focus of the current work is to establish the effects of self-consistency
originating from the transition form factors to the weak decays in a
quantitative manner. The self-consistency affects the numerical results of the
form factors $A_{0}(q^{2})$ and $A_{1}(q^{2})$, which in turn appear in the
semileptonic and nonleptonic decays of the $B_{c}$ meson. It is well known
that the coefficient of $A_{0}(q^{2})$ form factor is suppressed in the
semileptonic decay rates; therefore, semileptonic decays only provide a
comprehensive picture that corresponds to the effects originating from
$A_{1}(q^{2})$ form factor. Thus, to observe the effect of $A_{0}(q^{2})$ form
factor, we calculated the $B_{c}\to PV$ decays which involve $F_{1}(q^{2})$
and $A_{0}(q^{2})$ form factors. Therefore, we calculated the transition form
factors in CLFQM formalism in Tables 4 and 5. In the current work, we
thoroughly examined the appropriate $q^{2}$ formulations, especially for
bottom-conserving transitions involving $B_{c}\to V(P)$ form factors.
Therefore, we have analyzed two different $q^{2}$ formulations in Type-II
correspondence referred to as Type-II and Type-II* by using Eqs. (37) and
(38), respectively. We also compared our results with Type-I correspondence
for the $q^{2}$ formulation in Eq. (40) to assess the effects of self-
consistency quantitatively. Consequently, we observed their implications on
semileptonic and nonleptonic weak decays of $B_{c}$ meson. In addition, we
calculated the experimentally significant physical observables, namely, the FB
asymmetry, lepton-side convexity parameter, longitudinal (transverse)
polarization of the charged lepton, and asymmetry parameter. We list our major
conclusions as follows.

  * •

We reconfirmed that the form factors $A_{0}(q^{2})$ and $A_{1}(q^{2})$ in
CLFQM Type-I correspondence scheme acquire zero-mode contributions through
$B^{(i)}_{j}$ functions, which results in different numerical values for the
longitudinal and transverse polarization states. These issues are resolved
within Type-II correspondence, which ensures self-consistency and covariance
of matrix elements. It may be emphasized that the zero-mode contributions in
Type-II correspondence vanish numerically, though exist formally in the
analytical relations of the form factors. For bottom-conserving transitions,
the numerical results of the Type-II(*) form factors, $A_{0}(0)$ and
$A_{1}(0)$, show a significant change of $(70-90)\%$ and $\sim 23\%$,
respectively, as compared to those of the Type-I scheme. Similarly, for
bottom-changing transitions, we observed that the numerical values of the form
factor $A_{0}(0)(A_{1}(0))$ in Type-II correspondence, for both Eqs. (37) and
(38), roughly vary by $\sim 30\%\leavevmode\nobreak\ (10\%)$ as compared to
Type-I for $B_{c}\to D_{(s)}^{*}$ transitions. We also observe that these form
factors are sensitive to $q^{2}$ formulations, resulting in significantly
different slope parameters (coefficients). Therefore, we conclude that the
improvement in the numerical results of Type-II correspondence cannot simply
be determined from the variation of form factors at $q^{2}=0$; the
modification in the numerical values of slope parameters also plays a
significant role in the quantitative evaluation of these effects. Furthermore,
the Type-II correspondence influences $B_{c}\to J/\psi$ transition form
factors minimally, as compared to both bottom-conserving and other bottom-
changing transition form factors.

  * •

We also found that the $M^{{\prime}({\prime}{\prime})}\to
M_{0}^{{\prime}({\prime}{\prime})}$ transformation, in general, affects the
numerical values of all the transition form factors irrespective of the spin-
parity of the final state meson. Therefore, the numerical values of the form
factors which do not suffer from self-consistency issues have also been
modified. We found that the numerical results for the Type-II(*) form factors
$F^{B_{c}B_{(s)}}(q^{2})$ are in very good agreement with the LQCD
observations for both at $q^{2}=0$ and $q^{2}_{max}$. On the other hand, the
numerical values of the form factors
$F^{B_{c}D_{s}}(q^{2})\leavevmode\nobreak\ (F^{B_{c}D}(q^{2}))$ are consistent
with the LQCD predictions within $\sim 10\%$ ($\sim 25\%$).

  * •

We found that $\mathcal{B}(B_{c}\to B_{s}^{(*)}l\nu_{l})$ and
$\mathcal{B}(B_{c}\to J/\psi(\eta_{c})l\nu_{l})$ are the most dominant among
the $B_{c}\to V(P)l\nu_{l}$ semileptonic decays. Our results for
$\mathcal{B}(B_{c}^{+}\to B_{(s)}^{0}l^{+}\nu_{l})$ are in good agreement with
the recent LQCD predictions. In addition, the decay width ratios of
semileptonic decays with $B_{s}^{0}$ to $B^{0}$ in final state for Type-II*
match well with LCQD expectations. Furthermore, the decays involving the
$\tau$ lepton have the lowest branching ratios among all the decays because of
the significantly larger mass of the $\tau$ lepton. We quantified the effect
of self-consistency on the branching ratios of the semileptonic decay modes by
comparing our results with those of Type-I correspondence. We found that the
numerical results for the Type-II scheme are larger by $(50-60)\%$,
$(64-85)\%$, and $(24-28)\%$ as compared to the branching ratios in the Type-I
scheme involving $B_{c}\to B_{(s)}^{*}$, $B_{c}\to D^{*}$, and $B_{c}\to
J/\psi$ semileptonic decays, respectively. Furthermore, we found that our LFU
ratio involving $b\to c\tau\nu_{\tau}$ for $J/\psi$ in the final state match
well with LQCD and other theoretical models; however, are smaller than the
experimental measurement.

  * •

For the nonleptonic $B_{c}$ decays, branching ratios will be affected by the
self-consistency issues for decays involving $A_{0}(q^{2})$ transition form
factors. These decays presented an excellent opportunity to observe these
effects in a quantitative manner. Interestingly, the branching ratios of CKM
and color-favored $B_{c}\to PV$ decays are roughly affected by
$\sim(30-70)\%$. However, the color-favored CKM-suppressed and doubly-
suppressed modes are more seriously affected, where some of the branching
ratios are changing by more than $100\%$. Therefore, we conclude that the
self-consistency effects are predominant in $B_{c}\to PV$ decays.

In conclusion, the agreement between our predictions and the LQCD results
assures the reliability of our numerical results for $B_{c}$ meson decays. We
wish to remark that the we have ignored nonfactorizable processes, for
example, W-exchange, W-annihilation, penguin processes in our analysis of
nonleptonic $B_{c}$ weak decays. However, the study of nonfactorizable
contributions and CP-symmetries can more reliably be carried out in model-
independent manner that requires huge amount of experimental data. We hope
that the experimental observation of these $B_{c}$ weak decays can help to
shed some light on the underlying physics of the $B_{c}$ meson.

## Acknowledgment

The authors are pleased to express their thanks to H. Y. Cheng, C. K. Chua, H.
M. Choi, C. R. Ji, and Gautam Bhattacharyya for their helpful comments and
discussions. The author (RD) gratefully acknowledge the financial support by
the Department of Science and Technology (SERB:CRG/2018/002796), New Delhi.

## References

  * [1]

