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

Yu-Shuai Li1,2<EMAIL_ADDRESS>Xiang Liu1,2,3111Corresponding author
<EMAIL_ADDRESS>1School of Physical Science and Technology, Lanzhou
University, Lanzhou 730000, China
2Research Center for Hadron and CSR Physics, Lanzhou University and Institute
of Modern Physics of CAS, Lanzhou 730000, China
3Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics
of Gansu Province, and Frontier Science Center for Rare Isotopes, Lanzhou
University, Lanzhou 730000, China

###### Abstract

In this work, we calculate the branching ratios of the color-allowed two-body
nonleptonic decays of the bottom baryons, which include the
$\Xi_{b}\to\Xi_{c}^{(*)}$ and $\Omega_{b}\to\Omega_{c}^{(*)}$ weak transitions
by emitting a pseudoscalar meson ($\pi^{-}$, $K^{-}$, $D^{-}$, and
$D_{s}^{-}$) or a vector meson ($\rho^{-}$, $K^{*-}$, $D^{*-}$, and
$D_{s}^{*-}$). For achieving this aim, we adopt the three-body light-front
quark model with the support of hadron spectroscopy, where the spatial wave
functions of these heavy baryons involved in these weak decays are obtained by
a semirelativistic potential model associated with the Gaussian expansion
method. Our results show that these decays with the $\pi^{-}$, $\rho^{-}$, and
$D_{s}^{(*)-}$-emitted mode have considerable widths, which could be
accessible at the ongoing LHCb and Belle II experiments.

##  I introduction

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.

##  II The transition form factors of the bottom baryon to the charmed baryon

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.

##  III The semirelativistic potential model for calculating baryon wave
function

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.

Table 1: The parameters used in the semirelativistic potential model Li:2021qod . Parameters | Values | Parameters | Values
---|---|---|---
$m_{u}~{}(\text{GeV})$ | $0.220$ | $\epsilon^{\text{so}(s)}$ | $0.448$
$m_{d}~{}(\text{GeV})$ | $0.220$ | $\epsilon^{\text{so}(v)}$ | $-0.062$
$m_{s}~{}(\text{GeV})$ | $0.419$ | $\epsilon^{\text{tens}}$ | $0.379$
$m_{c}~{}(\text{GeV})$ | $1.628$ | $\epsilon^{\text{con}}$ | $-0.142$
$m_{b}~{}(\text{GeV})$ | $4.977$ | $\sigma_{0}~{}(\text{GeV})$ | $2.242$
$b~{}(\text{GeV}^{2})$ | $0.142$ | $s$ | $0.805$
$c~{}(\text{GeV})$ | $-0.302$ |  |

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.

Table 2: Spatial wave functions of the concerned $\Xi_{Q}$ and $\Omega_{Q}$ from the GI model and GEM. It is worth to mention that the masses for the neutral and charged states are degenerate here due to the same masses for $u$ and $d$ quarks. The second column denotes our theoretically prediction, while the third column denotes the experimental data quoted from the PDG ParticleDataGroup:2020ssz . Here, the first value in each row is the masses for the neutral baryon, while the second one is the mass for the charged state. The Gaussian bases $(n_{\rho},n_{\lambda})$ listed in the third column are arranged as $[(1,1),(1,2),\cdots,(1,n_{\lambda_{\rm{max}}}),(2,1),(2,2),\cdots,(2,n_{\lambda_{\rm{max}}}),\cdots,(n_{\rho_{\rm{max}}},1),(n_{\rho_{\rm{max}}},2),\cdots,(n_{\rho_{\rm{max}}},n_{\lambda_{\rm{max}}})]$. Baryon | This work (GeV) | Experiment (MeV) | Eigenvector
---|---|---|---
$\Xi_{b}\left(\frac{1}{2}^{+}\right)$ | 5.804 |  $5791.9\pm 0.5$ $5797.0\pm 0.6$ | $[-0.017,-0.040,-0.075,0.002,-0.003,0.001,-0.033,-0.026,-0.004,$
$-0.009,0.004,-0.001,0.005,-0.266,-0.267,0.013,-0.009,0.002,$
$0.008,0.017,-0.363,-0.041,0.007,-0.001,-0.006,0.004,-0.023,$
$-0.079,0.014,-0.003,0.002,0.001,0.010,0.007,-0.003,0.001]$
$\Omega_{b}\left(\frac{1}{2}^{+}\right)$ | 6.043 |  $6046.1\pm 1.7$ | $[0.002,0.004,0.011,-0.006,0.003,-0.001,0.075,-0.024,0.040,$
$0.002,0.000,-0.000,-0.034,0.361,0.096,0.002,-0.001,0.001,$
$-0.009,-0.022,0.588,-0.002,0.011,-0.003,0.009,-0.025,-0.046,$
$0.101,-0.025,0.006,-0.002,0.006,0.008,-0.013,0.005,-0.001]$
$\Xi_{c}\left(\frac{1}{2}^{+}\right)$ | 2.474 |  $2470.90^{+0.22}_{-0.29}$ $2467.94^{+0.17}_{-0.20}$ | $[-0.017,-0.027,-0.082,-0.010,-0.001,0.000,-0.028,-0.032,-0.010,$
$-0.011,0.004,-0.001,0.005,-0.192,-0.315,-0.032,-0.000,0.000,$
$0.002,0.037,-0.297,-0.116,0.020,-0.005,-0.004,-0.002,-0.010,$
$-0.082,0.010,-0.002,0.001,0.002,0.007,0.009,-0.003,0.001]$
$\Xi_{c}^{*}\left(\frac{1}{2}^{+}\right)$ | 2.947 |  $2970.9^{+0.4}_{-0.6}$ $2966.34^{+0.17}_{-1.00}$ | $[-0.023,-0.072,-0.098,0.147,-0.012,0.003,-0.039,-0.081,-0.007,$
$0.048,-0.004,0.001,0.015,-0.390,-0.469,0.501,-0.049,0.011,$
$0.011,0.013,-0.268,0.682,-0.023,0.005,-0.007,-0.005,-0.048,$
$0.314,0.056,-0.010,0.001,0.006,0.012,-0.044,0.005,-0.000]$
$\Omega_{c}\left(\frac{1}{2}^{+}\right)$ | 2.692 |  $2695.2\pm 1.7$ | $[0.006,-0.003,0.019,-0.008,0.004,-0.001,0.093,-0.027,0.045,$
$0.001,0.002,-0.000,-0.049,0.351,0.135,0.029,-0.010,0.003,$
$0.005,-0.078,0.527,0.075,-0.002,-0.001,0.004,-0.001,-0.071,$
$0.096,-0.021,0.005,-0.001,0.000,0.013,-0.014,0.005,-0.001]$
$\Omega_{c}^{*}\left(\frac{1}{2}^{+}\right)$ | 3.149 |  $-$ | $[0.022,-0.025,0.042,-0.016,0.007,-0.002,0.100,0.112,-0.022,$
$-0.060,0.003,-0.000,-0.043,0.412,0.494,-0.188,0.036,-0.008,$
$-0.002,0.032,0.068,-0.754,0.052,-0.011,-0.008,0.019,-0.076,$
$-0.375,-0.010,0.000,0.003,-0.008,0.021,0.036,-0.007,0.001]$

As collected in the PDG ParticleDataGroup:2020ssz , there are ten states in
the $\Xi_{c}$ family, where the ground states includes $\Xi_{c}^{+}$ and
$\Xi_{c}^{0}$ with the quark flavor $usc$ and $dsc$, respectively.
$\Xi_{c}^{+}$ was first reported by SPEC Biagi:1983en , and then confirmed in
Ref. FermilabE687:1992wmm  by analyzing the $\Xi^{-}\pi^{+}\pi^{+}$ final
state, while the neutral one $\Xi_{c}^{0}$ was first discovered by CLEO
CLEO:1988yda  in the $\Xi^{-}\pi^{+}$ mode. The masses fitted by the PDG are
$2467.71\pm 0.23$ and $2470.44\pm 0.28$ MeV for charged $\Xi_{c}^{+}$ and
neutral $\Xi_{c}^{0}$, respectively. And then, the Belle Collaboration found
$\Xi_{c}^{+}(2970)$ and $\Xi_{c}^{0}(2970)$ in the
$\Lambda_{c}^{+}K^{-}\pi^{+}$ and $\Lambda_{c}^{+}K_{S}^{0}\pi^{-}$ final
states Belle:2006edu , respectively, where the masses of the charged and
neutral $\Xi_{c}(2970)$ states are measured to be $2964.3\pm 1.5$ and
$2967.1\pm 1.7$ MeV, respectively. As indicated by our calculation shown in
Table 2, the observed $\Xi_{c}(2970)$ are good candidate of $\Xi_{c}^{*}(2S)$.
The ground $\Omega_{c}$ state, denoted as $\Omega_{c}(\frac{1}{2}^{+})$, was
firstly observed in the $\Xi^{-}K^{-}\pi^{+}\pi^{+}$ channel by WA62
Biagi:1984mu , and then was confirmed in ARGUS ARGUS:1992mwl  by checking the
same mode. Its mass was fitted as $2695.2\pm 1.7$ MeV by the PDG. Our result
given in Table 2 indeed supports this assignment since the calculated mass of
$\Omega_{c}(\frac{1}{2}^{+})$ is 2.692 GeV consistent with the experimental
data. For the $\Omega_{c}^{*}(\frac{1}{2}^{+})$ state, which is the first
radial excitation of $\Omega_{c}(\frac{1}{2}^{+})$, its mass is calculated to
be 3.149 GeV333In 2017, the LHCb Collaboration LHCb:2017uwr  announced that
five narrow excited $\Omega_{c}$ states, $\Omega_{c}(3000)$,
$\Omega_{c}(3050)$, $\Omega_{c}(3066)$, $\Omega_{c}(3090)$, and
$\Omega_{c}(3119)$, were found in the $\Xi_{c}^{-}K^{+}$ invariant mass
spectrum. Later, Belle Belle:2017ext  confirmed four narrow excited
$\Omega_{c}$ states in the same mode. The spin-parity of these excited strange
charmed baryons are not measured yet. In these five excited $\Omega_{c}$
states, the masses of $\Omega_{c}(3090)$ and $\Omega_{c}(3119)$ were measured
as $3090.0\pm 0.5$ and $3119.1\pm 1.0$ MeV, respectively. Their structures
were discussed by various theoretical approaches Chen:2017gnu ; Cheng:2017ove
; Chen:2017sci ; Wang:2017hej ; Agaev:2017jyt ; Debastiani:2018adr . Chen et
al. Chen:2017gnu  indicated that $\Omega_{c}(3119)$ cannot be a $2S$ candidate
by performing an analysis of the mass spectrum and decay behavior. Cheng et
al. Cheng:2017ove  assigned $\Omega_{c}(3090)$ and $\Omega_{c}(3119)$ as the
first radially excited states with $J^{P}=1/2^{+}$ and $3/2^{+}$,
respectively, by the analysis of the Regge trajectories and a direct
calculation of the mass via a quark-diquark model. Wang et al. Wang:2017hej
proposed that the $\Omega_{c}(3119)$ favors the $2S$ assignment by a study
with a constituent quark model. Agaev et al. Agaev:2017jyt  discussed the
favored assignment $\Omega_{c}(2S)$ state with $J^{P}=1/2^{+}$ and $3/2^{+}$
for $\Omega_{c}(3066)$ and $\Omega_{c}(3119)$ with QCD sum rules. Thus,
establishing $\Omega_{c}^{*}(\frac{1}{2}^{+})$ state is still ongoing. In this
work, we adopt the calculated result as mass input of the
$\Omega_{c}^{*}(\frac{1}{2}^{+})$ state..

