Possibility of searching for $B_{c}^{\ast}$ ${\to}$ $B_{u,d,s}V$, $B_{u,d,s}P$ decays

TL;DR

This paper explores the potential to detect specific $B_{c}^{ ext{*}}$ meson decays into $B_{u,d,s}$ mesons with vector or pseudoscalar mesons, using QCD factorization, and finds some decay modes with measurable branching ratios.

Contribution

It provides theoretical predictions for $B_{c}^{ ext{*}}$ decay branching ratios using QCD factorization and the Wirbel-Stech-Bauer model, highlighting potentially observable decay channels.

Findings

Branching ratios for $B_{c}^{ ext{*}} \to B_{s}\rho$ and $B_{s}\pi$ can reach ${\cal O}(10^{-7})$

Some decay modes are potentially measurable at future LHC experiments

The study offers theoretical support for experimental searches of $B_{c}^{\text{*}}$ decays.

Abstract

The $B_{c}^{\ast}$ ${\to}$ $B_{u,d,s}V$, $B_{u,d,s}P$ decays are investigated with the QCD factorization approach, where $V$ and $P$ denote the ground $SU(3)$ vector and pseudoscalar mesons, respectively. The $B_{c}^{\ast}$ ${\to}$ $B_{u,d,s}$ transition form factors are calculated with the Wirbel-Stech-Bauer model. It is found that branching ratios for the color-favored and Cabibbo-favored $B_{c}^{\ast}$ ${\to}$ $B_{s}{\rho}$, $B_{s}{\pi}$ decays can reach up to ${\cal O}(10^{-7})$, which might be measurable in the future LHC experiments.

Tables

Table 1. Table 1: The numerical values for the Wilson coefficients and a 1 , 2 subscript 𝑎 1 2 a_{1,2} for the B c ∗ superscript subscript 𝐵 𝑐 ∗ B_{c}^{\ast} → → {\to} B q ​ π subscript 𝐵 𝑞 𝜋 B_{q}{\pi} decay.

scale	LO		NLO		NF		QCDF
$\mu$	$C_1$	$C_2$	$C_1$	$C_2$	$a_1$	$a_2$	$a_1$	$a_2$
$0.8m_c$	$1.310$	$-0.553$	$1.253$	$-0.473$	$1.096$	$-0.055$	$1.235+i\mathrm{\hspace{0.17em}0.081}$	$-0.384-i\mathrm{\hspace{0.17em}0.192}$
$m_c$	$1.259$	$-0.479$	$1.209$	$-0.404$	$1.074$	$-0.001$	$1.193+i\mathrm{\hspace{0.17em}0.060}$	$-0.313-i\mathrm{\hspace{0.17em}0.157}$
$1.2m_c$	$1.227$	$-0.430$	$1.180$	$-0.358$	$1.061$	$0.035$	$1.167+i\mathrm{\hspace{0.17em}0.048}$	$-0.269-i\mathrm{\hspace{0.17em}0.137}$

Table 2. Table 2: The numerical values for the form factors in the B c ∗ superscript subscript 𝐵 𝑐 ∗ B_{c}^{\ast} → → {\to} B q subscript 𝐵 𝑞 B_{q} transition, where the uncertainties come from both m c subscript 𝑚 𝑐 m_{c} and m b subscript 𝑚 𝑏 m_{b} .

transition	$\omega$	$A_0\left(0\right)$	$A_1\left(0\right)$	$A_2\left(0\right)$	$V\left(0\right)$
	$0.4$ GeV	${0.540}_{-0.015}^{+0.015}$	${0.291}_{-0.002}^{+0.002}$	${3.286}_{-0.209}^{+0.210}$	${1.953}_{-0.040}^{+0.038}$
$B_c^{\ast }$ $\to$ $B_u$	$0.6$ GeV	${0.784}_{-0.008}^{+0.008}$	${0.429}_{-0.011}^{+0.011}$	${4.694}_{-0.222}^{+0.219}$	${2.877}_{-0.109}^{+0.111}$
	$m{\alpha }_s$	${0.944}_{-0.002}^{+0.002}$	${0.539}_{-0.017}^{+0.017}$	${5.403}_{-0.213}^{+0.208}$	${3.613}_{-0.156}^{+0.160}$
	$0.4$ GeV	${0.540}_{-0.015}^{+0.015}$	${0.291}_{-0.002}^{+0.002}$	${3.288}_{-0.209}^{+0.210}$	${1.954}_{-0.040}^{+0.038}$
$B_c^{\ast }$ $\to$ $B_d$	$0.6$ GeV	${0.784}_{-0.008}^{+0.008}$	${0.429}_{-0.011}^{+0.011}$	${4.696}_{-0.222}^{+0.219}$	${2.878}_{-0.109}^{+0.111}$
	$m{\alpha }_s$	${0.944}_{-0.002}^{+0.002}$	${0.539}_{-0.017}^{+0.017}$	${5.405}_{-0.213}^{+0.208}$	${3.614}_{-0.156}^{+0.160}$
	$0.4$ GeV	${0.609}_{-0.015}^{+0.015}$	${0.361}_{-0.003}^{+0.003}$	${3.618}_{-0.234}^{+0.234}$	${1.867}_{-0.061}^{+0.059}$
$B_c^{\ast }$ $\to$ $B_s$	$0.6$ GeV	${0.821}_{-0.007}^{+0.007}$	${0.494}_{-0.012}^{+0.012}$	${4.785}_{-0.247}^{+0.242}$	${2.554}_{-0.120}^{+0.122}$
	$m{\alpha }_s$	${0.954}_{-0.002}^{+0.002}$	${0.598}_{-0.017}^{+0.018}$	${5.268}_{-0.238}^{+0.232}$	${3.097}_{-0.159}^{+0.163}$

Table 3. Table 3: The numerical values for input parameters.