## References

* [2] S. S. Gershtein, V. V. Kiselev, A. K. Likhoded and A. V. Tkabladze, Phys. Usp. 38, 1-37 (1995) doi:10.1070/PU1995v038n01ABEH000063 [arXiv:hep-ph/9504319 [hep-ph]].
* [3] I. P. Gouz, V. V. Kiselev, A. K. Likhoded, V. I. Romanovsky and O. P. Yushchenko, Phys. Atom. Nucl. 67, 1559-1570 (2004) doi:10.1134/1.1788046 [arXiv:hep-ph/0211432 [hep-ph]].
* [4] G. Chen, C. H. Chang and X. G. Wu, Phys. Rev. D 97, no.11, 114022 (2018) doi:10.1103/PhysRevD.97.114022 [arXiv:1803.11447 [hep-ph]].
* [5] R. Aaij et al. [LHCb], JHEP 07, 123 (2020) doi:10.1007/JHEP07(2020)123 [arXiv:2004.08163 [hep-ex]].
* [6] R. Aaij et al. [LHCb], Phys. Lett. B 742, 29-37 (2015) doi:10.1016/j.physletb.2015.01.010 [arXiv:1411.6899 [hep-ex]].
* [7] R. Aaij et al. [LHCb], Phys. Rev. Lett. 122, no.23, 232001 (2019) doi:10.1103/PhysRevLett.122.232001 [arXiv:1904.00081 [hep-ex]].
* [8] A. M. Sirunyan et al. [CMS], Phys. Rev. Lett. 122, no.13, 132001 (2019) doi:10.1103/PhysRevLett.122.132001 [arXiv:1902.00571 [hep-ex]].
* [9] G. Aad et al. [ATLAS], Phys. Rev. Lett. 113, no.21, 212004 (2014) doi:10.1103/PhysRevLett.113.212004 [arXiv:1407.1032 [hep-ex]].
* [10] R. Aaij et al. [LHCb], Phys. Rev. Lett. 111, no.18, 181801 (2013) doi:10.1103/PhysRevLett.111.181801 [arXiv:1308.4544 [hep-ex]].
* [11] R. Aaij et al. [LHCb], JHEP 12, 117 (2021) doi:10.1007/JHEP12(2021)117 [arXiv:2109.00488 [hep-ex]].
* [12] R. Aaij et al. [LHCb], Nucl. Phys. B 930, 563-582 (2018) doi:10.1016/j.nuclphysb.2018.03.015 [arXiv:1712.04702 [hep-ex]].
* [13] R. L. Workman et al. [Particle Data Group], PTEP 2022, 083C01 (2022) doi:10.1093/ptep/ptac097
* [14] Y. S. Amhis et al. [HFLAV], Phys. Rev. D 107, no.5, 052008 (2023) doi:10.1103/PhysRevD.107.052008 [arXiv:2206.07501 [hep-ex]].
* [15] G. Aad et al. [ATLAS], JHEP 08, 087 (2022) doi:10.1007/JHEP08(2022)087 [arXiv:2203.01808 [hep-ex]].
* [16] R. Aaij et al. [LHCb], JHEP 09, 153 (2016) doi:10.1007/JHEP09(2016)153 [arXiv:1607.06823 [hep-ex]].
* [17] G. Aad et al. [ATLAS], Eur. Phys. J. C 76, no.1, 4 (2016) doi:10.1140/epjc/s10052-015-3743-8 [arXiv:1507.07099 [hep-ex]].
* [18] R. Aaij et al. [LHCb], Phys. Rev. Lett. 120, no.12, 121801 (2018) doi:10.1103/PhysRevLett.120.121801 [arXiv:1711.05623 [hep-ex]].
* [19] J. Harrison et al. [LATTICE-HPQCD], Phys. Rev. Lett. 125, no.22, 222003 (2020) doi:10.1103/PhysRevLett.125.222003 [arXiv:2007.06956 [hep-lat]].
* [20] L. J. Cooper et al. [HPQCD], Phys. Rev. D 102, no.1, 014513 (2020) [erratum: Phys. Rev. D 103, no.9, 099901 (2021)] doi:10.1103/PhysRevD.102.014513 [arXiv:2003.00914 [hep-lat]].
* [21] L. J. Cooper et al. [HPQCD], Phys. Rev. D 105, no.1, 014503 (2022) doi:10.1103/PhysRevD.105.014503 [arXiv:2108.11242 [hep-lat]].
* [22] V. V. Kiselev, A. E. Kovalsky and A. K. Likhoded, Nucl. Phys. B 585, 353-382 (2000) doi:10.1016/S0550-3213(00)00386-2 [arXiv:hep-ph/0002127 [hep-ph]].
* [23] V. V. Kiselev, O. N. Pakhomova and V. A. Saleev, J. Phys. G 28, 595-606 (2002) doi:10.1088/0954-3899/28/4/301 [arXiv:hep-ph/0110180 [hep-ph]].
* [24] V. V. Kiselev, [arXiv:hep-ph/0211021 [hep-ph]].
* [25] A. Abd El-Hady, M. A. K. Lodhi and J. P. Vary, Phys. Rev. D 59, 094001 (1999) doi:10.1103/PhysRevD.59.094001 [arXiv:hep-ph/9807225 [hep-ph]].
* [26] A. Abd El-Hady, J. H. Munoz and J. P. Vary, Phys. Rev. D 62, 014019 (2000) doi:10.1103/PhysRevD.62.014019 [arXiv:hep-ph/9909406 [hep-ph]].
* [27] X. J. Li, Y. S. Li, F. L. Wang and X. Liu, [arXiv:2308.07206 [hep-ph]].
* [28] R. N. Faustov, V. O. Galkin and X. W. Kang, Phys. Rev. D 106, no.1, 013004 (2022) doi:10.1103/PhysRevD.106.013004 [arXiv:2206.10277 [hep-ph]].
* [29] Z. Q. Zhang, Z. J. Sun, Y. C. Zhao, Y. Y. Yang and Z. Y. Zhang, Eur. Phys. J. C 83, no.6, 477 (2023) doi:10.1140/epjc/s10052-023-11576-x [arXiv:2301.11107 [hep-ph]].
* [30] W. Wang, Y. L. Shen and C. D. Lu, Phys. Rev. D 79, 054012 (2009) doi:10.1103/PhysRevD.79.054012 [arXiv:0811.3748 [hep-ph]].
* [31] W. Wang, Y. L. Shen and C. D. Lu, Eur. Phys. J. C 51, 841-847 (2007) doi:10.1140/epjc/s10052-007-0334-3 [arXiv:0704.2493 [hep-ph]].
* [32] D. Ebert, R. N. Faustov and V. O. Galkin, Eur. Phys. J. C 32, 29-43 (2003) doi:10.1140/epjc/s2003-01347-5 [arXiv:hep-ph/0308149 [hep-ph]].
* [33] D. Ebert, R. N. Faustov and V. O. Galkin, Phys. Rev. D 68, 094020 (2003) doi:10.1103/PhysRevD.68.094020 [arXiv:hep-ph/0306306 [hep-ph]].
* [34] M. A. Ivanov, J. G. Korner and P. Santorelli, Phys. Rev. D 73, 054024 (2006) doi:10.1103/PhysRevD.73.054024 [arXiv:hep-ph/0602050 [hep-ph]].
* [35] M. A. Ivanov, J. G. Korner and O. N. Pakhomova, Phys. Lett. B 555, 189-196 (2003) doi:10.1016/S0370-2693(03)00052-2 [arXiv:hep-ph/0212291 [hep-ph]].
* [36] M. A. Ivanov, J. G. Körner and P. Santorelli, doi:10.1007/978-88-470-0530-3_43 [arXiv:hep-ph/0609122 [hep-ph]].
* [37] S. Naimuddin, S. Kar, M. Priyadarsini, N. Barik and P. C. Dash, Phys. Rev. D 86, 094028 (2012) doi:10.1103/PhysRevD.86.094028
* [38] L. Nayak, P. C. Dash, S. Kar and N. Barik, Phys. Rev. D 105, no.5, 053007 (2022) doi:10.1103/PhysRevD.105.053007 [arXiv:2202.01167 [hep-ph]].
* [39] Z. Rui, Z. Zhitian and C. D. Lu, Phys. Rev. D 86, 074019 (2012) doi:10.1103/PhysRevD.86.074019 [arXiv:1203.2303 [hep-ph]].
* [40] Z. Rui and Z. T. Zou, Phys. Rev. D 90, no.11, 114030 (2014) doi:10.1103/PhysRevD.90.114030 [arXiv:1407.5550 [hep-ph]].
* [41] J. Sun, N. Wang, Q. Chang and Y. Yang, Adv. High Energy Phys. 2015, 104378 (2015) doi:10.1155/2015/104378 [arXiv:1504.01286 [hep-ph]].
* [42] H. Y. Cheng, C. Y. Cheung, C. W. Hwang and W. M. Zhang, Phys. Rev. D 57, 5598-5610 (1998) doi:10.1103/PhysRevD.57.5598 [arXiv:hep-ph/9709412 [hep-ph]].
* [43] W. Jaus, Phys. Rev. D 41, 3394 (1990) doi:10.1103/PhysRevD.41.3394
* [44] W. Jaus, Phys. Rev. D 44, 2851-2859 (1991) doi:10.1103/PhysRevD.44.2851
* [45] C. R. Ji, P. L. Chung and S. R. Cotanch, Phys. Rev. D 45, 4214-4220 (1992) doi:10.1103/PhysRevD.45.4214
* [46] H. Y. Cheng, C. Y. Cheung and C. W. Hwang, Phys. Rev. D 55, 1559-1577 (1997) doi:10.1103/PhysRevD.55.1559 [arXiv:hep-ph/9607332 [hep-ph]].
* [47] W. Jaus, Phys. Rev. D 60, 054026 (1999) doi:10.1103/PhysRevD.60.054026
* [48] E. E. Salpeter and H. A. Bethe, Phys. Rev. 84, 1232-1242 (1951) doi:10.1103/PhysRev.84.1232
* [49] E. E. Salpeter, Phys. Rev. 87, 328-342 (1952) doi:10.1103/PhysRev.87.328
* [50] H. M. Choi and C. R. Ji, Phys. Rev. D 58, 071901 (1998) doi:10.1103/PhysRevD.58.071901 [arXiv:hep-ph/9805438 [hep-ph]].
* [51] H. Y. Cheng, C. K. Chua and C. W. Hwang, Phys. Rev. D 69, 074025 (2004) doi:10.1103/PhysRevD.69.074025 [arXiv:hep-ph/0310359 [hep-ph]].
* [52] H. M. Choi and C. R. Ji, Phys. Rev. D 89, no.3, 033011 (2014) doi:10.1103/PhysRevD.89.033011 [arXiv:1308.4455 [hep-ph]].
* [53] W. Jaus, Phys. Rev. D 67, 094010 (2003) doi:10.1103/PhysRevD.67.094010 [arXiv:hep-ph/0212098 [hep-ph]].
* [54] B. L. G. Bakker, H. M. Choi and C. R. Ji, Phys. Rev. D 63, 074014 (2001) doi:10.1103/PhysRevD.63.074014 [arXiv:hep-ph/0008147 [hep-ph]].
* [55] B. L. G. Bakker, H. M. Choi and C. R. Ji, Phys. Rev. D 65, 116001 (2002) doi:10.1103/PhysRevD.65.116001 [arXiv:hep-ph/0202217 [hep-ph]].
* [56] B. L. G. Bakker, H. M. Choi and C. R. Ji, Phys. Rev. D 67, 113007 (2003) doi:10.1103/PhysRevD.67.113007 [arXiv:hep-ph/0303002 [hep-ph]].
* [57] H. M. Choi and C. R. Ji, Phys. Rev. D 72, 013004 (2005) doi:10.1103/PhysRevD.72.013004 [arXiv:hep-ph/0504219 [hep-ph]].
* [58] R. C. Verma, J. Phys. G 39, 025005 (2012) doi:10.1088/0954-3899/39/2/025005 [arXiv:1103.2973 [hep-ph]].
* [59] Q. Chang, X. L. Wang and L. T. Wang, Chin. Phys. C 44, no.8, 083105 (2020) doi:10.1088/1674-1137/44/8/083105 [arXiv:2003.10833 [hep-ph]].
* [60] Q. Chang, X. N. Li and L. T. Wang, Eur. Phys. J. C 79, no.5, 422 (2019) doi:10.1140/epjc/s10052-019-6949-3 [arXiv:1905.05098 [hep-ph]].
* [61] Q. Chang, X. N. Li, X. Q. Li, F. Su and Y. D. Yang, Phys. Rev. D 98, no.11, 114018 (2018) doi:10.1103/PhysRevD.98.114018 [arXiv:1810.00296 [hep-ph]].
* [62] A. Hazra, T. M. S., N. Sharma and R. Dhir, [arXiv:2309.03655 [hep-ph]].
* [63] H. M. Choi, Phys. Rev. D 103, no.7, 073004 (2021) doi:10.1103/PhysRevD.103.073004 [arXiv:2102.02015 [hep-ph]].
* [64] M. Wirbel, B. Stech and M. Bauer, Z. Phys. C 29, 637 (1985) doi:10.1007/BF01560299
* [65] W. Jaus, Phys. Rev. D 53, 1349 (1996) [erratum: Phys. Rev. D 54, 5904 (1996)] doi:10.1103/PhysRevD.53.1349
* [66] O. Heger, M. Gómez-Rocha and W. Schweiger, Phys. Rev. D 104, no.11, 116005 (2021) doi:10.1103/PhysRevD.104.116005 [arXiv:2109.07849 [hep-ph]].
* [67] D. Melikhov, Phys. Rev. D 53, 2460-2479 (1996) doi:10.1103/PhysRevD.53.2460 [arXiv:hep-ph/9509268 [hep-ph]].
* [68] J. D. Richman and P. R. Burchat, Rev. Mod. Phys. 67, 893-976 (1995) doi:10.1103/RevModPhys.67.893 [arXiv:hep-ph/9508250 [hep-ph]].
* [69] D. Becirevic and A. B. Kaidalov, Phys. Lett. B 478, 417-423 (2000) doi:10.1016/S0370-2693(00)00290-2 [arXiv:hep-ph/9904490 [hep-ph]].
* [70] D. Melikhov and B. Stech, Phys. Rev. D 62, 014006 (2000) doi:10.1103/PhysRevD.62.014006 [arXiv:hep-ph/0001113 [hep-ph]].
* [71] M. Bauer and M. Wirbel, Z. Phys. C 42, 671 (1989) doi:10.1007/BF01557675
* [72] R. J. Hill, Phys. Rev. D 73, 014012 (2006) doi:10.1103/PhysRevD.73.014012 [arXiv:hep-ph/0505129 [hep-ph]].
* [73] R. J. Hill, eConf C060409, 027 (2006) [arXiv:hep-ph/0606023 [hep-ph]].
* [74] D. Melikhov, Eur. Phys. J. direct 4, no.1, 2 (2002) doi:10.1007/s1010502c0002 [arXiv:hep-ph/0110087 [hep-ph]].
* [75] N. Gubernari, A. Khodjamirian, R. Mandal and T. Mannel, JHEP 05, 029 (2022) doi:10.1007/JHEP05(2022)029 [arXiv:2203.08493 [hep-ph]].
* [76] C. Bourrely, I. Caprini and L. Lellouch, Phys. Rev. D 79, 013008 (2009) [erratum: Phys. Rev. D 82, 099902 (2010)] doi:10.1103/PhysRevD.82.099902 [arXiv:0807.2722 [hep-ph]].
* [77] P. Colangelo and F. De Fazio, Phys. Rev. D 61, 034012 (2000) doi:10.1103/PhysRevD.61.034012 [arXiv:hep-ph/9909423 [hep-ph]].
* [78] M. A. Ivanov, J. G. Korner and P. Santorelli, Phys. Rev. D 71, 094006 (2005) [erratum: Phys. Rev. D 75, 019901 (2007)] doi:10.1103/PhysRevD.75.019901 [arXiv:hep-ph/0501051 [hep-ph]].
* [79] R. Dhir and C. S. Kim, Phys. Rev. D 88, no.3, 034024 (2013) doi:10.1103/PhysRevD.88.034024 [arXiv:1307.0216 [hep-ph]].
* [80] T. E. Browder and K. Honscheid, Prog. Part. Nucl. Phys. 35, 81-220 (1995) doi:10.1016/0146-6410(95)00042-H [arXiv:hep-ph/9503414 [hep-ph]].
* [81] M. Wirbel, Prog. Part. Nucl. Phys. 21, 33-98 (1988) doi:10.1016/0146-6410(88)90031-2
* [82] R. Dhir and R. C. Verma, Phys. Rev. D 79, 034004 (2009) doi:10.1103/PhysRevD.79.034004 [arXiv:0810.4284 [hep-ph]].
* [83] T. E. Browder, K. Honscheid and S. Playfer, doi:10.1142/9789814503846_0004 [arXiv:hep-ph/9404314 [hep-ph]].
* [84] F. Jugeau, A. Le Yaouanc, L. Oliver and J. C. Raynal, Phys. Rev. D 72, 094010 (2005) doi:10.1103/PhysRevD.72.094010 [arXiv:hep-ph/0504206 [hep-ph]].
* [85] R. Dhir and C. S. Kim, Phys. Rev. D 87, no.3, 034004 (2013) doi:10.1103/PhysRevD.87.034004 [arXiv:1210.7890 [hep-ph]].
* [86] H. Y. Cheng and C. W. Chiang, Phys. Rev. D 81, 074021 (2010) doi:10.1103/PhysRevD.81.074021 [arXiv:1001.0987 [hep-ph]].
* [87] H. Y. Cheng, C. W. Chiang and A. L. Kuo, Phys. Rev. D 91, no.1, 014011 (2015) doi:10.1103/PhysRevD.91.014011 [arXiv:1409.5026 [hep-ph]].
* [88] C. W. Hwang, Phys. Rev. D 81, 114024 (2010) doi:10.1103/PhysRevD.81.114024 [arXiv:1003.0972 [hep-ph]].
* [89] C. Y. Cheung, C. W. Hwang and W. M. Zhang, Z. Phys. C 75, 657-664 (1997) doi:10.1007/s002880050511 [arXiv:hep-ph/9602309 [hep-ph]].
* [90] H. M. Choi and C. R. Ji, Phys. Rev. D 80, 054016 (2009) doi:10.1103/PhysRevD.80.054016 [arXiv:0903.0455 [hep-ph]].
* [91] M. Neubert, V. Rieckert, B. Stech and Q. P. Xu, Adv. Ser. Direct. High Energy Phys. 10, 286-333 (1992) doi:10.1142/9789814503587_0005
* [92] J. N. Pandya, P. Santorelli and N. R. Soni, doi:10.1140/epjs/s11734-023-01006-0 [arXiv:2307.14245 [hep-ph]].
* [93] E. Hernandez, J. Nieves and J. M. Verde-Velasco, Phys. Rev. D 74, 074008 (2006) doi:10.1103/PhysRevD.74.074008 [arXiv:hep-ph/0607150 [hep-ph]].
* [94] A. Y. Anisimov, I. M. Narodetsky, C. Semay and B. Silvestre-Brac, Phys. Lett. B 452, 129-136 (1999) doi:10.1016/S0370-2693(99)00273-7 [arXiv:hep-ph/9812514 [hep-ph]].
* [95] R. Aaij et al. [LHCb], Phys. Rev. D 90, no.3, 032009 (2014) doi:10.1103/PhysRevD.90.032009 [arXiv:1407.2126 [hep-ex]].
* [96] R. J. Dowdall, C. T. H. Davies, G. P. Lepage and C. McNeile, Phys. Rev. D 88, 074504 (2013) doi:10.1103/PhysRevD.88.074504 [arXiv:1303.1670 [hep-lat]].
* [97] A. Bharucha, D. M. Straub and R. Zwicky, JHEP 08, 098 (2016) doi:10.1007/JHEP08(2016)098 [arXiv:1503.05534 [hep-ph]]
* [98] B. Chakraborty et al. [HPQCD], Phys. Rev. D 96, no.7, 074502 (2017) doi:10.1103/PhysRevD.96.074502 [arXiv:1703.05552 [hep-lat]].
* [99] V. Lubicz et al. [ETM], Phys. Rev. D 96, no.3, 034524 (2017) doi:10.1103/PhysRevD.96.034524 [arXiv:1707.04529 [hep-lat]].
* [100] M. Ablikim et al. [BESIII], Phys. Rev. Lett. 131, no.14, 141802 (2023) doi:10.1103/PhysRevLett.131.141802 [arXiv:2304.12159 [hep-ex]].

