Final State Interaction Effects on the 𝐵⁺→𝐽/𝜓𝜌⁺ Decay

Abstract

The exclusive decay of 𝐵⁺→𝐽/𝜓𝜌⁺ is studied in the framework of the QCD factorization (QCDF) method and final state interaction (FSI). A direct 𝐵⁺→𝐽/𝜓𝜌⁺ decay is only occurred via a tree and a penguin based on the quark diagram analysis. The result that is found by using the QCDF method is less than the experimental result, so, the role of FSI is considered. The intermediate states 𝐷⁺𝐷⁰, 𝐷^+∗𝐷^0∗, 𝐷^+∗𝐷⁰, and 𝐷⁺𝐷^0∗ via the exchange of 𝐷⁻ and 𝐷^−∗ are contributed to the 𝐵⁺→𝐽/𝜓𝜌⁺ decay. The above intermediate states is calculated by using the QCDF method. In the FSI effects the results of our calculations depend on “η” as the phenomenological parameter. The range of this parameter are selected from 1 to 2. For the exchanged particles 𝐷⁻ and 𝐷^−∗, it is found that if 𝜂=1.58∼1.83 is selected the numbers of the branching ratio are placed in the experimental range. The experimental branching ratio of 𝐵⁺→𝐽/𝜓𝜌⁺ decay is (5±0.8)×10⁻⁵, and our prediction number is (1.42±0.36)×10⁻⁵ in the absence of FSI effects, and it becomes (4.2∼5.8)×10⁻⁵ when FSI contributions are taken into account.

1. Introduction

The importance of FSI in weak nonleptonic 𝐵 meson decays is investigated by using a relativistic chiral unitary approach based on coupled channels [1–3]. The chiral Lagrangian approach is proved to be reliable for evaluating hadronic processes, but there are too many free parameters which are determined by fitting data, so that its applications are much constrained. Therefore, we have tried to look for some simplified models which can give rise to reasonable estimation of FSI [4, 5]. The FSI can be considered as a rescattering process of some intermediate two-body states with one particle exchange in the t-channel and computed via the absorptive part of the hadronic loop level (HLL) diagrams. The calculation with the single-meson-exchange scenario is obviously much simpler and straightforward. Moreover, some theoretical uncertainties are included in an off-shell form factor which modifies the effective vertices. Since the particle exchanged in the t-channel is off shell and since final state particles are hard, form factors or cutoffs must be introduced to the strong vertices to render the calculation meaningful in perturbation theory. If the intermediate two body mesons are hard enough, so that the perturbative calculation can make sense and work perfectly well, but the FSI can be modelled as the soft rescattering of the intermediate mesons. When one or two intermediate meson can reach a low-energy region where they are not sufficiently hard, one can be convinced that at this region the perturbative QCD approach fails or cannot result in reasonable values. If the intermediate mesons are soft, one can conjecture that at this region the nonperturbative QCD would dominate, and it could be attributed into the FSI effects. Because all FSI processes are concerning nonperturbative QCD [6], we have to rely on phenomenological models to analyze the FSI effects in certain reactions. In fact, after weak decays of heavy mesons, the particles produced can rescatter into other particle states through nonperturbative strong interaction. In the FSI the decay 𝐵⁺→𝐽/𝜓𝜌⁺ is a very interesting mode [7]. We calculated the 𝐵⁺→𝐽/𝜓𝜌⁺ decay according to QCDF method and selected the leading order Wilson coefficients at the scale 𝑚_𝑏 and obtained the BR(𝐵⁺→𝐽/𝜓𝜌⁺)=(1.42±0.36)×10⁻⁵. The FSI can give sizable corrections, and we can utilize it [8]. Rescattering amplitude can be derived by calculating the absorptive part of triangle diagrams. In this case, intermediate states are 𝐷⁺𝐷⁰, 𝐷^+∗𝐷^0∗, 𝐷^+∗𝐷⁰, and 𝐷⁺𝐷^0∗. Then, we calculated the 𝐵⁺→𝐽/𝜓𝜌⁺ decay according to HLL method. Taken FSI corrections into account the branching ratio of 𝐵⁺→𝐽/𝜓𝜌⁺ becomes (4.2∼5.8)×10⁻⁵ and the experimental result of this decay is (5±0.8)×10⁻⁵ [9].

This paper is organized as follows. We present the calculation of QCDF for 𝐵⁺→𝐽/𝜓𝜌⁺ decay in Section 2. In Section 3, we calculate the amplitudes of the intermediate states of 𝐵⁺→𝐷^+(∗)𝐷^0(∗) decays. Then, we present the calculation of HLL for 𝐵⁺→𝐽/𝜓𝜌⁺ decay in Section 4. In Section 5, we give the numerical results, and in the last section, we have a short conclusion.

2. 𝐵⁺→𝐽/𝜓𝜌⁺ Decay in QCD Factorization

A detailed discussion of the QCDF approach can be found in [10, 11]. Factorization is a property of the heavy-quark limit, in which we assume that the b quark mass is parametrically large. The QCDF formalism allows us to compute systematically the matrix elements of the effective weak Hamiltonian in the heavy-quark limit for certain two-body final states 𝐽/𝜓𝜌⁺. In this section, we obtain the amplitude of 𝐵⁺→𝐽/𝜓𝜌⁺ decay using QCDF method. In factorization approach, there are color-suppressed tree and allowed penguin diagrams to 𝐵⁺→𝐽/𝜓𝜌⁺ decay. We adopt leading order Wilson coefficients at the scale 𝑚_𝑏 for QCDF approach. The diagrams describing this decay are shown in Figure 1. According to the QCDF, the amplitude of 𝐵⁺→𝐽/𝜓𝜌⁺ decay is given by 𝐴_QCDF𝐵⁺⟶𝐽𝜓𝜌⁺𝐺=−𝑖^𝐹√2𝑓_𝐽/𝜓𝑚_𝐽/𝜓𝜖_𝐽/𝜓⋅𝜖_𝜌𝑚_𝐵+𝑚_𝜌𝐴₁^𝐵𝜌𝑚²_𝐽/𝜓−𝜖_𝐽/𝜓⋅𝑝_𝐵𝜖_𝜌⋅𝑝_𝐵2𝐴₂^𝐵𝜌𝑚²_𝐽/𝜓𝑚_𝐵+𝑚_𝜌−𝑖𝜀_{𝜇𝜈𝛼𝛽}𝜖^𝜇_𝐽/𝜓𝜖^𝜈_𝜌𝑝^𝛼_𝐵𝑝^𝛽₁×2𝑉^𝐵𝜌𝑚²_𝐽/𝜓𝑚_𝐵+𝑚_𝜌×𝑉_𝑐𝑏𝑉^∗_𝑐𝑑𝑎₂−𝑉_𝑡𝑏𝑉^∗_𝑡𝑑𝑎₃+𝑟_𝜒^𝐽/𝜓𝑎₅+𝑎₇+𝑎₉, (2.1) where 𝑟_𝜒^𝐽/𝜓=2𝑚_𝐽/𝜓𝑚_𝑏𝑓^⟂_𝐽/𝜓𝑓_𝐽/𝜓. (2.2) The effective coefficients 𝑎_𝑖, which are specific to the factorization approach, and defined as 𝑎_𝑖=𝑐^eff_𝑖+1𝑁_𝑐𝑐^eff_𝑖+1(𝑖=odd),𝑎_𝑖=𝑐^eff_𝑖+1𝑁_𝑐𝑐^eff_𝑖−1(𝑖=even), (2.3) where the quantities of 𝑐^eff_𝑖 are effective Wilson coefficients at the renormalization scale 𝜇 for the 𝑏→𝑑 transition. In the above amplitude the determination of 𝑎₂ in the 𝑏→𝑐 current-current transitions has received a lot of attention, the quantities of 𝑎₃, 𝑎₅ and 𝑎₇, 𝑎₉ are the QCD-penguin and electroweak-penguin coefficients, respectively. Numerical values of 𝑎_𝑖(𝑖=1,…,10) for representative value of the phenomenological parameter 𝑁_𝑐 are displayed in Section 5.