Wolfenstein parameters pdg
$\lambda$ $=$ $0.22506\pm 0.00050$,   $A$ $=$ $0.811\pm 0.026$,   $\overline{\rho }$ $=$ ${0.124}_{-0.018}^{+0.019}$,   $\overline{\eta }$ $=$ $0.356\pm 0.011$;
Mass of particles and QCD characteristic scale
$m_{B_c^{\ast }}$ $=$ $6332\pm 9$ MeV111Other predictions of the $B_c^{\ast }$ meson mass with different models can be found in Table II of Ref.prd93.074010 . prd86.094510 ,	$m_{{\pi }^{+}}$ $=$ $139.57$ MeV pdg ,	$m_{{\pi }^0}$ $=$ $134.98$ MeV pdg ,
$m_{B_u}$ $=$ $5279.31\pm 0.15$ MeV pdg ,	$m_{K^{+}}$ $=$ $493.677\pm 0.016$ MeV pdg ,	$m_{K^0}$ $=$ $497.611\pm 0.013$ MeV pdg ,
$m_{B_d}$ $=$ $5279.62\pm 0.15$ MeV pdg ,	$m_{\eta }$ $=$ $547.862\pm 0.017$ MeV pdg ,	$m_{{\eta }^{\prime }}$ $=$ $957.78\pm 0.06$ MeV pdg ,
$m_{B_s}$ $=$ $5366.82\pm 0.22$ MeV pdg ,	$m_{K^{\ast +}}$ $=$ $891.66\pm 0.26$ MeV pdg ,	$m_{K^{\ast 0}}$ $=$ $895.81\pm 0.19$ MeV pdg ,
$m_{\rho }$ $=$ $775.26\pm 0.25$ MeV pdg ,	$m_{\omega }$ $=$ $782.62\pm 0.12$ MeV pdg ,	$m_{\varphi }$ $=$ $1019.461\pm 0.019$ MeV pdg ,
$m_b$ $=$ ${4.18}_{-0.03}^{+0.04}$ GeV pdg ,	$m_c$ $=$ $1.27\pm 0.03$ GeV pdg ,	${\Lambda }_{\mathrm{QCD}}^{\left(5\right)}$ $=$ $210\pm 14$ MeV pdg ,
$m_s$ $=$ $0.51$ GeV uds ,	$m_{u,d}$ $=$ $0.31$ GeV uds ,	${\Lambda }_{\mathrm{QCD}}^{\left(4\right)}$ $=$ $292\pm 16$ MeV pdg ,
Decay constants
$f_{\pi }$ $=$ $130.2\pm 1.7$ MeV pdg ,	$f_K$ $=$ $155.6\pm 0.4$ MeV pdg ,	$f_{K^{\ast }}$ $=$ $220\pm 5$ MeV jhep.0703.069 ,
$f_{\rho }$ $=$ $216\pm 3$ MeV jhep.0703.069 ,	$f_{\omega }$ $=$ $187\pm 5$ MeV jhep.0703.069 ,	$f_{\varphi }$ $=$ $215\pm 5$ MeV jhep.0703.069 ,
$f_q$ $=$ $\left(1.07\pm 0.02\right)f_{\pi }$ prd.58.114006 ,	$f_s$ $=$ $\left(1.34\pm 0.06\right)f_{\pi }$ prd.58.114006 ,
Gegenbauer moments at the sacle of $\mu$ $=$ 1 GeV
$a_1^{\pi }$ $=$ $a_1^{{\eta }_q}$ $=$ $a_1^{{\eta }_s}$ $=$ $0$ jhep.0605.004 ,	$a_1^K$ $=$ $0.06\pm 0.03$ jhep.0605.004 ,	$a_1^{\rho }$ $=$ $a_1^{\omega }$ $=$ $a_1^{\varphi }$ $=$ $0$ jhep.0703.069 ,
$a_2^{\pi }$ $=$ $a_2^{{\eta }_q}$ $=$ $a_2^{{\eta }_s}$ $=$ $0.25\pm 0.15$ jhep.0605.004 ,	$a_2^K$ $=$ $0.25\pm 0.15$ jhep.0605.004 ,	$a_2^{\rho }$ $=$ $a_2^{\omega }$ $=$ $0.15\pm 0.07$ jhep.0703.069 ,
$a_1^{K^{\ast }}$ $=$ $0.03\pm 0.02$ jhep.0703.069 ,	$a_2^{K^{\ast }}$ $=$ $0.11\pm 0.09$ jhep.0703.069 ,	$a_2^{\varphi }$ $=$ $0.18\pm 0.08$ jhep.0703.069 .

Table 4. Table 4: Branching ratios for the B c ∗ superscript subscript 𝐵 𝑐 ∗ B_{c}^{\ast} → → {\to} B q ​ V subscript 𝐵 𝑞 𝑉 B_{q}V , B q ​ P subscript 𝐵 𝑞 𝑃 B_{q}P decays calculated with the scale of μ 𝜇 {\mu} = = m c subscript 𝑚 𝑐 m_{c} , where the parameters in the “CKM” (“ a i subscript 𝑎 𝑖 a_{i} ”) column give the CKM factors (coefficients) of the decay amplitude; the uncertainties come from mass m c subscript 𝑚 𝑐 m_{c} and m b subscript 𝑚 𝑏 m_{b} .

final	parameters			branching ratio
state	CKM	$a_i$	case	$\omega$ $=$ $0.4$ GeV	$\omega$ $=$ $0.6$ GeV	$\omega$ $=$ $m{\alpha }_s$	unit
$B_s^0{\rho }^{+}$	$V_{cs}^{\ast }{V}_{ud}$ $\sim$ $𝒪\left(1\right)$	$a_1$	1-I	${6.29}_{-0.02}^{+0.02}$	${11.67}_{-0.35}^{+0.36}$	${16.77}_{-0.68}^{+0.72}$	${10}^{-7}$
$B_s^0{\pi }^{+}$	$V_{cs}^{\ast }{V}_{ud}$ $\sim$ $𝒪\left(1\right)$	$a_1$	1-I	${4.00}_{-0.20}^{+0.21}$	${7.26}_{-0.15}^{+0.15}$	${9.82}_{-0.06}^{+0.06}$	${10}^{-7}$
$B_s^0{K}^{\ast +}$	$V_{cs}^{\ast }{V}_{us}$ $\sim$ $𝒪\left(\lambda \right)$	$a_1$	1-II	${2.21}_{-0.03}^{+0.02}$	${4.12}_{-0.17}^{+0.18}$	${6.01}_{-0.30}^{+0.32}$	${10}^{-8}$
$B_s^0{K}^{+}$	$V_{cs}^{\ast }{V}_{us}$ $\sim$ $𝒪\left(\lambda \right)$	$a_1$	1-II	${1.98}_{-0.10}^{+0.10}$	${3.60}_{-0.08}^{+0.08}$	${4.87}_{-0.03}^{+0.03}$	${10}^{-8}$
$B_d^0{\rho }^{+}$	$V_{cd}^{\ast }{V}_{ud}$ $\sim$ $𝒪\left(\lambda \right)$	$a_1$	1-II	${3.51}_{-0.05}^{+0.05}$	${7.51}_{-0.17}^{+0.17}$	${11.47}_{-0.41}^{+0.44}$	${10}^{-8}$
$B_d^0{\pi }^{+}$	$V_{cd}^{\ast }{V}_{ud}$ $\sim$ $𝒪\left(\lambda \right)$	$a_1$	1-II	${2.14}_{-0.13}^{+0.13}$	${4.50}_{-0.11}^{+0.11}$	${6.53}_{-0.05}^{+0.05}$	${10}^{-8}$
$B_d^0{K}^{\ast +}$	$V_{cd}^{\ast }{V}_{us}$ $\sim$ $𝒪\left({\lambda }^2\right)$	$a_1$	1-III	${1.45}_{-0.01}^{+0.01}$	${3.13}_{-0.11}^{+0.11}$	${4.84}_{-0.23}^{+0.24}$	${10}^{-9}$
$B_d^0{K}^{+}$	$V_{cd}^{\ast }{V}_{us}$ $\sim$ $𝒪\left({\lambda }^2\right)$	$a_1$	1-III	${1.15}_{-0.07}^{+0.07}$	${2.41}_{-0.06}^{+0.06}$	${3.50}_{-0.02}^{+0.02}$	${10}^{-9}$
$B_u^{+}{\overline{K}}^{\ast 0}$	$V_{cs}^{\ast }{V}_{ud}$ $\sim$ $𝒪\left(1\right)$	$a_2$	2-I	${5.59}_{-0.03}^{+0.03}$	${12.08}_{-0.36}^{+0.39}$	${18.80}_{-0.80}^{+0.85}$	${10}^{-8}$
$B_u^{+}{\overline{K}}^0$	$V_{cs}^{\ast }{V}_{ud}$ $\sim$ $𝒪\left(1\right)$	$a_2$	2-I	${3.48}_{-0.27}^{+0.29}$	${7.32}_{-0.32}^{+0.34}$	${10.60}_{-0.30}^{+0.31}$	${10}^{-8}$
$B_u^{+}{K}^{\ast 0}$	$V_{cd}^{\ast }{V}_{us}$ $\sim$ $𝒪\left({\lambda }^2\right)$	$a_2$	2-III	${1.59}_{-0.01}^{+0.01}$	${3.44}_{-0.10}^{+0.11}$	${5.34}_{-0.23}^{+0.24}$	${10}^{-10}$
$B_u^{+}{K}^0$	$V_{cd}^{\ast }{V}_{us}$ $\sim$ $𝒪\left({\lambda }^2\right)$	$a_2$	2-III	${9.82}_{-0.77}^{+0.81}$	${20.67}_{-0.90}^{+0.94}$	${29.95}_{-0.82}^{+0.85}$	${10}^{-11}$
$B_u^{+}{\rho }^0$	$V_{cd}^{\ast }{V}_{ud}$ $\sim$ $𝒪\left(\lambda \right)$	$a_2$	2-II	${1.84}_{-0.03}^{+0.04}$	${3.95}_{-0.07}^{+0.07}$	${6.08}_{-0.19}^{+0.20}$	${10}^{-9}$
$B_u^{+}\omega$	$V_{cd}^{\ast }{V}_{ud}$ $\sim$ $𝒪\left(\lambda \right)$	$a_2$	2-II	${1.36}_{-0.02}^{+0.02}$	${2.93}_{-0.05}^{+0.06}$	${4.51}_{-0.14}^{+0.15}$	${10}^{-9}$
$B_u^{+}\varphi$	$V_{cs}^{\ast }{V}_{us}$ $\sim$ $𝒪\left(\lambda \right)$	$a_2$	2-II	${1.36}_{-0.01}^{+0.01}$	${2.95}_{-0.12}^{+0.13}$	${4.64}_{-0.25}^{+0.26}$	${10}^{-9}$
$B_u^{+}{\pi }^0$	$V_{cd}^{\ast }{V}_{ud}$ $\sim$ $𝒪\left(\lambda \right)$	$a_2$	2-II	${9.24}_{-0.72}^{+0.77}$	${19.44}_{-0.85}^{+0.89}$	${28.17}_{-0.78}^{+0.81}$	${10}^{-10}$
$B_u^{+}\eta$	$V_{cd}^{\ast }{V}_{ud}$, $V_{cs}^{\ast }{V}_{us}$	$a_2$	2-II	${2.37}_{-0.19}^{+0.20}$	${4.98}_{-0.22}^{+0.23}$	${7.22}_{-0.20}^{+0.21}$	${10}^{-9}$
$B_u^{+}{\eta }^{\prime }$	$V_{cd}^{\ast }{V}_{ud}$, $V_{cs}^{\ast }{V}_{us}$	$a_2$		${6.62}_{-0.52}^{+0.55}$	${13.93}_{-0.61}^{+0.64}$	${20.18}_{-0.56}^{+0.58}$	${10}^{-11}$

