ψπ+ decay channel

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Nuclear Physics A ••• (••••) •••–•••

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www.elsevier.com/locate/nuclphysa

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Evaluation of the three body branching fraction of Bc+ → J /ψD 0 K + by applying the Bc+ → J /ψπ + decay channel

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Behnam Mohammadi, Akbar Abdisaray

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Department of Physics, Urmia University, Urmia, Iran

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Received 21 November 2018; received in revised form 15 March 2019; accepted 17 March 2019

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Abstract

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Three body Bc+ → J /ψD 0 K + decay, which has been recently observed by LHCb collaboration, and are calculated in the model which takes into account the intermediate resonances and nonresonant contributions. In this process, the Bc meson decays first into J /ψ and the quark pair c¯s , and then the quark pair hadronizes into DK components, which undergo final state interaction. The nonresonant contributions arise from the +(∗) two-body matrix elements of the weak current D 0 K + |Jμweak |0 transition. The Ds mesons are used for intermediate pseudoscalar and vector resonant contributions. The annihilation amplitude arising from the 3-body matrix element of J /ψD 0 K + |c¯s |0 is also associated. Their effects are described in terms of the Breit-Wigner formalism. Because of the Bc+ → J /ψD 0 K + branching fraction is calculated relative to the Bc+ → J /ψπ + decay, this decay mode is estimated separately, the ratio between the B(Bc+ → J /ψD 0 K + ) and B(Bc+ → J /ψπ + ) to be 0.382 ± 0.084 that is compatible with the experimental data 0.432 ± 0.136 ± 0.028. © 2019 Elsevier B.V. All rights reserved.

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Keywords: B meson decays; Weak interactions; QCD factorization; Resonances and nonresonant contribution

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E-mail addresses: [email protected] (B. Mohammadi), [email protected] (A. Abdisaray). https://doi.org/10.1016/j.nuclphysa.2019.03.007 0375-9474/© 2019 Elsevier B.V. All rights reserved.

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1. Introduction

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Bc meson is one of the most interesting mesons that can be studied at the Tevatron, the discovery of the Bc was reported by the CDF collaboration in the Bc → J /ψl ± ν¯ l process at Fermilab [1]. After that, the decay mode Bc± → J /ψπ ± has been observed by CDF and D0 collaboration significance of more than 8σ and 5σ respectively [2,3]. This decay mode and Bc± → J /ψDs± decay has also been observed by the LHCb collaboration at the LHC center-of-mass energy 7 TeV of proton-proton collisions [4,5]. Studies of Bc 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, Bc 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 Bs0 mesons, more than 70% of the Bc+ width is due to c-quark decays, in which c → s transition has been observed with Bc+ → Bs0 π + decays [7]. Around 20% of its width is due to the b-quark decays [8]. In charmless final states, the ¯ → W + → qq bc ¯ annihilation amplitudes account for only 10% of the Bc+ width [9]. In this work the spectator diagrams are expected for the Bc+ → J /ψD 0 K + decays such as (∗)+ decays [5]. The weak annihilation topology, unlike other B meson decays, the Bc+ → J /ψDs is not suppressed and can contribute to the amplitude of the Bc+ → J /ψD 0 K + decay, our calculations show that the contribution from the weak annihilation diagram is 11.54% of the total branching ratio of the Bc+ → J /ψD 0 K + decay. The resonance mesons related to this work in (∗)+ mesons, such as the ones that have been introduced in the DK system are categories of Ds [10], they have contributed either the color-favored tree diagram or the color suppressed tree diagram, they have found that the penguin diagram contribution is in the leading order as the tree contribution in decays Bc+ → D 0 K + . ∗+ , have been observed by CLEO, its The charmed strange J P = 2+ meson, designated Ds2 0 + ∗0 + allowed decay modes are D K and D K , both proceeding through a D-wave [11]. The Ds1 (2536)+ has been detected in its D ∗ K decay mode and analysis of the D ∗ decay angular distribution prefers J P = 1+ [12]. A narrow state, which have been tagged DsJ (2458)+ , has been observed by the BABAR collaboration, since the DsJ (2458) mass lies above the kinematic threshold for decay to DK (but not for D ∗ K), the narrow width suggests this ∗ (2317)+ is lower than the threshold DK so decay does not occur [13,14]. The state DsJ + 0 only decays to Ds π [14–19]. Using a sum rule and lattice QCD that reformulates Wein∗ (2317) contains a KD component in an amount berg compositeness condition the state Ds0 ∗ of about 70%, while the state Ds1 (2460) contains a similar amount of KD ∗ [20,21]. After that the BABAR collaboration observed a new Ds meson decaying into D 0 K + by describing economically as two s-wave states (Ds+ , Ds∗+ ) with J P = 0− , 1− , and four p-wave states ∗ (2317)+ , D (2460)+ , D (2536)+ , D (2573)+ ) with J P = 0+ , 1+ , 1+ , 2+ , though the (Ds0 s1 s1 s2 last two spin-parity assignments are not firmly established [22]. Another state related to our ∗ (2700)+ resonance that has been observed by the LHCb collaboration. Later, this work is the Ds1 observation was confirmed by BaBar collaboration [23]. In the present work, the D 0K + invariant mass distributions should be given in the decay Bc → J /ψD 0 K + , from which information on ∗ (2317)+ , D (2460)+ , D (2536)+ , D ∗ (2573)+ , D ∗ (2700)+ the internal structures of the Ds0 s1 s1 sJ s1 + and DsJ (2856) states will be obtained. Besides the weak decay of the Bc meson and hadronization of the quark-antiquark pair to two mesons, the final state interaction is involved. In order to describe the final state interaction, the chiral unitary approach which makes use of the onshell