Figure 1 

The Feynman diagrams contributing to the 𝐵+→𝐽/𝜓𝜌+ decay.

3. Amplitudes of Intermediate State

To estimate the 𝐵⁺→𝐷^+(∗)𝐷^0(∗) decays amplitudes of intermediate states [12], the QCDF method is again used. For 𝐵⁺→𝐷⁺𝐷⁰ decay, Feynman diagrams are shown in Figure 2, and the amplitude comes 𝐴𝐵⁺⟶𝐷⁺𝐷⁰𝐺=𝑖^𝐹√2𝑓_𝐷𝐹₀^𝐵𝐷𝑚²_𝐵−𝑚²_𝐷×𝑎₁+𝑎₂𝑉_𝑐𝑏𝑉^∗_𝑐𝑑+𝑎₄+𝑎₁₀+𝑟^𝐷_𝜒𝑎₆+𝑎₈𝜆_𝑝+𝑓_𝐵𝑓_𝐷𝑏₃+12𝑏_3,𝐸𝑊𝜆_𝑝, (3.1) where 𝜆_𝑝=_{𝑝=𝑢,𝑐}𝑉_𝑝𝑏𝑉^∗_𝑝𝑑,𝑏₃=𝐶_𝐹𝑁²_𝑐𝑐₃𝐴^𝑖₁+𝑐₅𝐴^𝑖₃+𝐴^𝑓₃+𝑁_𝑐𝑐₆𝐴^𝑓₃,𝑏_3,𝐸𝑊=𝐶_𝐹𝑁²_𝑐𝑐₉𝐴^𝑖₁+𝑐₇𝐴^𝑖₃+𝐴^𝑓₃+𝑁_𝑐𝑐₈𝐴^𝑓₃,𝐴^𝑖₁≈2𝜋𝛼_𝑠9𝑋_𝐴𝜋−4+²3+𝑟^𝐷+_𝜒𝑟^𝐷−_𝜒𝑋²_𝐴,𝐴^𝑖₃𝐴=0,_𝑓3≈6𝜋𝛼_𝑠𝑟^𝐷+_𝜒+𝑟^𝐷−_𝜒2𝑋²_𝐴−𝑋_𝐴,𝑋_𝐴=1+𝜌𝑒^𝑖𝜙𝑚ln^𝐵Λ_QCD. (3.2) There are also some similar diagrams such as 𝐵⁺→𝐷^+∗𝐷^0∗ and 𝐵⁺→𝐷^+∗𝐷⁰(𝐷⁺𝐷^0∗) decays, which have polarization vectors. In their amplitude, just contribution of 𝑎₁ is considered, which has the dominant contribution. So the amplitude of 𝐵⁺→𝐷^+∗𝐷^0∗ decay, read as 𝐴𝐵⁺⟶𝐷^+∗𝐷^0∗𝐺=−𝑖^𝐹√2𝑓_𝐷∗𝑚_𝐷∗𝑉_𝑐𝑏𝑉^∗_𝑐𝑑×𝜖^∗₁⋅𝜖^∗₂𝑚_𝐵+𝑚_𝐷∗𝐴^𝐵𝐷∗₁𝑚²_𝐷∗−𝜖₁⋅𝑝_𝐵𝜖₂⋅𝑝_𝐵2𝐴^𝐵𝐷∗₂𝑚²_𝐷∗𝑚_𝐵+𝑚_𝐷∗, (3.3) where 𝐴^𝐵𝐷∗₁ and 𝐴^𝐵𝐷∗₂ are form factors for 𝐵→𝐷^∗ [13], and the amplitude of 𝐵⁺→𝐷^+∗𝐷⁰ is given by 𝐴𝐵⁺⟶𝐷^+∗𝐷⁰=√2𝐺_𝐹𝑚_𝐷∗𝑉_𝑐𝑏𝑉^∗_𝑐𝑑𝜖_𝐷∗⋅𝑝_𝐷𝑓_𝐷∗𝐹₁^𝐵𝐷𝑚²_𝐷∗+𝑓_𝐷𝐴^𝐵𝐷∗₀𝑚²_𝐷. (3.4)

Figure 2 

𝐵+→𝐷+𝐷0 decay diagrams.