In Table 2, we also collected the numerical spatial wave functions
corresponding to these charmed baryons, which will be applied to the following
study.

##  IV The form factors and color-allowed two-body nonleptonic decays

###  IV.1 The weak transitions form factors

With the input of these obtained numerical wave functions of bottom (see Table
2) and charmed baryons, and the expressions of the form factors [see Eqs.
(2.7)-(2.8)], we present the numerical results for the weak transition form
factors of $\Xi_{b}\to\Xi_{c}^{(*)}(1/2^{+})$ and
$\Omega_{b}\to\Omega_{c}^{(*)}(1/2^{+})$ processes. Since the expressions of
form factors in Eqs. (2.5)-(2.8) are working in the spacelike region
($q^{2}<0$), we need to extend them to the timelike region ($q^{2}>0$). The
dipole form Li:2021qod ; Ke:2019smy ; Chua:2018lfa ; Chua:2019yqh

$F(q^{2})=\frac{F(0)}{(1-q^{2}/M^{2})[1-b_{1}(q^{2}/M^{2})+b_{2}(q^{2}/M^{2})^{2}]}$
(4.1)

is applied in this work, where $F(0)$ is the form factor at $q^{2}=0$,
$b_{1}$, and $b_{2}$ are obtained by computing each form factor by Eqs.
(2.7)-(2.8) from $q^{2}=-q_{\text{max}}^{2}$ to $q^{2}=0$, and fit them by Eq.
(4.1) .

With the spatial wave functions obtained in the last subsection, we can
calculate out the form factors numerically in the framework of the three-body
light-front quark model. In this way, all free parameters of the
semirelativistic potential model can be fixed by reproducing the mass spectrum
of observed heavy baryons. In the previous work Guo:2005qa ; Zhu:2018jet ;
Zhao:2018zcb ; Chua:2018lfa ; Chua:2019yqh ; Ke:2019smy  on baryon weak
transitions, simple hadronic oscillator wave function with the oscillator
parameter $\beta$ was widely used to simulate the baryon spatial wave
function. This treatment makes the results dependent on $\beta$ value. In this
work, our study is supported by hadron spectroscopy. Thus, we can avoid the
above uncertainty resulted by the selection of spatial wave functions of heavy
baryons involved in these discussed transitions.

The extended form factors of $\Xi_{b}\to\Xi_{c}^{(*)}$ are collected in Table
3. The $q^{2}$ dependence of $f^{V}_{1,2,3}$ and $g^{A}_{1,2,3}$ for the
$\Xi_{b}\to\Xi_{c}$ and $\Xi_{b}\to\Xi_{c}(2970)$ transitions are plotted in
Fig. 1.

Table 3: The form factors for the $\Xi_{b}\to\Xi_{c}^{(*)}$ transitions in the standard light front quark model. Here, we adopt the form defined in Eq. (4.1) for analyzing these form factors. | $F(0)$ | $F(q^{2}_{\text{max}})$ | $b_{1}$ | $b_{2}$
---|---|---|---|---
| $\Xi_{b}\rightarrow\Xi_{c}$
$f^{V}_{1}$ | $0.481$ | $1.015$ | $0.970$ | $0.233$
$f^{V}_{2}$ | $-0.127$ | $-0.312$ | $1.380$ | $0.578$
$f^{V}_{3}$ | $-0.046$ | $-0.097$ | $1.187$ | $0.875$
$g^{A}_{1}$ | $0.471$ | $0.978$ | $0.929$ | $0.226$
$g^{A}_{2}$ | $-0.026$ | $-0.068$ | $1.318$ | $0.122$
$g^{A}_{3}$ | $-0.154$ | $-0.377$ | $1.493$ | $0.947$
| $\Xi_{b}\rightarrow\Xi_{c}(2970)$
$f^{V}_{1}$ | $0.214$ | $0.200$ | $-1.146$ | $2.282$
$f^{V}_{2}$ | $-0.072$ | $-0.081$ | $-0.356$ | $1.600$
$f^{V}_{3}$ | $-0.111$ | $-0.221$ | $1.444$ | $0.168$
$g^{A}_{1}$ | $0.204$ | $0.186$ | $-1.269$ | $2.474$
$g^{A}_{2}$ | $-0.087$ | $-0.231$ | $1.867$ | $-0.907$
$g^{A}_{3}$ | $-0.095$ | $-0.113$ | $-0.022$ | $1.687$
---
Figure 1: The $q^{2}$ dependence of the form factors $f_{1,2,3}^{V}(q^{2})$
and $g_{1,2,3}^{A}(q^{2})$ for the $\Xi_{b}\to\Xi_{c}$ (left) and
$\Xi_{b}\to\Xi_{c}(2970)$ (right) transitions. Here, the solid and dashed
lines represent the vector-type and pseudoscalar-type form factors denoting by
the subscripts $V$ and $A$, respectively, while the blue, red, and purple
lines (both solid and dashed lines) represent the $i$th form factors denoting
by the subscripts respectively for each types.

For the $\Xi_{b}\to\Xi_{c}$ transition, the corresponding transition matrix
element can be rewritten as Georgi:1990ei ; Bowler:1997ej ; Chua:2019yqh

$\left<\Xi_{c}(1/2^{+})(\nu^{\prime})|\bar{c}_{\nu^{\prime}}\Gamma
b_{\nu}|\Xi_{b}(1/2^{+})(\nu)\right>=\zeta(\omega)\bar{u}(\nu^{\prime})\Gamma
u(\nu),$ (4.2)

in the heavy quark limit at the leading order, so the form factors have more
simple behaviors as

$\begin{split}&f^{V}_{1}(q^{2})=g^{A}_{1}(q^{2})=\zeta(\omega),\\\
&f^{V}_{2}=f^{V}_{3}=g^{A}_{2}=g^{A}_{3}=0,\end{split}$ (4.3)

where $\omega=\nu\cdot\nu^{\prime}=(M^{2}+M^{\prime 2}-q^{2})/(2MM^{\prime})$
with $\nu^{\prime}=p^{\prime}/M^{\prime}$ and $\nu=p/M$ denoting the four
velocities for $\Xi_{c}$ and $\Xi_{b}$, respectively. $\zeta(\omega)$ is the
well-known Isgur-Wise function (IWF) and usually expressed as a Taylor series
expansion as

$\zeta(\omega)=1-\zeta_{1}(\omega-1)+\frac{\zeta_{2}}{2}(\omega-1)^{2}+\cdots,$
(4.4)

where $\zeta_{1}=-\frac{d\zeta(\omega)}{d\omega}|_{\omega=1}$ and
$\zeta_{2}=\frac{d^{2}\zeta(\omega)}{d\omega^{2}}|_{\omega=1}$ are two shape
parameters depicting the IWF. The most obvious character is in the point
$q^{2}=q_{\rm{max}}^{2}=(M-M^{\prime})^{2}$ (or $\omega=1$),

$f^{V}_{1}(q_{\rm{max}}^{2})=g^{A}_{1}(q_{\rm{max}}^{2})=\zeta(1)=1.$

It provided one strong restriction for our result. Besides, when comparing our
results with the predictions in heavy quark limit (HQL), we can conclude that
our results can well match the requirement from heavy quark effective theory,
i.e.,

  1. 1.

$f^{V}_{1}$ and $g^{A}_{1}$ are close to each other, and dominate over
$f^{V}_{2,3}$ and $g^{A}_{2,3}$.

  2. 2.

At $q^{2}=q_{\rm{max}}^{2}$, $f^{V}_{1}(q_{\rm{max}}^{2})=1.015$ and
$g^{A}_{1}(q_{\rm{max}}^{2})=0.978$ are very approach to 1.

In addition we also extract the two IWF’s shape parameters $\xi_{1}$ and
$\xi_{2}$ in Eq. (4.4) by fitting $\zeta(\omega)$ from $f^{V}_{1}(q^{2})$ and
$g^{A}_{1}(q^{2})$, respectively. The concrete results and other theoretical
predictions are listed in Table 4.

Table 4: Our results for the IWF’s shape parameters of the $\Xi_{b}\to\Xi_{c}$ transition. The superscripts $[a]$ and $[b]$ in the second and third rows represent the fitting of $f_{1}^{V}$ and $g_{1}^{A}$, respectively. | $\zeta_{1}$ | $\zeta_{2}$
---|---|---
This work[a] | 1.97 | 3.28
This work[b] | 2.23 | 4.63
RQM Ebert:2006rp  | 2.27 | 7.74

For the $\Xi_{b}\to\Xi_{c}(2970)$ transition, the HQL requires
$f_{1}^{V}=g_{1}^{A}=0$ at $q^{2}=q_{\text{max}}^{2}$ since the wave functions
of the low-lying $\Xi_{b}$ and the radial excited state $\Xi_{c}^{*}(2S)$ are
orthogonal Chua:2019yqh . Evidently, our results well embody this prediction
according to Fig. 1.