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+ Fig. 1. Feynman diagrams for Bc+ → J /ψDsi → J /ψD 0 K + decay.

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version of the factorized Bethe-Salpeter equation are used which has successfully explained the existence of some resonances. In this research, the Bc+ → J /ψD 0 K + branching fraction is calculated relative to the Bc+ → J /ψπ + decay, these analyses are studied using a simple model based on the framework of the factorization approach [24]. In the study of the three body decay mechanism, double and single pole form factors for the Bc+ meson transitions are considered. The three-body meson decays are generally dominated by intermediate vector and scalar resonances, namely, they proceed via quasi-two-body decays [25] containing a resonance state and a pseudoscalar meson. Indeed, most of the quasi-two-body decays are extracted from the analysis of three-body decays using the Dalitz plot technique [26]. First evidence for the Bc± meson in the fully reconstructed decay channel Bc± → J /ψπ ± , with J /ψ → μ+ μ− and first observation of the decay Bc± → J /ψπ ± were reported by CDF collaboration [2,27] the branching fraction of Bc+ → J /ψπ + decay listed in the PDG is less than 8.2 × 10−5 [28]. The ratio between the branching fractions of Bc+ → J /ψD 0 K + and Bc+ → J /ψπ + decays to be B(Bc+

→ J /ψD 0 K + )

B(Bc+

→ J /ψπ + )

= 0.382 ± 0.084,

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(1)

that is compatible with the experimental data 0.432 ± 0.136 ± 0.028 [29].

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2. Decay amplitudes 2.1. Nonresonant and resonant amplitudes of the Bc+ → J /ψD 0 K + decay In this paper, we will discuss the decay mechanism of the Bc+ meson into J /ψD 0 K + and also + ∗ (2317)+ , D (2460)+ , D (2536)+ , D ∗ (2573)+ , where i denote sum over the Ds0 into J /ψDsi s1 s1 sJ ∗ (2700)+ and D (2856)+ resonance mesons. In the next step these resonance mesons are Ds1 sJ decaying into the pair of D 0 K + mesons, for this purpose the DK invariant mass is required. In hence the mass spectrum of D (∗) K (∗) resonances can be chosen from order mD (∗) K (∗) = 2358.52 + + MeV to 2906.22 MeV. According to Fig. 1 we need the effective vertices Bc+ → J /ψDsi , Dsi → 0 + + D K . Because the Bc meson decays due to the weak interactions, the impact factor of effective contributions to the decay amplitude is frighteningly large, especially when taking into account angular momentum due to its conservation, space parity is not conserved, this fact restricts the + number of possible Lorenz structures in the effective decay amplitudes. The coupling of Dsi with DK is considered to be due to the strong interactions, hence they conserve parity. We take into account the kinematics of the decay as Bc+ (p) → J /ψ(q) + D(k1 ) + K(k2 ), all particles in the final state have relatively low momenta. Indeed, the invariant mass of the DK pair varies in the range mD + mK ≤ mDK ≤ mBc − mJ /ψ , in which 2358.52 ≤ mDK ≤ 3180.08 MeV. When

