Quasi-two-body decays $B\to K\rho\to K\pi\pi$ in perturbative QCD approach

We analyze the quasi-two-body decays $B\to K\rho\to K\pi\pi$ in the perturbative QCD (PQCD) approach, in which final-state interactions between the pions in the resonant regions associated with the $P$-wave states $\rho(770)$ and $\rho^\prime(1450)$ 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 $B\to K\rho(770),K\rho^\prime(1450)\to K\pi\pi$ 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 $B\to K\rho(770)\to K\pi\pi$ decays supports the PQCD factorization framework for exclusive hadronic $B$ meson decays.

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 p B and the light spectator quark momentum k B as in the light-cone coordinates, with m B being the B meson mass and x B 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 p 3 (in the minus z direction) and the associated non-strange quark momentum k 3 as with the variable η = w 2 /m 2 B , w = p 2 being the invariant mass of the resonant state, and the momentum fractions z and x 3 . The momenta p 1 and p 2 for the two pions from the resonant state have the components [54] 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][62][63] φ I vν (z, ζ, w 2 ) = where N c is the number of colors, n − = (0, 1, 0 T ) is a dimensionless vector, T = τ 3 /2 is chosen for the isovector I = 1 state, ψ represents the u-d quark doublet, and f ⊥ 2π is a normalization constant. For I = 1, the P -wave is the leading partial wave, to which φ I=1 vν=− and φ I=1 tν=⊥ contribute at twist 2, and φ I=1 vν=⊥ , φ I=1 s , and φ I=1 tν=+ contribute at twist 3. With w 2 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 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 with the Gegenbauer polynomial C 3/2 2 (t) = 3 5t 2 − 1 /2 and the Legendre polynomial P 1 (2ζ − 1) = 2ζ − 1. In principle, the time-like form factors associated with the second Gegenbauer moments a 0,t,s 2 can differ from F π,s,t (w 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 a 0,t,s 2 will be regarded as free parameters and determined in this work. Up to the second Gegenbauer terms, the Legendre polynomial P 3 (2ζ − 1) also contributes. However, more unknown form factors will be introduced, and currently available data are not sufficient for their extraction.
The quasi-two-body decays B → Kρ → Kππ can be also analyzed in an alternative approach based on two-body decays: the quark pair qq from a hard decay kernel forms the ρ meson, followed by its BW propagator, and then by the ρ → ππ transition with the strength g ρππ . The equivalence between the framework with the ρ meson propagator and the present one with the two-pion distribution amplitudes hints the relation, where F ρ π represents the ρ component of Eq. (11), f ρ is the ρ meson decay constant, and D ρ 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, F ρ s,t (w 2 ) ≈ g ρππ wf T ρ /D ρ (w 2 ), in which the decay constant f T ρ 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 F s,t (w 2 ) 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] where the hard kernel H contains only one hard gluon exchange at leading order in the strong coupling α s as in the two-body formalism, the symbol ⊗ means convolutions in parton momenta, and the B meson (kaon, two-pion) distribution amplitude φ B (φ K , φ I=1 ππ ) absorbs nonperturbative dynamics in the decay processes. We then have their differential branching fractions [98] dB τ B being the B meson mean lifetime. The magnitudes of the pion and kaon momenta, | − → p π | and | − → p K |, are written, in the center-of-mass frame of the pion pair, as with the pion mass m π and the kaon mass m K . 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.
The decay constant f ρ has been extracted from the τ ± → ρ ± ν τ decay rate for the charged ρ ± meson and from ρ 0 → e + e − for the neutral ρ 0 meson. In this work we take their arithmetic average value f ρ = (0.216 ± 0.003) GeV [100,101]. The decay constant f T ρ has been computed in lattice QCD [102][103][104][105], for which we choose f T ρ = 0.184 GeV [102]. The ratio f T ρ /f ρ then determines the ratios F s,t /F π postulated in the previous section. The B meson and kaon distribution amplitudes are the same as widely adopted in the PQCD approach [54,[106][107][108].
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 a 0 2 = 0.25, a s 2 = 0.75, and a t 2 = −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 (A CP ) for the Table I. The theoretical uncertainties come from the variations of the shape parameter of the B meson distribution amplitude ω B = 0.40 ± 0.04 GeV, a t 2 = −0.60 ± 0.20, the chiral scale associate with the kaon m K 0 = 1.6 ± 0.1 GeV, a 0 2 = 0.25 ± 0.10, and a s 2 = 0.75 ± 0.25. The uncertainties from τ B ± , τ B 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 A CP 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 ± ρ 0 → 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 + ρ 0 → K + π + π − decay in the ranges of w, [m ρ − 0.5Γ ρ , m ρ + 0.5Γ ρ ] and [m ρ − Γ ρ , m ρ + Γ ρ ], 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 [2m π , 1.5 GeV] for this mode. Figure 2(b) displays the differential distributions of A CP for the four B → Kρ → Kππ modes, in which a falloff of A CP with w is seen for B + → K + ρ 0 → 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 m ρ , may be overestimated. The ascent of the differential distribution of A CP with w for B 0 → K 0 ρ 0 → 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 → r 2 ρ for the momenta in Eqs. (2) and (3), with the mass ratio r ρ = m ρ /m B . Employing the same Gegenbauer moments a 0,t,s 2 for the ρ meson distribution amplitudes, we obtain 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 A CP for the decays B + → K + ρ 0 → 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 F π . Because A CP in Table I agree better with the data, it may be more appropriate to treat B → Kρ as three-body decays.
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 F π 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 F π involved in the twopion distribution amplitudes accommodates both resonant and nonresonant contributions. Inputting F π 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 F π 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.

Appendix A: DECAY AMPLITUDES
The quasi-two-body B → Kρ → Kππ decay amplitudes are given, in the PQCD approach, by in which G F 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 = m K 0 /m B , the amplitudes from Fig. 1(a) are written as with the color factor C F = 4/3 and the kaon decay constant f K . The amplitudes from Fig. 1(b) are written as The amplitudes from Fig. 1(d) are written as The hard functions h iα , the hard scales t iα , and the evolution factors E iab and E icd , 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 P 1 (2ζ − 1) in the P -wave two-pion distribution amplitudes appears as an overall factor in decay amplitudes, the integration over ζ can be performed trivially, yielding a factor 1 0 dζ(2ζ − 1) 2 = 1/3 to branching ratios.