Table 5: The form factors for the $\Omega_{b}\to\Omega_{c}^{(*)}$ transitions in the standard light front quark model. We use a three parameter form defined in Eq. (4.1) for these form factors. | $F(0)$ | $F(q^{2}_{\text{max}})$ | $b_{1}$ | $b_{2}$
---|---|---|---|---
| $\Omega_{b}\to\Omega_{c}$
$f^{V}_{1}$ | $0.493$ | $1.232$ | $1.765$ | $1.272$
$f^{V}_{2}$ | $0.436$ | $1.075$ | $1.658$ | $1.001$
$f^{V}_{3}$ | $-0.255$ | $-0.620$ | $1.628$ | $1.005$
$g^{A}_{1}$ | $-0.161$ | $-0.329$ | $1.053$ | $0.337$
$g^{A}_{2}$ | $0.011$ | $0.018$ | $0.822$ | $1.526$
$g^{A}_{3}$ | $0.055$ | $0.137$ | $1.680$ | $1.052$
| $\Omega_{b}\to\Omega_{c}(2S)$
$f^{V}_{1}$ | $0.180$ | $0.163$ | $-1.135$ | $3.320$
$f^{V}_{2}$ | $0.133$ | $0.107$ | $-1.727$ | $4.270$
$f^{V}_{3}$ | $-0.150$ | $-0.215$ | $0.481$ | $0.239$
$g^{A}_{1}$ | $-0.058$ | $-0.047$ | $-1.701$ | $3.487$
$g^{A}_{2}$ | $0.029$ | $0.053$ | $1.455$ | $0.772$
$g^{A}_{3}$ | $0.023$ | $0.023$ | $-0.671$ | $2.407$
---
Figure 2: The $q^{2}$ dependence of the form factors $f_{1,2,3}^{V}(q^{2})$
and $g_{1,2,3}^{A}(q^{2})$ for $\Omega_{b}\to\Omega_{c}$ (left) and
$\Omega_{b}\to\Omega_{c}(2S)$ (right) transitions, in which the solid and
dashed lines represent the vector or pseudoscalar-types form factors denoting
by the subscripts $V$ and $A$, respectively, while the blue, red and purple
lines (both solid and dashed lines) represent the $i$th form factors denoting
by the subscripts, respectively, for each types.

Additionally, the extended form factors of $\Omega_{b}\to\Omega_{c}^{(*)}$ are
collected in Table 5. The $q^{2}$ dependence of $f^{V}_{1,2,3}$ and
$g^{A}_{1,2,3}$ for the $\Omega_{b}\to\Omega_{c}$ and
$\Omega_{b}\to\Omega_{c}(3090)$ transitions are plotted in Fig. 2. For the
$\Omega_{b}\to\Omega_{c}$ transition, the corresponding transition matrix
element can be rewritten as Georgi:1990ei ; Bowler:1997ej ; Chua:2019yqh

$\begin{split}\langle\Omega_{c}&(1/2^{+})(\nu^{\prime})|\bar{c}_{\nu^{\prime}}\Gamma
b_{\nu}|\Omega_{b}(1/2^{+})(\nu)\rangle=\\\
&-\frac{1}{3}(g^{\rho\sigma}\xi_{1}-v^{\rho}v^{\prime\sigma}\xi_{2})\bar{u}(v^{\prime})(\gamma_{\rho}-v^{\prime}_{\rho})\Gamma(\gamma_{\sigma}-v_{\sigma})u(v)\end{split}$
(4.5)

in HQL at the leading order. Thus, the form factors in HQL have more simple
behaviors as

$\begin{split}&f_{1}^{V}(q_{\text{max}}^{2})=\frac{1}{3}+\frac{1}{3}\frac{M^{2}+M^{\prime
2}}{MM^{\prime}}=1.23,\\\
&f_{2}^{V}(q_{\text{max}}^{2})=\frac{1}{3}\frac{M+M^{\prime}}{M^{\prime}}=1.08,\\\
&f_{3}^{V}(q_{\text{max}}^{2})=-\frac{1}{3}\frac{M-M^{\prime}}{M^{\prime}}=-0.41,\\\
&g_{1}^{A}(q_{\text{max}}^{2})=-\frac{1}{3},\\\
&g_{2}^{A}(q_{\text{max}}^{2})=g_{3}^{A}(q_{\text{max}}^{2})=0,\end{split}$
(4.6)

at $q^{2}=q_{\text{max}}^{2}$ point by substituting the involved masses.
Obviously, our results located in the third column of the Table 5 match well
the requirement from the HQL as shown in Eq. (4.6), which can be as a direct
test to the HQL.

###  IV.2 The color-allowed two-body nonleptonic decays

With the preparation of the obtained form factors, we further calculate the
color-allowed two-body nonleptonic decays of $\Xi_{b}$ and $\Omega$ with
emitting a pseudoscalar meson ($\pi^{-}$, $K^{-}$, $D^{-}$, and $D_{s}^{-}$)
or a vector meson ($\rho^{-}$, $K^{*-}$, $D^{*-}$, and $D_{s}^{*-}$). In this
work, the decay rates are investigated by the naïve factorization
approach444The naïve factorization approach works well for the color-allowed
dominated processes. But, there exists the case that the color-suppressed and
penguin dominated processes can not be explained by the naïve factorization,
which may show important nonfactorizable contributions to nonleptonic decays
Zhu:2018jet . As indicated in Refs. Lu:2009cm ; Chua:2018lfa ; Chua:2019yqh ,
the nonfactorizable contributions in bottom baryon nonleptonic decays are
cosiderable comparing with the factorized ones. Since a precise study of
nonfactorizable contributions is beyond the scope of the present work, we
still adopt the naïve factorization approximation..

Generally, in the naïve factorization assumption, the hadronic transition
matrix element is factorized into a product of two independent matrix elements
Ke:2019smy

$\begin{split}\langle\mathcal{B}_{c}^{(*)}&(P^{\prime},J_{z}^{\prime})\ M^{-}\
|\mathcal{H}_{\text{eff}}|\ \mathcal{B}_{b}(P,J_{z})\rangle\\\
=&\frac{G_{F}}{\sqrt{2}}V_{cb}V_{qq^{\prime}}^{*}\langle
M^{-}|\bar{q}^{\prime}\gamma_{\mu}(1-\gamma_{5})q|0\rangle\\\
&\times\langle\mathcal{B}_{c}^{(*)}(P^{\prime},J_{z}^{\prime})|\bar{c}\gamma^{\mu}(1-\gamma_{5})b|\mathcal{B}_{b}(P,J_{z})\rangle,\end{split}$
(4.7)

where the meson transition term is given by

$\begin{split}\langle
M|\bar{q}^{\prime}&\gamma_{\mu}(1-\gamma_{5})q|0\rangle=\left\\{\begin{array}[]{ll}if_{P}q_{\mu},&M=P\\\
if_{V}\epsilon_{\mu}^{*}m_{V},&M=V\end{array}.\right.\end{split}$ (4.8)

Here, $P$ and $V$ denote pseudoscalar and vector mesons, respectively. The
baryon transition term can be obtained by Eq. (2.1). The corresponding Feynman
diagram (taking the $\Xi_{b}^{-}\to\Xi_{c}^{0}M^{-}$ as an example here) is
displayed in Fig. 3.

Figure 3: The diagram for depicting the color-allowed two-body nonleptonic
decay $\Xi_{b}^{-}\to\Xi_{c}^{0}M^{-}$ in the tree level.

Finally the decay width and asymmetry parameter are given by Ke:2019smy

$\begin{split}\Gamma&=\frac{|p_{c}|}{8\pi}\left(\frac{(M+M^{\prime})^{2}-m^{2}}{M^{2}}|A|^{2}+\frac{(M-M^{\prime})^{2}-m^{2}}{M^{2}}|B|^{2}\right),\\\
\alpha&=\frac{2\kappa\text{Re}(A^{*}B)}{|A|^{2}+\kappa^{2}|B|^{2}},\end{split}$
(4.9) $\begin{split}\Gamma&=\frac{|p_{c}|(E^{\prime}+M^{\prime})}{4\pi
M}\left(2(|S|^{2}+|P_{2}|^{2})+\frac{E_{m}^{2}}{m^{2}}(|S+D|^{2}+|P_{1}|^{2})\right),\\\
\alpha&=\frac{4m^{2}\text{Re}(S^{*}P_{2})+2E_{m}^{2}\text{Re}(S+D)^{*}P_{1}}{2m^{2}(|S|^{2}+|P_{2}|^{2})+E_{m}^{2}(|S+D|^{2}+|P_{1}|^{2})},\end{split}$
(4.10)

for the cases involved in the pseudoscalar and vector meson final state,
respectively, where $p_{c}$ is the momentum of the daughter baryon in the rest
frame of the parent baryon and $\kappa=|p_{c}|/(E^{\prime}+M^{\prime})$.
Besides, $M(E)$ and $M^{\prime}(E^{\prime})$ are the masses (energies) of the
parent (daughter) baryons, respectively, while $m(E_{m})$ denotes the
mass(energy) of the meson in the final state.

