Nonleptonic charmless decays of $B_c\to TP, TV$ in the perturbative QCD approach

TL;DR

This paper investigates charmless nonleptonic $B_c$ decays involving tensor mesons using perturbative QCD, predicting decay rates and polarization fractions, and highlighting modes detectable at the LHC.

Contribution

First calculation of $B_c o TP, TV$ decays in pQCD, providing predictions for branching ratios and polarization fractions for the first time.

Findings

Several decay modes have large branching ratios around 10^{-6}.

Most $B_c o TV$ decays are dominated by longitudinal polarization.

Predicted polarization fractions range from 78% to 98%.

Abstract

Two-body charmless hadronic $B_c$ decays involving a light $1^3\!P_2$-tensor($T$) meson are investigated for the first time within the framework of perturbative QCD(pQCD) at leading order, in which the other meson is the lightest pseudoscalar($P$) or vector($V$) state. The concerned processes can only occur through the pure weak annihilation topology in the standard model. We predict the CP-averaged branching ratios and polarization fractions of those considered decays in Cabibbo-Kobayashi-Maskawa(CKM) favored and suppressed modes. Phenomenologically, several modes$-$such as the $B_c \to K_2^*(1430) K$ and the CKM-favored $B_c \to TV-$have large decay rates of $10^{-6}$, which are expected to be detected at Large Hadron Collider experiments in the near future. Moreover, all of the $B_c \to TV$ modes are governed by the longitudinal amplitudes in the pQCD calculations and the corresponding fractions vary around $78\%$-$98\%$. A confirmation of these results could prove the reliability of the pQCD approach used here and further shed some light on the annihilation decay mechanism.

Tables

Table 1. Table 1: Decay constants of the light tensor mesons (in GeV)  Cheng:2010hn

$f_{a_2}$	$f_{a_2}^T$	$f_{K_2^{\ast }}$	$f_{K_2^{\ast }}^T$	$f_{f_2}$	$f_{f_2}^T$	$f_{f_2^{\prime }}$	$f_{f_2^{\prime }}^T$
$0.107\pm 0.006$	$0.105\pm 0.021$	$0.118\pm 0.005$	$0.077\pm 0.014$	$0.102\pm 0.006$	$0.117\pm 0.025$	$0.126\pm 0.004$	$0.065\pm 0.012$

Table 2. Table 2: CP -averaged branching ratios of charmless decays B c → T ​ P → subscript 𝐵 𝑐 𝑇 𝑃 B_{c}\to TP in the pQCD approach.

Decay Modes($\Delta S=0$)	Branching ratios(${10}^{-7}$)	Decay modes($\Delta S=1$)	Branching ratios(${10}^{-8}$)
$B_c\to a_2^{+}{\pi }^0$	${5.13}_{-1.40-0.56-0.68-0.01}^{+1.61+0.59+0.94+0.00}$	$B_c\to K_2^{\ast 0}{\pi }^{+}$	${3.73}_{-1.18-0.31-0.44-0.02}^{+1.66+0.31+0.37+0.04}$
$B_c\to a_2^0{\pi }^{+}$	${5.13}_{-1.40-0.56-0.68-0.01}^{+1.61+0.59+0.94+0.00}$	$B_c\to K_2^{\ast +}{\pi }^0$	${1.87}_{-0.59-0.16-0.23-0.02}^{+0.82+0.15+0.18+0.01}$
$B_c\to a_2^{+}\eta$	${3.92}_{-0.31-0.70-0.28-0.34-0.11}^{+0.39+0.73+0.18+0.35+0.12}$	$B_c\to K_2^{\ast +}\eta$	${5.37}_{-0.93-0.55-0.34-0.05-0.15}^{+1.77+0.54+0.46+0.04+0.15}$
$B_c\to a_2^{+}{\eta }^{\prime }$	${2.56}_{-0.20-0.46-0.19-0.23-0.11}^{+0.25+0.48+0.11+0.22+0.11}$	$B_c\to K_2^{\ast +}{\eta }^{\prime }$	${6.51}_{-0.34-1.14-0.35-0.37-0.16}^{+0.90+1.21+0.42+0.39+0.14}$
$B_c\to f_2{\pi }^{+}$	${7.38}_{-0.39-1.53-0.31-0.70}^{+0.63+1.59+0.16+0.72}$	$B_c\to a_2^{+}{K}^0$	${9.04}_{-0.90-1.45-1.34-0.44}^{+1.52+1.62+1.11+0.52}$
$B_c\to K_2^{\ast +}{\overline{K}}^0$	${11.14}_{-0.15-1.45-1.07-0.33}^{+0.80+1.64+2.42+0.49}$	$B_c\to a_2^0{K}^{+}$	${4.52}_{-0.45-0.72-0.67-0.21}^{+0.76+0.82+0.57+0.26}$
$B_c\to {\overline{K}}_2^{\ast 0}{K}^{+}$	${10.46}_{-2.96-0.87-1.50-0.22}^{+4.82+0.93+0.84+0.20}$	$B_c\to f_2{K}^{+}$	${4.56}_{-0.42-0.84-0.65-0.26}^{+0.72+0.93+0.55+0.30}$
		$B_c\to f_2^{\prime }{K}^{+}$	${6.33}_{-1.82-0.39-0.86-0.15}^{+2.79+0.40+0.46+0.11}$

Table 3. Table 3: CP -averaged branching ratios and polarization fractions of CKM-favored B c → T ​ V → subscript 𝐵 𝑐 𝑇 𝑉 B_{c}\to TV modes in the pQCD approach.

