# Investigating the transition form factors of
$\Lambda_{b}\to\Lambda_{c}(2625)$ and $\Xi_{b}\to\Xi_{c}(2815)$ and the
corresponding weak decays with support from baryon spectroscopy

Yu-Shuai Li1,2<EMAIL_ADDRESS>Xiang Liu1,2,3<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 Frontiers Science Center for Rare Isotopes, Lanzhou
University, Lanzhou 730000, China

###### Abstract

We calculate the form factors of the $\Lambda_{b}\to\Lambda_{c}(2625)$ and
$\Xi_{b}\to\Xi_{c}(2815)$ transitions, and additionally evaluate the
corresponding semileptonic decays and the color-allowed two-body nonleptonic
decays. In order to obtain the concerned form factors, we use the three-body
light-front quark model with the support from baryon spectroscopy. In this
work, as important physical inputs, the spatial wave functions of concerned
baryons are obtained by the Gaussian expansion method with a semirelativistic
potential model. For the semileptonic processes, the branching ratios of the
electron and muon channels can reach up to the order of magnitude of $1\%$,
where our result
$\mathcal{B}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}(2625)\mu^{-}\nu_{\mu})=(1.641\pm
0.113)\%$ is consistent with the current experimental data. As for the
nonleptonic processes, the decays to $\pi^{-}$, $\rho^{-}$, and $D_{s}^{(*)-}$
final states have considerable widths. These discussed decay modes could be
accessible at the LHCb experiment.

##  I introduction

The investigation of bottom baryon weak decay is a fiery topic in heavy flavor
physics, which has drawn attentions at both theoretical and experimental
arenas. There exist abundant decay modes involved in bottom baryons due to
their higher mass. So the bottom baryon decay provides a superb platform to
test the Quantum Chromodynamics (QCD), and to search for new physics beyond
the Standard Model (SM) via detecting whether the lepton flavor universality
(LFU) is violated or not BaBar:2012obs ; BaBar:2013mob ; Belle:2015qfa ;
LHCb:2015gmp ; Belle:2016dyj ; Belle:2019rba ; FermilabLattice:2021cdg .
Besides, it is also helpful to discover the new exotic states including the
hidden-charm pentaquark states $P_{c}(4312)$, $P_{c}(4380)$, $P_{c}(4440)$,
and $P_{c}(4457)$ in $\Lambda_{b}\to J/\psi pK$ LHCb:2015yax ; LHCb:2019kea
mode, the $P_{c}(4337)$ in the $B_{s}^{0}\to J/\psi p\bar{p}$ LHCb:2021chn
mode, and the $P_{cs}(4459)$ in the $\Xi_{b}\to J/\psi\Lambda K$ LHCb:2020jpq
mode.

In theoretical aspect, the bottom baryons decaying into the $J^{P}=1/2^{+}$
ground charmed baryons by both semileptonic and nonleptonic processes have
been widely studied via various theoretical approaches, including lattice QCD
(LQCD) Gottlieb:2003yb ; Detmold:2015aaa , QCD sum rules Huang:2005mea ;
Azizi:2018axf ; Zhao:2020mod , light-cone sum rules Wang:2009yma ;
Duan:2022uzm ; Miao:2022bga , and various phenomenological quark models
Pervin:2005ve ; Ebert:2006rp ; Ke:2007tg ; Gutsche:2015mxa ; Faustov:2016pal ;
Gutsche:2018nks ; Chua:2018lfa ; Ke:2019smy ; Chua:2019yqh ; Rahmani:2020kjd ;
Geng:2020ofy ; Li:2021qod ; Li:2021kfb . However, compared with these studies
mentioned above, the investigation of the decays of bottom baryon into the
$P$-wave baryon state should still be paid more attention. In the past years,
some theoretical groups were dedicated to this issue. For example, Pervin et
al. studied the semileptonic decays of $\Lambda_{b}$ into $\Lambda_{c}$ baryon
with $J^{P}=(1/2^{\pm},3/2^{-})$ by a constituent quark model with both
nonrelativistic and semirelativistic Hamiltonians Pervin:2005ve . Gutsche et
al. also studied the same channels by a covariant confined quark model (CCQM)
Gutsche:2018nks . The heavy quark spin symmetry (HQSS) was also applied to
estimate the semileptonic decays
$\Lambda_{b}\to\Lambda_{c}(2595,2625)\ell^{-}\nu_{\ell}$ Nieves:2019kdh .
Besides, Meinel and Rendon performed the first LQCD calculation of the
$\Lambda_{b}\to\Lambda_{c}(2595,2625)\ell^{-}\nu_{\ell}$ decays Meinel:2021rbm
; Meinel:2021mdj . For the nonleptonic process, Chua calculated a series of
color-allowed decay of bottom baryon into $P$-wave charmed baryon by the
light-front quark model (LFQM) Chua:2019yqh . And Liang et al. evaluated the
nonleptonic $\Lambda_{b}\to\Lambda_{c}(2595,2625)\pi^{-}$ Liang:2016ydj ,
$\Lambda_{b}\to\Lambda_{c}(2595,2625)D_{s}^{-}$ Liang:2016ydj , and
semileptonic $\Lambda_{b}\to\Lambda_{c}(2625)\ell\nu_{\ell}$ Liang:2016exm
decays by assuming the $\Lambda_{c}(2595)$ and $\Lambda_{c}(2625)$ as a
dynamically generated resonance from the $DN$, $D^{*}N$ interaction and
coupled channels Liang:2016ydj ; Liang:2016exm . Pavao et al. carried out the
investigation of the $\Xi_{b}^{-}\to\Xi_{c}^{0}(2815)\pi^{-}(D_{s}^{-})$ and
$\Xi_{b}^{-}\to\Xi_{c}^{0}(2815)\ell^{-}\nu_{\ell}$ processes when treating
the $\Xi_{c}(2815)$ as dynamically generated resonance from the vector meson-
baryon interaction Pavao:2017cpt .

In our former work, we once studied the form factors and the semileptonic
decays into charmed baryons with $J^{P}=1/2^{-}$ by LFQM Li:2021qod , which is
supported by the baryon spectroscopy. This treatment is different from that
given in Ref. Chua:2019yqh . We should indicate that there exists difference
of the results of these discussed transition when adopting different
frameworks Pervin:2005ve ; Liang:2016ydj ; Liang:2016exm ; Pavao:2017cpt ;
Gutsche:2018nks ; Nieves:2019kdh ; Chua:2019yqh , which should be clarified by
further experimental measurement. In general, the issue around these weak
transitions of bottom baryons is still open.

As a continuation of the decays into charmed baryons with $J^{P}=1/2^{-}$
Li:2021qod , in this work, we investigate the weak transitions relevant to the
$J^{P}=3/2^{-}$ charmed baryons, which include the
$\Lambda_{b}\to\Lambda_{c}(2625)$ and $\Xi_{b}\to\Xi_{c}(2815)$ processes,
where the $\Lambda_{c}(2625)$ and $\Xi_{c}(2815)$ are treated as the
conventional $\rho$-mode excited $P$-wave charmed baryons111In this work, we
do not consider several other possible spin 3/2 $\Lambda_{c}$ and $\Xi_{c}$
resonances such as the $\Lambda_{c}(2860)$ LHCb:2017jym  and $\Xi_{c}(2645)$
Belle:2016lhy , which have positive parity different from that of the
$\Lambda_{c}(2625)$ and $\Xi_{c}(2815)$. Here, the $\Lambda_{c}(2860)$ and
$\Xi_{c}(2645)$ are good candidates of $D$-wave Chen:2017aqm  and $S$-wave
charmed baryons Chen:2016iyi , while the discussed $\Lambda_{c}(2625)$ and
$\Xi_{c}(2815)$ are assigned as $P$-wave charmed baryons.. Note that this
assignment is suitable since the experimental mass value of the
$\Lambda_{c}(2625)$ and $\Xi_{c}(2815)$ can be reproduced by the potential
models Chen:2015kpa ; Guo:2019ytq ; Yu:2022ymb ; Li:2022xtj ; Capstick:1985xss
. Although we adopt the similar way given by Ref. Li:2021qod , we still want
to emphasize the improvement made in this work. First of all, the involved
charmed baryons in the final state have $3/2^{-}$ quantum number, which makes
whole deduction framework become more complicated. Especially, at present the
study around the production of charmed baryons with $3/2^{-}$ is not enough
compared with that relevant to these low-lying charmed baryons. Our work is a
timely investigation of this issue. Secondly, in Ref. Li:2021qod , we focus on
the weak decays of the $\Lambda_{b}$ baryon into $1/2^{\pm}$ charmed baryons.
However, in the present work we study the $\Xi_{b}$ decays into $3/2^{-}$
charmed baryons, which is motivated by the experiment fact that the data of
the $\Xi_{b}$ bottom baryon can be also largely produced in the $pp$
collisions at Large Hadron Collider (LHC) LHCb:2019sxa . Obviously, the
present work is just at the right time, which may provide valuable hint to
future experimental search for these discussed decays. Especially, with the
high-luminosity upgrade to LHC, the LHCb experiment will have great potential
to explore these discussed transitions.

As indicated in Ref. Li:2021qod , the baryon spectroscopy can provide
important input to the spatial wave functions of these involved baryons when
estimating the weak transition matrix element or the corresponding form
factors. In the realistic calculation of baryon spectroscopy, we adopt the
three quarks treatment, which is different from the quark-diquark
approximation used by former theoretical works of weak decays Ke:2007tg ;
Wang:2017mqp ; Zhao:2018zcb ; Zhao:2018mrg ; Chua:2018lfa ; Ke:2017eqo ;
Zhu:2018jet ; Chua:2019yqh ; Wang:2022ias ; Zhao:2022vfr . With the support
from baryon spectroscopy, the dependence of the result on the $\beta$ value,
which is a parameter of the simple hadronic oscillator, can be avoided as
indicated in Refs. Li:2021qod ; Li:2021kfb . In the next section, we will
introduce more details of the deduction.

This paper is organized as follows. After the Introduction, the deduction of
the formulas of eight transition form factors of the
$\mathcal{B}_{b}(1/2^{+})\to\mathcal{B}_{c}(3/2^{-})$ process is given in Sec.
II. For obtaining the spatial wave functions of the involved baryons, we
introduce the semirelativistic potential model and adopt the Gaussian
expansion method (GEM) in Sec. III. And then, in Sec. IV we present the
results of the form factors of the $\Lambda_{b}\to\Lambda_{c}(2625)$ and
$\Xi_{b}\to\Xi_{c}(2815)$ transitions, and further evaluate the corresponding
semileptonic decays and color-allowed two-body nonleptonic decays. Finally,
this paper ends with the discussion and conclusion.

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

The $b\to c$ weak decay is usually dependent on the hadronic structure
reflected by the baryon to baryon weak transition matrix element
$\langle\mathcal{B}_{c}|\bar{c}\gamma^{\mu}(1-\gamma^{5})b|\mathcal{B}_{b}\rangle$.
In this section, we briefly introduce how to calculate the matrix element.
Since the constituent quarks are confined in hadron, the matrix element cannot
be calculated by the perturbative QCD. Usually, the matrix element can be
parameterized in terms of a series of dimensionless form factors Chua:2019yqh
, i.e.,

$\begin{split}\langle\mathcal{B}_{c}(&3/2^{-})|\bar{c}\gamma^{\mu}b|\mathcal{B}_{b}(1/2^{+})\rangle\\\
=&\bar{u}_{\alpha}(P^{\prime},J_{z}^{\prime})\bigg{[}g_{1}^{V}(q^{2})g^{\alpha\mu}+g_{2}^{V}(q^{2})\frac{P^{\alpha}}{M}\gamma^{\mu}+g_{3}^{V}(q^{2})\frac{P^{\alpha}P^{\prime\mu}}{MM^{\prime}}\\\
&+g_{4}^{V}(q^{2})\frac{P^{\alpha}P^{\mu}}{M^{2}}\bigg{]}u(P,J_{z}),\end{split}$
(2.1)
$\begin{split}\langle\mathcal{B}_{c}(&3/2^{-})|\bar{c}\gamma^{\mu}\gamma^{5}b|\mathcal{B}_{b}(1/2^{+})\rangle\\\
=&\bar{u}_{\alpha}(P^{\prime},J_{z}^{\prime})\bigg{[}f_{1}^{A}(q^{2})g^{\alpha\mu}+f_{2}^{A}(q^{2})\frac{P^{\alpha}}{M}\gamma^{\mu}+f_{3}^{A}(q^{2})\frac{P^{\alpha}P^{\prime\mu}}{MM^{\prime}}\\\
&+f_{4}^{A}(q^{2})\frac{P^{\alpha}P^{\mu}}{M^{2}}\bigg{]}\gamma^{5}u(P,J_{z}),\end{split}$
(2.2)

where $M$ and $M^{\prime}$ are the masses of the initial bottom baryon and the
final charmed baryon, respectively. $P$ and $P^{\prime}$ are the corresponding
three-momentum, and $J_{z}$ and $J_{z}^{\prime}$ are the third components of
the spins. Here, we ignore the spin quantum number since they are definite.