## Appendix

##  Appendix A Branching ratios of $B_{c}^{+}\to Pl^{+}\nu_{l}$ decays

We list the numerical values of $B_{c}^{+}\to Pl^{+}\nu_{l}$ semileptonic
decays using the form factors given in Tables 4 and 5, and the numerical
inputs are discussed in Sec. III.

Table 1: Branching ratios of $B_{c}^{+}\to Pl^{+}\nu_{l}$ decays. For the definitions of Type-II, Type-II*, and Type-I, refer to the caption of Table 4. Decay | Type-II | Type-II* | Type-I
---|---|---|---
$B_{c}^{+}\to B^{0}e^{+}\nu_{e}$ | $7.76\times 10^{-4}$ | $1.07\times 10^{-3}$ | $2.40\times 10^{-4}$
$B_{c}^{+}\to B^{0}\mu^{+}\nu_{\mu}$ | $7.46\times 10^{-4}$ | $1.01\times 10^{-3}$ | $2.17\times 10^{-4}$
$B_{c}^{+}\to B_{s}^{0}e^{+}\nu_{e}$ | $1.35\times 10^{-2}$ | $1.75\times 10^{-2}$ | $5.94\times 10^{-3}$
$B_{c}^{+}\to B_{s}^{0}\mu^{+}\nu_{\mu}$ | $1.28\times 10^{-2}$ | $1.64\times 10^{-2}$ | $5.37\times 10^{-3}$
$B_{c}^{+}\to D^{0}e^{+}\nu_{e}$ | $1.82\times 10^{-5}$ | $1.21\times 10^{-5}$ | $1.41\times 10^{-6}$
$B_{c}^{+}\to D^{0}\mu^{+}{\nu}_{\mu}$ | $1.82\times 10^{-5}$ | $1.21\times 10^{-5}$ | $1.39\times 10^{-6}$
$B_{c}^{+}\to D^{0}\tau^{+}{\nu}_{\tau}$ | $1.35\times 10^{-5}$ | $7.71\times 10^{-6}$ | $3.63\times 10^{-8}$
$B_{c}^{+}\to\eta_{c}e^{+}{\nu}_{e}$ | $7.56\times 10^{-3}$ | $6.43\times 10^{-3}$ | $3.06\times 10^{-3}$
$B_{c}^{+}\to\eta_{c}\mu^{+}{\nu}_{\mu}$ | $7.53\times 10^{-3}$ | $6.40\times 10^{-3}$ | $3.03\times 10^{-3}$
$B_{c}^{+}\to\eta_{c}\tau^{+}{\nu}_{\tau}$ | $2.25\times 10^{-3}$ | $1.74\times 10^{-3}$ | $2.80\times 10^{-4}$