4. Final State Interaction of 𝐵⁺→𝐽/𝜓𝜌⁺ Decay

In our approach, we consider only leading contributions in charmless FSI, when the FSI for decay is calculated, two-body intermediate states, such as, 𝐷⁺𝐷⁰, 𝐷^+∗𝐷^0∗, 𝐷^+∗𝐷⁰, and 𝐷⁺𝐷^0∗ via the exchange of 𝐷⁻ and 𝐷^−∗ are contributed to the 𝐵⁺→𝐽/𝜓𝜌⁺ decay. The quark model for 𝐵⁺→𝐷^+(∗)𝐷^0(∗)→𝐽/𝜓𝜌⁺ diagram is shown in Figure 3. This diagram just shows tree level (𝑎₁ contribution), while there are other contributions that were calculated in the last section. The diagrams which determine the HLL effects on the rate of 𝐵⁺→𝐽/𝜓𝜌⁺ decay are shown in Figure 4. At the HLL, the charge exchange reaction between the 𝐽/𝜓 and 𝜌 can proceed either by exchange of a 𝐷 or 𝐷^∗ meson. In order to calculate the various Feynman diagrams for the reactions 𝐷+𝐷→𝐽/𝜓+𝜌. And so forth (were calculated in the nonrelativistic quark model [14]) we need the construct the effective three mesons vertices, such as, 𝜓𝐷𝐷 and 𝜌𝐷𝐷. We begin with an SU(4) symmetry so as to include charm, and we introduce pseudoscalar (𝑃=𝜑_𝑎𝜆_𝑎) and vector (𝑉^𝜇=𝑣^𝜇_𝑎𝜆_𝑎) meson matrices, where 𝜑_𝑎 and 𝑣^𝜇_𝑎 are pseudoscalar and vector multiplets and the 𝜆_𝑠 are SU(4) generators. The free meson lagrangian is read [15] £₀=Tr𝜕^𝜇𝑃^†𝜕_𝜇𝑃−12Tr𝜕^𝜇𝑉^†,𝜈−𝜕^𝜈𝑉^†,𝜇𝜕_𝜇𝑉_𝜈−𝜕_𝜈𝑉_𝜇. (4.1) Interaction Lagrangians are generated by replacing the space-time derivative with a gauge covariant one 𝜕_𝜇𝑓→𝐷_𝜇𝑓=𝜕_𝜇𝑓+[𝐴_𝜇,𝑓], where 𝐴_𝜇=−𝑖𝑔/2𝑉_𝜇. The vector mesons are recognizable as playing roles of gauge bosons. As is usual in effective field theory strategies, to keep gauge consistency, we must collect terms of order 𝑔². The effective lagrangian is read £_int=𝑖𝑔Tr𝑃𝑉^𝜇𝜕_𝜇𝑃−𝜕^𝜇𝑃𝑉_𝜇𝑃+12𝑔²Tr𝑃𝑉^𝜇𝑉_𝜇𝑃−𝑃𝑉^𝜇𝑃𝑉_𝜇+𝑖𝑔Tr𝜕^𝜇𝑉^𝜈𝑉_𝜇,𝑉_𝜈+𝑉^𝜇,𝑉^𝜈𝜕_𝜇𝑉_𝜈+𝑔²Tr𝑉^𝜇𝑉^𝜈𝑉_𝜇,𝑉_𝜈. (4.2) The order 𝑔 terms deliver three-point vertices and the order 𝑔² terms are responsible for so-called contact terms or four-point couplings which are necessary for a gauge invariant theory. The interactions of interest for this study are the following [5]: £_𝐷𝐷𝜌=𝑖𝑔_𝐷𝐷𝜌𝐷𝜌^𝜇𝜕_𝜇𝐷−𝜕_𝜇𝐷𝜌^𝜇𝐷,£_{𝐷∗𝐷𝜌}=−𝑔_{𝐷∗𝐷𝜌}𝜖^{𝜇𝜈𝛼𝛽}𝐷𝜕_𝜇𝜌_𝜈𝜕_𝛼𝐷^∗_𝛽+𝜕_𝜇𝐷^∗_𝜈𝜕_𝛼𝜌_𝛽𝐷,£_{𝐷∗𝐷∗𝜌}=𝑖𝑔_{𝐷∗𝐷∗𝜌}𝜕_𝜇𝐷^∗_𝜈𝜌^𝜇𝐷^∗𝜈−𝐷^∗_𝜈𝜌_𝜇𝜕^𝜇𝐷^∗𝜈+𝐷^∗𝜈𝜕_𝜇𝜌_𝜈−𝜕_𝜇𝐷^∗_𝜈𝜌^𝜈𝐷^∗𝜇+𝐷^∗𝜇𝜌^𝜈𝜕_𝜇𝐷^∗_𝜈−𝜕_𝜇𝜌_𝜈𝐷^∗𝜈,£_𝜓𝐷𝐷=𝑖𝑔_𝜓𝐷𝐷𝜓_𝜇𝜕^𝜇𝐷𝐷−𝐷𝜕^𝜇𝐷,£_{𝜓𝐷∗𝐷}=−𝑔_{𝜓𝐷∗𝐷}𝜖^{𝜇𝜈𝛼𝛽}𝜕_𝜇𝜓_𝜈𝜕_𝛼𝐷^∗_𝛽𝐷+𝐷𝜕_𝛼𝐷^∗_𝛽,£_{𝜓𝐷∗𝐷∗}=−𝑖𝑔_{𝜓𝐷∗𝐷∗}𝜓^𝜇𝜕_𝜇𝐷^∗𝜈𝐷^∗_𝜈−𝐷^∗𝜈𝜕_𝜇𝐷^∗_𝜈+𝜕_𝜇𝜓_𝜈𝐷^∗𝜈−𝜓_𝜈𝜕_𝜇𝐷^∗𝜈𝐷^∗𝜇+𝐷^∗𝜇𝜓^𝜈𝜕_𝜇𝐷^∗_𝜈−𝜕_𝜇𝜓_𝜈𝐷^∗𝜈,£_{𝜓𝜌𝐷𝐷}=−𝑔_𝜌𝐷𝐷𝑔_𝜓𝐷𝐷𝜓^𝜇𝐷𝜌_𝜇𝐷,£_{𝜓𝜌𝐷∗𝐷∗}=𝑔_{𝜓𝐷∗𝐷∗}𝑔_{𝜌𝐷∗𝐷∗}2𝐷^∗𝜈𝜌^𝜇𝐷^∗_𝜈−𝐷^∗𝜈𝜌_𝜈𝐷^∗𝜇−𝐷^∗𝜇𝜌^𝜈𝐷^∗_𝜈. (4.3) Here 𝜖₀₁₂₃=+1 and we define the charm meson iso-doublets as 𝐷^𝑇=𝐷⁰,𝐷⁻,𝐷=𝐷⁰,𝐷⁺,𝐷^∗𝑇=𝐷^∗0,𝐷^∗−,𝐷^∗=𝐷^∗0,𝐷^∗+. (4.4) In this approach there are several coupling constants in (4.3). Methods for constraining them will be discussed below. The 𝑂(𝑔²) terms carry one power of coupling constant for each three-point vertices from which the contact term collapses. The coupling constants 𝑔_𝜓𝐷𝐷, 𝑔_𝜌𝐷𝐷, 𝑔_{𝜓𝐷∗𝐷∗}, and 𝑔_{𝜌𝐷∗𝐷∗} can be derived in the standard framework of the vector meson dominance model (VDM) [16–18]. By using VDM to the coupling of 𝛾𝐷𝐷 we have 𝛾_𝜌𝑔_𝜌𝐷𝐷=√𝜋𝛼_𝑒𝑚,𝛾_𝐽/𝜓𝑔_𝜓𝐷𝐷=43√𝜋𝛼_𝑒𝑚, (4.5) where 𝛾_𝜌=3Γ_{𝜌→𝑒+𝑒−}𝑚_𝜌𝛼_𝑒𝑚,𝛾_𝐽/𝜓=3Γ_{𝐽/𝜓→𝑒+𝑒−}𝑚_𝐽/𝜓𝛼_𝑒𝑚. (4.6) The same method can be used for the 𝛾𝐷^∗𝐷^∗ coupling. Vector dominance gives 𝑔_{𝜌𝐷∗𝐷∗}=𝑔_𝜌𝐷𝐷,𝑔_{𝜓𝐷∗𝐷∗}=𝑔_𝜓𝐷𝐷. (4.7) For the 𝜌𝐷^∗𝐷 and 𝜓𝐷^∗𝐷 couplings the VDM can be applied to the radiative decays of 𝐷^∗ into D, that is, 𝐷^∗→𝐷𝛾. By using the same technique we have 𝛾_𝜌𝑔_{𝜌𝐷∗𝐷}=13√𝜋𝛼_𝑒𝑚𝑔^±_𝑉+2𝑔⁰_𝑉,𝛾_𝐽/𝜓𝑔_𝜓𝐷𝐷=43√𝜋𝛼_𝑒𝑚𝑔^±_𝑉=43√𝜋𝛼_𝑒𝑚𝑔⁰_𝑉, (4.8) where 𝑔_𝑉^±,0 are calculated in the model prediction based on relativistic potential model that we follow [19]. With the above preparation we can write out the decay amplitude involving HLL contributions with the following formula: 𝐴𝐵𝑝_𝐵⟶𝑀𝑝₁𝑀𝑝₂⟶𝑀𝑝₃𝑀𝑝₄=12𝑑𝑝^3→₁2𝐸₁(2𝜋)³𝑑𝑝^3→₂2𝐸₂(2𝜋)³×(2𝜋)⁴𝛿⁴𝑝_𝐵−𝑝₁−𝑝₂𝑀𝐵⟶𝑀₁𝑀₂×𝐺𝑀₁𝑀₂⟶𝑀₃𝑀₄, (4.9) for which both intermediate mesons (𝑀₁,𝑀₂) are pseudoscalar. And 𝐴𝐵𝑝_𝐵⟶𝑀𝑝₁𝑀𝑝₂⟶𝑀𝑝₃𝑀𝑝₄𝐺=−𝑖^𝐹2√2𝑑𝑝^3→₁2𝐸₁(2𝜋)³𝑑𝑝^3→₂2𝐸₂(2𝜋)³×(2𝜋)⁴𝛿⁴𝑝_𝐵−𝑝₁−𝑝₂𝑓_𝐷∗𝑚_𝐷∗𝑉_𝑐𝑏𝑉^∗_𝑐𝑑×𝜖^∗₁⋅𝜖^∗₂𝑚_𝐵+𝑚₁𝐴^𝐵𝑀1₁𝑚²₂−𝜖^∗₁⋅𝑝_𝐵𝜖^∗₂⋅𝑝_𝐵2𝐴^𝐵𝑀1₂𝑚²₂𝑚_𝐵+𝑚₁×𝐺𝑀₁𝑀₂⟶𝑀₃𝑀₄, (4.10) in which both meson are vector. Also 𝐺(𝑀₁𝑀₂→𝑀₃𝑀₄) involves hadronic vertices factor, which is defined as 