Angular distributions for multi-body semileptonic charmed baryon decays

Abstract

We perform an analysis of angular distributions in semileptonic decays of charmed baryons $B_1^{(\prime )}\rightarrow B_2^{(\prime )}(\rightarrow B_3^{(\prime )}B_4^{(\prime )})\ell ^+\nu _{\ell }$, where the $B_1{=}(\Lambda _c^+,\Xi _c^{(0,+)})$ are the SU(3)-antitriplet baryons and $B_1'{=}\Omega _c^-$ is an SU(3) sextet. We will firstly derive analytic expressions for angular distributions using the helicity amplitude technique. Based on the lattice quantum chromodynamics (QCD) results for $\Lambda _c^+\rightarrow \Lambda$ and $\Xi _c^0\rightarrow \Xi ^-$ form factors and model calculation of the $\Omega _c^0\rightarrow \Omega ^-$ transition, we predict the branching fractions: ${\mathcal {B}}(\Lambda _{c}^{+} \rightarrow \Lambda (\rightarrow p \pi ^{-}) e^{+} \nu _{e})=2.48(15)\%$, ${\mathcal {B}}(\Lambda _{c}^+\rightarrow \Lambda (\rightarrow p \pi ^{-})\mu ^{+}\nu _{\mu })=2.50(14)\%$, ${\mathcal {B}}(\Xi _{c}^0\rightarrow \Xi ^-(\rightarrow \Lambda \pi ^{-})e^{+}\nu _{e})=2.40(30)\%$, ${\mathcal {B}}(\Xi _{c}^0\rightarrow \Xi ^-(\rightarrow \Lambda \pi ^{-})\mu ^{+}\nu _{\nu })=2.41(30)\%$, ${\mathcal {B}}(\Omega _{c}^0\rightarrow \Omega ^-(\rightarrow \Lambda K^{-})e^{+}\nu _{e})=\!0.362(14)\%$, ${\mathcal {B}}(\Omega _{c}^0\rightarrow \Omega ^-\!(\rightarrow \Lambda K^{-})\mu ^{+\!}\nu _{\nu })=0.350(14)\%$. We also predict the $q^2$ dependence and angular distributions of these processes, in particular the coefficients for the $\cos n\theta _{\ell }$ ($\cos n\theta _{h}$, $\cos n\phi$) $(n=0, 1, 2, \ldots )$ terms. This work can provide a theoretical basis for the ongoing experiments at BESIII, LHCb, and BELLE-II.

1 Introduction

Weak decays of heavy baryons play an important role in testing the standard model (SM), in which any significant deviation from SM expectation would indirectly provide definitive clues for new physics beyond the SM. Most previous analysis has concentrated on the heavy meson sector, such as B and D meson decays [1,2,3,4,5,6,7,8,9,10], although heavy baryon decays have recently received increasing attention by both experimental [11, 13,14,15,16,17] and theoretical researchers [18,19,20,21,22,23,24]. One of the most important motivations for studying the weak decays of heavy baryons is measuring the Cabibbo–Kobayashi–Maskawa (CKM) matrix elements which describe the quark mixing and strength of CP violation. From this viewpoint, the study of semileptonic decays of charmed baryons, which can provide an ideal way to determine the $|V_{cd}|$ and $|V_{cs}|$, and examine the CKM unitarity $\sum _{i=d,s,b}|V_{ci}|^2=1$, is of great value. In the singly charmed baryons with $q_1q_2c$, the two light quarks can be decomposed as an antitriplet and a sextet. Focusing on the ground states with $J^P=1/2^+$, only four baryonic states, $(\Lambda _c^+,\Xi _c^0,\Xi _c^+)$, and $\Omega _c^0$ baryons, can have measurable weak decays, while others such as $\Xi _c'$ and $\Sigma _c'$ have strong and electromagnetic decay modes [25,26,27,28,29].

Among various decay modes, semileptonic decays are the simplest [31,32,33], and in recent years, charmed baryon decays have attracted great interest from both theoretical and experimental researchers [34,35,36,37,38,39,40,41,42,43,44,45]. Semileptonic decays of $\Lambda ^{+}_{c}$ have been fruitfully studied in quark models and QCD sum rules [19, 46,47,48,49,50,51,52,53,54,55,56,57,58,59,60], and predictions for branching fractions differ substantially. A precise measurement of branching fractions of $\Lambda _{c}$ weak decays was recently reported by the BESIII collaboration: ${\mathcal {B}}(\Lambda _{c}\rightarrow \Lambda e^{+} \nu _{e})=0.0363$ and ${\mathcal {B}}(\Lambda _{c}\rightarrow \Lambda \mu ^{+} \nu _{\mu })=0.0363$ [37, 38]. For $\Xi _{c}$, the CLEO collaboration has measured the ratio of branching fractions ${\mathcal {B}}(\Xi ^{0}_{c}\rightarrow \Xi ^{-} e^{+}\nu _{e})/{\mathcal {B}}(\Xi ^{0}_{c}\rightarrow \Xi ^{-} \pi ^{+})$ [61, 62], and the Belle collaboration recently reported [13]: ${\mathcal {B}}(\Xi ^{0}_{c}\rightarrow \Xi ^{-} e^{+}\nu _{e})=1.31(04\pm 07\pm 38)\%$, ${\mathcal {B}}(\Xi ^{0}_{c}\rightarrow \Xi ^{-} \mu ^{+}\nu _{\mu })=1.27(06\pm 10\pm 37)\%$. On the theoretical side, a variety of models have been developed to analyze $\Xi _{c}$ weak decays [63,64,65,66,67], including a recent analysis of $\Xi _{c}\rightarrow \Xi$ transition form factors from lattice QCD [68]. Limited by low production rates and high background levels of current experiments, measurements of $\Omega _{c}$ decay branching ratios are not available. In theory, branching fractions of $\Omega _{c}$ weak decays are predicted in the light-front quark model: ${\mathcal {B}}(\Omega ^{0}_{c}\rightarrow \Omega ^{-}e^{+}\nu _{e})=5.4(\pm 0.2)\times 10^{-3}$ [69]. In this work, we will explore decay widths for semileptonic decays of charmed baryons with the lattice QCD results for form factors, and in particular we derive, for the first time, the angular distributions for four-body weak decays of $\Omega _{c}$. Feynman diagrams for these decay chains are shown in Fig. 1.

Fig. 1

Feynman diagram of charmed baryon weak decay of $B_1^{(\prime )}\rightarrow B_2^{(\prime )}(\rightarrow B_3^{(\prime )}B_4^{(\prime )})\ell ^+\nu _\ell$

The rest of this paper is organized as follows. In Sect. 2, we presen the theoretical framework for calculating the helicity amplitudes of charmed baryon decays, including the theoretical results of the Lorentz invariant leptonic and hadronic matrix elements. In Sect. 3, we list the differential decay widths of the three-body and four-body decay formulas. Integrating out the $q^2$, we obtain numerical results of partial decay width, as well as the illustration of momentum transfer and angular distributions of the decay width. A brief summary will be presented in the last section.

2 Theoretical framework

2.1 Formalism

We will focus on semileptonic four-body decays of both $\mathrm {SU}(3)$ ${\bar{\mathbf {3}}}$ and ${\mathbf {6}}$ ground states of singly charmed baryons, denoted as $B_1^{(\prime )}\rightarrow B_2^{(\prime )}(\rightarrow B_3^{(\prime )}B_4^{(\prime )})\ell ^+\nu _\ell$, where $\ell =e,\mu$ and $\nu _\ell$ are the charged and neutral leptons, respectively. For the antitriplet baryon $B_1=(\Lambda _c^+,\Xi _c^0,\Xi _c^+)$ decay, the intermediate states $B_2$ are spin 1/2 baryon with an SU(3) octet [22, 70, 71], while the sextet $B_1^{\prime }=\Omega _c^0$ decays weakly via the spin-3/2 decuplet intermediate baryons $B_2^{\prime } ~(\Omega ^-)$. $B_3^{(\prime )}$ and $B_4^{(\prime )}$ are the baryonic and mesonic final states, respectively. Examples of specific processes include:

$$\begin{aligned} \Lambda _c^+&\rightarrow \Lambda (\rightarrow p\pi ^-) \ell ^+\nu _\ell , \end{aligned}$$

(1)

$$\begin{aligned} \Xi _c^0&\rightarrow \Xi ^-(\rightarrow \Lambda \pi ^-)\ell ^+\nu _\ell , \end{aligned}$$

(2)

$$\begin{aligned} \Omega _c^0&\rightarrow \Omega ^-(\rightarrow \Lambda K^-)\ell ^+\nu _\ell . \end{aligned}$$

(3)

The effective weak Hamitonian for the semileptonic decays of charmed baryons can be written as

$$\begin{aligned} {\mathcal {H}}_{\mathrm {eff}}=\frac{G_F}{\sqrt{2}}V_{cs}\Big (\bar{s}\gamma ^{\mu }(1-\gamma _5)c\Big ) \Big ({\bar{\nu }}_{\ell }\gamma _{\mu }(1-\gamma _5)\ell \Big ), \end{aligned}$$

(4)

where $G_F$ is the Fermi constant, and $V_{cs}$ is the CKM matrix element. Based on the above effective Hamiltonian, we can obtain the decay amplitudes of $B_1^{(\prime )}\rightarrow B_3^{(\prime )}B_4^{(\prime )}\ell ^+\nu _\ell$

$$\begin{aligned} {\mathcal {M}} =&\,\, \frac{G_F}{\sqrt{2}}V_{cs}\left\langle B_{3}^{(\prime )} B_{4}^{({\prime })}\right| {\bar{s}} \gamma ^{\mu }\left( 1-\gamma _{5}\right) c\left| B_{1}^{(\prime )}\right\rangle \nonumber \\&\times \left\langle \ell ^{+} \nu _{\ell }\left| {\bar{\nu }}_{\ell }\gamma _{\mu }(1-\gamma _5)\ell \right| 0\right\rangle . \end{aligned}$$

(5)

With the decomposition of $g_{\mu \nu }$

$$\begin{aligned} g_{\mu \nu }=-\sum _{\lambda }\epsilon _{\mu }^*(\lambda )\epsilon _{\nu }(\lambda )+\frac{q_{\mu }q_{\nu }}{q^2}, \end{aligned}$$

(6)

the above amplitude can be decomposed into the Lorentz invariant hadronic and leptonic matrix elements:

$$\begin{aligned} {\mathcal {M}}=&\,\, \frac{G_F}{\sqrt{2}}V_{cs}\left\langle B_{3}^{(\prime )} B_{4}^{({\prime })}\right| {\bar{s}} \gamma ^{\mu }\left( 1-\gamma _{5}\right) c\left| B_{1}^{(\prime )}\right\rangle \nonumber \\&\times \left\langle \ell ^{+} \nu _{\ell }\left| {\bar{\nu }}_{\ell }\gamma ^{\nu }(1-\gamma _5)\ell \right| 0\right\rangle g_{\mu \nu } = \frac{G_F}{\sqrt{2}}V_{cs}\nonumber \\&\times \left( -\sum _{\lambda }H^{\mu }\epsilon _{\mu }^*(\lambda )\!\times \! L^{\nu }\epsilon _{\nu }(\lambda )\!+\!H^{\mu }\epsilon _{\mu }^*(t)\!\times \! L^{\nu }\epsilon _{\nu }(t) \right) , \end{aligned}$$

(7)

where the hadronic part $H^{\mu }\!\equiv \!\left\langle B_{3}^{(\prime )} B_{4}^{({\prime })}\right| {\bar{s}} \gamma ^{\mu }\left( 1\!-\!\gamma _{5}\right) c\left| B_{1}^{(\prime )}\right\rangle$ and leptonic part $L^{\nu }\equiv \left\langle \ell ^{+} \nu _{\ell }\left| {\bar{\nu }}_{\ell }\gamma ^{\nu }(1-\gamma _5)\ell \right| 0\right\rangle$, and $\epsilon$ is the polarization vector of the decomposed $W^+$ boson with helicity state $\lambda$ and t, in which $\epsilon _{\mu }(t)\equiv q_{\mu }/\sqrt{q^2}$.

Focusing on the hadronic part and inserting the one-particle completeness states of the intermediate baryon

$$\begin{aligned} 1&=\sum _{s_2}\int \frac{d^4{p}_2}{(2\pi )^4}\frac{i}{p_2^2-m_2^2+im_2\Gamma _{2}}\nonumber \\&\quad \times \left| B_2^{(\prime )}(p_2,s_2)\right\rangle \left\langle B_2^{(\prime )}(p_2,s_2)\right| , \end{aligned}$$

(8)

we find the $H^{\mu }$ given as:

$$\begin{aligned} H^{\mu }&=\sum _{s_{2}} \int \frac{d^{4} p_{2}}{(2 \pi )^{4}} \frac{i}{p_{2}^{2}-m_{2}^{2}+i m_{2} \Gamma _{2}} \nonumber \\&\quad \times \left\langle B_{3}^{(\prime )}(p_3,s_3) B_{4}^{({\prime })}(p_4)\right. \left| B_2^{(\prime )}(p_2,s_2)\right\rangle \nonumber \\&\quad \times \left\langle B_2^{(\prime )}(p_2,s_2)\right| {\bar{s}} \gamma ^{\mu }\left( 1-\gamma _{5}\right) c\left| B_{1}^{(\prime )}(p_1,s_1)\right\rangle . \end{aligned}$$

(9)

In the above, $p_2$, $m_2$, and $\Gamma _2$ are the four-momentum, mass, and decay rate of intermediate state $B_2^{(\prime )}$, respectively (Fig. 2). The $B_{1}^{(\prime )}\rightarrow B_{2}^{(\prime )}$ transition $H_1$ can be parameterized as hadronic form factors, while the $B_{2}^{(\prime )}\rightarrow B_{3}^{(\prime )}B_{4}^{(\prime )}$ transition, with one final-state $B_{4}^{(\prime )}$ as a pseudoscalar meson, can be reduced as

$$\begin{aligned}&\left\langle B_{3}^{(\prime )}(p_3,s_3) B_{4}^{({\prime })}(p_4) \right. \left| B_2^{(\prime )}(p_2,s_2)\right\rangle _{H}\nonumber \\&\quad =\! \left\langle B_{3}^{(\prime )}(p_3,s_3) B_{4}^{({\prime })}(p_4)\left| -\!i \int d^{4} x {\mathcal {H}}_{\mathrm {int}}(x)\right| B_2^{(\prime )}(p_2,s_2)\right\rangle _{I} \nonumber \\&\quad = \left\langle B_{3}^{(\prime )}(p_3,s_3) B_{4}^{({\prime })}(p_4)\left| -i \right. \right. \nonumber \\&\left. \left. \qquad \times \int d^{4} x \Big ( ig_{H_2}{\bar{\psi }}_{B_{3}}(x) \phi _{B_{4}}(x)\left( A-B \gamma _{5}\right) \psi _{B_{2}}(x) \Big )\right| \right. \nonumber \\&\quad \left. B_2^{(\prime )}(p_2,s_2)\right\rangle = (2 \pi )^{4} \delta ^{4}\left( p_{3}+p_{4}-p_{2}\right) g_{H_2}{\bar{u}}_{B_{3}}\nonumber \\&\qquad \times \left( p_{3}, s_{3}\right) \left( A-B \gamma _{5}\right) u_{B_2}\left( p_{2}, s_{2}\right) , \end{aligned}$$

(10)

where the subscripts “H” and “I” denote a Heisenberg and interaction representation matrix element, respectively, $g_{H_2}$ represents the coupling constant of hadronic vertex $B_{2}^{(\prime )}\rightarrow B_{3}^{(\prime )}B_{4}^{(\prime )}$, and factors A and B are constants weighting the contributions from scalar and pesudoscalar density operators [62].