Decay Modes($\Delta S=0$)	Branching ratios(${10}^{-6}$)	Polarization fractions $f_L$(%)	Polarization fractions $f_T$(%)
$B_c\to a_2^{+}{\rho }^0$	${1.29}_{-0.29-0.16-0.02-0.07}^{+0.33+0.18+0.04+0.08}$	${78.2}_{-6.3-1.0-0.2-1.2}^{+4.4+1.0+0.3+1.3}$	${21.8}_{-4.4-1.0-0.3-1.3}^{+6.3+1.0+0.2+1.2}$
$B_c\to a_2^0{\rho }^{+}$	${1.29}_{-0.29-0.16-0.02-0.07}^{+0.33+0.18+0.04+0.08}$	${78.2}_{-6.3-1.0-0.2-1.2}^{+4.4+1.0+0.3+1.3}$	${21.8}_{-4.4-1.0-0.3-1.3}^{+6.3+1.0+0.2+1.2}$
$B_c\to a_2^{+}\omega$	${0.87}_{-0.09-0.13-0.04-0.02}^{+0.11+0.12+0.03+0.02}$	${97.4}_{-0.3-0.3-0.1-0.1}^{+0.3+0.3+0.1+0.1}$	${2.6}_{-0.3-0.3-0.1-0.1}^{+0.3+0.3+0.1+0.1}$
$B_c\to f_2{\rho }^{+}$	${0.97}_{-0.11-0.16-0.04-0.02}^{+0.11+0.17+0.04+0.00}$	${98.0}_{-0.2-0.4-0.1-0.1}^{+0.2+0.3+0.1+0.0}$	${2.0}_{-0.2-0.3-0.1-0.0}^{+0.2+0.4+0.1+0.1}$
$B_c\to K_2^{\ast +}{\overline{K}}^{\ast 0}$	${1.75}_{-0.08-0.15-0.07-0.07}^{+0.02+0.15+0.08+0.10}$	${82.7}_{-0.7-0.7-0.2-0.6}^{+0.0+0.8+0.4+1.0}$	${17.3}_{-0.0-0.8-0.4-1.0}^{+0.7+0.7+0.2+0.6}$
$B_c\to {\overline{K}}_2^{\ast 0}{K}^{\ast +}$	${1.54}_{-0.45-0.13-0.07-0.12}^{+0.57+0.14+0.08+0.12}$	${81.2}_{-7.9-0.4-0.2-1.7}^{+5.2+0.1+0.1+1.3}$	${18.8}_{-5.2-0.1-0.1-1.3}^{+7.9+0.4+0.2+1.7}$

Table 4. Table 4: Same as Table  3 but for CKM-suppressed B c → T ​ V → subscript 𝐵 𝑐 𝑇 𝑉 B_{c}\to TV modes.

Decay Modes($\Delta S=1$)	Branching ratios(${10}^{-8}$)	Polarization fractions $f_L$(%)	Polarization fractions $f_T$(%)
$B_c\to K_2^{\ast 0}{\rho }^{+}$	${7.17}_{-2.11-0.62-0.13-0.47}^{+2.67+0.65+0.14+0.45}$	${84.8}_{-6.5-0.2-0.0-1.1}^{+4.3+0.2+0.1+0.9}$	${15.2}_{-4.3-0.2-0.1-0.9}^{+6.5+0.2+0.0+1.1}$
$B_c\to K_2^{\ast +}{\rho }^0$	${3.59}_{-1.06-0.31-0.07-0.24}^{+1.33+0.31+0.07+0.22}$	${84.8}_{-6.5-0.2-0.0-1.1}^{+4.3+0.2+0.1+0.9}$	${15.2}_{-4.3-0.2-0.1-0.9}^{+6.5+0.2+0.0+1.1}$
$B_c\to K_2^{\ast +}\omega$	${3.14}_{-0.92-0.27-0.09-0.20}^{+1.17+0.28+0.10+0.20}$	${84.4}_{-6.5-0.2-0.0-1.0}^{+4.4+0.2+0.1+1.0}$	${15.6}_{-4.4-0.2-0.1-1.0}^{+6.5+0.2+0.0+1.0}$
$B_c\to K_2^{\ast +}\varphi$	${11.80}_{-0.83-1.05-0.38-0.29}^{+0.15+1.11+0.41+0.26}$	${80.2}_{-1.4-1.2-0.3-0.5}^{+0.0+1.1+0.3+0.4}$	${19.8}_{-0.0-1.1-0.3-0.4}^{+1.4+1.2+0.3+0.5}$
$B_c\to a_2^{+}{K}^{\ast 0}$	${9.20}_{-0.68-1.06-0.38-0.38}^{+0.00+1.11+0.38+0.26}$	${87.4}_{-0.9-1.0-0.3-0.6}^{+0.0+0.9+0.2+0.4}$	${12.6}_{-0.0-0.9-0.2-0.4}^{+0.9+1.0+0.3+0.6}$
$B_c\to a_2^0{K}^{\ast +}$	${4.60}_{-0.34-0.53-0.19-0.19}^{+0.00+0.55+0.19+0.12}$	${87.4}_{-0.9-1.0-0.3-0.6}^{+0.0+0.9+0.2+0.4}$	${12.6}_{-0.0-0.9-0.2-0.4}^{+0.9+1.0+0.3+0.6}$
$B_c\to f_2{K}^{\ast +}$	${4.38}_{-0.30-0.56-0.20-0.19}^{+0.00+0.58+0.17+0.09}$	${88.6}_{-0.6-1.1-0.3-0.5}^{+0.0+1.0+0.2+0.2}$	${11.4}_{-0.0-1.0-0.2-0.2}^{+0.6+1.1+0.3+0.5}$
$B_c\to f_2^{\prime }{K}^{\ast +}$	${9.60}_{-2.68-0.58-0.43-0.69}^{+3.42+0.66+0.46+0.77}$	${79.4}_{-8.1-0.1-0.0-1.6}^{+5.6+0.3+0.1+1.5}$	${20.6}_{-5.6-0.3-0.1-1.5}^{+8.1+0.1+0.0+1.6}$

Equations

$$\displaystyle\left(\begin{array}[]{c}f_{2}(1270)\\ f_{2}^{\prime}(1525)\\ \end{array}\right)$$

