Restudy of the color-allowed two-body nonleptonic decays of bottom baryons ${\Xi_{b}}$ and ${\Omega_{b}}$ supported by hadron spectroscopy

The investigation of bottom baryon weak decay has aroused the attentions from both theorist and experimentalist. It is not only an important approach to deepen our understanding to the dynamics of the weak transition, but also is the crucial step of searching for new physics beyond the Standard Model (SM).

Taking this opportunity, we want to introduce several recent progresses. As we know, the lepton flavor universality (LFU) violation has been examined in various $b\to c$ weak transitions BaBar:2012obs ; BaBar:2013mob ; Belle:2015qfa ; LHCb:2015gmp ; Belle:2016dyj ; Belle:2019rba ; FermilabLattice:2021cdg in the past decade. The measurement of the ratio $R_{D^{(*)}}=\mathcal{B}(B\to D^{(*)}\tau\nu_{\tau})/\mathcal{B}(B\to D^{(*)}e(\mu)\nu_{e(\mu)})$ BaBar:2012obs ; BaBar:2013mob ; Belle:2015qfa ; LHCb:2015gmp ; Belle:2016dyj ; Belle:2019rba ; FermilabLattice:2021cdg shows the discrepancy with the prediction of the SM HFLAV:2019otj , which indicates the possible evidence of new physics. Inspired by the anomalies of $R_{D^{(*)}}$ existing in the $b\to c$ weak transitions, it is interesting to study the corresponding ratios for the bottom baryon weak decays like $\Xi_{b}\to\Xi_{c}\ell^{-}\nu_{\ell}$ and $\Omega_{b}\to\Omega_{c}\ell^{-}\nu_{\ell}$, where the key point is to calculate the form factors involved in the corresponding weak transition of the bottom baryon into the charmed baryon. For the nonleptonic decays of the bottom baryon, a series of intriguing measurements were performed, which include the observation of charmful and charmless modes CDF:2008llm ; LHCb:2014yin ; LHCb:2016rja ; ParticleDataGroup:2020ssz , the discovery of the hidden-charm pentaquark states $P_{c}(4312)$, $P_{c}(4380)$, $P_{c}(4440)$, and $P_{c}(4457)$ in the $\Lambda_{b}\to J/\psi pK$ process LHCb:2015yax ; LHCb:2019kea , and $P_{cs}(4459)$ in the $\Xi_{b}\to J/\psi\Lambda K$ process LHCb:2020jpq . These efforts make us gain a deeper understanding of the dynamics involved in the heavy-flavor baryon weak decays.

Although great progress had been made, continuing to explore new allowed decay modes of the bottom baryons is a research issue full of opportunity [see the Particle Data Group (PDG) ParticleDataGroup:2020ssz for learning the present experimental status]. With the accumulation of experimental data, the LHCb experiment shows its potential to explore the allowed decays of the bottom baryons like the $\Xi_{b}$ and $\Omega_{b}$ states, which is still missing in the PDG. Besides, with the KEKB upgrading to the SuperKEKB, the center-of-mass energy of the $e^{+}e^{-}$ collision may reach up to 11.24 GeV. The ongoing Belle II Belle-II:2018jsg should be a potential experiment to perform the study on the bottom-flavor physics. Facing this exciting status, we have reason to believe that it is suitable time to investigate the two-body nonleptonic decays of the $\Xi_{b}$ and $\Omega_{b}$ baryons, which is the main task of this work.

