Weak decays of the triply heavy baryons in the three-quark picture with the light-front quark model

Abstract

We investigate the weak decays of the triply heavy baryon $\Omega _{QQQ}$ in the light-front quark model. Since $\Omega _{QQQ}$ consists of three indistinguishable identical heavy quarks, the commonly adopted quark–diquark picture does not seem to be valid anymore. Instead, we employ the three-quark picture for baryons where the three quarks are regarded as individual quarks. We calculate the hadronic form factors for the transitions and give predictions for the decay widths of the semi-leptonic decay modes $\Omega _{ccc}\rightarrow \Xi _{cc}/ \Omega _{cc}+ l^+\nu _l$, $\Omega _{bbb}\rightarrow \Xi _{bb}+ l^-\bar{\nu }_l$ and the non-leptonic decay modes $\Omega _{ccc}\rightarrow \Xi _{cc}/\Omega _{cc}+ M$, $\Omega _{bbb}\rightarrow \Xi _{bb}+ M$. Our study can be a guide for future experiments to discover the triply heavy baryons.

1 Introduction

According to the conventional quark model, baryons consist of three valence quarks and mesons are composed of one quark and one antiquark. Compared to the two-body meson system, the theoretical study of the three-body baryon system is far more complicated, therefore the researches on baryons are much more behind than those on mesons, especially for the heavy flavor baryons. Experimentally, higher energy is necessary to produce the heavy baryons, and usually, the production rates are not very large. In the last century, the properties of the light baryons have been extensively studied in many models with great success. However, heavy baryon spectroscopy has been hampered by insufficient experimental data. As experimental techniques have advanced, more and more new data on the heavy baryons have been accumulated in recent years. So far, the baryons containing one heavy quark (b, c) have been extensively studied and experimentally observed. In 2017, the LHCb collaboration observed the doubly charmed baryon $\Xi _{cc}^{++}$ in the final state $\Lambda _c^{+}K^{-}\pi ^+\pi ^+$ [1], which was subsequently confirmed in decays $\Xi _{cc}^{++}\rightarrow \Xi _{c}^+\pi ^+$and $\Xi _{cc}^{++}\rightarrow \Xi _{c}^{\prime +}\pi ^+$ [2, 3]. This discovery has spurred extensive research on the doubly heavy baryons [4,5,6,7,8]. The triply heavy baryons, consisting of three heavy flavor quarks, have not yet been discovered. Nevertheless, theoretical studies on the triply heavy baryons have been ongoing, including their mass spectra [9,10,11,12,13,14,15,16] and pertinent decay modes [17,18,19,20,21]. In this work we focus on the the weak decays of the triply heavy baryon $\Omega _{QQQ}$, which is composed of three identical heavy quarks Q (b or c), to the doubly heavy baryon $\mathcal B_{QQ}$.

We employ the light front quark model (LFQM) to investigate the weak decay processes. LFQM is a relativistic quark model which was initially used to study the decays of mesons [22,23,24,25,26,27,28] and has been extended to deal with the transitions between two heavy baryons [29,30,31]. The most common way of dealing with the baryons is to transform a three-body problem into a two-body one in the quark–diquark picture ansatz where the spectator quarks are treated as the diquark. In Refs. [29,30,31,32,33,34,35,36], the transition between baryons was studied with the quark–diquark picture and the results are in good agreement with the experiment. In Refs. [19, 20], the authors employed the quark–diquark picture to study the weak decays of the triply heavy baryon $\Omega _{QQQ}$ within the LFQM framework. In general, light quark pair [$q_1q_2$] in the single-heavy baryon is regarded as the diquark as well as the heavy quark pair [$Q_1Q_2$] in the doubly heavy baryon. However, for the triply heavy baryon, as $\Omega _{QQQ}$ consists entirely of heavy quarks, the interactions between any two quarks are equally strong, quark–diquark picture does not seem to be so appropriate anymore. In addition, for the weak decay of the doubly charmed baryon $\Xi _{cc}^{++}$, the diquark in the initial state and that in the final state are not spectators, so there are also some difficulties in using the quark–diquark picture here [37]. As discussed in Refs. [38,39,40,41,42,43,44], the recipe is to use the three-quark picture, which can compensate the shortcoming of the quark–diquark picture. In this work, we also employ the three-quark picture and use the same approach as that in the Ref. [44] to explore the decays of $\Omega _{QQQ}$ within the LFQM framework.

This paper is organized as follows: in Sect. 2 we present the vertex functions of the heavy flavor baryons, and derive the form factors of the transitions $\Omega _{ccc}\rightarrow \Xi _{cc}/\Omega _{cc}$ and $\Omega _{bbb}\rightarrow \Xi _{bb}$ in the LFQM. In Sect. 3 we present numerical results for the transitions $\Omega _{ccc}\rightarrow \Xi _{cc}/\Omega _{cc}$ and $\Omega _{bbb}\rightarrow \Xi _{bb}$ along with all necessary input parameters, and then we calculate the form factors and the decay widths of related semi-leptonic and non-leptonic decays; Sect. 4 is devoted to the conclusions and discussions.

2 $\Omega _{QQQ}\rightarrow \mathcal B_{QQ}$ in LFQM

2.1 The vertex functions

In Ref. [44], under the three-quark picture, the vertex function of a baryon $\mathcal B_{QQ}$ ($\Xi _{cc}$, $\Omega _{cc}$ or $\Xi _{bb}$) with the total spin $S=1/2$ and total momentum P is

$$\begin{aligned}{} & {} |\mathcal B_{QQ}(P,S,S_z)\rangle \nonumber \\{} & {} \quad =\int \{d^3\tilde{p}_1\}\{d^3\tilde{p}_2\}\{d^3\tilde{p}_3\} \, 2(2\pi )^3\delta ^3(\tilde{P}-\tilde{p_1}-\tilde{p_2}-\tilde{p_3}) \nonumber \\{} & {} \quad \quad \times \sum _{\lambda _1,\lambda _2,\lambda _3}\Psi _{\mathcal B_{QQ}}^{SS_z}(\tilde{p}_1,\tilde{p}_2,\tilde{p}_3,\lambda _1,\lambda _2,\lambda _3) \mathcal {C}^{\alpha \beta \gamma }\mathcal {F}_{QQq}\nonumber \\{} & {} \quad \quad \times |4 Q_{\alpha }(p_1,\lambda _1)Q_{\beta }(p_2,\lambda _2)q_{ \gamma }(p_3,\lambda _3)\rangle , \end{aligned}$$

(1)

where q denotes light quark (u, d or s), $\lambda _i$ and $p_i\, (i=1,2,3)$ are helicities and light-front momenta of the quarks, $\mathcal {C}^{\alpha \beta \gamma }$ and $\mathcal {F}_{QQq}$ are the color and flavor wave functions.