In this work, we use the standard light-front quark model to calculate the
relevant form factors. The light-front quark model, which was proposed by
Terentev and Berestetsky in a relativistic quark model Terentev:1976jk ;
Berestetsky:1977zk  based on the light front formalism and light front
quantization of QCD, have been widely and successfully used in studying the
weak decay form factors (see Ref. Chang:2019obq  and its references). In this
work, we take the same framework Ke:2019smy ; Ke:2019lcf ; Ke:2021pxk  to
calculate the relevant form factors. In the concrete calculation, there exists
input of the spatial wave function of these discussed baryon states. Usually,
one takes a simple Harmonic Oscillator (SHO) wave function, which must result
in the calculated physical quantities dependent on the $\beta$ value, which is
the parameter in the SHO wave function. For avoiding such problem, we proposed
to directly adopt the numerical spatial wave function by solving the potential
model with the help of the Gaussian expansion method Li:2021qod ; Li:2021kfb .
In analog to Refs. Cheung:1995ub ; Cheng:1996if ; Geng:1997ws ; Cheng:2004cc ;
Wang:2017mqp ; Ke:2019smy ; Ke:2019lcf ; Geng:2022xpn , the vertex function of
a single heavy flavor baryon $\mathcal{B}_{Q}$ with spin $J$ and momentum $P$
can be written as

$\begin{split}|\mathcal{B}_{Q}(P,J&,J_{z})\rangle=\int\frac{d^{3}\tilde{p}_{1}}{2(2\pi)^{3}}\frac{d^{3}\tilde{p}_{2}}{2(2\pi)^{3}}\frac{d^{3}\tilde{p}_{3}}{2(2\pi)^{3}}2(2\pi)^{3}\\\
&\times\sum_{\lambda_{1},\lambda_{2},\lambda_{3}}\Psi^{J,J_{z}}(\tilde{p}_{i},\lambda_{i})C^{\alpha\beta\gamma}\delta^{3}(\tilde{P}-\tilde{p}_{1}-\tilde{p}_{2}-\tilde{p}_{3})\\\
&\times~{}F_{q_{1}q_{2}Q}~{}|q_{1\alpha}(\tilde{p}_{1},\lambda_{1})\rangle~{}|q_{2\beta}(\tilde{p}_{2},\lambda_{2})\rangle~{}|Q_{\gamma}(\tilde{p}_{3},\lambda_{3})\rangle,\end{split}$
(2.3)

where $C^{\alpha\beta\gamma}$ and $F_{q_{1}q_{2}q_{3}}$ represent the color
and flavor factors, respectively, 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

$\tilde{p}_{i}=(p_{i}^{+},\vec{p}_{i\bot}),\quad
p_{i}^{+}=p_{i}^{0}+p_{i}^{3},\quad\vec{p}_{i\bot}=(p_{i}^{1},p_{i}^{2}).$
(2.4)

For describing the motions of the constituents, we should introduce the
intrinsic variables $(x_{i},~{}\vec{k}_{i})$ ($i=1,2,3$)

$p_{i}^{+}=x_{i}P^{+},~{}~{}\vec{p}_{i\bot}=x_{i}\vec{P}_{i\bot}+\vec{k}_{i\bot},~{}~{}\sum_{i=1}^{3}\vec{k}_{i\bot}=0,~{}~{}\sum_{i=1}^{3}x_{i}=1,$
(2.5)

where $x_{i}$ are the light-front momentum fractions constrained by
$0<x_{i}<1$.

In this work, the spin-spatial wave functions for anti-triplet single heavy
baryon $\mathcal{B}_{Q}(\bar{3}_{f},J^{P}=1/2^{+})$ and
$\mathcal{B}_{Q}(\bar{3}_{f},J^{P}=3/2^{-})$ are written as Korner:1994nh ;
Hussain:1995xs ; Tawfiq:1998nk

$\begin{split}\Psi^{1/2,J_{z}}(\tilde{p}_{i},\lambda_{i})=&A_{0}\bar{u}(p_{1},\lambda_{1})[(\not{P}+M_{0})\gamma^{5}]v(p_{2},\lambda_{2})\\\
&\times\bar{u}_{Q}(p_{3},\lambda_{3})u(P,J_{z})\psi(x_{i},\vec{k}_{i}),\\\
\Psi^{3/2,J_{z}}(\tilde{p}_{i},\lambda_{i})=&B_{0}\bar{u}(p_{1},\lambda_{1})[(\not{P}+M_{0})\gamma^{5}]v(p_{2},\lambda_{2})\\\
&\times\bar{u}_{Q}(p_{3},\lambda_{3})K^{\alpha}u_{\alpha}(P,J_{z})\psi(x_{i},\vec{k}_{i}),\end{split}$
(2.6)

respectively.

As the fundamental inputs, the spatial wave functions $\psi$ should be
discussed here. Usually, the single heavy flavor baryon is regarded as a quasi
two-body bound state of the light quark cluster with heavy quark ($b\
(\text{or}\ c)$) to form the $\rho$-mode excitation. The spatial wave function
of a single heavy baryon can be written as Ke:2019smy ; Ke:2019lcf ;
Ke:2021pxk

$\begin{split}\psi(x_{i},\vec{k}_{i})=&N_{\psi}\sqrt{\frac{e_{1}e_{2}e_{3}}{x_{1}x_{2}x_{3}M_{0}}}\phi_{\rho}\Big{(}\frac{m_{1}\vec{k}_{2}-m_{2}\vec{k}_{1}}{m_{1}+m_{2}}\Big{)}\\\
&\times\phi_{\lambda}\Big{(}\frac{(m_{1}+m_{2})\vec{k}_{3}-m_{3}(\vec{k}_{1}+\vec{k}_{2})}{m_{1}+m_{2}+m_{3}}\Big{)},\end{split}$
(2.7)

where $\vec{k}_{i}=(\vec{k}_{i\bot},k_{iz})$ with Ke:2019smy

$k_{iz}=\frac{x_{i}M_{0}}{2}-\frac{m_{i}^{2}+\vec{k}_{i\bot}^{2}}{2x_{i}M_{0}}.$
(2.8)

The $\phi_{\rho(\lambda)}$ is the spatial wave function of
$\rho(\lambda)$-mode excitation.

The normalized factors in Eq. (2.6) are expressed as

$\begin{split}A_{0}&=\frac{1}{\sqrt{16P^{+}M_{0}^{3}(e_{1}+m_{1})(e_{2}+m_{2})(e_{3}+m_{3})}},\\\
B_{0}&=\frac{\sqrt{3}}{\sqrt{16P^{+}M_{0}^{3}(e_{1}+m_{1})(e_{2}+m_{2})(e_{3}-m_{3})(e_{3}+m_{3})^{2}}},\end{split}$

where the factor in Eq. (2.7) is $N_{\psi}=(4\pi^{3/2})^{2}$ for the ground
state and $N_{\psi}=(4\pi^{3/2})^{2}/\sqrt{3}$ for the $P$-wave state. These
factors are determined by the following normalizations:

$\begin{split}\sum_{J_{z},J_{z}^{\prime}}&\langle\mathcal{B}_{Q}(P^{\prime},J,J_{z}^{\prime})|\mathcal{B}_{Q}(P,J,J_{z})\rangle=\sum_{J_{z},J_{z}^{\prime}}2(2\pi)^{3}P^{+}\delta^{3}(\tilde{P}-\tilde{P}^{\prime})\delta_{J_{z},J_{z}^{\prime}},\end{split}$
(2.9)

and

$\begin{split}\int&\Bigg{(}\prod_{i=1}^{3}\frac{dx_{i}d^{2}\vec{k}_{i\bot}}{2(2\pi)^{3}}\Bigg{)}2(2\pi)^{3}\delta\Big{(}1-\sum_{i}x_{i}\Big{)}\\\
&\times\delta^{2}\Big{(}\sum_{i}\vec{k}_{i\bot}\Big{)}\psi^{*}(x_{i},\vec{k}_{i})\psi(x_{i},\vec{k}_{i})=1.\end{split}$
(2.10)

With the above vertex wave functions in the framework of LFQM, the general
expression of the weak transition matrix element can be expressed as

$\begin{split}\langle\mathcal{B}_{c}(P^{\prime},J_{z}^{\prime})|\bar{c}\Gamma^{\mu}_{i}b|\mathcal{B}_{b}(P,J_{z})\rangle=&\int\Big{(}\frac{dx_{1}d^{2}\vec{k}_{1\bot}}{2(2\pi)^{3}}\Big{)}\Big{(}\frac{dx_{2}d^{2}\vec{k}_{2\bot}}{2(2\pi)^{3}}\Big{)}\frac{\psi_{c}^{\ast}(x_{i}^{\prime},\vec{k}_{i\bot}^{\prime})\psi_{b}(x_{i},\vec{k}_{i\bot})}{(16/\sqrt{3})\sqrt{x_{3}x_{3}^{\prime}M_{0}^{3}M_{0}^{\prime
3}}}\\\
&\times\frac{\text{Tr}[(\not{P}^{\prime}-M_{0}^{\prime})\gamma^{5}(\not{p}_{1}+m_{1})(\not{P}+M_{0})\gamma^{5}(\not{p}_{2}-m_{2})]}{\sqrt{(e_{1}+m_{1})(e_{2}+m_{2})(e_{3}+m_{3})(e_{1}^{\prime}+m_{1}^{\prime})(e_{2}^{\prime}+m_{2}^{\prime})(e_{3}^{\prime}-m_{3}^{\prime})(e_{3}^{\prime}+m_{3}^{\prime})^{2}}}\\\
&\times\bar{u}_{\alpha}(P^{\prime},J_{z}^{\prime})K^{\prime\alpha}(\not{p}_{3}^{\prime}+m_{3}^{\prime})\Gamma^{\mu}_{i}(\not{p}_{3}+m_{3})u(P,J_{z}).\end{split}$
(2.11)

Here, the Lorentz structures is defined as
$\Gamma^{\mu}_{i}=\big{\\{}\gamma^{\mu},\gamma^{\mu}\gamma^{5}\big{\\}}$,
$K^{\prime}=\big{[}(m_{1}^{\prime}+m_{2}^{\prime})p_{3}^{\prime}-m_{3}^{\prime}(p_{1}^{\prime}+p_{2}^{\prime})\big{]}/\big{(}m_{1}^{\prime}+m_{2}^{\prime}+m_{3}^{\prime}\big{)}$
is the $\lambda$-mode momentum of the $P$-wave charmed baryon, and the
$\psi_{b}$ and $\psi_{c}$ are the spatial wave functions of the bottom baryon
and the charmed baryon, respectively.

Next, we should introduce how to extract the form factors by setting the
$q^{+}=0$ and $\vec{q}_{\bot}\neq 0$ condition. To extract the four form
factors of the vector current, we multiply
$\bar{u}(P,J_{z})\Gamma_{i}^{V,\mu\beta}u_{\beta}(P^{\prime},J_{z}^{\prime})$
on both sides of Eq. (2.11) with specifically setting
$\Gamma_{i}^{\mu}=\gamma^{\mu}$, and then sum over the polarizations of the
initial and the final baryons. The left side can be replaced by Eq. (2.1), and
the right side can be calculated by performing the traces and then the
integrations. The Lorentz structures are chosen as
$\Gamma_{i}^{V,\mu\beta}=\big{\\{}g^{\beta\mu},P^{\beta}\gamma^{\mu},P^{\beta}P^{\prime\mu},P^{\beta}P^{\mu}\big{\\}}$
Wang:2022ias ; Zhao:2022vfr . The complete expressions of the form factors of
the vector current are

$\begin{split}g_{1}^{V}(q^{2})=&-\frac{1}{2\tilde{Q}_{+}}G_{1}^{V}(q^{2})-\frac{M_{0}^{\prime}}{2\tilde{Q}_{-}\tilde{Q}_{+}}G_{2}^{V}(q^{2})+\frac{M_{0}^{2}+M_{0}M_{0}^{\prime}+M_{0}^{\prime
2}-q^{2}}{\tilde{Q}_{-}\tilde{Q}_{+}^{2}}G_{3}^{V}(q^{2})-\frac{M_{0}^{\prime
2}}{\tilde{Q}_{-}\tilde{Q}_{+}^{2}}G_{4}^{V}(q^{2}),\\\
g_{2}^{V}(q^{2})=&-\frac{MM_{0}^{\prime}}{2\tilde{Q}_{-}\tilde{Q}_{+}}G_{1}^{V}(q^{2})-\frac{2MM_{0}^{\prime
2}}{\tilde{Q}_{-}^{2}\tilde{Q}_{+}}G_{2}^{V}(q^{2})+\frac{MM_{0}^{\prime}(M_{0}^{2}+4M_{0}M_{0}^{\prime}+M_{0}^{\prime
2}-q^{2})}{\tilde{Q}_{-}^{2}\tilde{Q}_{+}^{2}}G_{3}^{V}(q^{2})+\frac{2MM_{0}^{\prime
3}}{\tilde{Q}_{-}^{2}\tilde{Q}_{+}^{2}}G_{4}^{V}(q^{2}),\\\
g_{3}^{V}(q^{2})=&\frac{MM^{\prime}(M_{0}^{2}+M_{0}M_{0}^{\prime}+M_{0}^{\prime
2}-q^{2})}{\tilde{Q}_{-}\tilde{Q}_{+}^{2}}G_{1}^{V}(q^{2})+\frac{MM^{\prime}M_{0}^{\prime}(M_{0}^{2}+4M_{0}M_{0}^{\prime}+M_{0}^{\prime
2}-q^{2})}{\tilde{Q}_{-}^{2}\tilde{Q}_{+}^{2}}G_{2}^{V}(q^{2})\\\
&-\frac{2MM^{\prime}(M_{0}^{4}+2M_{0}^{3}M_{0}^{\prime}+2M_{0}M_{0}^{\prime}(M_{0}^{\prime
2}-q^{2})+(M_{0}^{\prime 2}-q^{2})^{2}+2M_{0}^{2}(6M_{0}^{\prime
2}-q^{2}))}{\tilde{Q}_{-}^{2}\tilde{Q}_{+}^{3}}G_{3}^{V}(q^{2})\\\
&+\frac{4MM^{\prime}M_{0}^{\prime
2}(2M_{0}^{2}-M_{0}M_{0}^{\prime}+2(M_{0}^{\prime
2}-q^{2}))}{\tilde{Q}_{-}^{2}\tilde{Q}_{+}^{3}}G_{4}^{V}(q^{2}),\\\
g_{4}^{V}(q^{2})=&-\frac{M^{2}M_{0}^{\prime
2}}{\tilde{Q}_{-}\tilde{Q}_{+}^{2}}G_{1}^{V}(q^{2})+\frac{2M^{2}M_{0}^{\prime
3}}{\tilde{Q}_{-}^{2}\tilde{Q}_{+}^{2}}G_{2}^{V}(q^{2})+\frac{4M^{2}M_{0}^{\prime
2}(2M_{0}^{2}-M_{0}M_{0}^{\prime}+2(M_{0}^{\prime
2}-q^{2}))}{\tilde{Q}_{-}^{2}\tilde{Q}_{+}^{3}}G_{3}^{V}(q^{2})-\frac{20M^{2}M_{0}^{\prime
4}}{\tilde{Q}_{-}^{2}\tilde{Q}_{+}^{3}}G_{4}^{V}(q^{2}),\end{split}$ (2.12)

where $M$ and $M^{\prime}$ are the physical masses of the bottom and charmed
baryons, respectively, and $\tilde{Q}_{\pm}=(M_{0}\pm
M_{0}^{\prime})^{2}-q^{2}$ with