$A$ and $B$ in Eqs. (4.9) are given by

$\begin{split}A&=\frac{G_{F}}{\sqrt{2}}V_{cb}V_{qq^{\prime}}^{*}a_{i}f_{P}(M-M^{\prime})f_{1}^{V}(m^{2}),\\\
B&=-\frac{G_{F}}{\sqrt{2}}V_{cb}V_{qq^{\prime}}^{*}a_{i}f_{P}(M+M^{\prime})g_{1}^{A}(m^{2}),\end{split}$
(4.11)

and $S$, $P_{1,2}$ and $D$ in Eqs. (4.10) are expressed as

$\begin{split}S&=A_{1},\\\
P_{1}&=-\frac{|p_{c}|}{E_{m}}\left(\frac{M+M^{\prime}}{E^{\prime}+M^{\prime}}B_{1}+MB_{2}\right),\\\
P_{2}&=\frac{|p_{c}|}{E^{\prime}+M^{\prime}}B_{1},\\\
D&=\frac{|p_{c}|^{2}}{E_{m}(E^{\prime}+M^{\prime})}(A_{1}-MA_{2})\end{split}$
(4.12)

with

$\begin{split}A_{1}&=\frac{G_{F}}{\sqrt{2}}V_{cb}V_{qq^{\prime}}^{*}a_{i}f_{V}m_{V}\left(g_{1}^{A}(m^{2})+g_{2}^{A}(m^{2})\frac{M-M^{\prime}}{M}\right),\\\
A_{2}&=\frac{G_{F}}{\sqrt{2}}V_{cb}V_{qq^{\prime}}^{*}a_{i}f_{V}m_{V}\left(2g_{2}^{A}(m^{2})\right),\\\
B_{1}&=\frac{G_{F}}{\sqrt{2}}V_{cb}V_{qq^{\prime}}^{*}a_{i}f_{V}m_{V}\left(f_{1}^{V}(m^{2})-f_{2}^{V}(m^{2})\frac{M+M^{\prime}}{M}\right),\\\
B_{2}&=\frac{G_{F}}{\sqrt{2}}V_{cb}V_{qq^{\prime}}^{*}a_{i}f_{V}m_{V}\left(2f_{2}^{V}(m^{2})\right),\end{split}$
(4.13)

where $a_{1}=c_{1}+c_{2}/N\approx 1.018$ and $a_{2}=c_{2}+c_{1}/N\approx
0.170$ Chua:2019yqh .

With the naïve factorization, the color-allowed two-body nonleptonic decays by
emitting one pseudoscalar meson or vector meson are presented. The lifetimes
of $\Xi_{b}^{-,0}$ and $\Omega_{b}^{-}$ was reported by the LHCb LHCb:2014chk
; LHCb:2014wqn ; LHCb:2016coe  and CDF CDF:2014mon  collaborations. In this
work, we use the central values as

$\tau_{\Xi_{b}^{0}}=1.480~{}\rm{fs},~{}~{}\tau_{\Xi_{b}^{-}}=1.572~{}\rm{fs},~{}~{}\tau_{\Omega_{b}^{-}}=1.65~{}\rm{fs},$

averaged by the PDG ParticleDataGroup:2020ssz . Besides, the masses of the
concerned baryons are from the GEM calculation and the Cabibbo-Kobayashi-
Maskawa matrix elements

$\begin{split}&V_{cb}=0.0405,\ V_{ud}=0.9740,\ V_{us}=0.2265,\\\
&V_{cd}=0.2264,\ V_{cs}=0.9732,\end{split}$

are taken from the PDG ParticleDataGroup:2020ssz . The decay constants of
pseudoscalar and vector mesons include Cheng:2003sm ; Chua:2019yqh

$\begin{split}&f_{\pi}=130.2,\ f_{K}=155.6,\ f_{D}=211.9,\ f_{D_{s}}=249.0,\\\
&f_{\rho}=216,\ f_{K^{*}}=210,\ f_{D^{*}}=220,\ f_{D_{s}^{*}}=230,\end{split}$

in the unit of MeV.

By substituting our numerical results of the form factors from the three-body
light-front quark model and the presented decay parameters into Eqs.
(4.9)-(4.10), the branching ratios and asymmetry parameters can be further
obtained, which are collected in Tables 6-7 for 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 $D_{s}^{*-}$), respectively.

In Table 8, we compare our results of
$\mathcal{B}(\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}M^{-})$ and
$\mathcal{B}(\Omega_{b}^{-}\to\Omega_{c}^{0}M^{-})$ with other theoretical
results from the nonrelativistic quark model Cheng:1996cs , the relativistic
three-quark model Ivanov:1997hi ; Ivanov:1997ra , the light-front quark model
Zhao:2018zcb ; Chua:2019yqh , and the covariant confined quark model
Gutsche:2018utw . Our results are comparable with those calculated from other
approaches. We also notice that the concerned transitions with emitting
$\pi^{-}$, $\rho^{-}$, and $D_{s}^{(*)-}$ meson have considerable widths,
which are worthy to be explored in future experiment like LHCb and Belle II.

Table 6: The branching ratios and asymmetry parameters of the $\Xi_{b}\to\Xi_{c}^{(*)}M$ transitions with $M$ denoting a pseudoscalar or vector meson, where the branching ratios out of or in brackets correspond to the $\Xi_{b}^{0}\rightarrow\Xi_{c}^{+}$ and $\Xi_{b}^{-}\rightarrow\Xi_{c}^{0}$ transitions, respectively. Mode | $\mathcal{B}\ (\times 10^{-3})$ | $\alpha$ | Mode | $\mathcal{B}\ (\times 10^{-3})$ | $\alpha$
---|---|---|---|---|---
$\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}\pi^{-}$ | 4.04 (4.29) | -1.000 | $\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}\rho^{-}$ | 13.3 (14.1) | -0.792
$\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}K^{-}$ | 0.31 (0.33) | -1.000 | $\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}K^{*-}$ | 0.71 (0.76) | -0.737
$\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}D^{-}$ | 0.58 (0.62) | -0.983 | $\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}D^{*-}$ | 1.51 (1.60) | -0.239
$\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}D_{s}^{-}$ | 14.8 (15.7) | -0.978 | $\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}D_{s}^{*-}$ | 32.4 (34.4) | -0.206
$\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}(2970)\pi^{-}$ | 1.78 (1.89) | -0.999 | $\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}(2970)\rho^{-}$ | 2.78 (2.95) | -0.763
$\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}(2970)K^{-}$ | 0.04 (0.05) | -0.998 | $\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}(2970)K^{*-}$ | 0.09 (0.10) | -0.702
$\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}(2970)D^{-}$ | 0.04 (0.05) | -0.952 | $\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}(2970)D^{*-}$ | 0.12 (0.12) | -0.181
$\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}(2970)D_{s}^{-}$ | 1.05 (1.12) | -0.940 | $\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}(2970)D_{s}^{*-}$ | 2.30 (2.45) | -0.148
Table 7: The branching rates and asymmetry parameters of $\Omega_{b}\to\Omega_{c}^{(*)}M$ transitions with $M$ denoting a pseudoscalar or vector pmeson. Mode | $\mathcal{B}\ (\times 10^{-3})$ | $\alpha$ | Mode | $\mathcal{B}\ (\times 10^{-3})$ | $\alpha$
---|---|---|---|---|---
$\Omega_{b}^{-}\to\Omega_{c}^{0}\pi^{-}$ | 2.82 | 0.59 | $\Omega_{b}^{-}\to\Omega_{c}^{0}\rho^{-}$ | 7.92 | 0.61
$\Omega_{b}^{-}\to\Omega_{c}^{0}K^{-}$ | 0.22 | 0.58 | $\Omega_{b}^{-}\to\Omega_{c}^{0}K^{*-}$ | 0.41 | 0.62
$\Omega_{b}^{-}\to\Omega_{c}^{0}D^{-}$ | 0.52 | 0.49 | $\Omega_{b}^{-}\to\Omega_{c}^{0}D^{*-}$ | 0.48 | 0.69
$\Omega_{b}^{-}\to\Omega_{c}^{0}D_{s}^{-}$ | 13.5 | 0.47 | $\Omega_{b}^{-}\to\Omega_{c}^{0}D_{s}^{*-}$ | 9.73 | 0.70
$\Omega_{b}^{-}\to\Omega_{c}^{0}(2S)\pi^{-}$ | 0.30 | 0.58 | $\Omega_{b}^{-}\to\Omega_{c}^{0}(2S)\rho^{-}$ | 0.70 | 0.60
$\Omega_{b}^{-}\to\Omega_{c}^{0}(2S)K^{-}$ | 0.02 | 0.57 | $\Omega_{b}^{-}\to\Omega_{c}^{0}(2S)K^{*-}$ | 0.03 | 0.60
$\Omega_{b}^{-}\to\Omega_{c}^{0}(2S)D^{-}$ | 0.03 | 0.45 | $\Omega_{b}^{-}\to\Omega_{c}^{0}(2S)D^{*-}$ | 0.02 | 0.65
$\Omega_{b}^{-}\to\Omega_{c}^{0}(2S)D_{s}^{-}$ | 0.62 | 0.43 | $\Omega_{b}^{-}\to\Omega_{c}^{0}(2S)D_{s}^{*-}$ | 0.36 | 0.65
Table 8: Comparison of theoretical predictions for $\mathcal{B}(\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}M^{-})$ and $\mathcal{B}(\Omega_{b}^{-}\to\Omega_{c}^{0}M^{-})$. Here, all values should be multiplied by a factor of $10^{-3}$. | This work | Cheng Cheng:1996cs  | Ivanov et al. Ivanov:1997hi ; Ivanov:1997ra  | Zhao Zhao:2018zcb  | Gutsche et al. Gutsche:2018utw  | Chua Chua:2019yqh
---|---|---|---|---|---|---
$\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}\pi^{-}$ | 4.03 (4.29) | 4.9 (5.2) | 7.08 (10.13) | 8.37 (8.93) | $-$ |  $3.66^{+2.29}_{-1.59}$ ($3.88^{+2.43}_{-1.69}$)
$\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}\rho^{-}$ | 13.3 (14.1) | $-$ | $-$ | 24.0 (25.6) | $-$ |  $10.88^{+6.83}_{-4.74}$ ($11.56^{+7.25}_{-5.04}$)
$\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}K^{-}$ | 0.31 (0.33) | $-$ | $-$ | 0.667 (0.711) | $-$ |  $0.28^{+0.17}_{-0.12}$ ($0.29^{+0.18}_{-0.13}$)
$\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}K^{*-}$ | 0.71 (0.76) | $-$ | $-$ | 1.23 (1.31) | $-$ |  $0.56^{+0.35}_{-0.24}$ ($0.60^{+0.37}_{-0.26}$)
$\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}D^{-}$ | 0.58 (0.62) | $-$ | $-$ | 0.949 (1.03) | 0.45 |  $0.43^{+0.29}_{-0.20}$ ($0.45^{+0.31}_{-0.21}$)
$\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}D^{*-}$ | 1.51 (1.60) | $-$ | $-$ | 1.54 (1.64) | 0.95 |  $0.77^{+0.50}_{-0.35}$ ($0.82^{+0.53}_{-0.37}$)
$\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}D_{s}^{-}$ | 14.8 (15.7) | 14.6 | $-$ | 24.6 (26.2) | $-$ |  $10.87^{+7.51}_{-5.03}$ ($11.54^{+7.98}_{-5.34}$)
$\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}D_{s}^{*-}$ | 32.4 (34.4) | 23.1 | $-$ | 36.5 (39.0) | $-$ |  $16.24^{+10.54}_{-7.25}$ ($17.26^{+11.2}_{-7.70}$)
$\Omega_{b}^{-}\to\Omega_{c}^{0}\pi^{-}$ | 2.82 | 4.92 | 5.81 | 4.00 | 1.88 | $1.10^{+0.85}_{-0.55}$
$\Omega_{b}^{-}\to\Omega_{c}^{0}\rho^{-}$ | 7.92 | 12.8 | $-$ | 10.8 | 5.43 | $3.07^{+2.41}_{-1.53}$
$\Omega_{b}^{-}\to\Omega_{c}^{0}K^{-}$ | 0.22 | $-$ | $-$ | 0.326 | $-$ | $0.08^{+0.07}_{-0.04}$
$\Omega_{b}^{-}\to\Omega_{c}^{0}K^{*-}$ | 0.41 | $-$ | $-$ | 0.544 | $-$ | $0.16^{+0.12}_{-0.08}$
$\Omega_{b}^{-}\to\Omega_{c}^{0}D^{-}$ | 0.52 | $-$ | $-$ | 0.636 | $-$ | $0.15^{+0.14}_{-0.08}$
$\Omega_{b}^{-}\to\Omega_{c}^{0}D^{*-}$ | 0.48 | $-$ | $-$ | 0.511 | $-$ | $0.16^{+0.13}_{-0.08}$
$\Omega_{b}^{-}\to\Omega_{c}^{0}D_{s}^{-}$ | 13.5 | 17.9 | $-$ | 17.1 | $-$ | $4.03^{+3.72}_{-2.21}$
$\Omega_{b}^{-}\to\Omega_{c}^{0}D_{s}^{*-}$ | 9.73 | 11.5 | $-$ | 11.7 | $-$ | $3.18^{+2.69}_{-1.61}$