The bottom baryon weak decays have been widely studied by various approaches including the quark models Cheng:1996cs ; Ivanov:1997hi ; Ivanov:1997ra ; Albertus:2004wj ; Ebert:2006rp ; Gutsche:2018utw ; Faustov:2018ahb ; Geng:2020ofy , the flavor symmetry method Zhao:2018zcb , the light-front approach Chua:2018lfa ; Chua:2019yqh ; Ke:2019smy ; Zhu:2018jet ; Zhao:2018zcb ; Li:2021qod , and the quantum chromodynamics (QCD) sum rules Wang:2008sm ; Khodjamirian:2011jp ; Wang:2015ndk ; Zhao:2020mod . For these theoretical studies, how to estimate the form factors of the weak transition is the key issue. Additionally, for the bottom baryon weak decays, how to optimize the three-body problem is also a challenge. Usually, the quark-diquark scheme as an approximate treatment was widely used in previous theoretical works Guo:2005qa ; Zhu:2018jet ; Zhao:2018zcb ; Chua:2018lfa ; Chua:2019yqh ; Ke:2019smy . And the spatial wave functions of these hadrons involved in the bottom baryon weak decays are approximately taken as a simple harmonic oscillator wave function, which makes the results dependent on the parameter of the harmonic oscillator wave function. For avoiding the uncertainty from these approximate treatments mentioned above, in this work we calculate the weak transition form factors of the $\Xi_{b}\to\Xi_{c}^{(*)}$ and $\Omega_{b}\to\Omega_{c}^{(*)}$ transitions with emitting a pseudoscalar meson ($\pi^{-}$, $K^{-}$, $D^{-}$ and $D_{s}^{-}$) or a vector meson ($\rho^{-}$, $K^{*-}$, $D^{*-}$, and $D_{s}^{*-}$) in the three-body light-front quark model. Here, $\Xi_{c}^{(*)}$ denotes the ground state $\Xi_{c}$ or its first radial excited state $\Xi_{c}(2970)$, while $\Omega_{c}^{(*)}$ represents the ground state $\Omega_{c}$ or its first radial excited state $\Omega_{c}(2S)$. In the realistic calculation, we take the numerical spatial wave functions of these involved bottom and charmed baryons as input, where the semirelativistic potential model Capstick:1985xss ; Li:2021qod associated with the Gaussian expansion method (GEM) Hiyama:2003cu ; Yoshida:2015tia ; Hiyama:2018ivm ; Yang:2019lsg is adopted. By fitting the mass spectrum of these observed bottom and charmed baryons, the parameters of the adopted semirelativistic potential model can be fixed. Comparing with former approximation of taking a simple harmonic oscillator wave function Guo:2005qa ; Zhu:2018jet ; Zhao:2018zcb ; Chua:2018lfa ; Chua:2019yqh ; Ke:2019smy , the treatment given in this work can avoid the uncertainties resulting from the selection of the spatial wave function of the heavy baryon. Thus, the color-allowed two-body nonleptonic decays of bottom baryons ${\Xi_{b}}$ and ${\Omega_{b}}$ with the support of hadron spectroscopy as a development. In the following sections, more details will be illustrated.

This paper is organized as follows. After the introduction, the formula of the form factors of the weak transitions $\Xi_{b}\to\Xi_{c}^{(*)}$ and $\Omega_{b}\to\Omega_{c}^{(*)}$ is given in Sec. II. For getting the numerical spatial wave functions of these involved heavy baryons, we introduce the adopted semirelativistic potential model and GEM. With these results as input, the calculated concerned form factors are displayed. In Sec. III, we study the color-allowed two-body nonleptonic decays with emitting a pseudoscalar meson ($\pi^{-}$, $K^{-}$, $D^{-}$, and $D_{s}^{-}$) or vector meson ($\rho^{-}$, $K^{*-}$, $D^{*-}$, and $D_{s}^{*-}$) in the naïve factorization assumption. Finally, the paper ends with a short summary.

In this section, we briefly introduce how to calculate the form factors discussed in this work. Given that the quarks are confined in hadron, the weak transition matrix element cannot be calculated in the framework of perturbative QCD. Usually, the weak transition matrix element can be parametrized in terms of a series of dimensionless form factors Li:2021qod ; Ke:2019smy

$$\begin{split}\langle&\mathcal{B}_{c}(1/2^{+})(P^{\prime},J^{\prime}_{z})|\bar{c}\gamma^{\mu}(1-\gamma_{5})b|\mathcal{B}_{b}(1/2^{+})(P,J_{z})\rangle\\ &=\bar{u}(P^{\prime},J^{\prime}_{z})\left[f^{V}_{1}(q^{2})\gamma^{\mu}+i\frac{f^{V}_{2}(q^{2})}{M}\sigma^{\mu\nu}q_{\nu}+\frac{f^{V}_{3}(q^{2})}{M}q^{\mu}\right.\\ &\quad\left.-\left(g^{A}_{1}(q^{2})\gamma^{\mu}+i\frac{g^{A}_{2}(q^{2})}{M}\sigma^{\mu\nu}q_{\nu}+\frac{g^{A}_{3}(q^{2})}{M}q^{\mu}\right)\gamma_{5}\right]u(P,J_{z})\end{split}$$

(2.1)

for the transitions of the bottom baryon to the charmed baryon. Here, $M(P)$ and $M^{\prime}(P^{\prime})$ are the mass(four-momentum) for the initial and final baryons, respectively, $\sigma^{\mu\nu}=i[\gamma^{\mu},\gamma^{\nu}]/2$, and $q=P-P^{\prime}$ denotes the transferred momentum between the initial and final baryons.

The vertex function of a single heavy-flavor baryon $\mathcal{B}_{Q}$ ($Q=b,c$) with the spin $J=1/2$ and the momentum $P$ is

$$\begin{split}|\mathcal{B}_{Q}&(P,J,J_{z})\rangle=\int\frac{d^{3}\tilde{p}_{1}}{2(2\pi)^{3}}\frac{d^{3}\tilde{p}_{2}}{2(2\pi)^{3}}\frac{d^{3}\tilde{p}_{3}}{2(2\pi)^{3}}2(2\pi)^{3}\\ &\times\sum_{\lambda_{1},\lambda_{2},\lambda_{3}}\Psi^{J,J_{z}}(\tilde{p}_{i},\lambda_{i})C^{\alpha\beta\gamma}\delta^{3}(\tilde{\bar{P}}-\tilde{p}_{1}-\tilde{p}_{2}-\tilde{p}_{3})\\ &\times~{}F_{nnQ}~{}|n_{\alpha}(\tilde{p}_{1},\lambda_{1})\rangle~{}|n_{\beta}(\tilde{p}_{2},\lambda_{2})\rangle~{}|Q_{\gamma}(\tilde{p}_{3},\lambda_{3})\rangle.\end{split}$$