$${\varphi }_T(x)$$

$${\varphi }_T^t(x)$$

$${\varphi }_T^v(x)$$

$${\varphi }_{\parallel }(x)$$

$$h_{\parallel }^t(x)$$

$$g_{\perp }^v(x)$$

$$Br(B_c\to K_2^{\ast +}{\bar{K}}^0)$$

$$R_{{\bar{K}}^0/K^{+}}$$

$$R_{\eta /{\eta }^{\prime }}^{a_2}$$

$$R_{{\eta }^{\prime }/\eta }^{K_2^{\ast }}$$

$$Br(B_c\to {\bar{K}}_2^{\ast 0}{\pi }^{+})$$

$$Br(B_c\to a_2^{+}{K}^0)$$

$$R_{K/\pi }^{a_2}$$

$$R_{K/\pi }^{f_2}$$

$$R_{f_2/f_2^{\prime }}^K$$

$$m_{\rho }$$

$$f_{\rho }^T$$

$$m_{a_2}$$

$$f_{a_2}^T$$

$$R_{\omega /{\rho }^0}^{a_2}$$

$$R_{f_2/a_2^0}^{\rho }$$

$$R_{a_2\omega /f_2\rho }$$

$$Br(B_c\to K_2^{\ast +}{\bar{K}}^{\ast 0})$$

$$R_{{\bar{K}}^{\ast 0}/K^{\ast +}}$$

$$R_{f_2/f_2^{\prime }}^{K^{\ast }}$$

$$Br(B_c\to f_2{K}^{+})$$

$$Br(B_c\to K_2^{\ast 0}{\rho }^{+})$$

$$R_{\rho /{\bar{K}}^{\ast }}$$

$$R_{\rho /K^{\ast }}$$

$$R_{K^{\ast }/\rho }^{a_2}$$

$$R_{K^{\ast }/\rho }^{f_2}$$

Full text

††preprint: JSNU-PHY-HEP-2017-1

Heavy flavor physics has played an important role in the precision tests of the standard model(SM), as well as in investigating the properties of involved light hadrons after the advent of two $B$ factories, i.e., BABAR at SLAC and Belle at KEK. An increasing number of interesting mesons have been observed in the decay channels of the heavy mesons$-$specifically, $D_{(s)}$ mesons with a $c$ quark and $B_{(s)}$ mesons with a $b$ quark Olive:2016xmw $-$which provide a fertile ground for probing the perturbative and nonperturbative QCD dynamics in the SM. With the advent of the Large Hadron Collider(LHC) at CERN, a new territory has been developed since a great number of $B_{c}$ meson events can be observed. The properties of the $B_{c}$ meson and the dynamics involved in $B_{c}$ decays could be fully exploited through the precision measurements at the LHC with its high collision energy and high luminosity. Therefore, the $B_{c}$ meson decays will open a window to richer physics, which could start a new golden era of heavy flavor physics with the LHC experiments Xiao:2013lia ; Liu:2009qa .

Tensor mesons with quantum number $J^{P}=2^{+}$ have recently become a hot topic. On the one hand, experimentally, BABAR and Belle have measured several charmless hadronic $B$ decays involving a light tensor meson in the final states Olive:2016xmw ; Amhis:2016xyh . Furthermore, the measurements on the polarization fractions of $B\to\phi K_{2}^{*}(1430)$ decays showed that these two modes are dominated by the longitudinal polarization amplitudes, which is contrary to the same $b\to s\bar{s}s$-transition-induced $B\to\phi K^{*}$ processes. This phenomenology makes the well-known “polarization puzzle” more confusing. On the other hand, theoretically, the tensor meson cannot be produced through either local vector or axial-vector operators, or via the tensor current, which implies that large nonfactorizable amplitudes or annihilation diagrams would contribute to the tensor meson emitted modes with experimentally sizable branching ratios and the relevant investigations should go beyond the naive factorization. Of course, the polarization studies on the tensor-vector, tensor-axial-vector, and even tensor-tensor modes in heavy flavor decays can further shed light on the underlying helicity structure of the decay mechanism Cheng:2010yd . According to the counting rule, the annihilation contributions are usually power suppressed, compared to other spectator diagrams. Nevertheless, the annihilation contributions are not negligible and the size is still an important issue in $B$ meson physics (see, e.g., Refs. Keum:2000ph ; Lu:2000em ; Lu:2002iv ; Li:2004ep ; Kagan:2004uw ; Li:2004mp ; Hong:2005wj ; Ali:2007ff ; Chay:2007ep ; Liu:2009qa ; Xiao:2011tx ; Xiao:2013lia ; Zhu:2011mm ; Chang:2014rla ; Zou:2015iwa ). Indeed, the experiments have confirmed some large annihilation decay modes, for example, the well-known $B_{d}\to K^{+}K^{-}$ and $B_{s}\to\pi^{+}\pi^{-}$ decays Aaij:2016elb . Moreover, phenomenologically, the theoretical studies on the $B\to\phi K^{*}$  Kagan:2004uw ; Li:2004mp ; Zou:2015iwa and $B\to\phi K_{2}^{*}(1430)$ decays Cheng:2010yd ; Kim:2013cpa have provided important improvements in the explanation of the “polarization puzzle” by including the annihilation effects, though the authors claimed that $f_{L}(B_{d}\to\phi K_{2}^{*}(1430))\sim{\cal O}(1)$ with or without the annihilation effects Chen:2007qj .

Compared to the annihilation amplitudes in the charmless $B$ decays, the magnitude in the $B_{c}$ decays would be roughly enlarged by a factor $|V_{cb}/V_{ub}|\sim 11.5$, which would consequently result in a 100 times enhancement to the branching ratios. Therefore, the annihilation $B_{c}$ modes could possibly provide a promising and more appropriate platform to study the contributions from the annihilation diagrams, and even further uncover the annihilation decay mechanism. It is great to find that the measurements on the pure annihilation $B_{c}$ decay modes have been initiated by the LHCb Collaboration, for example, $B_{c}\to K^{+}\bar{K}^{0}$ Aaij:2013fja , $B_{c}\to K^{+}K^{-}\pi^{+}$ Aaij:2016xas , and $B_{c}\to p\bar{p}\pi^{+}$ Aaij:2016xxs , etc. Certainly, with the increasing number of $B_{c}$ events being collected, more and more annihilation types of $B_{c}$ decay channels will be opened. Sequentially, much more information on the annihilation decay mechanism must be obtained.

To date, an agreement on how to calculate the Feynman diagrams with annihilation topology reliably has not been achieved among the theorists. At least, the perturbative QCD(pQCD) approach Keum:2000ph ; Lu:2000em ; Li:2003yj and soft-collinear effective theory(SCET) Bauer:2000yr , as two popular tools for calculating hadronic matrix elements based on QCD dynamics, 111Another popular method is the QCD factorization approach Beneke:1999br ; Du:2000ff , which cannot make effective calculations on the annihilation diagrams since there exist end-point singularities in the integrals. However, data fitting has been broadly adopted in this approach to make theoretical predictions in the $B_{(s)}$ decays; see, for example, Refs.  Beneke:1999br ; Du:2000ff ; Beneke:2003zv ; Cheng:2009cn ; Li:2003hea have rather different viewpoints: the almost imaginary annihilation amplitudes with a large strong phase obtained through keeping the parton’s transverse momentum in the pQCD framework Chay:2007ep , and the almost real annihilation amplitudes with a tiny strong phase obtained by considering the zero-bin subtraction in the SCET framework Arnesen:2006vb . However, objectively speaking, the confirmation of the predicted branching ratios for the pure annihilation $B_{d}\to K^{+}K^{-}$ and $B_{s}\to\pi^{+}\pi^{-}$ decays provided by the CDF Ruffini:2013jea and LHCb Aaij:2012as ; Aaij:2016elb collaborations provided firm support to the current pQCD approach.