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the final J /ψ meson is at rest while the two D and K mesons move in opposite directions the maximum momentum of the D(K) is reached, and this gives |k1|max /mD = 0.52. The color-suppressed internal and color-allowed external W-emission tree diagrams of the selected Bc → J /ψDK decay are shown in Fig. 1(a) and 1(b), the weak annihilation topology is addressed in Fig. 1(c), that is not suppressed and can contribute significantly to the decay amplitude. According to Fig. 1 as an illustration, under the factorization approach, the Bc → J /ψDK nonresonant decay amplitude consists of three distinct factorizable terms: (i) the current-induced process with a meson emission, Bc → DK × 0 → J /ψ, (ii) the transition process, Bc → J /ψ × 0 → DK, and (iii) the annihilation process Bc → 0 × 0 → J /ψDK, where A → B denotes a A → B transition matrix element. For the nonresonant current-induced pro¯ V −A |Bc (p) has the general cess, the two-meson transition matrix element D(k1 )K(k2 )|(cb) expression as [30]

nr +iω− (k2

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D(k1 )K(k2 )|(c¯s )V −A |0 =

(k1 − k2 )μ (1 −

s+ 2χ

+ i  ∗χ )

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(3)

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(4)

,

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−[(p + q)μ −

(m2Bc

( · p) Bc →J /ψ + q ]A (s ), s+ μ 0

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( · p) B →J /ψ + q )]A c (s ) s+ μ 1 B →J /ψ + − m2J /ψ ) A2 c (s ) q ]( · p) μ s+ (mBc + mJ /ψ )

J /ψ(, q)|Vμ − Aμ |Bc (p) = [(mBc + mJ /ψ )(μ −

+[2mJ /ψ

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where ∗ = 200MeV, χ = 830MeV and the form factor of Bc (p) → J /ψ(, q) is defined as

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where s12 = (k1 + k2 )2 , s13 = (k1 + q)2 and s23 = (k2 + q)2 . Up to trivial angular rotations, two independent variables label the final state, which may usefully be taken as the Dalitz plot variables: s12 = s + , s23 = s − and s13 = s 0 , where the relation s + + s − + s 0 = m2Bc + m2J /ψ + m2D + m2K , tells that there are only two independent variables, say s + and s − . For the transition amplitude the matrix element can be expressed in terms of time-like current form factors as [33]

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(2)

¯ V −A |Bc (p)NR A1 ∝ J /ψ(q)|(cc) ¯ V −A |0D(k1 )K(k2 )|(cb) current−ind fJ /ψ 2 nr =− [(2mJ /ψ r + 2 nr nr + (s23 − s13 − m2K + m2D )ω− ], (m2Bc − s12 − m2J /ψ )ω+

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nr and ωnr are explicitly ¯ V −A ≡ bγ ¯ μ (1 − γ5 )c and the nonresonant form factors r nr , ω− where (cb) + defined in [31,32]. This leads to nonresonant amplitude of the current-induced process

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− k1 )μ ,

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nr ¯ V −A |Bc (p) = ir nr (p − k1 − k2 )μ + iω+ D(k1 )K(k2 )|(cb) (k2 + k1 )μ

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(5)

here qμ = (p − q)μ and under the Lorentz condition  · q = 0. In this work we make the extrapolation by using the formula in Ref. [34] f (q 2 ) = f (0)exp(σ1 q 2 + σ2 q 4 ), where the f (q 2 ) B →J /ψ + B →J /ψ + B →J /ψ + (s ), A1 c (s ) and A2 c (s ) and σ1 , σ2 and denote the weak form-factors A0 c f (0) are the parameters to be determined by the fitting procedure. So the nonresonant amplitude of the transition process is described by

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¯ V −A |Bc (p)NR A2 ∝ D(k1 )K(k2 )|(c¯s )V −A |0J /ψ(, q)|(cb) transition = 

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= 2m2Bc

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(9)