(2.2)

Here, $n=u,d,s$ is the light-flavor quark, $C^{\alpha\beta\gamma}$ and $F_{nnQ}$ represent the color and flavor factors, and $\lambda_{i}$ and $p_{i}$ ($i$=1,2,3) are the helicities and light-front momenta of the on-mass-shell quarks, respectively, defined as

$$\tilde{p}_{i}=(p_{i}^{+},p_{i\bot}),\quad p_{i}^{+}=p_{i}^{0}+p_{i}^{3},\quad p_{i\bot}=(p_{i}^{1},p_{i}^{2}).$$

(2.3)

As suggested in Ref. Tawfiq:1998nk , the spin and spatial wave functions for $\mathcal{B}_{Q}(\bar{3}_{f})$ and $\mathcal{B}_{Q}(6_{f})$ with the spin-parity quantum number $J^{P}=1/2^{+}$ are written as

$$\begin{split}\Psi^{J,J_{z}}(\tilde{p}_{i},\lambda_{i})=&A_{0}\bar{u}(p_{1},\lambda_{1})[(\not{\bar{P}}+M_{0})\gamma_{5}]v(p_{2},\lambda_{2})\\ &\times\bar{u}_{Q}(p_{3},\lambda_{3})u(\bar{P},J,J_{z})\phi(x_{i},k_{i\bot}),\\ \Psi^{J,J_{z}}(\tilde{p}_{i},\lambda_{i})=&A_{1}\bar{u}(p_{1},\lambda_{1})[(\not{\bar{P}}+M_{0})\gamma_{\bot\alpha}]v(p_{2},\lambda_{2})\\ &\times\bar{u}_{Q}(p_{3},\lambda_{3})\gamma_{\bot}^{\alpha}\gamma_{5}u(\bar{P},J,J_{z})\phi(x_{i},k_{i\bot})\end{split}$$

(2.4)

with

$$A_{0}=\sqrt{3}A_{1}=\frac{1}{\sqrt{16\bar{P}^{+}M_{0}^{3}(e_{1}+m_{1})(e_{2}+m_{2})(e_{3}+m_{3})}}$$

representing the normalization factor Ke:2019smy .

In the framework of the three-body light-front quark model, the general expressions are written as Li:2021qod ; Ke:2019smy

$$\begin{split}\langle\mathcal{B}&{}_{c}^{(*)}\left(\bar{3}_{f},1/2^{+}\right)(\bar{P}^{\prime},J^{\prime}_{z})|\bar{c}\gamma^{\mu}(1-\gamma_{5})b|\mathcal{B}_{b}\left(\bar{3}_{f},1/2^{+}\right)(\bar{P},J_{z})\rangle\\ =&\int\left(\frac{dx_{1}d^{2}\vec{k}_{1\bot}}{2(2\pi)^{3}}\right)\left(\frac{dx_{2}d^{2}\vec{k}_{2\bot}}{{2(2\pi)^{3}}}\right)\frac{\phi(x_{i},\vec{k}_{i\bot})\phi^{*}(x_{i}^{\prime},\vec{k}_{i\bot}^{\prime})}{16\sqrt{x_{3}x_{3}^{\prime}{M_{0}}^{3}{M_{0}^{\prime}}^{3}}}\frac{\text{Tr}[(\not{\bar{P}}^{\prime}-M_{0}^{\prime})\gamma_{5}(\not{p}_{1}+m_{1})(\not{\bar{P}}+M_{0})\gamma_{5}(\not{p}_{2}-m_{2})]}{\sqrt{(e_{1}+m_{1})(e_{2}+m_{2})(e_{3}+m_{3})(e_{1}^{\prime}+m_{1}^{\prime})(e_{2}^{\prime}+m_{2}^{\prime})(e_{3}^{\prime}+m_{3}^{\prime})}}\\ &\times\bar{u}(\bar{P}^{\prime},J_{z}^{\prime})(\not{p}_{3}^{\prime}+m_{3}^{\prime})\gamma^{\mu}(1-\gamma_{5})(\not{p}_{3}+m_{3})u(\bar{P},J_{z}),\end{split}$$

