Two-body charmed baryon decays involving vector meson with $SU(3)$ flavor symmetry

TL;DR

This paper uses $SU(3)$ flavor symmetry to predict branching ratios of two-body charmed baryon decays involving vector mesons, aligning well with existing data and providing new predictions for unmeasured modes.

Contribution

It offers the first theoretical predictions for several charmed baryon decay branching ratios involving vector mesons using $SU(3)$ symmetry.

Findings

Predicted branching ratios for $\Lambda_c^+$ decays agree with experimental bounds.

Provided new estimates for $\Xi_c^+$ decay modes, including the first measurement of $\Xi_c^+ o p \phi$.

Decays with branching ratios around $10^{-4}$ to $10^{-3}$ are within experimental reach.

Abstract

We study the two-body anti-triplet charmed baryon decays of ${\bf B}_c\to {\bf B}_n V$, with ${\bf B}_c=(\Xi_c^{0},-\Xi_c^{+},\Lambda_c^+)$ and ${\bf B}_n(V)$ the baryon (vector meson) states. Based on the $SU(3)$ flavor symmetry, we predict that ${\cal B}(\Lambda^{+}_{c}\to \Sigma^{+}\rho^{0},\Lambda^0 \rho^+)=(0.61\pm 0.46,0.74\pm 0.34)\%$, in agreement with the experimental upper bounds of $(1.7,6)\%$, respectively. We also find ${\cal B}(\Lambda^+_c \to \Xi^0 K^{*+},\Sigma^0 K^{*+},\Lambda^0 K^{*+}) =(8.7 \pm 2.7,1.2\pm 0.3,2.0\pm 0.5)\times 10^{-3}$ to be compatible with the pseudoscalar counterparts. For the doubly Cabibbo-suppressed decay $\Xi^+_c \to p\phi$, measured for the first time, we predict its branching ratio to be $(1.5\pm 0.7)\times 10^{-4}$, together with ${\cal B}(\Xi^+_c \to p \bar K^{*0},\Sigma^+ \phi) =(7.8 \pm 2.2,1.9\pm0.9)\times 10^{-3}$. The ${\bf B}_c\to{\bf B}_n V$ decays with ${\cal B}\simeq {\cal O}(10^{-4}-10^{-3})$ are accessible to the BESIII, BELLEII and LHCb experiments.

Tables

Table 1. Table 1: The T 𝑇 T -amps for the 𝐁 c → 𝐁 n ​ V → subscript 𝐁 𝑐 subscript 𝐁 𝑛 𝑉 {\bf B}_{c}\to{\bf B}_{n}V decays, where CF denotes the Cabibbo-favored processes, while SCS (DCS) the singly (doubly) Cabibbo-suppressed ones.

${\Xi }_c^0$	CF $T$-amp
${\Sigma }^{+}{K}^{\ast -}$	$2\left({\overline{a}}_2+\frac{{\overline{a}}_4+{\overline{a}}_7}{2}\right)$
${\Sigma }^0{\overline{K}}^{\ast 0}$	$-\sqrt{2}({\overline{a}}_2+{\overline{a}}_3$ $-\frac{{\overline{a}}_6-{\overline{a}}_7}{2})$
${\Xi }^0{\rho }^0$	$-\sqrt{2}({\overline{a}}_1-{\overline{a}}_3$ $-\frac{{\overline{a}}_4-{\overline{a}}_5}{2})$
${\Xi }^0\omega$	$\sqrt{2}({\overline{a}}_1-{\overline{a}}_3+2\overline{h}$ $+\frac{{\overline{a}}_4+{\overline{a}}_5+2{\overline{h}}^{\prime }}{2})$
${\Xi }^0\varphi$	${\overline{a}}_2+\overline{h}+\frac{{\overline{a}}_7+{\overline{h}}^{\prime }}{2}$
${\Xi }^{-}{\rho }^{+}$	$2\left({\overline{a}}_1+\frac{{\overline{a}}_5+{\overline{a}}_6}{2}\right)$
${\Lambda }^0{\overline{K}}^{\ast 0}$	$-\sqrt{\frac{2}{3}}(2{\overline{a}}_1-{\overline{a}}_2-{\overline{a}}_3$
	$+\frac{2{\overline{a}}_5-{\overline{a}}_6-{\overline{a}}_7}{2})$
${\Xi }_c^{+}$	CF $T$-amp
${\Sigma }^{+}{\overline{K}}^{\ast 0}$	$-2\left({\overline{a}}_3-\frac{{\overline{a}}_4+{\overline{a}}_6}{2}\right)$
${\Xi }^0{\rho }^{+}$	$2\left({\overline{a}}_3+\frac{{\overline{a}}_4+{\overline{a}}_6}{2}\right)$
${\Lambda }_c^{+}$	CF $T$-amp
${\Sigma }^0{\rho }^{+}$	$-\sqrt{2}({\overline{a}}_1-{\overline{a}}_2-{\overline{a}}_3$ $-\frac{{\overline{a}}_5-{\overline{a}}_7}{2})$
${\Sigma }^{+}{\rho }^0$	$\sqrt{2}({\overline{a}}_1-{\overline{a}}_2-{\overline{a}}_3$ $-\frac{{\overline{a}}_5-{\overline{a}}_7}{2})$
${\Sigma }^{+}\omega$	$\sqrt{2}(-{\overline{a}}_1-{\overline{a}}_2+{\overline{a}}_3-2\overline{h}$
	$+\frac{{\overline{a}}_5+{\overline{a}}_7+2{\overline{h}}^{\prime }}{2})$
${\Sigma }^{+}\varphi$	${\overline{a}}_4-2\overline{h}+{\overline{h}}^{\prime }$
${\Xi }^0{K}^{\ast +}$	$-2\left({\overline{a}}_2-\frac{{\overline{a}}_4+{\overline{a}}_7}{2}\right)$
$p{\overline{K}}^{\ast 0}$	$-2\left({\overline{a}}_1-\frac{{\overline{a}}_5+{\overline{a}}_6}{2}\right)$
${\Lambda }^0{\rho }^{+}$	$-\sqrt{\frac{2}{3}}({\overline{a}}_1+{\overline{a}}_2+{\overline{a}}_3$
	$-\frac{{\overline{a}}_5-2{\overline{a}}_6+{\overline{a}}_7}{2})$

