Resonance contributions from $\chi_{c0}$ in the charmless three-body hadronic $B$ meson decays

Within the framework of perturbative QCD factorization, we investigate the nonfactorizable contributions to these factorization-forbidden Quasi-two-body decays $B_{(s)}\rightarrow h\chi_{c0}\rightarrow h\pi^+\pi^-(K^+K^-)$ with $ h=\pi, K$. We compare our predicted branching ratios for the $B_{(s)}\rightarrow K\chi_{c0}\rightarrow K\pi^+\pi^-(K^+K^-)$ decay with available experiment data as well as predictions by other theoretical studies. The branching ratios of these decays are consistent with data and other theoretical predictions. In the Cabibbo-suppressed decays $B_{(s)}\rightarrow h\chi_{c0}\rightarrow h\pi^+\pi^-(K^+K^-)$ with $h=\bar{K}^0,\pi$, however, the values of the branching ratios are the order of $10^{-7}$ and $10^{-8}$. The ratio $R_{\chi_{c0}}$ between the decay $B^+\rightarrow \pi^+\chi_{c0}\rightarrow \pi^+\pi^+\pi^-$ and $B^+\rightarrow K^+\chi_{c0}\rightarrow K^+\pi^+\pi^-$ and the distribution of branching ratios for different decay modes in invariant mass are considered in this work.

Since Belle's measurement [5], numerous theoretical studies have been conducted to investigate the large nonfactorizable contributions, the decay characteristic in B + → K + χ c0 and other relevant decay modes. In the light-cone QCD sum rules approach, the nonfactorizable soft contributions in the B → Kη c , Kχ c0 decays were analyzed in the Ref. [8]. Within the perturbative QCD (PQCD) approach, the nonfactorizable contributions to the B meson decays into charmonia including B 0,+ → K ( * )0,+ χ c0 were calculated in the Refs. [9,10]. In the framework of QCD factorization (QCDF), the exclusive decays including the B → χ c0 K were studied in [11][12][13][14][15][16]. From these studies it was observed that infrared divergences resulting from nonfactorizable vertex corrections could not be eliminated [11,12]. Non-zero gluon mass was then employed to regularize the infrared divergences in vertex corrections [13]. While the authors of [16] found those infrared divergences can be subtracted consistently into the matrix elements of colour-octet operators in the exclusive B to P -wave charmonia decays. In Ref. [15], the B → Kχ c0,2 decays were investigated in QCDF by introducing a non-zero binding energy to regularize the infrared divergence of the vertex part and adopting a model dependent parametrization to remove the logarithmic and linear infrared divergences in the spectator diagrams.
The rescattering effects mediated by intermediate charmed mesons were studied in Refs. [17,18], the authors concluded that such effects could produce a large branching ratio for the decay B + → K + χ c0 .

II. FRAMEWORK
Under the factorization hypothesis, the decay amplitude for B → hχ c0 → hK + K − is given by where the denominator [7] are the pole mass and full width of the resonant state χ c0 , the s is invariant mass square for K + K − pair in the decay final state. L R is the spin of the resonances [27,29]. In the rest frame of the resonant state χ c0 , its daughter K + or K − has the magnitude of the momentum as q = 1 2 s − 4m 2 K , and q 0 in Γ(s) is the value of q at s = m 2 0 . The amplitude A(s) = hχ c0 |H eff |B for the concerned quasi-two-body decays in this work can be found in the Appendix. The mass-dependent coefficient C KK (s) is g χc0K + K − /D BW . We have the coupling constant g χc0K + K − from the relation [54,55] where the Γ χc0→K + K − is the partial width for χ c0 → K + K − . For the process B → hχ c0 → hπ + π − , we need the replacement K → π for the Eqs. (1)-(2) and the relevant parameters. The effective Hamiltonian H eff with the four-fermion operators are the same as in [9].
In the rest frame of the B meson, we choose its momentum p B , the momenta p 3 and p for the bachelor state h and χ c0 , as where x B , x 3 , and z are the corresponding momentum fractions, m B is the mass of B meson. The variable η is defined as η = s/m 2 B , with the invariant mass square s = p 2 . For the B +,0 and B 0 s in this work, we employ the same distribution amplitudes φ B/Bs as in Refs. [36,56]. The wave functions for the bachelor states π and K in this work are written as where m h 0 is the chiral mass, p and z are the momentum and corresponding momentum fraction of π and k. The distribution amplitudes (DAs) φ A (z), φ P (z), φ T (z) can be written as [57][58][59][60] where the Gegenbauer moments are chosen as a π 1 = 0, a K 1 = 0.06, a π,K 2 = 0.25 ± 0.15, a π 4 = −0.015 and the paraments follow The Gegenbauer polynomials are defined as where the variable t = 2z −1. The mass-dependent ππ or KK system, which comes from χ c0 , has the distribution amplitude [9] Φ ππ(KK) = 1 with the twist-2 and twist-3 distribution amplitudes φ v ππ(KK) (z, s) and φ s ππ(KK) (z, s) The timelike form factor F χc0 (z, s) is parametrized with the RBW line shape [61] and can be expressed as follows [62][63][64], where m 0 is the pole mass. The mass-dependent decay width Γ (s) is defined as L R is the spin of the resonances, and L R = 0 for the scalar intermediate state χ c0 .