Equations

$${\cal H}_{\rm eff}\,=\,\frac{G_{F}}{\sqrt{2}}\sum\limits_{q,q^{\prime}=s,d}V_{cq}^{\ast}V_{uq^{\prime}}\Big{\{}C_{1}({\mu})\,Q_{1}({\mu})\,+C_{2}({\mu})\,Q_{1}({\mu})\Big{\}}+{\rm h.c.},$$

$$V_{cs}^{\ast}V_{ud}\ =\ 1-{\lambda}^{2}-\frac{1}{2}\,A^{2}\,{\lambda}^{4}+\frac{1}{2}\,A^{2}\,{\lambda}^{6}\,\big{\{}1-{\rho}^{2}-{\eta}^{2}-2\,({\rho}-i\,{\eta})\big{\}}+{\cal O}({\lambda}^{8}),$$

$$V_{cs}^{\ast}V_{us}\ =\ {\lambda}-\frac{{\lambda}^{3}}{2}-\frac{{\lambda}^{5}}{8}-\frac{{\lambda}^{7}}{16}-\frac{1}{2}\,A^{2}\,{\lambda}^{5}+\frac{1}{2}\,A^{2}\,{\lambda}^{7}\,\big{\{}\frac{1}{2}-{\rho}^{2}-{\eta}^{2}-2\,({\rho}-i\,{\eta})\big{\}}+{\cal O}({\lambda}^{8}),$$

$$V_{cd}^{\ast }{V}_{ud}=-V_{cs}^{\ast }{V}_{us}-A^2{\lambda }^5(\rho -i\eta )+\mathcal{O}({\lambda }^8),$$

$$V_{cd}^{\ast}V_{us}\ =\ -{\lambda}^{2}+\frac{1}{2}\,A^{2}\,{\lambda}^{6}\,\big{\{}1-2\,({\rho}-i\,{\eta})\big{\}}+{\cal O}({\lambda}^{8}),$$

$$\vec{C}({\mu }_c)=U_4({\mu }_c,m_b)U_5(m_b,m_W)\vec{C}(m_W),$$

$$Q_{1}\,=\,\big{[}\bar{q}_{\alpha}\,{\gamma}_{\mu}\,(1-{\gamma}_{5})\,c_{\alpha}\big{]}\,\big{[}\bar{u}_{\beta}\,{\gamma}^{\mu}\,(1-{\gamma}_{5})\,q^{\prime}_{\beta}\big{]},$$

$$Q_{2}\,=\,\big{[}\bar{q}_{\alpha}\,{\gamma}_{\mu}\,(1-{\gamma}_{5})\,c_{\beta}\big{]}\,\big{[}\bar{u}_{\beta}\,{\gamma}^{\mu}\,(1-{\gamma}_{5})\,q^{\prime}_{\alpha}\big{]},$$

$${\langle}B_{q}M{|}Q_{i}{|}B_{c}^{\ast}{\rangle}\,=\,f_{M}\sum_{j}F^{B_{c}^{\ast}{\to}B_{q}}_{j}{\int}dx\,{\cal H}_{ij}(x)\,{\phi}(x)\,=\,f_{M}\sum_{j}F^{B_{c}^{\ast}{\to}B_{q}}_{j}\,\Big{\{}1+{\alpha}_{s}\,r_{j}+{\cdots}\Big{\}},$$

$${\phi}_{P}(x)\ =\ 6\,x\,\bar{x}\,\Big{\{}1+\sum\limits_{n=1}a_{n}^{P}\,C_{n}^{3/2}(x-\bar{x})\Big{\}},$$

$${\phi}_{V}(x)\,=\,6\,x\,\bar{x}\,\Big{\{}1+\sum\limits_{n=1}a_{n}^{V}\,C_{n}^{3/2}(x-\bar{x})\Big{\}},$$

$$C_1^{3/2}(z)=3z,C_2^{3/2}(z)=\frac{3}{2}(5z^2-1),\cdots$$

$$\mathcal{A}(B_c^{\ast }\to B_{q}M)=⟨B_{q}M|{\mathcal{H}}_{\mathrm{eff}}|B_c^{\ast }⟩=\frac{G_F}{\sqrt{2}}{V}_{cq}^{\ast }{V}_{u{q}^{\prime }}{a}_i⟨M|j^{\mu }|0⟩⟨B_q|j_{\mu }|B_c^{\ast }⟩.$$

$$a_1$$

$$a_2$$

$${\cal V}=6\,{\log}\Big{(}\frac{m_{c}^{2}}{{\mu}^{2}}\Big{)}-18-\Big{(}\frac{1}{2}+i\,3\,{\pi}\Big{)}+\Big{(}\frac{11}{2}-i\,3\,{\pi}\Big{)}\,a_{1}^{M}-\frac{21}{20}\,a_{2}^{M}+{\cdots},$$

$$⟨V(ϵ,p)|{\bar{q}}_1{\gamma }^{\mu }(1-{\gamma }_5)q_2|0⟩=f_V{m}_V{ϵ}^{\ast \mu },$$

$$⟨P(p)|{\bar{q}}_1{\gamma }^{\mu }(1-{\gamma }_5)q_2|0⟩=-i{f}_P{p}^{\mu },$$

$$⟨B_q(p_2)|\bar{q}{\gamma }_{\mu }(1-{\gamma }_5)c|B_c^{\ast }(p_1,ϵ)⟩$$

$$2m_{B_c^{\ast }}{A}_3(q^2)=(m_{B_c^{\ast }}+m_{B_q})A_1(q^2)+(m_{B_c^{\ast }}-m_{B_q})A_2(q^2).$$

$$\mathcal{A}(B_c^{\ast }\to B_{q}V)$$

$${\cal A}(B_{c}^{\ast}{\to}B_{q}P)\,=\,\sqrt{2}\,G_{F}\,m_{B_{c}^{\ast}}\,({\epsilon}_{B_{c}^{\ast}}{\cdot}p_{B_{q}})\,f_{P}\,A_{0}\,V_{cq}^{\ast}V_{uq^{\prime}}\,\big{\{}a_{1}\,{\delta}_{B_{q},B_{d,s}}+a_{2}\,{\delta}_{B_{q},B_{u}}\big{\}}.$$

