Contributions of the kaon pair from $\rho(770)$ for the three-body decays $B \to D K\bar{K}$

We study the contributions of the kaon pair originating from the resonance $\rho(770)$ for the three-body decays $B \to D K\bar{K}$ by employing the perturbative QCD approach. According to the predictions in this work, the contributions from the intermediate state $\rho(770)^0 $ are relative small for the three-body decays such as $B^0 \to \bar{D}^0 K^+ K^-$, $B_s^0 \to \bar{D}^0 K^+ K^-$ and $B^+ \to D_s^+ K^+K^-$, while a percent at about $20\%$ of the total three-body branching fraction for $B^+ \to \bar{D}^0 K^+ \bar{K}^0$ could possibly come from the subprocess $\rho(770)^+\to K^+ \bar{K}^0$. We also estimate the branching fractions for $\rho(770)^\pm$ decay into kaon pair to be about one percent and that for the neutral $\rho(770)$ into $K^+K^-$ or $K^0\bar{K}^0$ to be about $0.5\%$, which will be tested by the future experiments.

decays B → Dh 1 h 2 (h 1,2 stands for pion or kaon) have been studied within the PQCD approach based on the k T factorization theorem [90][91][92][93]. In this work, we shall focus on the contributions of the subprocesses ρ(770) → KK for the three-body decays B → DKK, where the symbolK means the kaons K + and K 0 and the symbol K means the kaons K − andK 0 . In view of the narrow decay width of ω(782) and the gap between its pole mass and the threshold of kaon pair, the branching fractions for the decays with the subprocess ω(782) → KK are small and negligible comparing with the contribution from ρ(770) → KK in the same decay mode [80]. Meanwhile, there are still disparities between the fitted coefficients of the time-like form factors for kaons from currently known experimental results [65,66,94], we will leave the possible subprocesses with those excited states of ρ(770) and ω(782) decay into KK to the future study.
The rest of this paper is organized as follows. In Sec. II, we give a brief review of the PQCD framework for the concerned decay processes. The numerical results and the phenomenological analyses are given in Sec. III. The 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, the momenta p B , p and p 3 for the B meson, the resonance ρ and the final state D, respectively, are chosen as where m B denotes the mass for B meson, the variable η is defined as η = s/(m 2 B − m 2 D ), the invariant mass square s = p 2 = m 2 KK for the kaon pair and the mass ratio r = m D /m B . The momentum of the light quark in the B meson, ρ and D meson are denoted as k B , k and k 3 with where the momentum fraction x B , z and x 3 run between zero and unity. The decay amplitude A for the quasi-two-body processes B → Dρ(770) → DKK in the PQCD approach can be expressed as the convolution of a hard kernel H containing one hard gluon exchange with the relevant hadron distribution amplitudes [21,95]  where the distribution amplitudes Φ B , Φ D and Φ KK for the initial and final state mesons absorb the nonperturbative dynamics. In this work, we employ the same distribution amplitudes for B and D mesons as those widely adopted in the studies of the hadronic B meson decays in the PQCD approach, one can find their explicit expressions and parameters in the Ref. [82] and the references therein. The P -wave KK system distribution amplitudes along with the subprocesses ρ(770) → KK are defined as [37,80] φ P -wave where z is the momentum fraction for the spectator quark, s is the squared invariant mass of the kaon pair, ǫ L and p are the longitudinal polarization vector and momentum for the resonance. The twist-2 and twist-3 distribution amplitudes φ 0 , φ s and φ t are parameterized as [80] φ 0 (z, s) with the Gegenbauer polynomial C 3/2 [24], a = 1 for ρ(770) 0 and a = 2 for ρ(770) ± . In the numerical calculation, we adopt f ρ = 0.216 GeV [96,97] and f T ρ = 0.184 GeV [98]. The Gegenbauer moments a 0,s,t 2 are the same as they in the distribution amplitudes for the intermediate state ρ(770) in Refs. [24,80]. The vector time-like form factors for kaons are written as [12] where F ρ , F ω and F φ come from the definitions of the electromagnetic form factors for the charged and neutral kaon [65,66] The symbol means the summation for the resonances ρ(770), ω(782) or φ(1020) and their corresponding excited states. The normalization factors c K V for resonances determined by fitting experimental data and the corresponding BW formula can be found in the Refs. [65,66,94]. It is not difficult to find that the corresponding coefficients c K V for ρ(770), ω(782) or φ(1020) are close to each other in [65,66,94], while those coefficients for the excited states have significant differences by comparing the fitted parameters in the Table 2 in Refs. [65,66] and Table 1 in [94]. In this work, we concern only the ρ(770) components of the vector kaon time-like form factors and the fitted values for the coefficients C K ρ(770) in the kaon form factors collected from the Refs. [65,66,94] have been listed in the TABLE I. The "Fit(1)", "Fit(2)" and "Model I", "Model II" represent the values parameterized with different constraints in each work. Due to the closeness of the coefficients C K ρ(770) in Refs. [65,66,94], we choose the value of "Fit(1)" in the Ref. [65] in our numerical calculation. The resonance shape for ρ(770) is described by the KS version of the BW formula [65,99] where the effective s-dependent width is given by with β(s, m) = 1 − 4m 2 /s. In addition, one has the time-like form factor for K +K 0 and K − K 0 from the relation [65,67] and keep only the ρ resonance contributions with isospin symmetry.
where G F = 1.16639 × 10 −5 GeV −2 , V ij are the CKM matrix elements, C 1,2 (µ) represent the Wilson coefficients at the renormalization scale µ and O 1,2 are the local four-quark operators. According to the typical Feynman diagrams for the concerned decays as shown in Figs. 1 and 2, the decay amplitudes for B (s) →D (s) ρ(770) with the subprocesses ρ(770) 0 → K + K − /K 0K 0 and ρ(770) + → K +K 0 are given as follows: while the decay amplitudes for B (s) → D (s) ρ(770) with the subprocesses ρ(770) 0 → K + K − /K 0K 0 , ρ(770) + → K +K 0 and ρ(770) − → K − K 0 can be written as: with the Wilson coefficients a 1 = C 1 + C 2 /3 and a 2 = C 2 + C 1 /3. The explicit expressions of individual amplitude F and M for the factorizable and nonfactorizable Feynman diagrams can be found in Appendix. The differential branching fractions (B) for the quasi-two-body decays B → Dρ(770) → DKK can be written as [7,37,80] The magnitudes of the momenta for K and D in the center-of-mass frame of the kaon pair are written as