Table 2. Table 2: The (ratios of) branching fractions of the 𝐁 c → 𝐁 n ​ V → subscript 𝐁 𝑐 subscript 𝐁 𝑛 𝑉 {\bf B}_{c}\to{\bf B}_{n}V decays. In column 2, the numbers are calculated with the extracted parameters, in comparison with the initial experimental inputs in column 3.

(Ratio of) Branching fraction	This work	Data
${10}^2ℬ\left({\Lambda }_c^{+}\to p{\overline{K}}^{\ast 0}\right)$	$1.9\pm 0.3$	$1.94\pm 0.27$ pdg
${10}^2ℬ\left({\Lambda }_c^{+}\to {\Sigma }^{+}\omega \right)$	$1.6\pm 0.7$	$1.69\pm 0.21$ pdg
${10}^3ℬ\left({\Lambda }_c^{+}\to {\Sigma }^{+}\varphi \right)$	$3.9\pm 0.6$	$3.8\pm 0.6$ pdg
$ℛ\left({\Lambda }_c^{+}\right)=\frac{ℬ\left({\Lambda }_c^{+}\to {\Sigma }^{+}{\rho }^0\right)}{ℬ\left({\Lambda }_c^{+}\to {\Sigma }^{+}\omega \right)}$	$0.4\pm 0.3$	$0.3\pm 0.2$ Kubota:1993pw
${10}^3ℬ\left({\Lambda }_c^{+}\to {\Sigma }^{+}{K}^{\ast 0}\right)$	$2.3\pm 0.6$	$3.4\pm 1.0$ pdg
${10}^4ℬ\left({\Lambda }_c^{+}\to p\omega \right)$	$11.4\pm 5.4$	$9.4\pm 3.9$ Aaij:2017nsd
${10}^4ℬ\left({\Lambda }_c^{+}\to p\varphi \right)$	$10.4\pm 2.1$	$10.6\pm 1.4$ pdg
${10}^4ℬ\left({\Xi }_c^0\to {\Lambda }^0\varphi \right)$	$8.4\pm 3.9$	$6.1\pm 2.2$ pdg ; Li:2018qak
${ℛ}_1\left({\Xi }_c^{+}\right)=\frac{ℬ\left({\Xi }_c^{+}\to p{\overline{K}}^{\ast 0}\right)}{ℬ\left({\Xi }_c^{+}\to {\Sigma }^{+}{\overline{K}}^{\ast 0}\right)}$	$\left(1.6\pm 0.2\right)s_c^2$	$\left(2.8\pm 1.0\right)s_c^2$ pdg
${ℛ}_2\left({\Xi }_c^{+}\right)=\frac{ℬ\left({\Xi }_c^{+}\to {\Sigma }^{+}\varphi \right)}{ℬ\left({\Xi }_c^{+}\to {\Sigma }^{+}{\overline{K}}^{\ast 0}\right)}$	$\left(0.4\pm 0.1\right)s_c^2$	$\left(1.7\pm 1.2\right)s_c^2$ pdg ; Link:2003cd

Table 3. Table 3: The numerical results of the 𝐁 c → 𝐁 n ​ V → subscript 𝐁 𝑐 subscript 𝐁 𝑛 𝑉 {\bf B}_{c}\to{\bf B}_{n}V decays, with ℬ 𝐁 n ​ V ≡ ℬ ​ ( 𝐁 c → 𝐁 n ​ V ) subscript ℬ subscript 𝐁 𝑛 𝑉 ℬ → subscript 𝐁 𝑐 subscript 𝐁 𝑛 𝑉 {\cal B}_{{\bf B}_{n}V}\equiv{\cal B}({\bf B}_{c}\to{\bf B}_{n}V) .