Fig. 2

Illustration of four-body decay process $B_1^{(\prime )}\rightarrow B_2^{(\prime )}(\rightarrow B_3^{(\prime )}B_4^{(\prime )})\ell ^+\nu _\ell$

Therefore, the decay amplitude in Eq. (5) can be expressed as a convolution of the Lorentz invariant leptonic part $L(s_{\ell }, s_{\nu }, s_W)$ and two hadronic parts, $H_1(s_1,s_2,s_W)$, $H_2(s_2,s_3)$:

$$\begin{aligned}&i{\mathcal {M}}\left( B_1^{(\prime )}\rightarrow B_3^{(\prime )}B_4^{(\prime )}\ell ^+\nu _\ell \right) \nonumber \\&\quad =\sum _{s_2} i{\mathcal {M}}\left( B_1^{(\prime )}\rightarrow B_2^{(\prime )}\ell ^+\nu _\ell \right) \nonumber \\&\quad \times i{\mathcal {M}}\left( B_2^{(\prime )}\rightarrow B_3^{(\prime )}B_4^{(\prime )}\right) {\times } \frac{i}{p_{2}^{2}-m_{2}^{2}{+}i m_{2} \Gamma _{2}} \nonumber \\&\quad =\!\!\sum _{s_2} \left( i\frac{G_F}{\sqrt{2}}V_{cs}\sum _{s_W\!=\!\{\pm ,0,t\}} H_1(s_1,s_2,s_W) \!\!\times \!\! L(s_{\ell }, s_{\nu },s_W) \right) \nonumber \\&\quad \times H_2(s_2,s_3)\times \frac{i}{p_{2}^{2}-m_{2}^{2}+i m_{2} \Gamma _{2}} \nonumber \\&\quad = -\frac{G_FV_{cs}}{\sqrt{2}}\frac{1}{p_{2}^{2}-m_{2}^{2}+i m_{2} \Gamma _{2}}\nonumber \\&\quad \times \sum _{s_2}\sum _{s_W{=}\{\pm ,0,t\}}H_1(s_1,s_2,s_W) L(s_{\ell }, s_{\nu },s_W) {\times } H_2(s_2,s_3), \end{aligned}$$

(11)

with

$$\begin{aligned} L(s_{\ell }, s_{\nu },s_W)\equiv&~{\bar{u}}_{\nu }(p_{\nu },s_{\nu }) \gamma ^{\nu }\left( 1-\gamma _{5}\right) v_{\ell }(p_{\ell },s_{\ell }) \varepsilon _{\nu }(s_W), \end{aligned}$$

(12)

$$\begin{aligned} H_1(s_1,s_2,s_W)\equiv&\,\left\langle B_2^{(\prime )}(p_2,s_2)\right| {\bar{s}} \gamma ^{\mu }\left( 1-\gamma _{5}\right) c\nonumber \\&\times \left| B_{1}^{(\prime )}(p_1,s_1)\right\rangle \epsilon _{\mu }^*(s_W), \end{aligned}$$

(13)

$$\begin{aligned} H_2(s_2,s_3)\equiv&\, ~ g_{H_2}{\bar{u}}_{B_{3}}\left( p_{3}, s_{3}\right) \left( A-B \gamma _{5}\right) u_{B_2}\left( p_{2}, s_{2}\right) . \end{aligned}$$

(14)

Averaging the spin of the initial state and summing over the spins of final states, we obtain the squared amplitude as

$$\begin{aligned}&\frac{1}{2}\overline{\left| i{\mathcal {M}}\left( B_1^{(\prime )}\rightarrow B_3^{(\prime )}B_4^{(\prime )}\ell ^+\nu _\ell \right) \right| ^2}\nonumber \\&\quad =\frac{G_F^2|V_{cs}|^2}{2}\frac{1}{(q_2^2-m_2^2)^2+m_2^2\Gamma _2^2}\frac{1}{2}\sum _{s_1}\sum _{s_3,s_l,s_{\nu }} \nonumber \\&\qquad \times \left| \sum _{s_2}\sum _{s_W=\{\pm ,0,t\}}H_1(s_1,s_2,s_W)\times L(s_{\ell }, s_{\nu },s_W)\right. \nonumber \\&\qquad \left. \times H_2(s_2,s_3)\right| ^2. \end{aligned}$$

(15)

2.2 Kinematics

With the abbreviations

$$\begin{aligned} s_{\pm }=(m_1\pm m_2)^2-q^2,~\lambda (m_1,m_2,q)=s_+s_-, \end{aligned}$$

(16)

the momentum of the initial and final states in the subprocesses can be set as follows.

• 1. $B_1^{(\prime )}\rightarrow B_2^{(\prime )}W^+$ subprocess: in the rest frame of initial-state $B_1^{(\prime )}$, suppose that $B_2^{(\prime )}$ moves along the positive z-direction ($\theta =0,\phi =0$) and $W^+$ boson along the negative z-direction ($\theta =\pi ,\phi =\pi$), so we have

  $$\begin{aligned} p_1^{\mu }&=(m_1,0,0,0),~p_2^{\mu }=(E_2,0,0,|\vec {p}_2|),\nonumber \\ q^{\mu }&=(E_W,0,0,-|\vec {p}_2|), \end{aligned}$$

  (17)

  with

  $$\begin{aligned} E_2&=\frac{m_1^2+m_2^2-q^2}{2m_1},~\nonumber \\ E_W&=\frac{m_1^2-m_2^2+q^2}{2m_1},~\nonumber \\ |\vec {p}_2|&=\frac{\sqrt{\lambda (m_1,m_2,q)}}{2m_1}. \end{aligned}$$

  (18)

• 2. $B_2^{(\prime )}\rightarrow B_3^{(\prime )}B_4^{(\prime )}$ subprocess: in the rest frame of $B_2^{(\prime )}$, suppose the final-state baryon $B_3^{(\prime )}$ moves along the ($\theta =\theta _h,\phi =0$) direction, and therefore the meson $B_4^{(\prime )}$ moves along the ($\theta =\pi -\theta _h,\phi =\pi$) direction. We thus have

  $$\begin{aligned} p_2^{\mu }&=(m_2,0,0,0), ~ \nonumber \\ p_3^{\mu }&=(E_3, |\vec {p}_3|\sin \theta _h, 0,|\vec {p}_3|\cos \theta _h),~ \nonumber \\ p_4^{\mu }&=(E_4, -|\vec {p}_3|\sin \theta _h, 0,-|\vec {p}_3|\cos \theta _h), \end{aligned}$$

  (19)

  with

  $$\begin{aligned} E_3&=\frac{m_2^2+m_3^2-m_4^2}{2m_2},\quad E_4=\frac{m_2^2-m_3^2+m_4^2}{2m_2},~\nonumber \\ |\vec {p}_3|&=\frac{\sqrt{\lambda (m_2,m_3,m_4)}}{2m_2}. \end{aligned}$$

  (20)

• 3. $W^+\rightarrow \ell ^+\nu _l$ subprocess: in the rest frame of $W^+$, suppose the charged lepton moves along the ($\theta =\theta _l,\phi =\phi$) direction, and thereby the neutrino along the ($\theta =\pi -\theta _l,\phi =\pi +\phi$) direction, we have

  $$\begin{aligned} q^{\mu }=&\,\,\left( \sqrt{q^2},0,0,0\right) , \nonumber \\ p_l^{\mu }=&\,\,(E_l, |\vec {p}_l|\sin \theta _l\cos \phi , |\vec {p}_l|\sin \theta _l\sin \phi , |\vec {p}_l|\cos \theta _l), \nonumber \\ p_{\nu }^{\mu }=&\,\,(E_\nu , -|\vec {p}_l|\sin \theta _l\cos \phi , -|\vec {p}_l|\sin \theta _l\sin \phi , \nonumber \\&\quad -|\vec {p}_l|\cos \theta _l), \end{aligned}$$

  (21)

  with

  $$\begin{aligned} E_l=\frac{q^2+m_l^2}{2\sqrt{q^2}},~E_\nu =\frac{q^2-m_l^2}{2\sqrt{q^2}}, ~|\vec {p}_l|=\frac{q^2-m_l^2}{2\sqrt{q^2}}. \end{aligned}$$

  (22)

For the phase space of n-body decays, the four-body one can be generated recursively by the two-body subprocesses. The two-body phase space can be expressed as

$$\begin{aligned} d\Phi _2(p\rightarrow p_1p_2)=\frac{1}{(2\pi )^5}\frac{|\vec {p}_1|d\cos \theta }{4\sqrt{{\hat{s}}}}, \end{aligned}$$

(23)

where $\sqrt{{\hat{s}}}=\sqrt{p^2}$ is the center-of-mass energy, and $\theta$ is the angle between the two final states. Based on this, the three-body phase space of $B_1^{(\prime )}\rightarrow B_2^{(\prime )}\ell ^+\nu _{\ell }$ can be written as

$$\begin{aligned}&d\Phi _3\left( B_1^{(\prime )}\rightarrow B_2^{(\prime )} \ell ^+\!\nu _{\ell }\right) \!=\!(2\pi )^3dq^2\times d\Phi _2(W^+\rightarrow \ell ^+\nu _{\ell })\nonumber \\&\quad \times d\Phi _2(B_1^{(\prime )}\rightarrow B_2^{(\prime )}W^+) \nonumber \\&=\frac{(1-{\hat{m}}_l^2)\sqrt{\lambda (m_1,m_2,q)}}{(2\pi )^{7}32m_1^2}dq^2d\cos \theta _l, \end{aligned}$$

(24)

where ${\hat{m}}_l=m_l/\sqrt{q^2}$. Similarly, the total four-body phase space is

$$\begin{aligned}&d\Phi _4\left( B_1^{(\prime )}\rightarrow B_3^{(\prime )}B_4^{(\prime )}\ell ^+\nu _\ell \right) \nonumber \\&\quad =(2\pi )^3dp_2^2\times d\Phi _3\left( B_1^{(\prime )}\rightarrow B_2^{(\prime )} \ell ^+\nu _{\ell }\right) \nonumber \\&\qquad \times d\Phi _2\left( B_2^{(\prime )}\rightarrow B_3^{(\prime )}B_4^{(\prime )}\right) \nonumber \\&\quad =\frac{(1-{\hat{m}}_l^2)\sqrt{\lambda (m_1,m_2,q)\lambda (m_2,m_3,m_4)}}{(2\pi )^{10}256m_1^2m_2^2}\nonumber \\&\qquad \times dq^2dp_2^2d\cos \theta _hd\cos \theta _ld\phi . \end{aligned}$$

(25)

Note that the integration variable $p_2^2$ is artificially introduced from the insertion of intermediate state $B_2^{(\prime )}$, and in the narrow-width limit, this integration will be conducted as

$$\begin{aligned} \int dq_2^2\frac{m_2\Gamma _2}{\pi }\frac{1}{(q_2^2-m_2^2)^2+m_2^2\Gamma _2^2}=1, \end{aligned}$$

(26)

while

$$\begin{aligned}&\int dq_2^2\frac{m_2\Gamma \left( B_2^{(\prime )}\rightarrow B_3^{(\prime )}B_4^{(\prime )}\right) }{\pi }\frac{1}{(q_2^2-m_2^2)^2+m_2^2\Gamma _2^2}\nonumber \\&\quad ={\mathcal {B}}\left( B_2^{(\prime )}\rightarrow B_3^{(\prime )}B_4^{(\prime )}\right) , \end{aligned}$$

(27)

where $\Gamma \left( B_2^{(\prime )}\rightarrow B_3^{(\prime )}B_4^{(\prime )}\right)$ and ${\mathcal {B}}\left( B_2^{(\prime )}\rightarrow B_3^{(\prime )}B_4^{(\prime )}\right)$ are the total width and branching fraction of the subprocess $B_2^{(\prime )}\rightarrow B_3^{(\prime )}B_4^{(\prime )}$, respectively.

Combining the four-body phase space in Eq. (25) and squared amplitude in Eq. (15), we can write the differential decay width of the four-body process $B_1^{(\prime )}\rightarrow B_3^{(\prime )}B_4^{(\prime )}\ell ^+\nu _{\ell }$:

$$\begin{aligned}&\frac{d\Gamma (B_1^{(\prime )}\rightarrow B_3^{(\prime )}B_4^{(\prime )}\ell ^+\nu _{\ell })}{dq^2d\cos \theta _hd\cos \theta _ld\phi } \nonumber \\&\quad =\frac{G_F^2|V_{cs}|^2(1-{\hat{m}}_l^2)\sqrt{\lambda (m_1,m_2,q)\lambda (m_2,m_3,m_4)}}{(2\pi )^{5}4096m_1^3m_2^3}\nonumber \\&\qquad \times \frac{{\mathcal {B}}\left( B_2^{(\prime )}\rightarrow B_3^{(\prime )}B_4^{(\prime )}\right) }{\Gamma \left( B_2^{(\prime )}\rightarrow B_3^{(\prime )}B_4^{(\prime )}\right) } \nonumber \\&\qquad \times \sum _{s_1}\sum _{s_3,s_l,s_{\nu }} \left| \sum _{s_2}\sum _{s_W=\{\pm ,0,t\}}H_1(s_1,s_2,s_W)\right. \nonumber \\&\qquad \left. \times L(s_{\ell }, s_{\nu },s_W) \times H_2(s_2,s_3)\right| ^2. \end{aligned}$$

(28)

2.3 Leptonic part

In this section, we will focus on the Lorentz invariant leptonic part defined in Eq. (12):

$$\begin{aligned} L(s_{\ell }, s_{\nu },s_W)\equiv&~{\bar{u}}_{\nu }(p_{\nu },s_{\nu }) \gamma ^{\nu }\left( 1-\gamma _{5}\right) v_{\ell }(p_{\ell },s_{\ell }) \varepsilon _{\nu }(s_W). \end{aligned}$$

(29)

Combining each component of spinors of leptons and polarization vectors of the $W^+$ boson, we can obtain the following nonvanishing matrix elements:

$$\begin{aligned}&L\left( s_{\ell } =+\frac{1}{2}, s_{\nu }=-\frac{1}{2}, s_{W}=0\right) \nonumber \\&\quad =-\sqrt{2} \sin \theta _{\ell } N_{\ell }, \end{aligned}$$

(30)

$$\begin{aligned}&L\left( s_{\ell }=+\frac{1}{2}, s_{\nu }=-\frac{1}{2}, s_{W}=+1\right) \nonumber \\&\quad =-e^{-i \phi }\left( 1-\cos \theta _{\ell }\right) N_{\ell }, \end{aligned}$$

(31)

$$\begin{aligned}&L\left( s_{\ell }=+\frac{1}{2}, s_{\nu }=-\frac{1}{2}, s_{W}=-1\right) \nonumber \\&\quad =-e^{i \phi }\left( 1+\cos \theta _{\ell }\right) N_{\ell }, \end{aligned}$$

(32)

$$\begin{aligned}&L\left( s_{\ell }=-\frac{1}{2}, s_{\nu }=-\frac{1}{2}, s_{W}=0\right) \nonumber \\&\quad =-\sqrt{2} {\hat{m}}_{\ell } \cos \theta _{\ell } N_{\ell } ,\ \end{aligned}$$

(33)

$$\begin{aligned}&L\left( s_{\ell }=-\frac{1}{2}, s_{\nu }=-\frac{1}{2}, s_{W}=+1\right) \nonumber \\&\quad =-{\hat{m}}_{\ell } e^{-i \phi } \sin \theta _{\ell } N_{\ell }, \end{aligned}$$

(34)

$$\begin{aligned}&L\left( s_{\ell }=-\frac{1}{2}, s_{\nu }=-\frac{1}{2}, s_{W}=-1\right) \nonumber \\&\quad ={\hat{m}}_{\ell } e^{i \phi } \sin \theta _{\ell } N_{\ell }, \end{aligned}$$

(35)

$$\begin{aligned}&L\left( s_{\ell }=-\frac{1}{2}, s_{\nu }=-\frac{1}{2}, s_{W}=t\right) \nonumber \\&\quad =\sqrt{2} {\hat{m}}_{\ell } N_{\ell }, \end{aligned}$$

(36)

where the factor $N_{\ell }=i\sqrt{2\left( q^2-m_l^2\right) }$ and ${\hat{m}}_l=m_l/\sqrt{q^2}$.

2.4 Hadronic part $H_1$ with spin-1/2 intermediate state

If $B_1$ is a singly charmed baryon antitriplet, the transition matrix elements with weak current $J^{\mu }=V^{\mu }-A^{\mu }={\bar{s}}\gamma ^{\mu }(1-\gamma _5)c$ in Eq. (13) can be parameterized as [57, 72]

$$\begin{aligned}&\left\langle B_{2}\left( p_{2}, s_{2}\right) \left| V^{\mu }\right| B_{1}\left( p_{1}, s_{1}\right) \right\rangle ={\bar{u}}\left( p_{2}, s_{2}\right) \nonumber \\&\quad \times \left[ \gamma ^{\mu } f_{1}\left( q^{2}\right) +i \sigma ^{\mu \nu } \frac{q_{\nu }}{m_{1}} f_{2}\left( q^{2}\right) +\frac{q^{\mu }}{m_{1}} f_{3}\left( q^{2}\right) \right] \nonumber \\&\qquad u\left( p_{1}, s_{1}\right) , \end{aligned}$$

(37)

$$\begin{aligned}&\quad \times \left\langle B_{2}\left( p_{2}, s_{2}\right) \left| A^{\mu }\right| B_{1}\left( p_{1}, s_{1}\right) \right\rangle ={\bar{u}}\left( p_{2}, s_{2}\right) \nonumber \\&\quad \times \left[ \gamma ^{\mu } g_{1}\left( q^{2}\right) +i \sigma ^{\mu \nu } \frac{q_{\nu }}{m_{1}} g_{2}\left( q^{2}\right) +\frac{q^{\mu }}{m_{1}} g_{3}\left( q^{2}\right) \right] \nonumber \\&\qquad \gamma _{5} u\left( p_{1}, s_{1}\right) , \end{aligned}$$

(38)

with the momentum transfer $q^{\mu }=p_{1}^{\mu }-p_{2}^{\mu }$ and $\sigma ^{\mu \nu }=i[\gamma ^{\mu },\gamma ^{\nu }]/2$. The form factors $f_i$ and $g_{i}$ are functions of $q^2$, and the relations between these form factors and other parameterizations are collected in the Appendix.