Table 2:  Constituent quark masses and the Gaussian parameters $\beta$ for P
and V mesons.111Note that here $q$ denotes either $u$ or $d$ quark.
Constituent quark masses (in GeV)
---
$m_{u}=m_{d}=0.27;$ | $m_{s}=0.45;$ | $m_{c}=1.60;$ | $m_{b}=4.80$
Gaussian parameters $\beta$ (in GeV)
${}^{2S+1}L_{J}$ | ${}^{1}S_{0}$ | ${}^{3}S_{1}$ | ${}^{2S+1}L_{J}$ | ${}^{1}S_{0}$ | ${}^{3}S_{1}$
$\beta_{c\bar{q}}$ | $0.4656$ | $0.4255$ | $\beta_{b\bar{q}}$ | $0.5547$ | $0.5183$
$\beta_{c\bar{s}}$ | $0.5358$ | $0.4484$ | $\beta_{b\bar{s}}$ | $0.6103$ | $0.5589$
$\beta_{c\bar{c}}$ | $0.7690$ | $0.6492$ | $\beta_{b\bar{c}}$ | $0.9207$ | $0.8451$
Table 3: Transition pole masses for $B_{c}\to P$ and $V$ form factors (in GeV).  Quark transition | $F_{1}(q^{2}),V(q^{2})$ | $F_{0}(q^{2})$ | $A_{0}(q^{2})$ | $A_{1}(q^{2}),A_{2}(q^{2})$
---|---|---|---|---
$J^{P}=1^{-}$ | $0^{+}$ | $0^{-}$ | $1^{+}$
Bottom-conserving transitions
$c\to d$ | $2.010$ | $2.308$ | $1.870$ | $2.422$
$c\to s$ | $2.112$ | $2.318$ | $1.968$ | $2.460$
Bottom-changing transitions
$b\to u$ | $5.325$ | $5.670$ | $5.279$ | $5.726$
$b\to s$ | $5.415$ | $5.762$ | $5.367$ | $5.829$
$b\to c$ | $6.473$ | $6.836$ | $6.274$ | $6.866$
Table 4: Form factors of bottom-conserving transitions. Note that Type-II (Type-II*) results represent the transition form factors calculated using $q^{2}$ dependence given in Eq. (37) (Eq. (38)) with transition pole masses as listed in Table 2. Type-I results represent the transition form factors calculated using $q^{2}$ dependence given in Eq. (40) with parent pole mass ($M_{B_{c}}$). Form Factor | Type-II (Type-II*) | Type-I
---|---|---
$F(0)$ | $F(q^{2}_{max})$ | $a$ | $b$ | $F(0)$ | $F(q^{2}_{max})$ | $a$ | $b$
$(a^{\prime}_{0})$ | $(a^{\prime}_{1})$ | $(a^{\prime}_{2})$
$B_{c}\to P$ transitions
$F_{0}^{B_{c}B}$ | $0.56$ | $0.72$ | $0.26$ | $0.68$ | $0.49$ | $0.46$ | $-2.05$ | $174.50$
($0.57$) |  ($0.79$) |  ($-32.11$) |  ($442.10$)
$F_{1}^{B_{c}B}$ | $0.56$ | $0.75$ | $0.24$ | $0.76$ | $0.49$ | $0.34$ | $-9.08$ | $318.23$
($0.57$) |  ($0.89$) |  ($-49.84$) |  ($1279.22$)
$F_{0}^{B_{c}B_{s}}$ | $0.66$ | $0.82$ | $0.26$ | $0.51$ | $0.61$ | $0.63$ | $1.33$ | $128.97$
($0.68$) |  ($0.86$) |  ($-32.72$) |  ($324.65$)
$F_{1}^{B_{c}B_{s}}$ | $0.66$ | $0.85$ | $0.37$ | $0.67$ | $0.61$ | $0.53$ | $-2.40$ | $234.47$
($0.68$) |  ($0.93$) |  ($-54.44$) |  ($1330.93$)
$B_{c}\to V$ transitions
$V^{B_{c}B^{*}}$ | $2.54$ | $3.24$ | $0.17$ | $0.97$ | $2.18$ | $1.42$ | $-14.64$ | $381.51$
($2.65$) |  ($3.98$) |  ($-253.95$) |  ($7055.58$)
$A_{0}^{B_{c}B^{*}}$ | $0.55$ | $0.75$ | $0.19$ | $0.54$ | $0.04$ | $0.01$ | $-138.36$ | $1150.18$
($0.56$) |  ($0.87$) |  ($-47.19$) |  ($1162.64$)
$A_{1}^{B_{c}B^{*}}$ | $0.53$ | $0.65$ | $0.43$ | $1.02$ | $0.41$ | $0.37$ | $-0.19$ | $158.12$
($0.54$) |  ($0.72$) |  ($-39.22$) |  ($864.12$)
$A_{2}^{B_{c}B^{*}}$ | $0.26$ | $0.28$ | $-0.40$ | $0.89$ | $0.24$ | $0.21$ | $-4.51$ | $117.58$
($0.26$) |  ($0.30$) |  ($1.28$) |  ($-630.49$)
$V^{B_{c}B_{s}^{*}}$ | $2.85$ | $3.51$ | $0.33$ | $0.93$ | $2.55$ | $2.04$ | $-7.67$ | $301.89$
($2.93$) |  ($3.96$) |  ($-269.90$) |  ($7368.86$)
$A_{0}^{B_{c}B_{s}^{*}}$ | $0.66$ | $0.85$ | $0.29$ | $0.54$ | $0.19$ | $0.07$ | $-87.95$ | $894.76$
($0.68$) |  ($0.93$) |  ($-55.12$) |  ($1349.55$)
$A_{1}^{B_{c}B_{s}^{*}}$ | $0.65$ | $0.78$ | $0.52$ | $0.86$ | $0.51$ | $0.50$ | $2.20$ | $129.87$
($0.66$) |  ($0.82$) |  ($-46.59$) |  ($1026.99$)
$A_{2}^{B_{c}B_{s}^{*}}$ | $0.42$ | $0.47$ | $-0.0001$ | $0.88$ | $0.40$ | $0.37$ | $-1.25$ | $122.92$
($0.42$) |  ($0.50$) |  ($-15.31$) |  ($-112.92$)
Table 5: Form factors of bottom-changing transitions. For the definitions of
Type-II, Type-II*, and Type-I, refer to the caption of Table 4.

Form Factor | Type-II (Type-II*) | Type-I
---|---|---
$F(0)$ | $F(q^{2}_{max})$ | $a$ | $b$ | $F(0)$ | $F(q^{2}_{max})$ | $a$ | $b$
$(a^{\prime}_{0})$ | $(a^{\prime}_{1})$ | $(a^{\prime}_{2})$
$B_{c}\to P$ transitions
$F_{0}^{B_{c}D}$ | $0.14$ | $0.52$ | $1.14$ | $1.02$ | $0.14$ | $0.15$ | $2.41$ | $4.61$
($0.14$) |  ($0.73$) |  ($-1.39$) |  ($4.07$)
$F_{1}^{B_{c}D}$ | $0.14$ | $0.52$ | $1.55$ | $1.94$ | $0.14$ | $0.07$ | $2.95$ | $10.40$
($0.14$) |  ($1.29$) |  ($-2.15$) |  ($10.90$)
$F_{0}^{B_{c}D_{s}}$ | $0.21$ | $0.70$ | $0.92$ | $0.67$ | $0.21$ | $0.33$ | $2.15$ | $3.00$
($0.21$) |  ($0.83$) |  ($-1.69$) |  ($3.13$)
$F_{1}^{B_{c}D_{s}}$ | $0.21$ | $0.89$ | $1.39$ | $1.33$ | $0.21$ | $0.17$ | $2.84$ | $7.18$
($0.21$) |  ($1.42$) |  ($-2.86$) |  ($12.83$)
$F_{0}^{B_{c}\eta_{c}}$ | $0.61$ | $0.94$ | $0.74$ | $0.40$ | $0.61$ | $0.91$ | $1.45$ | $0.95$
($0.61$) |  ($0.95$) |  ($-3.36$) |  ($1.71$)
$F_{1}^{B_{c}\eta_{c}}$ | $0.61$ | $1.15$ | $1.33$ | $0.87$ | $0.61$ | $1.01$ | $2.11$ | $2.46$
($0.61$) |  ($1.17$) |  ($-6.68$) |  ($26.17$)
$B_{c}\to V$ transitions
$V^{B_{c}D^{*}}$ | $0.17$ | $0.48$ | $1.68$ | $2.59$ | $0.17$ | $0.07$ | $2.94$ | $13.14$
($0.17$) |  ($1.41$) |  ($-3.02$) |  ($17.34$)
$A_{0}^{B_{c}D^{*}}$ | $0.18$ | $0.63$ | $1.49$ | $1.82$ | $0.12$ | $0.01$ | $-3.32$ | $56.88$
($0.18$) |  ($1.37$) |  ($-2.77$) |  ($14.31$)
$A_{1}^{B_{c}D^{*}}$ | $0.13$ | $0.41$ | $1.38$ | $1.55$ | $0.12$ | $0.11$ | $2.50$ | $5.59$
($0.13$) |  ($0.65$) |  ($-1.61$) |  ($6.29$)
$A_{2}^{B_{c}D^{*}}$ | $0.09$ | $0.21$ | $1.71$ | $2.76$ | $0.09$ | $0.05$ | $2.71$ | $9.84$
($0.09$) |  ($0.52$) |  ($-1.35$) |  ($6.83$)
$V^{B_{c}D_{s}^{*}}$ | $0.22$ | $0.63$ | $1.68$ | $2.37$ | $0.22$ | $0.11$ | $2.95$ | $11.38$
($0.22$) |  ($1.44$) |  ($-3.74$) |  ($21.15$)
$A_{0}^{B_{c}D_{s}^{*}}$ | $0.22$ | $0.78$ | $1.49$ | $1.69$ | $0.16$ | $0.02$ | $-0.23$ | $38.66$
($0.22$) |  ($1.38$) |  ($-3.41$) |  ($17.39$)
$A_{1}^{B_{c}D_{s}^{*}}$ | $0.17$ | $0.52$ | $1.39$ | $1.43$ | $0.15$ | $0.17$ | $2.46$ | $4.91$
($0.17$) |  ($0.72$) |  ($-2.05$) |  ($8.05$)
$A_{2}^{B_{c}D_{S}^{*}}$ | $0.12$ | $0.30$ | $1.76$ | $2.65$ | $0.12$ | $0.08$ | $2.73$ | $8.86$
($0.12$) |  ($0.59$) |  ($-1.81$) |  ($9.26$)
$V^{B_{c}J/\psi}$ | $0.76$ | $1.45$ | $1.65$ | $1.49$ | $0.76$ | $1.19$ | $2.35$ | $3.65$
($0.76$) |  ($1.49$) |  ($-10.56$) |  ($53.11$)
$A_{0}^{B_{c}J/\psi}$ | $0.71$ | $1.33$ | $1.37$ | $1.02$ | $0.59$ | $0.74$ | $2.52$ | $6.67$
($0.71$) |  ($1.36$) |  ($-8.72$) |  ($38.33$)
$A_{1}^{B_{c}J/\psi}$ | $0.62$ | $1.01$ | $1.19$ | $0.75$ | $0.56$ | $0.85$ | $1.74$ | $1.54$
($0.62$) |  ($1.03$) |  ($-5.59$) |  ($16.55$)
$A_{2}^{B_{c}J/\psi}$ | $0.45$ | $0.82$ | $1.73$ | $1.63$ | $0.45$ | $0.71$ | $2.18$ | $3.03$
($0.45$) |  ($0.84$) |  ($-5.92$) |  ($27.71$)

