The nonleptonic decays of b -ﬂavored mesons to S -wave charmonium and charm meson states

The detection of radially excited heavy meson states in recent years and measurement of heavy meson de-cays, particularly B + c → J / ψ D + s and B + c → J / ψ D ∗ + s , by the LHCb and ATLAS Collaborations, have aroused a lot of theoretical interest in the nonleptonic decays of b -ﬂavored mesons. In this paper, we study the exclusive two-body non-leptonic ¯ B 0 , ¯ B 0 s , B − and B − c -meson decays to two vector meson ( V 1 ( nS ) V 2 ) states. Assuming the factorization hypothesis, we calculate the weak-decay form factors from the over-lapping integrals of meson wave functions, in the framework of the relativistic independent quark (RIQ) model. We ﬁnd a few dominant decay modes: B − → D ∗ 0 ρ − , ¯ B 0 → D ∗ + ρ − , ¯ B 0 s → D ∗ + s ρ − , B − → J / ψ K ∗− and B − c → J / ψ D ∗− s with predicted branching fractions of 1.54, 1.42, 1.17, 0.53 and 0.52 (in %), which are experimentally accessible. The predicted branching fractions for corresponding decay modes to excited (2 S ) states, obtained in the order O ( 10 − 3 − 10 − 4 ) lie within the detection accuracy of the current experiments at LHCb and Tevatron. The sizeable CP -odd fractions predicted for B − c -meson decay to two charmful states: D ∗ 0 D ∗− ( s ) and ¯ D ∗ 0 D ∗− ( s ) indicate signiﬁcant CP -violation hinting at the so-called new physics beyond the standard model.


Introduction
The experimental probes over the last two decades in the b-flavored heavy meson (B, B s , B c ) sector have led to the discovery of many excited states which include the radially excited charmonium ψ(2S) and η c (2S) states by the Belle Collaborations [1].In the heavy-light meson sector, a number of charm meson states such as D * s1 (2710) ± [2] by Belle, D(2550) 0 [3], and D J (2580) 0 [4] by the LHCb, a e-mail: kalpalatadash982@gmail.com b e-mail: lopalmn95@gmail.comD * (2640) ± [5] by Delphi and D * (2650) 0 [4] and D * 1 (2680) 0 [6] by LHCb have also been discovered, which can be identified as D * s (2S) ± [7], D(2S) 0 [8], D * (2S) ± [8] and D * (2S) 0 state, respectively.Recently LHCb discovered D * 1 (2600) [3] which might be the same object as D * (2650) 0 and D * 1 (2650).The current RUN-II at Tevatron, RUN-III at CERN LHC, and the e + e − collider activities at Belle-II are all designed to boost the measurement scenario in heavy flavor physics.Specially designed detectors at BTeV and LHCb, dedicated to enhancing the event accumulation rates, are expected to yield high statistics b-flavored (B, B s ) events and the B cevents, in particular, at the rate ∼ 10 10 events per annum; providing a fascinating area of research in B-physics.The measurement of B, B s -meson decays by BaBar, Belle, and LHCb Collaborations [9][10][11] and recent measurement of B cmeson decays: B c → J/ψD s and B c → J/ψD * s , performed by LHCb [12] and ATLAS [13] Collaborations have aroused a great deal of theoretical interest in nonleptonic decays of heavy-flavored mesons.The study of nonleptonic decays of heavy mesons is important as it helps in probing the interplay of QCD and electroweak interactions, determining the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, testing predictions of the standard model (SM), and exploring new physics beyond SM.
The analysis of nonleptonic decays is notoriously nontrivial as it is strongly influenced by confining color forces and it involves matrix elements of local four-quark operators in the non-perturbative QCD approach, the mechanism of which is not yet clear in the SM framework.Ignoring the weak annihilation contribution, the transition amplitudes can be conveniently described in the so-called naive factorization approximation [17, 20-23, 27, 33-55], which works reasonably well in the nonleptonic b-flavored meson decays, where the quark-gluon sea is suppressed in the heavy quarkonium [47][48][49][50].In this approach, the transition matrix element of local four-quark operators is factorized into two single current matrix elements.One of the factorized amplitudes, in which the decaying parent meson is connected to one final meson state, can be covariantly expanded in terms of Lorentz invariant weak form factors as in the case of semileptonic decays.The other factorized amplitude, where the vacuum is connected to the second final meson state, can be parametrized in terms of meson decay constants that describe the leptonic decays.The description of the nonleptonic decay process is thus reduced to the calculation of weak decay form factors in the framework of a suitable phenomenological model.
