Study of B + c decays to the K + K − π + final state by using B 0 s , χ c 0 and D 0 resonances and weak annihilation nonresonant topologys

In this research the weak decay of B c decays to the K+K−π+ final state, which is observed by LHCb collaboration for the first time, is calculated in the quasi-two-body decays which takes into account the B s , χc0 and D resonances and weak annihilation nonresonant contributions. In this process, the B c meson decays first into B sπ, χc0π and Dπ intermediate states, and then the B s , χc0 andD resonances decay intoK+K− components, which undergo final state interaction. The mode of the B c → D(→ K−π+)K+ is also associated to the calculation, in this mode the intermediate resonance D decays to the K−π+ final mesons. The resonances B s , χc0 andD effects in the B c → B s (→ K+K−)π+, B c → χc0(→ K+K−)π+ and B c → D(→ K+K−)π+, D(→ K−π+)K+ decays are described in terms of the quasi-two-body modes. There is a weak annihilation nonresonant contribution in which B c decays to the K+K−π+ directly, so the point-like 3-body matrix element 〈K+K−π+|ud̄|0〉 is also considered. The decay mode of the B c → K̄∗0(892)K+ is contributed to the annihilation contribution. The branching ratios of quasi-two-body decays expand in the range from 1.98× 10−6 to 7.32× 10−6.


Introduction
B c meson is one of the most interesting mesons that can be studied at the Tevatron, the discovery of the B c was reported by the CDF collaboration in the B c → J/ψl ±ν l process at Fermilab [1]. After that, the decay mode B ± c → J/ψπ ± has been observed by CDF and D0 collaboration significance of more than 8σ and 5σ respectively [2, 3] . This decay mode addition B ± c → J/ψD ± s decay have also been observed by the LHCb collaboration at the LHC centerof-mass energy 7 TeV of proton-proton collisions [4,5]. Studies of B c properties are important, because it is made of two different heavy quarks, bottom-charm antiquark-quark pair. Each of the quarks can participate in a weak interaction in which other quark participates as a spectator. For this, B c is also the only meson in which decays of both heavy quarks compete with each other, therefore a wide range of decay channels are possible. However, a significant number of these channels has not been observed yet [6]. Unlike B 0 , B + and B 0 s mesons, more than 70% of the B + c width is due to c-quark decays, in which c → s transition has been observed with B + c → B 0 s π + decays [7]. Around 20% of its width is due to the b-quark decays [8]. In charmless final states, thebc → W + →qq annihilation amplitudes account for only 10% of the B + c width [9]. The mass of the B c meson has been predicted using a variety of theoretical techniques. Nonrelativistic potential models have been used to predict a mass of the B c in the range of 6247 -6286 MeV/c2 [10,11,12], by using a perturbative QCD calculation slightly higher value is found [13] and recent a B c mass prediction of 630412 +18 −0 MeV/c2 have been provided applying lattice QCD calculations [14], in which are heavier than that of other B mesons, this suggests an expected lifetime much shorter than those [15]. The decay of B + c meson to three light charged hadrons like K + , K − and π + , which is observed by LHCb collaboration for the first time [9], provide a good way to study standard model for which has a large available phase space. In this work the decay mode of B + c → K + K − π + is studied in which includes other processes such as B + c → B 0 s (→ K + K − )π + decay mediated by c → s transition, charmonium mode B + c → χ c0 (→ K + K − )π + mediated by theb →c transition, B + c → D 0 (→ K + K − )π + mediated by theb →ū andb →d transitions and finally the decay of B + c → D 0 (→ K − π + )K + mediated by theb →ū andb →s transitions. In the standard model, there is another process that can mediated via cb → W + → ud annihilation topology, in this mode the B + c decays with no charm and beauty particles in the final or intermediate states. In the B + c region 6.0 < m(K + K − π + ) < 6.5 GeV/c 2 , the signals were fitted by authors in Ref.
[9] separately for regions of the phase space corresponding to the different expected contributions: (a) the annihilation region, m(K − π + ) < 1.834 GeV/c 2 , they have claimed theK * 0 (892) meson can be in this region. The contribution of the mode B + c →K * 0 (892)(→ K − π + )K + and direct annihilation are obtained separately which the estimate ofK * 0 (892) → K − π + is lower than the direct annihilation calculation, (b) the D 0 → K − π + region, 1.834 < m(K − π + ) < 1.894 GeV/c 2 , (c) the B 0 s → K + K − region, 5.3 < m(K + K − ) < 5.4 GeV/c 2 , (d) the χ c0 → K + K − region, 3.38 < m(K + K − ) < 3.46 GeV/c 2 . A concentration of events was observed by [9] around m 2 (K + K − ) ∼ 11GeV 2 /c 4 , a one-dimensional projection of m(K + K − ) shows clustering near 3.41 GeV/c 2 , close to the mass of the charmonium state χ c0 where has the highest branching fraction into the K + K − final state [16]. The accumulation of events was also observed near m 2 (K + K − ) ∼ 29GeV 2 /c 4 close to the mass of the B 0 s meson. In this research the study of the B + c → K + K − π + via quasi-two-body decay was considered, this decay mode observed in the It is known that in the narrow width approximation, in the models we use to obtain the amplitudes of the decays, the 3-body decay rate obeys the factorization relation [17] with R being B 0 s , χ c0 and D 0 intermediate resonance mesons and M , M 1 and M 2 are K + , K − and π + final state mesons. The intermediate resonance effects are described in terms of the Breit-Wigner formalism. The Breit-Wigner resonant term associated to quasi two body R + M state which seems to play an important role as indicated by experiments. We have to calculate the branching ratios of the B(B + c → RM ) and B(R → M 1 M 2 ) by using the Feynman quark diagrams.