$M_{0}^{(\prime)2}=\frac{\vec{k}_{1\bot}^{(\prime)2}+m_{1}^{(\prime)2}}{x_{1}}+\frac{\vec{k}_{2\bot}^{(\prime)2}+m_{2}^{(\prime)2}}{x_{2}}+\frac{\vec{k}_{3\bot}^{(\prime)2}+m_{3}^{(\prime)2}}{x_{3}}$
(2.13)

being the invariant mass square Ke:2019smy . Besides,

$\begin{split}G_{(1,2,3,4)}^{V}(q^{2})=&\int\bigg{(}\frac{dx_{1}d^{2}\vec{k}_{1\bot}}{2(2\pi)^{3}}\bigg{)}\bigg{(}\frac{dx_{2}d^{2}\vec{k}_{2\bot}}{2(2\pi)^{3}}\bigg{)}\frac{\psi_{b}(x_{i},\vec{k}_{i\bot})\psi_{c}^{\ast}(x_{i}^{\prime},\vec{k}^{\prime}_{i\bot})}{\sqrt{x_{3}x_{3}^{\prime}}}A_{0}B_{0}^{\prime}\text{Tr}[\cdots]\\\
&\times\text{Tr}\big{[}(G_{\mathcal{B}_{c}})_{\beta\alpha}K^{\prime\alpha}(\not{p}_{3}^{\prime}+m_{3}^{\prime})\gamma^{\mu}(\not{p}_{3}+m_{3})(\not{P}+M_{0})\Gamma_{(1,2,3,4),\mu}^{V,\beta}\big{]}\end{split}$
(2.14)

with

$\displaystyle A_{0}$ $\displaystyle=$ $\displaystyle
1\big{/}{\sqrt{16M_{0}^{3}(e_{1}+m_{1})(e_{2}+m_{2})(e_{3}+m_{3})}},$ (2.15)
$\displaystyle B_{0}^{\prime}$ $\displaystyle=$
$\displaystyle\sqrt{3}\big{/}{\sqrt{16M_{0}^{\prime
3}(e_{1}^{\prime}+m_{1}^{\prime})(e_{2}^{\prime}+m_{2}^{\prime})(e_{3}^{\prime}-m_{3}^{\prime})(e_{3}^{\prime}+m_{3}^{\prime})^{2}}},$
(2.16) $\displaystyle\text{Tr}[\cdots]$ $\displaystyle=$
$\displaystyle\text{Tr}[(\not{P}^{\prime}-M_{0}^{\prime})\gamma^{5}(\not{p}_{1}+m_{1})(\not{P}-M_{0})\gamma^{5}(\not{p}_{2}-m_{2})],$
(2.17) $\displaystyle(G_{\mathcal{B}_{c}})^{\mu\nu}$ $\displaystyle=$
$\displaystyle-(\not{P}^{\prime}+M_{0}^{\prime})\Big{[}g^{\mu\nu}-\frac{1}{3}\gamma^{\mu}\gamma^{\nu}-\frac{2}{3M_{0}^{\prime
2}}P^{\prime\mu}P^{\prime\nu}-\frac{1}{3M_{0}^{\prime}}\big{(}\gamma^{\mu}P^{\prime\nu}-\gamma^{\nu}P^{\prime\mu}\big{)}\Big{]}.$
(2.18)

Analogously, the form factors of the axial-vector current can be extracted
with the structures
$\bar{u}(P,J_{z})\Gamma^{A,\mu\beta}_{i}u_{\beta}(P^{\prime},J_{z}^{\prime})$,
where
$\Gamma_{i}^{A,\mu\beta}=\big{\\{}g^{\beta\mu}\gamma^{5},P^{\beta}\gamma^{\mu}\gamma^{5},P^{\beta}P^{\prime\mu}\gamma^{5},P^{\beta}P^{\mu}\gamma^{5}\big{\\}}$
is defined. The complete expressions of the form factors of the axial-vector
current are expressed as

$\begin{split}f_{1}^{A}(q^{2})=&\frac{1}{2\tilde{Q}_{-}}F_{1}^{A}(q^{2})-\frac{M_{0}^{\prime}}{2\tilde{Q}_{-}\tilde{Q}_{+}}F_{2}^{A}(q^{2})-\frac{M_{0}^{2}-M_{0}M_{0}^{\prime}+M_{0}^{\prime
2}-q^{2}}{\tilde{Q}_{-}^{2}\tilde{Q}_{+}}F_{3}^{A}(q^{2})+\frac{M_{0}^{\prime
2}}{\tilde{Q}_{-}^{2}\tilde{Q}_{+}}F_{4}^{A}(q^{2}),\\\
f_{2}^{A}(q^{2})=&\frac{MM_{0}^{\prime}}{2\tilde{Q}_{-}\tilde{Q}_{+}}F_{1}^{A}(q^{2})-\frac{2MM_{0}^{\prime
2}}{\tilde{Q}_{-}\tilde{Q}_{+}^{2}}F_{2}^{A}(q^{2})-\frac{MM_{0}^{\prime}(M_{0}^{2}-4M_{0}M_{0}^{\prime}+M_{0}^{\prime
2}-q^{2})}{\tilde{Q}_{-}^{2}\tilde{Q}_{+}^{2}}F_{3}^{A}(q^{2})-\frac{2MM_{0}^{\prime
3}}{\tilde{Q}_{-}^{2}\tilde{Q}_{+}^{2}}F_{4}^{A}(q^{2}),\\\
f_{3}^{A}(q^{2})=&-\frac{MM^{\prime}(M_{0}^{2}-M_{0}M_{0}^{\prime}+M_{0}^{\prime
2}-q^{2})}{\tilde{Q}_{-}^{2}\tilde{Q}_{+}}F_{1}^{A}(q^{2})+\frac{MM^{\prime}M_{0}^{\prime}(M_{0}^{2}-4M_{0}M_{0}^{\prime}+M_{0}^{\prime
2}-q^{2})}{\tilde{Q}_{-}^{2}\tilde{Q}_{+}^{2}}F_{2}^{A}(q^{2})\\\
&+\frac{2MM^{\prime}(M_{0}^{4}-2M_{0}^{3}M_{0}^{\prime}-2M_{0}M_{0}^{\prime}(M_{0}^{\prime
2}-q^{2})+(M_{0}^{\prime 2}-q^{2})^{2}+2M_{0}^{2}(6M_{0}^{\prime
2}-q^{2}))}{\tilde{Q}_{-}^{3}\tilde{Q}_{+}^{2}}F_{3}^{A}(q^{2})\\\
&-\frac{4MM^{\prime}M_{0}^{\prime
2}(2M_{0}^{2}+M_{0}M_{0}^{\prime}+2(M_{0}^{\prime
2}-q^{2}))}{\tilde{Q}_{-}^{3}\tilde{Q}_{+}^{2}}F_{4}^{A}(q^{2}),\\\
f_{4}^{A}(q^{2})=&\frac{M^{2}M_{0}^{\prime
2}}{\tilde{Q}_{-}^{2}\tilde{Q}_{+}}F_{1}^{A}(q^{2})+\frac{2M^{2}M_{0}^{\prime
3}}{\tilde{Q}_{-}^{2}\tilde{Q}_{+}^{2}}F_{2}^{A}(q^{2})-\frac{4M^{2}M_{0}^{\prime
2}(2M_{0}^{2}+M_{0}M_{0}^{\prime}+2(M_{0}^{\prime
2}-q^{2}))}{\tilde{Q}_{-}^{3}\tilde{Q}_{+}^{2}}F_{3}^{A}(q^{2})+\frac{20M^{2}M_{0}^{\prime
4}}{\tilde{Q}_{-}^{3}\tilde{Q}_{+}^{2}}F_{4}^{A}(q^{2}),\\\ \end{split}$
(2.19)

where

$\begin{split}F_{(1,2,3,4)}^{V}(q^{2})=&\int\bigg{(}\frac{dx_{1}d^{2}\vec{k}_{1\bot}}{2(2\pi)^{3}}\bigg{)}\bigg{(}\frac{dx_{2}d^{2}\vec{k}_{2\bot}}{2(2\pi)^{3}}\bigg{)}\frac{\psi_{b}(x_{i},\vec{k}_{i\bot})\psi_{c}^{\ast}(x_{i}^{\prime},\vec{k}^{\prime}_{i\bot})}{\sqrt{x_{3}x_{3}^{\prime}}}A_{0}B_{0}^{\prime}\text{Tr}[\cdots]\\\
&\times\text{Tr}\big{[}(G_{\mathcal{B}_{c}})_{\beta\alpha}K^{\prime\alpha}(\not{p}_{3}^{\prime}+m_{3}^{\prime})\gamma^{\mu}\gamma^{5}(\not{p}_{3}+m_{3})(\not{P}+M_{0})\Gamma_{(1,2,3,4),\mu}^{A,\beta}\big{]}.\end{split}$
(2.20)

All the traces in Eq. (2.14), Eq. (2.17), and Eq. (2.20) are calculable with
the help of the FEYNCALC program Mertig:1990an ; Shtabovenko:2016sxi ;
Shtabovenko:2020gxv , where the following relations

$\begin{split}P\cdot P&=M_{0}^{2},~{}~{}~{}~{}P^{\prime}\cdot
P^{\prime}=M_{0}^{\prime 2},\\\ P\cdot P^{\prime}&=(M_{0}^{2}+M_{0}^{\prime
2}-q^{2})/2,\\\ p_{1}\cdot P&=e_{1}M_{0},~{}~{}~{}~{}p_{2}\cdot
P=e_{2}M_{0},\\\ p_{1}\cdot
P^{\prime}&=e_{1}^{\prime}M_{0}^{\prime},~{}~{}~{}~{}p_{2}\cdot
P^{\prime}=e_{2}^{\prime}M_{0}^{\prime},\\\ p_{1}\cdot
p_{2}&=(M_{0}^{2}+m_{3}^{2}-m_{1}^{2}-m_{2}^{2}-2e_{3}M_{0})/2,\end{split}$
(2.21)

are used. We also have $p_{i}^{(\prime)2}=m_{i}^{(\prime)2}$ with
$m_{i}^{(\prime)}$ being the mass of corresponding quark. Moreover,
$e_{i}^{(\prime)}$, the energy of $i^{(\prime)}$-th quark, is defined as

$e_{i}^{\prime}=\frac{1}{2}\Big{(}x_{i}^{(\prime)}M_{0}^{(\prime)}+\frac{m_{i}^{(\prime)2}+\vec{k}_{i\bot}^{(\prime)2}}{x_{i}^{(\prime)}M_{0}^{(\prime)}}\Big{)}.$
(2.22)

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

In Refs. Chua:2018lfa ; Ke:2019smy ; Chua:2019yqh ; Ke:2019lcf , the spatial
wave function of baryon is usually treated as a simple harmonic oscillator
form with a phenomenological parameter $\beta$, which results in the $\beta$
dependence of the calculated form factors. For avoiding the $\beta$ dependence
of result, we can take the numerical spatial wave function as input, which is
obtained by solving the three-body Schrödinger equation with the
semirelativistic potential model. So, in the present section, we should
introduce the semirelativistic potential model and the GEM.