Bjorken's intuitive argument on color transparency in his pioneering work [51], theoretical development based on the QCD approach in the 1 N c limit [52] and the heavy quark effective theory (HQET) [53], etc. justify the naive factorization approximation, where strong-interaction effects such as the final-state interaction, rescattering of the final state hadrons and the renormalization-point dependence of amplitudes are shown to be marginal [56].Discovery of excited charmonium and charm meson states and prediction of b-flavored meson decays to the ground and radially excited states by different theoretical approaches, inspired our group to predict energetic nonleptonic B and B c → PP, PV,V P de-cays [33,34] as well as their decays to two vector meson (VV ) ground states [35,36], within the framework of our relativistic independent quark (RIQ) model.Here P(V ) refers to a pseudoscalar (vector) meson state.The nonleptonic B cdecays to an S-wave charmonium and a light or charm meson state [55] have also been predicted by our group in good comparison with the experiment and other SM predictions.
Note here that the approach based on naive factorization approximation may be justified in the analysis of energetic nonleptonic decays of B F → PP, PV type, with the quark flavor F → d, u, s, c; where the strong interaction effects such as the final state interaction, rescattering of final state meson as well as the renormalization point dependence of factorized amplitude have been shown to be marginal [57].Such an approach, however, may not hold up well in the description of B F → V 1 (nS)V 2 decays, where both the final state mesons being heavy, are expected to be in the region close to zero recoil point.Here also both the longitudinal and transverse polarization components contribute to decay amplitude which can be measured experimentally.From the naive counting rules, the longitudinal polarization fraction in this sector is expected to dominate over transverse components which can be checked as well.The nonleptonic B F -decays to two charmful vector meson states are of special interest as they provide valuable information which is different from cases with light meson productions.For example, the evaluation of CP-asymmetries in B − c decays: provides an important clue in testing the SM predictions and exploring new physics beyond SM.
In this paper we would like to extend the applicability of our RIQ model to study, within factorization approximation, the nonleptonic B F → V 1 (nS)V 2 decays to Swave charmonium and charm vector meson states (nS) along with a light or a heavy-light meson state, where n = 1, 2, 3. We ignore the decay channels involving higher (4S) charmonium and charm meson states since their properties are still not understood well.We adopt here the general formalism used in Ref. [55].In the present study we consider the contribution of the current-current operators [58] only in calculating the treelevel diagram, expected to be dominant in The contribution of the penguin diagram may be significant in the evaluation of CP-violation and search for new physics beyond SM, but its contribution to these decay amplitudes is considered less significant.In fact, the QCD and electroweak penguin operators' contribution have been shown [59][60][61][62] negligible compared to the contribution of current-current operators in these decays due to serious suppression of CKM matrix elements.The Wilson's coefficients of penguin operators being very small, their contribution to decay amplitude is only relevant in rare decays, where the tree-level contribution is either strongly CKM-suppressed as in B → K * π or matrix elements of current-current operators do not contribute at all as in the case of rare decays: B → K * γ and B0 → K0 φ [58].
The rest of the paper is organized as follows.In the following section, we present a general remark on the factorization approximation and discussed the factorized amplitudes of the nonleptonic decay.In Section 3, we obtain the model expressions for invariant weak-decay form factors and the factorized transition amplitudes.Section 4 is devoted to the numerical results and discussion and Section 5 encompasses our brief summary and conclusion.A brief review of the model conventions, wave-packet representation of the meson state and momentum probability amplitudes of the constituent quarks inside the meson bound-state are given in the Appendix.