${\Xi }_c^0$	our results
${10}^3{ℬ}_{{\Sigma }^{+}{K}^{\ast -}}$	$9.3\pm 2.9$
${10}^3{ℬ}_{{\Sigma }^0{\overline{K}}^{\ast 0}}$	$2.7\pm 2.2$
${10}^2{ℬ}_{{\Xi }^0{\rho }^0}$	$1.4\pm 0.4$
${10}^3{ℬ}_{{\Xi }^0\omega }$	${1.0}_{-1.0}^{+8.6}$
${10}^4{ℬ}_{{\Xi }^0\varphi }$	${1.5}_{-1.5}^{+7.1}$
${10}^3{ℬ}_{{\Xi }^{-}{\rho }^{+}}$	$8.6\pm 1.2$
${10}^3{ℬ}_{{\Lambda }^0{\overline{K}}^{\ast 0}}$	$4.6\pm 2.1$
${\Xi }_c^{+}$	our results
${10}^2{ℬ}_{{\Sigma }^{+}{\overline{K}}^{\ast 0}}$	$10.1\pm 2.9$
${10}^2{ℬ}_{{\Xi }^0{\rho }^{+}}$	$9.9\pm 2.9$
${\Lambda }_c^{+}$	our results
${10}^3{ℬ}_{{\Sigma }^0{\rho }^{+}}$	$6.1\pm 4.6$
${10}^3{ℬ}_{{\Sigma }^{+}{\rho }^0}$	$6.1\pm 4.6$
${10}^3{ℬ}_{{\Xi }^0{K}^{\ast +}}$	$8.7\pm 2.7$
${10}^3{ℬ}_{{\Lambda }^0{\rho }^{+}}$	$7.4\pm 3.4$

Equations

$${\mathcal{H}}_{eff}$$

$$O_{\pm }=\frac{1}{2}\left[(\bar{u}d)(\bar{s}c)\pm (\bar{s}d)(\bar{u}c)\right],$$

$$O_{\pm }^q=\frac{1}{2}\left[(\bar{u}q)(\bar{q}c)\pm (\bar{q}q)(\bar{u}c)\right],$$

$$O_{\pm }^{\prime }=\frac{1}{2}\left[(\bar{u}s)(\bar{d}c)\pm (\bar{d}s)(\bar{u}c)\right],$$

$$O_{-(+)}\sim {\mathcal{O}}_{6(\overline{15})}=\frac{1}{2}(\bar{u}d\bar{s}\mp \bar{s}d\bar{u})c,$$

$$O_{-(+)}^q\sim {\mathcal{O}}_{6(\overline{15})}^q=\frac{1}{2}(\bar{u}q\bar{q}\mp \bar{q}q\bar{u})c,$$

$$O_{-(+)}^{\prime }\sim {{\mathcal{O}}^{\prime }}_{6(\overline{15})}=\frac{1}{2}(\bar{u}s\bar{d}\mp \bar{d}s\bar{u})c,$$

$${\mathcal{H}}_{eff}$$

$$({\mathbf{B}}_c{)}_i$$

$$({\mathbf{B}}_n{)}_j^i$$

$$(V{)}_j^i$$

$$\mathcal{A}({\mathbf{B}}_c\to {\mathbf{B}}_{n}V)=⟨{\mathbf{B}}_{n}V|{\mathcal{H}}_{eff}|{\mathbf{B}}_c⟩=\frac{G_F}{\sqrt{2}}T({\mathbf{B}}_c\to {\mathbf{B}}_{n}V),$$

$$T({\mathbf{B}}_c\to {\mathbf{B}}_{n}V)$$

$$T({\mathcal{O}}_6)$$

$$T({\mathcal{O}}_{\overline{15}})$$

$$\mathcal{B}({\mathbf{B}}_c\to {\mathbf{B}}_{n}V)=\frac{|{\vec{p}}_{cm}|{\tau }_{{\mathbf{B}}_{\mathbf{c}}}}{8\pi m_{{\mathbf{B}}_c}^2}|\mathcal{A}({\mathbf{B}}_c\to {\mathbf{B}}_{n}V){|}^2,$$

$$|{\vec{p}}_{cm}|=\frac{\sqrt{[(m_{{\mathbf{B}}_c}^2-(m_{{\mathbf{B}}_n}+m_V{)}^2][(m_{{\mathbf{B}}_c}^2-(m_{{\mathbf{B}}_n}-m_V{)}^2]}}{2m_{{\mathbf{B}}_c}},$$

$${\bar{a}}_1,{\bar{a}}_2{e}^{i{\delta }_{{\bar{a}}_2}},{\bar{a}}_3{e}^{i{\delta }_{{\bar{a}}_3}},\bar{h}{e}^{i{\delta }_{\bar{h}}},$$