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si

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si

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g Dsi →DK . m2D + − s + − imD + D + si

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+ ¯ + Dsi |cb|Bc 

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g Dsi →DK fD + 2 , si m − s + − imD + D + D+

+ where g Dsi →DK is the Dsi → DK strong coupling. Resonant mesons, with intermediate scalar pole contributions, can also contribute to the three-body matrix element as

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¯ c (p)R = D(k1 )K(k2 )|cb|B

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As we know the amplitude in the SM consists of the short-distance and long-distance contributions. For the resonant contributions we implement the effects of long-distance contributions + + from the decays Bc+ → J /ψDsi where Dsi is the resonant mesons. The two-body matrix element R DK|c¯s |0 receives resonant contribution is described by



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D(k1 )K(k2 )|c¯s |0 = (k1 + k2 )μ

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2 2 + − fB s + s − mJ /ψ − mD . − c fK m2B − m2K

R

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(m2Bc − m2K )(2 − 3m2Bc )

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(6)

m2Bc + m2J /ψ − s −

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¯ c (p)NR A3 ∝ J /ψ(q)D(k1 )K(k2 )|(cγ5 s¯ )|00|(cγ5 b)|B annihilaton

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The matrix elements involving final three mesons creation are given by

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¯ V −A |Bc (p) = −2 fBc (p · k2 ) . (8) J /ψ(q)D(k1 )K(k2 )|(c¯s )V −A |00|(cb) fK (m2Bc − m2K )

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m2K

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where p = pBc = q + k1 + k2 . It is easily seen that in the chiral limit

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¯ V −A |Bc (p) = ifBc pμ , 0|(cb)

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2i p · k2 (k2μ − 2 pμ ), fK (mBc − m2K )

J /ψ(q)D(k1 )K(k2 )|(c¯s )V −A |0 =

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(λ=0) = (|

The J /ψ meson q |, 0, 0, q 0 )/mJ /ψ and  (λ=±1) = √ polarization vectors become:  ∓(0, 1, ±i, 0)/ 2. Finally, the point-like 3-body matrix element and the vacuum state are presented as [33]

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 ( · p)(m2D − m2K )  Bc →J /ψ + × (mBc + mJ /ψ )  · (k1 − k2 ) − (s ) A1 s+  ((k − k ) · (p + q)) (m − m )(m2 − m2 )  Bc J /ψ 1 2 B →J /ψ + D K − (s ) ( · p)A2 c − mBc + mJ /ψ s+ 2mJ /ψ ( · p) Bc →J /ψ +  A0 (s ) . +(m2D − m2K ) s+

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si

However, there are additional resonant contributions to this three-body matrix element due to the intermediate vector mesons as

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¯ c (p)R D(k1 )K(k2 )|cb|B +∗   g Dsi →DK +∗ ¯ + ∗ = D +∗ · (k1 − k2 )Dsi |c b|Bc . 2 + − im +∗ +∗ si m − s D D i pol D +∗

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Hence the resonant amplitudes are described by

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Bc+ →J /ψ

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¯ c+ (p)R = 2mJ /ψ ( · p)A A4 ∝ D(k1 )K(k2 )|(c¯s )|0J /ψ(q)|(cb)|B 0

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×

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fD + g

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(s + )

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si

m2D + − s + − imD + D + si

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and

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= fJ /ψ

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+mJ /ψ

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×

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−

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g m2D +∗ − s + − imD +∗ D +∗ si

si

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si

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+∗  B + →Dsi · (k1 − k2 ) (J /ψ · D +∗ )(mBc+ + mD +∗ )A1 c (m2J /ψ ) si

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si

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(m2J /ψ ) 

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mBc+ + mD +∗ si

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∗ D +∗ si

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+ + B + →Dsi − m2D + )F0 c (m2J /ψ )g Dsi →DK si m2D + − s + − imD + D + si si si +∗ Dsi →DK

+∗ B + →Dsi

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 pol

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(m2B + c

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¯ c (p)J /ψ(q)|(c¯s )|0R A5 ∝ D(k1 )K(k2 )|(cb)|B