By combining each spin component of $B_1$, $B_2$, and $W^+$, we give the nonzero terms of $H_1(s_1,s_2,s_W)$ as

$$\begin{aligned}&H_{1 V}\left( s_1=-\frac{1}{2}, s_2=\frac{1}{2}, s_W=1\right) \nonumber \\&\quad =H_{1 V}\left( s_1=\frac{1}{2}, s_2=-\frac{1}{2}, s_W=-1\right) \nonumber \\&\quad =\sqrt{2 s_{-}}\left[ f_{1}(q^2)-\frac{m_{1}+m_{2}}{m_{1}} f_{2}(q^2)\right] , \end{aligned}$$

(39)

$$\begin{aligned}&H_{1 V}\left( s_1=\frac{1}{2}, s_2=\frac{1}{2}, s_W=0\right) \nonumber \\&\quad =H_{1 V}\left( s_1=-\frac{1}{2}, s_2=-\frac{1}{2}, s_W=0\right) \nonumber \\&\quad =\sqrt{\frac{s_{-}}{q^{2}}}\left[ \left( m_{1}+m_{2}\right) f_{1}(q^2)-\frac{q^{2}}{m_{1}} f_{2}(q^2)\right] , \end{aligned}$$

(40)

$$\begin{aligned}&H_{1 V}\left( s_1=\frac{1}{2}, s_2=\frac{1}{2}, s_W=t\right) \nonumber \\&\quad =H_{1 V}\left( s_1=-\frac{1}{2}, s_2=-\frac{1}{2}, s_W=t\right) \nonumber \\&\quad =\sqrt{\frac{s_{+}}{q^{2}}}\left[ \left( m_{1}-m_{2}\right) f_{1}(q^2)+\frac{q^{2}}{m_{1}} f_{3}(q^2)\right] , \end{aligned}$$

(41)

$$\begin{aligned}&H_{1 A}\left( s_1=-\frac{1}{2}, s_2=\frac{1}{2}, s_W=1\right) \nonumber \\&\quad =-H_{1 A}\left( s_1=\frac{1}{2}, s_2=-\frac{1}{2}, s_W=-1\right) \nonumber \\&\quad =\sqrt{2 s_{+}}\left[ g_{1}(q^2)+\frac{m_{1}-m_{2}}{m_{1}} g_{2}(q^2)\right] , \end{aligned}$$

(42)

$$\begin{aligned}&H_{1 A}\left( s_1=\frac{1}{2}, s_2=\frac{1}{2}, s_W=0\right) \nonumber \\&\quad =-H_{1 A}\left( s_1=-\frac{1}{2}, s_2=-\frac{1}{2}, s_W=0\right) \nonumber \\&\quad =\sqrt{\frac{s_{+}}{q^{2}}}\left[ \left( m_{1}-m_{2}\right) g_{1}(q^2)+\frac{q^{2}}{m_{1}} g_{2}(q^2)\right] , \end{aligned}$$

(43)

$$\begin{aligned}&H_{1 A}\left( s_1=\frac{1}{2}, s_2=\frac{1}{2}, s_W=t\right) \nonumber \\&\quad =-H_{1 A}\left( s_1=-\frac{1}{2}, s_2=-\frac{1}{2}, s_W=t\right) \nonumber \\&\quad =\sqrt{\frac{s_{-}}{q^{2}}}\left[ \left( m_{1}+m_{2}\right) g_{1}(q^2)-\frac{q^{2}}{m_{1}} g_{3}(q^2)\right] , \end{aligned}$$

(44)

in which the abbreviation $s_{\pm }$ has been defined in Eq. (16).

Reversing the helicities gives the same results for the vector current, but an opposite sign exists for the axial vector current. The total amplitudes are

$$\begin{aligned} H_{1}(s_1,s_2,s_W)= H_{1V}(s_1,s_2,s_W)- H_{1A}(s_1,s_2,s_W). \end{aligned}$$

(45)

2.5 Hadronic part $H_1$ with a spin-3/2 intermediate state

The charmed baryons $B_1'=\Omega _c^0$ will weakly decay into spin-3/2 baryon decuplets $\Omega ^-$, and the transition matrix elements for this process can be parametrized as

$$\begin{aligned}&\left\langle B_{2}^{\prime }\left( p_{2}, s_{2}\right) \left| V^{\mu }\right| B_{1}^{\prime }\left( p_{1}, s_{1}\right) \right\rangle \nonumber \\&\quad ={\bar{U}}_{\alpha }\left( p_{2}, s_2\right) \left[ \gamma ^{\mu } p_{1}^{\alpha } \frac{f_{1}\left( q^{2}\right) }{m_{1}}+\frac{f_{2}\left( q^{2}\right) }{m_{1}^{2}} p_{1}^{\alpha } p_{1}^{\mu } \right. \nonumber \\&\qquad \left. \left. +\frac{f_{3}\left( q^{2}\right) }{m_{1} m_{2}} p_{1}^{\alpha } p_{2}^{\mu }+f_{4}\left( q^{2}\right) g^{\alpha \mu }\right] \gamma ^{5} u\left( p_{1}, s_{1}\right) ,\right. \end{aligned}$$

(46)

$$\begin{aligned}&\qquad \left\langle B_{2}^{\prime }\left( p_{2}, s_2\right) \left| A^{\mu }\right| B_{1}^{\prime }\left( p_{1}, s_{1}\right) \right\rangle \nonumber \\&\quad ={\bar{U}}_{\alpha }\left( p_{2}, s_2\right) \left[ \gamma ^{\mu } p_{1}^{\alpha } \frac{g_{1}\left( q^{2}\right) }{m_{1}}+\frac{g_{2}\left( q^{2}\right) }{m_{1}^{2}} p_{1}^{\alpha } p_{1}^{\mu } \right. \nonumber \\&\qquad \left. \left. +\frac{g_{3}\left( q^{2}\right) }{s_2m_{1} m_{2}} p_{1}^{\alpha } p_{2}^{\mu }+g_{4}\left( q^{2}\right) g^{\alpha \mu }\right] u\left( p_{1}, s_{1}\right) .\right. \end{aligned}$$

(47)

The definition of vectorial spinor $U_{\alpha }(\vec {p},S_z)$ for the spin-3/2 baryon is shown in the Appendix. Therefore, the nonzero terms of $H_1(s_1,s_2,s_W)$ are collected as

$$\begin{aligned}&H_{1V}\left( s_{1}=\frac{1}{2}, s_2=\frac{3}{2}, s_{W}=1\right) \nonumber \\&\quad =-H_{1V}\left( s_{1}=-\frac{1}{2}, s_2=-\frac{3}{2}, s_{W}=-1\right) \nonumber \\&\quad =\sqrt{s_{-}} f_{4}\left( q^{2}\right) , \end{aligned}$$

(48)

$$\begin{aligned}&H_{1V}\left( s_{1}=-\frac{1}{2}, s_2=\frac{1}{2}, s_{W}=1\right) \nonumber \\&\quad =-H_{1V}\left( s_{1}=\frac{1}{2}, s_2=-\frac{1}{2}, s_{W}=-1\right) \nonumber \\&\quad =\sqrt{\frac{s_{-}}{3}}\left( \frac{s_{+}}{m_{1} m_{2}} f_{1}\left( q^{2}\right) -f_{4}\left( q^{2}\right) \right) , \end{aligned}$$

(49)

$$\begin{aligned}&H_{1V}\left( s_{1}=\frac{1}{2}, s_2=\frac{1}{2}, s_{W}=0\right) \nonumber \\&\quad =-H_{1V}\left( s_{1}=-\frac{1}{2}, s_2=-\frac{1}{2}, s_{W}=0\right) \nonumber \\&\quad = \sqrt{\frac{s_{-}}{6 q^{2}}}\left[ \frac{\left( m_{1}-m_{2}\right) s_{+}}{m_{1} m_{2}} f_{1}\left( q^{2}\right) \right. \nonumber \\&\qquad - \frac{\lambda \left( m_{1}, m_{2}, q\right) }{2 m_{1} m_{2}}\left( \frac{1}{m_{1}} f_{2}\left( q^{2}\right) +\frac{1}{m_{2}} f_{3}\left( q^{2}\right) \right) \nonumber \\&\qquad \left. -\frac{m_{1}^{2}-m_{2}^{2}-q^{2}}{m_{2}} f_{4}\left( q^{2}\right) \right] , \end{aligned}$$

(50)

$$\begin{aligned}&H_{1V}\left( s_{1}=\frac{1}{2}, s_2=\frac{1}{2}, s_{W}=t\right) \nonumber \\&\quad =-H_{1V}\left( s_{1}=-\frac{1}{2}, s_2=-\frac{1}{2}, s_{W}=t\right) \nonumber \\&\quad =\sqrt{\frac{s_{+}}{q^{2}}} \frac{s_{-}}{\sqrt{6} m_{2}}\left[ \frac{m_{1}+m_{2}}{m_{1}} f_{1}\left( q^{2}\right) \right. \nonumber \\&\qquad -\frac{m_{1}^{2}-m_{2}^{2}+q^{2}}{2 m_{1}^{2}} f_{2}\left( q^{2}\right) \nonumber \\&\qquad \left. -\frac{m_{1}^{2}-m_{2}^{2}-q^{2}}{2 m_{1} m_{2}} f_{3}\left( q^{2}\right) -f_{4}\left( q^{2}\right) \right] , \ \end{aligned}$$

(51)

$$\begin{aligned}&H_{1A}\left( s_{1}=\frac{1}{2}, s_2=\frac{3}{2}, s_{W}=1\right) \nonumber \\&\quad =H_{1A}\left( s_{1}=-\frac{1}{2}, s_2=-\frac{3}{2}, s_{W}=-1\right) \nonumber \\&\quad =-\sqrt{s_{+}} g_{4}\left( q^{2}\right) , \end{aligned}$$

(52)

$$\begin{aligned}&H_{1A}\left( s_{1}=-\frac{1}{2}, s_2=\frac{1}{2}, s_{W}=1\right) \nonumber \\&\quad =H_{1A}\left( s_{1}=\frac{1}{2}, s_2=-\frac{1}{2}, s_{W}=-1\right) \nonumber \\&\quad =\sqrt{\frac{s_{+}}{3}}\left( \frac{s_{-}}{m_{1} m_{2}} g_{1}\left( q^{2}\right) -g_{4}\left( q^{2}\right) \right) , \end{aligned}$$

(53)

$$\begin{aligned}&H_{1A}\left( s_{1}=\frac{1}{2}, s_2=\frac{1}{2}, s_{W}=0\right) \nonumber \\&\quad =H_{1A}\left( s_{1}=-\frac{1}{2}, s_2=-\frac{1}{2}, s_{W}=0\right) \nonumber \\&\quad =\sqrt{\frac{s_{+}}{6 q^{2}}}\left[ \frac{\left( m_{1}+m_{2}\right) s_{-}}{m_{1} m_{2}} g_{1}\left( q^{2}\right) \right. \nonumber \\&\qquad +\frac{\lambda \left( m_{1}, m_{2}, q\right) }{2 m_{1} m_{2}}\left( \frac{1}{m_{1}} g_{2}\left( q^{2}\right) +\frac{1}{m_{2}} g_{3}\left( q^{2}\right) \right) \nonumber \\&\qquad \left. +\frac{m_{1}^{2}-m_{2}^{2}-q^{2}}{m_{2}} g_{4}\left( q^{2}\right) \right] , \end{aligned}$$

(54)

$$\begin{aligned}&H_{1A}\left( s_{1}=\frac{1}{2}, s_2=\frac{1}{2}, s_{W}=t\right) \nonumber \\&\quad =H_{1A}\left( s_{1}=-\frac{1}{2}, s_2=-\frac{1}{2}, s_{W}=t \right) \nonumber \\&\quad =\sqrt{\frac{s_{-}}{q^{2}}} \frac{s_{+}}{\sqrt{6} m_{2}}\left[ \frac{m_{1}-m_{2}}{m_{1}} g_{1}\left( q^{2}\right) \right. \nonumber \\&\qquad +\frac{m_{1}^{2}-m_{2}^{2}+q^{2}}{2 m_{1}^{2}} g_{2}\left( q^{2}\right) \nonumber \\&\qquad +\left. \frac{m_{1}^{2}-m_{2}^{2}-q^{2}}{2 m_{1} m_{2}} g_{3}\left( q^{2}\right) +g_{4}\left( q^{2}\right) \right] , \end{aligned}$$

(55)

and the total transition amplitudes give

$$\begin{aligned} H_1\left( s_{1}, s_2, s_{W}\right) =H_{1A}\left( s_{1}, s_2, s_{W}\right) -H_{1V}\left( s_{1}, s_2, s_{W}\right) . \end{aligned}$$

(56)

2.6 Hadronic part $H_2$ with spin-1/2 intermediate state

The amplitude for a spin-1/2 charm baryon $B_2$ decaying into a spin-1/2 baryon $B_3$, and a spin-0 meson $B_4$ can be written in the form [29]

$$\begin{aligned} i{\mathcal {M}}=iG_Fm_4^2{\bar{u}}_{B_3}(p_3,s_3)(A-B\gamma _5)u_{B_2}(p_2,s_2), \end{aligned}$$

(57)

which corresponds to the hadronic coupling constant $g_H=G_Fm_4^2$ in Eq. (14) with low-energy effective theory. Here, $G_F$ is a standard model electroweak coupling constant. A and B are factors weighting the contributions from scalar and pesudoscalar operators [30]. It is convenient to introduce the asymmetry parameter

$$\begin{aligned} \alpha =\frac{2 {\text {Re}}\left[ s^{*} r\right] }{|s|^{2}+|r|^{2}}, \end{aligned}$$

(58)

with $s=A$ and $r=B\times |\vec {p}_3|/(E_3+m_3)$, and follow $(s\pm r)^2=(|s|^{2}+|r|^{2})(1\pm \alpha )$.

Therefore, combining each spin component of $B_2$ and $B_3$, we get the following hadronic matrix elements $H_2(s_2,s_3)$:

$$\begin{aligned} H_{2}\left( s_{2}=\frac{1}{2}, s_{3}=\frac{1}{2}\right)&=N_{h}(s+r) \cos \left( \theta _{h} / 2\right) , \end{aligned}$$

(59)

$$\begin{aligned} H_{2}\left( s_{2}=\frac{1}{2}, s_{3}=-\frac{1}{2}\right)&=-N_{h}(s-r) \sin \left( \theta _{h} / 2\right) , \end{aligned}$$

(60)

$$\begin{aligned} H_{2}\left( s_{2}=-\frac{1}{2}, s_{3}=\frac{1}{2}\right)&=N_{h}(s+r) \sin \left( \theta _{h} / 2\right) , \end{aligned}$$

(61)

$$\begin{aligned} H_{2}\left( s_{2}=-\frac{1}{2}, s_{3}=-\frac{1}{2}\right)&=N_{h}(s-r) \cos \left( \theta _{h} / 2\right) , \end{aligned}$$

(62)

with $N_h=G_Fm_{\pi }^2\sqrt{2m_2(E_3+m_3)}$. Using the two-body phase space and squared matrix elements, we obtain the spin-averaged decay width

$$\begin{aligned} \Gamma \left( B_{2} \rightarrow B_{3} B_{4}\right)&=\int _{-1}^{1} \frac{1}{2 m_{2}}(2 \pi )^{4} \frac{1}{(2 \pi )^{5}} \frac{\left| \vec {p}_{3}\right| d \cos \theta _{h}}{4 m_{2}}\nonumber \\&\quad \times \frac{1}{2} \overline{\left| {\mathcal {M}}\left( B_{2} \rightarrow B_{3} B_{4}\right) \right| ^{2}} \nonumber \\&=\frac{N_{h}^{2} \sqrt{\lambda \left( m_{2}, m_{3}, m_{4}\right) }}{16 \pi m_{2}^{3}}\left( |s|^{2}+|r|^{2}\right) . \end{aligned}$$

(63)

2.7 Hadronic part $H_2$ with spin-3/2 intermediate state

The transition for a spin-3/2 baryon $\Omega ^-$ decaying into a spin-1/2 baryon $\Lambda$ and a spin-0 pseudoscalar meson $K^-$ is parametrized as [73]

$$\begin{aligned} i{\mathcal {M}}=i\frac{g_{h}}{m_4}{\bar{u}}_{B'_3}(p_3,s_3)(A-B\gamma _5)U^{\alpha }_{B'_2}(p_2,s_2)p_{4\alpha }, \end{aligned}$$

(64)

where $g_h$ denotes the coupling constant of the hadronic vertex $\Omega \Lambda {K}$. This nonperturbative parameter is usually determined from the experimental partial decay width of the relevant process, but here we can absorb it into the spin-averaged decay width in Eq. (73) to avoid the uncertainties and model dependence of nonperturbative methods.