##  V Summary

With the accumulation of experimental data from LHCb and Belle II Belle-
II:2018jsg , experimental exploration of weak decay of the bottom baryons
$\Xi_{b}$ and $\Omega_{b}$ is becoming possible. Facing this opportunity, in
this work we study the color-allowed two-body nonleptonic decay of the bottom
baryons $\Xi_{b}$ and $\Omega_{b}$, i.e., the $\Xi_{b}\to\Xi_{c}^{(*)}M$ and
$\Omega_{b}\to\Omega_{c}^{(*)}M$ decay with emitting a pseudoscalar meson
($\pi^{-}$, $K^{-}$, $D^{-}$, and $D_{s}^{-}$) or a vector meson ($\rho^{-}$,
$K^{*-}$, $D^{*-}$, and $D_{s}^{*-}$).

We adopt the three-body light-front quark model to calculate these form
factors depicting these discussed bottom baryon to the charmed baryon
transitions under the naïve factorization framework. We also improve the
treatment of the spatial wave function of these involved heavy baryons in
these decays, where the semirelativistic three-body potential model
Capstick:1985xss ; Li:2021qod  is applied to calculate the numerical spatial
wave function of these heavy baryons with the help of the GEM Hiyama:2003cu ;
Yoshida:2015tia ; Hiyama:2018ivm ; Yang:2019lsg . We call that the study of
color-allowed two-body nonleptonic decay of bottom baryons $\Xi_{b}$ and
$\Omega_{b}$ is supported by hadron spectroscopy. Our result shows that these
color-allowed two-body nonleptonic decays $\Xi_{b}^{0,-}\to\Xi_{c}^{(*)+,0}$
and $\Omega_{b}^{-}\to\Omega_{c}^{(*)0}$ with the $\pi^{-}$, $\rho^{-}$, and
$D_{s}^{(*)-}$-emitted modes have considerable widths.

We suggest to measure these discussed color-allowed two-body nonleptonic decay
of the bottom baryons $\Xi_{b}$ and $\Omega_{b}$, which will be good chance
for the ongoing LHCb and Belle II experiments.

## ACKNOWLEDGMENTS

This work is supported by the China National Funds for Distinguished Young
Scientists under Grant No. 11825503, National Key Research and Development
Program of China under Contract No. 2020YFA0406400, the 111 Project under
Grant No. B20063, the National Natural Science Foundation of China under Grant
No. 12047501, and by the Fundamental Research Funds for the Central
Universities.