Decay amplitudes and branching fractions
In the standard factorization scheme, the decay amplitude is obtained by applying weak Hamiltonian which is given below: where the weak current J µ is given by: mixture of the d, s and b quarks, as given by the CKM matrix. Current matrix elements are defined as: P 1,2 (p 1,2 )|J µ |P (p) = (p+p 1,2 −q(m 2 P −m 2 p 1,2 )/q 2 ) µ F 1 (q 2 )+q µ (m 2 P −m 2 p 1,2 )/q 2 F 0 (q 2 ) with q µ = p µ − p (1,2)µ and P 2,1 (p 2,1 )|J µ |0 = if p 2,1 p µ 2,1 . It has been pointed out in the BSW2 model [18] that consistency with the heavy quark symmetry requires certain form factors such as F 0 and F 1 to have dipole q 2 dependence i.e. F P →P 1,2 0,1 In the following, we calculate the amplitudes of the all decay modes. Different with B u,d,s mesons, the B + c system consists of two heavy quarksb and c, which can decay individually. Here we will considerb decays while c acts as a spectator except B + c → B 0 s π + decay in which we have c → s transition. According to figure 1, at the quark level corresponding effective Hamiltonian is given by: where λ p is the CKM matrix elements, c i are the Wilson coefficients evaluated at the renormalization scale µ, Q p 1,2 are the left-handed current-current operators arising from W-boson exchange, Q 3,...,6 and Q 7,...,10 are QCD and electroweak penguin operators, and Q 7γ and Q 8g are the electromagnetic and chromomagnetic dipole operators. Because in the tree level diagrams (a 1 coefficient) bothb →ū andb →c transitions are available, so we have considered to current-current operators Q u 1 , Q u 2 , Q c 1 and Q c 2 as: The operators arise from the QCD-penguin diagrams (bothb →d andb →s transitions) which contribute in the a 4 coefficient we have considered: Here α and β are the SU(3) color indices and the subscript V −A represent the chiral projection 1 − γ 5 .
In the following, we calculate the amplitude of the B + c → B 0 s π + decay mediated by c → s transition. The amplitude of the B + c → B 0 s π + decay by using the color-allowed external W-emission tree diagram become: F denote the factorized hadronic matrix element, which has the same definition as that in the "nonfactorizable" approach. The form factor of the B 0 s (p 1 )|(cs) V −A |B + c (p) can be as follows: The expressions of decay amplitude for mediated byb →ū andb →d transitions; (e) and (f) B + c → D 0 (→ K − π + )K + channel mediated byb →ū andb →s transitions; (g) annihilation process.
B + c → B 0 s π + within the factorization framework can be written as where a 1 = c 1 + c 2 /3. The branching fraction of B + c → B 0 s π + in B + c meson rest frame can be written as in which where τ B + c is the lifetime of the B + c meson and | p| is the absolute value of the 3momentum of the B 0 s or π + mesons that can be calculated via: The same calculation, similar to the pervious calculation, can also be applied to the B + c → χ c0 π + mode, with this difference this state of decay mediated byb →c transition so the form factor of the The amplitude and branching fraction of the B + c → χ c0 π + is the same with the B + c → B 0 s π + one which are given by and to obtain the absolute value of the 3-momentum of the χ c0 or π + mesons, the χ c0 meson mass should be replaced instead of the B 0 s meson mass in the pervious calculation. The branching ratios for other decays These decays, in additionb →ū tree level a 1 coefficient, involve theb →d andb →s penguin amplitudes (a 4 coefficient) with the QCD penguins participating. The same amplitudes of the B + c → D 0 π + and B + c → D 0 K + decays are given by where a 4 = c 4 + c 3 /3.