The vertex function of $\Omega _{QQQ}$ with the total spin $S=3/2$ and total momentum P is

$$\begin{aligned}{} & {} |\Omega _{QQQ} (P,S,S_z)\rangle \nonumber \\{} & {} \quad =\int \{d^3\tilde{p}_1\}\{d^3\tilde{p}_2\}\{d^3\tilde{p}_3\} \, 2(2\pi )^3\delta ^3(\tilde{P}-\tilde{p_1}-\tilde{p_2}-\tilde{p_3}) \nonumber \\{} & {} \quad \quad \times \sum _{\lambda _1,\lambda _2,\lambda _3}\Psi _{\Omega _{QQQ}}^{SS_z}(\tilde{p}_1,\tilde{p}_2,\tilde{p}_3,\lambda _1,\lambda _2,\lambda _3) \mathcal {C}^{\alpha \beta \gamma }\mathcal {F}_{QQQ}\nonumber \\{} & {} \quad \quad \times | Q_{\alpha }(p_1,\lambda _1)Q_{\beta }(p_2,\lambda _2)Q_{\gamma }(p_3,\lambda _3)\rangle . \end{aligned}$$

(2)

where Q denotes heavy quark (b or c).

In the light-front quark model, the on-mass-shell light-front momentum p is defined as

$$\begin{aligned}{} & {} \tilde{p}=(p^{+},p_{\perp }), \quad p_{\perp } =(p^{1},p^{2}), \quad p^{-} =\frac{m^2+p^2_{\perp }}{p^{+}} \nonumber \\{} & {} \quad \{d^3p\} = \frac{dp^{+}d^2p_{\perp }}{2(2\pi )^3} . \end{aligned}$$

(3)

Internal variables $(x_i, k_{i\perp })$ are introduced to facilitate the calculation ( $i=1,2,3$) and the kinematics constituents are

$$\begin{aligned}{} & {} p^+_i=x_i P^+, \quad p_{i\perp }=x_i P_{\perp }+k_{i\perp }, \quad x_1+x_2+x_3=1, \nonumber \\{} & {} \quad k_{1\perp }+k_{2\perp }+k_{3\perp }=0, \end{aligned}$$

(4)

where $x_i$ is the momentum fraction and satisfies the relation $0<x_1, x_2, x_3<1$.

The spin-space wave functions $\Psi _{\Omega _{QQQ}}^{SS_z}$ and $\Psi _{\mathcal B_{QQ}}^{SS_z}$ [44] are

$$\begin{aligned} \Psi _{\Omega _{QQQ}}^{SS_z}(\tilde{p}_i,\lambda _i)= & {} A_0 \bar{u}(p_3,\lambda _3)[(\bar{P}\!\!\!\!/+M_0) \gamma _{\perp }^\alpha ]v(p_2,\lambda _2)\nonumber \\{} & {} \times \bar{u}(p_1,\lambda _1) u_{\alpha }(\bar{P},S_z)\phi (x_i,k_{i\perp }),\nonumber \\ \Psi _{\mathcal B_{QQ}}^{SS_z}(\tilde{p}_i,\lambda _i)= & {} A_1 \bar{u}(p_3,\lambda _3)[(\bar{P}\!\!\!\!/+M_0) \gamma _{\perp }^\beta ]v(p_2,\lambda _2)\nonumber \\{} & {} \times \bar{u}(p_1,\lambda _1) \gamma _{\perp \beta }\gamma _{5} u(\bar{P},S_z) \phi (x_i,k_{i\perp }), \end{aligned}$$

(5)

where $u_{\alpha }$ is the Rarita-Schwinger spinor for the spin 3/2 particle, $p_i (i=1,2,3)$ is the momentum of the constituent quark, $\lambda _i (i=1,2,3)$ is the helicity of the constituent quark, $\bar{P}$ ($\bar{P}=p_1+p_2+p_3$) is the sum of the momenta of the constituent quarks and satisfies the condition $\bar{P} ^2=M_0^2$, $M_0$ is the invariant mass of the baryon which is different from the baryon mass M $(M^2=P^2)$ and $\gamma _{\perp }^\alpha =\gamma ^{\alpha }-v\!\!\!/v^{\alpha }$. Note that we have used $\bar{u}=v^T C^{-1}$ to simplify $C \bar{u}^T$ in the spin wave function, where C is the charge conjugate matrix [45,46,47], so there is the anti-quark spinor $v(p_2,\lambda _2)$ in Eq. (5).

The factors $A_0$ and $A_1$ in Eq. (5) read

$$\begin{aligned} A_0{} & {} =\frac{1}{4\sqrt{2P^+M_0^3(m_1+e_1)(m_2+e_2)(m_3+e_3)}},\nonumber \\ A_1{} & {} =\frac{1}{4\sqrt{3P^+M_0^3(m_1+e_1)(m_2+e_2)(m_3+e_3)}}. \end{aligned}$$

(6)

The invariant mass square $M_0^2$ is defined as a function of the internal variables $x_i$ and $k_{i\perp }$

$$\begin{aligned} M_0^2=\frac{k_{1\perp }^2+m_1^2}{x_1}+ \frac{k_{2\perp }^2+m_2^2}{x_2}+\frac{k_{3\perp }^2+m_3^2}{x_3}, \end{aligned}$$

(7)

with the internal momentum

$$\begin{aligned} k_i= & {} (k_i^-,k_i^+,k_{i\bot })=(e_i-k_{iz},e_i+k_{iz},k_{i\bot })\nonumber \\= & {} (\frac{m_i^2+k_{i\bot }^2}{x_iM_0},x_iM_0,k_{i\bot }), \end{aligned}$$

(8)

and it is easy to obtain

$$\begin{aligned} e_i= & {} \frac{x_iM_0}{2}+\frac{m_i^2+k_{i\perp }^2}{2x_iM_0},\nonumber \\ k_{iz}= & {} \frac{x_iM_0}{2}-\frac{m_i^2+k_{i\perp }^2}{2x_iM_0}, \end{aligned}$$

(9)

where $e_i$ is the energy of the i-th constituent, and they obey the condition $e_1+e_2+e_3=M_0$. The transverse $k_{i\bot }$ and z direction $k_{iz}$ components constitute a momentum vector $\textbf{k}_i=(k_{i\bot }, k_{iz})$.

