ) = (2.4 ± 1.0 ± 0.6) × 10−6 from BaBar [117] by assuming B(ρ′ → ππ) ≈ 100% is 0 + − much larger than Eq. (24) based on B(ρ′ → ππ) = 4.56% ∼ 10.0% or 10.04+5.23 −2.61 %, and the data B(B → K ρ ) = −6 (6.6 ± 0.5 ± 0.8) × 10 in Ref [117]. The branching ratios and the direct CP asymmetries of the quasi-two-body decays B → K(ω, ρ′′ , ρ′′′ ) → Kππ can be predicted by singling out the corresponding components in the time-like form factor Fπ in principle, since the Gegenbauer moments of the P -wave two-pion distribution amplitudes have been determined. This is a merit of our PQCD formalism for three-body hadronic B meson decay. Besides, we can extract, for example, the B → Kω branching ratios from the predictions for the B → Kω → Kππ modes, given the ω → ππ branching fraction. We will leave the above observables to future studies. IV.
CONCLUSION
In this paper we have applied the PQCD approach to the quasi-two-body decays B → Kρ → Kππ, which were analyzed in both three-body and two-body factorization formalisms. In the former strong dynamics between the P -wave resonances and the pion pair, including two-pion final-state interactions, is parametrized into the two-pion distribution amplitudes. The advantage of this approach is that the time-like pion form factor Fπ involved in the twopion distribution amplitudes accommodates both resonant and nonresonant contributions. Inputting Fπ extracted from the e+ e− annihilation data, we have calculated the branching ratios and the direct CP asymmetries of the B → Kρ → Kππ modes, whose agreement with the data was achieved by tuning the Gegenbauer moments of the P -wave two-pion distribution amplitudes. The consistency between the three-body and two-body analyses of the B → Kρ → Kππ branching ratios was verified, which supports the PQCD approach to exclusive hadronic B meson decays. The comparison to the results from the two-body framework indicates that the direct CP asymmetries of the B → Kρ → Kππ modes have been moderated by the finite width of the ρ resonance, and become closer to the data. It suggests that the three-body framework is more appropriate for studying quasi-two-body hadronic B meson decays. The contribution from the ρ′ intermediate state was simply singled out from the given time-like form factor Fπ in our formalism. Using the determined Gegenbauer moments of the P -wave two-pion distribution amplitudes, we have predicted the branching ratios and the direct CP asymmetries of the B → Kρ′ → Kππ channels, and compared their differential branching ratios with the B → Kρ → Kππ ones. With the estimated ρ′ → ππ branching fraction, the two-body B → Kρ′ branching ratios have been extracted from the results for the B → Kρ′ → Kππ decays. All these predictions can be confronted with future data. The same framework is applicable straightforwardly to other channels B → K(ω, ρ′′ , ρ′′′ ) → Kππ in principle. Moreover, with both the S-wave and P -wave distribution amplitudes being ready, we will proceed to predictions for differential branching ratios and direct CP asymmetries of three-body hadronic B meson decays in various localized regions of two-pion phase space in a forthcoming paper. Acknowledgments
We thank Prof. H.-Y. Cheng for valuable discussions. This work was supported in part by National Science Foundation of China under Grant No. 11547038, and by the Ministry of Science and Technology of R.O.C. under Grant No. MOST-104-2112-M-001-037-MY3.