Factorization approximation and nonleptonic transition amplitude
The transition amplitude for two-body nonleptonic transition: where G F is the Fermi Coupling constant, λ i the CKM factor, C i is the Wilson coefficients and O i is the matrix element of local four-quark operators.In the factorization approximation, the matrix element of the local four-quark operator is factorized into two single-particle matrix elements of quark current as where J µ ≡ V µ − A µ is the vector-axial vector current.
The difficulty inherent in such an approach is that Wilson's coefficients C i (µ), which include the short distance QCD effect between µ = m N and µ = m b are µ scale and renormalization scheme dependent, whereas O i are µ scale and renormalization scheme independent.As a result, physical amplitude depends on the µ scale.However, in the naive factorization approach, the long-distance effects are disentangled from the short-distance effect assuming that the matrix element O at the µ scale contains nonfactorizable contributions.This results in the cancellation of the µ dependence and scheme dependence of C i (µ).
We neglect here the so-called W exchange and annihilation diagram, since in the limit M W → ∞ they are connected by Fiertz transformation and doubly suppressed by a kinematic factor of order ( ) [52].We also discard the color octet current which emerges after the Fiertz transformation of color singlet operators.Clearly, these currents violate factorization since they cannot provide transitions to vacuum states.Taking into account the Fiertz reordered contribution, the relevant coefficients are not C 1 (µ) and C 2 (µ) but the combination The factorization approximation, in general, works well in the description of two-body nonleptonic decays of heavy mesons in the limit of a large number of colors.Assuming a large N c limit to fix the QCD coefficients a 1 ≈ c 1 and a 2 ≈ c 2 at µ ≈ m 2 b , nonleptonic decays of heavy mesons have been analyzed in Refs.[15,24,[63][64][65][66][67].
The hadronic matrix element of the weak current J µ are covariantly expanded in terms of weak form factors as where ε * is the polarization of the vector meson V 1 .p and k represent the four-momentum of the parent meson B F and daughter meson V 1 , respectively.With the four-momentum transfer q = p − k ≡ (E, 0, 0, |q|) and mass m V 1 , the polarization of the daughter meson V 1 is taken in the form The matrix element of the current J µ between vacuum and vector-meson V 2 in the final state can be parametrized in terms of meson decay constant f V 2 as In the factorization approach, the nonleptonic transition amplitude can be calculated from one of the three possible treelevel diagrams shown in Fig. 1.The color-favored transitions, shown in quark level diagram in Fig. 1(a), represent "class I" transitions which are characterized by external emission of W -boson.In these transitions, the factorized amplitude coupled to the QCD factor a 1 only give the nonvanishing contribution.On the other hand, color-suppressed transitions shown in the diagram in Fig. 1(b) representing "class II" transitions are characterized by internal W emission.In such transitions, the nonvanishing contribution to the decay rate comes from factorized amplitude proportional to the QCD factor a 2 .Figure 1(c), however, represents "class III" transitions which are due to both color-favored and colorsuppressed diagrams.In such decays, the factorized amplitudes corresponding to a 1 and a 2 contribute coherently to give the transition amplitude.
For the color-favored general type tree-level transition B F → V 1 (nS)V 2 pertaining to "class I" transitions, the decay rate can be written as [35,36,46] where M and k represent the parent-meson mass and threemomentum of the recoiled daughter meson V 1 , respectively, in the parent-meson rest frame.|A | 2 is the sum of the polarized amplitude squared with We use the notation j = ±, ∓ or ll, where the first and second labels denote the helicity of the V 1 and V 2 meson, respectively.From the polarized amplitudes expressed in terms of the weak form factors f , g and a + and the decay constant f V 2 shown in Eqs.( 4)- (7), it is straightforward to find expressions for the positive, negative, and longitudinal polarizations, respectively, of the daughter meson V 1 as where The decay widths and branching fractions for B F → V 1 (nS)V 2 decays can be predicted from Eq.( 8) using the expressions in Eqs.(9)(10)(11) for the polarized amplitudes in terms of the weak form factors derivable in the framework of the RIQ model.