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+ Ds1 →DK

The strong coupling constants such as g and g are determined from the measured partial widths through the relations V = 2pc3 gV2 →DK /(12πm2V ) and S = 2 pc gS→DK /(8πm2S ) for vector and scalar mesons, respectively, where pc is the c.m. momentum. Then the decay amplitude including nonresonant and resonant contributions reads A(Bc+

+

→ J /ψ(, q)D (k1 )K (k2 )) GF = √ Vcb Vcs∗ [a2 (A1 + A5 ) + a1 (A2 + A3 + A4 )], 2

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0

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Bc+

→ J /ψD 0 K +

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(15)

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where a1 = c1 + c2 /3. The vertex corrections to the decays, denoted as fI in QCD factorization, have been calculated in the NDR scheme, and can be adopted directly. Their effects can be combined into the Wilson coefficients associated with the factorizable contributions as [35] c1 αs c1 mb a2 = c 2 + (16) + [−18 + 12ln( ) + fI ], 3 9π μ

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where

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Fig. 2. Feynman diagram for Bc+ → J /ψπ + decay.

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√ (1 − x) 2 6 (1 − 2x) dxφJL/ψ (x)[3 ], fI = ln(x) − 3πi + 3ln(1 − r 2 ) + 2r 2 fJ /ψ (1 − x) (1 − r 2 x)

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φJL/ψ (x) =

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The decay rate of

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9.58fJ /ψ 1−x x(1 − x)[x ]0.7 . √ 1 − 2.8x(1 − x) 2 6 Bc+

→ J /ψD 0 K +

(Bc+

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1 1 → J /ψD K ) = 3 (2π) 32m3B 0

+

where

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− = (mD + mK )2 , smax

+ smin,max (s − ) = m2D

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(18)

with

+ y2

2.2. Amplitude of the

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(19)

Bc+

+ z2

)2

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and

1 − − [(m2Bc − s − − m2J /ψ )(s − − m2K + m2D ) 2s

 ∓ λ(s − , mBc , mJ /ψ ) × λ(s − , mD , mK )],

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(20)

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decay

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we will consider b¯ decays while c acts as a spectator. According to Fig. 2, at the quark level, ¯ transition and the corresponding effective Bc+ → J /ψπ + decay is characterized by b¯ → (c¯du) Hamiltonian is given by  GF  p p =√ λp (c1 Q1 + c2 Q2 + ci Qi + c7γ q7γ + c8g Q8g ), 2 p=u,c i=3,...,10

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− 2(xy + xz + yz).

→ J /ψπ +

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In the following, we calculate the amplitude of the Bc+ → J /ψπ + decay. Different with Bu,d,s mesons, the Bc+ system consists of two heavy quarks b¯ and c, which can decay individually. Here

Heff

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smax smax |A(Bc+ → J /ψD 0 K + )|2 ds + ds − ,

= (mBc − mJ /ψ

+ m2J /ψ

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λ(x, y, z) = x 2

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c + − smin smin

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− smin

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(17)

is then given by

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here φJL/ψ (x) is the J /ψ meson asymptotic distribution amplitude which is given by [36]

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(21)

where λp is the CKM matrix elements, ci are the Wilson coefficients evaluated at the renorp malization scale μ, Q1,2 are the left-handed current-current operators arising from W-boson exchange, Q3,...,6 and Q7,...,10 are QCD and electroweak penguin operators, and Q7γ and Q8g are the electromagnetic and chromomagnetic dipole operators. Because only the tree level diagram exist for this decay mode, so we have considered to current-current operators Qc1 and Qc2 as: Qc1 = (b¯α cα )V −A (d¯β uβ )V −A and Qc2 = (b¯α uβ )V −A (d¯α cβ )V −A . Here α and β are the

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Fig. 3. Nonfactorizable diagrams for Bc+ → J /ψπ + decay.