8 Appendix A: DECAY AMPLITUDES
The quasi-two-body B → Kρ → Kππ decay amplitudes are given, in the PQCD approach, by C2 C1 GF LL LL ∗ LL + C + M + M + C Vub Vus FTLL + F + C2 FTLL A B + → K + [ρ0 →]π + π − = 1 1 Tρ Aρ K ρ Aρ 2 3 3 C9 C3 LL ∗ + C4 + + C10 FTLL + C2 MTLL ρ + FAρ K − Vtb Vts 3 3 C5 C7 LL LL SP + + C6 + + C8 FTSP ρ + FAρ + (C3 + C9 ) MT ρ + MAρ 3 3 3C7 C10 C8 3C9 LL LR F + FTLL + + + (C5 + C7 ) MTLR + M + TK K ρ Aρ 2 2 2 2 3C10 LL 3C8 SP + , (A1) MT K + MT K 2 2 GF C1 C3 C9 C10 ∗ LL LL A B + → K 0 [ρ+ →]π + π 0 = √ Vub Vus + C1 MAρ − Vtb∗ Vts FTLL + C2 FAρ + C4 − − ρ 3 3 6 2 2 C9 C7 C7 C8 C5 FTSP MTLL MTLR + C6 − − + ρ + C3 − ρ + C5 − ρ 3 6 2 2 2 C3 C5 C9 C7 LL SP LL + + + (C3 + C9 ) MAρ + C4 + + C10 FAρ + C6 + + C8 FAρ 3 3 3 3 LR , (A2) + (C5 + C7 ) MAρ C1 GF C3 C9 ∗ LL ∗ Vus A B 0 → K + [ρ− →]π 0 π − = √ Vub + C M − V V FTLL + C2 FTLL + C + + C 1 ts 4 10 ρ Tρ tb ρ 3 3 3 2 C7 C5 LL LR + C6 + + C8 FTSP + ρ + (C3 + C9 ) MT ρ + (C5 + C7 ) MT ρ 3 3 C3 C5 C9 C10 C7 C8 C9 LL SP LL + + C4 − − + C6 − − FAρ + FAρ + C3 − MAρ 3 6 2 3 6 2 2 C7 LR (A3) MAρ , + C5 − 2 1 A B 0 → K 0 [ρ0 →]π + π − = − √ A B + → K 0 [ρ+ →]π + π 0 + A B 0 → K + [ρ− →]π 0 π − 2 (A4) + A B + → K + [ρ0 →]π + π − ,
in which GF is the Fermi coupling constant, V ’s are the Cabibbo-Kobayashi-Maskawa matrix elements, and the amplitudes F (M ) denote the factorizable (nonfactorizable) contributions. It should be understood that the Wilson coefficients C and the amplitudes F and M appear in convolutions in momentum fractions and impact parameters b. With the ratio r = mK 0 /mB , the amplitudes from Fig. 1(a) are written as Z Z √ LL 4 FT ρ = 8πCF mB fK dxB dz bB dbB bdbφB (xB , bB )(1 − η) η(1 − 2z)(φs + φt ) + (1 + z)φ0 √ √ × E1ab (t1a )h1a (xB , z, bB , b) + η 2φs − ηφ0 E1ab (t1b )h1b (xB , z, bB , b) , (A5) Z Z √ √ 4 η(2 + z)φs − ηzφt + (1 + η(1 − 2z))φ0 FTSP dxB dz bB dbB bdbφB (xB , bB ) ρ = −16πCF mB rfK √ (A6) × E1ab (t1a )h1a (xB , z, bB , b) + 2 η(1 − xB + η)φs + (xB − 2η)φ0 E1ab (t1b )h1b (xB , z, bB , b) , Z Z p 4 MTLL dxB dzdx3 bB dbB b3 db3 φB (xB , bB )φA ρ = 32πCF mB / 2Nc K (1 − η) √ × ηz(φt − φs ) + ((1 − η)(1 − x3 ) − xB + zη)φ0 E1cd (t1c )h1c (xB , z, x3 , bB , b3 ) √ (A7) + z( η(φs + φt ) − φ0 ) − (x3 (1 − η) − xB )φ0 E1cd (t1d )h1d (xB , z, x3 , bB , b3 ) ,