$$\displaystyle\chi^{2}=\sum_{i}\bigg{(}\frac{{\cal B}^{i}_{th}-{\cal B}^{i}_{ex}}{\sigma_{ex}^{i}}\bigg{)}^{2}+\sum_{j}\bigg{(}\frac{{\cal R}^{j}_{th}-{\cal R}^{j}_{ex}}{\sigma_{ex}^{j}}\bigg{)}^{2}\,,$$

$$({\bar{a}}_1,{\bar{a}}_2,{\bar{a}}_3,\bar{h})=(0.22\pm 0.02,0.23\pm 0.04,0.39\pm 0.05,0.16\pm 0.01){\text{GeV}}^3,$$

$$({\delta }_{{\bar{a}}_2},{\delta }_{{\bar{a}}_3},{\delta }_{\bar{h}})=(-85.5\pm 13.0,78.4\pm 8.8,99.3\pm 7.7{)}^{\circ },$$

$${\chi }^2/n.d.f=6.3/3=2.1,$$

$$\mathcal{B}({\Lambda }_c^{+}\to {\Sigma }^{+}{\rho }^0,{\Lambda }^0{\rho }^{+})$$

$$\mathcal{B}({\Lambda }_c^{+}\to {\Xi }^0{K}^{\ast +},{\Sigma }^0{K}^{\ast +},{\Lambda }^0{K}^{\ast +})=(8.7\pm 2.7,1.2\pm 0.3,2.0\pm 0.5)\times 10^{-3},$$

$$T({\Lambda }_c^{+}\to {\Sigma }^0{\rho }^{+})+T({\Lambda }_c^{+}\to {\Sigma }^{+}{\rho }^0)=0,$$

$$T({\Lambda }_c^{+}\to {\Sigma }^{+}{K}^{\ast 0})-\sqrt{2}T({\Lambda }_c^{+}\to {\Sigma }^0{K}^{\ast +})=-2{\bar{a}}_5{s}_c,$$

$$T({\Lambda }_c^{+}\to n{\rho }^{+})-\sqrt{2}T({\Lambda }_c^{+}\to p{\rho }^0)=-(\frac{{\bar{a}}_4+{\bar{a}}_6}{2})s_c,$$

$$T({\Lambda }_c^{+}\to n{K}^{\ast +})+T({\Lambda }_c^{+}\to p{K}^{\ast 0})=-2({\bar{a}}_4+{\bar{a}}_5)s_c^2.$$

$$\mathcal{B}({\Lambda }_c^{+}\to {\Sigma }^0{\rho }^{+},{\Sigma }^{+}{\rho }^0)=(6.1\pm 4.6)\times 10^{-3},$$

$$\mathcal{B}({\Lambda }_c^{+}\to p{\rho }^0)=\frac{1}{2}\mathcal{B}({\Lambda }_c^{+}\to n{\rho }^{+})=(3.5\pm 2.9)\times 10^{-4},$$

$$\mathcal{B}({\Lambda }_c^{+}\to n{K}^{\ast +},p{K}^{\ast 0})=(1.6\pm 0.5)\times 10^{-4},$$

$$\frac{1}{\sqrt{2}}T({\Lambda }_c^{+}\to p{\bar{K}}^{\ast 0})-\frac{1}{s_c}T({\Lambda }_c^{+}\to p{\rho }^0)=T({\Lambda }_c^{+}\to {\Sigma }^0{\rho }^{+}),$$

$$\frac{1}{\sqrt{2}}T({\Lambda }_c^{+}\to p{\bar{K}}^{\ast 0})+\frac{1}{s_c}T({\Lambda }_c^{+}\to p{\rho }^0)=\sqrt{3}T({\Lambda }_c^{+}\to {\Lambda }^0{\rho }^{+}),$$

$$\mathcal{B}({\Lambda }_c^{+}\to p{\rho }^0)$$

$$\mathcal{B}({\Xi }_c^{+}\to {\Sigma }^{+}{\bar{K}}^{\ast 0},{\Xi }^0{\rho }^{+})=(10.1\pm 2.9,9.9\pm 2.9)\times 10^{-2},$$

$$\mathcal{B}({\Xi }_c^{+}\to p{\bar{K}}^{\ast 0},{\Sigma }^{+}\varphi )=(7.8\pm 2.2,1.9\pm 0.9)\times 10^{-3}.$$

$$\mathcal{B}({\Lambda }_c^{+}\to {\Sigma }^{+}{\rho }^0,{\rho }^0\to {\pi }^{+}{\pi }^{-})$$

$$\mathcal{B}({\Lambda }_c^{+}\to {\Sigma }^0{\rho }^{+},{\rho }^{+}\to {\pi }^{+}{\pi }^0)$$

$$\mathcal{B}({\Lambda }_c^{+}\to {\Lambda }^0{\rho }^{+},{\rho }^{+}\to {\pi }^{+}{\pi }^0)$$