In this work, we will study the two-body nonleptonic charmless $B_{c}$ decays involving a light tensor meson($T$) and a light pseudoscalar($P$) or vector meson($V$) in the final states by employing the pQCD approach at leading order. These considered decays can only occur through weak annihilation interactions in the SM. Here, the light pseudoscalar(vector) meson includes $\pi$, $K$, $\eta$, and $\eta^{\prime}$($\rho$, $K^{*}$, $\omega$, and $\phi$). In the quark model, the observed light tensor meson contains the isovector states $a_{2}(1320)$, the isodoublet states $K_{2}^{*}(1430)$, and the isoscalar singlet states $f_{2}(1270)$ and $f_{2}^{\prime}(1525)$, which have been well established in various processes Olive:2016xmw . Hereafter, for the sake of simplicity, we will adopt $a_{2}$, $K_{2}^{*}$, $f_{2}$, and $f_{2}^{\prime}$ to denote the light tensor mesons correspondingly, unless otherwise specified. It is worth mentioning that, just like the $\eta-\eta^{\prime}$ mixing in the pseudoscalar sector, the two isoscalar tensor states $f_{2}(1270)$ and $f_{2}^{\prime}(1525)$ also have a mixing as

[TABLE]

with $f_{2q}\equiv(u\bar{u}+d\bar{d})/\sqrt{2}$ and $f_{2s}\equiv s\bar{s}$. The angle between $f_{2}(1270)$ and $f_{2}^{\prime}(1525)$ mixing should be small due to a fact that the former(latter) predominantly decays into $\pi\pi(K\bar{K})$ Olive:2016xmw . Specifically, the mixing angle $\phi_{f_{2}}$ lies in the range $6^{\circ}$-$10^{\circ}$ Olive:2016xmw ; Li:2000zb ; Cheng:2011fk . Therefore, analogous to $\omega$ and $\phi$ mesons in the vector sector, we will first approximately assume $f_{2}(f_{2}^{\prime})$ as the pure $f_{2q}(f_{2s})$ state firstly. The mixture of $f_{2}-f_{2}^{\prime}$ with the mixing angle $\phi_{f_{2}}$ will be left for future studies associated with experimentally precise measurements.

As mentioned above, the pQCD approach is an appropriate tool to effectively calculate the hadronic matrix elements of annihilation topology in the nonleptonic weak $B$ meson decays. The most important feature of the pQCD approach is that it picks up the intrinsic transverse momentum $k_{T}$ of the valence quarks in light of the end-point divergences that exist in the collinear factorization. Then, based on the $k_{T}$ factorization theorem, by utilizing the technique of resummation the double logarithmic divergences factored out from the hard part can be grouped into a Sudakov factor($e^{-S}$) Botts:1989kf and a threshold factor [$S_{t}(x)$] Li:2001ay , which consequently make the pQCD approach more self-consistent. Then, the single logarithmic divergences separated from the hard kernel can be reabsorbed into the meson wave functions using the eikonal approximation Li:1994iu . The interested reader can refer to the review paper Li:2003yj for more details about this approach. Presently, many quantitative annihilation-type-diagram calculations have been made with this pQCD approach.

The Feynman diagrams for the nonleptonic charmless $B_{c}\to TP,TV$ decays in the pQCD approach at leading order are illustrated in Fig. 1: Figs. 1(a) and 1(b) use the factorizable annihilation topology, while Figs. 1(c) and 1(d) use the nonfactorizable annihilation topology. For a spin-2 tensor meson, the polarization can be specified by a symmetric and traceless tensor $\epsilon^{\mu\nu}_{(\lambda)}$ with helicity $\lambda$ that satisfies the relation $\epsilon_{(\lambda)}^{\mu\nu}P_{\mu}=\epsilon_{(\lambda)}^{\mu\nu}P_{\nu}=0$, with $P$ being its momentum. Furthermore, this polarization tensor can be constructed through the spin-1 polarization vector $\epsilon_{V}$ Berger:2000wt . Although a tensor meson contains five spin degrees of freedom, only $\lambda=0$ will give a nonzero contribution in the $B_{c}\to TP$ modes, since the mother $B_{c}$ meson is spinless and the daughter $T$ and $P$ mesons should obey the conservation law of angular momentum. Likewise, the $B_{c}\to TV$ decays will be contributed from $\lambda=0$ and $\lambda=\pm 1$ helicities. Then, one can intuitively postulate that the considered $B_{c}\to TP,TV$ decays appear more like $B_{c}\to VP,VV$ ones by elaborating a new polarization vector $\epsilon_{T}$ for the tensor meson Datta:2007yk ; Wang:2010ni . Actually, $\epsilon_{T}$ has been explicitly presented in the literature (see, for example, Refs. Datta:2007yk ; Wang:2010ni ; Cheng:2010yd ) with $\epsilon_{T}(L)=\sqrt{\frac{2}{3}}\epsilon_{V}(L)$ and $\epsilon_{T}(T)=\sqrt{\frac{1}{2}}\epsilon_{V}(T)$. 222Since only three helicities $\lambda=0,\pm 1$ contribute to the $B_{c}\to TV$ decays, the involved light tensor meson can be treated as a vector-like meson with tensor meson mass. Here, the capital $L$ and $T$ in the parentheses describe the longitudinal and transverse polarizations, respectively ( not to be confused with the abbreviation $T$ for the light tensor meson). The decay amplitudes of $B\to TP$ and $B\to TV$ modes presented in Refs. Cheng:2010yd ; Kim:2013cpa ; Zou:2012td have confirmed the above postulation. Therefore, the decay amplitudes of the $B_{c}\to TP,TV$ decays considered in this work can be straightforwardly obtained by replacing the polarization vector $\epsilon_{V}$ of the vector meson with the corresponding $\epsilon_{T}$ of the tensor one in the $B_{c}\to VP,VV$ Liu:2009qa modes. That is:

• (1) Equations $(28)-(31)$ in Ref. Liu:2009qa with a factor $\sqrt{\frac{2}{3}}$ will give the analytic Feynman amplitudes of the $B_{c}\to PT,TP$ decays with only longitudinal polarization, in which the vector meson mass and distribution amplitudes should be replaced with the tensor state.
• (2) Equations (49)$-$(50)[Eqs. (51)$-$(54)] in Ref. Liu:2009qa with a factor $\sqrt{\frac{2}{3}}[\sqrt{\frac{1}{2}}]$ can contribute to the analytic Feynman amplitudes of the $B_{c}\to VT,TV$ decays in longitudinal[transverse] polarizations, where the corresponding quantities of the tensor state will be substituted for those of one of the two vector mesons.