(2.5)

$$\begin{split}\langle\mathcal{B}&{}_{c}^{(*)}\left(6_{f},1/2^{+}\right)(\bar{P}^{\prime},J^{\prime}_{z})|\bar{c}\gamma^{\mu}(1-\gamma_{5})b|\mathcal{B}_{b}\left(6_{f},1/2^{+}\right)(\bar{P},J_{z})\rangle=\\ \int&\left(\frac{dx_{1}d^{2}\vec{k}_{1\bot}}{2(2\pi)^{3}}\right)\left(\frac{dx_{2}d^{2}\vec{k}_{2\bot}}{{2(2\pi)^{3}}}\right)\frac{\phi(x_{i},\vec{k}_{i\bot})\phi^{*}(x_{i}^{\prime},\vec{k}_{i\bot}^{\prime})}{48\sqrt{x_{3}x_{3}^{\prime}{M_{0}}^{3}{M_{0}^{\prime}}^{3}}}\frac{\text{Tr}[\gamma_{\bot}^{\alpha}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})(\not{p}_{1}+m_{1})(\not{\bar{P}}+M_{0})\gamma_{\bot}^{\beta}(\not{p}_{2}-m_{2})]}{\sqrt{(e_{1}+m_{1})(e_{2}+m_{2})(e_{3}+m_{3})(e_{1}^{\prime}+m_{1}^{\prime})(e_{2}^{\prime}+m_{2}^{\prime})(e_{3}^{\prime}+m_{3}^{\prime})}}\\ &\times\bar{u}(\bar{P}^{\prime},J_{z}^{\prime})\gamma_{\bot\alpha}\gamma_{5}(\not{p}_{3}^{\prime}+m_{3}^{\prime})\gamma^{\mu}(1-\gamma_{5})(\not{p}_{3}+m_{3})\gamma_{\bot\beta}\gamma_{5}u(\bar{P},J_{z}),\end{split}$$

(2.6)

for the $\mathcal{B}_{b}(\bar{3}_{f})\to\mathcal{B}_{c}(\bar{3}_{f})$ and $\mathcal{B}_{b}(6_{f})\to\mathcal{B}_{c}(6_{f})$ transitions, respectively. Here, $\bar{P}=p_{1}+p_{2}+p_{3}$ and $\bar{P}^{\prime}=p_{1}+p_{2}+p_{3}^{\prime}$ are the light-front momenta for initial and final baryons, respectively, considering $p_{1}=p_{1}^{\prime}$ and $p_{2}=p_{2}^{\prime}$ in the spectator scheme, while $\phi$ and $\phi^{*}$ represent the spatial wave functions for the initial bottom baryon and the final charmed baryon, respectively. In the previous references Guo:2005qa ; Zhu:2018jet ; Zhao:2018zcb ; Chua:2018lfa ; Chua:2019yqh ; Ke:2019smy , the wave functions for baryon are usually treated as a simple harmonic oscillator forms with the oscillator parameter $\beta$, which results in the $\beta$ dependence of the result. For avoiding this uncertainty, in this work, we adopt the numerical spatial wave functions for these involved baryons calculated by solving the three-body Schrödinger equation with the semirelativistic quark model.

To calculate the form factors defined in Eq. (2.1) from Eqs. (2.5)-(2.6), $V^{+}$, $A^{+}$, $\vec{q}_{\bot}\cdot\vec{V}$, $\vec{q}_{\bot}\cdot\vec{A}$, $\vec{n}_{\bot}\cdot\vec{V}$, and $\vec{n}_{\bot}\cdot\vec{A}$ are applied within a special gauge $q^{+}=0$. The details can be found in Ref. Chua:2019yqh . Finally, the form factors are expressed as Li:2021qod