𝐷𝑝₃𝜌𝜀₂,𝑝₂|𝑖£|𝐷𝑝₁=−𝑖𝑔_𝐷𝐷𝜌𝜖₂⋅𝑝₁+𝑝₃,𝐷^∗𝜖₃,𝑝₃𝜌𝜖₂,𝑝₂|𝑖£|𝐷𝑝₁=−𝑖√2𝑔_{𝐷∗𝐷𝜌}𝜖_{𝜇𝜈𝛼𝛽}𝜀^𝜇₂𝜖₃^∗𝜈𝑝^𝛼₁𝑝^𝛽₂,𝐷^∗𝜖₃,𝑝₃𝜌𝜖₂,𝑝₂|𝑖£|𝐷^∗𝜖₁,𝑝₁=−𝑖𝜖^𝛽₁𝜖^𝜈₂𝜖^𝜆₃2𝑝_1𝜈𝑔_𝛼𝛽−𝑝₁+𝑝_2𝛼𝑔_𝜈𝛽+2𝑝_2𝛽𝑔_𝛼𝜈,𝐷𝑝₃𝜓𝜖₂,𝑝₂|𝑖£|𝐷𝑝₁=−𝑖𝑔_𝜓𝐷𝐷𝜖₂⋅𝑝₁+𝑝₃,𝐷^∗𝜖₃,𝑝₃𝜓𝜖₂,𝑝₂|𝑖£|𝐷𝑝₁=−𝑖𝑔_{𝜓𝐷∗𝐷}𝜖_{𝜇𝜈𝛼𝛽}𝜖^𝜇₂𝜖₃^∗𝜈𝑝^𝛼₁𝑝^𝛽₂,𝐷^∗𝜖₃,𝑝₃𝜓𝜖₂,𝑝₂|𝑖£|𝐷^∗𝜖₁,𝑝₁=−𝑖𝜖^𝛽₁𝜖^𝜈₂𝜖^𝜆₃2𝑝_1𝜈𝑔_𝛼𝛽−𝑝₁+𝑝_2𝛼𝑔_𝜈𝛽+2𝑝_2𝛽𝑔_𝛼𝜈. (4.11) The dispersive part of the rescattering amplitude can be obtained from the absorptive part via the dispersion relation [6, 20]: Dis𝑀𝑚²_𝐵=1𝜋^∞_𝑠Abs𝑀𝑠^′𝑠′−𝑚²_𝐵𝑑𝑠′, (4.12) where 𝑠′ is the square of the momentum carried by the exchanged particle and s is the threshold of intermediate states, in this case 𝑠∼𝑚²_𝐵. Unlike the absorptive part, the dispersive contribution suffers from the large uncertainties arising from the complicated integrations. So the amplitudes of the mode 𝐵⁺→𝐷⁺(𝑝₁)𝐷⁰(𝑝₂)→𝐽/𝜓(𝑝₃,𝜖₃)𝜌⁺(𝑝₄,𝜖₄), where 𝐷⁻ and 𝐷^−∗ mesons are exchanged, respectively, are given by Abs(4a)=¹₋₁^→𝑝₁𝑑(cos𝜃)16𝜋𝑚_𝐵𝐴𝐵⁺⟶𝐷⁺𝐷⁰(−𝑖)𝑔_𝜓𝐷𝐷2𝜖₃⋅𝑝₁×(−𝑖)𝑔_𝐷𝐷𝜌2𝜖₄⋅𝑝₂𝐹²𝑞²,𝑚²_𝐷𝑞²−𝑚²_𝐷=−¹₋₁^→𝑝₁𝑑(cos𝜃)4𝜋𝑚_𝐵𝑔_𝜓𝐷𝐷𝑔_𝐷𝐷𝜌𝐴𝐵⁺⟶𝐷⁺𝐷⁰𝐹²𝑞²,𝑚²_𝐷𝑞²−𝑚²_𝐷𝐻₁, (4.13) where 𝜃 is the angel between ^→𝑝₁ and ^→𝑝₃, 𝑞 is the momentum of the exchange 𝐷 meson, and 𝐻₁=𝜖_3𝜇𝑝^𝜇₁𝜖_4𝜈𝑝^𝜈₂=𝑝⁰₁−𝑝^→₁𝑝⁰₂−𝑝^→₂,𝑞²=𝑚²₁+𝑚²₃−2𝐸₁𝐸𝑝₃^→𝑝₁^→₃cos𝜃, (4.14) and 𝐹(𝑞²,𝑚²_𝐷) is the form factor defined to take care of the off-shell of the exchange particles, which introduced as [6, 21] 𝐹𝑞²,𝑚²_𝐷=Λ²−𝑚²_𝐷Λ²−𝑞^2𝑛. (4.15) The form factor (i.e., 𝑛=1) normalized to unity at 𝑞²=𝑚²_𝐷. 𝑚_𝐷 and 𝑞 are the physical parameters of the exchange particle and Λ is a phenomenological parameter. It is obvious that for 𝑞²→0, 𝐹(𝑞²,𝑚²_𝐷) becomes a number. If Λ≫𝑚_𝑖 then 𝐹(𝑞²,𝑚²_𝐷) turns to be unity, whereas, as 𝑞²→∞ the form factor approaches to zero and the distance becomes small and the hadron interaction is no longer valid. Since Λ should not be far from the 𝑚_𝐷 and 𝑞, we choose Λ=𝑚_𝐷+𝜂Λ_QCD, (4.16) where 𝜂 is the phenomenological parameter that its value in the form factor is expected to be of the order of unity and can be determined from the measured rates, and Abs(4b)=¹₋₁^→𝑝₁𝑑(cos𝜃)16𝜋𝑚_𝐵𝐴𝐵⁺⟶𝐷⁺𝐷⁰(−𝑖)√2𝑔_{𝐷∗𝐷𝜌}𝜖_{𝜇𝜈𝛼𝛽}𝜖^𝜇₄𝑝^𝛼₂𝑝^𝛽₄×(−𝑖)𝑔_{𝜓𝐷∗𝐷}𝜖_{𝜌𝜎𝜆𝜂}𝜖^𝜌₃𝑝^𝜆₁𝑝^𝜂₃𝑔_𝜎𝜈−𝑞_𝜎𝑞_𝜈𝑚²_𝐷∗𝐹²𝑞²,𝑚²_𝐷∗𝑞²−𝑚²_𝐷∗=−¹₋₁^→𝑝₁𝑑(cos𝜃)16𝜋𝑚_𝐵𝑔_{𝜓𝐷∗𝐷}𝑔_{𝐷∗𝐷𝜌}𝐴𝐵⁺⟶𝐷⁺𝐷⁰𝐹²𝑞²,𝑚²_𝐷∗𝑞²−𝑚²_𝐷∗𝐻₂, (4.17) where 