Table 6: Branching ratios, relative decay widths and physical observables of $B_{c}^{+}\to Vl^{+}\nu_{l}$ decays in Type-II CLFQM.  Quark | Decay | $\mathcal{B}$ | $\frac{\Gamma_{L}}{\Gamma}$ | $\frac{\Gamma_{T}}{\Gamma}$ | $\frac{\Gamma_{L}}{\Gamma_{T}}$ | $\langle A_{FB}\rangle$ | $\langle C_{F}^{l}\rangle$ | $\langle P_{L}^{l}\rangle$ | $\langle P_{T}^{l}\rangle$ | $\alpha^{*}$
---|---|---|---|---|---|---|---|---|---|---
Transition
$c\to d$ | $B_{c}^{+}\to B^{*0}e^{+}\nu_{e}$ | $1.85\times 10^{-3}$ | $0.54$ | $0.46$ | $1.16$ | $-0.16$ | $-0.46$ | $1.00$ | $-1.08\times 10^{-3}$ | $-0.40$
$B_{c}^{+}\to B^{*0}\mu^{+}\nu_{\mu}$ | $1.75\times 10^{-3}$ | $0.53$ | $0.47$ | $1.13$ | $-0.19$ | $-0.37$ | $0.91$ | $-0.14$ | $-0.39$
$c\to s$ | $B_{c}^{+}\to B_{s}^{*0}e^{+}\nu_{e}$ | $3.20\times 10^{-2}$ | $0.54$ | $0.46$ | $1.17$ | $-0.13$ | $-0.46$ | $1.00$ | $-1.24\times 10^{-3}$ | $-0.40$
$B_{c}^{+}\to B_{s}^{*0}\mu^{+}\nu_{\mu}$ | $2.99\times 10^{-2}$ | $0.53$ | $0.47$ | $1.14$ | $-0.16$ | $-0.36$ | $0.90$ | $-0.16$ | $-0.39$
$b\to u$ | $B_{c}^{+}\to D^{*0}e^{+}\nu_{e}$ | $7.73\times 10^{-5}$ | $0.53$ | $0.47$ | $1.15$ | $-0.21$ | $-0.45$ | $1.00$ | $-1.71\times 10^{-4}$ | $-0.39$
$B_{c}^{+}\to D^{*0}\mu^{+}\nu_{\mu}$ | $7.72\times 10^{-5}$ | $0.53$ | $0.47$ | $1.15$ | $-0.21$ | $-0.45$ | $0.99$ | $-3.28\times 10^{-2}$ | $-0.39$
$B_{c}^{+}\to D^{*0}\tau^{+}\nu_{\tau}$ | $4.21\times 10^{-5}$ | $0.52$ | $0.48$ | $1.06$ | $-0.29$ | $-0.16$ | $0.61$ | $-0.17$ | $-0.36$
$b\to c$ | $B_{c}^{+}\to J/\psi e^{+}\nu_{e}$ | $2.60\times 10^{-2}$ | $0.54$ | $0.46$ | $1.16$ | $-0.17$ | $-0.46$ | $1.00$ | $-3.09\times 10^{-4}$ | $-0.40$
$B_{c}^{+}\to J/\psi\mu^{+}\nu_{\mu}$ | $2.59\times 10^{-2}$ | $0.54$ | $0.46$ | $1.16$ | $-0.17$ | $-0.44$ | $0.99$ | $-5.62\times 10^{-2}$ | $-0.40$
$B_{c}^{+}\to J/\psi\tau^{+}\nu_{\tau}$ | $6.41\times 10^{-3}$ | $0.46$ | $0.54$ | $0.87$ | $-0.26$ | $-0.07$ | $0.50$ | $-0.15$ | $-0.27$
Table 7: Branching ratios of $B_{c}^{+}\to Vl^{+}\nu_{l}$ decays. For the definitions of Type-II, Type-II*, and Type-I, refer to the caption of Table 4. Decay | Ours | [27] | [28] | [30] | [34]
---|---|---|---|---|---
Type-II (Type-II*) | Type-I
$B_{c}^{+}\to B^{*0}e^{+}\nu_{e}$ | $1.85\times 10^{-3}$ | $8.35\times 10^{-4}$ | $7.77\times 10^{-4}$ | $1.79\times 10^{-3}$ | $1.41\times 10^{-3}$ | $6.30\times 10^{-3}$
($2.06\times 10^{-3}$)
$B_{c}^{+}\to B^{*0}\mu^{+}\nu_{\mu}$ | $1.75\times 10^{-3}$ | $7.59\times 10^{-4}$ | $7.36\times 10^{-4}$ | $1.72\times 10^{-3}$ | $1.34\times 10^{-3}$ | -
($1.96\times 10^{-3}$)
$B_{c}^{+}\to B_{s}^{*0}e^{+}\nu_{e}$ | $3.20\times 10^{-2}$ | $1.63\times 10^{-2}$ | $1.42\times 10^{-2}$ | $2.30\times 10^{-2}$ | $1.96\times 10^{-2}$ | $2.37\times 10^{-2}$
($3.41\times 10^{-2}$)
$B_{c}^{+}\to B_{s}^{*0}\mu^{+}\nu_{\mu}$ | $2.99\times 10^{-2}$ | $1.47\times 10^{-2}$ | $1.32\times 10^{-2}$ | $2.20\times 10^{-2}$ | $1.83\times 10^{-2}$ | -
($3.19\times 10^{-2}$)
$B_{c}^{+}\to D^{*0}e^{+}\nu_{e}$ | $7.73\times 10^{-5}$ | $2.80\times 10^{-5}$ | $1.26\times 10^{-4}$ | $8.40\times 10^{-5}$ | $4.50\times 10^{-5}$ | $3.80\times 10^{-5}$
($1.15\times 10^{-4}$)
$B_{c}^{+}\to D^{*0}\mu^{+}{\nu}_{\mu}$ | $7.72\times 10^{-5}$ | $2.79\times 10^{-5}$ | $1.25\times 10^{-4}$ | $8.40\times 10^{-5}$ | $4.50\times 10^{-5}$ | -
($1.14\times 10^{-4}$)
$B_{c}^{+}\to D^{*0}\tau^{+}{\nu}_{\tau}$ | $4.21\times 10^{-5}$ | $1.05\times 10^{-5}$ | $6.01\times 10^{-5}$ | $5.50\times 10^{-5}$ | $2.70\times 10^{-5}$ | $2.20\times 10^{-5}$
($6.83\times 10^{-5}$)
$B_{c}^{+}\to J/\psi e^{+}{\nu}_{e}$ | $2.60\times 10^{-2}$ | $1.97\times 10^{-2}$ | $2.13\times 10^{-2}$ | $1.31\times 10^{-2}$ | $1.49\times 10^{-2}$ | $2.07\times 10^{-2}$
($2.62\times 10^{-2}$)
$B_{c}^{+}\to J/\psi\mu^{+}{\nu}_{\mu}$ | $2.59\times 10^{-2}$ | $1.96\times 10^{-2}$ | $2.12\times 10^{-2}$ | $1.30\times 10^{-2}$ | $1.49\times 10^{-2}$ | -
($2.61\times 10^{-2}$)
$B_{c}^{+}\to J/\psi\tau^{+}{\nu}_{\tau}$ | $6.41\times 10^{-3}$ | $4.65\times 10^{-3}$ | $4.89\times 10^{-3}$ | $3.70\times 10^{-3}$ | $3.70\times 10^{-3}$ | $4.90\times 10^{-3}$
($6.49\times 10^{-3}$)
Table 8: Decay constants for P and V mesons (in MeV)111Available experimental
values are listed. The numerical values in the parentheses are from LQCD. Note
that we only listed the central values (uncertainties are ignored).. Decay
Constants
---
$f_{\pi}$ |  $130.56$ [13] | $f_{\rho}$ |  $210$ [97]
$f_{K}$ |  $155.7$ [13] | $f_{K^{*}}$ |  $204$ [97]
$f_{\eta}$ |  $(181.14)$ [96] | $f_{\phi}$ |  $(228.5)$ [98]
$f_{D}$ |  $203.8$ [13] | $f_{D^{*}}$ |  $(223.5)$ [99]
$f_{D_{s}}$ |  $250.1$ [13] | $f_{{D_{s}}^{*}}$ |  $213$ [100]
$f_{\eta_{c}}$ |  $335$ [13] | $f_{J/\psi}$ |  $416$ [13]
Table 9: Branching ratios of $B_{c}\to PV$ bottom-conserving decays. For the definitions of Type-II, Type-II*, and Type-I, refer to the caption of Table 4. Decay | Type-II | Type-II* | Type-I
---|---|---|---
$N_{c}=\infty$ | $N_{c}=3$ | $N_{c}=\infty$ | $N_{c}=3$ | $N_{c}=\infty$ | $N_{c}=3$
CKM-favored ($\Delta b=0,\leavevmode\nobreak\ \Delta C=-1,\leavevmode\nobreak\
\Delta S=-1$)
$B_{c}^{+}\to\pi^{+}B_{s}^{*0}$ | $4.86\times 10^{-2}$ | $3.63\times 10^{-2}$ | $5.18\times 10^{-2}$ | $3.87\times 10^{-2}$ | $2.14\times 10^{-3}$ | $1.60\times 10^{-3}$
$B_{c}^{+}\to\overline{K}^{0}B^{*+}$ | $8.08\times 10^{-3}$ | $2.61\times 10^{-4}$ | $1.00\times 10^{-2}$ | $3.12\times 10^{-4}$ | $3.52\times 10^{-6}$ | $1.09\times 10^{-7}$
$B_{c}^{+}\to B^{+}\overline{K}^{*0}$ | $3.18\times 10^{-3}$ | $9.90\times 10^{-5}$ | $4.34\times 10^{-3}$ | $1.35\times 10^{-4}$ | $7.68\times 10^{-4}$ | $2.39\times 10^{-5}$
$B_{c}^{+}\to B_{s}^{0}\rho^{+}$ | $3.65\times 10^{-2}$ | $2.73\times 10^{-2}$ | $4.34\times 10^{-2}$ | $3.25\times 10^{-2}$ | $1.50\times 10^{-2}$ | $1.12\times 10^{-2}$
CKM-suppressed ($\Delta b=0,\leavevmode\nobreak\ \Delta
C=-1,\leavevmode\nobreak\ \Delta S=0$)
$B_{c}^{+}\to K^{+}B_{s}^{*0}$ | $2.64\times 10^{-3}$ | $1.98\times 10^{-3}$ | $3.01\times 10^{-3}$ | $2.25\times 10^{-3}$ | $3.20\times 10^{-5}$ | $2.40\times 10^{-5}$
$B_{c}^{+}\to\pi^{+}B^{*0}$ | $2.26\times 10^{-3}$ | $1.69\times 10^{-3}$ | $2.48\times 10^{-3}$ | $1.86\times 10^{-3}$ | $4.62\times 10^{-6}$ | $3.46\times 10^{-6}$
$B_{c}^{+}\to\pi^{0}B^{*+}$ | $1.85\times 10^{-4}$ | $5.77\times 10^{-6}$ | $2.03\times 10^{-4}$ | $6.32\times 10^{-6}$ | $3.88\times 10^{-7}$ | $1.21\times 10^{-8}$
$B_{c}^{+}\to\eta B^{*+}$ | $3.98\times 10^{-4}$ | $1.27\times 10^{-5}$ | $4.94\times 10^{-4}$ | $1.54\times 10^{-5}$ | $1.43\times 10^{-7}$ | $4.45\times 10^{-9}$
$B_{c}^{+}\to B^{+}\rho^{0}$ | $2.36\times 10^{-4}$ | $7.36\times 10^{-6}$ | $3.09\times 10^{-4}$ | $9.61\times 10^{-6}$ | $6.86\times 10^{-5}$ | $2.13\times 10^{-6}$
$B_{c}^{+}\to B^{+}\omega$ | $2.04\times 10^{-4}$ | $6.35\times 10^{-6}$ | $2.68\times 10^{-4}$ | $8.33\times 10^{-6}$ | $5.80\times 10^{-5}$ | $1.81\times 10^{-6}$
$B_{c}^{+}\to B^{0}\rho^{+}$ | $2.87\times 10^{-3}$ | $2.15\times 10^{-3}$ | $3.75\times 10^{-3}$ | $2.81\times 10^{-3}$ | $8.35\times 10^{-4}$ | $6.25\times 10^{-4}$
$B_{c}^{+}\to B_{s}^{0}K^{*+}$ | $5.89\times 10^{-5}$ | $4.40\times 10^{-5}$ | $7.20\times 10^{-5}$ | $5.39\times 10^{-5}$ | $2.09\times 10^{-5}$ | $1.56\times 10^{-5}$
CKM-doubly-suppressed ($\Delta b=0,\leavevmode\nobreak\ \Delta
C=-1,\leavevmode\nobreak\ \Delta S=1$)
$B_{c}^{+}\to K^{+}B^{*0}$ | $1.40\times 10^{-4}$ | $1.05\times 10^{-4}$ | $1.67\times 10^{-4}$ | $1.25\times 10^{-4}$ | $5.97\times 10^{-8}$ | $4.47\times 10^{-8}$
$B_{c}^{+}\to K^{0}B^{*+}$ | $2.28\times 10^{-5}$ | $7.11\times 10^{-7}$ | $2.73\times 10^{-5}$ | $8.51\times 10^{-7}$ | $9.58\times 10^{-9}$ | $2.98\times 10^{-10}$
$B_{c}^{+}\to B^{+}K^{*0}$ | $8.67\times 10^{-6}$ | $2.70\times 10^{-7}$ | $1.18\times 10^{-5}$ | $3.68\times 10^{-7}$ | $2.09\times 10^{-6}$ | $6.52\times 10^{-8}$
$B_{c}^{+}\to B^{0}K^{*+}$ | $5.26\times 10^{-5}$ | $3.93\times 10^{-5}$ | $7.17\times 10^{-5}$ | $5.37\times 10^{-5}$ | $1.27\times 10^{-5}$ | $9.52\times 10^{-6}$
Table 10: Branching ratios of CKM-favored $B_{c}\to PV$ bottom-changing decays. For the definitions of Type-II, Type-II*, and Type-I, refer to the caption of Table 4. Decay | Type-II | Type-II* | Type-I
---|---|---|---
$N_{c}=\infty$ | $N_{c}=3$ | $N_{c}=\infty$ | $N_{c}=3$ | $N_{c}=\infty$ | $N_{c}=3$
$\Delta b=-1,\leavevmode\nobreak\ \Delta C=-1,\leavevmode\nobreak\ \Delta S=0$
$B_{c}^{+}\to\pi^{+}J/\psi$ | $1.91\times 10^{-3}$ | $1.63\times 10^{-3}$ | $1.91\times 10^{-3}$ | $1.63\times 10^{-3}$ | $1.33\times 10^{-3}$ | $1.13\times 10^{-3}$
$B_{c}^{+}\to\overline{D}^{0}D^{*+}$ | $2.21\times 10^{-5}$ | $4.62\times 10^{-6}$ | $2.48\times 10^{-5}$ | $4.71\times 10^{-6}$ | $4.79\times 10^{-6}$ | $9.10\times 10^{-7}$
$B_{c}^{+}\to D^{+}\overline{D}^{*0}$ | $1.82\times 10^{-5}$ | $3.46\times 10^{-6}$ | $1.86\times 10^{-5}$ | $3.53\times 10^{-6}$ | $1.66\times 10^{-5}$ | $3.15\times 10^{-6}$
$B_{c}^{+}\to\eta_{c}\rho^{+}$ | $4.31\times 10^{-3}$ | $3.66\times 10^{-3}$ | $4.31\times 10^{-3}$ | $3.67\times 10^{-3}$ | $4.26\times 10^{-3}$ | $3.63\times 10^{-3}$
$\Delta b=-1,\leavevmode\nobreak\ \Delta C=0,\leavevmode\nobreak\ \Delta S=1$
$B_{c}^{+}\to\pi^{0}D_{s}^{*+}$ | $3.88\times 10^{-9}$ | $7.37\times 10^{-10}$ | $3.88\times 10^{-9}$ | $7.38\times 10^{-10}$ | $1.92\times 10^{-9}$ | $3.65\times 10^{-10}$
$B_{c}^{+}\to\eta D_{s}^{*+}$ | $2.52\times 10^{-9}$ | $4.77\times 10^{-10}$ | $2.52\times 10^{-9}$ | $4.79\times 10^{-10}$ | $1.14\times 10^{-9}$ | $2.16\times 10^{-10}$
$B_{c}^{+}\to K^{+}{D}^{*0}$ | $1.40\times 10^{-7}$ | $1.19\times 10^{-7}$ | $1.40\times 10^{-7}$ | $1.19\times 10^{-7}$ | $5.23\times 10^{-8}$ | $4.45\times 10^{-8}$
$B_{c}^{+}\to\eta\prime D_{s}^{*+}$ | $1.52\times 10^{-9}$ | $2.87\times 10^{-10}$ | $1.52\times 10^{-9}$ | $2.89\times 10^{-10}$ | $6.16\times 10^{-10}$ | $1.17\times 10^{-10}$
$B_{c}^{+}\to{D}^{0}K^{*+}$ | $1.86\times 10^{-7}$ | $1.58\times 10^{-7}$ | $1.87\times 10^{-7}$ | $1.59\times 10^{-7}$ | $1.80\times 10^{-7}$ | $1.53\times 10^{-7}$
$B_{c}^{+}\to D_{s}^{+}\rho^{0}$ | $1.12\times 10^{-8}$ | $2.13\times 10^{-9}$ | $1.13\times 10^{-8}$ | $2.14\times 10^{-9}$ | $1.06\times 10^{-8}$ | $2.02\times 10^{-9}$
$B_{c}^{+}\to D_{s}^{+}\omega$ | $7.00\times 10^{-10}$ | $1.33\times 10^{-10}$ | $7.03\times 10^{-10}$ | $1.34\times 10^{-10}$ | $6.63\times 10^{-10}$ | $1.26\times 10^{-10}$
$B_{c}^{+}\to D_{s}^{+}\phi$ | $1.10\times 10^{-8}$ | $2.08\times 10^{-9}$ | $1.10\times 10^{-8}$ | $2.09\times 10^{-9}$ | $1.03\times 10^{-8}$ | $1.96\times 10^{-9}$
$B_{c}^{+}\to D_{s}^{+}J/\psi$ | $3.00\times 10^{-3}$ | $3.83\times 10^{-3}$ | $3.00\times 10^{-3}$ | $3.84\times 10^{-3}$ | $1.99\times 10^{-3}$ | $2.67\times 10^{-3}$
$B_{c}^{+}\to\eta_{c}D_{s}^{*+}$ | $1.88\times 10^{-3}$ | $2.41\times 10^{-3}$ | $1.83\times 10^{-3}$ | $2.42\times 10^{-3}$ | $2.14\times 10^{-3}$ | $2.21\times 10^{-3}$
Table 11: Branching ratios of CKM-suppressed $B_{c}\to PV$ bottom-changing decays. For the definitions of Type-II, Type-II*, and Type-I, refer to the caption of Table 4. Decay | Type-II | Type-II* | Type-I
---|---|---|---
$N_{c}=\infty$ | $N_{c}=3$ | $N_{c}=\infty$ | $N_{c}=3$ | $N_{c}=\infty$ | $N_{c}=3$
$\Delta b=-1,\leavevmode\nobreak\ \Delta C=-1,\leavevmode\nobreak\ \Delta S=1$
$B_{c}^{+}\to K^{+}J/\psi$ | $1.45\times 10^{-4}$ | $1.23\times 10^{-4}$ | $1.45\times 10^{-4}$ | $1.23\times 10^{-4}$ | $1.01\times 10^{-4}$ | $8.58\times 10^{-5}$
$B_{c}^{+}\to\overline{D}^{0}D_{s}^{*+}$ | $1.77\times 10^{-6}$ | $3.66\times 10^{-7}$ | $1.96\times 10^{-6}$ | $3.72\times 10^{-7}$ | $5.90\times 10^{-7}$ | $1.12\times 10^{-7}$
$B_{c}^{+}\to D_{s}^{+}\overline{D}^{*0}$ | $2.09\times 10^{-6}$ | $3.97\times 10^{-7}$ | $2.11\times 10^{-6}$ | $4.01\times 10^{-7}$ | $1.90\times 10^{-6}$ | $3.61\times 10^{-7}$
$B_{c}^{+}\to\eta_{c}K^{*+}$ | $2.22\times 10^{-4}$ | $1.89\times 10^{-4}$ | $2.22\times 10^{-4}$ | $1.89\times 10^{-4}$ | $2.20\times 10^{-4}$ | $1.87\times 10^{-4}$
$\Delta b=-1,\leavevmode\nobreak\ \Delta C=0,\leavevmode\nobreak\ \Delta S=0$
$B_{c}^{+}\to\pi^{0}D^{*+}$ | $4.78\times 10^{-8}$ | $9.08\times 10^{-9}$ | $4.78\times 10^{-8}$ | $9.09\times 10^{-9}$ | $2.06\times 10^{-8}$ | $3.91\times 10^{-9}$
$B_{c}^{+}\to\pi^{+}{D}^{*0}$ | $1.78\times 10^{-6}$ | $1.51\times 10^{-6}$ | $1.78\times 10^{-6}$ | $1.51\times 10^{-6}$ | $7.64\times 10^{-7}$ | $6.50\times 10^{-7}$
$B_{c}^{+}\to\eta D^{*+}$ | $3.10\times 10^{-8}$ | $5.90\times 10^{-9}$ | $3.12\times 10^{-8}$ | $5.92\times 10^{-9}$ | $1.14\times 10^{-8}$ | $2.16\times 10^{-9}$
$B_{c}^{+}\to\eta\prime D^{*+}$ | $1.88\times 10^{-8}$ | $3.57\times 10^{-9}$ | $1.89\times 10^{-8}$ | $3.60\times 10^{-9}$ | $5.76\times 10^{-9}$ | $1.10\times 10^{-9}$
$B_{c}^{+}\to D^{+}\rho^{0}$ | $9.48\times 10^{-8}$ | $1.80\times 10^{-8}$ | $9.53\times 10^{-8}$ | $1.81\times 10^{-8}$ | $9.22\times 10^{-8}$ | $1.75\times 10^{-8}$
$B_{c}^{+}\to D^{+}\omega$ | $5.92\times 10^{-9}$ | $1.12\times 10^{-9}$ | $5.95\times 10^{-9}$ | $1.13\times 10^{-9}$ | $5.75\times 10^{-9}$ | $1.09\times 10^{-9}$
$B_{c}^{+}\to D^{+}\phi$ | $9.31\times 10^{-8}$ | $1.77\times 10^{-8}$ | $9.38\times 10^{-8}$ | $1.78\times 10^{-8}$ | $8.95\times 10^{-8}$ | $1.70\times 10^{-8}$
$B_{c}^{+}\to D^{+}J/\psi$ | $1.17\times 10^{-4}$ | $1.39\times 10^{-4}$ | $1.17\times 10^{-4}$ | $1.40\times 10^{-4}$ | $7.89\times 10^{-5}$ | $9.68\times 10^{-5}$
$B_{c}^{+}\to{D}^{0}\rho^{+}$ | $3.52\times 10^{-6}$ | $3.00\times 10^{-6}$ | $3.54\times 10^{-6}$ | $3.02\times 10^{-6}$ | $3.43\times 10^{-6}$ | $2.92\times 10^{-6}$
$B_{c}^{+}\to\eta_{c}D^{*+}$ | $1.02\times 10^{-4}$ | $1.22\times 10^{-4}$ | $9.93\times 10^{-5}$ | $1.22\times 10^{-4}$ | $1.16\times 10^{-4}$ | $1.11\times 10^{-4}$
Table 12: Branching ratios of CKM-doubly-suppressed $B_{c}\to PV$ bottom-changing decays. For the definitions of Type-II, Type-II*, and Type-I, refer to the caption of Table 4. Decay | Type-II | Type-II* | Type-I
---|---|---|---
$N_{c}=\infty$ | $N_{c}=3$ | $N_{c}=\infty$ | $N_{c}=3$ | $N_{c}=\infty$ | $N_{c}=3$
$\Delta b=-1,\leavevmode\nobreak\ \Delta C=1,\leavevmode\nobreak\ \Delta S=1$
$B_{c}^{+}\to{D}^{0}D_{s}^{*+}$ | $1.45\times 10^{-6}$ | $3.54\times 10^{-6}$ | $1.50\times 10^{-6}$ | $3.62\times 10^{-6}$ | $1.88\times 10^{-6}$ | $2.86\times 10^{-6}$
$B_{c}^{+}\to D_{s}^{+}{D}^{*0}$ | $3.31\times 10^{-6}$ | $6.14\times 10^{-6}$ | $3.39\times 10^{-6}$ | $6.26\times 10^{-6}$ | $2.29\times 10^{-7}$ | $1.45\times 10^{-6}$
$\Delta b=-1,\leavevmode\nobreak\ \Delta C=1,\leavevmode\nobreak\ \Delta S=0$
$B_{c}^{+}\to D^{+}{D}^{*0}$ | $1.30\times 10^{-7}$ | $2.08\times 10^{-7}$ | $1.33\times 10^{-7}$ | $2.12\times 10^{-7}$ | $1.29\times 10^{-8}$ | $4.94\times 10^{-8}$
$B_{c}^{+}\to{D}^{0}D^{*+}$ | $8.18\times 10^{-8}$ | $1.65\times 10^{-7}$ | $8.35\times 10^{-8}$ | $1.68\times 10^{-7}$ | $1.06\times 10^{-7}$ | $1.33\times 10^{-7}$
Table 13: Branching ratios of $B_{c}\to PV$ bottom-conserving decays of other
models.