Using the spinors and vectorial spinors, we can obtain the following terms:

$$\begin{aligned}&H_{2}\left( s_{2}=\frac{3}{2}, s_{3}=\frac{1}{2}\right) \nonumber \\&\quad =N_{h}^{\prime }\left| \vec {p}_{3}\right| (s+r) \sin \theta _{h} \cos \frac{\theta _{h}}{2}, \end{aligned}$$

(65)

$$\begin{aligned}&H_{2}\left( s_{2}=\frac{3}{2}, s_{3}=-\frac{1}{2}\right) \nonumber \\&\quad =-N_{h}^{\prime }\left| \vec {p}_{3}\right| (s-r) \sin \theta _{h} \sin \frac{\theta _{h}}{2}, \end{aligned}$$

(66)

$$\begin{aligned}&H_{2}\left( S_{\Omega }=-\frac{3}{2}, s_{3}=\frac{1}{2}\right) \nonumber \\&\quad =-N_{h}^{\prime }\left| \vec {p}_{3}\right| (s+r) \sin \theta _{h} \sin \frac{\theta _{h}}{2}, \end{aligned}$$

(67)

$$\begin{aligned}&H_{2}\left( s_{2}=-\frac{3}{2}, s_{3}=-\frac{1}{2}\right) \nonumber \\&\quad =-N_{h}^{\prime }\left| \vec {p}_{3}\right| (s-r) \sin \theta _{h} \cos \frac{\theta _{h}}{2}, \end{aligned}$$

(68)

$$\begin{aligned}&H_{2}\left( s_{2}=\frac{1}{2}, s_{3}=\frac{1}{2}\right) \nonumber \\&\quad =\frac{\sqrt{3}}{3} N_{h}^{\prime }\left| \vec {p}_{3}\right| (s+r) \cos \frac{\theta _{h}}{2}\left( 1-3 \cos \theta _{h}\right) , \end{aligned}$$

(69)

$$\begin{aligned}&H_{2}\left( s_{2}=\frac{1}{2}, s_{3}=-\frac{1}{2}\right) \nonumber \\&\quad =\frac{\sqrt{3}}{3} N_{h}^{\prime }\left| \vec {p}_{3}\right| (s-r) \sin \frac{\theta _{h}}{2}\left( 1+3 \cos \theta _{h}\right) , \end{aligned}$$

(70)

$$\begin{aligned}&H_{2}\left( s_{2}=-\frac{1}{2}, s_{3}=\frac{1}{2}\right) \nonumber \\&\quad =-\frac{\sqrt{3}}{3} N_{h}^{\prime }\left| \vec {p}_{3}\right| (s+r) \sin \frac{\theta _{h}}{2}\left( 1+3 \cos \theta _{h}\right) , \end{aligned}$$

(71)

$$\begin{aligned}&H_{2}\left( s_{2}=-\frac{1}{2}, s_{3}=-\frac{1}{2}\right) \nonumber \\&\quad =\frac{\sqrt{3}}{3} N_{h}^{\prime }\left| \vec {p}_{3}\right| (s-r) \cos \frac{\theta _{h}}{2}\left( 1-3 \cos \theta _{h}\right) , \end{aligned}$$

(72)

where $N_h'=g_h\sqrt{(E_3+m_3)m_2}/m_4$. The spin-averaged decay width of the process is given as

$$\begin{aligned} \Gamma \left( B_{2}^{\prime } \rightarrow B_{3}^{\prime } B_{4}^{\prime }\right)&=\int _{-1}^{1} \frac{1}{2 m_{2}}(2 \pi )^{4}\nonumber \\&\quad \times \frac{1}{(2 \pi )^{5}} \frac{\left| \vec {p}_{3}\right| d \cos \theta _{h}}{4 m_{2}} \nonumber \\&\quad \frac{1}{4}\overline{\left| {\mathcal {M}}\left( B_{2}^{\prime } \rightarrow B_{3}^{\prime } B_{4}^{\prime }\right) \right| ^{2}} \nonumber \\&=\frac{N_{h}^{\prime 2}\left| \vec {p}_{3}\right| ^{3}}{12 \pi m_{2}^{2}}\left( |s|^{2}+|r|^{2}\right) . \end{aligned}$$

(73)

3 Theoretical results of differential decay width

3.1 Differential decay width of $(\Lambda _c^+, \Xi _c^0)$

Combining the leptonic matrix elements in Eqs. (30)–(36) and hadronic ones $H_1$ in Eqs. (39)–(45), together with the three-body phase space in Eq. (24), we can obtain the three-body decay width of $B_1\rightarrow B_2\ell ^+\nu _{\ell }$ processes, in which $B_1=(\Lambda _c^+, \Xi _c^0)$ and corresponding $B_2=(\Lambda , \Xi ^-)$:

$$\begin{aligned}&\frac{d \Gamma \left( B_{1} \rightarrow B_{2} l^{+} \nu _{\ell }\right) }{d \cos \theta _{\ell } d q^{2}} =\frac{q^{2}\sqrt{\lambda \left( m_{1}, m_{2}, q\right) }}{1024 \pi ^{3}m_{1}^{3}}\nonumber \\&\qquad \times \left( 1-{\hat{m}}_{\ell }\right) ^{2}\frac{G_{F}^{2}}{2}\left| V_{\mathrm {cs}}\right| ^{2} \nonumber \\&\qquad \times \left\{ \left| H_{\frac{1}{2}, 1}\right| ^{2}\left[ \left( 1-\cos \theta _{\ell }\right) ^{2}+{\hat{m}}_{\ell }^{2} \sin ^{2} \theta _{\ell }\right] \right. \nonumber \\&\qquad +\left| H_{-\frac{1}{2},-1}\right| ^{2}\left[ \left( 1+\cos \theta _{\ell }\right) ^{2}+{\hat{m}}_{\ell }^{2} \sin ^{2} \theta _{\ell }\right] \nonumber \\&\qquad +\left( \left| H_{\frac{1}{2}, 0}\right| ^{2}+\left| H_{-\frac{1}{2}, 0}\right| ^{2}\right) 2\left( \sin ^{2} \theta _{\ell }+{\hat{m}}_{\ell } \cos ^{2} \theta _{\ell }\right) \nonumber \\&\qquad +\left( \left| H_{\frac{1}{2}, t}\right| ^{2}+\left| H_{-\frac{1}{2}, t}\right| ^{2}\right) 2 {\hat{m}}_{\ell }^{2}\nonumber \\&\qquad \left. +2\left( \left| H_{\frac{1}{2}, 0}H_{\frac{1}{2}, t}^{*}\right| +\left| H_{-\frac{1}{2}, 0}H_{-\frac{1}{2}, t}^{*}\right| \right) 2 {\hat{m}}_{\ell }^{2}\cos \theta _{l}\right\} , \end{aligned}$$

(74)

in which $H_{s_2,s_W}$ are short for the hadronic amplitudes $H(s_1, s_2, s_W)$ defined in Eqs. (39)–(44).

Additionally, taking the hadronic matrix elements $H_2$ from Eqs. (59)–(62) into account, we can obtain the four-body differential decay width for $B_1\rightarrow B_3B_4\ell ^+\nu _{\ell }$ defined in Eq. (28):

$$\begin{aligned}&\frac{d \Gamma \left( B_{1} \rightarrow B_{3} B_{4} l^{+} \nu _{\ell }\right) }{d \cos \theta _{\ell } d \cos \theta _{h} d \phi d q^{2}}\nonumber \\&\quad =\!\frac{G_{F}^{2}\left| V_{\mathrm {cs}}\right| ^{2} q^{2}\left( 1\!-\!{\hat{m}}_{\ell }^{2}\right) \sqrt{\lambda \left( m_{1}, m_{2}, q\right) }}{(2 \pi )^{4} 256 m_{1}^{3}} {\mathcal {B}}\left( B_{2} \rightarrow B_{3} B_{4}\right) \nonumber \\&\qquad \times \left\{ \left| H_{\frac{1}{2}, 1}\right| ^{2}\left( 1+\alpha \cos \theta _{h} \right) \left[ {\hat{m}}_{\ell }^{2} \sin ^{2} \theta _{\ell }+\left( 1-\cos \theta _{\ell }\right) ^{2}\right] \right. \nonumber \\&\qquad +\!\left| H_{\!-\!\frac{1}{2},\!-\!1}\right| ^{2}\left( 1-\alpha \cos \theta _{h} \right) \left[ {\hat{m}}_{\ell }^{2} \sin ^{2} \theta _{\ell }\!+\!\left( 1\!+\!\cos \theta _{\ell }\right) ^{2}\right] \nonumber \\&\qquad +2\left| H_{\frac{1}{2}, 0}\right| ^{2}\left( 1+\alpha \cos \theta _{h} \right) \left[ {\hat{m}}_{\ell }^{2} \cos ^{2} \theta _{\ell }+\sin ^{2} \theta _{\ell }\right] \nonumber \\&\qquad +2\left| H_{-\frac{1}{2}, 0}\right| ^{2}\left( 1-\alpha \cos \theta _{h} \right) \left[ {\hat{m}}_{\ell }^{2} \cos ^{2} \theta _{\ell }+\sin ^{2} \theta _{\ell }\right] \nonumber \\&\qquad +2\left| H_{\frac{1}{2}, t}\right| ^{2}\left( 1+\alpha \cos \theta _{h} \right) {\hat{m}}_{\ell }^{2} \nonumber \\&\qquad +2\left| H_{-\frac{1}{2}, t}\right| ^{2}\left( 1-\alpha \cos \theta _{h} \right) {\hat{m}}_{\ell }^{2} \nonumber \\&\qquad +2 \sqrt{2} \alpha \sin \theta _{h} \sin \theta _{\ell } \cos \phi \left| H_{\frac{1}{2}, 1} H_{-\frac{1}{2}, 0}^{*}\right| \nonumber \\&\qquad \times \left( 1-\cos \theta _{\ell }+{\hat{m}}_{\ell }^{2} \cos \theta _{\ell }\right) \nonumber \\&\qquad +2 \sqrt{2} \alpha \sin \theta _{h} \sin \theta _{\ell } \cos \phi \left| H_{-\frac{1}{2},-1} H_{\frac{1}{2}, 0}^{*}\right| \nonumber \\&\qquad \times \left( 1+\cos \theta _{\ell }-{\hat{m}}_{\ell }^{2} \cos \theta _{\ell }\right) \nonumber \\&\qquad +2 \sqrt{2} \alpha \sin \theta _{h} \sin \theta _{\ell } \cos \phi \mid H_{\frac{1}{2}, 1} H_{-\frac{1}{2}, t}^{*} {\hat{m}}_{\ell }^{2}\nonumber \\&\qquad -2 \sqrt{2} \alpha \sin \theta _{h} \sin \theta _{\ell } \cos \phi \left| H_{-\frac{1}{2},-1} H_{\frac{1}{2}, t}^{*}\right| {\hat{m}}_{\ell }^{2} \nonumber \\&\qquad +4 {\hat{m}}_{\ell }^{2} \cos \theta _{\ell }\left| H_{-\frac{1}{2}, 0} H_{-\frac{1}{2}, t}^{*}\right| \left( 1-\alpha \cos \theta _{h} \right) \nonumber \\&\qquad \left. +4 {\hat{m}}_{\ell }^{2} \cos \theta _{\ell }\left| H_{\frac{1}{2}, 0} H_{\frac{1}{2}, t}^{*}\right| \left( 1+\alpha \cos \theta _{h} \right) \right\} . \end{aligned}$$

(75)

By integrating out the parameters step by step, we can get the results of $q^2$-, $\theta _l$-, $\theta _h$- and $\phi$-dependence forms of total decay width, respectively:

• $q^2$-dependence:

  $$\begin{aligned} \frac{d \Gamma }{d q^{2}}&= \frac{\pi G_{F}^{2}\left| V_{\mathrm {cs}}\right| ^{2} q^{2}\left( 1-{\hat{m}}_{\ell }^{2}\right) \sqrt{\lambda \left( m_{1}, m_{2}, q\right) }}{(2 \pi )^{4} 48 m_{1}^{3}}\nonumber \\&\times {\mathcal {B}}\left( B_{2} \rightarrow B_{3} B_{4}\right) \nonumber \\&\quad \times \left[ \left( 2\!+\!{\hat{m}}_{\ell }^{2}\right) \left( \left| H_{-\frac{1}{2},-1}\right| ^{2}\!+\!\left| H_{\frac{1}{2}, 1}\right| ^{2}\!+\!\left| H_{-\frac{1}{2}, 0}\right| ^{2}\right. \right. \nonumber \\&\quad \left. \left. +\left| H_{\frac{1}{2}, 0}\right| ^{2}\right) +3 {\hat{m}}_{\ell }^{2}\left( \left| H_{-\frac{1}{2}, t}\right| ^{2}+\left| H_{\frac{1}{2}, t}\right| ^{2}\right) \right] , \end{aligned}$$

  (76)

• $\theta _l$-dependence:

  $$\begin{aligned}&\frac{d \Gamma }{d \cos \theta _{\ell }}\nonumber \\&=\int _{q_{\min }^{2}}^{q_{\max }^{2}} d q^{2} \frac{G_{F}^{2}\left| V_{\mathrm {cs}}\right| ^{2} q^{2}\left( 1-{\hat{m}}_{\ell }^{2}\right) \sqrt{\lambda \left( m_{1}, m_{2}, q\right) }}{(2 \pi )^{4} 256 m_{1}^{3}}\nonumber \\&\quad \times {\mathcal {B}}\left( B_{2} \rightarrow B_{3} B_{4}\right) \left( A_{\ell }+B_{\ell } \cos \theta _{\ell }+C_{\ell } \cos 2 \theta _{\ell }\right) , \end{aligned}$$

  (77)

  with

  $$\begin{aligned} A_{\ell }&=2 \pi \left[ \left( 3+{\hat{m}}_{\ell }^{2}\right) \left( \left| H_{-\frac{1}{2},-1}\right| ^{2}+\left| H_{\frac{1}{2}, 1}\right| ^{2}\right) \right. \nonumber \\&\quad \left. +2\left( 1+{\hat{m}}_{\ell }^{2}\right) \left( \left| H_{-\frac{1}{2}, 0}\right| ^{2}+\left| H_{\frac{1}{2}, 0}\right| ^{2}\right) \right. \nonumber \\&\quad \left. +4 {\hat{m}}_{\ell }^{2}\left( \left| H_{-\frac{1}{2}, t}\right| ^{2}+\left| H_{\frac{1}{2}, t}\right| ^{2}\right) \right] , \end{aligned}$$

  (78)
  $$\begin{aligned} B_{\ell }&=8 \pi \left[ \left| H_{-\frac{1}{2},-1}\right| ^{2}-\left| H_{\frac{1}{2}, 1}\right| ^{2}\right. \nonumber \\&\quad \left. +2 {\hat{m}}_{\ell }^{2}\left( \left| H_{-\frac{1}{2}, 0} H_{-\frac{1}{2}, t}^{*}\right| +\left| H_{\frac{1}{2}, 0} H_{\frac{1}{2}, t}^{*}\right| \right) \right] , \end{aligned}$$

  (79)
  $$\begin{aligned} C_{\ell }&=2 \pi \left( 1-{\hat{m}}_{\ell }^{2}\right) \left[ \left| H_{-\frac{1}{2},-1}\right| ^{2}+\left| H_{\frac{1}{2}, 1}\right| ^{2}\right. \nonumber \\&\quad \left. -2\left( \left| H_{-\frac{1}{2}, 0}\right| ^{2}+\left| H_{\frac{1}{2}, 0}\right| ^{2}\right) \right] , \end{aligned}$$

  (80)

• $\theta _h$-dependence:

  $$\begin{aligned}&\frac{d \Gamma }{d \cos \theta _{h}}\nonumber \\&=\int _{q_{\min }^{2}}^{q_{\max }^{2}} d q^{2} \frac{G_{F}^{2}\left| V_{\mathrm {cs}}\right| ^{2} q^{2}\left( 1-{\hat{m}}_{\ell }^{2}\right) \sqrt{\lambda \left( m_{1}, m_{2}, q\right) }}{(2 \pi )^{4} 256 m_{1}^{3}}\nonumber \\&\quad \times {\mathcal {B}}\left( B_{2} \rightarrow B_{3} B_{4}\right) \left( A_{h}+B_{h} \cos \theta _{h} \right) , \end{aligned}$$

  (81)

  with

  $$\begin{aligned} A_{h}= & {} \frac{8\pi }{3}\left[ \left( 2+{\hat{m}}_{\ell }^{2}\right) \left( \left| H_{-\frac{1}{2},-1}\right| ^{2}+\left| H_{\frac{1}{2}, 1}\right| ^{2}+\left| H_{-\frac{1}{2}, 0}\right| ^{2}\right. \right. \nonumber \\&\left. \left. +\left| H_{\frac{1}{2}, 0}\right| ^{2}\right) +3 {\hat{m}}_{\ell }^{2}\left( \left| H_{-\frac{1}{2}, t}\right| ^{2}+\left| H_{\frac{1}{2}, t}\right| ^{2}\right) \right] , \end{aligned}$$

  (82)
  $$\begin{aligned} B_{h}= & {} \frac{8 \pi }{3} \alpha \left[ \left( 2+{\hat{m}}_{\ell }^{2}\right) \left( -\left| H_{-\frac{1}{2},-1}\right| ^{2}+\left| H_{\frac{1}{2}, 1}\right| ^{2}\right. \right. \nonumber \\&\left. \left. -\left| H_{-\frac{1}{2}, 0}\right| ^{2}+\left| H_{\frac{1}{2}, 0}\right| ^{2}\right) +3 {\hat{m}}_{\ell }^{2}\left( -\left| H_{-\frac{1}{2}, t}\right| ^{2}+\left| H_{\frac{1}{2}, t}\right| ^{2}\right) \right] ,\nonumber \\ \end{aligned}$$