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15 16

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SU(3) color indices and the subscript V − A represent the chiral projection 1 − γ5 . The amplitude of the Bc+√ → J /ψπ + decay by using the color-allowed external W-emission tree diagram ∗ J /ψπ + |(c Qc + c Qc )|B +  , where J /ψπ + |(c Qc )|B +  is the become: (GF / 2)Vcb Vud 1 1 2 2 i i c F c F factorized hadronic matrix element, which has the same definition as that in the “nonfactorizable” approach. All the “nonfactorizable” effects (coming from the vertex-correction and hard spectator-scattering diagrams) are encoded in the coefficients a1 , which are process dependent and can be obtained perturbatively by calculating the diagrams in Fig. 3. The general form of the coefficients a1 at next-to-leading order in αs , can be written as c2 c2 αs 4π 2 a1 (J /ψπ ) = (c1 + )N (π + ) + (22) [V (π + ) + H (J /ψπ + )], 3 9π 3 where N(π + ) is the leading-order coefficient which will be considered N (π + ) = 1 [37,38]. The quantities V (π + ) account for one-loop vertex corrections and H (J /ψπ + ), the correction from hard gluon exchange between π + and the spectator quark are given by [37,38] +

31 32 33

V (π + ) =

34

dx6x(1 − x)[12ln( 0

37 38 39 40 41 42 43 44 45 46 47

fB fπ H (J /ψπ ) = 4π 2 c mBc F Bc →π +

18 19 20 21 22 23 24 25 26 27 28 29 30 31

1

35 36

17

1

2

0

mb 1 − 2x )+3 ln(x) − 3iπ], μ 1−x

Bc (ξ ) dξ ξ

1

33 34 35

dyL J /ψ 0

32

y , 1−y

(23)

here Bc (ξ ) is the wave function of Bc+ meson, we make use of

the same parameterizations as 2 2 2 2 2 in Ref. [39–41]: Bc (ξ ) = Nb ξ (1 − ξ ) exp − (ξ mBc )/(2ωb ) , in this research Nb = 8644.12 1 is a normalization factor ( 0 Bc (ξ )dξ = 1) that is related to the Bc+ meson decay constant fBc √ 1 via 0 Bc (ξ )dξ = fBc /(2 6). The ωb is the parameter in the wave function of the Bc+ meson and we take ωb = 400 MeV in numerical calculations. By applying these definitions we find: 1 1 0 dξ Bc (ξ )/ξ = 13.18 which is near to the 0 dξ Bc (ξ )/ξ = mBc /λb = 15.75. Since Bc (ξ ) has support only for ξ from order QCD /mBc , λb can be chosen from order QCD = 200 MeV to 500 MeV, we fix λb = 400 MeV.

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In this decay mode the J /ψ meson is placed in the form factor, the meson of π + is produced from the vacuum state, therefore the amplitude of this decay consist of Bc+ → J /ψ multiplied by 0 → π +  which is factorizable term. The expressions of decay amplitude for Bc+ → J /ψπ + within the QCDF framework can be written as √ B →J /ψ ∗ A(Bc+ → J /ψπ + ) = 2GF Vcb Vud fπ A0 c (m2π )mJ /ψ (J /ψ · pπ )a1 . (24) The decay rate of

8 9 10 11 12 13 14 15

9

Bc+

→ J /ψπ +

18 19 20

(25)

in which |p|

is the absolute value of the 3-momentum of the J /ψ or π + mesons that can be calculated via: (m2Bc + m2J /ψ − m2π )2 − 4m2Bc m2J /ψ /(2mBc ).

The branching ratio of the

31 32 33 34 35 36 37 38 39

G2F



42 43 44 45 46 47

9 10 11 12 13 14

18

2

19 20 21

−

22

(26)

and for the

Bc+

→ J /ψπ +

25

So the ratio

26

decay the branching fraction is calculated via

B(Bc+ → J /ψπ + )

2

2

2F 1 |p|G Bc →J /ψ ∗ 2 2 2 2 V m f a (m ) (J /ψ · pπ )2 . = V A cb π π ud J /ψ 1 0

Bc 4πm2B c B(Bc+

→ J /ψD 0 K + )

to the

23 24

+ − smin smin

B(Bc+

→ J /ψπ + )

27 28 29

(27)

can now be estimated as follows:

V ∗ 2 B(Bc+ → J /ψD 0 K + ) 1 cs = ∗ 128π 2 mBc |p|

Vud B(Bc+ → J /ψπ + )  2 +  s−  smax  + − max  a (A + A ) + a (A + A + A )   ds ds 2 1 1 2 3 4 5 + − smin smin . (28)