Decay |  RIQM [37] |  RCQM[34] |  RQM[32] |  QCDF [41]
---|---|---|---|---
$B_{c}^{+}\to\pi^{+}B_{s}^{*0}$ | $8.61\times 10^{-2}$ | $2.1\times 10^{-2}$ | $1.6\times 10^{-2}$ | $-$
$B_{c}^{+}\to\overline{K}^{0}B^{*+}$ | $2.26\times 10^{-2}$ | $8.8\times 10^{-4}$ | $1.1\times 10^{-3}$ | $-$
$B_{c}^{+}\to B^{+}\overline{K}^{*0}$ | $1.83\times 10^{-2}$ | $1.1\times 10^{-3}$ | $9.0\times 10^{-4}$ | $3.72\times 10^{-3}$
$B_{c}^{+}\to B_{s}^{0}\rho^{+}$ | $9.97\times 10^{-2}$ | $2.3\times 10^{-2}$ | $1.4\times 10^{-2}$ | $4.44\times 10^{-2}$
$B_{c}^{+}\to K^{+}B_{s}^{*0}$ | $4.99\times 10^{-3}$ | $1.3\times 10^{-3}$ | $1.1\times 10^{-3}$ | $-$
$B_{c}^{+}\to\pi^{+}B^{*0}$ | $5.55\times 10^{-3}$ | $5.7\times 10^{-4}$ | $2.6\times 10^{-4}$ | $-$
$B_{c}^{+}\to\pi^{0}B^{*+}$ | $4.70\times 10^{-4}$ | $2.0\times 10^{-5}$ | $1.0\times 10^{-5}$ | $-$
$B_{c}^{+}\to B^{+}\rho^{0}$ | $1.06\times 10^{-3}$ | $7.1\times 10^{-5}$ | $5.0\times 10^{-5}$ | $2.86\times 10^{-4}$
$B_{c}^{+}\to B^{+}\omega$ | $-$ | $-$ | $-$ | $2.05\times 10^{-4}$
$B_{c}^{+}\to B^{0}\rho^{+}$ | $1.30\times 10^{-2}$ | $2.0\times 10^{-3}$ | $1.3\times 10^{-3}$ | $5.32\times 10^{-3}$
$B_{c}^{+}\to B_{s}^{0}K^{*+}$ | $4.00\times 10^{-4}$ | $1.1\times 10^{-4}$ | $3.0\times 10^{-5}$ | $1.25\times 10^{-4}$
$B_{c}^{+}\to K^{+}B^{*0}$ | $3.90\times 10^{-4}$ | $3.6\times 10^{-5}$ | $4.0\times 10^{-5}$ | $-$
$B_{c}^{+}\to B^{+}K^{*0}$ | $-$ | $-$ | $-$ | $1.07\times 10^{-5}$
$B_{c}^{+}\to B^{0}K^{*+}$ | $3.20\times 10^{-5}$ | $4.8\times 10^{-5}$ | $4.0\times 10^{-5}$ | $1.06\times 10^{-4}$