9 Z Z p √ 4 T s t MTLR = −32πC rm / 2N ηz(φP dx dzdx bB dbB b3 db3 φB (xB , bB ) F c B 3 ρ B K − φK )(φ + φ ) √ T s t P T 0 P T 0 + η((1 − x3 )(1 − η) − xB )(φP K + φK )(φ − φ ) + ((1 − x3 )(1 − η) − xB )(φK + φK )φ + ηz(φK − φK )φ √ √ 0 T t s P T × E1cd (t1c )h1c (xB , z, x3 , bB , b3 ) + − ηz(φP K + φK )( ηφ + (φ + φ )) + (xB − x3 (1 − η))(φK − φK ) √ (A8) × ( η(φs − φt ) + φ0 ) E1cd (t1d )h1d (xB , z, x3 , bB , b3 ) ,
with the color factor CF = 4/3 and the kaon decay constant fK . The amplitudes from Fig. 1(b) are written as Z Z √ s t A 0 LL FAρ = 8πCF m4B fB dzdx3 bdbb3 db3 2r ηφP K ((2 − z)φ + zφ ) − (1 − η)(1 − z)φK φ E4ab (t4a ) √ s A 0 × h4a (z, x3 , b, b3 ) + 2r η[(1 − x3 )(1 − η)φTK − (1 + x3 + (1 − x3 )η)φP K ]φ + (x3 (1 − η) + η)(1 − η)φK φ × E4ab (t4b )h4b (z, x3 , b, b3 ) , (A9) Z Z √ SP s t P 0 FAρ = 16πCF m4B fB dzdx3 bdbb3 db3 η(1 − η)(1 − z)φA K (φ + φ ) − 2r(1 + (1 − z)η)φK φ E4ab (t4a ) 0 √ s P P T × h4a (z, x3 , b, b3 ) + 2 η(1 − η)φA K φ − r 2ηφK + x3 (1 − η)(φK − φK ) φ E4ab (t4b )h4b (z, x3 , b, b3 ) , (A10) Z Z p LL 0 MAρ = 32πCF m4B / 2Nc dxB dzdx3 bB dbB b3 db3 φB (xB , bB ) (η − 1)[x3 (1 − η) + xB + η(1 − z)]φA Kφ √ √ √ T s t P T s t P s T t + r η(x3 (1 − η) + xB + η)(φP K + φK )(φ − φ ) + r η(1 − z)(φK − φK )(φ + φ ) + 2r η(φK φ + φK φ ) √ 0 P T s t × E4cd (t4c )h4c (xB , z, x3 , bB , b3 ) + (1 − η 2 )(1 − z)φA K φ + r η(xB − x3 (1 − η) − η)(φK − φK )(φ + φ ) √ T s t (A11) − r η(1 − z)(φP K + φK )(φ − φ ) E4cd (t4d )h4d (xB , z, x3 , bB , b3 ) , Z Z p √ LR s t MAρ = −32πCF m4B / 2Nc dxB dzdx3 bB dbB b3 db3 φB (xB , bB ) η(1 − η)(1 + z)φA K (φ − φ ) + r(2 − xB √ T 0 P T 0 η(1 − η)(1 − z)φA − x3 (1 − η))(φP K K + φK )φ + rη(zφK − (2 + z)φK )φ E4cd (t4c )h4c (xB , z, x3 , bB , b3 ) + T P T 0 × (φs − φt ) + r((x3 (1 − η) − xB )(φP + φ ) + η((2 − z)φ + zφ ))φ E (t )h (x , z, x , b , b ) , 4cd 4d 4d B 3 B 3 K K K K (A12) with the B meson decay constant fB . The amplitudes from Fig. 1(c) are written as Z Z 4 P FTLL = 8πC m dx dx bB dbB b3 db3 φB (xB , bB ) (1 + x3 (1 − η))(1 − η)φA F B 3 K B K + r(1 − 2x3 )(1 − η)φK P + r(1 + η − 2x3 (1 − η))φTK E2ab (t2a )h2a (xB , x3 , bB , b3 ) + xB (η − 1)ηφA K + 2r(1 − η(1 − xB ))φK E2ab (t2b ) × h2b (xB , x3 , bB , b3 ) , (A13) Z Z p 4 dxB dzdx3 bB dbB bdbφB (xB , bB )φ0 (1 − xB − z)(1 − η 2 )φA MTLL K K = 32πCF mB / 2Nc T P T P − rx3 (1 − η)(φP K − φK ) + r(xB + z)η(φK + φK ) − 2rηφK E2cd (t2c )h2c (xB , z, x3 , bB , b) − (z − xB P T P T + x3 (1 − η))(1 − η)φA K + r(xB − z)η(φK − φK ) − rx3 (1 − η)(φK + φK ) E2cd (t2d )h2d (xB , z, x3 , bB , b) , (A14) Z Z p √ 4 MTLR dxB dzdx3 bB dbB bdbφB (xB , bB ) (1 − xB − z)(1 − η)(φs + φt )φA π = 32πCF mB η/ 2Nc K T s t P T + r(1 − xB − z)(φs + φt )(φP K − φK ) + r(x3 (1 − η) + η)(φ − φ )(φK + φK ) E2cd (t2c )h2c (xB , z, x3 , bB , b) s t P T s t P T − (z − xB )(1 − η)(φs − φt )φA K + r(z − xB )(φ − φ )(φK − φK ) + rx3 (1 − η)(φ + φ )(φK + φK ) E2cd (t2d ) × h2d (xB , z, x3 , bB , b) , (A15) Z Z p 4 dxB dzdx3 bB dbB bdbφB (xB , bB )φ0 (1 + η − xB − z + x3 (1 − η))(1 − η)φA MTSP K π = 32πCF mB / 2Nc T P T P + rη(xB + z)(φP K − φK ) − rx3 (1 − η)(φK + φK ) − 2rηφK E2cd (t2c )h2c (xB , z, x3 , bB , b) P T P T − (z − xB )(1 − η 2 )φA K − rx3 (1 − η)(φK − φK ) + rη(xB − z)(φK + φK ) E2cd (t2d )h2d (xB , z, x3 , bB , b) . (A16)
10 The amplitudes from Fig. 1(d) are written as Z Z LL 4 FAK = 8πCF mB fB dzdx3 bdbb3 db3 √ 0 P T P s × (x3 (1 − η) − 1)(1 − η)φA K φ + 2r η(x3 (1 − η)(φK − φK ) − 2φK )φ E3ab (t3a )h3a (z, x3 , b, b3 ) √ P 0 s t s t + z(1 − η)φA (A17) K φ + 2r ηφK ((1 − η)(φ − φ ) + z(φ + φ )) E3ab (t3b )h3b (z, x3 , b, b3 ) , Z Z SP FAπ = 16πCF m4B fB dzdx3 bdbb3 db3 √ s P T 0 P T 0 × 2 η(1 − η)φA K φ + r(1 − x3 )(φK + φK )φ + rη((1 + x3 )φK − (1 − x3 )φK )φ E3ab (t3a )h3a (z, x3 , b, b3 ) √ 0 A s t + 2r(1 − η(1 − z))φP (A18) K φ + z η((1 − η)φK (φ − φ ) E3ab (t3b )h3b (z, x3 , b, b3 ) , Z Z p 0 LL MAK = 32πCF m4B / 2Nc dxB dzdx3 bB dbB b3 db3 φB (xB , bB ) (η − 1)(−η + (1 + η)(xB + z))φA Kφ √ √ √ P s T s t P T s t + r η(x3 (1 − η) + η)(φP K + φK )(φ − φ ) + r η(1 − xB − z)(φK − φK )(φ + φ ) − 4r ηφK φ E3cd (t3c ) √ 0 P T s t × h3c (xB , z, x3 , bB , b3 ) + (1 − η)((1 − x3 )(1 − η) − η(xB − z))φA K φ − r η(xB − z)(φK + φK )(φ − φ ) √ T s t (A19) + r η(1 − η)(1 − x3 )(φP K − φK )(φ + φ ) E3cd (t3d )h3d (xB , z, x3 , bB , b3 ) , Z Z p √ LR s t MAπ = 32πCF m4B / 2Nc dxB dzdx3 bB dbB b3 db3 φB (xB , bB ) η(1 − η)(2 − xB − z)φA K (φ + φ ) − r(1 + x3 ) T 0 P T P T P 0 × (φP K − φK )φ − rη[(1 − xB − z)(φK + φK ) − x3 (φK − φK ) + 2φK ]φ E3cd (t3c )h3c (xB , z, x3 , bB , b3 ) √ √ T 0 A s t T 0 η(xB − z)[r η(φP − r(1 − η)(x3 − 1)(φP K + φK )φ + (1 − η)φK (φ + φ )] E3cd (t3d ) K − φK )φ + × h3d (xB , z, x3 , bB , b3 ) . (A20)
The hard functions hiα , the hard scales tiα , and the evolution factors Eiab and Eicd , with i = 1, 2, 3, 4 and α = a, b, c, d, have their explicit expressions in the Appendix of Ref. [54]. Since the Legendre polynomial P1 (2ζ − 1) in the P -wave two-pion distribution amplitudes appears as an overall factor in decay amplitudes, the integration over ζ can be R1 performed trivially, yielding a factor 0 dζ(2ζ − 1)2 = 1/3 to branching ratios.
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