The spatial wave function $\phi (x_i, k_{i\perp })$ in Eq. (5) is defined as

$$\begin{aligned}{} & {} \phi (x_1,x_2,x_3,k_{1\perp },k_{2\perp },k_{3\perp })\nonumber \\{} & {} \quad =\sqrt{\frac{e_1e_2e_3}{x_1x_2x_3M_0}} \varphi (\overrightarrow{k}_1,\beta _1)\varphi (\frac{\overrightarrow{k}_2-\overrightarrow{k}_3}{2},\beta _{23}) \end{aligned}$$

(10)

where $\varphi (\overrightarrow{k},\beta )=4(\frac{\pi }{\beta ^2})^{3/4}\textrm{exp}(\frac{-k_z^2-k^2_\perp }{2\beta ^2})$ is the phenomenological Gaussian form and $\beta$ is a non-perturbative parameter that describes the inner structure of baryon [48, 49].

2.2 The form factors of $\Omega _{QQQ}\rightarrow \mathcal B_{QQ}$

Fig. 1

The schematic diagram for the triply heavy baryon decaying into the doubly heavy baryon $\Omega _{QQQ}\rightarrow \mathcal {B}_{QQ}$. The black dot denotes the $V-A$ weak current vertex

The lowest order Feynman diagram responsible for the weak decay $\Omega _{QQQ}\rightarrow \mathcal B_{QQ}$ is shown in Fig. 1. Following the approach given in Refs. [29, 38, 44], the transition matrix element can be calculated with the vertex functions $|\Omega _{QQQ}(P,3/2,S_z) (\mid \Omega _{QQQ}\rangle$) and $|\mathcal B_{QQ}(P',1/2,S'_z)\rangle$ ($\mid \mathcal B_{QQ} \rangle$). The matrix element takes the form

$$\begin{aligned}{} & {} \langle \mathcal B_{QQ}\mid \bar{q} \gamma ^{\mu } (1-\gamma _{5}) Q \mid \Omega _{QQQ} \rangle \nonumber \\{} & {} \quad = \int \frac{\{d^3 \tilde{p}_2\}\{d^3 \tilde{p}_3\}\phi ^*(x^\prime _i,k^\prime _{i \perp }) \phi (x_i,k_{i \perp })\textrm{Tr}[\gamma _{\perp }^\beta (\bar{P^\prime }\!\!\!\!\!/+M_0^\prime )(p_3\!\!\!\!\!/+m_3) (\bar{P}\!\!\!\!/+M_0)\gamma _{\perp }^\alpha (p_2\!\!\!\!\!/-m_2)]}{16\sqrt{6p^+_1p^{\prime +}_1{P}^+P^{\prime +}M_0^3M_0^{\prime 3}(m_1+e_1) (m_2+e_2)(m_3+e_3)(m_1^\prime +e_1^\prime ) (m_2^\prime +e_2^\prime )(m_3^\prime +e_3^\prime )}}\nonumber \\{} & {} \qquad \times \bar{u}(\bar{P}^\prime ,S^\prime _z)\gamma _{\perp \beta }\gamma _{5} (p_1\!\!\!\!\!/^\prime +m^\prime _1)\gamma ^{\mu }(1-\gamma _{5}) (p_1\!\!\!\!\!/+m_1) u_{\alpha }(\bar{P},S_z), \end{aligned}$$

(11)

where

$$\begin{aligned}{} & {} m_1=m_Q, \qquad m_1^{\prime }=m_{q}, \qquad m_2=m_Q, \qquad m_3=m_Q, \\{} & {} \gamma _{\perp }^\alpha =\gamma ^{\alpha }-v\!\!\!/v^{\alpha }, \qquad \gamma _{\perp }^\beta =\gamma ^{\beta }-v^\prime \!\!\!\!/v^{\prime \beta } \end{aligned}$$

(12)

with v ($\bar{P}/M_0$) and $v^\prime$ ($\bar{P^\prime }/M_0^\prime$) being the velocities of the initial state $\Omega _{QQQ}$ and final state $\mathcal B_{QQ}$, respectively. $M(M^\prime )$ is the mass of $\Omega _{QQQ}$ ($\mathcal B_{QQ}$) and $P (P^\prime )$ is the four-momentum of $\Omega _{QQQ}$ ($\mathcal B_{QQ}$).

Setting $\tilde{p}_1-\tilde{p}^\prime _1=\tilde{q}^\prime$, $\tilde{p}^\prime _2=\tilde{p}_2$, $\tilde{p}_3=\tilde{p}^\prime _3$, we can get

$$\begin{aligned}{} & {} x^\prime _{1,2,3}=x_{1,2,3},\quad k^\prime _{1\perp }=k_{1\perp } - (1-x_1) q^\prime _{\perp },\nonumber \\{} & {} k^\prime _{2\perp }=k_{2\perp } + x_2 q^\prime _{\perp }, \quad k^\prime _{3\perp }=k_{3\perp } + x_3 q^\prime _{\perp }, \end{aligned}$$

(13)

where $q^\prime$ denotes the transfer momentum $q^\prime \equiv P-P^\prime$ with $q^{\prime +}=0$ in the LFQM and $p_i^{(\prime )2}=m_i^{(\prime )2}$.