Full text

I Introduction

The two-body ${\bf B}_{c}\to{\bf B}_{n}V$ decays have not been abundantly measured as the ${\bf B}_{c}\to{\bf B}_{n}M$ counterparts, where ${\bf B}_{c}=(\Xi_{c}^{0},-\Xi_{c}^{+},\Lambda_{c}^{+})$ are the anti-triplet charmed baryon states, together with ${\bf B}_{n}$ and $V(M)$ the baryon and vector (pseudo-scalar) meson states, respectively. For example, all Cabibbo-favored (CF) $\Lambda_{c}^{+}\to{\bf B}_{n}M$ decays have been measured pdg , including the recent BESIII observation for $\Lambda_{c}^{+}\to\Sigma^{+}\eta^{\prime}$ Ablikim:2018czr , whereas for the CF vector modes only $\Lambda_{c}^{+}\to p\bar{K}^{*0},\Sigma^{+}\omega,\Sigma^{+}\phi$ have absolute branching fractions pdg . In addition, the first absolute branching ratio for the $\Xi_{c}^{0}$ decays is $\Xi_{c}^{0}\to\Xi^{-}\pi^{+}$ Li:2018qak , instead of any $\Xi_{c}^{0}\to{\bf B}_{n}V$ decays.

Nevertheless, the ${\bf B}_{c}\to{\bf B}_{n}V$ decays are not less important than the ${\bf B}_{c}\to{\bf B}_{n}M$ counterparts. First, the participations of BESIII, BELLEII and LHCb Collaborations are expected to make more accurate measurements for ${\bf B}_{c}\to{\bf B}_{n}V$, such as $\Lambda^{+}_{c}\to\Sigma^{+}\rho^{0},\Lambda^{0}\rho^{+}$, presented as ${\cal B}(\Lambda_{c}^{+}\to\Sigma^{+}\rho^{0},\Lambda^{0}\rho^{+})<(1.7,6)\%$ due to the previous measurements pdg . Second, in the three-body ${\bf B}_{c}\to{\bf B}_{n}MM^{\prime}$ decays, the $MM^{\prime}$ meson pair is assumed to be mainly in the S-wave state Geng:2018upx . However, the resonant ${\bf B}_{c}\to{\bf B}_{n}V,V\to MM^{\prime}$ decay causes $MM^{\prime}$ to be in the P-wave state, of which the contribution to the total ${\cal B}({\bf B}_{c}\to{\bf B}_{n}MM^{\prime})$ needs clarification. Note that (S,P) denote $L=(0,1)$ as the quantum numbers for the orbital angular momentum between $M$ and $M^{\prime}$. Third, the three-body $\Xi_{c}^{+}$ decays can be measured as the ratios of ${\cal B}(\Xi_{c}^{+}\to{\bf B}_{n}V)/{\cal B}(\Xi_{c}^{+}\to{\bf B}_{n}MM^{\prime})$. Particularly, the doubly Cabibbo-suppressed $\Xi_{c}^{+}\to p\phi$ decay is observed for the first time, with ${\cal B}(\Xi_{c}^{+}\to p\phi)/{\cal B}(\Xi_{c}^{+}\to pK^{-}\pi^{+})=(19.8\pm 0.7\pm 0.9\pm 0.2)\times 10^{-3}$ Aaij:2019kss . The information of ${\cal B}(\Xi_{c}^{+}\to{\bf B}_{n}V)$ is hence helpful to determine ${\cal B}(\Xi_{c}^{+}\to{\bf B}_{n}MM^{\prime})$.

Since the study of ${\bf B}_{c}\to{\bf B}_{n}V$ is necessary, it is important to provide a corresponding theoretical approach. The factorization approach for the heavy hadron decays ali ; Geng:2006jt ; Hsiao:2014mua seems applicable to ${\bf B}_{c}\to{\bf B}_{n}V$. Nonetheless, it has been shown that, besides the factorizable effects, there exist significant non-factorizable contributions in ${\bf B}_{c}\to{\bf B}_{n}M$ Zhao:2018mov , such that the factorization approach fails to explain the data. In contrast, with both factorizable and non-factorizable effects He:2000ys ; Fu:2003fy ; Hsiao:2015iiu ; He:2015fwa ; He:2015fsa ; Savage:1989qr ; Savage:1991wu ; h_term ; He:2018joe , the $SU(3)$ flavor symmetry ($SU(3)_{f}$) approach can accommodate the measurements for ${\bf B}_{c}\to{\bf B}_{n}M$ Lu:2016ogy ; Wang:2017azm ; Wang:2017gxe ; Geng:2017esc ; Geng:2018bow ; Geng:2018plk ; Geng:2018rse , such as the purely non-factorizable $\Lambda_{c}^{+}\to\Xi^{0}K^{+}$ decay Ablikim:2018bir . In addition, the predicted values of ${\cal B}(\Lambda_{c}^{+}\to\Sigma^{+}\eta^{\prime})$ and ${\cal B}(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})$ are in agreement with the recent observations Ablikim:2018czr ; Li:2018qak ; Geng:2017esc ; Geng:2018plk . Therefore, we propose to extend the $SU(3)_{f}$ symmetry to ${\bf B}_{c}\to{\bf B}_{n}V$, while the existing observations have been sufficient for the numerical analysis. In this report, we will extract the $SU(3)_{f}$ amplitudes, and predict the to-be-measured ${\bf B}_{c}\to{\bf B}_{n}V$ branching fractions.

