Contributions for the kaon pair from $\rho(770)$, $\omega(782)$ and their excited states in the $B\to K\bar K h$ decays

We study the resonance contributions for the kaon pair originating from the intermediate states $\rho(770,1450,1700)$ and $\omega(782,1420,1650)$ for the three-body hadronic decays $B\to K\bar K h$ in the perturbative QCD approach, where $h=(\pi, K)$. The branching fractions of the virtual contributions for $K\bar K$ from the Breit-Wigner formula tails of $\rho(770)$ and $\omega(782)$ which have been ignored in experimental and theoretical studies for these decays are found larger than the corresponding contributions from the resonances $\rho(1450,1700)$ and $\omega(1420,1650)$. The differential branching fractions for $B\to \rho(770) h\to K\bar K h$ and $B\to\omega(782) h \to K\bar K h$ are found nearly unaffected by the quite different values of the full widths for $\rho(770)$ and $\omega(782)$ in this paper. The predictions in this work for the branching fractions of the quasi-two-body decays $B^+\to \pi^+ \rho(1450)^0\to \pi^+K^+K^-$ and $B^+\to \pi^+ \rho(1450)^0\to \pi^+\pi^+\pi^-$ meet the requirement of $SU(3)$ symmetry relation.


I. INTRODUCTION
Charmless three-body hadronic B meson decays provide us a field to investigate different aspects of weak and strong interactions. The underlying weak decay for b-quark is simple which can be described well by the effective Hamiltonian [1], but the strong dynamics in these three-body processes is very complicated, owing to the hadron-hadron interactions, the threebody effects [2,3] and the rescattering processes [4][5][6][7] in the final states, and also on account of the resonant contributions which are related to the scalar, vector and tensor resonances and are commonly described by the relativistic Breit-Wigner (BW) formula [8] as well as the nonresonant contributions which are the rest at the amplitude level for the relevant decay processes. The experimental efforts for the three-body B decays by employing Dalitz plot technique [9] within the isobar formalism [10][11][12] have revealed valuable information on involved strong and weak dynamics. But a priori model with all reliable and correct strong dynamical components is needed for the Dalitz plot analyses [13]. The expressions of the decay amplitudes for those three-body decays without or have wrong factors for certain intermediate states will have negative impacts on the observables such as the branching fractions and CP violations for the relevant decay processes.
This paper is organized as follows. In Sec. II, we review the kaon vector time-like form factors, which are the crucial inputs for the quasi-two-body framework within PQCD and decisive for the numerical results of this work. In Sec. III, we give a brief introduction of the theoretical framework for the quasi-two-body B meson decays within PQCD approach. In Sec. IV, we present our numerical results of the branching fractions and direct CP asymmetries for the quasi-two-body decays B → ρ(770, 1450, 1700)h → KKh and B → ω(782, 1420, 1650)h → KKh, along with some necessary discussions. Summary of this work is given in Sec. V. The wave functions and factorization formulae for the related decay amplitudes are collected in the Appendix.

II. KAON TIME-LIKE FORM FACTORS
The electromagnetic form factors for the charged and neutral kaon are important for the precise determination of the hadronic loop contributions to the anomalous magnetic moment of the muon and the running of the QED coupling to the Z boson mass [43,101,102] and are also valuable for the measurements of the resonance parameters [38,40,41,43,46,49,50]. The kaon electromagnetic form factors have been extensively studied in Refs. [54,[103][104][105][106] on the theoretical side. Up to now the experimental information on these form factors comes from the measurements of the reactions e + e − → K + K − [38,39,44] and e + e − → K + K − (γ) [41]. Since KK is not an eigenstate of isospin, both isospin 0 and 1 resonances need to be considered in components of the form factors of kaon [41]. The combined analysis of the e + e − → K + K − and e + e − → K S K L cross sections and the spectral function in the τ − → K − K 0 ν τ decay allows one to extract the isovector and isoscalar electromagnetic form factors for kaons [107].