  (83)

• $\phi$-dependence:

  $$\begin{aligned} \frac{d \Gamma }{d \phi }&\!=\!\int _{q_{\min }^{2}}^{q_{\max }^{2}} d q^{2} \frac{G_{F}^{2}\left| V_{\mathrm {cs}}\right| ^{2} q^{2}\left( 1\!-\!{\hat{m}}_{\ell }^{2}\right) \sqrt{\lambda \left( m_{1}, m_{2}, q\right) }}{(2 \pi )^{4} 256 m_{1}^{3}}\nonumber \\&\quad \times {\mathcal {B}}\left( B_{2} \rightarrow B_{3} B_{4}\right) \left( A_{\phi }+B_{\phi } \cos \phi \theta _{h} \right) , \end{aligned}$$

  (84)

  with

  $$\begin{aligned} A_{\phi }&=\frac{8}{3}\left[ \left( 2+{\hat{m}}_{\ell }^{2}\right) \left( \left| H_{-\frac{1}{2},-1}\right| ^{2}+\left| H_{\frac{1}{2}, 1}\right| ^{2}+\left| H_{-\frac{1}{2}, 0}\right| ^{2}\right. \right. \nonumber \\&\quad \left. \left. +\left| H_{\frac{1}{2}, 0}\right| ^{2}\right) +3 {\hat{m}}_{\ell }^{2}\left( \left| H_{-\frac{1}{2}, t}\right| ^{2}+\left| H_{\frac{1}{2}, t}\right| ^{2}\right) \right] , \end{aligned}$$

  (85)
  $$\begin{aligned} B_{\phi }&=\frac{\pi ^{2}}{\sqrt{2}} \alpha \left[ \left| H_{\frac{1}{2}, 1} H_{-\frac{1}{2}, 0}^{*}\right| +\left| H_{-\frac{1}{2},-1} H_{\frac{1}{2}, 0}^{*}\right| \right. \nonumber \\&\quad \left. +{\hat{m}}_{\ell }^{2}\left( \left| H_{\frac{1}{2}, 1} H_{-\frac{1}{2}, t}^{*}\right| -\left| H_{-\frac{1}{2},-1} H_{\frac{1}{2}, t}^{*}\right| \right) \right] . \end{aligned}$$

  (86)

3.2 Differential decay width of $\Omega _c^0$

Using the leptonic matrix elements in Eqs. (30)–(36) and hadron ones $H_1$ in Eqs. (48)–(55), together with the three-body phase space in Eq. (24), we can obtain the three-body decay width from the spin-1/2 charmed baryon $\Omega _c^0$ to spin-3/2 baryon $\Omega ^-$ and leptons:

$$\begin{aligned}&\frac{d \Gamma \left( \Omega _c^0 \rightarrow \Omega ^- l^{+} \nu _{\ell }\right) }{d \cos \theta _{\ell } d q^{2}}\nonumber \\&\quad =\frac{q^{2}\sqrt{\lambda \left( m_{\Omega _c}, m_{\Omega }, q\right) }}{1024 \pi ^{3} m_{\Omega _c}^{3}}\left( 1-{\hat{m}}_{\ell }^{2}\right) ^{2} \frac{G_{F}^{2}}{2}\left| V_{\mathrm {cs}}\right| ^{2} \nonumber \\&\qquad \times \left\{ \left( \left| H_{\frac{3}{2}, 1}\right| ^{2}+\left| H_{\frac{1}{2}, 1}\right| ^{2}\right) \left[ \left( 1-\cos \theta _{\ell }\right) ^{2}+{\hat{m}}_{\ell }^{2} \sin ^{2} \theta _{\ell }\right] \right. \nonumber \\&\qquad +\left( \left| H_{-\frac{3}{2},-1}\right| ^{2}+\left| H_{-\frac{1}{2}, -1}\right| ^{2} \right) \left[ \left( 1+\cos \theta _{\ell }\right) ^{2}+{\hat{m}}_{\ell }^{2} \sin ^{2} \theta _{\ell }\right] \nonumber \\&\qquad +\left( \left| H_{\frac{1}{2}, 0}\right| ^{2}+\left| H_{-\frac{1}{2}, 0}\right| ^{2}\right) 2\left( \sin ^{2} \theta _{\ell }+{\hat{m}}_{\ell } \cos ^{2} \theta _{\ell }\right) \nonumber \\&\qquad +\left( \left| H_{\frac{1}{2}, t}\right| ^{2}+\left| H_{-\frac{1}{2}, t}\right| ^{2}\right) 2 {\hat{m}}_{\ell }^{2} \nonumber \\&\qquad \left. +2\left( \left| H_{\frac{1}{2}, 0} H_{\frac{1}{2}, t} \right| ^{*}+\left| H_{-\frac{1}{2}, 0}H_{-\frac{1}{2}, t} \right| ^{*}\right) \left( 2{\hat{m}}^{2}_{\ell }\cos \theta _{l} \right) \right\} , \end{aligned}$$

(87)

in which $H_{s_2,s_W}$ are short for the hadronic parts in Eqs. (48)–(55).

Bringing the results of the leptonic part in Eqs. (30)–(36), hadronic part $H_1$ in Eqs. (48)–(55), and $H_2$ in Eqs. (65)–(72) together, the differential decay width of the four-body process $\Omega _c^0\rightarrow \Lambda K^-\ell ^+\nu _l$ can be expressed as

$$\begin{aligned}&\frac{d \Gamma \left( \Omega _c^0\rightarrow \Lambda K^-\ell ^+\nu _l\right) }{d q^{2} d \cos \theta _{h} d \cos \theta _{\ell } d \phi }\\&\quad = \frac{3 G_{F}^{2}\left| V_{\mathrm {cs}}\right| ^{2} q^{2}\left( 1-{\hat{m}}_{\ell }^{2}\right) ^{2} \sqrt{\lambda \left( m_{\Omega _{c}}, m_{\Omega }, q\right) } {\mathcal {B}}(\Omega ^- \rightarrow \Lambda K^-)}{512(2 \pi )^{4} m_{\Omega _{c}}^{3}}\\&\qquad \times \left\{ \left| H_{-\frac{3}{2},-1}\right| ^{2}\left[ {\hat{m}}_{\ell }^{2} \sin ^{2} \theta _{\ell }+\left( 1+\cos \theta _{\ell }\right) ^{2}\right] \right. \\&\qquad \times \sin ^{2} \theta _{h} \left( 1-\alpha \cos \theta _{h} \right) \\&\qquad +\left| H_{\frac{3}{2}, 1}\right| ^{2}\left[ {\hat{m}}_{\ell }^{2} \sin ^{2} \theta _{\ell }+\left( 1-\cos \theta _{\ell }\right) ^{2}\right] \\&\qquad \times \sin ^{2} \theta _{h} \left( 1+\alpha \cos \theta _{h} \right) \\&\qquad +\frac{1}{12}\left| H_{\frac{1}{2}, 1}\right| ^{2}\left[ {\hat{m}}_{\ell }^{2} \sin ^{2} \theta _{\ell }+\left( 1-\cos \theta _{\ell }\right) ^{2}\right] \end{aligned}$$

$$\begin{aligned}&\qquad \times \left[ 2\left( 5+3 \cos 2 \theta _{h} \right) +\alpha \left( 7 \cos \theta _{h} +9 \cos 3 \theta _{h} \right) \right] \\&\qquad +\frac{1}{12}\left| H_{-\frac{1}{2},-1}\right| ^{2}\left[ {\hat{m}}_{\ell }^{2} \sin ^{2} \theta _{\ell }+\left( 1+\cos \theta _{\ell }\right) ^{2}\right] \\&\qquad \times \left[ 2\left( 5+3 \cos 2 \theta _{h} \right) -\alpha \left( 7 \cos \theta _{h} +9 \cos 3 \theta _{h} \right) \right] \\&\qquad +\frac{1}{6}\left| H_{-\frac{1}{2}, 0}\right| ^{2}\left[ {\hat{m}}_{\ell }^{2} \cos ^{2} \theta _{\ell }+\sin ^{2} \theta _{\ell }\right] \\&\qquad \times \left[ 2\left( 5+3 \cos 2 \theta _{h} \right) -\alpha \left( 7 \cos \theta _{h} +9 \cos 3 \theta _{h} \right) \right] \\&\qquad +\frac{1}{6}\left| H_{\frac{1}{2}, 0}\right| ^{2}\left[ {\hat{m}}_{\ell }^{2} \cos ^{2} \theta _{\ell }+\sin ^{2} \theta _{\ell }\right] \\&\qquad \times \left[ 2\left( 5+3 \cos 2 \theta _{h} \right) +\alpha \left( 7 \cos \theta _{h} +9 \cos 3 \theta _{h} \right) \right] \\&\qquad +\frac{1}{6}\left| H_{-\frac{1}{2}, t}\right| ^{2} {\hat{m}}_{\ell }^{2}\\&\qquad \times \left[ 2\left( 5+3 \cos 2 \theta _{h} \right) -\alpha \left( 7 \cos \theta _{h} +9 \cos 3 \theta _{h} \right) \right] \\&\qquad +\frac{1}{6}\left| H_{\frac{1}{2}, t}\right| ^{2} {\hat{m}}_{\ell }^{2}\\&\qquad \times \left[ 2\left( 5+3 \cos 2 \theta _{h} \right) +\alpha \left( 7 \cos \theta _{h} +9 \cos 3 \theta _{h} \right) \right] \\&\qquad +\frac{1}{3} \mid H_{-\frac{1}{2}, 0} H_{-\frac{1}{2}, t}^{*} {\hat{m}}_{\ell }^{2} \cos \theta _{\ell }\\&\qquad \times \left[ 2\left( 5+3 \cos 2 \theta _{h} \right) -\alpha \left( 7 \cos \theta _{h} +9 \cos 3 \theta _{h} \right) \right] \\&\qquad +\frac{1}{3}\left| H_{\frac{1}{2}, 0} H_{\frac{1}{2}, t}\right| {\hat{m}}_{\ell }^{2} \cos \theta _{\ell }\\&\qquad \times \left[ 2\left( 5+3 \cos 2 \theta _{h} \right) +\alpha \left( 7 \cos \theta _{h} +9 \cos 3 \theta _{h} \right) \right] \\&\qquad +\frac{2 \sqrt{3}}{3}\left| H_{-\frac{3}{2},-1} H_{\frac{1}{2}, 1}^{*}\right| \left( {\hat{m}}_{\ell }^{2}-1\right) \\&\qquad \times \sin ^{2} \theta _{\ell } \cos 2 \phi \sin ^{2} \theta _{h} \left( 1-3 \alpha \cos \theta _{h} \right) \\&\qquad +\frac{2 \sqrt{3}}{3}\left| H_{\frac{3}{2}, 1} H_{-\frac{1}{2},-1}^{*}\right| \left( {\hat{m}}_{\ell }^{2}-1\right) \\&\qquad \times \sin ^{2} \theta _{\ell } \cos 2 \phi \sin ^{2} \theta _{h} \left( 1+3 \alpha \cos \theta _{h} \right) \\&\qquad +\frac{\sqrt{6}}{6}\left| H_{-\frac{3}{2},-1} H_{-\frac{1}{2}, 0}^{*}\right| \sin \theta _{\ell } \cos \phi \left\{ \left( \sin \theta _{h} -3 \sin 3 \theta _{h} \right) \right. \\&\qquad \times \left[ {\hat{m}}_{\ell }^{2} \cos \theta _{\ell }+\alpha \left( 1+\cos \theta _{\ell }\right) \right] \\&\qquad \left. +4 \sin 2 \theta _{h} \left( \alpha {\hat{m}}_{\ell }^{2} \cos \theta _{\ell }+1+\cos \theta _{\ell }\right) \right\} \\&\qquad -\frac{\sqrt{6}}{6}\left| H_{\frac{3}{2}, 1} H_{\frac{1}{2}, 0}^{*}\right| \left[ {\hat{m}}_{\ell }^{2} \cos \theta _{\ell }+\left( 1-\cos \theta _{\ell }\right) \right] \\&\qquad \times \sin \theta _{\ell } \cos \phi \left[ 4 \sin 2 \theta _{h} -\alpha \left( \sin \theta _{h} -3 \cos 3 \theta _{h} \right) \right] \\&\qquad +\frac{\sqrt{6}}{6}\left| H_{-\frac{3}{2},-1} H_{-\frac{1}{2}, t}^{*}\right| {\hat{m}}_{\ell }^{2} \\&\qquad \times \sin \theta _{\ell } \cos \phi \left[ \left( \sin \theta _{h} -3 \sin 3 \theta _{h} \right) +4 \alpha \sin 2 \theta _{h} \right] \\&\qquad -\frac{\sqrt{6}}{6}\left| H_{\frac{3}{2}, 1} H_{\frac{1}{2}, t}^{*}\right| {\hat{m}}_{\ell }^{2} \\&\qquad \times \sin \theta _{\ell } \cos \phi \left[ 4 \sin 2 \theta _{h} -\alpha \left( \sin \theta _{h} -3 \cos 3 \theta _{h} \right) \right] \\&\qquad -\frac{\sqrt{2}}{6}\left| H_{\frac{1}{2}, 1} H_{-\frac{1}{2}, 0}^{*}\right| \left[ \cos \theta _{\ell } {\hat{m}}_{\ell }^{2}-\alpha \left( 1-\cos \theta _{\ell }\right) \right] \\&\qquad \times \sin \theta _{\ell } \cos \phi \left( 5 \sin \theta _{h} +9 \sin 3 \theta _{h} \right) \\&\qquad +\frac{\sqrt{2}}{6}\left| H_{-\frac{1}{2},-1} H_{\frac{1}{2}, 0}^{*}\right| \left[ \left( 1+\cos \theta _{\ell }\right) -\alpha {\hat{m}}_{\ell }^{2} \cos \theta _{\ell }\right] \\&\qquad \times \sin \theta _{\ell } \cos \phi \left( 5 \sin \theta _{h} +9 \sin 3 \theta _{h} \right) \\&\qquad -\frac{\sqrt{2}}{6}\left| H_{\frac{1}{2}, 1} H_{-\frac{1}{2}, t}^{*}\right| {\hat{m}}_{\ell }^{2} \sin \theta _{\ell } \cos \phi \end{aligned}$$

$$\begin{aligned}&\qquad \times \left( 5 \sin \theta _{h} +9 \sin 3 \theta _{h} \right) -\frac{\sqrt{2}}{6}\left| H_{-\frac{1}{2},-1} H_{\frac{1}{2}, t}^{*}\right| \nonumber \\&\qquad \times \left. {\hat{m}}_{\ell }^{2} \sin \theta _{\ell } \cos \phi \alpha \left( 5 \sin \theta _{h} +9 \sin 3 \theta _{h} \right) \right\} . \end{aligned}$$

(88)

By integrating out the parameters step by step, we can get the results of $q^2$-, $\theta _l$-, $\theta _h$- and $\phi$-dependence form of total decay width, respectively:

• $q^2$-dependence:

  $$\begin{aligned} \frac{d \Gamma }{d q^{2}}&=\frac{32 \pi }{9} \left[ \left( 2+{\hat{m}}_{\ell }^{2}\right) \left( \left| H_{-\frac{3}{2},-1}\right| ^{2}+\left| H_{\frac{3}{2}, 1}\right| ^{2}\right. \right. \nonumber \\&\left. \quad +\left| H_{-\frac{1}{2},-1}\right| ^{2} +\left| H_{\frac{1}{2}, 1}\right| ^{2} +\left| H_{-\frac{1}{2}, 0}\right| ^{2}+\left| H_{\frac{1}{2}, 0}\right| ^{2}\right) \nonumber \\&\quad \left. +3 {\hat{m}}_{\ell }^{2}\left( \left| H_{-\frac{1}{2}, t}\right| ^{2}+\left| H_{\frac{1}{2}, t}\right| ^{2}\right) \right] \end{aligned}$$

  (89)

• $\theta _l$-dependence:

  $$\begin{aligned} \frac{d \Gamma }{d \cos \theta _{\ell }}&=\int _{q_{\mathrm {min}^2}}^{q_{\mathrm {max}^2}} dq^2 \frac{3 G_{F}^{2}\left| V_{\mathrm {cs}}\right| ^{2} q^{2}\left( 1-{\hat{m}}_{\ell }^{2}\right) ^{2} \sqrt{\lambda \left( m_{\Omega _c}, m_{\Omega }, q\right) }}{512(2 \pi )^{4} m_{\Omega _c}^{3}} \nonumber \\&\quad \times {\mathcal {B}}\left( B_{2}^{\prime } \rightarrow B_{3}^{\prime } B_{4}^{\prime }\right) \left( A_{\ell }^{\prime }+B_{\ell }^{\prime } \cos \theta _{\ell }+C_{\ell }^{\prime } \cos 2 \theta _{\ell }\right) , \end{aligned}$$