In the study of baryon spectroscopy, the baryon wave function and its mass can
be obtained by solving the Schrödinger equation

$\mathcal{H}|\Psi_{\mathbf{J},\mathbf{M_{J}}}\rangle=E|\Psi_{\mathbf{J},\mathbf{M_{J}}}\rangle$
(3.1)

with the Rayleigh-Ritz variational principle, where $\mathcal{H}$ is the
Hamiltonian and $E$ is the corresponding eigenvalue. In this calculation, the
semirelativistic potential, which was given in Ref. Capstick:1985xss , are
applied. The concerned Hamiltonian Capstick:1985xss ; Li:2021qod ; Li:2021kfb

$\mathcal{H}=K+\sum_{i<j}(S_{ij}+G_{ij}+V^{\text{so(s)}}_{ij}+V^{\text{so(v)}}_{ij}+V^{\text{ten}}_{ij}+V^{\text{con}}_{ij})$
(3.2)

includes the kinetic energy $K=\sum_{i=1,2,3}\sqrt{m_{i}^{2}+p_{i}^{2}}$, the
linear confinement term $S_{ij}$:

$\displaystyle S_{ij}$ $\displaystyle=$
$\displaystyle-\frac{3}{4}\left(br_{ij}\left[\frac{e^{-\sigma_{ij}^{2}r_{ij}^{2}}}{\sqrt{\pi}\sigma_{ij}r_{ij}}+\left(1+\frac{1}{2\sigma_{ij}^{2}r_{ij}^{2}}\right)\frac{2}{\sqrt{\pi}}\right.\right.$
(3.3)
$\displaystyle\left.\left.\times\int_{0}^{\sigma_{ij}r_{ij}}e^{-x^{2}}dx\right]\right)\mathbf{F_{i}}\cdot\mathbf{F_{j}}+\frac{c}{3}$

with

$\sigma_{ij}^{2}=\sigma_{0}^{2}\left[\frac{1}{2}+\frac{1}{2}\bigg{(}\frac{4m_{i}m_{j}}{(m_{i}+m_{j})^{2}}\bigg{)}^{4}+s^{2}\bigg{(}\frac{2m_{i}m_{j}}{m_{i}+m_{j}}\bigg{)}^{2}\right],$
(3.4)

the Coulomb-like potential $G_{ij}$:

$G_{ij}=\sum_{k}\frac{\alpha_{k}}{r_{ij}}\left[\frac{2}{\sqrt{\pi}}\int_{0}^{\tau_{k}r_{ij}}e^{-x^{2}}dx\right]\mathbf{F_{i}}\cdot\mathbf{F_{j}},$
(3.5)

the scalar typed spin-orbit interaction $V^{\text{so}(s)}$:

$V^{\text{so}(s)}_{ij}=-\frac{\mathbf{r_{ij}}\times\mathbf{p_{i}}\cdot\mathbf{S_{i}}}{2m_{i}^{2}}\frac{1}{r_{ij}}\frac{\partial
S_{ij}}{\partial
r_{ij}}+\frac{\mathbf{r_{ij}}\times\mathbf{p_{j}}\cdot\mathbf{S_{j}}}{2m_{j}^{2}}\frac{1}{r_{ij}}\frac{\partial
S_{ij}}{\partial r_{ij}},$ (3.6)

the vector typed spin-orbit interaction $V^{\text{so}(v)}$:

$\begin{split}V^{\text{so}(v)}_{ij}=&\frac{\mathbf{r_{ij}}\times\mathbf{p_{i}}\cdot\mathbf{S_{i}}}{2m_{i}^{2}}\frac{1}{r_{ij}}\frac{\partial
G_{ij}}{\partial
r_{ij}}-\frac{\mathbf{r_{ij}}\times\mathbf{p_{j}}\cdot\mathbf{S_{j}}}{2m_{j}^{2}}\frac{1}{r_{ij}}\frac{\partial
G_{ij}}{\partial r_{ij}}\\\
&-\frac{\mathbf{r_{ij}}\times\mathbf{p_{j}}\cdot\mathbf{S_{i}}-\mathbf{r_{ij}}\times\mathbf{p_{i}}\cdot\mathbf{S_{j}}}{m_{i}~{}m_{j}}\frac{1}{r_{ij}}\frac{\partial
G_{ij}}{\partial r_{ij}},\end{split}$ (3.7)

the tensor potential $V^{\text{tens}}$:

$\begin{split}V^{\text{tens}}_{ij}=&-\frac{1}{m_{i}m_{j}}\left[\left(\mathbf{S_{i}}\cdot\mathbf{\hat{r}_{ij}}\right)\left(\mathbf{S_{j}}\cdot\mathbf{\hat{r}_{ij}}\right)-\frac{\mathbf{S_{i}}\cdot\mathbf{S_{j}}}{3}\right]\\\
&\times\left(\frac{\partial^{2}G_{ij}}{\partial r_{ij}^{2}}-\frac{\partial
G_{ij}}{r_{ij}\partial r_{ij}}\right),\end{split}$ (3.8)

and the spin-dependent contact potential $V^{\text{con}}$:

$V^{\text{con}}_{ij}=\frac{2\mathbf{S_{i}}\cdot\mathbf{S_{j}}}{3m_{i}m_{j}}\nabla^{2}G_{ij},$
(3.9)

where $m_{i}$ stands for the mass of constituent quark $i$, and the
$\mathbf{S_{i}}$ is the corresponding spin operator.
$\langle\mathbf{F_{i}}\cdot\mathbf{F_{j}}\rangle=-2/3$ is for quark-quark
interaction Godfrey:1985xj . It is worthy to note that the running coupling
constant $\alpha_{s}$ is defined as Godfrey:1985xj ; Capstick:1985xss

$\alpha_{s}(r)=\sum_{k=1}^{3}\alpha_{k}\frac{2}{\sqrt{\pi}}\int_{0}^{\gamma_{k}r}e^{-x^{2}}dx$
(3.10)

in Eq. (3.5). Here,
$\\{\alpha_{1},\alpha_{2},\alpha_{3}\\}=\\{0.25,0.15,0.20\\}$, and

$\tau_{k}=\frac{\gamma_{k}\sigma_{ij}}{\sqrt{\gamma_{k}^{2}+\sigma_{ij}^{2}}}$
(3.11)

with
$\\{\gamma_{1},\gamma_{2},\gamma_{3}\\}=\\{1/2,\sqrt{10}/2,\sqrt{1000}/2\\}$.
The remaining parameters are collected into Table 1.

For partially compensating relativistic effect in the non-relativistic limit,
the following transformation Godfrey:1985xj ; Capstick:1985xss

$\begin{split}&G_{ij}\to\left(1+\frac{p^{2}}{E_{i}E_{j}}\right)^{1/2}G_{ij}\left(1+\frac{p^{2}}{E_{i}E_{j}}\right)^{1/2},\\\
&\frac{V^{k}_{ij}}{m_{i}m_{j}}\to\left(\frac{m_{i}m_{j}}{E_{i}E_{j}}\right)^{1/2+\epsilon_{k}}\frac{V^{k}_{ij}}{m_{i}m_{j}}\left(\frac{m_{i}m_{j}}{E_{i}E_{j}}\right)^{1/2+\epsilon_{k}}\end{split}$
(3.12)

should be made, where $E_{i}=\sqrt{p^{2}+m_{i}^{2}}$ is the energy of $i$-th
constituent quark, the subscript $k$ are used to distinguish the contributions
from the contact, tensor, vector spin-orbit, and scalar spin-orbit terms, and
the $\epsilon_{k}$ are used to represent the relevant modification parameters.

By fitting the mass spectrum of the single charmed and single bottom baryon,
the model parameters in the semirelativistic potential model are obtained as
collected in Table 1.

Table 1: The parameters adopted in the semirelativistic potential model Li:2022nim . Besides, the quark masses are chosen as $m_{u}=220\ \text{MeV}$, $m_{d}=220\ \text{MeV}$, $m_{s}=419\ \text{MeV}$, $m_{c}=1628\ \text{MeV}$, and $m_{b}=4977\ \text{MeV}$ Godfrey:1985xj ; Capstick:1985xss . Parameters | Values | Parameters | Values
---|---|---|---
$b~{}(\text{GeV}^{2})$ | $0.1466\pm 0.0007$ | $\epsilon^{\text{so}(s)}$ | $0.5000\pm 0.0762$
$c~{}(\text{GeV})$ | $-0.3490\pm 0.0050$ | $\epsilon^{\text{so}(v)}$ | $-0.1637\pm 0.0131$
$\sigma_{0}~{}(\text{GeV})$ | $1.7197\pm 0.0304$ | $\epsilon^{\text{tens}}$ | $-0.3790\pm 0.5011$
$s$ | $0.5278\pm 0.0718$ | $\epsilon^{\text{con}}$ | $-0.1612\pm 0.0015$

The total wave function of a baryon can be written as

$\begin{split}\Psi_{\mathbf{J},\mathbf{M_{J}}}=&\sum_{\alpha}C^{(\alpha)}\Psi_{\mathbf{J},\mathbf{M_{J}}}^{(\alpha)},\\\
\Psi_{\mathbf{J},\mathbf{M_{J}}}^{(\alpha)}=&\chi^{\text{color}}\left\\{{\chi^{\text{spin}}}_{\mathbf{S},\mathbf{M_{S}}}\psi^{\text{partial}}_{\mathbf{L},\mathbf{M_{L}}}\right\\}_{\mathbf{J},\mathbf{M_{J}}}\psi^{\text{flavor}},\end{split}$
(3.13)

which is composed of color, spin, spatial, and flavor terms, where
$C^{(\alpha)}$ denotes the coefficient with $\alpha$ being the possible
quantum number. The color wave function $\chi^{\text{color}}=(rgb-rbg+gbr-
grb+brg-bgr)/\sqrt{6}$ is universal for any baryons. In the SU(2) flavor
symmetry, the flavor wave function is expressed as
$\psi_{\Lambda_{Q}}^{\text{flavor}}=(ud-du)Q/\sqrt{2}$ for the
$\Lambda_{Q}$-typed baryon, while the flavor wave function is
$\psi_{\Xi_{Q}}^{\text{flavor}}=(ns-sn)Q/\sqrt{2}$ with $n=u\ (\text{or}\ d)$
and $Q=b\ (\text{or}\ c)$ for a $\Xi_{Q}$-typed baryon. The subscripts S and L
represent the total spin and total orbital angular momentum, respectively. And
$\psi^{\text{spatial}}_{\mathbf{L},\mathbf{M_{L}}}$ is the spatial wave
function of $\rho$-mode and $\lambda$-mode excitation

$\psi^{\text{spatial}}_{\mathbf{L},\mathbf{M_{L}}}=\left\\{\phi_{\boldsymbol{l_{\rho}},\boldsymbol{ml_{\rho}}}\phi_{\boldsymbol{l_{\lambda}},\boldsymbol{ml_{\lambda}}}\right\\}_{\mathbf{L},\mathbf{M_{L}}},$
(3.14)

where the subscripts $\boldsymbol{l_{\rho}}$ and $\boldsymbol{l_{\lambda}}$
are the orbital angular momentum for the $\rho$ and $\lambda$-mode excitation,
respectively. The single heavy baryon can be regarded as a bound state of
light quark cluster and heavy quark. Here, the $\rho$-mode indicates the
radial excitation between two light quarks, while the $\lambda$-mode stands
for the redial excitation between the light quark cluster and heavy quark. For
the concerned bottom and charmed baryons, the internal Jacobi coordinates can
be chosen as

$\vec{\rho}=\vec{r}_{2}-\vec{r}_{1},~{}~{}~{}\vec{\lambda}=\vec{r}_{3}-\frac{m_{1}\vec{r}_{1}+m_{2}\vec{r}_{2}}{m_{1}+m_{2}}.$
(3.15)

For easily illustrating this point, we take the $\Lambda_{c}$ resonance as an
example and present the definitions of the $\rho$-mode and $\lambda$-mode as
displayed in Fig. 1.

Figure 1: The definitions of internal Jacobi coordinates $\vec{\rho}$ and
$\vec{\lambda}$ when taking the $\Lambda_{c}$ baryon as an example.

In the realistic calculation, the Gaussian basis Hiyama:2003cu ;
Hiyama:2012sma ; Yoshida:2015tia

$\begin{split}\phi_{nlm}^{G}(\vec{r})=&\phi^{G}_{nl}(r)~{}Y_{lm}(\hat{r})\\\
=&\sqrt{\frac{2^{l+2}(2\nu_{n})^{l+3/2}}{\sqrt{\pi}(2l+1)!!}}\lim_{\varepsilon\rightarrow
0}\frac{1}{(\nu_{n}\varepsilon)^{l}}\sum_{k=1}^{k_{\text{max}}}C_{lm,k}e^{-\nu_{n}(\vec{r}-\varepsilon\vec{D}_{lm,k})^{2}}\end{split}$
(3.16)

is adopted to expand the spatial wave functions
$\phi_{\boldsymbol{l_{\rho}},\boldsymbol{ml_{\rho}}}$ and
$\phi_{\boldsymbol{l_{\lambda}},\boldsymbol{ml_{\lambda}}}$
($n=1,2,\cdots,n_{max}$), where the Gaussian size parameter $\nu_{n}$ can be
settled as a geometric progression Luo:2022cun

$\nu_{n}=1/r^{2}_{n},~{}~{}~{}r_{n}=r_{min}~{}a^{n-1}$ (3.17)

with

$a=\left(\frac{r_{max}}{r_{min}}\right)^{\frac{1}{n_{max}-1}}.$ (3.18)

The Gaussian basis in the momentum space $\phi_{nlm}^{G}(\vec{k})$ can be
obtained by the replacement $\vec{r}\to\vec{k}$ and $\nu_{n}\to 1/(4\nu_{n})$
in Eq. (3.16). In our calculation, the values of $\rho_{min}$ and $\rho_{max}$
are set as $0.2$ fm and $2.0$ fm, respectively, and $n_{\rho_{max}}=6$. In the
meantime, the same Gaussian sized parameters are also applied to the
$\lambda$-mode excitation.