## References

  * (1) J. P. Lees et al. [BaBar], Evidence for an excess of $\bar{B}\to D^{(*)}\tau^{-}\bar{\nu}_{\tau}$ decays, Phys. Rev. Lett. 109 (2012), 101802.
  * (2) J. P. Lees et al. [BaBar], Measurement of an Excess of $\bar{B}\to D^{(*)}\tau^{-}\bar{\nu}_{\tau}$ Decays and Implications for Charged Higgs Bosons, Phys. Rev. D 88 (2013) no.7, 072012.
  * (3) M. Huschle et al. [Belle], Measurement of the branching ratio of $\bar{B}\to D^{(\ast)}\tau^{-}\bar{\nu}_{\tau}$ relative to $\bar{B}\to D^{(\ast)}\ell^{-}\bar{\nu}_{\ell}$ decays with hadronic tagging at Belle, Phys. Rev. D 92 (2015) no.7, 072014.
  * (4) R. Aaij et al. [LHCb], Measurement of the ratio of branching fractions $\mathcal{B}(\bar{B}^{0}\to D^{*+}\tau^{-}\bar{\nu}_{\tau})/\mathcal{B}(\bar{B}^{0}\to D^{*+}\mu^{-}\bar{\nu}_{\mu})$, Phys. Rev. Lett. 115 (2015) no.11, 111803 [erratum: Phys. Rev. Lett. 115 (2015) no.15, 159901].
  * (5) S. Hirose et al. [Belle], Measurement of the $\tau$ lepton polarization and $R(D^{*})$ in the decay $\bar{B}\to D^{*}\tau^{-}\bar{\nu}_{\tau}$, Phys. Rev. Lett. 118 (2017) no.21, 211801.
  * (6) G. Caria et al. [Belle], Measurement of $\mathcal{R}(D)$ and $\mathcal{R}(D^{*})$ with a semileptonic tagging method, Phys. Rev. Lett. 124 (2020) no.16, 161803.
  * (7) A. Bazavov et al. [Fermilab Lattice and MILC], Semileptonic form factors for $B\to D^{\ast}\ell\nu$ at nonzero recoil from 2 + 1-flavor lattice QCD, [arXiv:2105.14019 [hep-lat]].
  * (8) Y. S. Amhis et al. [HFLAV], Averages of b-hadron, c-hadron, and $\tau$-lepton properties as of 2018, Eur. Phys. J. C 81 (2021) no.3, 226.
  * (9) T. Aaltonen et al. [CDF], Observation of New Charmless Decays of Bottom Hadrons, Phys. Rev. Lett. 103 (2009), 031801.
  * (10) R. Aaij et al. [LHCb], Searches for $\Lambda^{0}_{b}$ and $\Xi^{0}_{b}$ decays to $K^{0}_{\rm S}p\pi^{-}$ and $K^{0}_{\rm S}pK^{-}$ final states with first observation of the $\Lambda^{0}_{b}\rightarrow K^{0}_{\rm S}p\pi^{-}$ decay, JHEP 04 (2014), 087.
  * (11) R. Aaij et al. [LHCb], Observations of $\Lambda_{b}^{0}\to\Lambda K^{+}\pi^{-}$ and $\Lambda_{b}^{0}\to\Lambda K^{+}K^{-}$ decays and searches for other $\Lambda_{b}^{0}$ and $\Xi_{b}^{0}$ decays to $\Lambda h^{+}h^{\prime-}$ final states, JHEP 05 (2016), 081.
  * (12) P. A. Zyla et al. [Particle Data Group], Review of Particle Physics, PTEP 2020 (2020) no.8, 083C01.
  * (13) R. Aaij et al. [LHCb], Observation of $J/\psi p$ Resonances Consistent with Pentaquark States in $\Lambda_{b}^{0}\to J/\psi K^{-}p$ Decays, Phys. Rev. Lett. 115 (2015), 072001.
  * (14) R. Aaij et al. [LHCb], Observation of a narrow pentaquark state, $P_{c}(4312)^{+}$, and of two-peak structure of the $P_{c}(4450)^{+}$, Phys. Rev. Lett. 122 (2019) no.22, 222001.
  * (15) R. Aaij et al. [LHCb], Evidence of a $J/\psi\Lambda$ structure and observation of excited $\Xi^{-}$ states in the $\Xi^{-}_{b}\to J/\psi\Lambda K^{-}$ decay, Sci. Bull. 66 (2021), 1278-1287.
  * (16) E. Kou et al. [Belle-II], The Belle II Physics Book, PTEP 2019 (2019) no.12, 123C01 [erratum: PTEP 2020 (2020) no.2, 029201].
  * (17) R. N. Faustov and V. O. Galkin, Relativistic description of the $\Xi_{b}$ baryon semileptonic decays, Phys. Rev. D 98 (2018) no.9, 093006.
  * (18) C. Q. Geng, C. W. Liu and T. H. Tsai, Nonleptonic two-body weak decays of $\Lambda_{b}$ in modified MIT bag model, Phys. Rev. D 102 (2020) no.3, 034033.
  * (19) C. Albertus, E. Hernandez and J. Nieves, Nonrelativistic constituent quark model and HQET combined study of semileptonic decays of $\Lambda_{b}$ and $\Xi_{b}$ baryons, Phys. Rev. D 71 (2005), 014012.
  * (20) D. Ebert, R. N. Faustov and V. O. Galkin, Semileptonic decays of heavy baryons in the relativistic quark model, Phys. Rev. D 73 (2006), 094002.
  * (21) H. Y. Cheng, Nonleptonic weak decays of bottom baryons, Phys. Rev. D 56 (1997), 2799-2811 [erratum: Phys. Rev. D 99 (2019) no.7, 079901].
  * (22) M. A. Ivanov, J. G. Korner, V. E. Lyubovitskij and A. G. Rusetsky, Exclusive nonleptonic bottom to charm baryon decays including nonfactorizable contributions, Mod. Phys. Lett. A 13 (1998), 181-192.
  * (23) M. A. Ivanov, J. G. Korner, V. E. Lyubovitskij and A. G. Rusetsky, Exclusive nonleptonic decays of bottom and charm baryons in a relativistic three quark model: Evaluation of nonfactorizing diagrams, Phys. Rev. D 57 (1998), 5632-5652.
  * (24) T. Gutsche, M. A. Ivanov, J. G. Körner and V. E. Lyubovitskij, Nonleptonic two-body decays of single heavy baryons $\Lambda_{Q}$, $\Xi_{Q}$, and $\Omega_{Q}$ $(Q=b,c)$ induced by $W$ emission in the covariant confined quark model, Phys. Rev. D 98 (2018) no.7, 074011.
  * (25) Z. X. Zhao, Weak decays of heavy baryons in the light-front approach, Chin. Phys. C 42 (2018) no.9, 093101.
  * (26) C. K. Chua, Color-allowed bottom baryon to charmed baryon nonleptonic decays, Phys. Rev. D 99 (2019) no.1, 014023.
  * (27) C. K. Chua, Color-allowed bottom baryon to $s$-wave and $p$-wave charmed baryon nonleptonic decays, Phys. Rev. D 100 (2019) no.3, 034025.
  * (28) H. W. Ke, N. Hao and X. Q. Li, Revisiting $\Lambda_{b}\rightarrow\Lambda_{c}$ and $\Sigma_{b}\rightarrow\Sigma_{c}$ weak decays in the light-front quark model, Eur. Phys. J. C 79 (2019) no.6, 540.
  * (29) J. Zhu, Z. T. Wei and H. W. Ke, Semileptonic and nonleptonic weak decays of $\Lambda_{b}^{0}$, Phys. Rev. D 99 (2019) no.5, 054020.
  * (30) Y. S. Li, X. Liu and F. S. Yu, Revisiting semileptonic decays of $\Lambda$b(c) supported by baryon spectroscopy, Phys. Rev. D 104 (2021) no.1, 013005.
  * (31) Y. m. Wang, Y. Li and C. D. Lu, Rare Decays of $\Lambda_{b}\to\Lambda+\gamma$ and $\Lambda_{b}\to\Lambda\ell^{+}\ell^{-}$ in the Light-cone Sum Rules, Eur. Phys. J. C 59 (2009), 861-882.
  * (32) A. Khodjamirian, C. Klein, T. Mannel and Y. M. Wang, Form Factors and Strong Couplings of Heavy Baryons from QCD Light-Cone Sum Rules, JHEP 09 (2011), 106.
  * (33) Y. M. Wang and Y. L. Shen, Perturbative Corrections to $\Lambda_{b}\to\Lambda$ Form Factors from QCD Light-Cone Sum Rules, JHEP 02 (2016), 179.
  * (34) Z. X. Zhao, R. H. Li, Y. L. Shen, Y. J. Shi and Y. S. Yang, The semi-leptonic form factors of $\Lambda_{b}\to\Lambda_{c}$ and $\Xi_{b}\to\Xi_{c}$ in QCD sum rules, Eur. Phys. J. C 80 (2020) no.12, 1181.
  * (35) P. Guo, H. W. Ke, X. Q. Li, C. D. Lu and Y. M. Wang, Diquarks and the semi-leptonic decay of $\Lambda_{b}$ in the hyrid scheme, Phys. Rev. D 75 (2007), 054017.
  * (36) S. Capstick and N. Isgur, Baryons in a Relativized Quark Model with Chromodynamics, AIP Conf. Proc. 132 (1985), 267-271.
  * (37) E. Hiyama and M. Kamimura, Study of various few-body systems using Gaussian expansion method (GEM), Front. Phys. (Beijing) 13 (2018) no.6, 132106.
  * (38) E. Hiyama, Y. Kino and M. Kamimura, Gaussian expansion method for few-body systems, Prog. Part. Nucl. Phys. 51 (2003), 223-307.
  * (39) T. Yoshida, E. Hiyama, A. Hosaka, M. Oka and K. Sadato, Spectrum of heavy baryons in the quark model, Phys. Rev. D 92 (2015) no.11, 114029.
  * (40) G. Yang, J. Ping, P. G. Ortega and J. Segovia, Triply heavy baryons in the constituent quark model, Chin. Phys. C 44 (2020) no.2, 023102.
  * (41) S. Tawfiq, P. J. O’Donnell and J. G. Korner, Charmed baryon strong coupling constants in a light front quark model, Phys. Rev. D 58 (1998), 054010.
  * (42) S. Godfrey and N. Isgur, Mesons in a Relativized Quark Model with Chromodynamics, Phys. Rev. D 32 (1985), 189-231.
  * (43) Q. T. Song, D. Y. Chen, X. Liu and T. Matsuki, Charmed-strange mesons revisited: mass spectra and strong decays, Phys. Rev. D 91 (2015), 054031.
  * (44) C. Q. Pang, J. Z. Wang, X. Liu and T. Matsuki, A systematic study of mass spectra and strong decay of strange mesons, Eur. Phys. J. C 77 (2017) no.12, 861.
  * (45) J. Z. Wang, Z. F. Sun, X. Liu and T. Matsuki, Higher bottomonium zoo, Eur. Phys. J. C 78 (2018) no.11, 915.
  * (46) J. Z. Wang, D. Y. Chen, X. Liu and T. Matsuki, Constructing $J/\psi$ family with updated data of charmoniumlike $Y$ states, Phys. Rev. D 99 (2019) no.11, 114003.
  * (47) M. X. Duan and X. Liu, Where are 3P and higher P-wave states in the charmonium family?, Phys. Rev. D 104 (2021) no.7, 074010.
  * (48) C. Chen, X. L. Chen, X. Liu, W. Z. Deng and S. L. Zhu, Strong decays of charmed baryons, Phys. Rev. D 75 (2007), 094017.
  * (49) B. Chen, S. Q. Luo and X. Liu, Universal behavior of mass gaps existing in the single heavy baryon family, Eur. Phys. J. C 81 (2021) no.5, 474.
  * (50) S. F. Biagi, M. Bourquin, A. J. Britten, R. M. Brown, H. J. Burckhart, A. A. Carter, C. Dore, P. Extermann, M. Gailloud and C. N. P. Gee, et al. Observation of a Narrow State at $2.46\ \text{GeV}/c^{2}$: A Candidate for the Charmed Strange Baryon $A^{+}$, Phys. Lett. B 122 (1983), 455.
  * (51) P. L. Frabetti et al. [Fermilab E687], Measurement of the mass and lifetime of the $\Xi_{c}^{+}$, Phys. Rev. Lett. 70 (1993), 1381-1384.
  * (52) P. Avery et al. [CLEO], Observation of the Charmed Strange Baryon $\Xi_{c}^{0}$, Phys. Rev. Lett. 62 (1989), 863.
  * (53) R. Chistov et al. [Belle], Observation of new states decaying into $\Lambda_{c}^{+}K^{-}\pi^{+}$ and $\Lambda_{c}^{+}\to K_{S}^{0}\pi^{-}$, Phys. Rev. Lett. 97 (2006), 162001.
  * (54) S. F. Biagi, M. Bourquin, A. J. Britten, R. M. Brown, H. J. Burckhart, A. A. Carter, C. Doré, P. Extermann, M. Gailloud and C. N. P. Gee, et al. Properties of the Charmed Strange Baryon A+ and Evidence for the Charmed Doubly Strange Baryon $T^{0}$ at $2.74\ \text{GeV}/c^{2}$, Z. Phys. C 28 (1985), 175.
  * (55) H. Albrecht et al. [ARGUS], Evidence for the production of the charmed, doubly strange baryon $\Omega_{c}$ in $e^{+}e^{-}$ annihilation, Phys. Lett. B 288 (1992), 367-372.
  * (56) R. Aaij et al. [LHCb], Observation of five new narrow $\Omega_{c}^{0}$ states decaying to $\Xi_{c}^{+}K^{-}$, Phys. Rev. Lett. 118 (2017) no.18, 182001.
  * (57) J. Yelton et al. [Belle], Observation of Excited $\Omega_{c}$ Charmed Baryons in $e^{+}e^{-}$ Collisions, Phys. Rev. D 97 (2018) no.5, 051102.
  * (58) B. Chen and X. Liu, New $\Omega_{c}^{0}$ baryons discovered by LHCb as the members of $1P$ and $2S$ states, Phys. Rev. D 96 (2017) no.9, 094015.
  * (59) H. Y. Cheng and C. W. Chiang, Quantum numbers of $\Omega_{c}$ states and other charmed baryons, Phys. Rev. D 95 (2017) no.9, 094018.
  * (60) H. X. Chen, Q. Mao, W. Chen, A. Hosaka, X. Liu and S. L. Zhu, Decay properties of $P$-wave charmed baryons from light-cone QCD sum rules, Phys. Rev. D 95 (2017) no.9, 094008.
  * (61) S. S. Agaev, K. Azizi and H. Sundu, On the nature of the newly discovered $\Omega$ states, EPL 118 (2017) no.6, 61001.
  * (62) K. L. Wang, L. Y. Xiao, X. H. Zhong and Q. Zhao, Understanding the newly observed $\Omega_{c}$ states through their decays, Phys. Rev. D 95 (2017) no.11, 116010.
  * (63) V. R. Debastiani, J. M. Dias, W. H. Liang and E. Oset, $\Omega_{b}^{-}\to(\Xi_{c}^{+}\,K^{-})\,\pi^{-}$ and the $\Omega_{c}$ states, Phys. Rev. D 98 (2018) no.9, 094022.
  * (64) H. Georgi, B. Grinstein and M. B. Wise, $\Lambda_{b}$ semileptonic decay form-factors for $m_{c}$ does not equal infinity, Phys. Lett. B 252 (1990), 456-460.
  * (65) K. C. Bowler et al. [UKQCD], First lattice study of semileptonic decays of $\Lambda_{b}$ and $\Xi_{b}$ baryons, Phys. Rev. D 57 (1998), 6948-6974.
  * (66) C. D. Lu, Y. M. Wang, H. Zou, A. Ali and G. Kramer, Anatomy of the pQCD Approach to the Baryonic Decays $\Lambda_{b}\to p\pi,pK$, Phys. Rev. D 80 (2009), 034011.
  * (67) R. Aaij et al. [LHCb], Precision measurement of the mass and lifetime of the $\Xi_{b}^{0}$ baryon, Phys. Rev. Lett. 113 (2014), 032001.
  * (68) R. Aaij et al. [LHCb], Measurement of the $\Xi_{b}^{-}$ and $\Omega_{b}^{-}$ baryon lifetimes, Phys. Lett. B 736 (2014), 154-162.
  * (69) R. Aaij et al. [LHCb], Measurement of the mass and lifetime of the $\Omega_{b}^{-}$ baryon, Phys. Rev. D 93 (2016) no.9, 092007.
  * (70) T. A. Aaltonen et al. [CDF], Mass and lifetime measurements of bottom and charm baryons in $p\bar{p}$ collisions at $\sqrt{s}=1.96$ TeV, Phys. Rev. D 89 (2014) no.7, 072014.
  * (71) H. Y. Cheng, C. K. Chua and C. W. Hwang, Covariant light front approach for $S$ wave and $P$ wave mesons: Its application to decay constants and form-factors, Phys. Rev. D 69 (2004), 074025.