  (90)

  with

  $$\begin{aligned} A_{\ell }^{\prime }=&\, \frac{4 \pi }{3}\left[ \left( 3+{\hat{m}}_{\ell }^{2}\right) \left( \left| H_{\frac{3}{2}, 1}\right| ^{2}\right. \right. \nonumber \\&\left. \left. +\left| H_{-\frac{3}{2},-1}\right| ^{2}+\left| H_{\frac{1}{2}, 1}\right| ^{2}+\left| H_{-\frac{1}{2},-1}\right| ^{2}\right) \right. \nonumber \\&+2\left( 1+{\hat{m}}_{\ell }^{2}\right) \left( \left| H_{\frac{1}{2}, 0}\right| ^{2}+\left| H_{-\frac{1}{2}, 0}\right| ^{2}\right) \nonumber \\&\left. +4 {\hat{m}}_{\ell }^{2}\left( \left| H_{\frac{1}{2}, t}\right| ^{2}+\left| H_{-\frac{1}{2}, t}\right| ^{2}\right) \right] , \end{aligned}$$

  (91)
  $$\begin{aligned} B_{\ell }^{\prime }=&\,\frac{16 \pi }{3} \left[ \left( \left| H_{-\frac{3}{2},-1}\right| ^{2}-\left| H_{\frac{3}{2}, 1}\right| ^{2}+\left| H_{-\frac{1}{2},-1}\right| ^{2}-\left| H_{\frac{1}{2}, 1}\right| ^{2}\right) \right. \nonumber \\&\left. +2 {\hat{m}}_{\ell }^{2}\left( \left| H_{-\frac{1}{2}, 0} H_{-\frac{1}{2}, t}^{*}\right| +\left| H_{\frac{1}{2}, 0} H_{\frac{1}{2}, t \mid }^{*}\right| \right) \right] , \end{aligned}$$

  (92)
  $$\begin{aligned} C_{\ell }^{\prime }=&\,\frac{4 \pi }{3}(1\left. -{\hat{m}}_{\ell }^{2}\right) \left[ \left( \left| H_{\frac{3}{2}, 1}\right| ^{2}+\left| H_{-\frac{3}{2},-1}\right| ^{2}+\left| H_{\frac{1}{2}, 1}\right| ^{2}\right. \right. \nonumber \\&\left. \left. +\left| H_{-\frac{1}{2},-1}\right| ^{2}\right) -2\left( \left| H_{\frac{1}{2}, 0}\right| ^{2}+\left| H_{-\frac{1}{2}, 0}\right| ^{2}\right) \right] , \end{aligned}$$

  (93)

• $\theta _h$-dependence:

  $$\begin{aligned}&\frac{d \Gamma }{d \cos \theta _{h}}=\int _{q_{\min }^{2}}^{q_{\max }^{2}} d q^{2} \frac{3 G_{F}^{2}\left| V_{\mathrm {cs}}\right| ^{2} q^{2}\left( 1-{\hat{m}}_{\ell }^{2}\right) ^{2} \sqrt{\lambda \left( m_{\Omega _c}, m_{\Omega }, q\right) }}{512(2 \pi )^{4} m_{\Omega _c}^{3}}\nonumber \\&\quad \times {\mathcal {B}}\left( B_{2}^{\prime } \rightarrow B_{3}^{\prime } B_{4}^{\prime }\right) \nonumber \\&\quad \times \left( A_{h}^{\prime }+B_{h}^{\prime } \cos \theta _{h}+C_{h}^{\prime } \cos 2 \theta _{h}+D_{h}^{\prime } \cos 3 \theta _{h}\right) , \end{aligned}$$

  (94)

  with

  $$\begin{aligned} A_{h}^{\prime }&=\frac{4 \pi }{9}\left\{ \left( 2+{\hat{m}}_{\ell }^{2}\right) \left[ 3\left( \left| H_{\frac{3}{2}, 1}\right| ^{2}+\left| H_{-\frac{3}{2},-1}\right| ^{2}\right) \right. \right. \nonumber \\&\left. \quad +5\left( \left| H_{\frac{1}{2}, 1}\right| ^{2}+\left| H_{-\frac{1}{2},-1}\right| ^{2}+\left| H_{\frac{1}{2}, 0}\right| ^{2}+\left| H_{-\frac{1}{2}, 0}\right| ^{2}\right) \right] \nonumber \\&\quad \left. +15 {\hat{m}}_{\ell }^{2}\left( \left| H_{\frac{1}{2}, t}\right| ^{2}+\left| H_{-\frac{1}{2}, t}\right| ^{2}\right) \right\} , \end{aligned}$$

  (95)
  $$\begin{aligned} B_{h}^{\prime }&=\frac{2 \pi \alpha }{9} \left\{ \left( 2+{\hat{m}}_{\ell }^{2}\right) \left[ 3\left( \left| H_{\frac{3}{2}, 1}\right| ^{2}-\left| H_{-\frac{3}{2},-1}\right| ^{2}\right) \right. \right. \nonumber \\&\left. \quad +7\left( \left| H_{\frac{1}{2}, 1}\right| ^{2}-\left| H_{-\frac{1}{2},-1}\right| ^{2}+\left| H_{\frac{1}{2}, 0}\right| ^{2}-\left| H_{-\frac{1}{2}, 0}\right| ^{2}\right) \right] \nonumber \\&\quad \left. +21 {\hat{m}}_{\ell }^{2}\left( \left| H_{\frac{1}{2}, t}\right| ^{2}-\left| H_{-\frac{1}{2}, t}\right| ^{2}\right) \right\} , \end{aligned}$$

  (96)
  $$\begin{aligned} C_{h}^{\prime }&=\frac{4 \pi }{3}\left[ \left( 2+{\hat{m}}_{\ell }^{2}\right) \left( -\left| H_{\frac{3}{2}, 1}\right| ^{2}-\left| H_{-\frac{3}{2},-1}\right| ^{2}\right. \right. \nonumber \\&\left. \quad +\left| H_{\frac{1}{2}, 1}\right| ^{2}+\left| H_{-\frac{1}{2},-1}\right| ^{2}+\left| H_{\frac{1}{2}, 0}\right| ^{2}+\left| H_{-\frac{1}{2}, 0}\right| ^{2}\right) \nonumber \\&\quad \left. +3 {\hat{m}}_{\ell }^{2}\left( \left| H_{\frac{1}{2}, t}\right| ^{2}+\left| H_{-\frac{1}{2}, t}\right| ^{2}\right) \right] , \end{aligned}$$

  (97)
  $$\begin{aligned} D_{h}^{\prime }&=\frac{2 \pi \alpha }{3} \left\{ \left( 2+{\hat{m}}_{\ell }^{2}\right) \left[ \left( \left| H_{-\frac{3}{2},-1}\right| ^{2}-\left| H_{\frac{3}{2}, 1}\right| ^{2}\right) \right. \right. \nonumber \\&\left. \quad +3\left( \left| H_{\frac{1}{2}, 1}\right| ^{2}-\left| H_{-\frac{1}{2},-1}\right| ^{2}+\left| H_{\frac{1}{2}, 0}\right| ^{2}-\left| H_{-\frac{1}{2}, 0}\right| ^{2}\right) \right] \nonumber \\&\quad \left. -9 {\hat{m}}_{\ell }^{2}\left( \left| H_{-\frac{1}{2}, t}\right| ^{2}-\left| H_{\frac{1}{2}, t}\right| ^{2}\right) \right\} , \end{aligned}$$

  (98)

• $\phi$-dependence:

  $$\begin{aligned} \frac{d \Gamma }{d \phi }&=\int _{q_{\min }^{2}}^{q_{\max }^{2}} d q^{2} \frac{3 G_{F}^{2}\left| V_{\mathrm {cs}}\right| ^{2} q^{2}\left( 1-{\hat{m}}_{\ell }^{2}\right) ^{2} \sqrt{\lambda \left( m_{\Omega _c}, m_{\Omega }, q\right) }}{512(2 \pi )^{4} m_{\Omega _c}^{3}}\nonumber \\&\quad \times {\mathcal {B}}\left( B_{2}^{\prime } \rightarrow B_{3}^{\prime } B_{4}^{\prime }\right) \left( A_{\phi }^{\prime }+B_{\phi }^{\prime } \cos \phi +C_{\phi }^{\prime } \cos 2 \phi \right) , \end{aligned}$$

  (99)

  with

  $$\begin{aligned} A_{\phi }^{\prime }&= \frac{16}{9}\left[ \left( 2+{\hat{m}}_{\ell }^{2}\right) \left( \left| H_{-\frac{3}{2},-1}\right| ^{2}+\left| H_{\frac{3}{2}, 1}\right| ^{2}+\left| H_{-\frac{1}{2},-1}\right| ^{2}\right. \right. \nonumber \\&\quad \left. \left. +\left| H_{\frac{1}{2}, 1}\right| ^{2}+\left| H_{-\frac{1}{2}, 0}\right| ^{2}+\left| H_{\frac{1}{2}, 0}\right| ^{2}\right) \right. \nonumber \\&\quad \left. +3 {\hat{m}}_{\ell }^{2}\left( \left| H_{-\frac{1}{2}, t}\right| ^{2}+\left| H_{\frac{1}{2}, t}\right| ^{2}\right) \right] , \end{aligned}$$

  (100)
  $$\begin{aligned} B_{\phi }^{\prime }&=\frac{\pi ^{2}}{12 \sqrt{2}}\left[ \sqrt{3} \alpha \left( \left| H_{-\frac{3}{2},-1} H_{-\frac{1}{2}, 0}^{*}\right| +\left| H_{\frac{3}{2}, 1} H_{\frac{1}{2}, 0}^{*}\right| \right) \right. \nonumber \\&\quad +\sqrt{3} {\hat{m}}_{\ell }^{2}\left( \left| H_{-\frac{3}{2},-1} H_{-\frac{1}{2}, t}^{*}\right| +\alpha \left| H_{\frac{3}{2}, 1} H_{\frac{1}{2}, t}^{*}\right| \right) \nonumber \\&\quad +5\left( \left| H_{-\frac{1}{2},-1} H_{\frac{1}{2}, 0}^{*}\right| +\alpha \left| H_{\frac{1}{2}, 1} H_{-\frac{1}{2}, 0}^{*}\right| \right) \nonumber \\&\quad \left. -5 {\hat{m}}_{\ell }^{2}\left( \alpha \left| H_{-\frac{1}{2},-1} H_{\frac{1}{2}, t}^{*}\right| +\left| H_{\frac{1}{2}, 1} H_{-\frac{1}{2}, t}^{*}\right| \right) \right] , \end{aligned}$$

  (101)
  $$\begin{aligned} C_{\phi }^{\prime }&=-\frac{32\left( 1-{\hat{m}}_{\ell }\right) ^{2}}{9 \sqrt{3}}\left( \left| H_{-\frac{3}{2},-1} H_{\frac{1}{2}, 1}^{*}\right| +\left| H_{\frac{3}{2}, 1} H_{-\frac{1}{2},-1}^{*}\right| \right) . \end{aligned}$$

  (102)

4 Numerical analysis

4.1 Input

Table 1 The masses (m) and lifetimes ($\tau$) of baryons [29]

Table 2 The $\Lambda _{c}\rightarrow \Lambda$ form factors calculated on the lattice [74]

In this section, all parameters used in the calculations will be collected, including the baryon masses and CKM matrix (Table 1). In addition, the lepton mass $m_{e}=0.005$ GeV, $m_{\mu }=0.1134$ GeV, and Fermi constant $G_F=1.166 \times 10^{-5}$ GeV$^{-2}$. The CKM matrix element $V_{cs}=0.973$. The decay widths of the baryons can be obtained from the reciprocal of lifetimes: $\Gamma (\mathrm {MeV})=1/\tau \times 6.582\times 10^{-22}$. In the calculation of the heavy baryon four-body decays, the asymmetry parameters and branch ratios are collected as [29]

$$\begin{aligned}&\alpha (\Lambda ^{0}\rightarrow p \pi ^{-})=0.732 , ~\alpha (\Xi ^{-} \rightarrow \Lambda ^{0} \pi ^{-})\nonumber \\&\quad =-0.401,~\alpha (\Omega ^{-} \rightarrow \Lambda ^{0} K^{-})=0.0157,\nonumber \\&\quad {\mathcal {B}}(\Lambda ^{0}\rightarrow p \pi ^{-})=63.900\%,~{\mathcal {B}}(\Xi ^{-} \rightarrow \Lambda ^{0} \pi ^{-})\nonumber \\&\quad =99.887\%,~{\mathcal {B}}(\Omega ^{-} \rightarrow \Lambda ^{0} K^{-})=67.800\%. \end{aligned}$$

(103)

4.2 Numerical results and discussions

In the hadronic part with a spin-1/2 intermediate state, we use the lattice QCD calculation of the $\Lambda _{c}^+\rightarrow \Lambda$ [74] and $\Xi _{c}^0\rightarrow \Xi ^-$ [68] for the decay processes of $\Lambda _{c}^+\rightarrow \Lambda \ell ^+\nu _{\ell }$ and $\Xi _{c}^0\rightarrow \Xi ^-\ell ^+\nu _{\ell }$, respectively. In order to access the $q^2$ distribution, we use the modified z expansions in the physical limits [75], and the fit functions are shown as

$$\begin{aligned} f(q^2)=\frac{1}{1-q^2/m^{2}_{\mathrm {pole}}}\sum ^{n_{\mathrm {max}}}_{n=0} a^{f}_{n}[z(q^{2})]^{n}\;. \end{aligned}$$

(104)

The expansion variable is defined as

$$\begin{aligned} z(q^{2})=\frac{\sqrt{t_{+}-q^2}-\sqrt{t_{+}-t_{0}}}{\sqrt{t_{+}-q^2}+\sqrt{t_{+}-t_{0}}}, \end{aligned}$$

(105)

with $t_{0}=q^{2}_{\mathrm {max}}=(m_{\Xi _{c}}-m_{\Xi })^{2}$ for $\Xi _{c}\rightarrow \Xi$ ,$t_{0}=q^{2}_{\mathrm {max}}=(m_{\Lambda _{c}}-m_{\Lambda })^{2}$ for $\Lambda _{c}\rightarrow \Lambda$, and $t_{+} =(m_{D}+m_{K})^{2}$. The pole masses in the form factors are used as $m^{f_{+},f_{\perp }}_{\mathrm {pole}}=~2.12$ GeV,$m^{f_{0}}_{\mathrm {pole}}=~2.318$ GeV,$m^{g_{+},g_{\perp }}_{\mathrm {pole}}=~2.460$ GeV, and $m^{g_{0}}_\mathrm{pole}=~1.968$ GeV.

• $\Lambda _{c}$ decays. We collect the fitted form factor parameters from [74] in Table 2. The resulting SM predictions for the $\Lambda _{c}^+\rightarrow \Lambda \ell ^+\nu _{\ell }$ and $\Lambda _{c}^+\rightarrow p\pi ^-\ell ^+\nu _{\ell }$ decay widths and branching fractions with corresponding error estimates are listed as

  $$\begin{aligned}&\Lambda _{c}^+\rightarrow \Lambda e^+\nu _{e}: ~~ \Gamma =~1.262 (75) \times 10^{-13} s^{-1}, \nonumber \\&\quad {\mathcal {B}}=~3.88(23)\%, \end{aligned}$$

  (106)
  $$\begin{aligned}&\Lambda _{c}^+\rightarrow \Lambda \mu ^+\nu _{\mu }: ~~ \Gamma =~1.272 (72) \times 10^{-13} s^{-1}, ~\nonumber \\&\quad {\mathcal {B}}=~3.91(22)\%, \end{aligned}$$

  (107)
  $$\begin{aligned}&\Lambda _{c}^+\rightarrow p \pi ^{-}e^{+}\nu _{e}:~~ \Gamma =~8.061(482)\times 10^{-14} s^{-1} , ~~\nonumber \\&\quad {\mathcal {B}}=~2.48(15)\%, \end{aligned}$$

  (108)
  $$\begin{aligned}&\Lambda _{c}^+\rightarrow p \pi ^{-}\mu ^{+}\nu _{\mu }:~~ \Gamma =~8.126(462)\times 10^{-14} s^{-1}, ~~\nonumber \\&\quad {\mathcal {B}}=~2.50(14)\%. \end{aligned}$$

  (109)

  The experimental measurements of the $\Lambda _c^+\rightarrow \Lambda \ell ^+\nu _{\ell }$ branching fractions have been reported by the BESIII collaboration as ${\mathcal {B}}(\Lambda _{c}^+\rightarrow \Lambda e^+\nu _{e})=3.63(38)(20)$ and ${\mathcal {B}}(\Lambda _{c}^+\rightarrow \Lambda \mu ^+\nu _{\mu })=3.49(46)(27)$ [37, 38], which are highly consistent with our results in Eqs. (106)–(107). For the processes of four-body decays, based on the form factors, we predict the differential decay widths for $\Lambda _{c}^+\rightarrow p \pi ^{-}\ell ^{+}\nu _{\ell }$ as a function of $q^{2}$ in Fig. 3a. Note that the increasing errors in the small-$q^{2}$ region come from the uncertainties of form factors at large momentum transfer in the lattice calculations (Fig. 4). In Eqs. (76–84), we show the theoretical results of $\theta _{l}$-, $\theta _{h}$- and $\phi$-dependence of the total four-body decay width with different final-state leptons, in which the coefficients are functions of $q^{2}$ only. After we integrate out $q^{2}$, the angular distributions with $\cos \theta _{l}$, $\cos \theta _{h}$, and $\phi$ are shown as