The form factors for the transition $\Omega _{QQQ}\rightarrow \mathcal B_{QQ}$ $(3/2^+\rightarrow 1/2^+)$ are defined as

$$\begin{aligned}{} & {} \langle \mathcal B_{QQ}\mid \bar{q} \gamma ^{\mu } (1-\gamma _{5}) Q \mid \Omega _{QQQ}\rangle \nonumber \\{} & {} \quad =\bar{u}(P',S'_z)\Bigg [ \frac{f_{1}(q'^{2})\gamma ^{\mu } P'^\alpha }{M}+ \frac{f_{2}(q'^{2})P'^\alpha P^\mu }{M(M-M')}\nonumber \\{} & {} \quad +\frac{f_{3}(q'^{2})P'^\alpha P'^\mu }{M(M-M')} +f_4(q'^{2})g^{\alpha \mu }\Bigg ] \gamma _{5} u_{\alpha }(P,S_z)\nonumber \\{} & {} \qquad -\bar{u}(P',S'_z) \Bigg [ \frac{g_{1}(q'^{2})\gamma ^{\mu } P'^\alpha }{M} + \frac{g_{2}(q'^{2})P'^\alpha P^\mu }{M(M-M')}\nonumber \\{} & {} \qquad +\frac{g_{3}(q'^{2})P'^\alpha P'^\mu }{M(M-M')} +g_4(q'^{2})g^{\alpha \mu }\Bigg ] u_{\alpha }(P,S_z). \end{aligned}$$

(14)

The momentum P $(P')$ satisfies the relation $P^{(\prime )2}=M^{(\prime )2}$, while the $\bar{P}$ ($\bar{P}^\prime$) is the sum of the momenta of the constituent quarks and does not obey the on-shell condition. Nonetheless, we can still take the approximation $P^{(\prime )}\simeq \bar{P}^{(\prime )}$.

In order to get the form factors, we can multiply the following terms $\bar{u}_{\xi }(\bar{P},S_z)\gamma _{\mu }\bar{P}'^\xi \gamma _{5} u(\bar{P}',S'_z)$, $\bar{u}_{\xi }(\bar{P},S_z)\bar{P}'_{\mu }\bar{P}'^\xi \gamma _{5} u ( \bar{P}',S'_z)$ and $\bar{u}_{\xi }(\bar{P},S_z) \bar{P}_{\mu }\bar{P}'^\xi \gamma _{5} u(\bar{P}',S'_z)$, $\bar{u}_{\xi }(\bar{P},S_z) g_{\mu }^{\xi } \gamma _{5} u (\bar{P}',S'_z)$ to the right sides of both Eqs. (11) and (14) and then summing over the polarizations of all states. Then we have four algebraic equations, each of which contains the form factors $f_{1}$, $f_{2}$, $f_{3}$ and $f_{4}$. Solving these equations, we obtain the explicit expressions of the form factors $f_{i}$($i=1,2,3,4$) (See Appendix A for details).

Similarly, multiplying the expressions $\bar{u}_{\xi }(\bar{P},S_z) \gamma _{\mu }\bar{P}'^\xi u (\bar{P}',S'_z)$, $\bar{u}_{\xi } (\bar{P},S_z)\bar{P}'_{\mu }\bar{P}'^\xi u (\bar{P}',S'_z)$, $\bar{u}_{\xi }(\bar{P},S_z) \bar{P}_{\mu }\bar{P}'^\xi u (\bar{P}',S'_z)$, $\bar{u}_{\xi }(\bar{P},S_z) g_{\mu }^{\xi } u(\bar{P}',S'_z)$ to the right sides of both Eqs. (11) and (14), we can get form factors $g_{i}$($i=1,2,3,4$).

The flavor wave function of $\Omega _{QQQ}$ is

$$\begin{aligned} \Omega _{QQQ}= & {} \frac{1}{\sqrt{3}}([Q_1Q_2]_AQ_3\nonumber \\{} & {} +[Q_1Q_3]_AQ_2+[Q_2Q_3]_AQ_1). \end{aligned}$$

(15)

Because the baryon $\Omega _{QQQ}$ is composed entirely of heavy quark (c or b), there is an overlap factor $\sqrt{3}$.

Table 1 The masses of the involved quarks and baryons (in units of GeV)

Table 2 The values of the parameter $\beta$ (in units of GeV)

3 Numerical results

3.1 The $\Omega _{QQQ}\rightarrow \mathcal B_{QQ}$ form factors

In order to calculate the form factors numerically, firstly it is necessary to determine the relevant parameters in the model. We adopt the quark mass values given in Ref. [26]. For the mass of $\Xi _{cc}$, the experimental value is adopted [50]. For the other heavy baryons, we adopt the values given in Refs. [10, 13]. The explicit parameter values are listed in Table 1.

The reciprocal of $\beta$ is generally related to the charge radius of two constituents. The rule for selecting $\beta$ is derived from Refs. [37, 44]. Since $\Omega _{QQQ}$ is composed entirely of heavy quark Q, the distances between any two quarks are equal. We choose $\beta _1=\beta _{Q[QQ]}=\frac{2}{\sqrt{3}}\beta _{QQ}$ in the initial state. For the doubly heavy baryon of the final state, we set $\beta _{d[cc]}=\sqrt{2}\beta _{c\bar{d}}$, $\beta _{s[cc]}=\sqrt{2}\beta _{c\bar{s}}$. However, since we know little about the structure of the triply heavy baryons, we make predictions with different $\beta _{QQ}$:

(1) Case I: Assuming that the spectators QQ in the initial state remains unchanged in the final state, which implies the distance between quarks is unchanged, then we set $\beta _{QQ}=2\beta _{Q\bar{Q}}$ for both the initial and final states.

(2) Case II: Assuming that the distance between the two spectators in the initial state is normal but two Q quarks approach each other to form a diquark during the decay process, then we set $\beta _{QQ}=\sqrt{2} \beta _{Q\bar{Q}}$ in the initial state and $\beta _{QQ}=2\beta _{Q\bar{Q}}$ in the final state.

The values of the parameter $\beta$ for different states are taken from Ref. [28] and listed in Table 2. With these parameters we calculate the form factors and make theoretical predictions about the transition rates. Anyway, the $\beta$ value is still model dependent, we hope in the future by comparing the theoretical predictions with experimental data, the parameter value can be determined better.