$$\begin{split}f^{V}_{1}(q^{2})&=\int\mathcal{DS}_{0}\frac{1}{8\bar{P}^{+}\bar{P}^{\prime+}}\text{Tr}[(\not{\bar{P}}+M_{0})\gamma^{+}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})(\not{p}_{3}^{\prime}+m_{3}^{\prime})\gamma^{+}(\not{p}_{3}+m_{3})],\\ f^{V}_{2}(q^{2})&=\int\mathcal{DS}_{0}\frac{iM}{8\bar{P}^{+}\bar{P}^{\prime+}q^{2}}\text{Tr}[(\not{\bar{P}}+M_{0})\sigma^{+\mu}q_{\mu}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})(\not{p}_{3}^{\prime}+m_{3}^{\prime})\gamma^{+}(\not{p}_{3}+m_{3})],\\ f^{V}_{3}(q^{2})&=\frac{M}{M+M^{\prime}}\left(f_{1}^{V}(q^{2})(1-\frac{2\bar{P}\cdot q}{q^{2}})+\int\frac{\mathcal{DS}_{0}}{4\sqrt{\bar{P}^{+}\bar{P}^{\prime+}}q^{2}}\text{Tr}[(\not{\bar{P}}+M_{0})\gamma^{+}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})(\not{p}_{3}^{\prime}+m_{3}^{\prime})\not{q}\not{p}_{3}+m_{3})]\right),\\ g^{A}_{1}(q^{2})&=\int\mathcal{DS}_{0}\frac{1}{8\bar{P}^{+}\bar{P}^{\prime+}}\text{Tr}[(\not{\bar{P}}+M_{0})\gamma^{+}\gamma_{5}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})(\not{p}_{3}^{\prime}+m_{3}^{\prime})\gamma^{+}\gamma_{5}(\not{p}_{3}+m_{3})],\\ g^{A}_{2}(q^{2})&=\int\mathcal{DS}_{0}\frac{-iM}{8\bar{P}^{+}\bar{P}^{\prime+}q^{2}}\text{Tr}[(\not{\bar{P}}+M_{0})\sigma^{+\mu}q_{\mu}\gamma_{5}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})(\not{p}_{3}^{\prime}+m_{3}^{\prime})\gamma^{+}\gamma_{5}(\not{p}_{3}+m_{3})],\\ g^{A}_{3}(q^{2})&=\frac{M}{M-M^{\prime}}\left(g^{A}_{1}(q^{2})(-1+\frac{2\bar{P}\cdot q}{q^{2}})+\int\frac{-\mathcal{DS}_{0}}{4\sqrt{\bar{P}^{+}\bar{P}^{\prime+}}q^{2}}\text{Tr}[(\not{\bar{P}}+M_{0})\gamma^{+}\gamma_{5}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})(\not{p}_{3}^{\prime}+m_{3}^{\prime})~{}\not{q}~{}\gamma_{5}~{}(\not{p}_{3}+m_{3})]\right),\\ \mathcal{DS}_{0}&=\frac{dx_{1}d^{2}\vec{k}_{1\bot}dx_{2}d^{2}\vec{k}_{2\bot}}{2(2\pi)^{3}2(2\pi)^{3}}\frac{\phi^{*}(x_{i}^{\prime},\vec{k}_{i\bot}^{\prime})\phi(x_{i},\vec{k}_{i\bot})}{16\sqrt{x_{3}x_{3}^{\prime}M_{0}^{3}M_{0}^{\prime 3}}}\frac{\text{Tr}[(\not{\bar{P}}^{\prime}-M_{0}^{\prime})\gamma_{5}(\not{p}_{1}+m_{1})(\not{\bar{P}}+M_{0})\gamma_{5}(\not{p}_{2}-m_{2})]}{\sqrt{(e_{1}+m_{1})(e_{2}+m_{2})(e_{3}+m_{3})(e_{1}^{\prime}+m_{1}^{\prime})(e_{2}+m_{2}^{\prime})(e_{3}^{\prime}+m_{3}^{\prime})}},\end{split}$$

(2.7)

and

$$\begin{split}f^{V}_{1}(q^{2})&=\int\mathcal{DS}_{1}\frac{1}{8\bar{P}^{+}\bar{P}^{\prime+}}\text{Tr}[(\not{\bar{P}}+M_{0})\gamma^{+}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})\gamma_{\bot\alpha}\gamma_{5}(\not{p}_{3}^{\prime}+m_{3}^{\prime})\gamma^{+}(\not{p}_{3}+m_{3})\gamma_{\bot\beta}\gamma_{5}],\\ f^{V}_{2}(q^{2})&=\int\mathcal{DS}_{1}\frac{iM}{8\bar{P}^{+}\bar{P}^{\prime+}q^{2}}\text{Tr}[(\not{\bar{P}}+M_{0})\sigma^{+\mu}q_{\mu}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})\gamma_{\bot\alpha}\gamma_{5}(\not{p}_{3}^{\prime}+m_{3}^{\prime})\gamma^{+}(\not{p}_{3}+m_{3})\gamma_{\bot\beta}\gamma_{5}],\\ f^{V}_{3}(q^{2})&=\frac{M}{M+M^{\prime}}\left(f_{1}^{V}(q^{2})(1-\frac{2\bar{P}\cdot q}{q^{2}})+\int\frac{\mathcal{DS}_{1}}{4\sqrt{\bar{P}^{+}\bar{P}^{\prime+}}q^{2}}\text{Tr}[(\not{\bar{P}}+M_{0})\gamma^{+}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})\gamma_{\bot\alpha}\gamma_{5}(\not{p}_{3}^{\prime}+m_{3}^{\prime})\not{q}\not{p}_{3}+m_{3})\gamma_{\bot\beta}\gamma_{5}]\right),\\ g^{A}_{1}(q^{2})&=\int\mathcal{DS}_{1}\frac{1}{8\bar{P}^{+}\bar{P}^{\prime+}}\text{Tr}[(\not{\bar{P}}+M_{0})\gamma^{+}\gamma_{5}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})\gamma_{\bot\alpha}\gamma_{5}(\not{p}_{3}^{\prime}+m_{3}^{\prime})\gamma^{+}\gamma_{5}(\not{p}_{3}+m_{3})\gamma_{\bot\beta}\gamma_{5}],\\ g^{A}_{2}(q^{2})&=\int\mathcal{DS}_{1}\frac{-iM}{8\bar{P}^{+}\bar{P}^{\prime+}q^{2}}\text{Tr}[(\not{\bar{P}}+M_{0})\sigma^{+\mu}q_{\mu}\gamma_{5}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})\gamma_{\bot\alpha}\gamma_{5}(\not{p}_{3}^{\prime}+m_{3}^{\prime})\gamma^{+}\gamma_{5}(\not{p}_{3}+m_{3})\gamma_{\bot\beta}\gamma_{5}],\\ g^{A}_{3}(q^{2})&=\frac{M}{M-M^{\prime}}\left(g^{A}_{1}(q^{2})(-1+\frac{2\bar{P}\cdot q}{q^{2}})+\int\frac{-\mathcal{DS}_{1}}{4\sqrt{\bar{P}^{+}\bar{P}^{\prime+}}q^{2}}\text{Tr}[(\not{\bar{P}}+M_{0})\gamma^{+}\gamma_{5}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})\gamma_{\bot\alpha}\gamma_{5}(\not{p}_{3}^{\prime}+m_{3}^{\prime})~{}\not{q}~{}\gamma_{5}~{}(\not{p}_{3}+m_{3})\gamma_{\bot\beta}\gamma_{5}]\right),\\ \mathcal{DS}_{1}&=\frac{dx_{1}d^{2}\vec{k}_{1\bot}dx_{2}d^{2}\vec{k}_{2\bot}}{2(2\pi)^{3}2(2\pi)^{3}}\frac{\phi^{*}(x_{i}^{\prime},\vec{k}_{i\bot}^{\prime})\phi(x_{i},\vec{k}_{i\bot})}{48\sqrt{x_{3}x_{3}^{\prime}M_{0}^{3}M_{0}^{\prime 3}}}\frac{\text{Tr}[\gamma_{\bot}^{\alpha}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})(\not{p}_{1}+m_{1})(\not{\bar{P}}+M_{0})\gamma_{\bot}^{\beta}(\not{p}_{2}-m_{2})]}{\sqrt{(e_{1}+m_{1})(e_{2}+m_{2})(e_{3}+m_{3})(e_{1}^{\prime}+m_{1}^{\prime})(e_{2}+m_{2}^{\prime})(e_{3}^{\prime}+m_{3}^{\prime})}},\end{split}$$

