Resonance contributions $\phi(1020,1680)\to K\bar K$ for the three-body decays $B\to K\bar K h$

We study the contributions for the $K^+K^-$ and $K^0\bar K^0$ originated from the intermediate states $\phi(1020)$ and $\phi(1680)$ in the charmless three-body decays $B\to K\bar K h$, with $h=(\pi, K)$, in the perturbative QCD approach. The subprocesses $\phi(1020,1680)\to K\bar K$ are introduced into the distribution amplitudes of $K\bar K$ system via the kaon electromagnetic form factor with the coefficients in which adopted from the fitted results. The predictions for the branching fractions of the decays $B\to\phi(1680)h \to K^+ K^- h$ are about $6\%$-$8\%$ of the corresponding results for the decays $B\to\phi(1020)h \to K^+ K^- h$ in this work, and the quasi-two-body decay mode with the subprocess $\phi(1680)\to K^0\bar K^0$ has the same branching fraction of its corresponding mode with $\phi(1680)\to K^+K^-$.

The intermediate states of the quasi-two-body decays B → φ(1020, 1680)h → KKh are generated in the hadronization of the quark-antiquark pair ss as demonstrated in the Fig. 1, in which the factorizable and nonfactorizable diagrams have been merged for the sake of simplicity, symbol B in the diagrams stands for the mesons B + , B 0 and B 0 s , and the inclusion of charge-conjugate processes throughout this work is implied. The subprocesses φ(1020, 1680) → KK which can not be calculated in the PQCD approach, will be introduced into the distribution amplitudes of the KK system by the vector meson dominance kaon electromagnetic form factor. The PQCD approach has been adopted in Refs. [38][39][40][41] for the tree-body B decays, and the quasi-two-body framework based on PQCD has been discussed in detail in [4] which has been followed by the works [42][43][44][45][46][47] for the charmless quasi-two-body B meson decays recently. Parallel analyses of three-body B decays with the QCD factorization (QCDF) can be found in Refs. [48][49][50][51][52][53][54][55][56][57][58], and the relevant works within the symmetries are referred to Refs [59][60][61][62][63][64][65][66][67][68].
This work is organized as follows. In Sec. II, we give a brief introduction for the theoretical framework involves the vector time-like form factors for kaon, the P -wave KK system distribution amplitudes and the differential branching fractions. In Sec. III, we provide numerical results for the concerned decay processes and give some necessary discussions. Summary of this work is presented in Sec. IV. The relevant quasi-two-body decay amplitudes are collected in the Appendix.