The vector time-like form factors for charged and neutral kaons are defined by the matrix elements [85,108] with the invariant mass square s = p 2 and the KK system momentum p = p 1 + p 2 . These two form factors F q K + K − and F q K 0K 0 can be related to kaon electromagnetic form factors F K + and F K 0 , which are defined by [104] and have the forms [104] F K + (s) = + 1 2 ι=ρ,ρ ′ ,...
with the electromagnetic current j em µ = 2 3ū γ µ u − 1 3d γ µ d − 1 3s γ µ s carried by the light quarks u, d and s [109]. The BW formula in F K + (s) and F K 0 (s) has the form [20,110] where the s-dependent width is given by The Blatt-Weisskopf barrier factor [111] with barrier radius r R BW = 4.0 GeV −1 [20] is given by The magnitude of the momentum and the | − → q 0 | is | − → q | at s = m 2 R . One should note thatcγ µ c can also contribute to F K + and F K 0 in the high-mass region [41,112,113] and the BW formula for the ρ family could be replaced with the Gounaris-Sakurai (GS) model [114] as in Refs. [104,106,115]. The F K + and F K 0 can be separated into the isospin I = 0 and I = 1 components as [70,104]. When concern only the contributions for K + K − and K 0K 0 from the resonant states ι = ρ(770, 1450, 1700) and ς = ω(782, 1420, 1650), we have [85] For the K +K 0 and K 0 K − pairs which have no contribution from the neutral resonances ω(782, 1420, 1650), we have [54,103,104] One should note that the different constants in Eqs. The c K R (with R = ι, ς, κ) is proportional to the coupling constant g RKK , and the coefficients have the constraints [107] ι=ρ,ρ ′ ,...
The results for c K ρ(1700) vary dramatically in Table I, from −0.015∓0.022 [104] to −0.234± 0.013 [107]. A reliable reference value should come from the measurements of F π rather than the result deduced from Eq. (16) since ρ(1700) is believed to be a 1 3 D 1 state in ρ family [15,122,124]. With Eq. (17) and the replacement ρ(1450) → ρ(1700) one has |c K ρ(1700) | ≈ 0.081 with the result |c ρ ′′ | = 0.068 for F π in [115]. The difference between the |c K ρ(1700) | and |c ρ ′′ | is induced by the differences of the BW and GS models and the different definitions for them. Then we adopt the fitted result −0.083 ∓ 0.019 for c K ρ(1700) [104] in the numerical calculation in this work. As for the coefficient c K ω(1650) , we employ the value −0.083 ∓ 0.019 of the constrained fits in [104] because of insufficiency of the knowledge for the properties of ω(1650).

III. KINEMATICS AND DIFFERENTIAL BRANCHING FRACTION
In the light-cone coordinates, the momentum p B for the initial state B + , B 0 or B 0 s with the mass m B is written as p B = m B √ 2 (1, 1, 0 T ) in the rest frame of B meson. In the same coordinates, the bachelor state pion or kaon in the concerned processes has the momentum p 3 = m B √ 2 (1 − ζ, 0, 0 T ), and its spectator quark has the momentum For the resonances ρ, ω and their excited states, and the KK system generated from them by the strong interaction, we have the momentum p = m B x, k T ) before and after it pass through the hard gluon vertex. The x B , x and x 3 , which run from zero to one in the numerical calculation, are the momentum fractions for the B meson, the resonances and the bachelor final state, respectively.