(2.8)

for the $\mathcal{B}_{b}(\bar{3}_{f})\to\mathcal{B}_{c}(\bar{3}_{f})$ and $\mathcal{B}_{b}(6_{f})\to\mathcal{B}_{c}(6_{f})$ transitions, respectively.

In this section, we illustrate how to obtain the concerned spatial wave functions by the semirelativistic quark model with the help of the GEM. Different from the meson system, baryon is a typical three-body system. Thus, its wave function can be extracted by solving the three-body Schrödinger equation. Here, the semirelativistic potentials were given in Refs. Godfrey:1985xj ; Capstick:1985xss which are applied to the realistic calculation of this work. The involved Hamiltonian includes Li:2021qod

$$\mathcal{H}=K+\sum_{i<j}(S_{ij}+G_{ij}+V^{\text{so(s)}}_{ij}+V^{\text{so(v)}}_{ij}+V^{\text{ten}}_{ij}+V^{\text{con}}_{ij})$$

(3.1)

with $K$, $S$, $G$, $V^{\text{so}(s)}$, $V^{\text{so}(v)}$,$V^{\text{tens}}$ and $V^{\text{con}}$ representing the kinetic energy, the spin-independent linear confinement piece, the Coulomb-like potential, the scalar type-spin-orbit interaction, the vector type-spin-orbit interaction, the tensor potential, and the spin-dependent contact potential, respectively. Their concrete expressions are listed here Godfrey:1985xj ; Capstick:1985xss ; Song:2015nia ; Pang:2017dlw :

	$\displaystyle K$	$\displaystyle=$	$\displaystyle\sum_{i=1,2,3}\sqrt{m_{i}^{2}+p_{i}^{2}},$		(3.2)
	$\displaystyle S_{ij}$	$\displaystyle=$	$\displaystyle-\frac{3}{4}\left(br_{ij}\left[\frac{e^{-\sigma^{2}r_{ij}^{2}}}{\sqrt{\pi}\sigma r_{ij}}+\left(1+\frac{1}{2\sigma^{2}r_{ij}^{2}}\right)\frac{2}{\sqrt{\pi}}\right.\right.$		(3.3)
			$\displaystyle\left.\left.\times\int_{0}^{\sigma r_{ij}}e^{-x^{2}}dx\right]\right)\mathbf{F_{i}}\cdot\mathbf{F_{j}}+\frac{c}{3},$
	$\displaystyle G_{ij}$	$\displaystyle=$	$\displaystyle\sum_{k}\frac{\alpha_{k}}{r_{ij}}\left[\frac{2}{\sqrt{\pi}}\int_{0}^{\tau_{k}r_{ij}}e^{-x^{2}}dx\right]\mathbf{F_{i}}\cdot\mathbf{F_{j}}$		(3.4)

for the spin-independent terms with

$$\sigma^{2}=\sigma_{0}^{2}\left[\frac{1}{2}+\frac{1}{2}\left(\frac{4m_{i}m_{j}}{(m_{i}+m_{j})^{2}}\right)^{4}+s^{2}\left(\frac{2m_{i}m_{j}}{m_{i}+m_{j}}\right)^{2}\right],$$