With above preparation, we can calculate the kinematic, the potential, and the
normalize matrix elements as

$\begin{split}T^{\alpha^{\prime},\alpha}=&\langle\Psi_{\mathbf{J},\mathbf{M_{J}}}^{(\alpha^{\prime})}|K|\Psi_{\mathbf{J},\mathbf{M_{J}}}^{(\alpha)}\rangle,\\\
V^{\alpha^{\prime},\alpha}=&\langle\Psi_{\mathbf{J},\mathbf{M_{J}}}^{(\alpha^{\prime})}|V|\Psi_{\mathbf{J},\mathbf{M_{J}}}^{(\alpha)}\rangle,\\\
N^{\alpha^{\prime},\alpha}=&\langle\Psi_{\mathbf{J},\mathbf{M_{J}}}^{(\alpha^{\prime})}|\Psi_{\mathbf{J},\mathbf{M_{J}}}^{(\alpha)}\rangle.\end{split}$
(3.19)

Then, the Schrödinger equation in Eq. (3.1) can be solved by the Rayleigh-Ritz
variational principle as

$\Big{(}T^{\alpha^{\prime},\alpha}+V^{\alpha^{\prime},\alpha}\Big{)}C^{(\alpha)}=EN^{\alpha^{\prime},\alpha}C^{(\alpha)}.$
(3.20)

For clarity, we take the $\Lambda_{c}(2625)$ as an example to illustrate the
detailed matrix element defined in Eq. (3.19). The quantum numbers of
$\Lambda_{c}(2625)$ are
$(\alpha)=(l_{\rho},l_{\lambda},L,S_{\rho},S,J)=(0,1,1,0,1/2,3/2)$. By
expanding the wave function in Eq. (3.13) with $n_{\rho_{\text{max}}}\times
n_{\lambda_{\text{max}}}=6\times 6=36$ Gaussian bases in Eq. (3.16), the
matrix element $T^{\alpha^{\prime},\alpha}$ can be written as

$T^{\alpha^{\prime},\alpha}=\begin{bmatrix}T_{1,1,1,1}&\cdots&\cdots&\cdots\\\
\vdots&\ddots&&\\\
\vdots&&T_{n_{\rho}^{\prime},n_{\lambda}^{\prime},n_{\rho},n_{\lambda}}&\\\
\vdots&&&\ddots\\\ \end{bmatrix}_{36\times 36},$ (3.21)

where

$\begin{split}T_{n_{\rho}^{\prime},n_{\lambda}^{\prime},n_{\rho},n_{\lambda}}=&\langle\chi^{\text{spin}}_{S^{\prime},M_{S}^{\prime}}\Big{\\{}\phi_{n_{\rho}^{\prime}l_{\rho}^{\prime}m_{l_{\rho}}^{\prime}}^{G}(\vec{p}_{\rho})\phi_{n_{\lambda}^{\prime}l_{\lambda}^{\prime}m_{l_{\lambda}}^{\prime}}^{G}(\vec{p}_{\lambda})\Big{\\}}_{L^{\prime},M_{L}^{\prime}}|K|\chi^{\text{spin}}_{S,M_{S}}\\\
&\times\Big{\\{}\phi_{n_{\rho}l_{\rho}m_{l_{\rho}}}^{G}(\vec{p}_{\rho})\phi_{n_{\lambda}l_{\lambda}m_{l_{\lambda}}}^{G}(\vec{p}_{\lambda})\Big{\\}}_{L,M_{L}}\rangle.\end{split}$
(3.22)

Here, we neglect the contributions of the color and flavor wave functions,
since their overlap equals to 1. The matrix elements
$V^{\alpha^{\prime},\alpha}$ and $N^{\alpha^{\prime},\alpha}$ can also be
obtained in similar method.

Now, we can handle the Schrödinger equation to obtain the eigenvectors and
eigenvalues, which correspond to the baryon wave functions and the masses,
respectively. In Table 2, we present our results of the masses and the radial
components of the spatial wave functions of the concerned baryons. It is
obvious that the calculated masses are consistent with the experimental values
ParticleDataGroup:2022pth . It also indicates that we can well reproduce the
charmed and bottom baryon spectrum by the adopted potential model, and the
obtained numerical wave functions are as input when getting the form factors
of these discussed weak transitions.

Table 2: The comparisons of the masses by our calculations and the PDG values ParticleDataGroup:2022pth , and the radial components of spatial wave functions of the concerned bottomed baryons $\Lambda_{b}$ and $\Xi_{b}$, as well as the $P$-wave charmed baryons $\Lambda_{c}(2625)$ and $\Xi_{c}(2815)$ from the GI model and GEM. The Gaussian bases $(n_{\rho},n_{\lambda})$ listed in the forth column are arranged as $[(1,1),(1,2),\cdots,(1,n_{\lambda_{max}}),(2,1),(2,2),\cdots,(2,n_{\lambda_{max}}),\cdots,(n_{\rho_{max}},1),(n_{\rho_{max}},2),\cdots,(n_{\rho_{max}},n_{\lambda_{max}})]$. For the masses of the $\Xi_{b}$ and $\Xi_{c}(2815)$, the values for the neutral and charged states are degenerated in our calculation since the same mass for the $u$ and $d$ quarks is applied in the potential model. States | This work (GeV) | Experiment (MeV) ParticleDataGroup:2022pth  | Eigenvectors
---|---|---|---
$\Lambda_{b}^{0}$ | $5.621\pm 0.005$ |  $5619.60\pm 0.17$ | $\Big{[}0.0068\pm 0.0007,0.0442\pm 0.0014,0.0732\pm 0.0016,0.0032\pm 0.0003,$
$0.0011\pm 0.0001,-0.0004\pm 0.0000,0.0270\pm 0.0012,0.0204\pm 0.0010,$
$0.0273\pm 0.0022,0.0067\pm 0.0004,-0.0027\pm 0.0001,0.0007\pm 0.0000,$
$-0.017\pm 0.0002,0.2541\pm 0.0058,0.2427\pm 0.0006,0.0005\pm 0.0002,$
$0.0060\pm 0.0001,-0.0017\pm 0.0000,-0.0037\pm 0.0003,-0.0426\pm 0.0010,$
$0.4052\pm 0.0028,0.0253\pm 0.0025,-0.0023\pm 0.0007,0.0004\pm 0.0002,$
$0.0071\pm 0.0001,-0.0052\pm 0.0008,0.0105\pm 0.0008,0.1224\pm 0.0015,$
$-0.0246\pm 0.0001,0.0054\pm 0.0000,-0.0020\pm 0.0000,0.0010\pm 0.0003,$
$-0.0112\pm 0.0003,-0.0139\pm 0.0001,0.0086\pm 0.0001,-0.0017\pm
0.0000\Big{]}$
$\Xi_{b}^{0,-}$ | $5.809\pm 0.004$ |  $5791.9\pm 0.5$ $5797.0\pm 0.6$ | $\Big{[}0.0069\pm 0.0008,0.0293\pm 0.0012,0.0543\pm 0.0016,-0.0002\pm 0.0003,$
$0.0014\pm 0.0001,-0.0004\pm 0.0000,0.0231\pm 0.0013,0.0397\pm 0.0003,$
$0.0278\pm 0.0018,0.0114\pm 0.0003,-0.0037\pm 0.0000,0.0009\pm 0.0000,$
$-0.0093\pm 0.0003,0.2285\pm 0.0053,0.2601\pm 0.0007,-0.0165\pm 0.0004,$
$0.0100\pm 0.0000,-0.0026\pm 0.0000,-0.0043\pm 0.0005,-0.0094\pm 0.0001,$
$0.3992\pm 0.0037,0.0525\pm 0.0026,-0.0092\pm 0.0006,0.0019\pm 0.0001,$
$0.0048\pm 0.0001,-0.0108\pm 0.0005,0.0095\pm 0.0005,0.0813\pm 0.0015,$
$-0.0145\pm 0.0002,0.0033\pm 0.0000,-0.0011\pm 0.0000,0.0011\pm 0.0002,$
$-0.0052\pm 0.0002,-0.0070\pm 0.0001,0.0034\pm 0.0001,-0.0007\pm
0.0000\Big{]}$
$\Lambda_{c}^{+}(2625)$ | $2.623\pm 0.007$ |  $2628.11\pm 0.19$ | $\Big{[}0.0012\pm 0.0001,0.0148\pm 0.0007,0.0760\pm 0.0021,0.0359\pm 0.0004,$
$-0.0044\pm 0.0001,0.0010\pm 0.0000,0.0066\pm 0.0003,0.0059\pm 0.0002,$
$0.0376\pm 0.0018,0.0183\pm 0.0015,-0.0034\pm 0.0002,0.0008\pm 0.0000,$
$-0.0027\pm 0.0000,0.0767\pm 0.0022,0.2861\pm 0.0039,0.1060\pm 0.0001,$
$-0.0126\pm 0.0001,0.0027\pm 0.0000,0.0031\pm 0.0002,-0.0383\pm 0.0002,$
$0.2926\pm 0.0013,0.2054\pm 0.0037,-0.0346\pm 0.0004,0.0082\pm 0.0001,$
$0.0028\pm 0.0001,-0.0030\pm 0.0001,-0.0008\pm 0.0012,0.1395\pm 0.0009,$
$-0.0074\pm 0.0003,0.0018\pm 0.0001,-0.0010\pm 0.0000,0.0017\pm 0.0000,$
$-0.0077\pm 0.0004,-0.0222\pm 0.0001,0.0072\pm 0.0000,-0.0013\pm
0.0000\Big{]}$
$\Xi_{c}^{0,+}(2815)$ | $2.811\pm 0.006$ |  $2819.79\pm 0.30$ $2816.51\pm 0.25$ | $\Big{[}0.0012\pm 0.0001,0.0100\pm 0.0005,0.0553\pm 0.0020,0.0231\pm 0.0005,$
$-0.0029\pm 0.0001,0.0006\pm 0.0000,0.0065\pm 0.0003,0.0119\pm 0.0001,$
$0.0432\pm 0.0013,0.0252\pm 0.0011,-0.0043\pm 0.0001,0.0010\pm 0.0000,$
$-0.0017\pm 0.0000,0.0715\pm 0.0018,0.2859\pm 0.0038,0.0892\pm 0.0000,$
$-0.0110\pm 0.0000,0.0023\pm 0.0000,0.0042\pm 0.0001,-0.0307\pm 0.0000,$
$0.3138\pm 0.0014,0.2328\pm 0.0038,-0.0377\pm 0.0004,0.0089\pm 0.0001,$
$0.0017\pm 0.0000,-0.0036\pm 0.0001,-0.0046\pm 0.0008,0.1014\pm 0.0011,$
$-0.0049\pm 0.0002,0.0012\pm 0.0000,-0.0005\pm 0.0000,0.0011\pm 0.0000,$
$-0.0032\pm 0.0002,-0.0122\pm 0.0000,0.0031\pm 0.0000,-0.0006\pm
0.0000\Big{]}$

##  IV The form factors and weak decays

###  IV.1 The weak transition form factors

In the following, we calculate these involved form factors of the
$\Lambda_{b}\to\Lambda_{c}(2625)$ and $\Xi_{b}\to\Xi_{c}(2815)$ transitions
numerically. The masses of baryons are quoted from the Particle Data Group
(PDG) ParticleDataGroup:2022pth , and the spatial wave functions illustrated
in Sec. III are shown in Table 2.

Eq. (2.14) and Eq. (2.20) are worked in spacelike region ($q^{2}<0$), since we
have set the $q^{+}=0$ condition. We need to extrapolate the obtained form
factors to the timelike region ($q^{2}>0$). To do the extrapolation, we take
advantage of the $z$-series parameterization as

$f(q^{2})=\frac{1}{1-q^{2}/(m_{\text{pole}}^{f})^{2}}\Big{[}a_{0}+a_{1}z^{f}(q^{2})\Big{]},$
(4.1)

where $a_{0}^{f}$ and $a_{1}^{f}$ are free parameters needed to be fitted in
spacelike region, and we have Lellouch:1995yv ; Bourrely:2005hp ;
Bourrely:2008za ; Bharucha:2015bzk ; Huang:2022lfr ; Aliev:2022gxi

$z^{f}(q^{2})=\frac{\sqrt{t_{+}^{f}-q^{2}}-\sqrt{t_{+}^{f}-t_{0}}}{\sqrt{t_{+}^{f}-q^{2}}+\sqrt{t_{+}^{f}-t_{0}}}$
(4.2)

with $t_{\pm}^{f}=(M\pm M^{\prime})^{2}$. The parameter $t_{0}$ is chosen as
Bharucha:2015bzk ; Aliev:2022gxi

$0\leqslant t_{0}=t_{+}\bigg{(}1-\sqrt{1-\frac{t_{-}}{t_{+}}}\bigg{)}\leqslant
t_{-}.$ (4.3)

The pole masses are chosen as $m_{B_{c}}=6.275\ \text{GeV}$
ParticleDataGroup:2022pth  for $g_{(1,3,4)}^{V}$, $m_{B_{c}^{\ast}}=6.338\
\text{GeV}$ Godfrey:2004ya  for $g_{2}^{V}$, $m_{B_{c0}}=6.706\ \text{GeV}$
Godfrey:2004ya  for $f_{(1,3,4)}^{A}$, and $m_{B_{c1}}=6.741\ \text{GeV}$
Godfrey:2004ya  for $f_{2}^{A}$. In order to fix the free parameters
$a_{0}^{f}$ and $a_{1}^{f}$, we numerically compute 24 points for each form
factors from $q^{2}=-q_{\text{max}}^{2}$ to $q^{2}=-0.01\ \text{GeV}^{2}$ in
the spacelike region, and then fit them with the MINUIT program. The fitted
parameters are collected in Table 3, and the $q^{2}$ dependence of the form
factors of $\Lambda_{b}\to\Lambda_{c}(2625)$ and $\Xi_{b}\to\Xi_{c}(2815)$
transitions are displayed in Fig. 2.