$$\mathcal{A}(B_c^{\ast }\to B_{q}V)=a({ϵ}_{B_c^{\ast }}\cdot {ϵ}_V^{\ast })+\frac{b}{m_{B_c^{\ast }}{m}_V}({ϵ}_{B_c^{\ast }}\cdot p_V)(p_{B_c^{\ast }}\cdot {ϵ}_V^{\ast })+\frac{ic}{m_{B_c^{\ast }}{m}_V}{ϵ}_{\mu \nu \alpha \beta }{ϵ}_{B_c^{\ast }}^{\mu }{ϵ}_V^{\ast \nu }{p}_{B_c^{\ast }}^{\alpha }{p}_V^{\beta },$$

$$a=\mathcal{F}(m_{B_c^{\ast }}+m_{B_q})A_1,$$

$$b=\mathcal{F}\frac{2m_{B_c^{\ast }}{m}_V}{m_{B_c^{\ast }}+m_{B_q}}{A}_2,$$

$$c=\mathcal{F}\frac{2m_{B_c^{\ast }}{m}_V}{m_{B_c^{\ast }}+m_{B_q}}V,$$

$${\cal F}\,=\,-i\,\frac{G_{F}}{\sqrt{2}}\,m_{V}\,f_{V}\,V_{cq}^{\ast}V_{uq^{\prime}}\,\big{\{}a_{1}\,{\delta}_{B_{q},B_{d,s}}+a_{2}\,{\delta}_{B_{q},B_{u}}\big{\}}.$$

$$H_{\pm }=a\pm c\sqrt{y^2-1},$$

$$H_0=-ay-b(y^2-1),$$

$$y=\frac{p_{B_c^{\ast }}\cdot p_V}{m_{B_c^{\ast }}{m}_V}=\frac{m_{B_c^{\ast }}^2-m_{B_q}^2+m_V^2}{2m_{B_c^{\ast }}{m}_V},$$

$$p_{\mathrm{cm}}=\frac{\sqrt{[m_{B_c^{\ast }}^2-(m_{B_q}+m_V{)}^2][m_{B_c^{\ast }}^2-(m_{B_q}-m_V{)}^2]}}{2m_{B_c^{\ast }}},$$

Full text

I Introduction

The vector $B_{c}^{{\ast}}$ meson, a spin-triplet ground state, consists of two heavy quarks with different flavor numbers $B$ $=$ $C$ $=$ ${\pm}1$, i.e., $\bar{b}c$ for $B_{c}^{{\ast}+}$ meson and $b\bar{c}$ for $B_{c}^{{\ast}-}$ meson. With nonzero bottom and charm numbers, the bottom and charm quarks of the $B_{c}^{\ast}$ meson cannot annihilate into gluons and photons via the strong and electromagnetic interactions, respectively, unlike the decay modes of the unflavored $J/{\psi}(1S)$ and ${\Upsilon}(1S)$ mesons. The $B_{c}^{\ast}$ meson serves as a unique object in studying the heavy quark dynamics, which is inaccessible through both charmonium and bottomonium.

The $B_{c}^{\ast}$ meson lies below the $B_{q}D_{q}$ ($q$ $=$ $u$, $d$, $s$) meson pair threshold. And the mass splitting $m_{B_{c}^{\ast}}$ $-$ $m_{B_{c}}$ ${\approx}$ 50 MeV prd86.094510 is less than the pion mass. Hence, the $B_{c}^{\ast}$ meson decays via the strong interaction are strictly forbidden. The electromagnetic transition process, $B_{c}^{\ast}$ ${\to}$ $B_{c}{\gamma}$, dominates the $B_{c}^{\ast}$ meson decays, but suffers seriously from a compact phase space suppression, which results in a lifetime of ${\tau}_{B_{c}^{\ast}}$ ${\sim}$ ${\cal O}(10^{-17}\,{\rm s})$ epja52.90 . Besides, the $B_{c}^{\ast}$ meson decays via the weak interaction, although with very small decay rates, are allowable within the standard model.

The $B_{c}^{\ast}$ meson has a relatively large mass. In addition, both constituent quarks $b$ and $c$ of the $B_{c}^{\ast}$ meson can decay individually. Therefore, the $B_{c}^{\ast}$ meson has rich weak decay channels. The $B_{c}^{\ast}$ meson weak decays, similar to the pseudoscalar $B_{c}$ meson weak decays zpc51 ; prd49 ; usp38 ; prd77.074013 ; prd89.114019 ; ahep2015.104378 ; qwg , can be divided into three classes: (1) the $c$ quark decay with the spectator $b$ quark, (2) the $b$ quark decay with the $c$ quark as a spectator, and (3) the $b$ and $c$ quarks annihilation into a virtual $W^{\pm}$ boson. This property makes the $B_{c}^{\ast}$ meson another potentially fruitful laboratory for studying the weak decay mechanism of heavy flavor hadrons.

The study of $B_{c}^{\ast}$ weak decays might be interesting, but has not really started yet. One of the major reasons is the extraordinary difficulty of producing the $B_{c}^{\ast}$ meson. The production cross section for the $B_{c}^{\ast}$ meson in hadronic collisions via the dominant process of $g$ $+$ $g$ ${\to}$ $B_{c}^{\ast}$ $+$ $b$ $+$ $\bar{c}$ qwg ; plb355 ; plb364 ; prd54.4344 ; epjc38.267 ; prd72.114009 is at least at the order of ${\alpha}_{s}^{4}$. The nature of QCD’s asymptotic freedom implies a much small possibility of creating two heavy quark pairs ($b\bar{b}$ and $c\bar{c}$) from the vacuum at the ultrahigh energy. Fortunately, the high luminosities of the running LHC and the future Super proton proton Collider (S$pp$C, which is still under discussion today) will promisingly improve this situation. It is expected that a huge amount of the $B_{c}^{\ast}$ data samples would be accumulated, and offer a valuable opportunity to investigate the $B_{c}^{\ast}$ weak decays.

As is well known, there exist some hierarchical structures among the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. The CKM coupling strength for the bottom quark weak decay is proportional to ${|}V_{cb}{|}$ ${\sim}$ ${\cal O}({\lambda}^{2})$ or ${|}V_{ub}{|}$ ${\sim}$ ${\cal O}({\lambda}^{3})$, while the CKM coupling strength for the charm quark weak decay is proportional to ${|}V_{cs}{|}$ ${\sim}$ ${\cal O}(1)$ or ${|}V_{cd}{|}$ ${\sim}$ ${\cal O}({\lambda})$, with the Wolfenstein parameter ${\lambda}$ ${\approx}$ $0.2$ pdg . The $B_{q}$ ($q$ $=$ $u$, $d$, $s$) weak decays are induced dominantly by the bottom quark decay with the phenomenological spectator scheme. The $B_{c}^{\ast}$ ${\to}$ $B_{q}V$, $B_{q}P$ decays are actually induced by the charm quark weak decay, where $V$ and $P$ denote respectively the lightest 9-pelts $SU(3)$ vector and pseudoscalar mesons. With respect to the $B_{q}$ weak decays, the $B_{c}^{\ast}$ ${\to}$ $B_{q}V$, $B_{q}P$ decays are favored by the CKM matrix elements. In this paper, we will study the $B_{c}^{\ast}$ ${\to}$ $B_{u,d,s}V$, $B_{u,d,s}P$ weak decays with the QCD factorization (QCDF) approach prl83.1914 ; npb591.313 ; npb606.245 ; plb488.46 ; plb509.263 ; prd64.014036 ; npb774.64 ; npb832.109 ; plb750.348 , in order to provide an available reference for the future experimental investigation. There is a more than $2.0\,{\sigma}$ discrepancy between the value for CKM matrix element ${|}V_{cs}{|}$ obtained from semileptonic $D$ decays and that from leptonic $D_{s}$ decays111The value for CKM matrix element ${|}V_{cs}{|}$ is ${|}V_{cs}{|}$ $=$ $0.953{\pm}0.008{\pm}0.024$ from semileptonic $D$ decays, and ${|}V_{cs}{|}$ $=$ $1.008{\pm}0.021$ from leptonic $D_{s}$ decays pdg . pdg . The $B_{c}^{(\ast)}$ ${\to}$ $B_{s}V$, $B_{s}P$ decays, together with semileptonic $D$ decays and leptonic $D_{s}$ decays, will provide ${|}V_{cs}{|}$ with more stringent constraints.