Because no $f_{2}-f_{2}^{\prime}$ mixing is considered, the $B_{c}\to\pi^{+}f_{2}^{\prime}$ and $B_{c}\to\rho^{+}f_{2}^{\prime}$ decays will naturally be absent. Also forbidden is the $B_{c}\to a_{2}^{+}\phi$ mode as a result of not including $\omega-\phi$ mixing effects. Therefore, here we will not present the factorization formulas and the expressions for total decay amplitudes explicitly for those considered decays. The readers can refer to Ref. Liu:2009qa for details.

Now we can turn to the numerical calculations of these $B_{c}\to TP,TV$ decays in the pQCD approach. Before proceeding, some comments on the input quantities are in order:

• (1) For the lightest pseudoscalar and vector mesons, the (chiral) masses, the decay constants, the QCD scale, and the light-cone distribution amplitudes including Gegenbauer moments are same as those used in Liu:2009qa . Please refer to the Appendix A of Ref. Liu:2009qa for detail.
• (2) For the $B_{c}$ meson, the distribution amplitude and the decay constant are the same as those adopted in Ref. Liu:2009qa but with the up-to-date mass $m_{B_{c}}=6.275$ GeV and lifetime $\tau_{B_{c}}=0.507$ ps, which have been updated in the latest version of the Review of Particle Physics Olive:2016xmw .
• (3) For the Cabibbo-Kobayashi-Maskawa(CKM) matrix elements, we also adopt the Wolfenstein parametrization at leading order, but with the updated parameters $A=0.811$ and $\lambda=0.22506$ Olive:2016xmw .
• (4) For the light tensor meson, the decay constants with longitudinal and transverse polarizations are collected in Table 1.

The related masses are $m_{a_{2}}=1.318$ GeV, $m_{K_{2}^{*}}=1.426$ GeV, $m_{f_{2}}=1.275$ GeV, and $m_{f_{2}^{\prime}}=1.525$ GeV. Again, the $f_{2}-f_{2}^{\prime}$ mixing is not considered in this work. Therefore, the mass of the pure flavor $f_{2q}(f_{2s})$ state is taken as that of the physical $f_{2}(f_{2}^{\prime})$ for convenience.

The related light-cone distribution amplitudes have been recently investigated in the QCD sum rules Cheng:2010hn . Analogous to the light vector meson, the asymptotic forms of the tensor meson distribution amplitudes are adopted. Here, we present the expressions of the light-cone distribution amplitudes for the light tensor mesons following Ref. Wang:2010ni :

[TABLE]

with

[TABLE]

where the Gegenbauer moment $a_{1}=\frac{5}{3}$ for the first rough estimates and the Gegenbauer polynomial $C_{1}^{3/2}(t)=3t$ with $t=2x-1$. It is worth commenting that, in principle, the Gegenbauer moments for different meson distribution amplitudes should usually be different due to the expected $SU(3)$ flavor symmetry-breaking effects. Therefore, the larger Gegenbauer moment $a_{1}$ adopted here will demand further improvements resultant from the near-future relevant measurements with good precision.

The pQCD predictions of the CP-averaged branching ratios in the $B_{c}\to TP,TV$ decays and of the polarization fractions in the $B_{c}\to TV$ modes collected in Tables 2, 3, and 4, respectively.

• (1) For the $B_{c}\to TP$ decays, the main four errors arise from the uncertainties of the charm-quark mass $m_{c}=1.5\pm 0.15$ GeV in the $B_{c}$ meson distribution amplitude, of the combined decay constants of $f_{T}$ and $f_{T}^{T}$ in the tensor meson distribution amplitudes, of the combined Gegenbauer moments $a_{1}$ and/or $a_{2}$ in the pseudoscalar meson distribution amplitudes, and of the chiral mass $m_{0}^{P}$ of the pseudoscalar mesons. 333In order to estimate the theoretical uncertainties induced by the meson chiral mass, here we consider 10% variations of the central values for simplicity. Of course, for the $B_{c}\to TP$ modes involving $\eta$ and $\eta^{\prime}$ states, we also take the the variations of the mixing angle $\phi_{P}=39.3^{\circ}\pm 1.0^{\circ}$ into account as the fifth error.
• (2) For the $B_{c}\to TV$ channels, the major four errors are induced by the uncertainties of the charm-quark mass $m_{c}=1.5\pm 0.15$ GeV in the $B_{c}$ meson distribution amplitude, of the combined decay constants of $f_{T}$ and $f_{T}^{T}$ in the tensor meson distribution amplitudes, of the combined decay constants $f_{V}$ and $f_{V}^{T}$ in the vector meson distribution amplitudes, and of the combined Gegenbauer moments of $a_{1}^{\parallel(\perp)}$ and/or $a_{2}^{\parallel(\perp)}$ in the vector meson distribution amplitudes.

Here, we will specify the decay modes into two types: the CKM-favored channels with $\Delta S=0$(no strange or two strange mesons in the final states) and the CKM-suppressed modes with $\Delta S=1$(only one strange meson in the final states) for clarifications.

Based on the pQCD predictions of the CP-averaged branching ratios for the considered decay channels $B_{c}\to TP$ presented in Table 2, one can find the following results:

• (1) Relative to the suppressed CKM matrix element $V_{us}\sim 0.22506$ Olive:2016xmw in the $\Delta S=1$ modes, the enhanced one $V_{ud}\sim 0.97434$ in the $\Delta S=0$ modes makes their decay rates generally much larger around one order, which can be clearly seen in Table 2.
• (2) Generally speaking, the nonleptonic charmless $B_{c}\to TP$ modes have decay rates from $10^{-7}$(e.g., $B_{c}\to f_{2}\pi^{+}$) to $10^{-8}$(e.g., $B_{c}\to K_{2}^{*+}\pi^{0}$) in the pQCD framework, except for the two $B_{c}\to K_{2}^{*}K$ processes with large branching ratios,

[TABLE]

which are expected to be tested in the near future since, as argued in Ref. DescotesGenon:2009ja , the $B_{c}$ decays with the branching ratios of $10^{-6}$ can be measured at the LHC experiments. In light of the still large theoretical errors in these two modes, we usually provide a more precise ratio between these two CP-averaged branching ratios $Br(B_{c}\to K_{2}^{*+}\bar{K}^{0})$ and $Br(B_{c}\to\bar{K}_{2}^{*0}K^{+})$ as