III. RESULTS
The differential branching ratios (B) for the decay processes B → hπ where τ B is the lifetime of B meson. The q h is the magnitude momentum for the bachelor h in the rest frame of χ c0 : The QCD scale follows Λ  [7]. For the shape parameter uncertainty of B (s) meson we use ω B = 0.4 ± 0.04 GeV and ω Bs = 0.5 ± 0.05 GeV, which contributed the largest error for the branching fractions. The second one is from the Gegenbauer moments a h 2 in the bachelor meson DAs. The other two error comes from decay width of the resonance χ c0 and the chiral mass m h 0 of bachelor meson, which have a smaller impact to the uncertainties in our approach. There are further errors which are tiny and can be ignored safely, such as minor and disregarded parameters in the bachelor meson (π/K) distribution amplitudes and Wolfenstein parameters.

Mode
Unit Branching ratios Data[7] B + → K + χc0 → K + π + π − (10 −6 ) 0.81 +0.21 We calculate the branching ratios for the decays of B → hχ c0 → hπ + π − (K + K − ) in Table (I), by using the differential branching ratios in Eq. (11), and the decay amplitudes in the Appendix. Compare our numerical results with current world average values from the PDG [7] and the various theoretical predictions in PQCD, LCSR and QCDF in Table (II), and we do some analyses.
We contrast the various theoretical predictions for the B → Kχ c0 cases of the investigated quasi-two-body and two-body decays. The LCSR calculations mainly focus on B + → K + χ c0 and the prediction value is (1.0 ± 0.6) × 10 −4 [8]. Compared with previous PQCD calculations [9,10], we update the charmonium distribution amplitudes and some of the input parameters in this study. Our predictions are smaller than those of [9] and closer to [10]. The QCDF suffers endpoint divergences caused by spectator amplitudes and infrared divergences resulting from vertex diagrams. The different treatment of these divergences as mentioned in the Introduction in [14][15][16] lead to different numerical results. Both our results in this work and the computations above are in excellent agreement with the available data for B + → K + χ c0 and B 0 → K 0 χ c0 .
For the quasi-two-body processes B + → π + χ c0 → π + π + π − and B + → K + χ c0 → K + π + π − , which have an identical step χ c0 → π + π − , the difference of these two decay modes originated from the bachelor particles pion and kaon. Assuming factorization and flavor-SU (3) symmetry, the ratio R χc0 for the branching fractions of these two processes is With the result in Review of Particle Physics [7], one has R χc0 ≈ 0.036. It still fits expectations from our PQCD anticipated ratio In Fig. 2, we show the distribution of branching ratios for decays modes B + → K + χ c0 → K + K + K − . The mass of χ c0 is visible as a narrow peaks near 3.414 GeV. We find that the central portion of the branching ratios lies in the region around the pole mass of the χ c0 resonance as shown by the distribution of the branching ratios in the ππ invariant mass.

IV. CONCLUSION
We studied the nonfactorizable contributions to these factorization-forbidden quasi-two-body decays B → Kχ c0 → Kππ(KK), B s →K 0 χ c0 →K 0 ππ(KK), and B s → πχ c0 → πππ(KK) in PQCD approach in this work. Our predictions for the branching ratios are summarized in Table I and compared with other theoretical results. The obtained branching ratios of B → Kχ c0 decay are essentially consistent with the current data. For the decay involving π orK in the final state not yet measured, the calculated branching ratios will be further tested by experiments in the near future. By utilizing the flavor-SU (3) symmetry to examine quasi-two-body decays with the same intermediate step, we were able to establish the ratio R χc0 for processes B + → π + χ c0 → π + π + π − and B + → K + χ c0 → K + π + π − . The ratio R χc0 is predicted by PQCD to be 0.049, which is close to the value 0.036 reported in Review of Particle Physics. We also display the distribution of branching ratios for various decay modes in invariant mass, and we discover that the majority of the branching ratios are located in the vicinity of the χ c0 resonance's pole mass.

V. ACKNOWLEDGEMENTS
Many thanks to Wen-Fei Wang, Da-Cheng Yan and Jun Hua for valuable discussions.

Appendix A: Decay amplitudes
The concerned quasi-two-body decay amplitudes are given in the PQCD approach by where G F is the Fermi coupling constant, V , s are the Cabibbo-Kobayashi-Maskawa matrix elements, and c i is Wilson coefficients. The amplitudes appeared in above equations are written as M LL eK(π) = −16 M SP eK(π) = −16 with the r c = m c /m B and r 3 = m h 0 /m B . The evolution factors in above formulas are given by The hard functions h a(b) , the hard scales t a(b) , and factor S ab (t) have their explicit expressions in the Appendix of [66].