The form factors $f_{i}$ ($i=1,2,3,4$) and $g_{i}$ ($i=1,2,3,4$) is calculated in the frame $q^{\prime +}=0$, i.e., in the space-like region $q^{\prime 2}=-q^{\prime 2}_{\perp }\le 0$, we can use the same polynomial as that in Ref. [44] to extend the factors into the time-like region. The form factor takes the form

$$\begin{aligned} F(q'^2)=F(0)+a\frac{q'^2}{M_{\Omega _{QQQ}}^2}+b\left( \frac{q'^2}{M_{\Omega _{QQQ}}^2}\right) ^2, \end{aligned}$$

(16)

and the $F(q'^2)$ represents any of the form factors $f_i$ and $g_i$. Using the form factors calculated numerically in the space-like region we fit the parameters a, b and F(0) in the unphysical region and then extrapolate to the physical region with $q'^2\ge 0$ through Eq. (16). The fitted values of a,  b and F(0) for the form factors $f_{i}$ and $g_{i}$ are presented in Tables 3 and 4 for two cases. The dependence of form factors on $q'^2$ for Case I is depicted in Figs. 2 and 3. Due to the similarity in the shape of the curves we leave out the graphs for Case II for brevity.

From Figs. 2 and 3, one can find that the absolute values of the form factors $f_3(q'^2)$ and $g_1(q'^2)$ are close to 0 and that of $f_2(q'^2)$ changes slowly. The curves of $g_2(q'^2)$ and $g_3(q'^2)$ are close to each other.

Table 3 The $\Omega _{ccc}\rightarrow \Xi _{cc}$ form factors given in the polynomial form (Case I in the left side and Case II in the right side)

Fig. 2

a The form factors $f_i\; (i=1,2,3,4)$ and b the form factors $g_i\; (i=1,2,3,4)$ of $\Omega _{ccc}\rightarrow \Xi _{cc}$ in case I

Table 4 The $\Omega _{bbb}\rightarrow \Xi _{bb}$ form factors given in the polynomial form (Case I in the left side and Case II in the right side)

Fig. 3

a The form factors $f_i\; (i=1,2,3,4)$ and b the form factors $g_i\; (i=1,2,3,4)$ of $\Omega _{bbb}\rightarrow \Xi _{bb}$ in case I

3.2 Semi-leptonic decays of $\Omega _{ccc}\rightarrow \mathcal B_{cc}+ l^+\nu _l$ ($\Omega _{bbb}\rightarrow \mathcal B_{bb}+ l^-\bar{\nu }_l$)

In terms of the form factors obtained in the last subsection, we can calculate the decay rates of $\Omega _{ccc}\rightarrow \Xi _{cc}/\Omega _{cc} l^+\nu _l$ and $\Omega _{bbb}\rightarrow \Xi _{bb}l^-\bar{\nu }_l$ in two cases. The explicit amplitudes are shown in Appendix B. We also evaluate the total decay widths and the ratio of the longitudinal to transverse decay rates R. The results are listed in Table 5. In Figs. 4 and 5, we also plot the $\omega$-dependence of the differential decay rates for $\Omega _{ccc}\rightarrow \Xi _{cc}l^+\nu _l$ and $\Omega _{bbb}\rightarrow \Xi _{bb}l^-\bar{\nu }_l$ in Case I. We ignore the graph about the differential decay rates depending on $\omega$ of $\Omega _{ccc}\rightarrow \Omega _{cc} l^+\nu _l$ since it is similar to Fig. 4, except that its peak value is about 23 times higher than that in Fig. 4.

Our predictions on the ratios of longitudinal to transverse decay widths R for $\Omega _{ccc}\rightarrow \Xi _{cc} l^+\nu _l$ and $\Omega _{ccc}\rightarrow \Omega _{cc} l^+\nu _l$ are close to 1, which are different from the results in Refs. [19, 20]. However, the R value for $\Omega _{bbb}\rightarrow \Xi _{bb} l^-\bar{\nu }_l$ is close to that in Ref. [20] in two cases. The difference in R of $\Omega _{ccc}\rightarrow \mathcal B_{cc}l^+\nu _l$ and $\Omega _{bbb}\rightarrow \Xi _{bb} l^-\bar{\nu }_l$ in our results is due to the mass difference of the initial and final states. Our predictions on $\Gamma (\Omega _{ccc}\rightarrow \Xi _{cc}l^+\nu _l)$ and $\Gamma (\Omega _{ccc}\rightarrow \Omega _{cc} l^+\nu _l)$ for Case I are close to those in Ref. [19] where the spectator quark pairs cc are regarded as the diquark. Their assumption is consistent with ours in Case I. However, $\Gamma (\Omega _{bbb}\rightarrow \Xi _{bb}l^-\bar{\nu }_l)$ for Case I is an order of magnitude smaller than that in Ref. [20]. The main reason comes from the difference of the physical pictures and the parameters. The decay width $\Gamma (\Omega _{bbb}\rightarrow \Xi _{bb}l^-\bar{\nu }_l)$ for Case I is larger than that for Case II.

3.3 Non-leptonic decays of $\Omega _{QQQ}\rightarrow \mathcal B_{QQ} + M$

For the non-leptonic decay modes, we adopt the theoretical framework of factorization assumption which ignores the interactions between hadrons, so the hadronic matrix element can be written as the product of two independent matrix elements. It is referred to Ref. [44] for more details.

In Table 6, we show the results of $\Omega _{ccc}\rightarrow \Xi _{cc}+ M$ ($\pi , K$), $\Omega _{ccc}\rightarrow \Omega _{cc}+ M$ ($\pi , K$) and $\Omega _{bbb}\rightarrow \Xi _{bb} + M$ ($\pi , K, D, D_{s}$) for two cases. From Table 6, one may notice that the results for Case II is about 1.5 times larger than those for Case I. Our predictions on $\Gamma ( \Omega _{bbb}\rightarrow \Xi _{bb}+M)$ in Case I are about 1.5 times as large as those in Ref. [20], while those on $\Gamma ( \Omega _{ccc}\rightarrow \Xi _{cc}+M)$ and $\Gamma ( \Omega _{ccc}\rightarrow \Omega _{cc}+ M)$ are about half as large. Besides, our results on $\Gamma ( \Omega _{ccc}\rightarrow \Omega _{cc}+ M)$ for Case I are close to those in [19], but the decay widths on $\Omega _{ccc}\rightarrow \Xi _{cc}+M$ are twice as large as those in [19].