In addition, some of the $B_{c}$ weak decays, for example, the $B_{c}$ ${\to}$ $B_{s}{\pi}$ decay prl.111 , have been measured now. One possible background might come from the $B_{c}^{\ast}$ decays, due to a slightly larger production cross section ${\sigma}(B_{c}^{\ast})$ than ${\sigma}(B_{c})$ in hadronic collisions plb364 ; prd54.4344 ; epjc38.267 ; prd72.114009 , and the nearly equal mass $m_{B_{c}^{\ast}}$ ${\simeq}$ $m_{B_{c}}$ prd86.094510 . Hence, the study of the $B_{c}^{\ast}$ ${\to}$ $B_{u,d,s}V$, $B_{u,d,s}P$ decays will be helpful to the experimental analysis on the $B_{c}$ ${\to}$ $B_{u,d,s}V$, $B_{u,d,s}P$ decays.

This paper is organized as follows. The theoretical framework and decay amplitudes will be presented in Section II. Section III is the numerical results and discussion. The last section is a summary.

II theoretical framework

II.1 The effective Hamiltonian

Using the operator product expansion and the renormalization group (RG) method, the low-energy effective weak Hamiltonian describing the $B_{c}^{\ast}$ ${\to}$ $B_{u,d,s}V$, $B_{u,d,s}P$ decays has the following general structure 9512380 ,

[TABLE]

where the Fermi coupling constant $G_{F}$ ${\simeq}$ $1.166{\times}10^{-5}\,{\rm GeV}^{-2}$ pdg ; $V_{cq}^{\ast}V_{uq^{\prime}}$ is a product of the CKM matrix elements. Using the Wolfenstein parameterization, there are pdg

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where the values for these Wolfenstein parameters $A$, ${\lambda}$, ${\rho}$ and ${\eta}$ are given in Table 3.

The renormalization scale ${\mu}$ separates the physical contributions into two parts. The hard contributions above the scale ${\mu}$ are summarized into the Wilson coefficients $C_{i}(\mu)$. With the RG equation for $C_{i}(\mu)$, the Wilson coefficients at an appropriate scale ${\mu}_{c}$ ${\sim}$ ${\cal O}(m_{c})$ for the charm quark decay are given by 9512380

[TABLE]

where $m_{W}$, $m_{b}$ and $m_{c}$ are the mass of the $W$ boson, $b$ quark and $c$ quark, respectively. Here $U_{f}(m_{2},m_{1})$ denotes the RG evolution matrix for $f$ active flavors. The initial values for the Wilson coefficients $\vec{C}(m_{W})$ at scale ${\mu}_{W}$ $=$ $m_{W}$ to a desired order in ${\alpha}_{s}$ can be calculated with perturbation theory. The expressions for the RG evolution matrix $U_{f}(m_{2},m_{1})$ and Wilson coefficients $\vec{C}(m_{W})$, including both leading order (LO) and next-to-leading order (NLO) corrections, have been presented in Ref.9512380 . The contributions below the scale ${\mu}$ are included in the hadronic matrix elements (HME) where the local four-quark operators $Q_{i}$ are sandwiched between the initial and final states. The expressions for the four-quark operators in question are

[TABLE]

[TABLE]

where the subscripts ${\alpha}$ and ${\beta}$ are color indices. It should be pointed out that (1) because the contributions from the penguin operators and annihilation topologies are proportional to the CKM factor $V_{cb}^{\ast}V_{ub}$ ${\sim}$ ${\cal O}({\lambda}^{5})$ and therefore negligible in the actual calculation of branching ratio prd89.114019 , only the contributions of tree operators are considered here. (2) The participation of the strong interaction, especially, the nonperturbative QCD effects, makes the theoretical treatment of HME very complicated. The main problem at this stage is how to effectively factorize HME into hard and soft parts, and how to evaluate HME properly.

II.2 Hadronic matrix elements

Hadronic matrix elements might be the most intricate part in the calculation of heavy flavor weak decay, due to the entanglement of perturbative and nonperturbative contributions. Phenomenologically, one has to turn to some approximation and assumption, which bring uncertainties and model dependence to theoretical predictions. A simple approximation is the naive factorization ansatz (NF) according to Bjorken’s color transparency argument, which says that the colorless energetic hadron has flown away from the weak interaction point during the formation time of the emission hadron npb11.325 . With the NF approach, HME is parameterized as a product of decay constants and hadron transition form factors plb73.418 ; npb133.315 ; zpc29.637 ; zpc34.103 . A major flaw of the NF approach is the disappearance of scale dependence and strong phases from HME, which results directly in a scale-sensitive nonphysical prediction and none of $CP$ violation for nonleptonic meson weak decays. In order to overcome these shortcomings of the NF approach, nonfactorizable contributions to HME should be carefully considered, as commonly recognized. Some QCD-inspired models, such as, the QCDF approach prl83.1914 ; npb591.313 ; npb606.245 ; plb488.46 ; plb509.263 ; prd64.014036 ; npb774.64 ; npb832.109 ; plb750.348 , the soft and collinear effective theory prd63.014006 ; prd63.114020 ; plb516.134 ; prd65.054022 ; prd66.014017 ; npb643.431 ; plb553.267 ; npb685.249 , the perturbative QCD approach pqcd1 ; pqcd2 ; pqcd3 , and so on, have been developed recently, based on the Lepage-Brodsky treatment on exclusive processes prd22 and some power counting rules in the expansion in ${\alpha}_{s}$ and ${\Lambda}_{\rm QCD}/m_{Q}$, where ${\alpha}_{s}$ is the strong coupling, ${\Lambda}_{\rm QCD}$ is the QCD characteristic scale, and $m_{Q}$ is the mass of a heavy quark. In these QCD-inspired models, HME is generally written as a convolution integral of hadron’s distribution amplitudes (DAs) and hard rescattering kernels. A virtue of the QCDF approach is that the NF’s result can be reproduced, if both the nonfactorizable contributions and the power suppressed contributions are neglected prl83.1914 ; npb591.313 ; npb606.245 ; plb488.46 ; plb509.263 ; prd64.014036 .

For the $B_{c}^{\ast}$ ${\to}$ $B_{q}V$, $B_{q}P$ decays ($q$ $=$ $u$, $d$, $s$), the spectator quark is a heavy quark — the bottom quark. It is generally assumed that the bottom quark in both the $B_{c}^{\ast}$ and $B_{q}$ mesons is nearly on shell, and that the gluon exchanged between the heavy spectator quark and other quarks is soft. The virtuality of emission gluon from the spectator quark is of order ${\Lambda}_{\rm QCD}^{2}$. The contributions of spectator scattering are power suppressed relative to the leading order contributions npb591.313 . In addition, it is supposed that the recoiled $B_{q}$ meson should move slowly in the rest frame of the $B_{c}^{\ast}$ meson. There should be a large overlap between the $B_{c}^{\ast}$ and $B_{q}$ mesons. The recoiled $B_{q}$ meson cannot be clearly factorized from the $B_{c}^{\ast}B_{q}$ system due to the soft and nonperturbative contributions. The $B_{c}^{\ast}B_{q}$ system should be parameterized by some physical from factors. Hence, with the QCDF approach, up to leading power corrections of order ${\Lambda}_{\rm QCD}/m_{Q}$, hadronic matrix elements have the following structure npb591.313 ,

[TABLE]

where $f_{M}$ is the decay constant for the light final $M$ (${\equiv}$ $V$ and $P$) meson; $F^{B_{c}^{\ast}{\to}B_{q}}_{j}$ is a transition form factor; ${\cal H}_{ij}(x)$ is a hard rescattering kernel; ${\phi}(x)$ is a DA of parton momentum fraction $x$. For the light pseudoscalar $P$ and longitudinally polarized vector $V$ mesons, the leading twist DAs are expanded in terms of the Gegenbauer polynomials jhep.0605.004 ; jhep.0703.069