2 B →J /ψ m2J /ψ fπ2 a12 A0 c (m2π ) (J /ψ · pπ )2

30 31 32 33 34 35 36 37 38 39 40

40 41

6

17

decay is then given by

1 Vcb Vcs∗

Bc 512π 3 m3B c +

25

30

→ J /ψD 0 K +

smax smax 2   × a2 (A1 + A5 ) + a1 (A2 + A3 + A4 ) ds + ds − ,

24

29

Bc+

B(Bc+ → J /ψD 0 K + ) =

23

28

5

16

22

27

4

15

3. Branching fractions and numerical results

21

26

3

8

16 17

2

7

in Bc meson rest frame can be written as

1 |p|

(Bc+ → J /ψπ + ) = |A(Bc+ → J /ψπ + )|2 , 8π m2B c

1

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 c1 and c2 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 = mb scale, which have been obtained in the NDR scheme [37]: c1 = 1.081 and c2 = −0.190.

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The CKM matrix elements, the CKM matrix elements required are [28]: Vud = 0.97420 ± 0.00021, Vcs = 0.997 ± 0.017 and Vcb = 0.0422 ± 0.0008. Masses and decay constants (in units of MeV), the meson masses and decay constants needed in our calculations are taken as [28]: mBc = 6274.9 ± 0.8, mJ /ψ = 3096.900 ± 0.006, mD 0 = 1864.83 ± 0.05, mK + = 493.677 ± 0.016, mπ + = 139.57061 ± 0.00024, fJ /ψ = 418 ± 9, fK = 159.8 ± 1.84 and fBc = 434 ± 15 [42]. Form factors, for the parameters σ1 , σ2 and f (0) used in transition weak form factors we take B →J /ψ + (s ); f (0) = 0.46, σ1 = 0.038, σ2 = [43]: f (0) = 0.59, σ1 = 0.047, σ2 = 0.0017 for A0 c Bc →J /ψ + B →J /ψ + (s ) and f (0) = 0.64, σ1 = 0.064, σ2 = 0.0041 for A2 c (s ); for the 0.0015 for A1 Bc →J /ψ B →J /ψ c form factor involving the A0 (m2π ) transition, we use A0 (m2π ) = 0.59. For the branching fraction of the Bc+ → J /ψπ + decay we obtain B(Bc+ → J /ψπ + ) = (2.44 ± 0.38) × 10−3 for which is in good agreement with the upper limit of the span the range [0.34, 2.9] × 10−3 [44], and for the B(Bc+ → J /ψD 0 K + ) the value of the (9.319 ± 1.454) × 10−4 is obtained. The relative branching fraction of the Bc+ decays is calculated to be

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

(29)

39 40 41 42 43 44 45 46 47

3 4 5 6 7 8 9 10 11 12 13 14 15

17 18 19

4. Conclusion

20

In this study, we have presented a comprehensive calculation of the Bc+ → J /ψD 0 K + and + Bc → J /ψπ + decays. In fact, we have been interested in examining the branching ratio of the Bc+ → J /ψD 0 K + decay relative to the Bc+ → J /ψπ + decay. In the calculation of the Bc+ → J /ψD 0 K + decay, we have considered the nonresonant and resonant contributions, the nonresonant contribution contains of three factorizable terms: (i) the current-induced process with a meson emission, (ii) the transition process and (iii) the annihilation process. For the resonant contribution we implement the effects of long-distance contributions from the de+ ∗ (2317)+ , D (2460)+ , D (2536)+ , where i denote sum over the Ds0 cays Bc+ → J /ψDsi s1 s1 ∗ ∗ + + + DsJ (2573) , Ds1 (2700) and DsJ (2856) resonance mesons. Under the QCD factorization approach, the amplitude of the Bc+ → J /ψπ + decay has been calculated by using the colorallowed external W-emission a1 tree diagram, all the “nonfactorizable” effects (coming from the vertex-correction and hard spectator-scattering diagrams) are encoded in the coefficients a1 , which are process dependent and can be obtained perturbatively. The ratio between the B(Bc+ → J /ψD 0 K + ) and B(Bc+ → J /ψπ + ) is 0.382 ± 0.084 that is compatible with the experimental data 0.432 ± 0.136 ± 0.028 [29].

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

37 38

2

16

B(Bc+ → J /ψD 0 K + ) = 0.382 ± 0.084. B(Bc+ → J /ψπ + )

19 20

1

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