B
In the factorization approach the amplitudes of the B 0 s → K + K − , D 0 → K + K − and D 0 → K − π + decays have the following form

Annihilation topology
In this subsection we offer two ways for annihilation topology, the first is two body pure annihilation topology for χ c0 → K + K − decay, this decay mode proceeds only through the Wannihilation diagram and the second is nonresonant three body B + c → K + K − π + annihilation contribution. The contribution of the mode B + c →K * 0 (892)K + in the annihilation processes could be also prominent.

Pure annihilation
The χ c0 → K + K − decay is a pure annihilation decay channel, this process only occurs via annihilation between c and s quarks. In the factorization method, Feynman diagram for the χ c0 → K + K − decay is shown in figure 1. When all the basic building blocks equations are solved, for the case that both mesons are pseudoscalar, it is found that weak annihilation Kernels exhibit endpoint divergence. Divergence terms are determined by 1 0 dx/x and 1 0 dy/y. For the liberation of the divergence, a small of Λ QCD /m χ c0 order was added to the denominator. So the answer to the integral becomes ln(1 + )/ form, which is shown with X A . Specifically, we treat X A = (1 + ρ A e iϕ A )ln(m χ c0 /Λ QCD ) as a arbitrary parameter obtained by using ρ A = 0.5 and a strong phase ϕ A = −55 • [19]. The factorization amplitude when both mesons are pseudoscalar is given by where b 1 is the building block of the non-singlet annihilation coefficient which is given by: (8/27)πα s c 1 9(X A − 4 + π 2 /3) + r K + χ r K − χ X 2 A . The light-cone expansion implies that only leading-twist distribution amplitudes are needed in the heavy-quark limit. There exist however a number of subleading quark-antiquark distribution amplitudes of twist 3, which have large normalization factors for pseudoscalar mesons, e.g. r K χ for the kaon we use 2m 2 K / (m c − m u )(m u + m s ) 2.5.2 Effects of B + c →K * 0 (892)K + decay to the annihilation processes According to the panel (g) of the figure 1, the decay mode of the B + c →K * 0 (892)K + could be contributed to the annihilation processes. The amplitude of the B + c →K * 0 (892)K + annihilation decay reads when one of the final state meson is vector and another is pseudoscalar, the building block of the non-singlet annihilation coefficient b 2 become: (8/9)πα s c 2 3(X A − 4 + π 2 /3) + rK * 0 2.5.3 Nonresonant annihilation contribution for B + c → K + K − π + decay As for the three-body nonresonant annihilation matrix element K + (p 1 )K − (p 2 )π + (p 3 )|(ud) V −A |0 , we can show that it vanishes in the chiral limit owing to the helicity suppression. To prove this clime, we first assume that the kaon and pion mesons are soft.Tthe point-like 3-body matrix element is chiral-realization dependent, this realization dependence should be compensated by the pole contribution, so the three-body nonresonant annihilation matrix element become: (2i/f π ) p 2µ − p µ (p · p 2 )/(p 2 − m 2 K ) . It is worth stressing again that the above matrix element is valid only for low-momentum pseudoscalars. It is easily seen that in the chiral limit the is needed to accommodate the fact that the final-state pseudoscalars are energetic rather than soft in which is assumed to be MeV is the chiral-symmetry breaking scale. The direct three body weak annihilation amplitude reads To obtain the p 1 · p 3 , we consider the decays of B + c meson (with 4-momentum of p and m mass) into three K + , K − and π + particles. Denote their masses m 1 , m 2 and m 3 and 4-momenta by p 1 , p 2 and p 3 , respectively. Energy-momentum conservation is expressed by p = p 1 + p 2 + p 3 . Define the following invariants s 12 = (p 1 + p 2 ) 2 = (p − p 3 ) 2 , s 13 = (p 1 + p 3 ) 2 = (p − p 2 ) 2 , s 23 = (p 2 + p 3 ) 2 = (p − p 1 ) 2 . The three invariants s 12 , s 13 and s 23 are not independent, it follows from their definitions together with 4-momentum conservation that s 12 + s 13 + s 23 = m 2 +m 2 1 +m 2 2 +m 2 3 . We take s 12 = s and s 23 = t, so we have s 13 = m 2 +m 2 1 +m 2 2 +m 2 3 −s−t. With these definitions, we obtain multiplying of the 4-momentum as: p 1 · p 3 = (1/2)(m 2 + m 2 2 − s − t). The decay width of a three-body process is given by where t min,max (s) = m 2 1 +m 2 3 −(1/(2s)) (m 2 −s−m 2 3 )(s−m 2 2 +m 2 1 )∓ λ(s, m 2 , m 2 3 ) λ(s, m 2 1 , m 2 2 ) , s min = (m 1 + m 2 ) 2 , s max = (m − m 3 ) 2 and λ(x, y, z) = x 2 + y 2 + z 2 − 2(xy + xz + yz).