[TABLE]

in which the uncertainties induced by the hadronic inputs could be greatly canceled. Of course, the largest error of the ratio $R_{\bar{K}^{0}/K^{+}}$ arising from the charm-quark mass in the $B_{c}$ meson distribution amplitude $\phi_{B_{c}}$ indicates that much more effort should be devoted to better understanding the nonperturbative QCD dynamics involved in the $B_{c}$ meson, which will be helpful to further provide theoretical predictions with good precision for experiments. Compared to the $B_{c}\to K_{2}^{*}K$ modes, it is worth noticing the different phenomenologies exhibited in the $B_{c}\to K^{*}K$ decays Liu:2009qa . The decay rate of $B_{c}\to\bar{K}^{*0}K^{+}$ is much larger than that of $B_{c}\to\bar{K}^{0}K^{*+}$ by a factor of about 5.5, in terms of the central values. The underlying reason is that, relative to the antisymmetric $K_{2}^{*}$ light-cone distribution amplitudes [see Eq. (11)] in the $SU(3)$ limit Cheng:2010hn , the significant $SU(3)$ flavor symmetry-breaking effects have been included in both $K$ and $K^{*}$ mesons, which can be seen evidently from the $a_{1}$ terms in their leading-twist distribution amplitudes Liu:2009qa .

• (3) The $B_{c}\to TP$ modes involving $\eta-\eta^{\prime}$ mixing effects [i.e., $B_{c}\to a_{2}^{+}(\eta,\eta^{\prime})$ and $B_{c}\to K_{2}^{*+}(\eta,\eta^{\prime})$ decays] show different interferences between $\eta_{q}$ and $\eta_{s}$ flavor states. That is, there is constructive(destructive) interference in the $B_{c}\to a_{2}^{+}\eta(B_{c}\to a_{2}^{+}\eta^{\prime})$ mode, while the opposite occurs in the $B_{c}\to K_{2}^{*+}\eta(B_{c}\to K_{2}^{*+}\eta^{\prime})$ channel. Furthermore, one can deduce the dominance of $\eta_{q}(\eta_{s})$ contributions in the $B_{c}\to a_{2}^{+}\eta^{(\prime)}(B_{c}\to K_{2}^{*+}\eta^{(\prime)})$ modes based on the numerical results of the branching ratios displayed in Table 2. Similar interferences have also been observed in the $B_{c}\to\rho^{+}(\eta,\eta^{\prime})$ and $B_{c}\to K^{*+}(\eta,\eta^{\prime})$ decays Liu:2009qa . We explicitly present four interesting ratios among the above-mentioned $B_{c}\to(\rho,K^{*},a_{2},K_{2}^{*})(\eta,\eta^{\prime})$ decays:

[TABLE]

where various errors in the ratios have been added in quadrature. The good isospin symmetry makes the $R_{\eta/\eta^{\prime}}^{a_{2}}$ approximately equal to the $R_{\eta/\eta^{\prime}}^{\rho}$; however, the significant $SU(3)$ flavor symmetry-breaking effects in the $K^{*}$ meson makes the $R_{\eta^{\prime}/\eta}^{K^{*}}$ quite different from the $R_{\eta^{\prime}/\eta}^{K_{2}^{*}}$. It is expected that future precise measurements of these ratios might be helpful to investigate the possible pseudoscalar glueball in the $\eta^{\prime}$ state Cheng:2008ss ; Liu:2012ib .

• (4) As far as the $B_{c}\to(a_{2}^{0},f_{2})(\pi^{+},K^{+})$ channels are concerned, one can find that, according to the pQCD predictions for the branching ratios, the constructive (destructive) interferences between $u\bar{u}$ and $d\bar{d}$ components in the $f_{2}(a_{2}^{0})$ meson with the same (opposite) sign result in a slightly larger (smaller) $Br(B_{c}\to f_{2}\pi^{+})=7.38^{+1.86}_{-1.75}\times 10^{-7}[Br(B_{c}\to a_{2}^{0}\pi^{+})=5.13^{+1.96}_{-1.65}\times 10^{-7}]$. On the other hand, due to only the $u\bar{u}$ component in both $a_{2}^{0}$ and $f_{2}$ states giving contributions, the almost equivalent branching ratios $Br(B_{c}\to a_{2}^{0}K^{+})\approx Br(B_{c}\to f_{2}K^{+})$ can be obtained, which are more like that seen in the $B_{c}\to(\rho^{0},\omega)K^{+}$ modes Liu:2009qa . The negligibly tiny deviations between the $B_{c}\to a_{2}^{0}K^{+}$ and $B_{c}\to f_{2}K^{+}$ decays arise from the slightly different decay constants and hadron masses of the $a_{2}^{0}$ and $f_{2}$ states, as well as from the same QCD behavior at leading twist. Likewise, the similar phenomenologies of the branching ratios and polarization fractions can be seen clearly from the processes of $B_{c}\to K_{2}^{*+}(\rho^{0},\omega)$ and $B_{c}\to(a_{2}^{0},f_{2})K^{*+}$ in Table 4.
• (5) Some simple relations and many other interesting ratios, which can shed light on the (non)validity of $SU(3)$ flavor symmetry in the considered decays, are given as follows:

[TABLE]

[TABLE]

Moreover, the ratio between $Br(B_{c}\to f_{2}K^{+})$ and $Br(B_{c}\to f^{\prime}_{2}K^{+})$ when confronted with the future precision data can provide useful hints for the $f_{2}-f^{\prime}_{2}$ mixing, though the ideal mixing is assumed in this work,

[TABLE]

where the largest error of the ratio is also induced by the variations of the $B_{c}$ meson distribution amplitude. Therefore, an in-depth understanding of the hadronization of the involved meson is the key to provide precise predictions in the pQCD approach for future experimental measurements.