In Table 3, we also present the $\chi^{2}$ values, which is defined by

$\chi^{2}=\frac{1}{n(n-1)}\sum_{i=1}^{n}\Bigg{(}\frac{f^{cal}(q_{i}^{2})-f^{ana}(q_{i}^{2})}{\delta
f^{cal}(q_{i}^{2})}\Bigg{)}^{2},$ (4.4)

to characterize the analytical continuation, where $n=24$, the $f^{cal}$ and
$f^{ana}$ represent the calculated value by quark model and the value of
analytical continuation. The $\delta f^{cal}$ is the error of $f^{cal}$.
Considering that the z-series parameterization have been widely used to
perform the analytical continuation, and the $\chi^{2}$ value in our fitting
is suitable, in this work we take the $z$-series form to deal with the
parameterization.

Table 3: The fitted parameters for the form factors of the $\Lambda_{b}\to\Lambda_{c}(2625)$ and $\Xi_{b}\to\Xi_{c}(2815)$ transitions in Eq. (4.1). $f$ | $a_{0}$ | $a_{1}$ | $\chi^{2}$ | $f$ | $a_{0}$ | $a_{1}$ | $\chi^{2}$
---|---|---|---|---|---|---|---
$\Lambda_{b}\to\Lambda_{c}(2625)$
$g_{1}^{V}$ | $0.0409\pm 0.0002$ | $-0.3224\pm 0.0053$ | $0.015$ | $f_{1}^{A}$ | $-0.0555\pm 0.0003$ | $0.4855\pm 0.0074$ | $0.026$
$g_{2}^{V}$ | $1.0889\pm 0.0016$ | $-10.5808\pm 0.0459$ | $0.460$ | $f_{2}^{A}$ | $0.7205\pm 0.0005$ | $-6.8173\pm 0.0168$ | $1.240$
$g_{3}^{V}$ | $-0.0763\pm 0.0002$ | $0.8045\pm 0.0046$ | $0.310$ | $f_{3}^{A}$ | $0.1093\pm 0.0003$ | $-1.2327\pm 0.0075$ | $0.363$
$g_{4}^{V}$ | $-0.2360\pm 0.0004$ | $2.5599\pm 0.0127$ | $0.473$ | $f_{4}^{A}$ | $-0.2749\pm 0.0006$ | $3.1018\pm 0.0169$ | $0.455$
$\Xi_{b}\to\Xi_{c}(2815)$
$g_{1}^{V}$ | $0.0397\pm 0.0002$ | $-0.3688\pm 0.0065$ | $0.016$ | $f_{1}^{A}$ | $-0.0549\pm 0.0003$ | $0.5577\pm 0.0091$ | $0.028$
$g_{2}^{V}$ | $1.1750\pm 0.0016$ | $-13.1039\pm 0.0538$ | $0.564$ | $f_{2}^{A}$ | $0.7629\pm 0.0006$ | $-8.3589\pm 0.0203$ | $1.403$
$g_{3}^{V}$ | $-0.0884\pm 0.0002$ | $1.0549\pm 0.0064$ | $0.297$ | $f_{3}^{A}$ | $0.1281\pm 0.0003$ | $-1.6232\pm 0.0103$ | $0.361$
$g_{4}^{V}$ | $-0.2719\pm 0.0004$ | $3.3205\pm 0.0139$ | $0.710$ | $f_{4}^{A}$ | $-0.3167\pm 0.0006$ | $3.9975\pm 0.0189$ | $0.625$
---
Figure 2: The $q^{2}$ dependent form factors of the
$\Lambda_{b}\to\Lambda_{c}(2625)$ (top panels) and $\Xi_{b}\to\Xi_{c}(2815)$
(bottom panels) transitions. Here, the uncertainties are also added. However,
they are not obvious when we present the corresponding results.

In Table 4, we compare our results of the form factors
$g_{(1,2,3,4)}^{V}(q^{2})$ and $f_{(1,2,3,4)}^{A}(q^{2})$ at the $q^{2}=0$ and
$q^{2}=q_{\text{max}}^{2}$ endpoints of the $\Lambda_{b}\to\Lambda_{c}(2625)$
transition with the other theoretical predictions by LFQM Chua:2019yqh  and
LQCD Meinel:2021mdj . The results of LQCD are reproduced with TABLE IX in Ref.
Meinel:2021mdj . In particular, the central value $O$ and the corresponding
statistical uncertainty $\sigma_{O,\text{stat}}$ are reproduced by the so-
called “nominal-order” fitting, i.e.,

$\begin{split}f(q^{2})&=F^{f}+A^{f}\big{(}\omega(q^{2})-1\big{)},\\\
\omega(q^{2})&=\frac{M^{2}+M^{\prime 2}-q^{2}}{2MM^{\prime}},\end{split}$
(4.5)

and the systematic uncertainty can be obtained by

$\sigma_{O,\text{syst}}=\text{max}\Big{(}|O_{\text{HO}}-O|,\sqrt{|\sigma_{O,\text{HO},\text{stat}}^{2}-\sigma_{O,\text{stat}}^{2}|}\Big{)},$
(4.6)

where $O_{\text{HO}}$ and $\sigma_{O,\text{HO},\text{stat}}$ are the central
value and the corresponding statistical uncertainty in the “higher-order”
fitting:

$f_{\text{HO}}(q^{2})=F_{\text{HO}}^{f}+A_{\text{HO}}^{f}\big{(}\omega(q^{2})-1\big{)}.$
(4.7)

Finally, the total uncertainty can be obtained by adding the systematic and
statistical uncertainties in quadrature as

$\sigma_{O,\text{total}}=\sqrt{\sigma_{O,\text{syst}}^{2}-\sigma_{O,\text{stat}}^{2}}.$
(4.8)

The definitions of the form factors used in LQCD Meinel:2021rbm ;
Meinel:2021mdj  can be converted into the present forms by the relations in
the Appendix 2 of Ref. Meinel:2021rbm  combined with

$\begin{split}g_{1}^{V}=&F_{1}^{V},~{}~{}~{}g_{2}^{V}=F_{2}^{V},~{}~{}~{}g_{3}^{V}=F_{3}^{V}-M^{\prime}/MF_{4}^{V},~{}~{}~{}g_{4}^{V}=F_{4}^{V},\\\
f_{1}^{A}=&F_{1}^{A},~{}~{}~{}f_{2}^{A}=F_{2}^{A},~{}~{}~{}f_{3}^{A}=F_{3}^{A}-M^{\prime}/MF_{4}^{A},~{}~{}~{}f_{4}^{A}=F_{4}^{A}.\end{split}$
(4.9)

We emphasize that only the $q^{2}_{\text{max}}$ endpoint values are presented,
since the LQCD’s results are limited to small kinematic region near
$q_{\text{max}}^{2}$. Our results of $g_{1,2}^{V}(q_{\text{max}}^{2})$ and
$f_{1,2}^{A}(q_{\text{max}}^{2})$ are comparable with the LQCD’s results,
while others show some deviations. We expect more theoretical works on these
form factors to further enrich our knowledge on these weak decays.

Table 4: The theoretical predictions for the form factors $g_{(1,2,3,4)}^{V}(q^{2})$ and $f_{(1,2,3,4)}^{A}(q^{2})$ at $q^{2}=0$ and $q^{2}=q_{\text{max}}^{2}$ endpoints of the $\Lambda_{b}\to\Lambda_{c}(2625)$ transition using different approaches. | $g_{1}^{V}(0)$ | $g_{2}^{V}(0)$ | $g_{3}^{V}(0)$ | $g_{4}^{V}(0)$
---|---|---|---|---
This Work | $0.0352\pm 0.0002$ | $0.9023\pm 0.0018$ | $-0.0621\pm 0.0002$ | $-0.1909\pm 0.0005$
LFQM Chua:2019yqh  | $-0.007^{+0.037}_{-0.026}$ | $0.509^{+0.184}_{-0.173}$ | $0.088^{+0.039}_{-0.043}$ | $0.004^{+0.058}_{-0.053}$
| $f_{1}^{A}(0)$ | $f_{2}^{A}(0)$ | $f_{3}^{A}(0)$ | $f_{4}^{A}(0)$
This Work | $-0.0469\pm 0.0003$ | $0.6003\pm 0.0006$ | $0.0876\pm 0.0003$ | $-0.2202\pm 0.0007$
LFQM Chua:2019yqh  | $0.028^{+0.065}_{-0.032}$ | $0.545^{+0.111}_{-0.104}$ | $0.022^{+0.033}_{-0.091}$ | $-0.005^{+0.104}_{-0.068}$
| $g_{1}^{V}(q_{\text{max}}^{2})$ | $g_{2}^{V}(q_{\text{max}}^{2})$ | $g_{3}^{V}(q_{\text{max}}^{2})$ | $g_{4}^{V}(q_{\text{max}}^{2})$
This Work | $0.0603\pm 0.0003$ | $1.6412\pm 0.0023$ | $-0.1171\pm 0.0003$ | $-0.3639\pm 0.0006$
LFQM Chua:2019yqh  | $-0.009^{+0.046}_{-0.033}$ | $0.737^{+0.267}_{-0.251}$ | $0.115^{+0.051}_{-0.056}$ | $0.005^{+0.072}_{-0.066}$
LQCD Meinel:2021mdj  | $0.0692\pm 0.0045$ | $1.1340\pm 0.1556$ | $-0.7977\pm 0.3646$ | $0.2117\pm 0.2795$
| $f_{1}^{A}(q_{\text{max}}^{2})$ | $f_{2}^{A}(q_{\text{max}}^{2})$ | $f_{3}^{A}(q_{\text{max}}^{2})$ | $f_{4}^{A}(q_{\text{max}}^{2})$
This Work | $-0.0800\pm 0.0004$ | $1.0470\pm 0.0007$ | $0.1636\pm 0.0004$ | $-0.4115\pm 0.0008$
LFQM Chua:2019yqh  | $0.035^{+0.082}_{-0.040}$ | $0.756^{+0.154}_{-0.144}$ | $0.027^{+0.041}_{-0.114}$ | $-0.006^{+0.131}_{-0.086}$
LQCD Meinel:2021mdj  | $-0.0660\pm 0.0280$ | $0.8310\pm 0.0978$ | $1.3386\pm 3.4803$ | $0.1795\pm 2.9081$

Besides, by Heavy Quark Effective Theory (HQET), one can rewrite the weak
transition matrix element of the concerned
$\mathcal{B}_{b}(\bar{3}_{f},1/2^{+})\to\mathcal{B}_{c}(\bar{3}_{f},3/2^{-})$
transition as Chua:2019yqh

$\langle\mathcal{B}_{c}(v^{\prime})|j_{V-A}^{\mu}|\mathcal{B}_{b}(v)\rangle=-\sigma(\omega)\bar{u}_{\alpha}(v^{\prime})v^{\alpha}\gamma^{\mu}(1-\gamma^{5})u(v),$
(4.10)

where $v=P/M$ and $v^{\prime}=P^{\prime}/M^{\prime}$ are the velocities of the
initial and the final baryons, respectively. Thus, the form factors have
simpler behavior Chua:2019yqh

$g_{2}^{V}=f_{2}^{A}=\sigma(\omega),~{}~{}~{}g_{1,2,3}^{V}=f_{1,2,3}^{A}=0.$
(4.11)

As shown in Fig. 2, obviously our results of $g_{2}^{V}$ and $f_{2}^{A}$ are
apparently larger than those of $g_{(1,3,4)}^{V}$ and $f_{(1,3,4)}^{A}$, which
is consistent with the expectation from HQET.