Transition amplitude and weak form factors
As discussed in the preceding section, the nonleptonic transition amplitude for the process B F → V 1 (nS)V 2 can be calculated from the tree-level diagram shown in Fig. 1.The class-I type decay modes, depicted in Fig 1(a), are induced by the b-quark transition to the daughter quark q with the emission of W -boson.The daughter quark q and the antiquark q of the decaying parent meson state |B F (p, S B F ) hadronize to form a vector meson state |V 1 (k, S V 1 ) .The externally emitted W -boson first decays to a quark-antiquark pair (q i q j ), which subsequently hadronizes to other vector meson state |V 2 (q, S V 2 ) .
The decay process, in fact, occurs physically in the momentum eigenstate of participating mesons.Therefore, a fieldtheoretic description of a decay process demands mesonbound states to be represented by appropriate momentum wave packets reflecting momentum and spin distribution between the quark constituents in the meson core.A brief discussion of the wave-packet representation of meson bound state in the RIQ model is given in the Appendix.Using the wave-packet representation (A.9-A.11) of participating meson states, the residual dynamics responsible for the decay process can, therefore, be described at the constituent level by the otherwise unbound quark and antiquark using the usual Feynman technique.The constituent-level S-matrix element S b→q q i q j f i obtained from the appropriate Feynman diagram when operated upon by the bag-like operator Λ (p, S B F ) in the wave packet representation can give rise to the mesoniclevel S-matrix element in the form Using the wave packet representation of the parent and daughter meson state, |B F (p, S B F ) and |V 1 (k, S V 1 ) , respectively, we calculate the Feynman Diagram Fig. 1(a) and obtain the S-matrix element in the parent meson rest frame in the general form: where the invariant transition amplitude M f i is obtained in the form: with The terms E b (p b ) and E q (p b + k) in ( 15) stand for the energy of the non-spectator quark of the parent and daughter meson, p b and k represent three momentum of the nonspectator constituent quark b and the daughter meson V 1 , respectively and q = p − k is the four-momentum transfer.Finally, S V |J µ |S B F is the symbolical representation of the spin matrix elements of the effective vector-axial vector current; which can be written in the explicit form: Here, u i stands for free Dirac spinor.ζB F (λ b , λ q) and ζ V 1 (λ q , λ q) are the appropriate SU( 6) spin flavor coefficients corresponding to the parent and daughter meson, respectively.It may be pointed out here that, in our description of the decay process, B F → V 1 (nS)V 2 the three momentum conservation is ensured explicitly via δ (3) (p b + p q − p) and δ (3) (p i + p j − k) in the participating meson states.However, energy conservation in such a scheme is not ensured so explicitly.This is in fact a typical problem in all potential model descriptions of mesons as bound states of valence quarks and antiquarks interacting via some instantaneous potential.This problem has been addressed in the previous analysis in this model in the context of radiative leptonic decays of heavy flavored meson B, B c , D, D s [68][69][70] and also in the QCD relativistic quark model [71,72], where the effective momentum distribution function G B F (p b , p q) that embodies bound-state characteristics of the meson ensures energy conservation in an average sense satisfying In view of this, we take the energy conservation constraint M = E b (p b ) + E q(−p b ) in the parent meson rest frame.This along with the three momentum conservation via appropriate δ (3) (p b + p q −p) in the meson state ensures the required four-momentum conservation δ (4) (p − k − q) at the mesonic level, which is pulled out of the quark-level integration to obtain the Smatrix element in the standard form (13).This has been discussed elaborately in earlier works [33][34][35][36]55].
Using usual spin algebra the spacelike and timelike components of the spin matrix elements S V 1 |J µ (0)|S B F (0) corresponding to vector and axial vector current are obtained in the form are, respectively, the energy of the non-spectator quark b and daughter quark q .The spacelike component of the hadronic matrix element H µ obtained from Eq.( 15) via Eqs.( 18) and ( 19) are compared with the corresponding expressions from Eqs. ( 4) and ( 5), which lead to the model expressions of the weak form factors g(q 2 ) and f (q 2 ) in the form: where The timelike component of hadronic amplitude obtained from Eq. ( 15) via Eq.( 20), when compared with the corresponding expression from Eq. ( 4) yields an expression of the form factor a + (q 2 ) in the form: Then it is straightforward to get the model expression for the polarized amplitude squared |A j | 2 using Eqs.(21)(22)(23)(24)(25). Summing over possible polarization states and integrating over the final-state particle momenta, the decay width is obtained in the parent-meson rest frame from the generic expression The two-body nonleptonic decay (B F → V 1 (nS)V 2 ), described so far in this section refers to the color-favored "class I" decays involving external emission of W -boson.Similarly, class II and III type B F → V 1 (nS)V 2 decays can be calculated from the corresponding Feynman diagrams shown in Figs.1(b) and 1(c), respectively.The model expressions for relevant form factors and decay rates for such decays (class II and class III) can be obtained by suitable replacement of appropriate flavor degree of freedom, quark masses, quark binding energies, QCD factors a 1 , a 2 , and the meson decay constants.