II. FRAMEWORK
In the light-cone coordinates, with the mass m B , the momenta p B for the B meson and k B for its light spectator quark are written as in the rest frame of B meson. For the kaon pair which generated from the intermediate state φ(1020) or φ(1680) by the strong interaction, one has the momentum p = mB √ 2 (ζ, 1, 0 T ) and the longitudinal polarization vector ǫ L = 1 √ 2 (− √ ζ, 1/ √ ζ, 0 T ), with the variable ζ = s/m 2 B and the invariant mass square s = m 2 KK ≡ p 2 . The spectator quark comes out from B meson and goes into resonance in the hadronization as shown in Fig. 1 (a) has the momentum k = (0, mB √ 2 z, k T ). For the bachelor final state pion or kaon and its spectator quark, we define their momenta p 3 and k 3 as The x B , z and x 3 above, which run from zero to one, are the momentum fractions for the B meson, resonance and the bachelor final state, respectively. The vector time-like form factors F K + (s) and F K 0 (s) for the charged and neutral kaons are related to the electromagnetic form factors for K + and K 0 , respectively, which are defined as [69] with the squared invariant mass s = (p 1 + p 2 ) 2 , the constraints F K + (0) = 1 and F K 0 (0) = 0, and 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 [70]. The form factors F K + and F K 0 can be separated into the isospin I = 1 and I = 0 components as [5,69].
With the BW formula for the resonances ω and φ and the Gounaris-Sakurai (GS) model [71] for ρ, we have the electromagnetic form factors [69,72] 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. [71,73].
Phenomenologically, the vector time-like form factor for kaon can also be defined by [54] When considering only the resonance contributions, we have Then the electromagnetic form factors can be expressed by [54]. The expressions for F ρ , F ω and F φ and their parameters can be found in [54,74]. It's easy to check that We concern only the φ component of the vector kaon time-like form factors in this work. Rather, for simplicity, we employ F K to stands for F s K + K − and F s K 0K 0 in the following discussions. For the subprocesses φ(1020, 1680) → KK, the P -wave KK system distribution amplitudes are organized into [11,75] φ P -wave with the momentum p = p 1 + p 2 . We have the distribution amplitudes with the Gegenbauer polynomial C [4] with the ratio f T φ /f φ = 0.75 at the scale µ = 2 GeV [76]. The Gegenbauer moment a φ 2 for φ 0 (z, s) are the same as it in the distribution amplitudes of the light vector meson φ in [75] for the two-body B meson decays.
The CP averaged differential branching fractions (B) for the quasi-two-body decays B → φ(1020, 1680)h → KKh are written as [11,50,77] where τ B being the B meson mean lifetime. The magnitudes of the momenta q and q h for the kaon and the bachelor h in the rest frame of the resonances φ(1020, 1680) are written as with m h the mass for the bachelor meson pion or kaon. The direct CP asymmetry A CP is defined as The Lorentz invariant decay amplitudes for the quasi-two-body decays B → φ(1020, 1680)h → KKh are collected in the Appendix.
Utilizing the differential branching fraction the Eq. (17) and the decay amplitudes collected in Appendix A, we obtain the concerned direct CP asymmetries and the CP averaged branching fractions for the quasi-two-body decays B → φ(1020)h → KKh in Table II Table III. Only the modes B + → φ(1020, 1680)K + and B 0 s → φ(1020, 1680)π 0 with φ(1020, 1680) decay into K + K − or K 0K 0 , which contain the contributions from the current-current operators of the weak effective Hamiltonian [87], have the direct CP asymmetries in Tables II, III.  The first error of these results in Tables II, III comes from the uncertainty of the shape parameters ω B = 0.40 ± 0.04 for B + and B 0 and ω B = 0.50 ± 0.05 for B 0 s , the second error is induced by the chiral masses m π 0 = 1.40 ± 0.10 GeV, m K 0 = 1.60 ± 0.10 GeV and the Gegenbauer moment a π,K 2 = 0.25 ± 0.15 for π and K as in [88], the third one is contributed by the Gegenbauer moment a φ 2 = 0.18 ± 0.08 [75] and the fourth error in Table III comes from the variation of the coefficient c K φ(1680) of the form factor F K , which will not change the direct CP asymmetries. There are other errors come from the uncertainties of the masses and the decay constants of the initial and final states, the other parameters in the distribution amplitudes of the bachelor pion or kaon, the Wolfenstein parameters of the CKM matrix, etc. are small and have been neglected.
The decay mode with the subprocess φ(1680) → K 0K 0 has the same branching fraction and direct CP asymmetry of its corresponding decay with φ(1680) → K + K − .
Decay modes Quasi-two-body results The two-body branching fractions for B → φh can be extracted from the quasi-two-body predictions with the relation where λ(a, b, c) = a 2 +b 2 +c 2 −2ab−2ac−2bc, theq h is the expression of Eq. (19) in the rest frame of B meson and fixed at s = m 2 φ . As an example, we have η ≈ 1.07 for the decays B 0 → φ(1020)K 0 and B 0 → φ(1020)K 0 → K + K − K 0 with the branching fraction B(φ(1020) → K + K − ) = 0.492 [27]. It means that the violation of the Eq. (23) is small when neglecting the effect of the squared invariant mass s in the decay amplitudes of the quasi-two-body decays. As the verification of Eq. (24), we calculate the decay B 0 → φ(1020)K 0 in the two-body framework of the PQCD approach with the same parameters and obtain its branching fraction B(B 0 → φ(1020)K 0 ) ≈ 7.21 × 10 −6 , which is about 98.0% of the result 7.36 × 10 −6 in Table IV extracted with the corresponding quasi-two-body result in Table II with the factorization relation.