𝐻₂=−2𝑝₁⋅𝑝₄𝑝₂⋅𝑝₃−𝑝₁⋅𝑝₂𝑝₃⋅𝑝₄+𝑝⁰₃𝑝⁰₄𝑝₁⋅𝑝₂−𝑝⁰₂−𝑝^→₂𝑝₁⋅𝑝₄+𝑝⁰₁−𝑝^→₁𝑝⁰₂−𝑝^→₂𝑝₃⋅𝑝₄−𝑝⁰₄𝑝₂⋅𝑝₃. (4.18) For diagrams (4c) and (4d) in Figure 4 the absorptive part of the amplitude corresponding to the process of 𝐵⁺→𝐷^+∗(𝑝₁,𝜖₁)𝐷^0∗(𝑝₂,𝜖₂)→𝐽/𝜓(𝑝₃,𝜖₃)𝜌⁺(𝑝₄,𝜖₄), where 𝐷⁻ and 𝐷^−∗ mesons are exchanged, respectively, is read as Abs(4c)𝐺=−𝑖^𝐹2√2𝑑𝑝^3→₁2𝐸₁(2𝜋)³𝑑𝑝^3→₂2𝐸₂(2𝜋)³(2𝜋)⁴𝛿⁴𝑝_𝐵−𝑝₁−𝑝₂×(−𝑖)𝑔_{𝜓𝐷∗𝐷}𝜖_{𝜇𝜈𝛼𝛽}𝜖^𝜇₃𝑝^𝜈₃𝜖^𝛼₁𝑝^𝛽₁−𝑖√2𝑔_{𝐷∗𝐷𝜌}𝜖_{𝜌𝜎𝜆𝜂}𝜖^𝜌₄𝑝^𝜎₄𝜖^𝜆₂𝑝^𝜂₂𝐹²𝑞²,𝑚²_𝐷𝑞²−𝑚²_𝐷×𝑓_𝐷∗𝑚_𝐷∗𝑉_𝑐𝑏𝑉^∗_𝑐𝑑𝜖^∗₁⋅𝜖^∗₂𝑚_𝐵+𝑚_𝐷∗𝐴^𝐵𝐷∗₁𝑚²_𝐷∗−𝜖₁⋅𝑝_𝐵𝜖₂⋅𝑝_𝐵2𝐴^𝐵𝐷∗₂𝑚²_𝐷∗𝑚_𝐵+𝑚_𝐷∗𝐺=𝑖^𝐹16𝜋𝑚_𝐵𝑓_𝐷∗𝑚_𝐷∗𝑔_{𝜓𝐷∗𝐷}𝑔_{𝐷∗𝐷𝜌}𝑉_𝑐𝑏𝑉^∗_𝑐𝑑×𝑝¹₋₁^→₁𝑑()𝐹cos𝜃²𝑞²,𝑚²_𝐷𝑞²−𝑚²_𝐷×𝑚_𝐵+𝑚_𝐷∗𝐴^𝐵𝐷∗₁𝑚²_𝐷∗𝐻₃−2𝐴^𝐵𝐷∗₂𝑚²_𝐷∗𝑚_𝐵+𝑚_𝐷∗𝐻^′₃, (4.19) where 𝐻₃=𝜖_{𝜇𝜈𝛼𝛽}𝜖_{𝜌𝜎𝜆𝜂}𝜖^𝜇₃𝑝^𝜈₃𝑝^𝛽₁𝜖^𝜌₄𝑝^𝜎₄𝑝^𝜂₂−𝑔^𝛼_𝜒+𝑝^𝛼₁𝑝_1𝜒𝑚²_𝐷∗−𝑔^𝜆𝜒+𝑝^𝜆₂𝑝^𝜒₂𝑚²_𝐷∗=−2𝑝₁⋅𝑝₂𝑝₃⋅𝑝₄−𝑝₁⋅𝑝₄𝑝₂⋅𝑝₃+𝑝⁰₄𝑝⁰₁−𝑝^→₁𝑝₂⋅𝑝₃−𝑝⁰₃𝑝₁⋅𝑝₂+𝑝⁰₂−𝑝^→₂𝑝⁰₃𝑝₁⋅𝑝₄−𝑝⁰₁−𝑝^→₁𝑝₃⋅𝑝₄,𝐻^′₃=𝜖_{𝜇𝜈𝛼𝛽}𝜖_{𝜌𝜎𝜆𝜂}𝜖^𝜇₃𝑝^𝜈₃𝑝^𝛽₁𝜖^𝜌₄𝑝^𝜎₄𝑝^𝜂₂−𝑝^𝛼_𝐵+𝑝₁⋅𝑝_𝐵𝑝^𝛼₁𝑚²_𝐷∗−𝑝^𝜆_𝐵+𝑝₂⋅𝑝_𝐵𝑝^𝜆₂𝑚²_𝐷∗=𝑚²_𝐵𝑝⁰₁−𝑝^→₁𝑝⁰₄𝑝₂⋅𝑝₃−𝑝⁰₂−𝑝^→₂𝑝₃⋅𝑝₄−2𝑝₁⋅𝑝₂𝑝₃⋅𝑝₄−𝑝₁⋅𝑝₄𝑝₂⋅𝑝₃+𝑝⁰₃𝑝⁰₂−𝑝^→₂𝑝₁⋅𝑝₄−𝑝⁰₄𝑝₁⋅𝑝₂+𝑝₃⋅𝑝₄−𝑝⁰₃𝑝⁰₄,𝑝⁰₁𝑝⁰₂−𝑝^→₂+𝑝⁰₂𝑝⁰₁−𝑝^→₁−𝑝₁⋅𝑝₂+3𝑝⁰₄𝑝⁰₃𝑝₁⋅𝑝₂−𝑝⁰₁𝑝₂⋅𝑝₃+𝑝₁⋅𝑝₄−𝑝⁰₄𝑝⁰₁−𝑝^→₁𝑝₂⋅𝑝₃−𝑝⁰₃𝑝⁰₂−𝑝^→₂−2𝑝⁰₂𝑝⁰₃𝑝₁⋅𝑝₄−𝑝⁰₁𝑝₃⋅𝑝₄+𝑝⁰₂𝑝⁰₃𝑝⁰₁𝑝⁰₄−𝑝₁⋅𝑝₄,Abs(4d)𝐺=−𝑖^𝐹2√2𝑑𝑝^3→₁2𝐸₁(2𝜋)³𝑑𝑝^3→₂2𝐸₂(2𝜋)³(2𝜋)⁴𝛿⁴𝑝_𝐵−𝑝₁−𝑝₂×(−𝑖)𝑔_{𝜓𝐷∗𝐷∗}𝜖^𝜇₁𝜖^𝜌₃2𝑝_1𝜌𝑔_𝜇𝛽−𝑝₁+𝑝_3𝛽𝑔_𝜇𝜌+2𝑝_3𝜇𝑔_𝛽𝜌−𝑔^𝛼𝛽+𝑞^𝛼𝑞^𝛽𝑚²_𝐷∗×(−𝑖)𝑔_{𝐷∗𝐷∗𝜌}𝜖^𝜈₂𝜖^𝜆₄2𝑝_2𝜆𝑔_𝛼𝜈−𝑝₂+𝑝_4𝛼𝑔_𝜈𝜆+2𝑝_4𝜈𝑔_𝛼𝜆𝐹²𝑞²,𝑚²_𝐷∗𝑞²−𝑚²_𝐷∗×𝑓_𝐷∗𝑚_𝐷∗𝑉_𝑐𝑏𝑉^∗_𝑐𝑑𝜖^∗₁⋅𝜖^∗₂𝑚_𝐵+𝑚_𝐷∗𝐴^𝐵𝐷∗₁𝑚²_𝐷∗−𝜖₁⋅𝑝_𝐵𝜖₂⋅𝑝_𝐵2𝐴^𝐵𝐷∗₂𝑚²_𝐷∗𝑚_𝐵+𝑚_𝐷∗𝐺=𝑖^𝐹16√2𝜋𝑚_𝐵𝑓_𝐷∗𝑚_𝐷∗𝑉_𝑐𝑏𝑉𝑝^∗_𝑐𝑑¹₋₁^→₁𝑑()𝐹cos𝜃²𝑞²,𝑚²_𝐷∗𝑞²−𝑚²_𝐷∗×𝑚_𝐵+𝑚_𝐷∗𝐴^𝐵𝐷∗₁𝑚²_𝐷∗𝐻₄−2𝐴^𝐵𝐷∗₂𝑚²_𝐷∗𝑚_𝐵+𝑚_𝐷∗𝐻^′₄, (4.20) where 𝐻₄=−𝑔^𝛼𝛽+𝑞^𝛼𝑞^𝛽𝑚²_𝐷∗−𝑔^𝜇𝜒+𝑝^𝜇₁𝑝^𝜒₁𝑚²_𝐷∗−𝑔^𝜈_𝜒+𝑝^𝜈₂𝑝_2𝜒𝑚²_𝐷∗𝜖^𝜌₃𝜖^𝜆₄×2𝑝_1𝜌𝑔_𝜇𝛽−𝑝₁+𝑝_3𝛽𝑔_𝜇𝜌+2𝑝_3𝜇𝑔_𝛽𝜌2𝑝_2𝜆𝑔_𝛼𝜈−𝑝₂+𝑝_4𝛼𝑔_𝜈𝜆+2𝑝_4𝜈𝑔_𝛼𝜆,𝐻^′₄=−𝑔^𝛼𝛽+𝑞^𝛼𝑞^𝛽𝑚²_𝐷∗−𝑝^𝜇_𝐵+𝑝₁⋅𝑝_𝐵𝑝^𝜇₁𝑚²_𝐷∗−𝑝^𝜈_𝐵+𝑝₂⋅𝑝_𝐵𝑝^𝜈₂𝑚²_𝐷∗𝜖^𝜌₃𝜖^𝜆₄×2𝑝_1𝜌𝑔_𝜇𝛽−𝑝₁+𝑝_3𝛽𝑔_𝜇𝜌+2𝑝_3𝜇𝑔_𝛽𝜌2𝑝_2𝜆𝑔_𝛼𝜈−𝑝₂+𝑝_4𝛼𝑔_𝜈𝜆+2𝑝_4𝜈𝑔_𝛼𝜆. (4.21) For diagrams (4e) and (4f) in Figure 4 the absorptive part of the amplitude corresponding to the process of 𝐵⁺→𝐷^+∗(𝑝₁,𝜖₁)D⁰(𝑝₂)→𝐽/𝜓(𝑝₃,𝜖₃)𝜌⁺(𝑝₄,𝜖₄), where 𝐷⁻ and 𝐷^−∗ mesons are exchanged, respectively, becomes