###  IV.2 The semileptonic decays

In this section, we further calculate the semileptonic decays
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}(2625)\ell^{-}\nu_{\ell}$ and
$\Xi_{b}^{0,-}\to\Xi_{c}(2815)^{+,0}\ell^{-}\nu_{\ell}$
$(\ell^{-}=e^{-},\mu^{-},\tau^{-})$. The differential decay width of the
semileptonic decay can be obtained by

$\begin{split}\frac{d^{2}\Gamma}{dq^{2}d\cos\theta_{\ell}}=&\Big{|}\frac{G_{F}}{\sqrt{2}}V_{cb}\Big{|}^{2}\frac{\sqrt{Q_{+}Q_{-}}q^{2}(1-\hat{m_{\ell}}^{2})^{2}}{512\pi^{3}M^{3}}\\\
&\times\Big{(}L_{1}+L_{2}\cos\theta_{\ell}+L_{3}\cos
2\theta_{\ell}\Big{)},\end{split}$ (4.12)

where $Q_{\pm}=(M\pm M^{\prime})^{2}-q^{2}$,
$\hat{m_{\ell}}^{2}=m_{\ell}^{2}/q^{2}$ and the angular coefficients $L_{1}$,
$L_{2}$, and $L_{3}$ are given as

$\begin{split}L_{1}=&\frac{1}{2}(3+\hat{m_{\ell}}^{2})\Big{(}H_{-3/2,-1}^{2}+H_{-1/2,-1}^{2}+H_{+1/2,+1}^{2}+H_{+3/2,+1}^{2}\Big{)}\\\
&+(1+\hat{m_{\ell}}^{2})\Big{(}H_{+1/2,0}^{2}+H_{-1/2,0}^{2}\Big{)}\\\
&+2\hat{m_{\ell}}^{2}\Big{(}H_{+1/2,t}^{2}+H_{-1/2,t}^{2}\Big{)},\end{split}$
(4.13)
$\begin{split}L_{2}=&2\Big{(}H_{-3/2,-1}^{2}+H_{-1/2,-1}^{2}-H_{+1/2,+1}^{2}-H_{+3/2,+1}^{2}\Big{)}\\\
&-2\hat{m_{\ell}}^{2}\Big{(}\text{Re}[H_{+1/2,0}^{\dagger}H_{+1/2,t}]+\text{Re}[H_{-1/2,0}^{\dagger}H_{-1/2,t}]\\\
&+\text{Re}[H_{+1/2,0}H_{+1/2,t}^{\dagger}]+\text{Re}[H_{-1/2,0}H_{-1/2,t}^{\dagger}]\Big{)},\end{split}$
(4.14)
$\begin{split}L_{3}=&-(1-\hat{m_{\ell}}^{2})\Big{(}H_{+1/2,0}^{2}+H_{-1/2,0}^{2}\Big{)}\\\
&+\frac{1-\hat{m_{\ell}}^{2}}{2}\Big{(}H_{-3/2,-1}^{2}+H_{-1/2,-1}^{2}+H_{+1/2,+1}^{2}+H_{+3/2,+1}^{2}\Big{)}.\end{split}$
(4.15)

The helicity amplitude $H_{\lambda^{\prime},\lambda_{W}}$ is defined as

$H_{\lambda_{\mathcal{B}_{c}},\lambda_{W}}=\epsilon^{*}_{\mu}(\lambda_{W})\langle\mathcal{B}_{c}(P^{\prime},\lambda_{\mathcal{B}_{c}})|V^{\mu}-A^{\mu}|\mathcal{B}_{b}(P,\lambda_{\mathcal{B}_{b}})\rangle$
(4.16)

with $\lambda$, $\lambda^{\prime}$, and $\lambda_{W}$ denoting the helicities
of the initial state $\mathcal{B}_{b}$, the final state $\mathcal{B}_{c}$, and
the off-shell $W$ boson, respectively. We have the relation
$\lambda=\lambda^{\prime}-\lambda_{W}$. Their concerned expressions are
Gutsche:2017wag

$\begin{split}H^{V}_{1/2,t}=&\sqrt{\frac{2}{3}\frac{Q_{-}}{q^{2}}}\frac{Q_{+}}{2MM^{\prime}}\bigg{(}g_{1}^{V}(q^{2})M+g_{2}^{V}(q^{2})M_{-}\\\
&+g_{3}^{V}(q^{2})\frac{M_{+}M_{-}-q^{2}}{2M^{\prime}}+g_{4}^{V}(q^{2})\frac{M_{+}M_{-}+q^{2}}{2M}\bigg{)},\\\
H^{V}_{1/2,0}=&\sqrt{\frac{2}{3}\frac{Q_{+}}{q^{2}}}\bigg{(}g_{1}^{V}(q^{2})\frac{M_{+}M_{-}-q^{2}}{2M^{\prime}}+g_{2}^{V}(q^{2})\frac{Q_{-}M_{+}}{2MM^{\prime}}\\\
&+g_{3}^{V}(q^{2})\frac{Q_{+}Q_{-}}{4MM^{\prime
2}}+g_{4}^{V}(q^{2})\frac{Q_{+}Q_{-}}{4M^{2}M^{\prime}}\bigg{)},\\\
H^{V}_{1/2,1}=&\sqrt{\frac{Q_{+}}{3}}\bigg{(}g_{1}^{V}(q^{2})-g_{2}^{V}(q^{2})\frac{Q_{-}}{MM^{\prime}}\bigg{)},\\\
H^{V}_{3/2,1}=&\sqrt{Q_{+}}g_{1}^{V}(q^{2}),\end{split}$ (4.17)
$\begin{split}H^{A}_{1/2,t}=&-\sqrt{\frac{2}{3}\frac{Q_{+}}{q^{2}}}\frac{Q_{-}}{2MM^{\prime}}\bigg{(}f_{1}^{A}(q^{2})M-f_{2}^{A}(q^{2})M_{+}\\\
&+f_{3}^{A}(q^{2})\frac{M_{+}M_{-}-q^{2}}{2M^{\prime}}+f_{4}^{A}(q^{2})\frac{M_{+}M_{-}+q^{2}}{2M}\bigg{)},\\\
H^{A}_{1/2,0}=&-\sqrt{\frac{2}{3}\frac{Q_{-}}{q^{2}}}\bigg{(}f_{1}^{A}(q^{2})\frac{M_{+}M_{-}-q^{2}}{2M^{\prime}}-f_{2}^{A}(q^{2})\frac{Q_{+}M_{-}}{2MM^{\prime}}\\\
&+f_{3}^{A}(q^{2})\frac{Q_{+}Q_{-}}{4MM^{\prime
2}}+f_{4}^{A}(q^{2})\frac{Q_{+}Q_{-}}{4M^{2}M^{\prime}}\bigg{)},\\\
H^{A}_{1/2,1}=&\sqrt{\frac{Q_{-}}{3}}\bigg{(}f_{1}^{A}(q^{2})-f_{2}^{A}(q^{2})\frac{Q_{+}}{MM^{\prime}}\bigg{)},\\\
H^{A}_{3/2,1}=&-\sqrt{Q_{-}}f_{1}^{A}(q^{2}),\end{split}$ (4.18)

for the vector current and the axial-vector current, respectively, with
$M_{\pm}=M\pm M^{\prime}$. The negative terms can be obtained by the relations

$H^{V}_{-\lambda^{\prime},-\lambda_{W}}=+H^{V}_{\lambda^{\prime},\lambda_{W}},~{}~{}H^{A}_{-\lambda^{\prime},-\lambda_{W}}=-H^{A}_{\lambda^{\prime},\lambda_{W}},$
(4.19)

and the total helicity amplitudes can be obtained by

$H_{\lambda^{\prime},\lambda_{W}}=H^{V}_{\lambda^{\prime},\lambda_{W}}-H^{A}_{\lambda^{\prime},\lambda_{W}}.$
(4.20)

After performing the integral of the angle $\theta_{\ell}$, the differential
decay width can be obtained by

$\begin{split}\frac{d\Gamma}{dq^{2}}=&\Big{|}\frac{G_{F}}{\sqrt{2}}V_{cb}\Big{|}^{2}\frac{\sqrt{s_{+}s_{-}}q^{2}(1-\hat{m_{\ell}}^{2})^{2}}{512\pi^{3}M^{3}}\Big{(}2L_{1}-\frac{2}{3}L_{3}\Big{)}.\end{split}$
(4.21)

And then, the decay width can be obtained by carrying out the integral of
$q^{2}$ in the range $m_{\ell}^{2}$ to $q^{2}_{\text{max}}$.

---
Figure 3: The differential branching ratios of $\Lambda_{b}^{0}\to\Lambda_{c}^{+}(2625)\ell^{-}\nu_{\ell}$ and $\Xi_{b}^{0}\to\Xi_{c}^{+}(2815)\ell^{-}\nu_{\ell}$ with $\ell^{-}=e^{-},\mu^{-},\text{or}\ \tau^{-}$. Table 5: The comparison of our numerical results and the experimental measurement, as well as other theoretical results of the absolute branching ratios of $\Lambda_{b}^{0}\to\Lambda_{c}^{+}(2625)\ell^{-}\nu_{\ell}$ and $\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}(2815)\ell^{-}\nu_{\ell}$ with $\ell=e,\mu,\tau$, where the branching ratios out of or in brackets in the second column correspond to the $\Xi_{b}^{0}\to\Xi_{c}(2815)^{+}$ and $\Xi_{b}^{-}\to\Xi_{c}(2815)^{0}$ transitions, respectively. Here, all values are given as a percent $(\%)$. Mode | This work | Expt. ParticleDataGroup:2022pth  | CCQM Gutsche:2018nks  | HQSS Nieves:2019kdh  | CQM Pervin:2005ve
---|---|---|---|---|---
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}(2625)e^{-}\nu_{e}$ | $1.653\pm 0.114$ | - | $0.17\pm 0.03$ | - | $(0.88-1.40)$
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}(2625)\mu^{-}\nu_{\mu}$ | $1.641\pm 0.113$ | $1.3^{+0.6}_{-0.5}$ | $0.17\pm 0.03$ | $3.5^{+1.3}_{-1.2}$ | $(0.88-1.40)$
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}(2625)\tau^{-}\nu_{\tau}$ | $0.1688\pm 0.0116$ | - | $0.018\pm 0.004$ | $0.38^{+0.09}_{-0.08}$ | $(0.18-0.22)$
$\Xi_{b}^{0(-)}\to\Xi_{c}^{+(0)}(2815)e^{-}\nu_{e}$ | $1.698\pm 0.122\ (1.803\pm 0.132)$ | - | - | - | -
$\Xi_{b}^{0(-)}\to\Xi_{c}^{+(0)}(2815)\mu^{-}\nu_{\mu}$ | $1.685\pm 0.121\ (1.789\pm 0.131)$ | - | - | - | -
$\Xi_{b}^{0(-)}\to\Xi_{c}^{+(0)}(2815)\tau^{-}\nu_{\tau}$ | $0.1758\pm 0.0126\ (0.1868\pm 0.0137)$ | - | - | - | -

Taking the form factors obtained by the light-front quark model as input, we
calculate the semileptonic decays of the $\Lambda_{b}\to\Lambda_{c}(2625)$ and
$\Xi_{b}\to\Xi_{c}(2815)$ processes. The masses of baryons and leptons are
taken from the PDG ParticleDataGroup:2022pth , and the lifetimes of
$\Lambda_{b}^{0}$ and $\Xi_{b}^{-,0}$ are fixed to be

$\begin{split}\tau_{\Lambda_{b}^{0}}=&(1.471\pm 0.009)\ \text{fs},\\\
\tau_{\Xi_{b}^{-}}=&(1.572\pm 0.040)\ \text{fs},\\\
\tau_{\Xi_{b}^{0}}=&(1.480\pm 0.030)\ \text{fs},\end{split}$

respectively, averaged by the PDG ParticleDataGroup:2022pth . Besides, the
involved CKM matrix element is $V_{cb}=(40.8\pm 1.4)\times 10^{-3}$
ParticleDataGroup:2022pth .

The $q^{2}$ dependence of the differential branching ratios are shown in Fig.
3. Since the ones of $\Xi_{b}^{-}\to\Xi_{c}^{0}(2815)\ell^{-}\nu_{\ell}$ act
similar with the neutral one, we would not display them here. In the meantime,
we also present the branching ratios, and compare our results with the
experimental data and other theoretical results, including the CCQM
Gutsche:2018nks , the HQSS Nieves:2019kdh , and the constituent quark model
(CQM) Pervin:2005ve , in Table 5.

Obviously, our result
$\mathcal{B}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}(2625)\mu^{-}\nu_{\mu})=(1.617\pm
0.003)\%$ is consistent with the current experimental data
$(1.3^{+0.6}_{-0.5})\%$ ParticleDataGroup:2022pth . Moreover, the predicted
branching ratios of the electron and muon channels can reach up to the
magnitude of $1\%$, which are accessible at LHCb. From Table 5, we notice that
our results of $\Lambda_{b}\to\Lambda_{c}(2625)\ell^{-}\nu_{\ell}$ are
consistent with the estimate from the HQSS Nieves:2019kdh  and the CQM
Pervin:2005ve , but are larger than the CCQM results Gutsche:2018nks .
Besides, we also find that there exists difference of our result of
$\mathcal{B}(\Xi_{b}\to\Xi_{c}(2815)\ell^{-}\nu_{\ell})$ and that given by
Ref. Pavao:2017cpt , where the $\Xi_{c}(2815)$ resonance is assumed as the
dynamically effect Pavao:2017cpt . It shows that the branching ratios of these
discussed transitions are dependent on the different structure assignments to
the $\Lambda_{c}(2625)$ and $\Xi_{c}(2815)$. We expect more theoretical
studies and the ongoing experiment to explore them, which will be a crucial
test to our result. What is more important is that different structure
assignments to the $\Lambda_{c}(2625)$ and $\Xi_{c}(2815)$ can be further
distinguished.