Numerical results and Discussion
In this section, we present our numerical results in comparison with other model predictions and the available experimental data.For numerical calculation, we use the model parameters (a,V 0 ), quark mass m q and quark binding energies E q , which have been fixed from hadron spectroscopy by fitting the data of heavy and heavy-light flavored mesons in their ground state as [73][74][75] (a,V 0 ) = (0.017166GeV 3 , −0.1375 GeV ), (m b , m c ) = (4.77659,1.49276) GeV, The description of the decay process involving radially excited meson states, the constituent quarks in the meson-bound states are expected to have higher binding energies compared to their ground-state binding energies.For this, we solve the cubic equation representing the binding energy condition (A.5) for respective constituent quarks (c, s, u = d) in radially excited 2S and 3S states of the ( cc), ( cu), ( cd) systems as Using the above input parameters (27), wide-ranging hadronic phenomena have been described within the framework of the RIQ model, which includes the two body nonleptonic decays of B and B c mesons to ground state mesons in the charmonium, charm, strange and non-strange light flavor sectors [33][34][35][36].For CKM parameters and the lifetime of decaying mesons, we take their respective central values from the Particle Data Group (2022) [82]: and respectively.For the mass and decay constant of the participating mesons, considered as phenomenological inputs in the numerical calculation, we take the central values of the available observed data from Ref. [5,6,82].In the absence of the observed data on the mass of excited (2S and 3S) charmed and strange-charmed mesons and the meson decay constants, we take the corresponding predicted data from established theoretical approaches [83][84][85][86].Accordingly, the updated meson masses and decay constants used in the present study are listed in Table -1.
It may be mentioned here that, in the prediction of nonleptonic decay, uncertainties mostly creep into the calculation through input parameters: potential parameter (a,V 0 ), quark mass (m q ) and quark binding energy (E q ), CKM parameters, meson decay constants and QCD coefficients (a 1 , a 2 ) etc.As mentioned above, the potential parameters, 3193 [83] ψ(3S) 4039.1 [82] 319 [86] quark masses and quark binding energies (27,28) have already been fixed at the static level application of RIQ model by fitting the mass spectra of heavy and heavy-light mesons.
In order to avoid uncertainty in our model predictions, we take the central values of the CKM parameter as well as the observed value of the decay constants.As such we do not have the liberty to use any free parameter in our calculation which could be fine-tuned from time to time to predict any hadronic phenomena.In that sense, we perform almost a parameter-free calculation in our studies.As regards the QCD coefficients: (a 1 , a 2 ), different sets of data for decays induced by the b-quark transition at the quark level, are used in the literature.For example, Colangelo and De Fazio, in Ref. [27] use QCD coefficients, Set 1: (a b 1 , a b 2 ) = (1.12,−0.26), as fixed in Refs.[87,88].In most earlier calculations, the authors use a different set of QCD coefficients, Set 2: 2 ) = (0.93, −0.27).We use all three sets of the Wilson coefficients in our calculation.
Before using the above input parameters in our numerical analysis, it is pertinent to elaborate a bit on the energy conservation ansatz mentioned in Sec.3.The present analysis based on the energy conservation constraints M = E b (p b ) + E q (−p b ) in the parent meson rest frame might lead to spurious kinematic singularities at the quark-level integration appearing in the decay amplitude.This problem has already been addressed previously in their QCD rela-tivistic quark model approach [71,72] and later by our group in the study of radiative leptonic decays of heavy and heavylight flavored meson sector [68][69][70], by assigning a running mass m b to the non-spectator quark that satisfies the relation: as an outcome of the energy conservation ansatz, while retaining definite mass m q of the spectator quark q.This leads In fact, the quark momentum distribution obtained in this model [68][69][70] is similar to the prediction of the QCD relativistic quark model analysis [71,72].The rms value of the active quark , the expectation value of the binding energies of the active quark b, and spectator q and the sum of the binding energy of quark and antiquark pair , respectively, calculated in the framework of RIQ model, are presented in Table 2.
It is noteworthy to discuss four important aspects of our present approach.(1) The rms value of the quark momentum in the meson-bound state is much less than the corresponding upper bound |p b | max , as expected.(2) The average energies of a constituent quark of the same flavor in different meson-bound states do not exactly match.This is because the kinematics and binding energy conditions for constituent quarks due to the color forces involved are different from one meson-bound state to other.The constituent quarks in the meson-bound state are considered to be free particles of definite momenta, each associated with its momentum probability amplitude derivable in this model via momentum space projection of the respective quark eigenmodes.On the other hand, the energies shown in Eq. (27,28), which are the energy eigenvalues of the corresponding bound quarks with no definite momenta of their own, are obtained from respective quark orbitals by solving the Dirac equation in this model.This makes a marginal difference between the energy eigenvalues (27,28) and the average energy of constituent quarks shown in Table 2. (3) The expectation values of the sum of the energy of a constituent quark and antiquark in the meson-bound state are obtained in good agreement with the corresponding observed meson masses as shown in Table 2.These important aspects of our results lend credence to our energy conservation ansatz in an average sense through the effective momentum distribution function like G B F (p b , −p b ) in the meson-bound state |B F (0) .This ansatz along with three momentum conservation in the meson-bound state (A.1) ensures the required energy-momentum conservation in our description of several decay processes pointed out earlier.In the absence of any rigorous field theoretic description of the meson-bound states, invoking such an ansatz is no doubt a reasonable approach for a constituent-level description of hadronic phenomena.( 4) Finally, in a self-consistent dynamic approach, we extract the form factors from the overlapping integrals of meson wave functions, where the q 2 dependence of the decay amplitude is automatically encoded.This is in contrast to some model approaches cited in the literature where the form factors are determined only at one kinematic point, i.e., either at q 2 → 0 or q 2 → q 2 max , and then extrapolated to the entire kinematic range using some phenomenological ansatz (mainly dipole or Gaussian form).
to higher and higher excited states.This is due to different kinematics and four-momentum transfer involved in different decay modes.
Our predicted form factors at q 2 → 0 (maximum recoil) and q 2 → q 2 max (minimum recoil) point for the transition to 1S, 2S and 3S charmonium and charm meson states are shown, respectively in Table (3,4,5).Before predicting the physical quantities of interest: decay width and branching fraction etc., it is interesting to go for a qualitative assessment of transition probabilities for transition to different Swave states.For this, we study the radial quark momentum distribution amplitude |p q |G B F (p q , −p q ) of the parent and daughter mesons over the physical range of respective quark momentum p q , for each such decay mode.From the plot shown in Figs.(9)(10) the overlap region between the momentum distribution profile of the parent and daughter meson in the transition to the 1S state is found to be maximum and it decreases for transitions to higher excited 2S and 3S Fig. 6: q 2 -dependence of form factors in B − → D * 0 (nS) type decays.
Table 5: Predicted values of form factors for decays to 3S state.
We calculate the decay width from the expression (26) via ( 14) and (9-10) and our predicted decay widths for general values of QCD coefficients (a 1 , a 2 ) of the operator product expansion are listed in Table 6 to facilitate a comparison with other dynamical model predictions.Our predicted branching fractions (BFs) for B − c , B0 s , B − , B0decays to 1S, 2S and 3S charmonium and charm meson states, are listed in Table (7,8,9), respectively, in reasonable agreement with available experimental data [82] and other model predictions.Our results for BFs of decays to 1S, 2S and 3S states corresponding to 3 sets of QCD parameters are listed in the second column of each table.As expected, our predicted branching fractions are obtained in the hierarchy:

B(B
Our results for transitions to 2S and 3S states are obtained two and three orders of magnitude down compared to those obtained for transition to 1S state.The node structure of the 2S wave function is responsible for small BFs.Since there is no node for the initial wave function, the contribution from the positive and negative parts of the final wave function cancel each other out yielding small BFs.In the case of the transition to 3S states, there are even more serious cancellations; leading to still smaller BFs.As expected, the tighter phase space and the q 2 -dependence of the form factors typical to the decay mode lead to smaller BFs for transitions to higher excited 2S and still smaller for the transition of 3S states.The BFs of the decay modes, considered in the present study, are obtained in a wide range of ∼ 10 −2 − 10 −6 .For nonleptonic B F -meson decays to 1S, 2S, and 3S charmonium and charm meson states, BFs range from ∼ 10 −2 − 10 −3 , ∼ 10 −4 − 10 −5 and ∼ 10 −5 − 10 −6 , respectively.The dominant decay modes:  tude O(∼ 10 −4 ), lie within the detection accuracy of current experiments.The neutral B-meson decay in the present study is found to have smaller BFs than those of charged Bmeson decays, as expected.This may be due to the spectator interaction effects of d and u quarks.As discussed earlier, the nonleptonic B F -decays to two charmful vector meson (V 1 V 2 ) states are of special interest as they help to evaluate CP-asymmetry factors, which provide the clue for testing SM predictions and exploring possible new physics beyond SM.The predicted decay widths (in 10 −15 GeV) for general values of QCD parameters (a 1 , a 2 ) for B − , B0 and B0 s decays to two charmful (1S, 2S, 3S) states along with their BFs, obtained in order of magnitude ∼ 10 −2 − 10 −4 , are shown in Table 10.However, the predicted BFs for B − c -decays: , obtained in the order of magnitude O(10 −6 ), are shown in Table 11 and 12, which cannot be measured in current experiments.Our predicted BFs for B0 , B − , B − c decays to two charmful states, however, are found somewhat underestimated compared to predictions to Ref. [23] and available experimental data [82].The relative size of BFs for nonleptonic decays is broadly estimated from a power counting of QCD factors: (a 1 , a 2 ) in the Wolfenstein parameterization [90].Accordingly, class I decays determined by a 1 are found to have comparatively large BFs as shown in Table.(7,8,9,10).On the other hand, class II decay modes, determined by a 2 , are found to have relatively small BFs Table.(7,8,9,11), as expected, except for decay modes: B − → J/ψK * − and B − → ψ(2S)K * − , characterized by a product of CKM factors: V bc V cs , which have BFs ∼ 0.31% and 0.12%, respectively.These modes should be measured at high luminosity hadron colliders.In class III decay modes that are characterized by Pauli interference, the BFs are determined by the relative value of a 1 with respect to a 2 .Considering positive values of a 1 = 1.12 and negative value of a 2 = −0.26 in Set 1, for example, which leads to destructive interference, the decay modes are suppressed compared to the case where interference is switched off.However, at a qualitative level, where the ratio a 2 a 1 , a function of running coupling constant α S evaluated at the factorization scale, is shown to be positive in the case of b-flavored meson decays corresponding to small coupling [32] s find enhancement by a factor of ∼ 3 and ∼ 20, respectively, over that obtained with a 2 = −0.26.For decay modes to 3S states, the enhancement is still more significant.
In the spirit of the experimental data favoring a constructive interference of the color-favored and color-suppressed b-flavored meson decays, the effect of Pauli interference inducing enhancement of BFs can be further probed by casting the decay width (Γ ) in the form: Γ = Γ 0 + ∆Γ , where x 2 a 1 a 2 and then evaluating ∆Γ Γ 0 in each case as done in [33][34][35][36][37][38][39][40]55].We find that the absolute values of ∆Γ Γ 0 for , and ∼ 88, respectively.For B c -decays to two charmful states: ), the enhancement in % is found to be ∼ 63, ∼ 96, ∼ 44, and ∼ 100, respectively.This indicates that interference is more significant in B − c -decays to two charmful states: B − c → D * 0 (1S, 2S)D * − (s) compared to other decay modes.This is particularly important since such de-Table 10: Decay widths in units of 10 −15 GeV and branching fractions in % for values of Wilson coefficient a 1 and a 2 .
Branching Fraction [82] Decay modes Decay width a 1 =1.12 a 1 =1.14 a 1 =0.93 Branching Fraction [23] Decay modes Decay width a 1 =1.12 a 1 =1.14 a 1 =0.93  cay modes have been proposed [91][92][93][94] for extraction of the CKM angle γ through amplitude relation.Another area of interest is CP-violation in nonleptonic decay of b-flavored mesons to two charmful states.The evaluation of the CP-odd fraction R ⊥ indeed indicates the degree of CP-violation in a decay process.We predict the longitudinal polarization fraction R L and CP-odd fraction R ⊥ from their expressions in terms of positive, negative and longitudinal polarization (10) as:  13 and Table 14, respectively.In all decay modes to two charmful states considered here, the longitudinal polarization fractions(R L ) dominate over the transverse polarization fraction (R ⊥ ).However the CP-odd fractions (R ⊥ ) in nonleptonic B c -decays to two charmful states are obtained here one order of magnitude higher than that in such other b-flavored meson decays; which indicates that CP-violation is more pronounced in B − c → D * 0 D * − (s) decays compared to that obtained in corresponding decays of ( B0 , B0 s and B − ) mesons.It is also found that R ⊥ in B − c -sector gets enhanced going from decays to ground state to corresponding decays to higher excited states (2S and 3S).Apart from significant CP-odd fraction R ⊥ predicted in B − c -decays to two charmful states, its noticeable enhancement is also predicted for B − c and B − decays to charmonium states: 15.
For color-favored B − c → D * (D * − , D * − s ) decays, the effect arising due to the short-distance non-spectator contribution is shown to be marginal [95].However, the longdistance (LD) nonfactorizable contributions from rescattering effects, final-state interactions, etc., may not be negligible.If a significant LD effect exists, one expects a large CP-odd fraction in these decays.The predicted longitudinal and transverse helicity amplitudes and the form factor g(q 2 ) yield R ⊥ values for different B F → V 1 (nS)V 2 decays, which are shown in Table 15.In particular, the predicted R ⊥ values for the transitions with two charmful final states indicate nonvanishing LD contributions, which lead to CP-violation in B − c → D * 0 D * − (s) and D * 0 D * − (s) decays.