Table 5 The widths (in unit $10^{10} s^{-1}$) of $\Omega _{QQQ}\rightarrow \mathcal B_{QQ}+ l^+\nu _l$

Fig. 4

Differential decay rates $d\Gamma /d\omega$ for the decay $\Omega _{ccc}\rightarrow \Xi _{cc}l^+\nu _l$ in case I

Fig. 5

Differential decay rates $d\Gamma /d\omega$ for the decay $\Omega _{bbb}\rightarrow \Xi _{bb}l^-\bar{\nu }_l$ in case I

Table 6 The widths (in unit $10^{10} s^{-1}$) of $\Omega _{QQQ}\rightarrow \mathcal B_{QQ} + M$

4 Summary

Inspired by the observation of the doubly charmed baryon and great potential on the triply heavy baryon at the LHCb experiments, we study the weak decay of the triply heavy baryon $\Omega _{QQQ}$ to the doubly heavy baryon $\mathcal B_{QQ}$ in this work. For the $\Omega _{QQQ}$ baryon, we employ the three-quark picture where three identical quarks are treated as independent individuals. In the decay process, a heavy quark Q in the initial state decays to a light quark q in the final state, while the other two quarks QQ are roughly considered as spectators. The transition matrix elements and the corresponding form factors are calculated within the LFQM framework. For the triply heavy baryon $\Omega _{QQQ}$ with three heavy identical quarks, we know very little about its inner structure, i.e. the distance between two QQ is normal or extraordinary, so we make predictions for two cases with different $\beta _{QQ}$, which is a model parameter describing the distance between two heavy quarks. The form factors are calculated in the space-like region and extended to the time-like region. Using these form factors, some semi-leptonic and non-leptonic decay widths are predicted. We found that the decay rates for Case II are larger than those for Case I for all channels we analyzed here. Within the quark–diquark picture, the authors in Refs. [19, 20] also studied the weak decays of triply heavy baryons in the LFQM. The predictions on $\Gamma (\Omega _{ccc}\rightarrow \Xi _{cc}l^+\nu _l)$ and $\Gamma (\Omega _{ccc}\rightarrow \Omega _{cc} l^+\nu _l)$ in Ref. [19] are close to our results in Case I, but the values of $\Gamma (\Omega _{ccc}\rightarrow \Xi _{cc}/\Omega _{cc} l^+\nu _l)$ and $\Gamma (\Omega _{bbb}\rightarrow \Omega _{bb} l^-\bar{\nu }_l)$ in Ref. [20] are significantly different from ours by an order of magnitude. For the non-leptonic decay, our results on $\Gamma (\Omega _{ccc}\rightarrow \Omega _{cc} +M)$ in Case I are close to those in Ref. [19], but $\Gamma (\Omega _{ccc} \rightarrow \Xi _{cc} +M)$ are about twice as large as those in Ref. [19]. Compared with the results in Ref. [20], our predictions on $\Gamma (\Omega _{ccc}\rightarrow \Xi _{cc}+M)$ and $\Gamma (\Omega _{ccc}\rightarrow \Omega _{cc}+M)$ in Case I are about half as large. For the decay $\Omega _{bbb}\rightarrow \Xi _{bb}+M$, the widths $\Gamma ( \Omega _{bbb}\rightarrow \Xi _{bb}+M)$ we obtained in Case I are around 1.5 times larger than those in Ref. [20]. Since we employ the three-quark picture instead of the normal diquark picture, different predictions between models can be understood. We hope that future experiments will measure more data to help us determine which option is more plausible so that we can learn more about the inner structure of the triply heavy baryon.

Appendices

Appendix A: The form factors of $\Omega _{QQQ}\rightarrow \mathcal B_{QQ}$

$\bar{u}_{\xi }(\bar{P},S_z)\gamma _{\mu }\bar{P}'^\xi \gamma _{5} u(\bar{P}',S'_z)$, $\bar{u}_{\xi }(\bar{P},S_z)\bar{P}'_{\mu }\bar{P}'^\xi \gamma _{5} u (\bar{P}',S'_z)$, $\bar{u}_{\xi }(\bar{P},S_z) \bar{P}_{\mu }\bar{P}'^\xi \gamma _{5} u(\bar{P}',S'_z)$, $\bar{u}_{\xi }(\bar{P},S_z) g_{\mu }^{\xi } \gamma _{5} u(\bar{P}',S'_z)$ are multiplied to the right side of Eq.(11), and then we have