II Formalism

To obtain the amplitudes for the two-body ${\bf B}_{c}\to{\bf B}_{n}V$ decays, where ${\bf B}_{c(n)}$ is the singly charmed (charmless) baryon state and $V$ the vector meson, we present the relevant effective Hamiltonian $({\cal H}_{eff})$ for the tree-level $c$ quark decays, given by Buras:1998raa

[TABLE]

with $q=d$ or $s$, where $G_{F}$ is the Fermi constant, $V_{ij}$ are the CKM matrix elements, and $c_{\pm}$ the scale-dependent Wilson coefficients. In Eq. (1), $O_{\pm}^{(q,\prime)}$ are the four-quark operators:

[TABLE]

with $(\bar{q}_{1}q_{2})\equiv\bar{q}_{1}\gamma_{\mu}(1-\gamma_{5})q_{2}$. By neglecting the Lorentz indices, the operator of $(\bar{q}_{1}q_{2})(\bar{q}_{3}c)$ transforms as $(\bar{q}^{i}q_{k}\bar{q}^{j})c$ under the $SU(3)_{f}$ symmetry, where $q_{i}=(u,d,s)$ represent the triplet of $3$. The operator can be decomposed as irreducible forms, which is accordance with $(\bar{3}\times 3\times\bar{3})c=(\bar{3}+\bar{3}^{\prime}+6+\overline{15})c$. One hence has Savage:1989qr ; Savage:1991wu

[TABLE]

with the subscripts $(6,\overline{15})$ denoting the two irreducible $SU(3)_{f}$ operators. By substituting ${\cal O}_{6(\overline{15})}^{(q,\prime)}$ for $O_{-(+)}^{(q,\prime)}$, the effective Hamiltonian in Eq. (1) becomes

[TABLE]

where the tensor notations of $1/2\epsilon_{ijl}H(6)^{lk}$ and $H(\overline{15})_{ij}^{k}$ contain ${\cal O}_{6}^{(q,\prime)}$ and ${\cal O}_{\overline{15}}^{(q,\prime)}$, respectively. In terms of $(V_{cs}V_{ud},V_{cd}V_{ud},V_{cs}V_{us},V_{cd}V_{us})=(1,-s_{c},s_{c},-s_{c}^{2})$ with $s_{c}\equiv\sin\theta_{c}$, where $\theta_{c}$ represents the well-known Cabbibo angle, we have $H_{22}(6)=2$, $H^{23,32}(6)=-2s_{c}$, $H^{2}_{12,21}(\overline{15})=-H^{3}_{13,31}(\overline{15})=s_{c}$, $H^{33}(6)=2s_{c}^{2}$, and $H^{3}_{12,21}(\overline{15})=-s_{c}^{2}$ as the non-zero entries Savage:1989qr . Note that $n=0,1$ and 2 in $s_{c}^{n}$ correspond to the Cabibbo-flavored (CF), singly Cabibbo-suppressed (SCS), and doubly Cabibbo-suppressed (DCS) decays, respectively. We also need ${\bf B}_{c}$ and ${\bf B}_{n}$ ($V$) to be in the irreducible representation of the $SU(3)_{f}$ symmetry, given by

[TABLE]

Subsequently, ${\cal H}_{eff}$ in Eq. (4) is enabled to be connected to the initial and final states in Eq. (II), such that we derive the amplitudes of ${\bf B}_{c}\to{\bf B}_{n}V$ as

[TABLE]

instead of introducing the details of the QCD calculations for the hadronization. Explicitly, the $T$ amplitudes ($T$-amps) are given by Savage:1989qr ; Savage:1991wu

[TABLE]

where $T_{ij}\equiv({\bf B}_{c})_{k}\epsilon^{ijk}$, and $(c_{-},c_{+})$ have been absorbed into the $SU(3)$ parameters $(\bar{a}_{i},\bar{h}^{(\prime)})$.

With the full expansion of $T$-amps in Table 1, the two-body ${\bf B}_{c}\to{\bf B}_{n}V$ decays are presented with the $SU(3)_{f}$ parameters. Since $\omega=(u\bar{u}+d\bar{d})/\sqrt{2}$ and $\phi=s\bar{s}$ actually mix with $\omega_{1}=(u\bar{u}+d\bar{d}+s\bar{s})/\sqrt{3}$ and $\omega_{8}=(u\bar{u}+d\bar{d}-2s\bar{s})/\sqrt{6}$, the $(\bar{h},\bar{h}^{\prime})$ terms that are related to $(V)^{l}_{l}=\sqrt{2}\omega+\phi=\sqrt{3}\omega_{1}$ can contribute to the decays with $(\omega,\phi)$ only. In terms of the equation for the two-body decays, given by pdg

