Will the subprocesses $\rho(770,1450)^0\to K^+K^-$ contribute large branching fractions for $B^\pm \to \pi^\pm K^+K^-$ decays?

We analyze the quasi-two-body decays $B^+\to\pi^+\rho(770,1450)^0 \to\pi^+ K^+K^-$ in the perturbative QCD approach. The results in this work do not support that large branching fractions contributed by the resonances $\rho(770,1450)^0$ in the $B^\pm \to \pi^\pm K^+K^-$ decays. The virtual contribution for $K^+K^-$ from the tail of the resonance $\rho(770)^0$ which has been ignored in the experimental studies is about $1.5$ times of the $\rho(1450)^0 \to K^+K^-$ contribution, with the predicted branching fractions ${\mathcal B}_v=(1.31\pm0.27)\times10^{-7}$ and ${\mathcal B}=(8.96\pm2.61)\times10^{-8}$, respectively, for these two subprocesses in the $B^\pm \to \pi^\pm K^+K^-$ decays. The absence of $\rho(770)^0\to K^+K^-$ for the decay amplitude of a three-body hadronic $B$ decay involving charged kaon pair could probably result in a larger proportion for the resonance $\rho(1450)^0$ in the experimental analysis.


I. INTRODUCTION
Strong dynamics of the charmless three-body hadronic B meson decays is related to the short-distance processes like in the two-body cases, and also involves the hadronhadron interactions, the three-body effects [1,2] and the rescattering processes [3,4] in the final states. In experimental studies, the strong interactions along with the weak processes of a certain three-body B meson decay are always described in the decay amplitude as the coherent sum of the resonant and nonresonant contributions in the isobar formalism [5][6][7]. The isobar expression with or without certain resonances would certainly have impacts on the fit fractions, and then influence the observables such as the branching ratios and CP violations for the resonant and the nonresonant contributions of the threebody process in view of the explicit distribution of the experimental events in the Dalitz plot [8].
We will analyze the ρ(770, 1450) 0 → K + K − contributions for the three-body decays B ± → π ± K + K − with the quasi-two-body framework based on the perturbative QCD (PQCD) approach [32][33][34][35]. The final state interaction effect was found to be suppressed for the ρ 0 → K + K − process [36] and will be neglected in the calculation in this work. For the quasi-twobody decays B ± → π ± ρ 0 → π ± K + K − , the intermediate states ρ(770) 0 and ρ(1450) 0 are generated in the hadronization of the light quark-antiquark pair qq, with q=(u, d), as demonstrated in the Fig. 1. The subprocesses ρ(770, 1450) 0 → K + K − , which can not be calculated in the PQCD approach, could be introduced into the distribution amplitudes of the kaon pair system by the time-like form factor of kaon. The quasi-two-body framework based on PQCD approach has been discussed in detail in [37], and has been adopted in some works for the quasi-two-body B meson decays [38][39][40][41][42][43] recently. This paper is organized as follows: In Sec. II, we briefly review the electromagnetic time-like form factor for the charged kaon, we introduce the P -wave K + K − system distribution amplitudes and give the expression of differential branching fraction for the quasi-two-body decays B ± → π ± ρ 0 → π ± K + K − . In Sec. III, we provide numerical results for the concerned decay processes and give some necessary discussions. The conclusions are presented in Sec. IV.
FIG. 1. Feynman diagrams for the processes B + → π + ρ 0 → π + K + K − , with ⊗ is the weak vertex, × denotes possible attachments for the hard gluons and the rectangle represents the resonance ρ(770) 0 and its excited states.