Summary and conclusion
In this work, we study the exclusive two-body nonleptonic decays of b-flavored ( B0 , B0 s , B − and B − c ) mesons to S-wave charmonium and charm meson 1 − states, in the framework of relativistic independent quark (RIQ) model.The weak decay form factors representing decay amplitude and their q 2 -dependence are extracted from the overlapping integrals of the meson wave functions obtainable in the RIQ model.The predicted branching fractions for different decay modes are obtained in a wide range, from O(10 −6 ) for B − c decays to two charmful states to as high as ∼ 1.54%, ∼ 1.42% and ∼ 1.17% for B − → D * 0 ρ − , B0 → D * + ρ − and B0 s → D * + s ρ − , respectively.Our results are in general agreement with the available experimental data and other SM predictions.The decay modes with predicted branching fractions in the order: 3S)ρ − with predicted branching fractions upto ∼ 10 −4 may be accesible at high luminosity hadron colliders in near future.Other decay modes and especially B c -decay to two charmful states with predicted branching factions in the order O(10 −6 ) can not reach the detection ability of the current experiments.As expected, our predicted branching fractions are obtained in the hierarchy: This is due to i) the nodal structure of the participating daughter mesons in their excited states, ii) tighter phase space and iii) typical q 2 -dependence of the weak decay form factors for decay modes to higher excited (2S and 3S) states in comparison to that for the corresponding decay modes to the corresponding ground (1S) state.
The relative size of branching fractions is broadly estimated from a power counting of QCD factors: (a 1 , a 2 ) in the Wolfenstein parametrization.The class I decay modes characterized by a 1 are found to have large branching fractions, as expected; compared to those obtained for class II decays which are determined by a 2 .The branching fractions of class III decays characterized by Pauli interference for B c -decays to two charmful states in particular, obtained in the order of magnitude O(10 −6 ) can not be measured in current experiments.
In view of experimental data favoring a constructive interference of the color-favored and color-suppressed bflavored meson decays, the effect of Pauli interference is studied in different decay modes, by evaluating the enhancement factor in each such decay mode.For B , the enhancement (in %) is found to be ∼ 63, ∼ 96, ∼ 44 and ∼ 100, respectively.This shows that the Pauli interference is more significant in B − c -decays to two charmful states: s) compared to other decay modes.This is particularly important since such decay modes have been proposed for extracting the CKM angle γ through amplitude relations.
We predict the longitudinal polarization fraction (R L ) and transverse polarization (CP-odd) fraction (R ⊥ ).We find that predicted R L dominates in all decay modes to two charmful states; considered in the present study.The CP-odd fraction (R ⊥ ) in nonleptonic B − c -decays to two charmful states are obtained one order magnitude higher than that in such other b-flavored meson decays; which indicates the CPviolation is more significant in B − c → D * 0 D * − (s) compared to that obtained in ( B0 , B0 s and B − )-meson decays.For colorfavored B − c → D * (D * − , D * − s ) decays, the effect arising due to the short-distance non-spectator contribution is shown to be marginal.However, the long-distance (LD) nonfactorizable contributions from rescattering effects, final-state interactions, etc., may not be negligible.If a significant LD effect exists, one expects a large CP-odd fraction in these decays.The predicted longitudinal and transverse helicity amplitudes and the form factor g(q  In the RIQ model, a meson is picturized as a color-singlet assembly of a quark and an antiquark independently confined by an effective and average flavor-independent potential in the form: U(r) = 1 2 (1 + γ 0 )(ar 2 + V 0 ), where (a, V 0 ) are the potential parameters.It is believed that the zeroth-order quark dynamics generated by the phenomenological confining potential U(r) taken in equally mixed scalar-vector harmonic form can provide an adequate tree-level description of the decay process being analyzed in this work.With the interaction potential U(r) put into the zeroth-order quark lagrangian density, the ensuing Dirac equation admits a static solution of positive and negative energy as: where, ξ = (nl j) represents a set of Dirac quantum numbers specifying the eigenmodes.χ l jm j (r) and χl jm j (r) are the spin angular parts given by, , χl jm j (r) = (−1) j+m j −l χ l j−m j (r).(A.2) With the quark binding energy E q and quark mass m q written in the form E q = (E q − V 0 /2), m q = (m q + V 0 /2) and ω q = E q + m q , one can obtain solutions to the resulting radial equation for g ξ (r) and f ξ (r) in the form: where, r nl = (aω q ) −1/4 is a state independent length parameter, N nl is an overall normalization constant given by (ω nl /r nl ) (3E q + m q ) , (A.4) and L l+1/2 n−1 (r 2 /r 2 nl ) etc. are associated Laguerre polynomials.The radial solutions yield an independent quark bound-state condition in the form of a cubic equation: (ω q /a)(E q − m q ) = (4n + 2l − 1). (A.5) From the solution of the cubic equation (A.5), the zerothorder binding energies of the confined quark and antiquark are obtained for all possible eigenmodes.In the relativistic independent particle picture of this model, the relativistic constituent quark and antiquark are thought to move independently inside the meson bound-state |B F (p, S B F ) with their momentum p b and p q, respectively.In order to study the decay process which takes place in the momentum eigenstates of participating mesons, we Fourier transform the quark orbitals (A.1) to momentum space and obtain the momentum probability amplitude of the quark and antiquark of participating mesons in the following forms: G q(p q) = − iπN q 2α qω q (E p q + m q) E p q (E p q + E q) × exp − p 2 q 4α q .(A. G q(p q) = − iπN q 2α qω q (E p q + m q) E p q (E p q + E q) × p 2 q 2α q − 3 2 exp − p 2 q 4α q .
(A. G q(p q) = − iπN q 2α qω q (E p q + m q) E p q (E p q + E q) × p q4 8α 2 q − 5p 2 q 4α q + 15 8 exp (− p 2 q 4α q ).(A.8) With the momentum probability amplitudes of quark constituents, we construct an effective momentum distribution function for the meson state in the form G B F (p b , p q) = G b (p b )G q(p q) in the light of an ansatz of Margolis and Mendel in their bag model description of the meson bound state [96].
Here the effective momentum distribution function G B F (p b , p q), which in fact, embodies the bound-state characteristics of |B F (p, S B F ) , defines the meson bound state at definite momentum p and spin projection S B F in the form: b † q(p q, λ q)|0 , (A.9) where |(p b , λ b ); (p q, λ q) is the Fock-space representation of unbound quark and antiquark in a color-singlet configuration with respective momentum and spin: (p b , λ b ) and (p q, λ q).b † b (p b , λ b ) and b † q(p q, λ q) are the quark and antiquark creation operators and Λ (p, S B F ) is a bag-like integral operator taken in the form: (A.11)
2), fixed by Buras et al. [52] in the mid-1980s, whereas Dubnicka et al. [89] use a different set of numerical value, i.e., Set 3: (a b 1 , a b to an upper bound on the quark momentum |p b | < M 2 −m 2 q 2M in order to retain m 2 b (|p b |) the positive definite.The upper limit |p b | max would have no other bearing to seriously affect the calculation which is apparent from the shape of the radial quark momentum distribution |p b |G (p b , −p b ).