  • 1. $\Lambda _{c}^+\rightarrow p \pi ^{-} e^{+}\nu _{e}$

    $$\begin{aligned} \frac{d\Gamma }{\Gamma d\cos \theta _{l}}&=0.4448(254)+0.1992(172)\cos \theta _{l}\nonumber \\&\quad -0.1657(204)\cos 2\theta _{l}\;, \end{aligned}$$

    (110)
    $$\begin{aligned} \frac{d\Gamma }{\Gamma d\cos \theta _{h}}&=0.5000(299)-0.3188(210)\cos \theta _{h}, \end{aligned}$$

    (111)
    $$\begin{aligned} \frac{d\Gamma }{\Gamma d\phi }&=0.1592(95)-0.0276(35)\cos \theta _{\phi }. \end{aligned}$$

    (112)

  • 2. $\Lambda _{c}^+\rightarrow p \pi ^{-} \mu ^{+}\nu _{\mu }$

    $$\begin{aligned} \frac{d\Gamma }{\Gamma d\cos \theta _{l}}&=0.455(57)+0.265(28)\cos \theta _{l}\nonumber \\&\quad -0.135(24)\cos 2\theta _{l}\;, \end{aligned}$$

    (113)
    $$\begin{aligned} \frac{d\Gamma }{\Gamma d\cos \theta _{h}}&=0.500(63)+0.162(19)\cos \theta _{h}, \end{aligned}$$

    (114)
    $$\begin{aligned} \frac{d\Gamma }{\Gamma d\phi }&=0.159(20)+0.012(3)\cos \theta _{\phi }. \end{aligned}$$

    (115)

• $\Xi _{c}^0$ decays Based on the recently announced $\Xi _{c}\rightarrow \Xi$ form factors calculated by lattice QCD [68], where the z-expansion parameters of helicity-based form factors are collected in Table 3, we perform the predictions for $\Xi _{c}^0\rightarrow \Xi ^-\ell ^+\nu _{\ell }$ and $\Xi _{c}^0\rightarrow \Lambda \pi ^{-} \ell ^+\nu _{\ell }$ decay widths and branching fractions with corresponding uncertainties

  $$\begin{aligned}&\Xi _{c}^{0}\rightarrow \Xi ^{-}e^{+}\nu _{e}: ~~ \Gamma =1.032(129)\times 10^{-13} s^{-1} , \nonumber \\&\quad {\mathcal {B}}=2.40(30)\%, \end{aligned}$$

  (116)
  $$\begin{aligned}&\Xi _{c}^{0}\rightarrow \Xi ^{-}\mu ^{+}\nu _{\mu }: ~~ \Gamma =1.038(131)\times 10^{-13} s^{-1} , \nonumber \\&\quad {\mathcal {B}}=2.41(30)\%, \end{aligned}$$

  (117)
  $$\begin{aligned}&\Xi _{c}^{0}\rightarrow \Lambda \pi ^{-}e^{+}\nu _{e}:~~ \Gamma =1.031(129)\times 10^{-13} s^{-1} , \nonumber \\&\quad {\mathcal {B}}=2.40(30)\%, \end{aligned}$$

  (118)
  $$\begin{aligned}&\Xi _{c}^{0}\rightarrow \Lambda \pi ^{-}\mu ^{+}\nu _{\nu }:~ \Gamma =1.037(131)\times 10^{-13} s^{-1}, ~~\nonumber \\&\quad {\mathcal {B}}=2.41(30)\%. \end{aligned}$$

  (119)

  Very recently, the ALICE [12] and Belle [13] collaborations reported their measurements of branching fractions for $\Xi _{c}^{0}\rightarrow \Xi ^{-}\ell ^{+}\nu _{\ell }$ respectively: ${\mathcal {B}}_{\mathrm {ALICE}}(\Xi _{c}^{0}\rightarrow \Xi ^{-}e^{+}\nu _{e})=2.43(25)(35)(72)\%$, ${\mathcal {B}}_{\mathrm {Belle}}(\Xi _{c}^{0}\rightarrow \Xi ^{-}e^{+}\nu _{e}) =1.31(4)(7)(38)\%$ and ${\mathcal {B}}_{\mathrm {Belle}}(\Xi _{c}^{0}\rightarrow \Xi ^{-}\mu ^{+}\nu _{\mu })=1.27(6)(10)(37)\%$.       We also predict the differential decay widths for $\Xi _{c}^0\rightarrow \Lambda \pi ^{-}\ell ^{+}\nu _{\ell }$ as a function of $q^{2}$ in Fig. 5; the errors in plots mainly come from the uncertainties of form factors extracted from lattice QCD, and its behavior is the same as $\Lambda _{c}^+\rightarrow p \pi ^{-}\ell ^{+}\nu _{\ell }$. The results of the $\theta _{l}$-, $\theta _{h}$- and $\phi$-dependence of differential decay widths with different leptonic final states are listed as follows

  • 1. $\Xi _{c}^0\rightarrow \Lambda \pi ^{-} e^{+}\nu _{e}$

    $$\begin{aligned} \frac{d\Gamma }{\Gamma d\cos \theta _{l}}&=0.4478(557) +0.2363(249)\cos \theta _{l}\nonumber \\&\quad -0.1565(273)\cos 2\theta _{l},\nonumber \\ \frac{d\Gamma }{\Gamma d\cos \theta _{h}}&=0.5000(626)+0.1616(191)\cos \theta _{h},\nonumber \\ \frac{d\Gamma }{\Gamma d\phi }&=0.1592(199)+0.0123(30)\cos \theta _{\phi }. \end{aligned}$$

    (120)

  • 2. $\Xi _{c}^0\rightarrow \Lambda \pi ^{-} \mu ^{+}\nu _{\mu }$

    $$\begin{aligned} \frac{d\Gamma }{\Gamma d\cos \theta _{l}}&=0.4549(569) +0.2650(280)\cos \theta _{l}\nonumber \\&\quad -0.1354(240)\cos 2\theta _{l},\nonumber \\ \frac{d\Gamma }{\Gamma d\cos \theta _{h}}&=0.500( 630)+0.1617(193)\cos \theta _{h},\nonumber \\ \frac{d\Gamma }{\Gamma d\phi }&=0.1592(200)+0.0122(29)\cos \theta _{\phi }, \end{aligned}$$

    (121)

  and the corresponding plots of angular distributions are collected in Fig. 5.

• $\Omega _{c}^0$ decays Due to the absence of the lattice QCD calculation, we use $\Omega _{c}^0\rightarrow \Omega ^{-}$ transition form factors calculated by the light-front quark model [69]. The form factors can be expressed as the following double-pole form:

  $$\begin{aligned} F(q^{2})=\frac{F(0)}{1-a(q^2/m^{2}_{F})+b(q^4/m^{4}_{F})}\;, \end{aligned}$$

  (122)

  where F(0) is the value of the form factors at $q^2=0$ with $m_{F}=1.86$ GeV, $\delta \equiv \delta {m_c}/m_c=\pm 0.04$. The numerical results from the light-front quark model are collected in Table 4. Through the decay width of $\Omega _{c}$ in Eq. (88) and integrating out all variables, we can obtain the numerical results of decay widths and branching fractions with errors:

  $$\begin{aligned}&\Omega _{c}^0\rightarrow \Omega ^{-}e^{+}\nu _{e}:~~ \Gamma =1.311(51)\times 10^{-14} s^{-1} , \nonumber \\&\quad {\mathcal {B}}=0.534(21)\%, \end{aligned}$$

  (123)
  $$\begin{aligned}&\Omega _{c}^0\rightarrow \Omega ^{-}\mu ^{+}\nu _{\mu }:~~ \Gamma =1.294(51)\times 10^{-14} s^{-1} , \nonumber \\&\quad {\mathcal {B}}=0.516(21)\%, \end{aligned}$$

  (124)
  $$\begin{aligned}&\Omega _{c}^0\rightarrow \Lambda K^{-}e^{+}\nu _{e}:~~ \Gamma =8.889(344)\times 10^{-15} s^{-1} , \nonumber \\&\quad {\mathcal {B}}=0.362(14)\%, \end{aligned}$$

  (125)
  $$\begin{aligned}&\Omega _{c}^0\rightarrow \Lambda K^{-}\mu ^{+}\nu _{\nu }:~~ \Gamma =8.771(343)\times 10^{-15} s^{-1}, \nonumber \\&\quad {\mathcal {B}}=0.350(14)\%. \end{aligned}$$

  (126)

  Figure 5a shows the differential decay widths of $\Omega _{c}^0\rightarrow \Lambda K^{-}\ell ^{+}\nu _{\ell }$ as a function of $q^{2}$. We also give the results of angular distributions of total decay width; the numerical results are listed in Eqs. (127, 128), and in Fig. 5, the illustration of the angular distributions with $\phi$, $\cos \theta _{l}$ and $\cos \theta _{h}$ are displayed.

  • 1. $\Omega _{c}^0\rightarrow \Lambda K^{-} e^{+}\nu _{e}$

    $$\begin{aligned} \frac{d\Gamma }{\Gamma d\cos \theta _{l}}&= 0.472(2)+0.244(3)\cos \theta _{l}\nonumber \\&\quad -0.085(14)\cos 2\theta _{l},\nonumber \\ \frac{d\Gamma }{\Gamma d\cos \theta _{h}}&= 0.548(24)-0.002(0)\cos \theta _{h}\nonumber \\&\quad +0.143(13)\cos 2\theta _{h}-0.001(0)\cos 3\theta _{h},\nonumber \\ \frac{d\Gamma }{\Gamma d\phi }&= 0.159(6)-0.0233(8)\cos \theta _{\phi }\nonumber \\&\quad -0.0173(4)\cos 2\theta _{\phi }. \end{aligned}$$

    (127)

  • 2. $\Omega _{c}^0\rightarrow \Lambda K^{-} \mu ^{+}\nu _{\mu }$

    $$\begin{aligned} \frac{d\Gamma }{\Gamma d\cos \theta _{l}}&= 0.479(16)\nonumber \\&\quad +0.273(5)\cos \theta _{l}-0.064(11)\cos 2\theta _{l},\nonumber \\ \frac{d\Gamma }{\Gamma d\cos \theta _{h}}&= 0.5479(238)-0.0021(6)\cos \theta _{h}\nonumber \\&\quad +0.1436(131)\cos 2\theta _{h}\nonumber \\&\quad -0.0014(0)\cos 3\theta _{h},\nonumber \\ \frac{d\Gamma }{\Gamma d\phi }&=0.1596(62)-0.0224(8)\cos \theta _{\phi }\nonumber \\&\quad -0.0120(2)\cos 2\theta _{\phi }. \end{aligned}$$

    (128)

Fig. 3

Differential decay width for $\Lambda _{c}^+\rightarrow p \pi ^{-}\ell ^{+}\nu _{\ell }$ as a function of $q^{2}$ (a), $\cos \theta _{l}$ (b), $\cos \theta _{h}$ c, and $\phi$ d, respectively. It should be noted that in b and c, the errors increase with $\theta _{l}$ from -1 to 1 and $\theta _{h}$ from 1 to $-1$, as a result of the remarkable uncertainties of form factors at the small-$q^2$ region

Fig. 4

Differential decay width for $\Xi _{c}^0\rightarrow \Lambda \pi ^{-}\ell ^{+}\nu _{\ell }$ as a function of $q^{2}$ (a), $\cos \theta _{l}$ (b), $\cos \theta _{h}$ (c), and $\phi$ (d). Note that in b and c, the errors increase with $\theta _{l}$ from $-1$ to 1 and $\theta _{h}$ from 1 to $-1$, as a result of the remarkable uncertainties of form factors at the small-$q^2$ region

Table 3 Results for the z-expansion parameters describing $\Xi _{c}\rightarrow \Xi$ form factors with statistical errors

Fig. 5

Differential decay width for $\Omega _{c}\rightarrow \Lambda K^{-}\ell ^{+}\nu _{\ell }$ as a function of $q^{2}$ (a), $\cos \theta _{l}$ (b), $\cos \theta _{h}$ (c), and $\phi$ (d). Note that in b and c, the errors increase with $\theta _{l}$ from $-1$ to 1 and $\theta _{h}$ from 1 to $-1$, as a result of the remarkable uncertainties of form factors at the small-$q^2$ region.

Table 4 $\Omega ^{0}_{c}\rightarrow \Omega ^{-}$ transition form factors with F(0) at $q^{2}=0$

5 Summary

In summary, we have studied charmed baryon four-body semileptonic decays including the antitriplet charmed baryons ($\Lambda _{c}^{+}$, $\Xi ^{0}_{c}$) and sextet $\Omega ^{0}_{c}$. With the form factors of the $\Lambda ^{+}_{c}\rightarrow \Lambda$ and $\Xi ^{0}_{c}\rightarrow \Xi ^{-}$ calculated by lattice QCD and the light-front quark model calculation of $\Omega ^{0}_{c}\rightarrow \Omega ^{-}$ form factors, we have predicted the decay widths and branching fractions using the helicity amplitude technique for three-body and four-body decays of the charmed baryons:

$$\begin{aligned}&\Gamma (\Lambda _{c}^+\rightarrow \Lambda e^+\nu _{e})=1.262 (75) \times 10^{-13} s^{-1},~~\\&\quad {\mathcal {B}}(\Lambda _{c}^+\rightarrow \Lambda e^+\nu _{e})=3.88(23)\% ,\\&\quad \Gamma (\Lambda _{c}^+\rightarrow \Lambda \mu ^+\nu _{\mu })=1.272 (72) \times 10^{-13} s^{-1}, ~~\\&\quad {\mathcal {B}}(\Lambda _{c}^+\rightarrow \Lambda \mu ^+\nu _{\mu })=3.91(22)\%, \\&\quad \Gamma (\Xi _{c}\rightarrow \Xi ^{-}e^{+}\nu _{e})=1.032(129)\times 10^{-13} s^{-1},~~\\&\quad {\mathcal {B}}(\Xi _{c}\rightarrow \Xi ^{-}e^{+}\nu _{e})=2.40(30)\%, \\&\quad \Gamma (\Xi _{c}\rightarrow \Xi ^{-}\mu ^{+}\nu _{\mu })=1.038(131)\times 10^{-13} s^{-1} ,~~\\&\quad {\mathcal {B}}(\Xi _{c}\rightarrow \Xi ^{-}\mu ^{+}\nu _{\mu })=2.41(30)\%, \\&\quad \Gamma (\Omega _{c}^0\rightarrow \Omega ^{-}e^{+}\nu _{e})=1.311(51)\times 10^{-14} s^{-1},~~\\&\quad {\mathcal {B}}(\Omega _{c}^0\rightarrow \Omega ^{-}e^{+}\nu _{e})=0.534(21)\%, \\&\quad \Gamma (\Omega _{c}^0\rightarrow \Omega ^{-}\mu ^{+}\nu _{\mu })=1.294(51)\times 10^{-14} s^{-1} ,~~\\&\quad {\mathcal {B}}(\Omega _{c}^0\rightarrow \Omega ^{-}\mu ^{+}\nu _{\mu })=0.516(21)\%, \\&\quad \Gamma (\Lambda _{c}^{+} \rightarrow p \pi ^{-} e^{+} \nu _{e})=8.061(482)\times 10^{-14} s^{-1},~~\\&\quad {\mathcal {B}}(\Lambda _{c}^{+} \rightarrow p \pi ^{-} e^{+} \nu _{e})=2.48(15)\%, \\&\quad \Gamma (\Lambda _{c}^+\rightarrow p \pi ^{-}\mu ^{+}\nu _{\mu })=8.126(462)\times 10^{-14} s^{-1}, ~~\\&\quad {\mathcal {B}}(\Lambda _{c}^+\rightarrow p \pi ^{-}\mu ^{+}\nu _{\mu })=2.50(14)\%, \\&\quad \Gamma (\Xi _{c}\rightarrow \Lambda \pi ^{-}e^{+}\nu _{e})=1.031(129)\times 10^{-13} s^{-1}, ~~\\&\quad {\mathcal {B}}(\Xi _{c}\rightarrow \Lambda \pi ^{-}e^{+}\nu _{e})=2.40(30)\%, \\&\quad \Gamma (\Xi _{c}\rightarrow \Lambda \pi ^{-}\mu ^{+}\nu _{\nu })=1.037(131)\times 10^{-13} s^{-1}, ~~\\&\quad {\mathcal {B}}(\Xi _{c}\rightarrow \Lambda \pi ^{-}\mu ^{+}\nu _{\nu })=2.41(30)\%, \\&\quad \Gamma (\Omega _{c}\rightarrow \Lambda K^{-}e^{+}\nu _{e})=8.889(344)\times 10^{-15} s^{-1}, ~~\\&\quad {\mathcal {B}}(\Omega _{c}\rightarrow \Lambda K^{-}e^{+}\nu _{e})=0.362(14)\%, \\&\quad \Gamma (\Omega _{c}\rightarrow \Lambda K^{-}\mu ^{+}\nu _{\nu })=8.771(343)\times 10^{-15} s^{-1}, ~~\\&\quad {\mathcal {B}}(\Omega _{c}\rightarrow \Lambda K^{-}\mu ^{+}\nu _{\nu })=0.350(14)\%. \end{aligned}$$

In addition, we give the angular distributions of all the processes with different angular $\cos \theta _{l}$,$\cos \theta _{h}$ and $\phi$. In the future, we expect to study the more general cases of semileptonic charmed baryon decays by calculating form factors such as $\Xi _c\rightarrow \Lambda$ and so on. This work can provide a theoretical basis for the ongoing experiments at BESIII, LHCb, and BELLE-II.