For the P -wave KK system along with the subprocesses ρ → KK and ω → KK, the distribution amplitudes are organized into [21,68,71] where F K is employed as the abbreviation of the vector time-like form factors in Eqs. (11)-(13) and gain different component for different resonance contribution from to the expressions of the Eqs. (11)- (13) in the concerned decay processes. Moreover, we have factored out the normalisation constant C X to make sure the the proper normalizations for the time-like form factors for kaon, and C X are given by The Gegenbauer polynomial C 3/2 2 (χ) = 3 (5χ 2 − 1) /2 for the distribution amplitudes φ 0 and φ t , and the Gegenbauer moments have been catered to the data in Ref. [68] for the quasi-two-body decays B → Kρ → Kππ. Within flavour SU(2) symmetry, we adopt the same Gegenbauer moments for the P -wave KK system originating from the intermediate states ω and ρ in this work. The vector time-like form factors F t K and F s K for the twist-3 distribution amplitudes are deduced from the relations F t,s [68] with the result f T ρ /f ρ = 0.687 at the scale µ = 2 GeV [125]. The relation [116] is employed because of the lack of a lattice QCD determination for f T ω . In PQCD approach, the factorization formula for the decay amplitude A of the quasitwo-body decays B → ρh → KKh and B → ωh → KKh is written as [126,127] according to Fig. 1 at leading order in the strong coupling α s . The hard kernel H here contains only one hard gluon exchange, and the symbol ⊗ means convolutions in parton momenta. For the B meson and bachelor final state h in this work, their distribution amplitudes φ B and φ h are the same as those widely adopted in the PQCD approach, we attach their expressions and parameters in the Appendix A.
For the CP averaged differential branching fraction (B), one has the formula [15,21,84] where τ B is the mean lifetime for B meson. The magnitude of the momentum | − → q h | for the state h in the rest frame of the intermediate states is written as with m h the mass for the bachelor meson pion or kaon. When m K = mK, the Eq. (10) has a simpler form Note that the cubic | − → q | and | − → q h | in Eq. (24) are caused by the introduction of the Zemach tensor −2 − → q · − → q h which is employed to describe the angular distribution for the decay of spin 1 resonances [128]. The direct CP asymmetry A CP is defined as The Lorentz invariant decay amplitudes according to Fig. 1 for the decays B → ρh → KKh and B → ωh → KKh are given in the Appendix B.
Utilizing the differential branching fractions the Eq. (24) and the decay amplitudes collected in the Appendix B, we obtain the CP averaged branching fractions and the direct CP asymmetries in Tables III, IV Table II. The uncertainties of c K ρ(770) = 1.247 ± 0.019, c K ω(782) = 1.113 ± 0.019, c K ρ(1450) = −0.156 ± 0.015, c K ω(1420) = −0.117 ± 0.013 and c K ω(1650),ρ(1700) = −0.083 ± 0.019 result in the fifth error for the predicted branching fractions in this work, while these coefficients c K R which exist only in the kaon time-like form factors will not change the direct CP asymmetries for the relevant decay processes. There are other errors for the PQCD predictions in this work, which come from the masses and the decay constants of the initial and final states, from the parameters in the distribution amplitudes for bachelor pion or kaon, from the uncertainties of the Wolfenstein parameters λ andη, etc., are small and have been neglected.