[TABLE]

[TABLE]

where $\bar{x}$ $=$ $1$ $-$ $x$; $a_{n}^{P,V}$ is a nonperturbative parameter, also called the Gegenbauer moment. The expressions for the Gegenbauer polynomials $C_{n}^{3/2}(z)$ are

[TABLE]

II.3 Decay amplitudes

The typical Feynman diagrams for the $B_{c}^{{\ast}}$ ${\to}$ $B_{s}{\pi}$ decay within the QCDF framework are shown in Fig.1, where no hard gluons are exchanged between the spectator quark and other partons. There is no gluon exchange in factorizable topology of Fig.1(a), so the emitted hadron matrix element is entirely separated from that of the $B_{c}^{{\ast}}B_{s}$ system. In this approximation, the hard rescattering kernel ${\cal H}_{ij}$ $=$ $1$ and the integral in Eq.(9) reduces to the normalization condition for distribution amplitude. According to the QCDF power counting rules, the leading order contributions come from the factorizable topology of Fig.1(a), and recover the NF’s results at the order of ${\alpha}_{s}^{0}$. For the radiative correction diagrams in Fig.1(b-e), hard gluons are exchanged between the emission meson and the $B_{c}^{{\ast}}B_{s}$ system. The hard rescattering kernel ${\cal H}_{ij}$ and $x$-integral in Eq.(9) are nontrivial. It has already been shown npb591.313 ; npb606.245 ; plb488.46 ; plb509.263 ; prd64.014036 that although both collinear and soft divergences exist for each of diagrams in Fig.1(b-e), infrared divergences cancel after summing up the vertex corrections. The strong phases could then come from HME. The renormalization scale ${\mu}$ dependence of HME is recuperated from the nonfactorizable contributions, which will reduce partly the ${\mu}$-dependence of Wilson coefficients.

After a straightforward calculation using the QCDF master formula Eq.(9), the amplitudes for the $B_{c}^{{\ast}}$ ${\to}$ $B_{q}M$ decays ($q$ $=$ $u$, $d$, $s$) are written as

[TABLE]

With the naive dimensional regularization scheme, the effective coefficients are npb591.313 ; npb606.245 ; plb488.46 ; plb509.263 ; prd64.014036 :

[TABLE]

[TABLE]

where $N_{c}$ $=$ $3$ and $C_{F}$ $=$ $4/3$; $C_{1,2}^{\rm NLO,LO}$ are Wilson coefficients containing NLO or LO contributions; $a_{i}^{M}$ is a Gegenbauer moment. For the transversely polarized vector meson, the vertex factor ${\cal V}$ $=$ [math] beyond the leading twist DAs. For convenience, the numerical values for $a_{1,2}$ of the $B_{c}^{\ast}$ ${\to}$ $B_{q}{\pi}$ decay are listed in Table 1.

There are some comments on the coefficients $a_{1,2}$. (1) The first two terms on the right hand side of Eq.(14) and Eq.(15) correspond to the leading order contributions. The third terms correspond to nonfactorizable contributions. The NF scenario follows when one neglects the nonfactorizable contributions, i.e., ${\cal V}$ $=$ [math]. (2) Nonfactorizable vertex corrections to HME are of order ${\alpha}_{s}$. They include the dependence on the renormalization scale. It is shown prd64.014036 that with the RG equations for the Wilson coefficients at leading order logarithm approximation, one can obtain $\displaystyle{\mu}\frac{{\rm d}}{{\rm d}{\mu}}a_{1,2}\,=\,0$. In principle, the residual scale dependence could be compensated by higher order corrections to HME. (3) Compared with the LO contributions, nonfactorizable contributions are generally suppressed by ${\alpha}_{s}$ and the factor $1/N_{c}$ (see Eq.(14) and Eq.(15)). Because the LO contributions of $a_{2}$ are color-suppressed, vertex corrections multiplied by the large Wilson coefficient $C_{1}^{\rm LO}$ could be sizable to branching rates of the $a_{2}$-dominated heavy flavor decays. The coefficients $a_{1,2}$ contain strong phases via the imaginary parts of vertex corrections. Correspondingly, strong scattering phase of $a_{1}$ ($a_{2}$) is small (large). This argument is also confirmed by the numerical results for $a_{1,2}$ in Table 1. (4) With the QCDF approach, nonfactorizable radiative corrections to HME occur first at order ${\alpha}_{s}$ as well as the leading strong phases at order ${\alpha}_{s}$. In addition, it should be pointed out that nonfactorizable power corrections beyond leading order are neglected here. For the charm quark decay, power ${\Lambda}_{\rm QCD}/m_{c}$ is comparable to ${\alpha}_{s}$. The strong phases due to soft (hard) interactions are of order ${\Lambda}_{\rm QCD}/m_{c}$ (${\alpha}_{s}$). One should not expect these phases to have great precision, as stated in Ref.npb591.313 . (5) With the QCDF approach, the values for $a_{1,2}$ are close to those for the charm quark decay npb268.16 ; plb252.690 ; ijmpa14.937 ; epjc55.607 ; prd81.074021 , ${|}a_{1,2}{|}$ ${\approx}$ ${|}C_{1,2}{|}$, and basically consistent with those of the large-$N_{c}$ approach npb268.16 .

The hadronic matrix elements of diquark current operators are defined as zpc29.637 :

[TABLE]

[TABLE]

[TABLE]

where $f_{V}$ and $f_{P}$ are the decay constants of vector $V$ and pseudoscalar $P$ mesons, respectively; $q$ $=$ $p_{1}$ $-$ $p_{2}$; ${\epsilon}$ is the polarization vector of vector mesons; $V(q^{2})$ and $A_{0,1,2,3}(q^{2})$ are the $B_{c}^{\ast}$ ${\to}$ $B_{q}$ transition form factors. To eliminate singularities at the pole of $q^{2}$ $=$ [math], a relation, $A_{0}(0)$ $=$ $A_{3}(0)$, is required, with $A_{3}(q^{2})$ given by zpc29.637 :

[TABLE]

In the bottom conservation transition $B_{c}^{\ast}$ ${\to}$ $B_{q}$, both the initial and final mesons contain a heavy bottom quark. After a sudden kick, the $B_{q}$ meson would move slowly, even remain nearly intact, with respect to the $B_{c}^{\ast}$ meson. Therefore, the zero-recoil configuration ($q^{2}$ $=$ [math]) would be a good approximation. Simultaneously, the emission meson would take up most of the energy available and fly rapidly away from the interaction point. This fact not only reproduces the NF scenario (Fig.1(a)) but also requires the exchanged gluon in vertex corrections (Fig.1(b-e)) to be hard. Due to the large virtuality of gluon exchanged between the emitted light meson and the $B_{c}^{\ast}B_{q}$ system, perturbative calculation of nonfactorizable vertex corrections with the QCDF approach should be applicable and reliable.

With the form factors given above, the decay amplitudes are expressed as

[TABLE]

[TABLE]

The $B_{c}^{\ast}$ ${\to}$ $B_{q}V$ decay amplitude is a sum of $S$-, $P$-, $D$-wave amplitudes prd39.3339 ; prd45.193 , i.e.,

[TABLE]

with $a$, $b$, $c$, the $S$-, $D$- and $P$-wave amplitudes respectively, in the notation of prd45.193 ,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

From the above expressions, one can find that the $P$- and $D$-wave amplitudes are suppressed by a factor of $\frac{2\,m_{B_{c}^{\ast}}\,m_{V}}{(m_{B_{c}^{\ast}}+m_{B_{q}})^{2}}$ relative to the $S$-wave amplitude. The relations among the helicity amplitudes and the $S$-, $P$-, $D$-wave amplitudes are prd45.193

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $p_{\rm cm}$ is the common momentum of final states in the rest frame of the $B_{c}^{\ast}$ meson.