[TABLE]

we can compute the branching ratio with the $SU(3)_{f}$ amplitudes, where $\tau_{\bf{B}_{c}}$ denotes the $\bf{B}_{c}$ lifetime. The $SU(3)_{f}$ amplitudes are accounted to be 9 complex numbers, leading to 17 independent parameters to be extracted, whereas there exist 10 data points for the numerical analysis. To have a practical fit, we follow Refs. Lu:2016ogy ; Geng:2017esc ; Geng:2018plk ; Geng:2018rse to reduce the parameters. In ${\cal H}_{eff}\propto c_{-}H(6)+c_{+}H(\overline{15})$, since the QCD calculation at the scale $\mu=1$ GeV leads to $(c_{+},c_{-})=(0.76,1.78)$ in the naive dimensional regularization (NDR) scheme Li:2012cfa ; Fajfer:2002gp , the ratio of $(c_{-}/c_{+})^{2}\simeq 0.17$ indicates the suppression of $H(\overline{15})$. We hence ignore $(\bar{a}_{4,5,...,7},h^{\prime})$. On the other hand, $(\bar{a}_{1,2,3},h)$ from $H(6)$ are kept for the fit, represented as

[TABLE]

with the phases $\delta_{\bar{a}_{2,3},\bar{h}}$, and $\bar{a}_{1}$ set to be relatively real.

III Numerical analysis

For the numerical analysis, we collect (the ratios of) the branching fractions for the observed ${\bf B}_{c}\to{\bf B}V$ decays in Table 2, where ${\cal B}(\Xi^{+}_{c}\to p\bar{K}^{*0},\Sigma^{+}\phi,\Sigma^{+}\bar{K}^{*0})$ are in fact measured to be relative to ${\cal B}(\Xi_{c}^{+}\to\Xi^{-}\pi^{+}\pi^{+})$ pdg ; Link:2003cd , recombined as ${\cal R}_{1,2}(\Xi^{+}_{c})$. We obtain ${\cal B}(\Xi^{0}_{c}\to\Lambda^{0}\phi)$ from ${\cal B}(\Xi^{0}_{c}\to\Lambda^{0}\phi)/{\cal B}(\Xi^{0}_{c}\to\Xi^{-}\pi^{+})$ pdg , with the input of ${\cal B}(\Xi^{0}_{c}\to\Xi^{-}\pi^{+})$ measured by BELLE Li:2018qak . In addition, the ratio of ${\cal R}(\Lambda_{c}^{+})=(\Lambda_{c}^{+}\to\Sigma^{+}\rho^{0})/{\cal B}(\Lambda_{c}^{+}\to\Sigma^{+}\omega)$ comes from the data events in Ref. Kubota:1993pw . Besides, $s_{c}=0.22453\pm 0.00044$ pdg is the theoretical input for the CKM matrix elements. By using the equation of Zhao:2018mov

[TABLE]

we are able to obtain the minimum $\chi^{2}$ value, such that the $SU(3)_{f}$ parameters can be extracted with the best fit. Note that ${\cal B}^{i}({\cal R}^{j})$ represents (the ratio of) the branching fraction, with the subscript $th$ ($ex$) denoting the theoretical (experimental) input, while $\sigma_{ex}^{i(j)}$ stands for the experimental error. As the inputs in Eq. (17), ${\cal B(R)}_{th}$ come from the $T$-amps in Table 1, while ${\cal B(R)}_{ex}$ and $\sigma_{ex}$ the data points in Table 2.

Subsequently, the global fit gives

[TABLE]

where $n.d.f$ represents the number of the degree of freedom. With the fit results in Eq. (III), we calculate the branching ratios, ${\cal R}(\Lambda_{c}^{+})$ and ${\cal R}_{1,2}(\Xi_{c}^{+})$ to be compared to their data inputs in Table 2. Moreover, we predict the branching fractions for the ${\bf B}_{c}\to{\bf B}_{n}V$ decays, given in Table 3.

IV Discussions and Conclusions

With $\chi^{2}/n.d.f\simeq 2$ to present a reasonable fit, the approach based the $SU(3)_{f}$ symmetry is demonstrated to be reliable for ${\bf B}_{c}\to{\bf B}_{n}V$. Besides, our prediction

[TABLE]

agrees with the experimental upper bounds of $(1.7,6)\%$, respectively pdg . We also find

[TABLE]

to be compatible with the pseudo-scalar counterparts. According to Table 1, we obtain

[TABLE]