Now we turn to the analyses of the branching ratios and polarization fractions of the $B_{c}\to TV$ decays in the pQCD approach. As stressed previously, due to the angular moment conservation, the $B_{c}\to TV$ decays contain three helicities, which are more like the $B_{c}\to VV$ ones. Then the definitions of the related helicity amplitudes, polarization fractions, and relative phases are also the same as those of $B_{c}\to VV$ modes (see Ref. Liu:2009qa for details). It should be noted that, as this is a first investigation of the nonleptonic charmless $B_{c}\to TV$ decays, only CP-averaged branching ratios and polarization fractions (whose values are collected in Tables 3 and 4) are presented in this work. Moreover, we specify the polarization fractions as longitudinal $f_{L}$ and transverse $f_{T}(=1-f_{L})$, not those $f_{L}$, $f_{\parallel}$, and $f_{\perp}$ adopted previously Liu:2009qa . Some remarks are in order:

• (1) Differently from the $B_{c}\to TP$ decays, all of the CKM-favored $B_{c}\to TV$ modes (which contain three polarization contributions with larger decay constants and hadron masses of vector mesons) have decay rates of $10^{-6}$ within theoretical errors in the pQCD approach at leading order. It is believed that the predictions of these large branching ratios can be confirmed soon by the LHC experiments at CERN DescotesGenon:2009ja . The CP-averaged branching ratios of the CKM-suppressed $B_{c}\to TV$ modes are nearly $10^{-8}-10^{-7}$, which may have to await future tests with much larger data samples. Nevertheless, one can easily find that all of the $B_{c}\to TV$ modes are governed by the longitudinal decay amplitudes, which result in the large polarization fractions in the range of $78\%-98\%$, as presented in Tables 3 and 4.
• (2) As shown in Table 3, the CP-averaged branching ratios and the polarization fractions of $B_{c}\to a_{2}^{+}\omega$ and $B_{c}\to f_{2}\rho^{+}$ channels are close to each other. The reason is that, on the one hand, the pure $\frac{u\bar{u}+d\bar{d}}{\sqrt{2}}$ component for the $f_{2}(1270)$ state is assumed which is same as the $\omega$ meson with ideal mixing, and on the other hand, the adopted decay constants and masses of the involved tensor and vector mesons are similar in magnitude with only small differences. More specifically,

[TABLE]

for the light vector $\rho$ and $\omega$ mesons, and

[TABLE]

for the light tensor $a_{2}$ and $f_{2}$ states. Therefore, it is also understandable that these two decay rates are a bit smaller than that of the $B_{c}\to\rho^{+}\omega$ channel Liu:2009qa .

• (3) It is interesting to note that the $B_{c}\to a_{2}^{+}\rho^{0}$ and $B_{c}\to a_{2}^{+}\omega$ decay rates indicate different interferences between the $u\bar{u}$ and $d\bar{d}$ components in the $\rho^{0}$ and $\omega$ mesons. As can be seen in Table 3, it is evident that the constructive(destructive) interferences contribute to the former(latter) mode. Similar phenomenologies also appear in the $B_{c}\to a_{2}^{0}\rho^{+}$ and $B_{c}\to f_{2}\rho^{+}$ decays. Moreover, one can easily observe that the numerical results of the branching ratios are sensitive to the hadronic parameters such as the charm-quark mass, the decay constants of the light tensor meson, etc. We thus define some ratios among the branching ratios as follows:

[TABLE]

where the uncertainties arising from the errors of the inputs have been greatly canceled, though these parameters involved in the meson wave functions are not factored out. These ratios and the detectable decay rates could be helpful to further explore the QCD behavior of the $a_{2}$ and $f_{2}$ states.

• (4) Analogous to the $B_{c}\to K_{2}^{*}K$ decays, the $B_{c}\to K_{2}^{*+}\bar{K}^{*0}$ and $B_{c}\to\bar{K}_{2}^{*0}K^{*+}$ modes also have branching ratios that are close to each other for the same reason. More interestingly, the ratio arising from $Br(B_{c}\to K_{2}^{*+}\bar{K}^{*0})$ over $Br(B_{c}\to\bar{K}_{2}^{*0}K^{*+})$ in the pQCD approach is approximately equal to that [see Eq. (15)] obtained in the $B_{c}\to K_{2}^{*}K$ decays, although these two branching ratios induced by three polarizations are clearly larger than the $B_{c}\to K_{2}^{*}K$ ones only from longitudinal polarization. The related branching ratios and ratio are,

[TABLE]

and

[TABLE]

The conservation law of angular momentum results in the tensor $K_{2}^{*}$ state contributing to the $B_{c}\to K_{2}^{*}K^{*}$ decays with only three helicities, $\lambda=0$ and $\pm 1$, which makes it behave more like a vector meson. Since the smaller decay constants (as shown in Table 1) of both longitudinal and transverse polarizations of the $K_{2}^{*}$ meson are adopted, the decay rates and polarization fractions of the $B_{c}\to K_{2}^{*}K^{*}$ modes are basically consistent with those of the $B_{c}\to\bar{K}^{*0}K^{*+}$ one within errors Liu:2009qa , though $m_{K_{2}^{*}}$ is nearly 2 times larger than $m_{K^{*}}$.

• (5) As reported by the BABAR Collaboration, the fact that $f_{L}/f_{T}\gg 1$ for $B\to\phi K_{2}^{*}$ decays Aubert:2008bc while $f_{L}/f_{T}\sim 1$ for $B\to\omega K_{2}^{*}$ decays Aubert:2009sx make the well-known “polarization puzzle” more confusing, although both of them have the same helicity structure as the $B\to\phi K^{*}$ modes with the penguin-dominated contributions. Furthermore, the branching ratio of the $B^{+}\to\omega K_{2}^{*+}$ channel is much larger than that of the $B^{+}\to\phi K_{2}^{*+}$ one by a factor of around 2.5, which is contrary to the ratio of the $B^{+}\to\omega K^{*+}$ and $B^{+}\to\phi K^{*+}$ decay rates Olive:2016xmw ; Amhis:2016xyh . The current theoretical studies on these anomalous phenomena cannot give satisfactory explanations, which means that more investigations on the light tensor $K_{2}^{*}$ meson are demanded. The CP-averaged branching ratios and polarization fractions of $B_{c}\to K_{2}^{*}(\rho,\omega,\phi)$ channels are given in the pQCD approach at leading order are presented in Table 4. One can find that these four modes are dominated by the longitudinal decay amplitudes, and the $B_{c}\to K_{2}^{*0}\rho^{+}$ and $B_{c}\to K_{2}^{*+}\phi$ decay rates are on the order of $10^{-7}$ within errors; these are expected to be tested by the LHC Run-II experiments at CERN in the near future.
• (6) Likewise, some interesting ratios of $B_{c}\to TV$ decays can also provide useful hints about the QCD dynamics involved in the light tensor mesons, as well as in the related decay channels. For example, future precise measurements can tell us the mixing information about $f_{2}(1275)$ and $f^{\prime}_{2}(1525)$ states through the ratio $R^{K^{*}}_{f_{2}/f^{\prime}_{2}}$,