(3.5)

and

$$\begin{split}V^{\text{so}(s)}_{ij}=&-\frac{\mathbf{r_{ij}}\times\mathbf{p_{i}}\cdot\mathbf{S_{i}}}{2m_{i}^{2}}\frac{1}{r_{ij}}\frac{\partial S_{ij}}{r_{ij}}+\frac{\mathbf{r_{ij}}\times\mathbf{p_{j}}\cdot\mathbf{S_{j}}}{2m_{j}^{2}}\frac{1}{r_{ij}}\frac{\partial S_{ij}}{\partial r_{ij}},\\ V^{\text{so}(v)}_{ij}=&\frac{\mathbf{r_{ij}}\times\mathbf{p_{i}}\cdot\mathbf{S_{i}}}{2m_{i}^{2}}\frac{1}{r_{ij}}\frac{\partial G_{ij}}{r_{ij}}-\frac{\mathbf{r_{ij}}\times\mathbf{p_{j}}\cdot\mathbf{S_{j}}}{2m_{j}^{2}}\frac{1}{r_{ij}}\frac{\partial G_{ij}}{r_{ij}}\\ &-\frac{\mathbf{r_{ij}}\times\mathbf{p_{j}}\cdot\mathbf{S_{i}}-\mathbf{r_{ij}}\times\mathbf{p_{i}}\cdot\mathbf{S_{j}}}{m_{i}~{}m_{j}}\frac{1}{r_{ij}}\frac{\partial G_{ij}}{\partial r_{ij}},\\ V^{\text{tens}}_{ij}=&-\frac{1}{m_{i}m_{j}}\left[\left(\mathbf{S_{i}}\cdot\mathbf{\hat{r}_{ij}}\right)\left(\mathbf{S_{j}}\cdot\mathbf{\hat{r}_{ij}}\right)-\frac{\mathbf{S_{i}}\cdot\mathbf{S_{j}}}{3}\right]\left(\frac{\partial^{2}G_{ij}}{\partial r^{2}}-\frac{\partial G_{ij}}{r_{ij}\partial r_{ij}}\right),\\ V^{\text{con}}_{ij}=&\frac{2\mathbf{S_{i}}\cdot\mathbf{S_{j}}}{3m_{i}m_{j}}\nabla^{2}G_{ij}\end{split}$$

for the spin-dependent terms, where $m_{i}$ and $m_{j}$ are the masses of quark $i$ and $j$, respectively. And, we take $\langle\mathbf{F_{i}}\cdot\mathbf{F_{j}}\rangle=-2/3$ for quark-quark interaction.

In the following, a general potential which relies on the center-of-mass of interacting quarks and momentum are made up for the loss of relativistic effects in the nonrelativistic limit Godfrey:1985xj ; Capstick:1985xss ; Wang:2018rjg ; Wang:2019mhs ; Duan:2021alw , that is,

$$\begin{split}&G_{ij}\to\left(1+\frac{p^{2}}{E_{i}E_{j}}\right)^{1/2}G_{ij}\left(1+\frac{p^{2}}{E_{i}E_{j}}\right)^{1/2},\\ &\frac{V^{k}_{ij}}{m_{i}m_{j}}\to\left(\frac{m_{i}m_{j}}{E_{i}E_{j}}\right)^{1/2+\epsilon_{k}}\frac{V^{k}_{ij}}{m_{i}m_{j}}\left(\frac{m_{i}m_{j}}{E_{i}E_{j}}\right)^{1/2+\epsilon_{k}}\end{split}$$

(3.6)

with $E_{i}=\sqrt{p^{2}+m_{i}^{2}}$, where subscript $k$ was applied to distinguish the contributions from the contact, tensor, vector spin-orbit, and scalar spin-orbit terms. In addition, $\epsilon_{k}$ represents the relevant modification parameters, which are collected in Table 1.

The total wave function of the single heavy baryon is composed of color, flavor, spatial, and spin wave functions, i.e.,

$$\Psi_{\mathbf{J},\mathbf{M_{J}}}=\chi^{c}\left\{\chi^{s}_{\mathbf{S},\mathbf{M_{S}}}\psi^{p}_{\mathbf{L},\mathbf{M_{L}}}\right\}_{\mathbf{J},\mathbf{M_{J}}}\psi^{f},$$