Abs(4e)=𝐺_𝐹√2𝑑𝑝^3→₁2𝐸₁(2𝜋)³𝑑𝑝^3→₂2𝐸₂(2𝜋)³(2𝜋)⁴𝛿⁴𝑝_𝐵−𝑝₁−𝑝₂×𝑉_𝑐𝑏𝑉^∗_𝑐𝑑𝑚_𝐷∗(−𝑖)𝑔_{𝜓𝐷∗𝐷}𝜖_{𝜇𝜈𝛼𝛽}𝜖^𝜇₃𝜖^𝜈₁𝑝^𝛼₁𝑝^𝛽₃(−𝑖)𝑔_𝐷𝐷𝜌𝜖₄⋅𝑞𝜖₁⋅𝑝₂×𝑓_𝐷∗𝐹₁^𝐵𝐷𝑚²_𝐷∗+𝑓_𝐷𝐴^𝐵𝐷∗₀𝑚²_𝐷𝐹²𝑞²,𝑚²_𝐷𝑞²−𝑚²_𝐷𝐺=−^𝐹4√2𝜋𝑚_𝐵𝑓_𝐷∗𝐹₁^𝐵𝐷𝑚²_𝐷∗+𝑓_𝐷𝐴^𝐵𝐷∗₀𝑚²_𝐷𝑉_𝑐𝑏𝑉^∗_𝑐𝑑𝑚_𝐷∗𝑔_{𝜓𝐷∗𝐷}𝑔_𝐷𝐷𝜌×𝑝¹₋₁^→₁𝑑()𝐹cos𝜃²𝑞²,𝑚²_𝐷𝑞²−𝑚²_𝐷𝐻₅, (4.22) where 𝐻₅=𝜖_{𝜇𝜈𝛼𝛽}𝜖^𝜇₃𝑝^𝛼₁𝑝^𝛽₃𝜖₄⋅𝑝₂−𝑝^𝜈₂+𝑝₁⋅𝑝₂𝑝^𝜈₁𝑚²_𝐷∗,Abs(4f)=𝐺_𝐹√2𝑑𝑝^3→₁2𝐸₁(2𝜋)³𝑑𝑝^3→₂2𝐸₂(2𝜋)³(2𝜋)⁴𝛿⁴𝑝_𝐵−𝑝₁−𝑝₂×𝑉_𝑐𝑏𝑉^∗_𝑐𝑑𝑚_𝐷∗(−𝑖)𝑔_{𝜓𝐷∗𝐷∗}𝜖^𝜇₁𝜖^𝜈₃𝜖^𝛼2𝑝_1𝜈𝑔_𝜇𝛽−𝑝₁+𝑝_3𝛽𝑔_𝜇𝜈+2𝑝_3𝜇𝑔_𝛽𝜈×(𝑖)√2𝑔_{𝐷∗𝐷𝜌}𝜖_{𝜌𝜎𝜆𝜂}𝜖^𝜌₄𝜖^𝜎𝑝^𝜆₂𝑝^𝜂₄𝑓_𝐷∗𝐹₁^𝐵𝐷𝑚²_𝐷∗+𝑓_𝐷𝐴^𝐵𝐷∗₀𝑚²_𝐷𝐹²𝑞²,𝑚²_𝐷∗𝑞²−𝑚²_𝐷∗=𝐺_𝐹8𝜋𝑚_𝐵𝑓_𝐷∗𝐹₁^𝐵𝐷𝑚²_𝐷∗+𝑓_𝐷𝐴^𝐵𝐷∗₀𝑚²_𝐷𝑉_𝑐𝑏𝑉^∗_𝑐𝑑𝑚_𝐷∗𝑔_{𝜓𝐷∗𝐷∗}𝑔_{𝐷∗𝐷𝜌}×𝑝¹₋₁^→₁𝑑()𝐹cos𝜃²𝑞²,𝑚²_𝐷∗𝑞²−𝑚²_𝐷∗𝐻₆, (4.23) where 𝐻₆=𝜖_{𝜌𝜎𝜆𝜂}𝜖^𝜌₄𝜖^𝜈₃𝑝^𝜆₂𝑝^𝜂₄2𝑝_1𝜈𝑔_𝜇𝛽−𝑝₁+𝑝_3𝛽𝑔_𝜇𝜈+2𝑝_3𝜇𝑔_𝛽𝜈×−𝑔^𝛼𝜎+𝑞^𝛼𝑞^𝜎𝑚²_𝐷∗−𝑝^𝜇₂+𝑝₁⋅𝑝₂𝑝^𝜇₁𝑚²_𝐷∗. (4.24) For diagrams (4g) and (4h) in Figure 4 the absorptive part of the amplitude corresponding to the process of 𝐵⁺→𝐷⁺(𝑝₁)𝐷^0∗(𝑝₂,𝜖₂)→𝐽/𝜓(𝑝₃,𝜖₃)𝜌⁺(𝑝₄,𝜖₄), where 𝐷⁻ and 𝐷^−∗ mesons are exchanged, respectively, is given Abs4g=𝐺_𝐹√2𝑑𝑝^3→₁2𝐸₁(2𝜋)³𝑑𝑝^3→₂2𝐸₂(2𝜋)³(2𝜋)⁴𝛿⁴𝑝_𝐵−𝑝₁−𝑝₂×𝑉_𝑐𝑏𝑉^∗_𝑐𝑑𝑚_𝐷∗𝑖√2𝑔_{𝐷∗𝐷𝜌}𝜖_{𝜇𝜈𝛼𝛽}𝜖^𝜇₄𝜖^𝜈₂𝑝^𝛼₂𝑝^𝛽₄(−𝑖)𝑔_𝜓𝐷𝐷𝜖₃⋅𝑞𝜖₂⋅𝑝₁×𝑓_𝐷∗𝐹₁^𝐵𝐷𝑚²_𝐷∗+𝑓_𝐷𝐴^𝐵𝐷∗₀𝑚²_𝐷𝐹²𝑞²,𝑚²_𝐷𝑞²−𝑚²_𝐷=𝐺_𝐹4𝜋𝑚_𝐵𝑓_𝐷∗𝐹₁^𝐵𝐷𝑚²_𝐷∗+𝑓_𝐷𝐴^𝐵𝐷∗₀𝑚²_𝐷𝑉_𝑐𝑏𝑉^∗_𝑐𝑑𝑚_𝐷∗𝑔_𝜓𝐷𝐷𝑔_{𝐷∗𝐷𝜌}×𝑝¹₋₁^→₁𝑑()𝐹cos𝜃²𝑞²,𝑚²_𝐷𝑞²−𝑚²_𝐷𝐻₇, (4.25) where 𝐻₇=𝜖_{𝜇𝜈𝛼𝛽}𝜖^𝜇₄𝑝^𝛼₂𝑝^𝛽₄𝜖₃⋅𝑝₁−𝑝^𝜈₁+𝑝₁⋅𝑝₂𝑝^𝜈₂𝑚²_𝐷∗,Abs(4h)=𝐺_𝐹√2𝑑𝑝^3→₁2𝐸₁(2𝜋)³𝑑𝑝^3→₂2𝐸₂(2𝜋)³(2𝜋)⁴𝛿⁴𝑝_𝐵−𝑝₁−𝑝₂×𝑉_𝑐𝑏𝑉^∗_𝑐𝑑𝑚_𝐷∗(−𝑖)𝑔_{𝐷∗𝐷∗𝜌}𝜖^𝜇₂𝜖^𝜈₄𝜖^𝛽2𝑝_2𝜇𝑔_𝛼𝜈−𝑝₂+𝑝_4𝛼𝑔_𝜇𝜈+2𝑝_4𝜈𝑔_𝛼𝜇×(−𝑖)𝑔_{𝜓𝐷∗𝐷}𝜖_{𝜌𝜎𝜆𝜂}𝜖^𝜌₃𝜖^𝜎𝑝^𝜆₁𝑝^𝜂₃𝑓_𝐷∗𝐹₁^𝐵𝐷𝑚²_𝐷∗+𝑓_𝐷𝐴^𝐵𝐷∗₀𝑚²_𝐷𝐹²𝑞²,𝑚²_𝐷∗𝑞²−𝑚²_𝐷∗=𝐺_𝐹8√2𝜋𝑚_𝐵𝑓_𝐷∗𝐹₁^𝐵𝐷𝑚²_𝐷∗+𝑓_𝐷𝐴^𝐵𝐷∗₀𝑚²_𝐷𝑉_𝑐𝑏𝑉^∗_𝑐𝑑𝑚_𝐷∗𝑔_{𝜓𝐷∗𝐷}𝑔_{𝐷∗𝐷∗𝜌}×𝑝¹₋₁^→₁𝑑()𝐹cos𝜃²𝑞²,𝑚²_𝐷∗𝑞²−𝑚²_𝐷∗𝐻₈, (4.26) where 𝐻₈=𝜖_{𝜌𝜎𝜆𝜂}𝜖^𝜌₃𝜖^𝜇₄𝑝^𝜆₁𝑝^𝜂₃2𝑝_2𝜇𝑔_𝛼𝜈−𝑝₂+𝑝_4𝛼𝑔_𝜇𝜈+2𝑝_4𝜈𝑔_𝛼𝜇×−𝑔^𝛽𝜎+𝑞^𝛽𝑞^𝜎𝑚²_𝐷∗−𝑝^𝜈₁+𝑝₁⋅𝑝₂𝑝^𝜈₂𝑚²_𝐷∗. (4.27) As the bridge between the dispersive part of FSI amplitude and the absorptive part, the dispersion relation is Dis4𝑚²_𝐵=1𝜋^∞_𝑠Abs_(4a)𝑠^′+Abs_(4b)𝑠^′+Abs_(4c)𝑠^′⋯Abs_(4h)𝑠^′𝑠′−𝑚²_𝐵𝑑𝑠′. (4.28) According to the FSI, the amplitude of 𝐵⁺→𝐽/𝜓𝜌⁺ decay is given by 𝐴_FSI𝐵⁺⟶𝐷^+(∗)𝐷^0(∗)⟶𝐽𝜓𝜌⁺=Abs(4a)+Abs(4b)+Abs(4c)+Abs(4d)+Abs(4e)+Abs(4f)+Abs4g+Abs(4h)+Dis4𝑚²_𝐵. (4.29) Concerning the QCDF amplitude and FSI correction, then the total decay amplitude turns to 𝐴𝐵⁺⟶𝐽𝜓𝜌⁺=𝐴_QCDF+𝑖𝐴_FSI. (4.30)