# Quantum Limits on the Capacity of Multispan Links with Phase-Sensitive
Amplification
††thanks: This work is a part of the project “Quantum Optical Technologies”
carried out within the International Research Agendas programme of the
Foundation for Polish Science co-financed by the European Union under the
European Regional Development Fund.

Karol Łukanowski124, Marcin Jarzyna1, and Konrad Banaszek12  1Centre for
Quantum Optical Technologies, CeNT, University of Warsaw, Banacha 2c, 02-097
Warszawa, Poland
2Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warszawa, Poland
4E-mail<EMAIL_ADDRESS>
###### Abstract

The capacity of a linear attenuating optical channel with the signal
regenerated using quantum-limited phase-sensitive amplifiers is analyzed for
conventional and generalized detection scenarios. The continuous model of
distributed amplification determines the attainable capacity for long-haul
links under the total power constraint.

###### Index Terms:

Optical fiber communication, optical amplifiers, low-noise amplifiers,
information rates, optical losses

##  I Introduction

The development of quantum technologies provides a strong driving force to
improve the performance of photonic components and devices [1]. While the
principal motivation is to reach performance levels that would enable new
functionalities with the flagship example of quantum key distribution [2],
these advances may greatly benefit also regular optical communications. It is
therefore vital to investigate how the availability of improved photonic
components and devices would affect the design and the modelling of optical
networks and to quantify the attainable enhancements. The complete analysis
needs to be based on the quantum theory of electromagnetic radiation [3] which
includes fundamental sources of quantum noise that determine the ultimate
performance limits of optical communication systems [4].

Recent years have witnessed an impressive progress in the technology of phase-
sensitive amplifiers [5, 6, 7]. The favorable noise figure of phase-sensitive
amplification (PSA) makes it a promising approach to increase the transmission
rates in long-haul optical communication systems. The purpose of this
contribution is to analyze theoretically the capacity limits of a multispan
point-to-point link with PSA regeneration taking fully into account quantum
mechanical fluctuations of the propagating optical field [8]. It is shown that
these fluctuations introduce a fundamental noise penalty that causes a decline
in the capacity of the resulting communication channel even for ideal quantum-
limited PSA with a 0 dB noise figure. The analysis is carried out under the
total power constraint for the propagating field and assumes one of two
detection scenarios at the channel output [4]: either a shot-noise-limited
(SNL) single-quadrature measurement, or the most general optimized measurement
strategy permitted by quantum mechanics. The efficiency of the latter is
characterized by the Gordon-Holevo capacity limit adapted to phase-sensitive
Gaussian channels [9, 10]. An approximate analytical solution is derived in
the continuous limit of distributed regeneration which facilitates comparison
with the phase-insensitive amplification (PIA) scenario discussed previously
[11].

##  II Channel model

We start by presenting the theoretical model used in numerical simulations of
a multispan optical communication link. Let $x^{I}$ and $x^{Q}$ denote two
conjugate $I$ and $Q$ quadratures of a single mode of the electromagnetic
field. It will be convenient to decompose $x^{I}=x_{S}^{I}+x_{N}^{I}$ and
$x^{Q}=x_{S}^{Q}+x_{N}^{Q}$, where $x_{S}^{I}$ and $x_{S}^{Q}$ are the mean
values that carry the modulated symbols, while zero-mean variables $x_{N}^{I}$
and $x_{N}^{Q}$ describe quantum fluctuations. Units are chosen such that
vacuum field fluctuations correspond to the variance
$\textrm{Var}[x_{N}^{I}]=\textrm{Var}[x_{N}^{Q}]=1/2$. Formally, the variables
$x^{I}$ and $x^{Q}$ defined above parameterize the quantum mechanical Wigner
function of the electromagnetic field mode under consideration [8]. For
Gaussian signal modulation and Gaussian noise it is helpful to introduce the
signal power and the noise power in individual quadratures:

$\begin{IEEEeqnarraybox}[]{[}][c]{rCl"rCl}S^{I}&=&\textrm{Var}[x_{S}^{I}],&N^{I}&=&\textrm{Var}[x_{N}^{I}],\\\
S^{Q}&=&\textrm{Var}[x_{S}^{Q}],&N^{Q}&=&\textrm{Var}[x_{N}^{Q}].\end{IEEEeqnarraybox}$
(1)

The Heisenberg uncertainty relation requires that $N^{I}\cdot N^{Q}\geq 1/4$.
The total energy per mode expressed as the mean photon number $\bar{n}$ is
given by

$\bar{n}=(S^{I}+S^{Q}+N^{I}+N^{Q})/2-1/2.$ (2)

The negative $-1/2$ term is needed to subtract vacuum fluctuations when
converting the quadrature variance budget to the optical energy carried by the
mode.

Consider now a link shown in Fig. 1(a) in the form of $R+1$ attenuating spans
with respective power transmissions $\tau_{i}$, $i=1,2,\ldots,R+1$ and the
total length $L$. The signal is regenerated using $R$ phase-sensitive
amplifiers with gains $G_{i}$. For quantum-limited PSA the recursive relation
for the optical field parameters over the $i$th span followed by an amplifier
reads

$\begin{IEEEeqnarraybox}[]{[}][c]{rCl^{\prime}rCl}S_{i}^{I}&=&G_{i}\tau_{i}S_{i-1}^{I},&N_{i}^{I}&=&G_{i}[\tau_{i}N_{i-1}^{I}+(1-\tau_{i})/2],\\\\[3.0pt]
S_{i}^{Q}&=&G_{i}^{-1}\tau_{i}S_{i-1}^{Q},&N_{i}^{Q}&=&G_{i}^{-1}[\tau_{i}N_{i-1}^{Q}+(1-\tau_{i})/2].\end{IEEEeqnarraybox}$
(3)

where $i=0$ denotes the channel input and $i=R+1$ specifies the channel
output. Equations (3) can be used also for the last unamplified span by
setting $G_{R+1}=1$. Importantly, the terms $(1-\tau_{i})/2$ in the
expressions for $N_{i}^{I}$ and $N_{i}^{Q}$ stem from vacuum fluctuations that
are added in the course of propagation through an attenuating medium [8]. In
the following indices ‘in’ for $i=0$ and ‘out’ for $i=R+1$ will be used for
clarity.

Figure 1: (a) A multispan link with $R+1$ segments with transmissions
$\tau_{i}$, $i=1,\ldots,R+1$. The signal is regenerated using $R$ phase-
sensitive amplifiers with gains $G_{i}$. (b) Channel capacity as a function of
distance for $\bar{n}=100$ input mean photon number optimized over the
locations and gains of $R$ phase-sensitive amplifiers under the total power
constraint. Solid lines depict the conventional SNL single-quadrature
detection scenario, whereas dashed-dotted lines correspond to the generalized
scenario described by the Gordon-Holevo (GH) expression adapted to phase-
sensitive Gaussian channels. The case of distributed amplification is labeled
as $R=\infty$. The standard value of $0.2~{}\textrm{dB}/\textrm{km}$ for
channel attenuation has been used. (c) Conventional-detection Shannon capacity
as a function of distance for distributed amplification and the input mean
photon number $\bar{n}=50$ (purple), $100$ (red), and $200$ (orange). The
numerical solutions of (4) describing the PSA scenario (solid lines) are
compared with the approximation (6) (dotted lines) and the PIA regeneration
scenario analyzed in [11] (dashed lines).