(3.7)

where $\chi^{c}=(rgb-rbg+gbr-grb+brg-bgr)/\sqrt{6}$ is the color wave function, which is universal for the baryon. For the $\Xi^{(*)}_{Q}$ baryon, its flavor wave function is $\psi^{\text{flavor}}_{\Xi^{(*)}_{Q}}=(ns-sn)Q/\sqrt{2}$, while for the $\Omega_{Q}^{(*)}$ baryon, its flavor wave function denotes $\psi^{\text{flavor}}_{\Omega^{(*)}_{Q}}=ssQ$, where $Q=b,c$ and $n=u,d$²²2 A brief introduction about the classification of the single heavy baryons is helpful to the reader to understand how to construct their wave functions. The single heavy baryons with one heavy-flavor quark and two light-flavor quarks belong to the symmetric $6_{\text{F}}$ or antisymmetric $\bar{3}_{\text{F}}$ flavor representations based on the flavor SU(3) symmetry. The total color-flavor-spin wave functions for the $S$-wave members must be antisymmetric. Considering the color wave function is antisymmetric invariably, hence the spin of the two light quarks is $S=1$ for $6_{\text{F}}$ (e.g. $\Sigma_{Q}$, $\Xi_{Q}^{\prime}$ and $\Omega_{Q}$) or $S=0$ for $\bar{3}_{\text{F}}$ (e.g. $\Lambda_{Q}$ and $\Xi_{Q}$). More details about the classification of the single heavy baryons can be found in Refs. Chen:2007xf ; Chen:2021eyk . For $\Xi_{Q}^{\prime}$, its flavor wave function is $\psi^{\text{flavor}}_{\Xi^{\prime}_{Q}}=(ns+sn)Q/\sqrt{2}$. . Besides, S denotes the total spin and L is the total orbital angular momentum. $\psi^{p}_{\mathbf{L},\mathbf{M_{L}}}$ is the spatial wave function which is composed of $\rho$ mode and $\lambda$ mode, that is,

$$\psi^{p}_{\mathbf{L},\mathbf{M_{L}}}(\vec{\rho},\vec{\lambda})=\left\{\phi_{\boldsymbol{l_{\rho}},\boldsymbol{ml_{\rho}}}(\vec{\rho})\phi_{\boldsymbol{l_{\lambda}},\boldsymbol{ml_{\lambda}}}(\vec{\lambda})\right\}_{\mathbf{L},\mathbf{M_{L}}},$$

(3.8)

where the subscripts $\boldsymbol{l_{\rho}}$ and $\boldsymbol{l_{\lambda}}$ are the orbital angular momentum quanta for $\rho$ and $\lambda$ mode, respectively, and the internal Jacobi coordinates are chosen as

$$\vec{\rho}=\vec{r}_{1}-\vec{r}_{2},~{}~{}\vec{\lambda}=\vec{r}_{3}-\frac{m_{1}\vec{r}_{1}+m_{2}\vec{r}_{2}}{m_{1}+m_{2}}.$$

(3.9)

In this work, the Gaussian basis Hiyama:2003cu ; Yoshida:2015tia ; Yang:2019lsg ,

$$\begin{split}\phi_{nlm}^{G}(\vec{r})=&\phi^{G}_{nl}(r)~{}Y_{lm}(\hat{r})\\ =&\sqrt{\frac{2^{l+2}(2\nu_{n})^{l+3/2}}{\sqrt{\pi}(2l+1)!!}}\lim_{\varepsilon\rightarrow 0}\frac{1}{(\nu_{n}\varepsilon)^{l}}\sum_{k=1}^{k_{\text{max}}}C_{lm,k}e^{-\nu_{n}(\vec{r}-\varepsilon\vec{D}_{lm,k})^{2}},\end{split}$$

(3.10)

is adopted to expand the spatial wave functions $\phi_{\boldsymbol{l_{\rho}},\boldsymbol{ml_{\rho}}}$ and $\phi_{\boldsymbol{l_{\lambda}},\boldsymbol{ml_{\lambda}}}$ ($n=1,2,\cdots,n_{\rm{max}}$). Here, a freedom parameter $n_{\rm{max}}$ should be chosen from positive integers, and the Gaussian size parameter $\nu_{n}$ is settled as a geometric progression as

$$\nu_{n}=1/r^{2}_{n},~{}r_{n}=r_{\rm{min}}~{}a^{n-1}$$

(3.11)

with

$$a=\left(\frac{r_{max}}{r_{\rm{min}}}\right)^{\frac{1}{n_{\rm{max}}-1}}.$$

Meanwhile, in our calculation the values of $\rho_{\rm{min}}$ and $\rho_{\rm{max}}$ are chosen as $0.2$ and $2.0$ fm, respectively, and $n_{\rho_{\rm{max}}}=6$. For $\lambda$ mode, we also use the same Gaussian sized parameters.

The Rayleigh-Ritz variational principle is used in this work to solve the three-body Schrödinger equation

$$\mathcal{H}\Psi_{\mathbf{J},\mathbf{M_{J}}}=E\Psi_{\mathbf{J},\mathbf{M_{J}}}.$$

(3.12)

Finally, by solving Schrödinger equation, the masses and wave functions of the baryons are obtained, which are collected in Table 2.