Table 14: Branching ratios of $B_{c}^{+}\to D_{(s)}^{(*)+}\overline{D}^{(*)0}$ and $D_{(s)}^{(*)+}{D}^{(*)0}$ bottom-changing decays of other models.  Decay |  RIQM [37] |  pQCD [39] |  RCQM[34] |  PDG [13]
---|---|---|---|---
$B_{c}^{+}\to{D}^{0}D_{s}^{*+}$ | $4.18\times 10^{-7}$ | $2.5\times 10^{-6}$ | $9.3\times 10^{-6}$ | $<9.0\times 10^{-4}$
$B_{c}^{+}\to D_{s}^{+}{D}^{*0}$ | $2.25\times 10^{-7}$ | $1.9\times 10^{-6}$ | $1.3\times 10^{-6}$ | $<6.6\times 10^{-4}$
$B_{c}^{+}\to D^{+}{D}^{*0}$ | $8.00\times 10^{-9}$ | $7.0\times 10^{-8}$ | $5.2\times 10^{-8}$ | $<3.7\times 10^{-4}$
$B_{c}^{+}\to{D}^{0}D^{*+}$ | $1.80\times 10^{-8}$ | $9.0\times 10^{-8}$ | $4.4\times 10^{-7}$ | $<2.0\times 10^{-4}$
$B_{c}^{+}\to\overline{D}^{0}D_{s}^{*+}$ | $3.15\times 10^{-7}$ | $7.0\times 10^{-8}$ | $6.5\times 10^{-7}$ | $<5.3\times 10^{-4}$
$B_{c}^{+}\to D_{s}^{+}\overline{D}^{*0}$ | $4.91\times 10^{-7}$ | $2.6\times 10^{-7}$ | $2.4\times 10^{-6}$ | $<4.6\times 10^{-4}$
$B_{c}^{+}\to\overline{D}^{0}D^{*+}$ | $1.60\times 10^{-6}$ | $1.2\times 10^{-6}$ | $8.8\times 10^{-6}$ | $<3.8\times 10^{-4}$
$B_{c}^{+}\to D^{+}\overline{D}^{*0}$ | $2.61\times 10^{-5}$ | $3.4\times 10^{-6}$ | $3.8\times 10^{-5}$ | $<6.5\times 10^{-4}$
Table 15: Branching ratios of $B_{c}\to PV$ bottom-changing decays of other models.  Decay |  CLFQM (Type-I)[29] |  RIQM [38] |  pQCD [40] |  RCQM[34]
---|---|---|---|---
$B_{c}^{+}\to D_{s}^{+}J/\psi$ | $6.09\times 10^{-3}$ | $1.15\times 10^{-3}$ | $8.05\times 10^{-3}$ | $3.4\times 10^{-3}$
$B_{c}^{+}\to D^{+}J/\psi$ | $2.00\times 10^{-4}$ | $3.69\times 10^{-5}$ | $2.80\times 10^{-4}$ | $1.5\times 10^{-4}$
$B_{c}^{+}\to K^{+}J/\psi$ | $1.60\times 10^{-4}$ | $3.00\times 10^{-5}$ | $1.90\times 10^{-4}$ | $1.3\times 10^{-4}$
$B_{c}^{+}\to\pi^{+}J/\psi$ | $1.97\times 10^{-3}$ | $3.80\times 10^{-4}$ | $2.33\times 10^{-3}$ | $1.7\times 10^{-3}$
$B_{c}^{+}\to\eta_{c}D_{s}^{*+}$ | $6.97\times 10^{-3}$ | $2.16\times 10^{-3}$ | $1.65\times 10^{-2}$ | $3.7\times 10^{-3}$
$B_{c}^{+}\to\eta_{c}D^{*+}$ | $3.10\times 10^{-4}$ | $7.60\times 10^{-5}$ | $5.80\times 10^{-4}$ | $1.9\times 10^{-4}$
$B_{c}^{+}\to\eta_{c}K^{*+}$ | $3.40\times 10^{-4}$ | $6.30\times 10^{-5}$ | $5.70\times 10^{-4}$ | $2.5\times 10^{-4}$
$B_{c}^{+}\to\eta_{c}\rho^{+}$ | $6.01\times 10^{-3}$ | $1.20\times 10^{-3}$ | $9.83\times 10^{-3}$ | $4.5\times 10^{-3}$