By ignoring the parameters in $H(\overline{15})$, we obtain

[TABLE]

which respect the isospin symmetry. We also get

[TABLE]

which lead to

[TABLE]

where the pre-factors (3.6,1.3,1.1) have taken into account the differences for $|\vec{p}_{cm}|$ in Eq. (II). It is interesting to note that the $\Lambda_{c}^{+}\to p\pi^{0}$ decay has a similar relation to that in Eq. (24), where $(\rho,\bar{K}^{*0})$ are replaced by $(\pi,\bar{K}^{0})$. However, the relation for $\Lambda_{c}^{+}\to p\pi^{0}$ causes ${\cal B}(\Lambda_{c}^{+}\to p\pi^{0})\simeq 5\times 10^{-4}$, disapproved by the data pdg . This indicates that, even though the ignoring of $H(\overline{15})$ is viable, the possible interferences between $H(6)$ and $H(\overline{15})$ might give sizeable contributions to some decay modes Geng:2018rse . In this work, since the fit still accommodates the data, it is not clear which of the $\Lambda_{c}^{+}\to{\bf B}_{n}V$ decays receives sizeable interferences between $H(6)$ and $H(\overline{15})$. Like the ${\cal B}(\Lambda_{c}^{+}\to p\pi^{0})$ case, the precise measurement of ${\cal B}(\Lambda_{c}^{+}\to p\rho^{0})$ can test the ignoring of $H(\overline{15})$. For the $\Xi_{c}^{+}$ decays, we obtain

[TABLE]

With $f_{\tau_{{\bf B}_{c}}}\equiv\tau_{\Xi_{c}^{+}}/\tau_{\Lambda_{c}^{+}}\simeq 2.2$, ${\cal B}(\Xi^{+}_{c}\to\Sigma^{+}\bar{K}^{*0},\Xi^{0}\rho^{+})\simeq(2-4)f_{\tau_{{\bf B}_{c}}}{\cal B}(\Lambda_{c}^{+}\to p\bar{K}^{*0})$ is found to be in accordance with $|\bar{a}_{3}|^{2}\simeq(2-4)|\bar{a}_{1}|^{2}$, which can be tested by more accurate measurements.

By means of ${\cal B}({\bf B}_{c}\to{\bf B}_{n}V,V\to MM^{\prime})={\cal B}({\bf B}_{c}\to{\bf B}_{n}V){\cal B}(V\to MM^{\prime})$, the resonant contribution to the total ${\cal B}({\bf B}_{c}\to{\bf B}_{n}MM^{\prime})$ can be investigated, where $MM^{\prime}$ from the vector meson decay are in the P-wave state. On the other hand, the theoretical study of ${\bf B}_{c}\to{\bf B}_{n}MM^{\prime}$ needs $MM^{\prime}$ to be mainly in the S-wave state Geng:2018upx . Using ${\cal B}(\rho^{0(+)}\to\pi^{+}\pi^{-(0)})\simeq 100\%$ pdg and the predictions for ${\cal B}(\Lambda_{c}^{+}\to\Sigma\rho,\Lambda^{0}\rho^{+})$, we obtain

[TABLE]

which are within the total branching ratios of $(4.42\pm 0.28,2.2\pm 0.8,7.0\pm 0.4)\times 10^{-2}$ pdg , respectively, showing that the P-wave contributions from $V\to MM^{\prime}$ are indeed minor to these decays. By putting ${\cal B}(\Xi^{+}_{c}\to p\phi)=(1.5\pm 0.7)\times 10^{-4}$ into the measured ratio of ${\cal B}(\Xi^{+}_{c}\to p\phi)/{\cal B}(\Xi^{+}_{c}\to pK^{-}\pi^{+})=(19.8\pm 0.7\pm 0.9\pm 0.2)\times 10^{-3}$ Aaij:2019kss , we obtain ${\cal B}(\Xi^{+}_{c}\to pK^{-}\pi^{+})=(0.8\pm 0.4)\%$, which is a little smaller than the predicted value of $(1.7\pm 0.5)\%$ Wang:2019dls .

In sum, within the framework of the $SU(3)_{f}$ symmetry, we have studied the ${\bf B}_{c}\to{\bf B}_{n}V$ decays. We have predicted ${\cal B}(\Lambda^{+}_{c}\to\Sigma^{+}\rho^{0},\Lambda^{0}\rho^{+})=(0.61\pm 0.46,0.74\pm 0.34)\%$, in agreement with the experimental upper bounds of $(1.7,6)\%$, respectively. It has also been shown that ${\cal B}(\Lambda^{+}_{c}\to\Xi^{0}K^{*+},\Sigma^{0}K^{*+},\Lambda^{0}K^{*+})=(8.7\pm 2.7,1.2\pm 0.3,2.0\pm 0.5)\times 10^{-3}$. For the $\Xi_{c}^{+}$ decays, we have obtained ${\cal B}(\Xi^{+}_{c}\to\Sigma^{+}\bar{K}^{*0},\Xi^{0}\rho^{+})=(10.1\pm 2.9,9.9\pm 2.9)\times 10^{-2}$, ${\cal B}(\Xi^{+}_{c}\to p\bar{K}^{*0},\Sigma^{+}\phi)=(7.8\pm 2.2,1.9\pm 0.9)\times 10^{-3}$ and ${\cal B}(\Xi^{+}_{c}\to p\phi)=(1.5\pm 0.7)\times 10^{-4}$. The predicted ${\cal B}({\bf B}_{c}\to{\bf B}_{n}V)$ can be compared to the future measurements by BESIII, BELLEII and LHCb.

ACKNOWLEDGMENTS

This work was supported by National Science Foundation of China (11675030).