$$\begin{aligned} F_1= & {} \int \frac{ d x_2 d^2 k^2_{2\perp }}{2(2\pi )^3}\frac{ d x_3 d^2 k^2_{3\perp }}{2(2\pi )^3}\frac{{\phi ^*(x'_i,k'_{i\perp }) \phi (x_i,k_{i\perp })}Tr[\gamma _{\perp }^\beta (\bar{P'}\!\!\!\!\!/+M_0')(p_3\!\!\!\!\!/+m_3) (\bar{P}\!\!\!\!/+M_0)\gamma _{\perp }^\alpha (p_2\!\!\!\!\!/-m_2)]}{A} \nonumber \\{} & {} \times \sum _{S_z,S'_z}\textrm{Tr}[u(\bar{P'},S'_z)\bar{u}(\bar{P}',S'_z) \gamma _{\perp \beta }\gamma _{5} (p_1\!\!\!\!\!/'+m'_1)\gamma ^{\mu } (p_1\!\!\!\!\!/+m_1) {u}_{\alpha }(\bar{P},S_z)\bar{u}_{\xi }(\bar{P},S_z)\gamma _{\mu }\bar{P}'^\xi \gamma _{5}],\\F_2= & {} \int \frac{ d x_2 d^2 k^2_{2\perp }}{2(2\pi )^3}\frac{ d x_3 d^2 k^2_{3\perp }}{2(2\pi )^3}\frac{{\phi ^*(x'_i,k'_{i\perp }) \phi (x_i,k_{i\perp })}Tr[\gamma _{\perp }^\beta (\bar{P'}\!\!\!\!\!/+M_0')(p_3\!\!\!\!\!/+m_3) (\bar{P}\!\!\!\!/+M_0)\gamma _{\perp }^\alpha (p_2\!\!\!\!\!/-m_2)]}{A} \nonumber \\{} & {} \times \sum _{S_z,S'_z}\textrm{Tr}[u(\bar{P'},S'_z)\bar{u}(\bar{P}',S'_z) \gamma _{\perp \beta }\gamma _{5} (p_1\!\!\!\!\!/'+m'_1)\gamma ^{\mu } (p_1\!\!\!\!\!/+m_1) {u}_{\alpha }(\bar{P},S_z)\bar{u}_{\xi }(\bar{P},S_z) \bar{P}'_{\mu }\bar{P}'^\xi \gamma _{5}],\\F_3= & {} \int \frac{ d x_2 d^2 k^2_{2\perp }}{2(2\pi )^3}\frac{ d x_3 d^2 k^2_{3\perp }}{2(2\pi )^3}\frac{{\phi ^*(x'_i,k'_{i\perp }) \phi (x_i,k_{i\perp })}Tr[\gamma _{\perp }^\beta (\bar{P'}\!\!\!\!\!/+M_0')(p_3\!\!\!\!\!/+m_3) (\bar{P}\!\!\!\!/+M_0)\gamma _{\perp }^\alpha (p_2\!\!\!\!\!/-m_2)]}{A} \nonumber \\{} & {} \times \sum _{S_z,S'_z}\textrm{Tr}[u(\bar{P'},S'_z)\bar{u}(\bar{P}',S'_z) \gamma _{\perp \beta }\gamma _{5} (p_1\!\!\!\!\!/'+m'_1)\gamma ^{\mu } (p_1\!\!\!\!\!/+m_1) {u}_{\alpha }(\bar{P},S_z)\bar{u}_{\xi }(\bar{P},S_z)\bar{P}_{\mu }\bar{P}'^\xi \gamma _{5}],\\F_4= & {} \int \frac{ d x_2 d^2 k^2_{2\perp }}{2(2\pi )^3}\frac{ d x_3 d^2 k^2_{3\perp }}{2(2\pi )^3}\frac{{\phi ^*(x'_i,k'_{i\perp }) \phi (x_i,k_{i\perp })}Tr[\gamma _{\perp }^\beta (\bar{P'}\!\!\!\!\!/+M_0')(p_3\!\!\!\!\!/+m_3) (\bar{P}\!\!\!\!/+M_0)\gamma _{\perp }^\alpha (p_2\!\!\!\!\!/-m_2)]}{A} \nonumber \\{} & {} \times \sum _{S_z,S'_z}\textrm{Tr}[u(\bar{P'},S'_z)\bar{u}(\bar{P}',S'_z) \gamma _{\perp \beta }\gamma _{5} (p_1\!\!\!\!\!/'+m'_1)\gamma ^{\mu } (p_1\!\!\!\!\!/+m_1) {u}_{\alpha }(\bar{P},S_z)\bar{u}_{\xi }(\bar{P},S_z)g_{\mu }^{ \xi } \gamma _{5}], \end{aligned}$$

(A1)

with

$$A=16\sqrt{6x_1x'_1M_0^3M_0'^3(m_1+e_1)(m_2+e_2)(m_3+e_3)(m_1'+e_1') (m_2'+e_2')(m_3'+e_3')}.$$

Simultaneously, $\bar{u}_{\xi }(\bar{P},S_z)\gamma _{\mu }\bar{P}'^\xi \gamma _{5} u(\bar{P}',S'_z)$, $\bar{u}_{\xi }(\bar{P},S_z) \bar{P}'_{\mu }\bar{P}'^\xi \gamma _{5} u (\bar{P}',S'_z)$, $\bar{u}_{\xi }(\bar{P},S_z) \bar{P}_{\mu }\bar{P}'^\xi \gamma _{5} u(\bar{P}',S'_z)$, $\bar{u}_{\xi }(\bar{P},S_z) g_{\mu }^{\xi } \gamma _{5} u(\bar{P}',S'_z)$ are multiplied to the right side of Eq.(14), one can obtain

$$\begin{aligned} F_1= & {} \textrm{Tr}\left\{ u(\bar{P'},S'_z)\bar{u}(\bar{P}',S'_z) \left[ \frac{f_{1}(q'^{2})\gamma ^{\mu } P'^\alpha }{M} + \frac{f_{2}(q'^{2})P'^\alpha P^\mu }{M(M-M')}+ \frac{f_{3}(q'^{2})P'^\alpha P'^\mu }{M(M-M')} +f_4(q'^{2})g^{\alpha \mu } \right] \right. \nonumber \\{} & {} \left. \gamma _{5} {u}_{\alpha }(\bar{P},S_z)\bar{u}_{\xi }(\bar{P},S_z)\gamma _{\mu }\bar{P}'^{\xi }\gamma _{5}\right\} ,\\F_2= & {} \textrm{Tr}\left\{ u(\bar{P'},S'_z)\bar{u}(\bar{P}',S'_z) \left[ \frac{f_{1}(q'^{2})\gamma ^{\mu } P'^\alpha }{M} + \frac{f_{2}(q'^{2})P'^\alpha P^\mu }{M(M-M')} + \frac{f_{3}(q'^{2})P'^\alpha P'^\mu }{M(M-M')} +f_4(q'^{2})g^{\alpha \mu } \right] \right. \nonumber \\{} & {} \left. \gamma _{5} {u}_{\alpha }(\bar{P},S_z)\bar{u}_{\xi }(\bar{P},S_z)\bar{P}'_{\mu }\bar{P}'^{\xi } \gamma _{5} \right\} ,\\F_3= & {} \textrm{Tr}\left\{ u(\bar{P'},S'_z)\bar{u}(\bar{P}',S'_z) \left[ \frac{f_{1}(q'^{2})\gamma ^{\mu } P'^\alpha }{M} + \frac{f_{2}(q'^{2})P'^\alpha P^\mu }{M(M-M')} + \frac{f_{3}(q'^{2})P'^\alpha P'^\mu }{M(M-M')} +f_4(q'^{2})g^{\alpha \mu } \right] \right. \nonumber \\{} & {} \left. \gamma _{5} {u}_{\alpha }(\bar{P},S_z)\bar{u}_{\xi }(\bar{P},S_z)\bar{P}_{\mu }\bar{P}'^{\xi } \gamma _{5} \right\} ,\\F_4= & {} \textrm{Tr}\left\{ u(\bar{P'},S'_z)\bar{u}(\bar{P}',S'_z) \left[ \frac{f_{1}(q'^{2})\gamma ^{\mu } P'^\alpha }{M} + \frac{f_{2}(q'^{2})P'^\alpha P^\mu }{M(M-M')} + \frac{f_{3}(q'^{2})P'^\alpha P'^\mu }{M(M-M')} +f_4(q'^{2})g^{\alpha \mu } \right] \right. \nonumber \\{} & {} \left. \gamma _{5} {u}_{\alpha }(\bar{P},S_z)\bar{u}_{\xi }(\bar{P},S_z)g_{\mu }^{\xi } \gamma _{5} \right\} . \end{aligned}$$