Appendix

1.1 A. Definitions of spinors, polarization vectors, and vectorial spinors

In this part, we will introduce the spinors and polarization vectors of fermions and the vector boson [76]. The spinors for spin-1/2 fermions are

$$\begin{aligned} u\left( \vec {p},+\frac{1}{2}\right)&=N_{p}\left( \begin{array}{c} e^{-i \phi / 2} \cos \left( \theta / 2\right) \\ e^{i \phi / 2} \sin \left( \theta / 2\right) \\ \frac{\left| \vec {p}\right| }{E+m} e^{-i \phi / 2} \cos \left( \theta / 2\right) \\ \frac{\left| \vec {p}\right| }{E+m} e^{i \phi / 2} \sin \left( \theta / 2\right) \end{array}\right) , \end{aligned}$$

(129)

$$\begin{aligned} u\left( \vec {p},-\frac{1}{2}\right)&=N_{p}\left( \begin{array}{c} -e^{-i \phi / 2} \sin \left( \theta / 2\right) \\ e^{i \phi / 2} \cos \left( \theta / 2\right) \\ \frac{\left| \vec {p}\right| }{E+m} e^{-i \phi / 2} \sin \left( \theta / 2\right) \\ -\frac{\left| \vec {p}\right| }{E+m} e^{i \phi / 2} \cos \left( \theta / 2\right) \end{array}\right) , \end{aligned}$$

(130)

where $N_{p}=\sqrt{E+m}$ and $p^{\mu }=(p^{0},\vec {p}\sin \theta \cos \phi ,\vec {p}\sin \theta \sin \phi ,\vec {p}\cos \theta )$. The spinors for anti-fermions can be obtained by ${\bar{v}}(\vec {p},s)=-i \gamma ^2 u^{\star }(\vec {p},s)$. The polarization vectors can be expressed as

$$\begin{aligned} \epsilon ^{\mu }(+1)&=\frac{1}{\sqrt{2}}(0,-\cos \theta \cos \phi + i \sin \phi ,-\cos \theta \sin \phi \nonumber \\&\quad -i\cos \phi , \sin \theta ), \nonumber \\ \epsilon ^{\mu }(-1)&=\frac{1}{\sqrt{2}}(0,\cos \theta \cos \phi + i \sin \phi ,\cos \theta \sin \phi \nonumber \\&\quad -i\cos \phi , -\sin \theta ), \nonumber \\ \epsilon ^{\mu }(0)=&\,\frac{1}{\sqrt{m}}(\left| \vec {p}\right| ,p^{0}\sin \theta \cos \phi , \nonumber \\&p^{0}\sin \theta \sin \phi , p^{0}\cos \theta ). \end{aligned}$$

(131)

The vectorial spinor $U_{\alpha }(\vec {p},S_z)$ for the spin-3/2 baryon is given as

$$\begin{aligned} U_{\alpha }\left( \vec {p}, S_{z}\right)&=\sum _{s_{z}, s_{z}^{\prime }}\left\langle s=1, s_{z} ; s^{\prime }=\frac{1}{2}, s_{z}^{\prime } \mid s=1, s^{\prime }=\frac{1}{2} ;\right. \nonumber \\&\left. S=\frac{3}{2}, S_{z}\right\rangle \varepsilon _{\alpha }\left( \vec {p}, s_{z}\right) u\left( \vec {p}, s_{z}^{\prime }\right) , \end{aligned}$$

(132)

where $u(\vec {p},s_z')$ is a wave function for spin-1/2, and $\epsilon _{\alpha }(\vec {p},s_z)$ for spin-1. The Clebsch–Gordan coefficients

$$\begin{aligned} S_{z}&=\frac{3}{2}:~~~ \left\langle s_{z}=1, s_{z}^{\prime }=\frac{1}{2} \mid S_{z}=\frac{3}{2}\right\rangle =1, \end{aligned}$$

(133)

$$\begin{aligned} S_{z}&=\frac{1}{2}:~~~ \left\langle s_{z}=1, s_{z}^{\prime }=-\frac{1}{2} \mid S_{z}=\frac{1}{2}\right\rangle \nonumber \\&=\frac{1}{\sqrt{3}}, \quad \left\langle s_{z}=0, s_{z}^{\prime }=\frac{1}{2} \mid S_{z}=\frac{1}{2}\right\rangle =\sqrt{\frac{2}{3}}, \end{aligned}$$

(134)

$$\begin{aligned} S_{z}&=-\frac{1}{2} :~~~ \left\langle s_{z}=0, s_{z}^{\prime }=-\frac{1}{2} \mid S_{z}=-\frac{1}{2}\right\rangle \nonumber \\&=\sqrt{\frac{2}{3}}, \quad \left\langle s_{z}=-1, s_{z}^{\prime }=\frac{1}{2} \mid S_{z}=-\frac{1}{2}\right\rangle =\frac{1}{\sqrt{3}}, \ \end{aligned}$$

(135)

$$\begin{aligned} S_{z}&=-\frac{3}{2}:~~~ \left\langle s_{z}=-1, s_{z}^{\prime }=-\frac{1}{2} \mid S_{z}=-\frac{3}{2}\right\rangle =1. \end{aligned}$$

(136)

1.2 B. Transformations of various-based form factors

Apart from the generalized parametrization in Eqs. (37, 38) for the spin-1/2-to-spin-1/2 case, another common treatment is called helicity-based form factors,

$$\begin{aligned}&\left\langle B_{2}\left( p_{2}, s_{2}\right) \left| V^{\mu }\right| B_{1}\left( p_{1}, s_{1}\right) \right\rangle \nonumber \\&\quad ={\bar{u}}(\left. p_{2}, s_{2}\right) \left[ \left( m_{1}-m_{2}\right) \frac{q^{\mu }}{q^{2}} f_{0}\left( q^{2}\right) \right. \nonumber \\&\qquad +\frac{m_{1}+m_{2}}{s_{+}}\left( p_{1}^{\mu }+p_{2}^{\mu }-\left( m_{1}^{2}-m_{2}^{2}\right) \frac{q^{\mu }}{q^{2}}\right) f_{+}\left( q^{2}\right) \nonumber \\&\qquad \left. +\left( \gamma ^{\mu }-\frac{2 m_{2}}{s_{+}} p_{1}^{\mu }-\frac{2 m_{1}}{s_{+}} p_{2}^{\mu }\right) f_{\perp }\left( q^{2}\right) \right] u\left( p_{1}, s_{1}\right) , \end{aligned}$$

(137)

$$\begin{aligned}&\left\langle B_{2}\left( p_{2}, s_{2}\right) \left| A^{\mu }\right| B_{1}\left( p_{1}, s_{1}\right) \right\rangle \nonumber \\&\qquad =-{\bar{u}} \left( p_{2}, s_{2}\right) \gamma _{5}\left[ \left( m_{1}+m_{2}\right) \frac{q^{\mu }}{q^{2}} g_{0}\left( q^{2}\right) \right. \nonumber \\&\qquad +\frac{m_{1}-m_{2}}{s_{-}}\left( p_{1}^{\mu }+p_{2}^{\mu }-\left( m_{1}^{2}-m_{2}^{2}\right) \frac{q^{\mu }}{q^{2}}\right) g_{+}\left( q^{2}\right) \nonumber \\&\qquad \left. +\left( \gamma ^{\mu }-\frac{2 m_{2}}{s_{-}} p_{1}^{\mu }-\frac{2 m_{1}}{s_{-}} p_{2}^{\mu }\right) g_{\perp }\left( q^{2}\right) \right] u\left( p_{1}, s_{1}\right) , \end{aligned}$$

(138)

where one can easily obtain the helicity-based form factors from the generalized ones under the following transformation:

$$\begin{aligned}&\left( \begin{array}{c} f_{\perp }\left( q^{2}\right) \\ f_{0}\left( q^{2}\right) \\ f_{+}\left( q^{2}\right) \end{array}\right) =\left( \begin{array}{ccc} 1 &{} -\frac{m_{1}+m_{2}}{m_{1}} &{} 0 \\ 1 &{} 0 &{} \frac{q^{2}}{m_{1}\left( m_{1}-m_{2}\right) } \\ 1 &{} -\frac{q^{2}}{m_{1}\left( m_{1}+m_{2}\right) } &{} 0 \end{array}\right) \left( \begin{array}{c} f_{1}\left( q^{2}\right) \\ f_{2}\left( q^{2}\right) \\ f_{3}\left( q^{2}\right) \end{array}\right) , \end{aligned}$$

(139)

$$\begin{aligned}&\left( \begin{array}{c} g_{\perp }\left( q^{2}\right) \\ g_{0}\left( q^{2}\right) \\ g_{+}\left( q^{2}\right) \end{array}\right) =\left( \begin{array}{ccc} 1 &{} \frac{m_{1}-m_{2}}{m_{1}} &{} 0 \\ 1 &{} 0 &{} -\frac{q^{2}}{m_{1}\left( m_{1}+m_{2}\right) } \\ 1 &{} \frac{q^{2}}{m_{1}\left( m_{1}-m_{2}\right) } &{} 0 \end{array}\right) \left( \begin{array}{l} g_{1}\left( q^{2}\right) \\ g_{2}\left( q^{2}\right) \\ g_{3}\left( q^{2}\right) \end{array}\right) . \end{aligned}$$

(140)

The helicity-based parametrized form factors of $\Omega _c^0\rightarrow \Omega ^-$ are given as

$$\begin{aligned}&\langle B^{\prime }_{2}(p_2,s_2)|{\bar{s}}\gamma ^{\mu }c|B^{\prime }_{1}(p_1,s_1)\rangle \nonumber \\&\quad =-{\bar{u}}(p_2,s_2)\gamma _{5}\left[ f_{0}\frac{m_{2}}{s_{-}}\frac{(m_{2}+m_{1})p_{1}^{\alpha }q^{\mu }}{q^2}\right. \nonumber \\&\qquad +f_{+}\frac{m_{1}}{s_{+}}\frac{(m_{1}-m_{2})p^{\alpha }\big (q^2(p_{1}^{\mu }+p_{2}^{\mu })-(m_{2}^{2}-m_{1}^{2})q^{\mu }\big )}{q^2 s_{+}}\nonumber \\&\qquad +f_{\perp }\frac{m_{1}}{s_{+}}\left( p^{\alpha }\gamma ^{\mu }-\frac{2p^{\alpha }(m_{2}p_{2}^{\mu }-m_{1}p_{1}^{\mu })}{s_{+}}\right) \nonumber \\&\qquad +f_{\perp ^{'}}\frac{m_{1}}{s_{+}}\left( p^{\alpha }\gamma ^{\mu }+\frac{2p_{1}^{\alpha }p_{2}^{\mu }}{m_{1}}+\frac{2p_{1}^{\alpha }(m_{1}p_{2}^{\mu }-m_{2}p^{\mu })}{s_{+}}\right) \nonumber \\&\left. \qquad -\frac{s_{+}g^{\alpha \mu }}{m_{2}}\right] u(p_1,s_1)\; ,\nonumber \\&\langle B_{2}^{\prime }(p_2,s_2)|{\bar{s}}\gamma ^{\mu }\gamma _{5}c|B_{1}^{\prime }(p_1,s_1)\rangle \nonumber \\&\quad ={\bar{u}}(p_2,s_2)\left[ g_{0}\frac{m_{2}}{s_{+}}\frac{(m_{2}-m_{1})p_{1}^{\alpha }q^{\mu }}{q^2}\right. \nonumber \\&\qquad +g_{+}\frac{m_{2}}{s_{-}}\frac{(m_{2}+m_{1})p^{\alpha }\big (q^2(p_{1}^{\mu }+p_{2}^{\mu })-(m_{1}^{2}-m_{2}^{2})q^{\mu }\big )}{q^2 s_{-}}\nonumber \\&\qquad +g_{\perp }\frac{m_{2}}{s_{-}}\left( p^{\alpha }\gamma ^{\mu }-\frac{2p^{\alpha }(m_{1}p_{2}^{\mu }+m_{2}p_{1}^{\mu })}{s_{+}}\right) \nonumber \\&\qquad +g_{\perp ^{'}}\frac{m_{2}}{s_{-}}\left( p^{\alpha }\gamma ^{\mu }-\frac{2p_{1}^{\alpha }p_{2}^{\mu }}{m_{2}}\right. \nonumber \\&\left. \left. \qquad +\frac{2p_{1}^{\alpha }(m_{1}p_{2}^{\mu }+m_{2}p^{\mu })}{s_{-}}\right) +\frac{s_{-}g^{\alpha \mu }}{m_{2}}\right] u(p_1,s_1). \end{aligned}$$

(141)

Similarly, we can obtain the helicity-based form factors from the generalized ones in Eqs. (46, 47) under the following transformation

$$\begin{aligned}&\left( \begin{array}{c} f_{\perp }\left( q^{2}\right) \\ f^{\prime }_{\perp }\left( q^{2}\right) \\ f_{0}\left( q^{2}\right) \\ f_{+}\left( q^{2}\right) \end{array}\right) =\left( \begin{array}{cccc} \frac{s_{+}}{m_{1}m_{2}} &{}0 &{} 0 &{}-1 \\ 0 &{} 0 &{} 0 &{}1 \\ \frac{\left( m_{1}-m_{2}\right) ^{2}-q^{2}}{m_{1}m_{2}} &{} -\frac{\left( m_{1}^{2}-m_{2}^{2}+q^{2}\right) s_{-}}{2m^{2}_{1}m_{2}(m_{1}+m_{2})} &{} \frac{\left( -m_{1}^{2}+m_{2}^{2}+q^{2}\right) s_{-}}{2m_{1}m^{2}_{2}(m_{1}+m_{2})} &{} \frac{-2\left( m_{1}-m_{2}\right) ^{2}+2q^{2}+\frac{(-m^{2}_{1}+m^{2}_{2}+q^{2})s_{-}}{m_{2}(m_{1}+m_{2})}}{s_{+}} \\ \frac{s_{+}}{m_{1}m_{2}} &{} -\frac{s_{-}s_{+}}{2m^{3}_{1}m_{2}-2m^{2}_{1}m^{2}_{2}} &{} -\frac{s_{-}s_{+}}{2m^{2}_{1}m^{2}_{2}-2m_{1}m^{3}_{2}} &{}-2-\frac{s_{-}}{\left( m_{1}-m_{2}\right) m_{2}} \end{array}\right) \left( \begin{array}{c} f_{1}\left( q^{2}\right) \\ f_{2}\left( q^{2}\right) \\ f_{3}\left( q^{2}\right) \\ f_{4}\left( q^{2}\right) \end{array}\right) , \end{aligned}$$

(142)

$$\begin{aligned}&\left( \begin{array}{c} g_{\perp }\left( q^{2}\right) \\ g^{\prime }_{\perp }\left( q^{2}\right) \\ g_{0}\left( q^{2}\right) \\ g_{+}\left( q^{2}\right) \end{array}\right) =\left( \begin{array}{cccc} \frac{s_{-}}{m_{1}m_{2}} &{}0 &{} 0 &{}-1 \\ 0 &{} 0 &{} 0 &{}1 \\ \frac{\left( m_{1}+m_{2}\right) ^{2}-q^{2}}{m_{1}m_{2}} &{} \frac{\left( m_{1}^{2}-m_{2}^{2}+q^{2}\right) s_{+}}{2m^{2}_{1}m_{2}(m_{1}-m_{2})} &{} \frac{\left( m_{1}^{2}-m_{2}^{2}-q^{2}\right) s_{-}}{2m_{1}m^{2}_{2}(m_{1}-m_{2})} &{} \frac{-2\left( m_{1}+m_{2}\right) ^{2}+2q^{2}+\frac{(m^{2}_{1}-m^{2}_{2}-q^{2})s_{+}}{m_{2}(m_{1}-m_{2})}}{s_{-}} \\ \frac{s_{-}}{m_{1}m_{2}} &{} \frac{s_{-}s_{+}}{2m^{3}_{1}m_{2}+2m^{2}_{1}m^{2}_{2}} &{} \frac{s_{-}s_{+}}{2m^{2}_{1}m^{2}_{2}+2m_{1}m^{3}_{2}} &{}-2+\frac{s_{+}}{\left( m_{1}+m_{2}\right) m_{2}} \end{array}\right) \left( \begin{array}{l} g_{1}\left( q^{2}\right) \\ g_{2}\left( q^{2}\right) \\ g_{3}\left( q^{2}\right) \\ g_{4}\left( q^{2}\right) \end{array}\right) . \end{aligned}$$

(143)