Fig. 9 :
Fig. 9: Overlap of momentum distribution amplitudes of the initial and final meson states.

Fig. 10 :
Fig. 10: Overlap of momentum distribution amplitudes of the initial and final meson states.
Our predicted R L and R ⊥ for (B − c → D * 0 (nS)D * − (s) ) and ( B0 → D * + (nS)D * − (s) , B − → D * 0 (nS)D * − (s) , B0 s → D * + s (nS)D * − ) are shown in Table decays.In particular, the predicted R ⊥ values for the transitions with two charmful final states indicate nonvanishing LD contributions, which lead to a significant CP-violation in B − c → D * 0 D * − (s) and D * 0 D * − (s) decays.In conclusion, the present analysis shows that the factorization approximation works reasonably well in describing the exclusive nonleptonic B F → V 1 (nS)V 2 decays in the framework of the RIQ model.

)
For the excited meson state:(n = 2,l = 0), G b (p b ) = iπN b 2α b ω b (E p b + m b ) E p b (E p b + E b )

Λ 3 N
(p, S B F ) = √ B F (p) ∑ λ b ,λ q ζ B F b, q d 3 p b d 3 p q δ (3) (p b + p q − p) G B F (p b , p q).(A.10)Here, √ 3 is the effective color factor and ζ B F b, q is the SU(6) spin-flavor coefficients for the B F -meson state.Imposing the normalization condition in the formB F (p)|B F (p ) = δ (3) (p − p ), the meson state normalization N B F (p) is obtainable in an integral form N B F (p) = d 3 p b |G B F (p b , p q)| 2 .

Table 1 :
The masses and decay constants of mesons.

Table 2 :
The rms values of quark momentum, expectation values of the momentum of quark and antiquark, and the sum of the momentum of quark and antiquark in the meson states.

Table 3 :
Predicted values of form factors for decays to 1S state.

Table 4 :
Predicted values of form factors for decays to 2S state.

Table 6 :
Decay widths in units of 10 −15 GeV in terms of Wilson coefficients a 1 and a 2 .

2 2 Table 7 :
Branching fractions in % for values of Wilson coefficients a 1 and a 2 .

Table 8 :
Branching fractions in 10 −4 for values of Wilson coefficients a 1 and a 2 .
−2) which should be experimentally accessible.For such decays to corresponding 2S modes, the predicted BFs, of 2.35, 10.45, 11.17, 11.81 and 35.18in the order of magni-

Table 9 :
Branching fractions in 10 −5 for values of Wilson coefficients a 1 and a 2 .

Table 12 :
Decay widths in units of 10 −15 GeV and branching fractions of order 10 −6 for general values of the Wilson coefficients a 1 and a 2 .

Table 15 :
Predicted longitudinal fraction (R L ) and CP-odd fraction (R ⊥ ) of state.− decays to two charmful mesons in their ground state, should be experimentally accessible.The decay modes to 2S and 3S char-monium states such as B O(10 −2 ), which include the B c -meson decays to 1S charmonium and charm meson states as well as B0 s , B0 , B