Numerical results and conclusion
The theoretical predictions depend on many input parameters such as Wilson coefficients, the CKM matrix elements, masses, lifetimes, decay constants, form factors, and so on. We present all the relevant input parameters as follows: Wilson coefficients, the Wilson coefficients c 1 , c 2 , c 3 and c 4 in the effective weak Hamiltonian have been reliably evaluated to the next-to-leading logarithmic order. To proceed, we use the following numerical values at µ = m b scale, which have been obtained in the NDR scheme [19]: 3) × 10 −6 obtained from Ref. [24]. Using the predictions listed in Ref. [25] for B(B + c → J/ψπ + ), which span the range (0.34 ∼ 2.9) × 10 −3 , we obtain σ(B + c )/σ(B + ) = (0.23 ∼ 2.15)%, so the experimental branching fractions become: B(B + c → B 0 s π + ) = (8.46 ∼ 128.69)% and B(B + c → χ c0 π + ) = (0.30 ∼ 6.09) × 10 −3 which in very good agreement with our prediction.
For the branching fraction of the B + c → D 0 K + and B + c → D 0 π + decays our calculations become from 10 −7 order, as have been obtained in [26]. The experimental result available for (f c /f u )×B(B + c → D 0 K + ) is (9.3 +2.8 −2.5 ±0.6)×10 −7 and for (f c /f u )×B(B + c → D 0 π + ) is less than 3.9 × 10 −7 [27], which implies f c /f u values in the range 0.004-0.012, the experimental branching fractions of them become from 10 −4 order in which our result a thousand times smaller than experimental one. Maybe for this reason for the mode B + c → D 0 (→ K − π + )K + , no significant deviation from the background-only hypothesis is observed [9].
For the B 0 s → K + K − mode, the decay amplitude is calculated at leading power in Λ QCD /m b and at next-to-leading order in α s using the QCD factorization approach. The calculation of the relevant hard-scattering kernels is completed. Important classes of power corrections, including "chirally-enhanced" terms and weak annihilation contributions, are estimated and included in the phenomenological analysis as given in Refs. [28,29]. Applying other contributions from the Feynman graph to this decay (such as the contribution of tree level a 2 in addition to the a 1 mentioned in the text, penguin diagrams a 3 and a 4 and weak annihilation of b 1 ) we obtain B(B 0 s → K + K − ) = 2.16 × 10 −5 , this is while its experimental value is B(B 0 s → K + K − ) = (2.50 ± 0.17) × 10 −5 [16].
For the pure annihilation mode of the χ c0 → K + K − theoretical calculation via factorization approach become from 10 −11 order, while the experimental value of that is from 10 −3 order. This is a strange result because the χ c0 meson has very small lifetime of 10 −23 s. The fact is that the branching fraction of the pure annihilation decays by using the factorization approaches become smaller than experimental one [30,31], from the theoretical calculation it seems that in the decay of χ c0 → K + K − , before the K + and K − mesons are produced in the final states, the pare mesons such as D + s , D − s and D 0 ,D 0 are produced in the intermediate state so the final state interaction effects are needed.
Finally, by using the obtained theoretical branching ratios the results for the quasi-two-body decay of B + c → K + K − π + become from 1.98 × 10 −6 to 7.32 × 10 −6 .