We assume that the vector mesons are ideally mixed in the singlet-octet basis, i.e., ${\phi}$ $=$ $s\bar{s}$ and ${\omega}$ $=$ $(u\bar{u}+d\bar{d})/\sqrt{2}$. As for the pseudoscalar ${\eta}$ and ${\eta}^{\prime}$ mesons, they are usually written as a linear superposition of states in either flavor basis or the singlet-octet basis. Here, we adopt the quark flavor basis description proposed in Ref. prd.58.114006 , i.e.,

[TABLE]

where ${\eta}_{q}$ $=$ $(u\bar{u}+d\bar{d})/{\sqrt{2}}$ and ${\eta}_{s}$ $=$ $s\bar{s}$; the mixing angle ${\phi}$ ${\approx}$ $(39.3{\pm}1.0)^{\circ}$ prd.58.114006 . Due to the symmetric flavor configurations of both ${\eta}_{q}$ and ${\eta}_{s}$ states, we assume that DAs for ${\eta}_{q}$ and ${\eta}_{s}$ states are similar to DAs for pion. It should be pointed out that the contributions from possible $c\bar{c}$ and gluonium compositions are not considered in our calculation for the moment, because (1) the final states with $B_{q}$ meson and $c\bar{c}$ or gluonium states lie above the $B_{c}^{\ast}$ meson mass; (2) the fraction of gluonium components in ${\eta}$ and ${\eta}^{\prime}$ is rather tiny jhep0705.006 . Thus, the amplitudes for the $B_{c}^{\ast}$ ${\to}$ $B_{u}{\eta}$, $B_{u}{\eta}^{\prime}$ decays are written as

[TABLE]

II.4 Form factors

The hadron transition form factors are the basic input parameters for decay amplitudes [see Eq.(21) and Eq.(22)]. It is assumed npb591.313 that form factors come mainly from soft contributions, and form factors are generally regarded as nonperturbative parameters in the QCDF master formula of Eq.(9). Fortunately, form factors are universal. Form factors determined by other means or extracted from data can be employed here to make predictions. Phenomenologically, form factors are written as overlap integrals of wave functions.

Here, we will employ the Wirbel-Stech-Bauer model zpc29.637 for evaluating the form factors. With a factorization of spin and spatial motion, wave function is written as

[TABLE]

where $\vec{k}_{\perp}$ and $x$ are the transverse momentum and longitudinal momentum fraction, respectively; $j$ ($s$) is the total angular momentum (spin); $j_{z}$ ($s_{z}$) is the magnetic quantum number; $s_{1}$ and $s_{2}$ are spins of valence quarks. $j$ $=$ $s$ $=$ $1$ for the ground vector $B_{c}^{\ast}$ meson, and [math] for the ground pseudoscalar $B_{u,d,s}$ meson. The spatial wave function of a relativistic scalar harmonic oscillator potential is given by zpc29.637

[TABLE]

where parameter ${\omega}$ determines the average transverse momentum of partons, i.e., ${\langle}\vec{k}_{\perp}^{2}{\rangle}$ $=$ ${\omega}^{2}$; $m$ is the mass of the concerned meson; $m_{q_{1}}$ ($m_{q_{2}}$) is the constituent mass of the decaying (spectator) quark carrying a gluon cloud; $N_{m}$ is a normalization factor determined by

[TABLE]

The form factors at zero momentum transfer are given by zpc29.637

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where ${\sigma}_{z,y}^{(1)}$ are Pauli matrixes acting on the spin indices of the decaying quark $q_{1}$.

It has been shown zpc29.637 that the form factors are sensitive to the choice of parameter ${\omega}$. And it is argued zpc29.637 that parameter ${\omega}$ is not expected to be largely different for various mesons due to the flavor independence of the QCD interactions. Thus the same ${\omega}$ might be applied to all mesons with the same spectator quark. The motion of the spectator (bottom) quark is nearly nonrelativistic in the $B_{c}^{\ast}$ ${\to}$ $B_{q}$ transition. Thus, nonrelativistic QCD (NRQCD) effective theory prd46.4052 ; prd51.1125 ; rmp77.1423 could be used to deal with both $B_{c}^{\ast}$ and $B_{q}$ mesons. According to the NRQCD power counting rules, the average transverse momentum is the order of ${\omega}$ ${\approx}$ $m\,{\alpha}_{s}$. In order to see the parameter ${\omega}$ effects on the form factors, we explore two scenarios. One is the same parameter ${\omega}$ for both the $B_{c}^{\ast}$ and $B_{q}$ mesons, and the other is ${\omega}$ $=$ $m\,{\alpha}_{s}$, i.e., ${\omega}$ ${\approx}$ $1.24$ GeV for the $B_{c}^{\ast}$ meson, $1.10$ GeV for the $B_{s}$ meson, and $1.09$ GeV for the $B_{u,d}$ mesons. The numerical results for form factors are shown in Table 2.

There are some comments on the form factors. (1) From the expressions in Eq.(41) and Eq.(42), it is seen that due to the factor $\frac{m_{c}-m_{q}}{m_{B_{c}^{\ast}}-m_{B_{q}}}$ ${\approx}$ $1$ and $\frac{m_{c}+m_{q}}{m_{B_{c}^{\ast}}+m_{B_{q}}}$ ${\ll}$ $1$, one can obtain a relation, $A_{1}(0)$ $<$ $V(0)$. (2) Compared with the integrand in Eq.(39), there is a factor $1/x$ for the integrand in Eq.(40) with longitudinal momentum fraction [math] $<$ $x$ $<$ $1$. Thus, it is expected to have in general $A_{0,3}(0)$ $<$ $V(0)$. (3) With the relation of form factors in Eq.(20), $A_{2}(0)$ is significantly enhanced by a factor of $\frac{2\,m_{B_{c}^{\ast}}}{m_{B_{c}^{\ast}}-m_{B_{q}}}$ (or $\frac{m_{B_{c}^{\ast}}+m_{B_{q}}}{m_{B_{c}^{\ast}}-m_{B_{q}}}$) relative to $A_{3}(0)$ (or $A_{1}(0)$). These relations are comprehensively verified by the numerical results for form factors in Table 2.

In addition, from the numbers in Table 2, it is seen that (1) the form factors increase as parameter ${\omega}$ increases, due to the fact that the overlap between wave functions of $B_{c}^{\ast}$ and $B_{q}$ mesons increases as parameter ${\omega}$ increases, as shown in Fig.2. (2) The flavor symmetry breaking effects on form factors are small, but the isospin symmetry is basically held. (3) The values for $A_{2}(0)$ ($V(0)$) are about ten (five) times as large as those for $A_{1}(0)$, as explained above. The large values for $A_{2}$ and $V$ would enhance the contributions from the $D$- and $P$-wave amplitudes (see Eq.(25) and Eq.(26)).

III Numerical results and discussion

In the rest frame of the $B_{c}^{\ast}$ meson, branching ratios are defined as

[TABLE]

[TABLE]

where ${\Gamma}_{B_{c}^{\ast}}$ is the full width of the $B_{c}^{\ast}$ meson.