IV. SUMMARY
In this work, we studied the contributions for the K + K − and K 0K 0 which originated from the intermediate states φ(1020) and φ(1680) in the charmless three-body decays B → KKh in PQCD approach. The subprocesses φ(1020, 1680) → KK were introduced into the distribution amplitudes of KK system via the kaon electromagnetic form factor with the coefficients c K φ(1020) and c K φ(1680) in which are adopted from the fitted results. With c K φ(1020) = 1.038 and c K φ(1680) = −0.150 ± 0.009 we predicted the branching fractions for the quasi-two-body decays B → φ(1020)h → KKh and B → φ(1680)h → K + K − h and the direct CP asymmetries for the decay modes B + → φ(1020, 1680)K + and B 0 s → φ(1020, 1680)π 0 with φ(1020, 1680) decay into K + K − or K 0K 0 . The predictions for the branching fractions of the decays B → φ(1680)h → K + K − h are about 6%-8% of the corresponding results for B → φ(1020)h → K + K − h in this work. The branching fraction for the decay φ(1680) → K 0K 0 is equal to that for φ(1680) → K + K − , and the decay mode with the subprocess φ(1680) → K 0K 0 has the same branching fraction of its corresponding mode with φ(1680) → K + K − for B → φ(1680)h → KKh. We defined a parameter η to measure the violation of the factorization relation for the decays B → φh and B → φh → KKh and found the violation is quite small. With the factorization relation, we extracted the branching fractions for the two-body decays B 0,+ → φ(1020)K 0,+ and B 0,+ → φ(1020)π 0,+ . The predictions for the decays B 0 → φ(1020)K 0 and B + → φ(1020)K + are agree with the existing data. And our results for B 0,+ → φ(1020)π 0,+ consistent with the theoretical results in literature.

Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grants No. 11505148, No. 11547038 and No. 11575110. Y.Y. Fan was also supported by the Nanhu Scholars Program for Young Scholars of XYNU. W.F. Wang thank Ai-Jun Ma for valuable discussions.

Appendix A: DECAY AMPLITUDES
The Lorentz invariant decay amplitude A for the quasi-two-body processes B → φ(1020, 1680)h → KKh is given by [4,38] in the PQCD approach, according to Feynman diagrams the Fig. 1. The symbol ⊗ means convolutions in parton momenta, the hard kernel H contains one hard gluon exchange at the leading order in strong coupling α s . The distribution amplitudes Φ B , Φ h and Φ KK absorb the nonperturbative dynamics in the relevant processes. The Φ B and Φ h for B meson and the bachelor final state h in this work are the same as those widely employed in the studies of the hadronic B meson decays in the PQCD approach, one can find their expressions and parameters in the Appendix of [44] and the references therein.
With the subprocesses φ → K + K − ,K 0 K 0 , and φ is the φ(1020) or φ(1680), the concerned quasi-two-body decay amplitudes are given as follows: where G F is the Fermi coupling constant, V 's are the CKM matrix elements. The combinations a i for the Wilson coefficients are defined as It should be understood that the Wilson coefficients C i , the amplitudes F and M for the factorizable and nonfactorizable Feynman diagrams, respectively, appear in convolutions in momentum fractions and impact parameters b.
The amplitudes from Fig. 1 (a) are written as where the color factor C F = 4/3 and the ratio r = m h 0 /m B . The amplitudes from Fig. 1 (b) are written as The amplitudes from Fig. 1 (c) are written as The amplitudes from Fig. 1(d) are written as For the errors induced by the parameter P ± ∆P for the B and A CP in the numerical calculation of this work, we employ the formulas [44] ∆B = ∂B ∂P ∆P, ∆A CP = 2(B∆B − B∆B) (B + B) 2 . (A32) The PQCD functions which appear in the factorization formulas, the Eqs. (A9)-(A31), can be found in the Appendix B of [44].