The PQCD predictions are omitted in Tables III, IV, V for those quasi-two-body decays with the subprocesses ρ(770, 1450, 1700) 0 → K 0K 0 and ω(782, 1420, 1650) → K 0K 0 . The variations caused by the small mass difference between K ± and K 0 for the branching fraction and direct CP asymmetry of a decay mode with one of these intermediate states decaying into K 0K 0 or K + K − are tiny. As the examples, we calculate the the branching fractions for the decays B + → π + ρ(770) 0 , B + → K + ρ(770) 0 , B + → π + ω(782) and B + → K + ω(782) with B(B + → π + ω(782) → π + K 0K 0 ) = 4.14 +1.64+1.02+0.07+0.20+0.14 −1.32−0.94−0.08−0.16−0.14 × 10 −8 , It's easy to check that these branching fractions are very close to the results in Table III for the corresponding decay modes with ρ(770) 0 and ω(782) decaying into K + K − . The impacts from the mass difference of K ± and K 0 for the direct CP asymmetries for the relevant processes are even smaller, which could be inferred from the comparison of the results in Table III with    Table IV for the corresponding decay processes with ρ(1450) 0 → K + K − . In view of the large errors for the predictions in Tables III, IV, V, we set the concerned decays with the subprocess ρ(770, 1450, 1700) 0 → K 0K 0 or ω(782, 1420, 1650) → K 0K 0 have the same results as their corresponding decay modes with the resonances decaying into K + K − . It should be stressed that the K 0K 0 with the P -wave resonant origin in the final state of B → KKh decays can not generate the K 0 S K 0 S system because of the Bose-Einstein statistics. From the branching fractions in Tables III, IV, one can find that the virtual contributions for KK from the BW tails of the intermediate states ρ(770) and ω(782) in those quasitwo-body decays which have been ignored in experimental and theoretical studies are all larger than the corresponding results from ρ(1450) and ω(1420). Specifically, the branching fractions in Table III with the resonances ρ(770) 0 and ρ(770) ± are about 1.2-1.8 times of the corresponding results in Table IV for the decays with ρ(1450) 0 and ρ(1450) ± , while the six predictions for the branching fractions in Table III with ω(782) in the quasi-two-body decay processes are about 2.2-2.9 times of the corresponding values for the decays with the V: PQCD predictions of the CP averaged branching ratios and the direct CP asymmetries for the quasi-two-body B → ρ(1700)h → KKh and B → ω(1650)h → KKh decays. The decays with the subprocess ρ(1700) 0 → K 0K 0 or ω(1650) → K 0K 0 have the same results as their corresponding decay modes with ρ(1700) 0 → K + K − or ω(1650) → K + K − . resonance ω(1420) in Table IV. The difference of the multiples between the results of the branching fractions with the resonances ρ and ω in Table III and Table IV should mainly be attributed to the relatively small value for the c K ω(1420) adopted in this work comparing with c K ρ(1450) . It is remarkable for these virtual contributions in Table III that their differential branching fractions are nearly unaffected by the full widths of ρ(770) and ω(782), which could be concluded from the Fig. 2. In this figure, the lines in the left diagram for B + → π + [ρ(770) 0 → ]K + K − and in the right diagram for B + → π + [ω(782) →]K + K − have very similar shape although there is a big difference between the values for the widths of ρ(770) and ω(782) as listed in Table II. The best explanation for Fig. 2 is that the imaginary part of the denominator in the BW formula the Eq. (7) which hold the energy dependent width for the resonances ρ(770) or ω(782) becomes unimportant when the invariant mass square s is large enough even if one employs the effective mass defined by the ad hoc formula [26,131] to replace the m 2 R in | − → q 0 | in Eq. (8) or calculates the energy dependent width with the partial widths and the branching ratios for the intermediate state as in Refs. [39,41,43,50]. At this point, the BW expression for ρ(770) or ω(782) is charged by the coefficient c K R in the time-like form factors for kaons and the gap between the invariant mass square s for kaon pair and the squared mass of the resonance. Although the threshold of kaon pair is not far from the pole masses of ρ(770) and ω(782), thanks to the strong suppression from the factor | − → q | 3 in Eq. (24), the differential branching fractions for those processes with ρ(770) or ω(782) decaying into kaon pair will reach their peak at about 1.35 GeV as shown in Fig. 2.