III. RESULTS
In the numerical calculations, the input parameters, such as masses and decay constants (in units of GeV) and B meson lifetimes (in units of ps), are adopted as follows [100]: For the Wolfenstein parameters of the CKM mixing matrix, we use the values A = 0.790 +0.017 −0.012 , λ = 0.22650 ± 0.00048, ρ = 0.141 +0.016 −0.017 ,η = 0.357 ± 0.011 as listed in Ref. [100]. In Tables II and III,   The PQCD predictions of the branching fractions for the B (s) →D (s) ρ(770) →D (s) KK decays. The decay mode with the subprocess ρ(770) 0 → K 0K 0 has the same branching fraction of its corresponding mode with ρ(770) 0 → K + K − .

Decay modes
Unit Quasi-two-body results The decay mode with the subprocess ρ(770) 0 → K 0K 0 has the same branching fraction of its corresponding decay with ρ(770) 0 → K + K − .
In Ref. [55], LHCb presented the first observation of the decay B + → D + s K + K − and the branching fraction was determined to be (7.1±0.5±0.6±0.7)×10 −6 . Utilize our prediction B(B + → D + s ρ(770) 0 → D + s K + K − ) = (6.26 +3.18 −1.59 )× 10 −8 , where the individual errors have been added in quadrature, we obtain the ratio 26 % which is quite small as expected. In addition, they also gave a branching fraction for B + → D + s φ(1020) decay of (1.2 +1.6 −1.4 ± 0.8 ± 0.1) × 10 −7 and set an upper limit as 4.9(4.2) × 10 −7 at 95%(90%) confidence level, which is roughly one order smaller than their previous result in [58]. By adopting B(φ(1020) → K + K − ) = 0.492 [100] and the relation between the quasi-body decay and corresponding two-body decay we find that B(B + → D + s ρ(770) 0 → D + s K + K − ) predicted in this work has the same magnitude with the branching ratio for B + → D + s φ(1020) → D + s K + K − measured by LHCb within large uncertainties, while B(B + → D + s φ(1020) → D + s K + K − ) was predicted to be (1.53 ± 0.23) × 10 −7 within the PQCD approach in Ref. [88]. In Fig. 3, we show the differential branching fraction of the decay mode B(B + → D + s ρ(770) 0 → D + s K + K − ) with the invariant mass in the range of [2m K , 3 GeV]. The bump in the curve is caused by the strong depression of the phase space factors q and q D in Eqs. (32) and (33) near the K + K − threshold. It is this depression near the threshold along with the similar mass between K ± and K 0 ,K 0 make the decay channel with the subprocess ρ(770) 0 → K 0K 0 has the same branching fraction of its corresponding decay mode with ρ(770) 0 → K + K − .

Decay modes
Bexp For comparison, we list the available experimental measurements for the branching fractions of the two-body B → Dρ(770) decays from the Review of Particle Physics [100] in Table IV, together with the PQCD predictions for the branching ratios of the relevant decay modes with the subprocess ρ(770) → KK shown in Tables II and III. The ratios between the relevant branching fractions are Due to the suppression from the phase space, the predicted branching fractions of the quasi-two-decays 9% of the experimental data for the corresponding two-body cases, while a ratio near 0.4% for B 0 →D 0 ρ(770) 0 →D 0 K + K − is found.

IV. SUMMARY
In this work, we analyzed the contributions for the kaon pair originating from the intermediate state ρ(770) for the three-body decays B → DKK in the PQCD approach. By the numerical evaluations and the phenomenological analyses, we found the following points: (ii) Our predictions for the corresponding branching fractions of the decay modes with the subprocess ρ(770) 0 → K + K − are much less than the measured branching fractions for the three-body decays while the percent at about 20% of the total three-body branching fraction for the quasi-two-body decay B + →D 0 ρ(770) + →D 0 K +K 0 was predicted in this work.
The expressions for amplitudes from Figs. 1 (e 1 -h 1 ) are written as: The expressions for amplitudes from Figs. 1 (m 1 -p 1 ) are written as: + (r − 1)(η + r)(x B + (r 2 − 1)z) +η(η − (1 + r − r 2 )r)x 3 E n (t p )h p (x B , z, The expressions for amplitudes from Figs. 2 (a 2 -d 2 ) are written as: The expressions for amplitudes from Figs. 2 (e 2 -h 2 ) are written as: In the formulae above, the symbolη = 1 − η, the mass ratio r = mD mB and r c = mc mB are adopted. The b B , b and b 3 are the conjugate variable of the transverse momenta of the light quarks in the B meson, resonance ρ(770) and D meson. The explicit expressions for the hard functions h i , the evolution factors E(t i ) and the threshold resummation factor S t can be found in Ref. [82].