Because the electromagnetic radiation process $B_{c}^{\ast}$ ${\to}$ $B_{c}{\gamma}$ dominates the $B_{c}^{\ast}$ meson decay, to a good approximation, ${\Gamma}_{B_{c}^{\ast}}$ ${\simeq}$ ${\Gamma}(B_{c}^{\ast}{\to}B_{c}{\gamma})$. However, there is still no experimental information about the partial width ${\Gamma}(B_{c}^{\ast}{\to}B_{c}{\gamma})$ now, because the photon from the $B_{c}^{\ast}$ ${\to}$ $B_{c}{\gamma}$ process is too soft to be easily identified. The information on ${\Gamma}(B_{c}^{\ast}{\to}B_{c}{\gamma})$ comes mainly from theoretical estimation on the magnetic dipole (M1) transition, i.e., epja52.90

[TABLE]

where ${\alpha}_{\rm em}$ is the fine-structure constant of electromagnetic interaction; $k_{\gamma}$ is the photon momentum in the rest frame of initial state; ${\mu}_{h}$ is the M1 moment of $B_{c}^{\ast}$ meson. There are plenty of theoretical predictions on ${\Gamma}(B_{c}^{\ast}{\to}B_{c}{\gamma})$, for example, the numbers in Tables 3 and 6 in Ref.epja52.90 . However, these estimations still suffer from large uncertainties due to our lack of a precise value for ${\mu}_{h}$. To give a quantitative evaluation, ${\Gamma}_{B_{c}^{\ast}}$ $=$ $50$ eV will be fixed in our calculation for the moment. The value of $50$ eV seems reasonable since it is close to the value given by the potential model (PM) which produces good agreement with experiment for the measured $J/{\psi}$ ${\to}$ ${\eta}_{c}{\gamma}$ decay rate. The value for the charm quark magnetic moment ${\mu}_{c}$ obtained from the charmonium M1 decay width can now be used to predict the $B_{c}^{\ast}$ ${\to}$ $B_{c}{\gamma}$ decay width, with a very small $b$ quark magnetic moment ${\mu}_{b}$ $=$ $-0.06\,{\mu}_{N}$ given in Ref.epja52.90 .

The numerical values for other input parameters are listed in Table 3. Unless otherwise stated, their central values will be fixed as the default inputs. Our numerical results are presented in Table 4. The following are some comments.

(1) According to the relative sizes of coefficients $a_{1,2}$ and CKM factors, the $B_{c}^{\ast}$ ${\to}$ $B_{q}V$, $B_{q}P$ decays could be classified into six cases (see Table 4). There is a clear hierarchical relation among branching ratios, i.e., ${\cal B}r(\text{case 1-I})$ ${\sim}$ ${\cal O}(10^{-7})$, ${\cal B}r(\text{case 1-II})$ ${\sim}$ ${\cal O}(10^{-8})$, ${\cal B}r(\text{case 1-III})$ ${\sim}$ ${\cal O}(10^{-9})$, and ${\cal B}r(\text{case 2-I})$ ${\sim}$ ${\cal O}(10^{-8})$, ${\cal B}r(\text{case 2-II})$ ${\sim}$ ${\cal O}(10^{-9})$, ${\cal B}r(\text{case 2-III})$ ${\sim}$ ${\cal O}(10^{-10})$.

(2) Branching ratios for the $B_{c}^{\ast}$ ${\to}$ $B_{q}V$ decays are generally larger than those for the $B_{c}^{\ast}$ ${\to}$ $B_{q}P$ decays with the same final $B_{q}$ meson, where $V$ and $P$ have the same quark components. There are two reasons for this. One is the decay constant relation $f_{V}$ $>$ $f_{P}$, and the other is three partial wave contributions to the $B_{c}^{\ast}$ ${\to}$ $B_{q}V$ decays rather than only the $P$-wave contributions to the $B_{c}^{\ast}$ ${\to}$ $B_{q}P$ decays.

It should be pointed out that although the $P$- and $D$-wave amplitudes for the $B_{c}^{\ast}$ ${\to}$ $B_{q}V$ decays are enhanced by large values for the form factors $V$ and $A_{2}$, they are suppressed by a factor of $\frac{2\,m_{B_{c}^{\ast}}\,m_{V}}{(m_{B_{c}^{\ast}}+m_{B_{q}})^{2}}$ relative to the $S$-wave amplitude, as discussed above. In addition, the $P$- and $D$-wave contributions to helicity amplitudes $H_{\pm}$ in Eq.(28) and $H_{0}$ in Eq.(29) are future suppressed respectively by factors of $\sqrt{y^{2}-1}$ and $(y^{2}-1)/y$ relative to the $S$-wave contribution. Take the $B_{c}^{\ast}$ ${\to}$ $B_{s}{\rho}$ decay for example, $\frac{2\,m_{B_{c}^{\ast}}\,m_{\rho}}{(m_{B_{c}^{\ast}}+m_{B_{s}})^{2}}$ ${\approx}$ 7%, $\sqrt{y^{2}-1}$ ${\approx}$ $0.7$ and $(y^{2}-1)/y$ ${\approx}$ $0.4$, resulting in the polarization fractions $f_{0}$ ${\approx}$ 60%, $f_{+}$ ${\approx}$ 30% and $f_{-}$ ${\approx}$ 10% with $f_{0,+,-}$ ${\equiv}$ $\frac{{|}H_{0,+,-}{|}^{2}}{{|}H_{0}{|}^{2}+{|}H_{+}{|}^{2}+{|}H_{-}{|}^{2}}$.

(3) The branching ratios for the $B_{c}^{\ast}$ ${\to}$ $B_{s}{\rho}$, $B_{s}{\pi}$ decays can reach up to ${\cal O}(10^{-7})$. With the estimated production cross section of the $B_{c}^{\ast}$ meson ${\sim}$ $30$ $nb$ at LHCprd72.114009 , it is expected to have more than $10^{10}$ $B_{c}^{\ast}$ mesons per ${\rm ab}^{-1}$ data at LHC, corresponding to more than $10^{3}$ events of the $B_{c}^{\ast}$ ${\to}$ $B_{s}{\rho}$, $B_{s}{\pi}$ decays. Therefore, even with the identification efficiency, the $B_{c}^{\ast}$ ${\to}$ $B_{s}{\rho}$, $B_{s}{\pi}$ decays might be measurable in the future.

(4) Branching ratios for the $B_{c}^{\ast}$ ${\to}$ $B_{q}V$, $B_{q}P$ decays are several orders of magnitude smaller, especially for the $a_{1}$ dominant decays, than those for the $B_{c}$ ${\to}$ $B_{q}P$, $B_{q}V$ decays ahep2015.104378 . This fact might imply that possible background from the $B_{c}^{\ast}$ ${\to}$ $BV$, $BP$ decays could be safely neglected for an analysis of the $B_{c}$ ${\to}$ $B_{q}P$, $B_{q}V$ decays, but not vice versa, i.e., one of main pollution for the $B_{c}^{\ast}$ ${\to}$ $B_{q}V$, $B_{q}P$ decays would likely come from the $B_{c}$ decays.

(5) It is seen clearly that the numbers in Table 4 are very sensitive to the choice of the parameter ${\omega}$. In addition, with a different value for ${\Gamma}_{B_{c}^{\ast}}$, branching ratios in Table 4 should be multiplied by a factor of ${50\,{\rm eV}}/{{\Gamma}_{B_{c}^{\ast}}}$. Of course, many factors, such as the choice of scale ${\mu}$, higher order corrections to HME, $q^{2}$-dependence of form factors, final state interactions, etc., are not carefully considered in detail here, but have effects on the estimation and deserve more dedicated study in the future.

IV Summary

With the running and upgrading of the LHC, there are certainly huge amounts of the $B_{c}^{\ast}$ mesons. This would provide us with a possibility of searching for the $B_{c}^{\ast}$ weak decays in the future. In this paper, the $B_{c}^{\ast}$ ${\to}$ $B_{q}V$, $B_{q}P$ decays ($q$ $=$ $u$, $d$ and $s$), induced by the charm quark weak decay, are studied phenomenologically with the QCDF approach. The form factors for the $B_{c}^{\ast}$ ${\to}$ $B$ transitions are calculated using the Wirbel-Stech-Bauer model. The nonfactorizable contributions from the vertex radiative corrections are considered at the order of ${\alpha}_{s}$. It is found that (1) form factors and branching ratios are sensitive to models of wave functions; (2) the color-favored and CKM-allowed $B_{c}^{\ast}$ ${\to}$ $B_{s}{\rho}$, $B_{s}{\pi}$ decays have large branching ratios of ${\cal O}(10^{-7})$, and might be accessible in the future LHC experiments.

Acknowledgments

The work is supported by the National Natural Science Foundation of China (Grant Nos. U1632109, 11547014 and 11475055). We thank the referees for their constructive suggestions, and Ms. Nan Li (HNU) for polishing this manuscript.