Figure 3 

Quark level diagram for 𝐵+→𝐷+(∗)𝐷0(∗)→𝐽/𝜓𝜌+.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 4 

HLL diagrams for long-distance t-channel contribution to 𝐵+→𝐽/𝜓𝜌+.

5. Numerical Results

Numerical values of effective coefficients 𝑎_𝑖 for 𝑏→𝑑 transition at 𝑁_𝑐=3 are given by [22] 𝑎₁=1.05,𝑎₂𝑎=0.053,₃=0.0048,𝑎₄𝑎=−0.0461−0.0124𝑖,₅=−0.0045,𝑎₆𝑎=−0.0597−0.0124𝑖,₇=0.00003−0.00018𝑖,𝑎₈𝑎=0.00045−0.00006𝑖,₉=−0.0095−0.00018𝑖,𝑎₁₀=−0.0014−0.00006𝑖. (5.1) The Cabibbo-Kobayashi-Maskawa (CKM) matrix is a 3×3 unitary matrix as [9] 𝑉𝐕=^𝑢𝑑𝑉_𝑢𝑠𝑉_𝑢𝑏𝑉_𝑐𝑑𝑉_𝑐𝑠𝑉_𝑐𝑏𝑉_𝑡𝑑𝑉_𝑡𝑠𝑉_𝑡𝑏. (5.2) The elements of the CKM matrix can be parameterized by three mixing angles 𝐴,𝜆,𝜌 and a CP-violating phase 𝜂𝑉_𝑢𝑑𝜆=1−²2,𝑉_𝑢𝑠=𝜆,𝑉_𝑢𝑏=𝐴𝜆³,𝑉𝜌−𝑖𝜂_𝑐𝑑=−𝜆,𝑉_𝑐𝑠𝜆=1−²2,𝑉_𝑐𝑏=𝐴𝜆²,𝑉_𝑡𝑑=𝐴𝜆³1−𝜌−𝑖𝜂,𝑉_𝑡𝑠=−𝐴𝜆²,𝑉_𝑡𝑏=1. (5.3) The results for the Wolfenstein parameters are 𝜆=0.2257±0.001,𝐴=0.814±0.02,𝜌=0.135±0.023,𝜂=0.349±0.016. (5.4) We use the central values of the Wolfenstein parameters and obtain 𝑉_𝑢𝑑=0.9745,𝑉_𝑢𝑠=0.2257,𝑉_𝑢𝑏𝑉=0.0035,_𝑐𝑑=0.2257,𝑉_𝑐𝑠=0.9745,𝑉_𝑐𝑏𝑉=0.0415,_𝑡𝑑=0.0087,𝑉_𝑡𝑠=0.0415,𝑉_𝑡𝑏=1. (5.5) The meson masses and decay constants needed in our calculations are taken as (in units of Mev) 𝑚_𝐵+=5278±0.3,𝑚_𝐷±𝑚=1869±0.2,_𝐷0=1864.84±0.17,𝑚_𝐷±∗𝑚=2010.27±0.17,_𝐷0∗=2006.97±0.19,𝑚_𝐽/𝜓𝑚=3096.916±0.011,_𝑏=4.20±0.12,𝑚_𝜌𝑓=775.49±0.34,_𝐵+=176±42,𝑓_𝐽/𝜓𝑓=0.416,_⟂𝐽/𝜓=0.405,𝑓_𝐷±=𝑓_𝐷⁰𝑓=222.6±19.5,_𝐷±∗=𝑓_𝐷^0∗=230±20, (5.6) The Borel mass squares 𝑀²₁ and 𝑀²₂ and continuum thresholds 𝑠₀ and 𝑠^′₀ are auxiliary parameters, hence the physical quantities should be independent of them. The parameters 𝑠₀ and 𝑠^′₀ are determined from the conditions that guarantee the sum rules for form factors to have the best stability in the allowed 𝑀²₁ and 𝑀²₂ region. The working regions for 𝑀²₁ and 𝑀²₂ as well as the values for continuum thresholds are determined in [13]. They choose the values 𝑠₀=35±5GeV², 𝑠^′₀=7±1GeV², 𝑀²₁=17.0±2.5GeV², 𝑀²₂=7±1GeV² from those working values for auxiliary parameters. The values of the form factors are given by 𝐹₀^𝐵𝐷𝑚²_𝐷=0.86±0.22,𝐹₁^𝐵𝐷𝑚²_𝐷∗𝐴=0.95±0.23,_𝐵𝐷∗0𝑚²_𝐷=2.50±0.60,𝐴^𝐵𝐷∗₁𝑚²_𝐷∗𝐴=1.09±0.26,_𝐵𝐷∗2𝑚²_𝐷∗=1.82±0.46[13],𝐴₁^𝐵𝜌=𝐴₂^𝐵𝜌𝑉=0.26,_𝐵𝜌=0.33[10]. (5.7) Nonuniversal annihilation phases are [23] 𝜙=−55(𝑃𝑃),𝜙=−20^∘(𝑃𝑉),𝜙=−70^∘(𝑉𝑃). (5.8) The signs of these phases are not predicted. As a result in [23], their predictions for the signs of CP asymmetries must be taken with caution. We use other input parameters as follows: Λ_QCD=0.225Gev,𝐶_𝐹=43𝐺,𝑁=3,_𝐹=1.166×10⁻⁵,𝛼_𝑠=0.2244,𝜌=0.5,𝑟^𝐷±_𝜒𝑔=1,_𝐷𝐷𝜌=𝑔_{𝐷∗𝐷∗𝜌}=2.52,𝑔_𝜓𝐷𝐷=𝑔_{𝜓𝐷∗𝐷∗}𝑔=7.71,_{𝐷∗𝐷𝜌}=2.82,𝑔_{𝜓𝐷∗𝐷}=8.64[5]. (5.9) By using the input parameters and according to the QCDF method of 𝐵⁺→𝐽/𝜓𝜌⁺ decay in Section 2, we get BR𝐵⁺⟶𝐽𝜓𝜌⁺=(1.42±0.36)×10⁻⁵. (5.10) We note that our estimate of branching ratio of 𝐵⁺→𝐽/𝜓𝜌⁺ decay according to QCDF method seems less than the experimental result. Before calculating the 𝐵⁺→𝐽/𝜓𝜌⁺ decay amplitude via FSI, we have to compute the intermediate state amplitude, for the 𝐵⁺→𝐷⁺𝐷⁰ decay amplitude we get 𝐴𝐵⁺⟶𝐷⁺𝐷⁰=(2.31±0.27)×10⁻⁷. (5.11) Now, using the above amplitude and according to the FSI, we are able to calculate the branching ratio of 𝐵⁺→𝐽/𝜓𝜌⁺ decay with different values of 𝜂 which are shown in Table 1 and Figure 5.

Table 1 

The branching ratio of 𝐵⁺→𝐽/𝜓𝜌⁺ decay with 𝜂=1∼2 and experimental data (in units of 10⁻⁵).

Figure 5 

The variation of the branching ratio of 𝐵+→𝐽/𝜓𝜌+ with 𝜂=1∼2.2.

6. Conclusion

In this work, we have calculated the contribution of the t-channel FSI, that is, inelastic rescattering processes to the branching ratio of 𝐵⁺→𝐽/𝜓𝜌⁺ decay. For evaluating the FSI effects, we have only considered the absorptive part of the HLL because both hadrons which produced via the weak interaction are on their mass shells.

We have calculated the branching ratio of 𝐵⁺→𝐽/𝜓𝜌⁺ decay by using FSI. The experimental result of this decay is BR(𝐵⁺→𝐽/𝜓𝜌⁺)=(5±0.8)×10⁻⁵ and according to QCDF method and assuming the FSI corrections into account, our results of the branching ratio are (1.42±0.36)×10⁻⁵ and (4.2∼5.8)×10⁻⁵, respectively.

There exist some phenomenological parameters in our calculations on FSI, such as, 𝜂 in (4.16). We have introduced the phenomenological parameter 𝜂 that its value in the form factor is expected to be of order unity and can be determined from the measured rates. For a given exchanged particle, we have used 𝜂=1∼2, and the branching ratios are 1.88∼7.32. We have considered that the factor of 𝜂 has important role to obtain of branching ratio. Note that [6] for the exchange particles 𝐷^∗ and D they have fixed 𝜂=2.2 and according to Figure 5, we have found that if 𝜂=1.58∼1.83 is selected (which is in good agreement with the reference [6] selection), the numbers of the branching ratio of the 𝐵⁺→𝐽/𝜓𝜌⁺ decay are placed in the experimental range.