##  III Numerical optimization

The capacity of the link described above has been optimized with respect to
the locations $l_{i}$ and gains $G_{i}$ of the amplifiers under the total
power constraint, i.e., that at no point along the link the right hand side of
(2) exceeds the input mean photon number $\bar{n}$. Two communication
scenarios have been considered. In the conventional scenario, information is
encoded only in the $I$ quadrature of ideal laser light with
$N^{I}_{\textrm{in}}=N^{Q}_{\textrm{in}}=1/2$ at the input, and SNL detection
of that quadrature is implemented at the output. The capacity is then given by
the standard Shannon expression ${\sf
C}_{\textrm{PSA}}=\frac{1}{2}\log_{2}(1+S^{I}_{\textrm{out}}/N^{I}_{\textrm{out}})$.
The second, generalized scenario permits an arbitrary distribution of the
input power between the $I$ and $Q$ quadratures at the input and allows using
non-classical squeezed states [8] for which either $N^{I}_{\textrm{in}}$ or
$N^{Q}_{\textrm{in}}$ is reduced below $1/2$. Furthermore, the generalized
scenario assumes optimization over all measurements of the optical field at
the channel output that are permitted by quantum mechanics. In such a case the
capacity ${\sf C}_{\textrm{GH}}$ is given by the Gordon-Holevo (GH) formula
extended to the case of phase-sensitive Gaussian channels [9, 10] that need to
be considered here.

The results of optimizing the channel capacity with PSA regeneration for the
input mean photon number $\bar{n}=100$ are shown in Fig. 1(b). The standard
$0.2~{}\textrm{dB}/\textrm{km}$ figure for the attenuation has been taken to
calculate transmission of individual spans
$\tau_{i}=\exp[-\alpha(l_{i}-l_{i-1})]$. It is seen that for very short
distances the generalized strategy offers advantage which however virtually
does not depend on the ability to regenerate the signal. The observed
advantage effectively stems from using both field quadratures to transmit
information over a loss-only channel [4]. For longer distances PSA becomes
useful to boost the link capacity. In this regime the difference between the
capacities attainable in the conventional and in the generalized scenarios
becomes negligible. Overall, the capacity is seen to decline with the distance
despite assuming ideal, quantum-limited PSA in (3). This effect can be
attributed to the amplification of vacuum fluctuations added in the course of
propagation through an attenuating medium.

##  IV Distributed amplification

The minimum decline rate for the channel capacity under PSA can be found by
considering densely spaced amplifiers at equidistant locations $l_{i}=i\Delta
l$. One can then approximate $\tau_{i}\approx 1-\alpha\Delta l$, where
$\alpha$ is the attenuation of the propagation medium, and take $G_{i}\approx
1+\gamma(l_{i})\Delta l$. In the continuous limit of distributed amplification
$\Delta l\rightarrow 0$ this gives a set of differential equations

$\begin{IEEEeqnarraybox}[]{[}][c]{rCl}\frac{\textrm{d}S^{I}}{\textrm{d}l}&=&[\gamma(l)-\alpha]S^{I},\\\
\frac{\textrm{d}N^{I}}{\textrm{d}l}&=&[\gamma(l)-\alpha]N^{I}+\alpha/2,\\\
\frac{\textrm{d}S^{Q}}{\textrm{d}l}&=&[-\gamma(l)-\alpha]S^{Q},\\\
\frac{\textrm{d}N^{Q}}{\textrm{d}l}&=&[-\gamma(l)-\alpha]N^{Q}+\alpha/2.\end{IEEEeqnarraybox}$
(4)

In the conventional communication scenario, when only the $I$ quadrature is
modulated one has $S^{Q}=0$ and can approximate $N^{Q}=1/2$. The simplified
total power constraint ${\textrm{d}}(S^{I}+N^{I})/\textrm{d}l=0$ yields the
amplification factor equal to $\gamma=\alpha[1-(4\bar{n}+1)^{-1}]$. It is seen
that full compensation of signal losses is not possible due to the buildup of
excess noise through amplification of vacuum fluctuations. The resulting
simplified solution for the $I$ quadrature at the channel output reads

$\begin{IEEEeqnarraybox}[]{[}][c]{rCl}S^{I}(L)&=&2\bar{n}\exp\left(-\frac{\alpha
L}{4\bar{n}+1}\right),\\\ N^{I}(L)&=&2\bar{n}\left[1-\exp\left(-\frac{\alpha
L}{4\bar{n}+1}\right)\right]+\frac{1}{2}.\end{IEEEeqnarraybox}$ (5)

If the vacuum fluctuations level $1/2$ can be neglected compared to $\alpha L$
and $\bar{n}$, the conventional-detection one-quadrature capacity is well
approximated by

$\displaystyle{\sf C}_{\textrm{PSA}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\log_{2}\left(1+\frac{S^{I}(L)}{N^{I}(L)}\right)$ (6)
$\displaystyle\approx$
$\displaystyle-\frac{1}{2}\log_{2}\left(1-\exp\left(-\frac{\alpha
L}{4\bar{n}}\right)\right).$

As seen in Fig. 1(c), the capacity calculated using the exact solution of the
set of differential equations (4) is matched well by the approximate
expression derived in (6). It is instructive to compare this result with the
case of phase-insensitive amplification (PIA) analyzed in [11]. The dashed
curves in Fig. 1(c) represent the two-quadrature capacity ${\sf
C}_{\textrm{PIA}}$ under conventional SNL coherent detection, which
analogously to (6) can be approximated by

${\sf C}_{\textrm{PIA}}\approx-\log_{2}\left(1-\exp\left(-\frac{\alpha
L}{\bar{n}}\right)\right).$ (7)

It is seen that while for short distances the benefit of using both
quadratures to encode information prevails, PSA becomes advantageous for long-
haul links. A comparison of the approximate expressions for ${\sf
C}_{\textrm{PSA}}$ and ${\sf C}_{\textrm{PIA}}$ quantifies the advantage as a
four times lower decline rate of the capacity with the distance. This effect
can be attributed to the fact that whereas the performance of a quantum-
limited PSA link is degraded only by amplification of vacuum fluctuations
contributed by the process of signal attenuation, PIA inevitably adds excess
noise to both field quadratures in the amount corresponding to at least 3 dB
noise figure.

##  V Conclusions

The capacity of a multispan link with PSA regeneration has been studied under
the total power constraint taking fully into account quantum mechanical
fluctuations of the propagating optical field. It has been found that for
long-haul links unconventional detection strategies described by the Gordon-
Holevo capacity expression do not offer a meaningful benefit compared to
conventional SNL single-quadrature detection. A simplified analysis of the
continuous limit of distributed amplification indicates that the ultimate
advantage of PSA regeneration is four times slower decline rate of the
capacity with the distance compared to the PIA scenario.

## Acknowledgment

Insightful discussions with P. Andrekson, M. Karlsson, J. Schröder, and J. P.
Turkiewicz are gratefully acknowledged.

## References

  * [1] E. Pelucchi, G. Fagas, I. Aharonovich, D. Englund, E. Figueroa, Q. Gong, H. Hannes, J. Liu, C.-Y. Lu, N. Matsuda, J.-W. Pan, F. Schreck, F. Sciarrino, C. Silberhorn, J. Wang, and K. D. Jöns, “The potential and global outlook of integrated photonics for quantum technologies,” _Nature Reviews Physics_ , 2021. [Online]. Available: https://doi.org/10.1038/s42254-021-00398-z
  * [2] S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. L. Pereira, M. Razavi, J. S. Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, “Advances in quantum cryptography,” _Adv. Opt. Photon._ , vol. 12, no. 4, pp. 1012–1236, Dec 2020. [Online]. Available: https://doi.org/10.1364/AOP.361502
  * [3] J. H. Shapiro, “The quantum theory of optical communications,” _IEEE J. Sel. Top. Quant. Electr._ , vol. 15, no. 6, pp. 1547–1569, 2009. [Online]. Available: https://doi.org/10.1109/JSTQE.2009.2024959
  * [4] K. Banaszek, L. Kunz, M. Jachura, and M. Jarzyna, “Quantum limits in optical communications,” _J. Lightwave Technol._ , vol. 38, no. 10, pp. 2741–2754, May 2020. [Online]. Available: https://doi.org/10.1109/JLT.2020.2973890
  * [5] J. Kakande, A. Bogris, R. Slavík, F. Parmigiani, D. Syvridis, M. Sköld, M. Westlund, P. Petropoulos, and D. J. Richardson, “QPSK phase and amplitude regeneration at 56 Gbaud in a novel idler-free non-degenerate phase sensitive amplifier,” in _Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2011_. Optical Society of America, 2011, p. OMT4. [Online]. Available: https://doi.org/10.1364/OFC.2011.OMT4
  * [6] T. Umeki, M. Asobe, H. Takara, Y. Miyamoto, and H. Takenouchi, “Multi-span transmission using phase and amplitude regeneration in PPLN-based PSA,” _Opt. Express_ , vol. 21, no. 15, pp. 18 170–18 177, Jul 2013. [Online]. Available: https://doi.org/10.1364/OE.21.018170
  * [7] S. L. Olsson, H. Eliasson, E. Astra, M. Karlsson, and P. A. Andrekson, “Long-haul optical transmission link using low-noise phase-sensitive amplifiers,” _Nature Communications_ , vol. 9, no. 1, p. 2513, Jun 2018. [Online]. Available: https://doi.org/10.1038/s41467-018-04956-5
  * [8] C. Gerry and P. Knight, _Introductory Quantum Optics_. Cambridge University Press, 2004.
  * [9] J. Schäfer, E. Karpov, R. García-Patrón, O. V. Pilyavets, and N. J. Cerf, “Equivalence relations for the classical capacity of single-mode Gaussian quantum channels,” _Phys. Rev. Lett._ , vol. 111, p. 030503, Jul 2013. [Online]. Available: https://doi.org/10.1103/PhysRevLett.111.030503
  * [10] J. Schäfer, E. Karpov, O. V. Pilyavets, and N. J. Cerf, “Classical capacity of phase-sensitive Gaussian quantum channels,” _arXiv:1609.04119 [quant-ph]_ , 2016. [Online]. Available: https://arxiv.org/abs/1609.04119
  * [11] M. Jarzyna, R. García-Patrón, and K. Banaszek, “Ultimate capacity limit of a multi-span link with phase-insensitive amplification,” in _45th European Conference on Optical Communication_ , Dublin, Ireland, Sep. 2019. [Online]. Available: https://doi.org/10.1049/cp.2019.0742