II. FRAMEWORK
In the light-cone coordinates, the momentum p B and light spectator quark's momentum k B for B + are defined as in the rest frame of B meson, with m B the mass and x B the momentum fraction. For the resonance ρ, its momentum p = mB √ 2 (η, 1, 0) with η = s/m 2 B and s = p 2 . The light spectator quark comes from B + and goes into resonance in the hadronization of ρ as shown in Fig. 1 (a) got the momentum k = (0, mB √ 2 z, k T ). For the bachelor final state pion and its spectator quark, the momenta p 3 and k 3 have the definitions as With the momentum fractions x B , x 3 and z run from zero to one. The time-like form factor F K for the charged kaon is related to the electromagnetic form factor and defined by [44] with the squared invariant mass s = (p 1 + p 2 ) 2 for kaon pair and the constraint F K (0) = 1. The electromagnetic current j em γ µ s is carried by the light quarks [45]. With the BW formula for the resonances ω and φ and the Gounaris-Sakurai (GS) model [46] for ρ, we have the time-like form factor [44] where the means the summation for the resonances ρ, ω or φ and their corresponding excited states, the explicit expressions and auxiliary functions for BW and GS are referred to Refs. [46,47]. The parameters c K ρ(ω,φ) have been fitted to the data in Refs [44,48,49].
For the concerned subprocesses ρ(770, 1450) 0 → K + K − , the P -wave K + K − system distribution amplitudes are organized into [50] φ P -wave The momentum p = p 1 + p 2 , and ǫ L is the longitudinal polarization vector for the resonances. We have the distribution amplitudes as [37] φ where the F ρ K is the ρ term of Eq. (4), the Gegenbauer polynomial C  [37] for the channel ρ 0 → π + π − . One has the differential branching fractions (B) for the quasi-two-body decays B ± → π ± ρ 0 → π ± K + K − as [22,51,52] where τ B being the B meson mean lifetime. It should be noted that the Eqs. (6)- (8) and Eq. (9) are slightly different from the corresponding expressions in [37]. These differences are caused by the introduction of the Zemach tensor −2 − → qπ · − → qK [53] in this work as did in Refs. [51,52], this tensor is employed to describe angular distribution for the decay of the spin 1 resonances. The magnitudes of the kaon and pion momenta | − → q K | and | − → q π | are written, in the center-of-mass frame of the kaon pair, as with the pion mass m π and the kaon mass m K .
The decay amplitudes A for the quasi-two-body decays B + → π + ρ(770, 1450) 0 → π + K + K − dependent only on the quark structures of the hadronic matrix elements for B + to π + ρ(770, 1450) 0 transitions and have the same expressions as the decays B + → π + ρ(770, 1450) 0 → π + π + π − in [40,41] except the replacement of the form factor F π → F K . The amplitudes in A according to the diagrams in Fig. 1 could be found in the Appendix of [37].

III. RESULTS
In the numerical calculation, we adopt the decay constant f B = 0.189 GeV [54] and the mean lifetime τ = (1.638 ± 0.004) × 10 −12 s [10] for the B + meson. The masses and the decay constant for the relevant particles in the numerical calculation in this work, the full widths for ρ(770) and ρ(1450), and the Wolfenstein parameters of the CKM matrix are presented in Table I.  The ratio between the f T ρ and f ρ for ρ(770) has been computed in lattice QCD, we choose the result f T ρ /f ρ = 0.687 at the scale µ = 2 GeV [55] for our numerical calculation. The decay constant f ρ for ρ(770) can be extracted from the processes τ − → ρ − ν τ and ρ 0 → e + e − , we take the value 0.216 GeV [56]. The c K ρ(770) was fitted to be 1.139 ± 0.010 and 1.195 ± 0.009 with the unconstrained and constrained fit procedure in [44], respectively, which are consistent with the values 1.138 ± 0.011 and 1.120 ± 0.007 in [48]. We employ the result 1.139 ± 0.010 for the quasi-two-body decay B + → π + ρ(770) 0 → π + K + K − . As for the c K for ρ(1450), we adopt the value −0.124 ± 0.012 [44] for the subprocess ρ(1450) 0 → K + K − , which is close to the constrained fit result−0.112 ± 0.010 in [44] and the unconstrained fit result −0.107 ± 0.010 in [48].
Utilizing the differential branching fractions the Eq. (9), we have the branching fractions The subscript v for B of B ± → π ± ρ(770) 0 → π ± K + K − means the virtual contribution [21,22] which is integrated of the Eq. (9) from the threshold of kaon pair. The first error for the two branching fractions above comes from the uncertainty of the Gegenbauer moments a s 2 , a 0 2 and a t 2 in the K + K − system distribution amplitudes, the second error is induced by the shape parameter ω B = 0.40 ± 0.04 for B + , the third one is contributed by the chiral mass m π 0 = 1.40 ± 0.10 GeV and the Gegenbauer moment a π 2 = 0.25 ± 0.15 for pion and the fourth one due to the variation of c K in the form factor F K . There are other errors come from the uncertainties of the Wolfenstein parameters of the CKM matrix, the parameters in the distribution amplitudes of bachelor pion, the masses and the decay constants of the initial and final states, etc. are small and have been neglected.
The virtual contribution from the BW tail of ρ(770) 0 for K + K − , which has not been taken into the decay amplitudes of the charmless three-body B meson decays involving kaon pair, was found about 1.5 times of the contribution from the resonance ρ(1450) 0 in this work, and is roughly 2.5% of the total branching fraction for the three-body decays B ± → π ± K + K − .
We found a peak, which is generated mainly by the tail of the BW formula and the phase space factors, at about 1.35 GeV of the invariant mass of kaon pair for B ± → π ± ρ(770) 0 → π ± K + K − . And this peak will disappear and the enhancement round 1.4 GeV will be the result when we investigated the total contributions from ρ(1450) 0 together with ρ(770) 0 . This means that the absence of ρ(770) 0 → K + K − in the decay amplitude of a three-body B decays could probably result in a larger proportion for the resonance ρ(1450) 0 in experimental amplitude analysis.