(A2)

Appendix B: Semi-leptonic decay of $\Omega _{ccc}\rightarrow \mathcal B_{cc}+ l^+\nu _l$ ($\Omega _{bbb}\rightarrow \mathcal B_{bb}+ l^-\bar{\nu }_l$)

The helicity amplitudes are given by the form factors for $\Omega _{QQQ}\rightarrow \mathcal B_{QQ}$ [19]

$$\begin{aligned} H^{V/A}_{1/2,0}= & {} {\pm }\frac{\sqrt{2MM'(w\mp 1)}}{2 \sqrt{6 q'^2} M^2} [4MM'(w{\pm }1) (M' {\mp } M) \mathcal{N}^{V,A}_1(w)\nonumber \\{} & {} -2M(M^2-M'^2+q'^2)\mathcal{N}^{V,A}_4(w)\nonumber \\{} & {} + \frac{4M^2M'^2}{M-M'} (w^2-1)(\mathcal{N}^{V,A}_2(w)+\mathcal{N}^{V,A}_3(w))],\nonumber \\ H^{V/A}_{1/2,1}= & {} - \sqrt{\frac{2MM'(w{\mp }1)}{3}}[ \mathcal{N}^{V,A}_1(w) \frac{2MM'(w{\pm }1)}{M^2}{\pm } \mathcal{N}^{V,A}_4(w)], \nonumber \\ H^{V/A}_{{1/2,-1}}= & {} {\mp }\sqrt{ 2MM'(w{\mp }1)} \mathcal{N}^{V,A}_4(w), \end{aligned}$$

(B1)

where again the upper (lower) sign corresponds to V(A), $\mathcal{N}^V_i\equiv f_i$, $\mathcal{N}^A_i\equiv g_i$ ($i=1,2,3,4$) and the $q'^2$ is the lepton pair invariant mass with $q'^2=M^2+M'^2-2\,MM' w$. The remaining helicity amplitudes can be obtained using the relation

$$\begin{aligned}{} & {} H^{V,A}_{-\lambda ',\,-\lambda _W}=\mp H^{V,A}_{\lambda ',\, \lambda _W}.\\{} & {} H_{\lambda ',\,\lambda _W}= H^{V}_{\lambda ',\, \lambda _W}- H^{A}_{\lambda ',\, \lambda _W}.\end{aligned}$$

Partial differential decay rates can be represented in the following form

$$\begin{aligned} \frac{d\Gamma _T}{dw}= & {} \frac{G_F^2 |V_{Qq}|^2}{(2\pi )^3}\frac{q'^2 M'^2\sqrt{w^2-1}}{192 \pi ^3M} [|H_{-1/2,\, 1}|^2 \nonumber \\{} & {} + |H_{-1/2,\, -1}|^2+|H_{1/2,\, 1}|^2+ |H_{1/2,\, -1}|^2],\\\frac{d\Gamma _L}{dw}= & {} \frac{G_F^2 |V_{Qq}|^2}{(2\pi )^3} \frac{q'^2 M'^2\sqrt{w^2-1}}{192 \pi ^3M} \nonumber \\{} & {} [|H_{1/2,\, 0}|^2+ |H_{-1/2,\, 0}|^2]. \end{aligned}$$

(B2)

The differential decay width of the $\Omega _{ccc}\rightarrow \mathcal B_{cc}+ l^+\nu _l$ ($\Omega _{bbb}\rightarrow \mathcal B_{bb}+ l^-\bar{\nu }_l$) can be written as

$$\begin{aligned} \frac{d\Gamma }{dw}= & {} \frac{d\Gamma _T}{dw}+\frac{d\Gamma _L}{dw}. \end{aligned}$$

(B3)

Integrating over the parameter $\omega$, we can obtain the total decay width

$$\begin{aligned} \Gamma =\int _1^{\omega _\textrm{max}}d\omega \frac{d\Gamma }{dw}, \end{aligned}$$

(B4)

where $\omega =v\cdot v'$ and the upper bound of the integration $\omega _\textrm{max}=\frac{1}{2}(\frac{M}{M'}+\frac{M'}{M})$ is the maximal recoil.

The ratio of the longitudinal to transverse decay rates R is defined by

$$\begin{aligned} R=\frac{\Gamma _L}{\Gamma _T}=\frac{\int _1^{\omega _\textrm{max}}d\omega ~\left[ |H_{\frac{1}{2},0}|^2+|H_{-\frac{1}{2},0}|^2 \right] }{\begin{array}{c}\int _1^{\omega _\textrm{max}}d\omega ~ \big [ |H_{-1/2,\, 1}|^2+ |H_{1/2,\, -1}|^2\\ +|H_{1/2,\, 1}|^2+ |H_{-1/2,\, -1}|^2\big ]\end{array}}. \end{aligned}$$

(B5)