Decay modes
As we have stated in Ref. [21], the bumps in Fig. 2 for B + → π + [ρ(770) 0 →]K + K − and B + → π + [ω(782) 0 →]K + K − are generated by the tails of the BW formula for the resonances ρ(770) and ω(782) along with the phase space factors in Eq. (24) and should not be taken as the evidence for a new resonant state at about 1.35 GeV. When we compare the curves for the differential branching fractions for B + → π + [ρ(770) 0 →]K + K − and B + → π + [ρ(770) 0 → ]π + π − , we can understand this point well. In order to make a better contrast, the differential branching fraction for B + → π + [ρ(770) 0 →]K + K − is magnified 10 times in the big one of the left diagram of Fig. 2. The dash-dot line for B + → π + [ρ(770) 0 →]π + π − shall climb to its peak at about the pole mass of ρ(770) 0 and then descend as exhibited in Fig. 2. While this pattern is inapplicable for the decay process of B + → π + [ρ(770) 0 →]K + K − , its curve can only show the existence from the threshold of kaon pair where the √ s has already crossed the peak of BW for ρ(770) 0 . As √ s becoming larger, the effect of the full width for ρ(770) fade from the stage, the ratio between the differential branching fractions for the quasi-twobody decays B + → π + [ρ(770) 0 →]K + K − and B + → π + [ρ(770) 0 →]π + π − will tend to be a constant which is proportional to the value of |g ρ(770)K + K − /g ρ(770)π + π − | 2 if the phase space for the decay process is large enough. This conclusion can also be demonstrated well from the curve of the ratio for the decays B + → π + [ρ(1450) 0 →]K + K − and B + → π + [ρ(1450) 0 →]π + π − in Fig. 3. The solid line which stands for the B + → π + [ρ(1450) 0 →]K + K − decay and has been magnified 10 times will arise at the threshold of kaon pair in Fig. 3 and contribute the zero for R ρ(1450) because of the factor | − → q | 3 in Eq. (24), and the following for R ρ(1450) is a rapid rise to the value about 0.1 in the region where the main portion of the branching fractions for is going to the value |g ρ(1450)K + K − /g ρ(1450)π + π − | 2 as the rise of s.
The branching fractions of the virtual contributions for KK in this work from the BW tails of the intermediate states ρ(770) and ω(782) in the concerned decays which have been ignored in experimental and theoretical studies were found larger than the corresponding results from ρ(1450, 1700) and ω(1420, 1650). A remarkable phenomenon for the virtual contributions discussed in this work is that the differential branching fractions for B → ρ(770)h → KKh and B → ω(782)h → KKh are nearly unaffected by the quite different values of the full widths for ρ(770) and ω(782). The definition of the partial decay width such as Γ ρ(770)→K + K − = Γ ρ(770) B ρ(770) 0 →K + K − for the virtual contribution are invalid. This conclusion can be extended to other strong decay processes with the virtual contributions come from the tails of the resonances. The bumps of the lines for the differential branching fractions for those virtual contributions, which are generated by the phase space factors and the tails of the BW formula of ρ(770) or ω(782), should not be taken as the evidence for a new resonant state at about 1.35 GeV.

Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grants No. 11547038 and No. 11575110.
where G F is the Fermi coupling constant, V 's are the CKM matrix elements. The combinations a i with i = 1-10 are defined as a 1 = C 2 + C 1 /3, a 2 = C 1 + C 2 /3, a 3 = C 3 + C 4 /3, a 4 = C 4 + C 3 /3, a 5 = C 5 + C 6 /3, a 6 = C 6 + C 5 /3, a 7 = C 7 + C 8 /3, a 8 = C 8 + C 7 /3, a 9 = C 9 + C 10 /3, a 10 = C 10 + C 9 /3, for the Wilson coefficients. The general amplitudes for the quasi-two body decays B → ρh → KKh and B → ωh → KKh in the decay amplitudes Eqs. (B1)-(B20) are given according to Fig. 1, the typical Feynman diagrams for the PQCD approach. The symbols LL, LR and SP are employed to denote the amplitudes from the (V − A)(V − A), (V − A)(V + A) and (S − P )(S + P ) operators, respectively. The emission diagrams are depicted in Fig. 1 (a) and (c), while the annihilation diagrams are shown by Fig. 1 (b) and (d). For the factorizable diagrams in Fig. 1, we name their expressions with F , while the others are nonfactorizable diagrams, we name their expressions with M. The specific expressions for these general amplitudes are the same as in the appendix of [71] but with the replacements φ → ρ and φ → ω for their subscripts for the subprocesses ρ → KK and ω → KK, respectively, in this work. It should be understood that the Wilson coefficients C and the amplitudes F and M for the factorizable and nonfactorizable contributions, respectively, appear in convolutions in momentum fractions and impact parameters b.