[TABLE]

where the $B_{c}\to f^{\prime}_{2}K^{*+}$ branching ratio reaches $10^{-7}$. If the future measured ratios $R^{K}_{f_{2}/f^{\prime}_{2}}$ and $R^{K^{*}}_{f_{2}/f^{\prime}_{2}}$ deviate from those predicted in Eqs. (22) and (30), then the mixture of the $f_{2q}$ and $f_{2s}$ flavor states should be included for the $f_{2}$ and $f^{\prime}_{2}$ mesons. It is noted that the $R^{K}_{f_{2}/f^{\prime}_{2}}$ is a bit larger than the $R^{K^{*}}_{f_{2}/f^{\prime}_{2}}$ by a factor of around 1.5, since $Br(B_{c}\to f_{2}K^{+})\sim Br(B_{c}\to f_{2}K^{*+})$ while $Br(B_{c}\to f^{\prime}_{2}K^{*+})\sim 1.5\times Br(B_{c}\to f^{\prime}_{2}K^{+})$. More data are demanded on the $f_{2}$ and $f^{\prime}_{2}$ states to further understand these phenomenologies, in particular, the approximately equal decay rates between $B_{c}\to f_{2}K^{+}$ and $B_{c}\to f_{2}K^{*+}$ modes,

[TABLE]

After all, the latter process receives contributions from three helicities.

• (7) The isospin symmetry can be observed in the pQCD calculations for $B_{c}\to K_{2}^{*}\rho$ and $B_{c}\to a_{2}K^{*}$ modes, that is,

[TABLE]

However, the $SU(3)$ flavor symmetry cannot be easily seen in the $B_{c}\to K_{2}^{*}K^{*}$ and $B_{c}\to K_{2}^{*}\rho$ decays [Eq. (18)], since these two decays have three helicity structures with different decay constants and different Gegenbauer moments of the vector $K^{*}$ and $\rho$ mesons in longitudinal and transverse polarizations, respectively. Nevertheless, we can still present the ratios between $Br(B_{c}\to K_{2}^{*}\rho^{+})$ and $Br(B_{c}\to K_{2}^{*}K^{*})$, which can be used to show the (non)validity of the $SU(3)$ flavor symmetry by combining future precise measurements,

[TABLE]

Two more relations can also be written as follows:

[TABLE]

where, by combining the ratios $R^{a_{2}}_{K/\pi}$ and $R^{f_{2}}_{K/\pi}$ in Eqs. (20) and (21), $Br(B_{c}\to a_{2}^{0}\rho^{+})\sim 2.5\times Br(B_{c}\to a_{2}^{0}\pi^{+})$ but $Br(B_{c}\to a_{2}^{0}K^{*+})\simeq Br(B_{c}\to a_{2}^{0}K^{+})$ result in the relation $R^{a_{2}}_{K/\pi}>R^{a_{2}}_{K^{*}/\rho}$. However, $Br(B_{c}\to f_{2}\rho^{+})\sim 1.3\times Br(B_{c}\to f_{2}\pi^{+})$ but $Br(B_{c}\to f_{2}K^{*+})\simeq Br(B_{c}\to f_{2}K^{+})$ leads to the relation $R^{f_{2}}_{K^{*}/\rho}<R^{f_{2}}_{K/\pi}$. It is worth mentioning that the $R^{a_{2}}_{K/\pi}$ is a bit larger than the $R^{f_{2}}_{K/\pi}$ in the $B_{c}\to(a_{2}^{0},f_{2})(\pi^{+},K^{+})$ decays, while the $R^{a_{2}}_{K^{*}/\rho}$ is slightly smaller than the $R^{f_{2}}_{K^{*}/\rho}$ in the $B_{c}\to(a_{2}^{0},f_{2})(\rho^{+},K^{*+})$ modes, which could be tested and further clarified by the related experiments with good precision in the future.

• (8) Recently, the three-body( or quasi-two-body) $B$ meson decays have attracted more and more attention, since the two $B$ factories and the LHC experiments have collected lots of data on the related channels. It is suggested that the considered light tensor states in this work can also be studied through the resonant contributions in the relevant three-body modes Aaij:2016qnm ; for example, the $f_{2}(1270)$ and $f^{\prime}_{2}(1525)$ mesons can be investigated in the $B_{c}\to f_{2}(1270)(\to\pi\pi)(\pi,K)$ and $B_{c}\to f^{\prime}_{2}(1525)(\to KK)(\pi,K)$ channels, respectively, which can play important roles in exploring the QCD dynamics of the light tensor mesons. These studies can also help to further deepen our understanding of the three-body decay mechanism.

In summary, we have analyzed the nonleptonic charmless $B_{c}\to TP,TV$ decays in the pQCD approach. Due to the angular momentum conservation, the light tensor meson can only contribute with one($\lambda=0$) or three helicities($\lambda=0,\pm 1$). By properly redefining the polarization tensor, the new polarization vector $\epsilon_{T}$ of the light tensor meson can be obtained, which is slightly different than the $\epsilon_{V}$ of the vector meson with coefficients $\sqrt{\frac{2}{3}}$ and $\sqrt{\frac{1}{2}}$ for longitudinal and transverse polarizations, respectively. Therefore, the decay amplitudes can be easily presented with appropriate replacements from the $B_{c}\to PV,VV$ decay modes. The CP-averaged branching ratios and polarization fractions for the considered channels have been predicted in the pQCD approach. Most of the CKM-favored processes have decay rates of $10^{-6}$, which are expected to be measured soon by the LHC experiments at CERN. Numerically, all of the $B_{c}\to TV$ modes are governed by the longitudinal contributions. Many interesting ratios among the branching ratios have been derived, as well as some simple relations that can be used to exhibit the (non)validity of the $SU(3)$ flavor symmetry. The predictions about these concerned $B_{c}\to TP,TV$ decays in the pQCD approach can be confronted with measurements in the (near) future, which are expected to shed some light on the annihilation decay mechanism in the related decay channels.

This work is supported by the National Natural Science Foundation of China under Grants No. 11205072, No. 11235005, No. 11447032, No. 11505098, and No. 11647310 and by the Research Fund of Jiangsu Normal University under Grant No. HB2016004, by the Doctoral Scientific Research Foundation of Inner Mongolia University, and by the Natural Science Foundation of Shandong Province under Grant No. ZR2014AQ013.