((a)) $B_{c}\to B$ transition

((b)) $B_{c}\to B_{s}$ transition

Figure 2: $q^{2}$ dependence of bottom-conserving $B_{c}\to P$ form factors in
Type-II (Type-II*) CLFQM using Eq. (37) (Eq. (38)).

((a)) $B_{c}\to B^{*}$ transition

((b)) $B_{c}\to B_{s}^{*}$ transition

Figure 3: $q^{2}$ dependence of bottom-conserving $B_{c}\to V$ form factors in
Type-II (Type-II*) CLFQM using Eq. (37) (Eq. (38)).

((a)) $B_{c}\to D$ transition

((b)) $B_{c}\to D_{s}$ transition

((c)) $B_{c}\to\eta_{c}$ transition

Figure 4: $q^{2}$ dependence of bottom-changing $B_{c}\to P$ form factors in
Type-II (Type-II*) CLFQM using Eq. (37) (Eq. (38)).

((a)) $B_{c}\to D^{*}$ transition

((b)) $B_{c}\to D_{s}^{*}$ transition

((c)) $B_{c}\to J/\psi$ transition

Figure 5: $q^{2}$ dependence of bottom-changing $B_{c}\to V$ form factors in
Type-II (Type-II*) CLFQM using Eq. (37) (Eq. (38)).

((a)) $B_{c}$ and $B^{*}$ (Overlap area $=0.668$)

((b)) $B_{c}$ and $B_{s}^{*}$ (Overlap area $=0.775$)

Figure 6: Overlap plots of $B_{c}$ and $B^{*},\leavevmode\nobreak\ B_{s}^{*}$
light-front wave function using Eq. (5), in Type-II CLFQM. Note that overlap
plots of $B_{c}$ and $B,\leavevmode\nobreak\ B_{s}$ wave function are similar
to $B^{*},\leavevmode\nobreak\ B_{s}^{*}$, with roughly $10\%$ increase in
overlap area.

((a)) $B_{c}$ and $D^{*}$ (Overlap area $=0.292$)

((b)) $B_{c}$ and $D_{s}^{*}$ (Overlap area $=0.339$)

((c)) $B_{c}$ and $J/\psi$ (Overlap area $=0.833$)

Figure 7: Overlap plots of $B_{c}$ and $D^{*},\leavevmode\nobreak\
D_{s}^{*},\leavevmode\nobreak\ J/\psi$ light-front wave function using Eq.
(5), in Type-II CLFQM. Note that overlap plots of $B_{c}$ and
$D,\leavevmode\nobreak\ D_{s},\leavevmode\nobreak\ \eta_{c}$ wave function
will be similar; however, we notice approximately $15\%$ and $28\%$ change
between $B_{c}$ and $D,\leavevmode\nobreak\ \eta_{c}$, as well as between
$B_{c}$ and $D_{s}$, respectively.

((a)) $F_{0}(x_{1})$ versus $x_{1}$

((b)) $F_{1}(x_{1})$ versus $x_{1}$

Figure 8: Dependence of form factor, $F(x_{1})$ on $x_{1}$ for $B_{c}\to P$
transition at $q^{2}=0\leavevmode\nobreak\ \text{GeV}^{2}$, in Type-II CLFQM
using Eq. (27).

((a)) $V(x_{1})$ versus $x_{1}$

((b)) $A_{0}(x_{1})$ versus $x_{1}$

((c)) $A_{1}(x_{1})$ versus $x_{1}$

((d)) $A_{2}(x_{1})$ versus $x_{1}$

Figure 9: Dependence of form factor, $F(x_{1})$ on $x_{1}$ for $B_{c}\to V$
transition at $q^{2}=0\leavevmode\nobreak\ \text{GeV}^{2}$, in Type-II CLFQM
using Eq. (27).

((a)) $B_{c}^{+}\to B^{*}l^{+}\nu_{l}$

((b)) $B_{c}^{+}\to B_{s}^{*}l^{+}\nu_{l}$

((c)) $B_{c}^{+}\to D^{*}l^{+}\nu_{l}$

((d)) $B_{c}^{+}\to J/\psi l^{+}\nu_{l}$

Figure 10: $q^{2}$ variation of differential decay rates of $B_{c}^{+}\to
Vl^{+}\nu_{l}$ decays in Type-II CLFQM using Eq. (50).

((a)) $B_{c}^{+}\to B^{*}l^{+}\nu_{l}$

((b)) $B_{c}^{+}\to B_{s}^{*}l^{+}\nu_{l}$

((c)) $B_{c}^{+}\to D^{*}l^{+}\nu_{l}$

((d)) $B_{c}^{+}\to J/\psi l^{+}\nu_{l}$

Figure 11: $q^{2}$ variation of forward-backward asymmetries of $B_{c}^{+}\to
Vl^{+}\nu_{l}$ decays in Type-II CLFQM using Eq. (50).