Additionally, other important physical observables, including the leptonic
forward-backward asymmetry ($A_{FB}$), the final hadron polarization
($P_{B}$), and the lepton polarization ($P_{\ell}$) are also investigated in
this work. The leptonic forward-backward asymmetry $A_{FB}$ can be obtained by

$\begin{split}A_{FB}(q^{2})=\frac{\Big{(}\int_{0}^{1}-\int_{-1}^{0}\Big{)}d\cos\theta_{\ell}\frac{d^{2}\Gamma}{dq^{2}d\cos\theta_{\ell}}}{\Big{(}\int_{0}^{1}+\int_{-1}^{0}\Big{)}d\cos\theta_{\ell}\frac{d^{2}\Gamma}{dq^{2}d\cos\theta_{\ell}}}=\frac{3L_{2}}{6L_{1}-2L_{3}}.\end{split}$
(4.22)

The final hadron polarization $P_{B}$ is defined as

$\begin{split}P_{B}(q^{2})=\frac{d\Gamma^{\lambda^{\prime}=(+3/2,+1/2)}/dq^{2}-d\Gamma^{\lambda^{\prime}=(-3/2,-1/2)}/dq^{2}}{d\Gamma/dq^{2}}\end{split}$
(4.23)

with $\lambda^{\prime}$ representing the polarization of the final charmed
hadron, and

$\begin{split}\frac{d\Gamma^{\lambda^{\prime}=(+3/2,+1/2)}}{dq^{2}}=&\frac{4}{3}\Bigg{(}(2+\hat{m_{\ell}}^{2})\Big{(}H_{1/2,0}^{2}+H_{1/2,1}^{2}+H_{3/2,1}^{2}\Big{)}\\\
&+3\hat{m_{\ell}}^{2}H_{1/2,t}^{2}\Bigg{)},\end{split}$ (4.24)
$\begin{split}\frac{d\Gamma^{\lambda^{\prime}=(-3/2,-1/2)}}{dq^{2}}=&\frac{4}{3}\Bigg{(}(2+\hat{m_{\ell}}^{2})\Big{(}H_{-1/2,0}^{2}+H_{-1/2,-1}^{2}\\\
&+H_{-3/2,-1}^{2}\Big{)}+3\hat{m_{\ell}}^{2}H_{-1/2,t}^{2}\Bigg{)}.\end{split}$
(4.25)

And the lepton polarization $P_{\ell}$ can be obtained by

$\begin{split}P_{\ell}(q^{2})=\frac{d\Gamma^{\lambda_{\ell}=+1/2}/dq^{2}-d\Gamma^{\lambda_{\ell}=-1/2}/dq^{2}}{d\Gamma/dq^{2}},\end{split}$
(4.26)

with $\lambda_{\ell}$ denoting the polarization of the lepton $\ell^{-}$, and

$\begin{split}\frac{d\Gamma^{\lambda_{\ell}=+1/2}}{dq^{2}}=&\frac{4}{3}\hat{m_{\ell}}^{2}\Bigg{(}H_{+1/2,0}^{2}+H_{-1/2,0}^{2}+H_{1/2,1}^{2}+H_{3/2,1}^{2}\\\
&+H_{-1/2,-1}^{2}+H_{-3/2,-1}^{2}+3H_{1/2,t}^{2}+3H_{-1/2,t}^{2}\Bigg{)},\end{split}$
(4.27)
$\begin{split}\frac{d\Gamma^{\lambda_{\ell}=-1/2}}{dq^{2}}=&\frac{8}{3}\Bigg{(}H_{1/2,0}^{2}+H_{-1/2,0}^{2}+H_{1/2,1}^{2}+H_{3/2,1}^{2}\\\
&+H_{-1/2,-1}^{2}+H_{-3/2,-1}^{2}\Bigg{)}.\end{split}$ (4.28)

Here, we neglect the common term

$\begin{split}\Big{|}\frac{G_{F}}{\sqrt{2}}V_{cb}\Big{|}^{2}\frac{\sqrt{s_{+}s_{-}}q^{2}(1-\hat{m_{\ell}}^{2})^{2}}{512\pi^{3}M^{3}}\end{split}$
(4.29)

for abbreviation.

The $q^{2}$ dependence of the leptonic forward-backward asymmetries
($A_{FB}$), the final hadron polarizations ($P_{B}$), and the lepton
polarizations ($P_{\ell}$) of the concerned semileptonic decays is displayed
in Fig. 4, Fig. 5, and Fig. 6, respectively. The future experimental
measurement of these physical observables may provide valuable information to
these discussed weak decays.

---
Figure 4: The leptonic forward-backward asymmetries ($A_{FB}$) of
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}(2625)\ell^{-}\nu_{\ell}$ and
$\Xi_{b}^{0}\to\Xi_{c}^{+}(2815)\ell^{-}\nu_{\ell}$ with
$\ell^{-}=e^{-},\mu^{-},\text{or}\ \tau^{-}$. Here, the uncertainties are also
added. However, they are not obvious when we present the corresponding
results.
---
Figure 5: The final hadron polarizations ($P_{B}$) of
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}(2625)\ell^{-}\nu_{\ell}$ and
$\Xi_{b}^{0}\to\Xi_{c}^{+}(2815)\ell^{-}\nu_{\ell}$ with
$\ell^{-}=e^{-},\mu^{-},\text{or}\ \tau^{-}$. Here, the uncertainties are also
added. However, they are not obvious when we present the corresponding
results.
---
Figure 6: The lepton polarizations ($P_{\ell}$) of
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}(2625)\ell^{-}\nu_{\ell}$ and
$\Xi_{b}^{0}\to\Xi_{c}^{+}(2815)\ell^{-}\nu_{\ell}$ with
$\ell^{-}=e^{-},\mu^{-},\text{or}\ \tau^{-}$. Here, the uncertainties are also
added. However, they are not obvious when we present the corresponding
results.

Moreover, we are also interested in the ratios of branching fractions

$\displaystyle
R_{\Lambda_{c}(2625)}=\frac{\mathcal{B}(\Lambda_{b}\to\Lambda_{c}(2625)\tau^{-}\nu_{\tau})}{\mathcal{B}(\Lambda_{b}\to\Lambda_{c}(2625)\ell^{-}\nu_{\ell})}\approx
0.10,$ $\displaystyle
R_{\Xi_{c}(2815)}=\frac{\mathcal{B}(\Xi_{b}\to\Xi_{c}(2815)\tau^{-}\nu_{\tau})}{\mathcal{B}(\Xi_{b}\to\Xi_{c}(2815)\ell^{-}\nu_{\ell})}\approx
0.10$

with $\ell^{-}=e^{-}\ \text{or}\ \mu^{-}$, which reflect the LFU. Our result
of $R_{\Lambda_{c}(2625)}$ is also consistent with $0.11\pm 0.02$ estimated by
the CCQM Gutsche:2018nks .

###  IV.3 The color-allowed two-body nonleptonic decays

In this subsection, we further evaluate the color-allowed two-body nonleptonic
decays $\Lambda_{b}^{0}\to\Lambda_{c}^{+}(2625)M^{-}$ and
$\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}(2815)M^{-}$ with $M^{-}$ being a pseudoscalar
meson ($\pi^{-}$, $K^{-}$, $D^{-}$, or $D_{s}^{-}$) or a vector meson
($\rho^{-}$, $K^{*-}$, $D^{*-}$, or $D_{s}^{*-}$). Based on the naïve
factorization assumption, the hadronic transition matrix element can be
factorized into a product of two independent matrix elements

$\begin{split}\langle\mathcal{B}_{c}(P^{\prime},&J_{z}^{\prime})M^{-}|\mathcal{H}_{\text{eff}}|\mathcal{B}_{b}(P,J_{z})\rangle\\\
=&\frac{G_{F}}{\sqrt{2}}V_{cb}V_{qq^{\prime}}^{\ast}\langle
M^{-}|\bar{q}^{\prime}\gamma_{\mu}(1-\gamma_{5})q|0\rangle\\\
&\times\langle\mathcal{B}_{c}(P^{\prime},J_{z}^{\prime})|\bar{c}\gamma^{\mu}(1-\gamma_{5})b|\mathcal{B}_{b}(P,J_{z})\rangle,\end{split}$
(4.30)

where the meson part is determined by a decay parameter as

$\begin{split}\langle
M|\bar{q}^{\prime}&\gamma_{\mu}(1-\gamma_{5})q|0\rangle=\left\\{\begin{array}[]{ll}if_{P}q_{\mu},&M\in\text{pseudoscalar\
meson}\\\ f_{V}\epsilon_{\mu}^{*}m_{V},&M\in\text{vector\
meson}\end{array}.\right.\end{split}$ (4.31)

Frankly speaking, the naïve factorization assumption works well for the color-
allowed dominated decays. However, there exists the case, where the color-
suppressed and penguin dominated processes can not be explained by the naïve
factorization, which may show important nonfactorizable contribution to
nonleptonic decays Zhu:2018jet . As shown in Refs. Lu:2009cm ; Chua:2018lfa ;
Chua:2019yqh , the nonfactorizable contributions in bottom baryon decays are
cosiderable comparing with the factorized ones. But the precise study of
nonfactorizable contributions is beyond the scope of the present work, we
still take the approximation of using the naïve factorization assumption.

In our calculation, the decay constants of these involved pseudoscalar and
vector mesons include Cheng:2003sm ; Chua:2019yqh ; Li:2021kfb

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

in the unit of MeV.

On the other hand, the decay amplitudes of the
$\mathcal{B}_{b}\to\mathcal{B}_{c}P$ and $\mathcal{B}_{b}\to\mathcal{B}_{c}V$
processes can be parameterized as

$\mathcal{A}\big{[}\mathcal{B}_{b}\to\mathcal{B}_{c}P\big{]}=iq_{\mu}\bar{u}^{\mu}(C+D\gamma_{5})u,$
(4.32)
$\begin{split}\mathcal{A}\big{[}\mathcal{B}_{b}\to\mathcal{B}_{c}V\big{]}=&\epsilon^{\ast\mu}\bar{u}^{\nu}\big{[}g_{\nu\mu}(C_{1}+D_{1}\gamma_{5})+q_{\nu}\gamma_{\mu}(C_{2}+D_{2}\gamma_{5})\\\
&+q_{\nu}P_{\mu}(C_{3}+D_{3}\gamma_{5})\big{]}u,\end{split}$ (4.33)

respectively, with $P(V)$ denoting the pseudoscalar (vector) meson, where the
parity-violated and parity-conserved amplitudes are written as

$\begin{split}C=&\frac{G_{F}}{\sqrt{2}}V_{cb}V_{qq^{\prime}}a_{1}f_{P}\bigg{[}g_{1}^{V}(m_{P}^{2})+(M-M^{\prime})\frac{g_{2}^{V}(m_{P}^{2})}{M}\\\
&+\frac{1}{2}(M^{2}-M^{\prime
2}-m_{P}^{2})\big{(}\frac{g_{3}^{V}(m_{P}^{2})}{MM^{\prime}}+\frac{g_{4}^{V}(m_{P}^{2})}{M^{2}}\big{)}-m_{P}^{2}\frac{g_{3}^{V}(m_{P}^{2})}{MM^{\prime}}\bigg{]},\\\
D=&-\frac{G_{F}}{\sqrt{2}}V_{cb}V_{qq^{\prime}}a_{1}f_{P}\bigg{[}f_{1}^{A}(m_{P}^{2})-(M+M^{\prime})\frac{f_{2}^{A}(m_{P}^{2})}{M}\\\
&+\frac{1}{2}(M^{2}-M^{\prime
2}-m_{P}^{2})\big{(}\frac{f_{3}^{A}(m_{P}^{2})}{MM^{\prime}}+\frac{f_{4}^{A}(m_{P}^{2})}{M^{2}}\big{)}-m_{P}^{2}\frac{g_{3}^{V}(m_{P}^{2})}{MM^{\prime}}\bigg{]},\\\
\end{split}$ (4.34)

$\begin{split}C_{1}=&\frac{G_{F}}{\sqrt{2}}V_{cb}V_{qq^{\prime}}a_{1}f_{V}g_{1}^{V}(m_{V}^{2}),\\\
D_{1}=&-\frac{G_{F}}{\sqrt{2}}V_{cb}V_{qq^{\prime}}a_{1}f_{V}f_{1}^{A}(m_{V}^{2}),\\\
C_{2}=&\frac{G_{F}}{\sqrt{2}}V_{cb}V_{qq^{\prime}}a_{1}f_{V}\frac{g_{2}^{V}(m_{V}^{2})}{M},\\\
D_{2}=&-\frac{G_{F}}{\sqrt{2}}V_{cb}V_{qq^{\prime}}a_{1}f_{V}\frac{f_{2}^{A}(m_{V}^{2})}{M},\\\
C_{3}=&\frac{G_{F}}{\sqrt{2}}V_{cb}V_{qq^{\prime}}a_{1}f_{V}\bigg{(}\frac{g_{3}^{V}(m_{V}^{2})}{MM^{\prime}}+\frac{g_{4}^{V}(m_{V}^{2})}{M^{2}}\bigg{)},\\\
D_{3}=&-\frac{G_{F}}{\sqrt{2}}V_{cb}V_{qq^{\prime}}a_{1}f_{V}\bigg{(}\frac{f_{3}^{A}(m_{V}^{2})}{MM^{\prime}}+\frac{f_{4}^{A}(m_{V}^{2})}{M^{2}}\bigg{)}.\end{split}$
(4.35)

The $m_{P}(m_{V})$ is the mass of the emitted pseudoscalar (vector) meson, and
$a_{1}=c_{1}+c_{2}/N\approx 1.018$ Chua:2019yqh . Besides, the CKM matrix
elements are ParticleDataGroup:2022pth

$\begin{split}&V_{cb}=(40.8\pm 1.4)\times 10^{-3},\ V_{ud}=0.97373\pm
0.00031,\\\ &V_{us}=0.2243\pm 0.0008,\ V_{cd}=0.221\pm 0.004,\\\
&V_{cs}=0.975\pm 0.006.\end{split}$

Finally, the decay width and asymmetry parameter can be evaluated by

$\begin{split}\Gamma=&\frac{|\vec{p}_{c}|^{3}}{12\pi}\bigg{[}\frac{(M+M^{\prime})^{2}-m_{P}^{2}}{M^{\prime
2}}|C|^{2}+\frac{(M-M^{\prime})^{2}-m_{P}^{2}}{M^{\prime
2}}|D|^{2}\bigg{]},\\\
\alpha=&-\frac{2\kappa\text{Re}[C^{\ast}D]}{|C|^{2}+\kappa^{2}|D|^{2}}\end{split}$
(4.36)

with $\kappa=|\vec{p}_{c}|/(E^{\prime}+M^{\prime})$, and

$\begin{split}\Gamma=&\frac{|\vec{p}_{c}|}{32\pi
M^{2}}\sum_{\lambda_{V}}\Big{(}|h_{\lambda_{V}+1/2,\lambda_{V};1/2}^{\text{PV}}|^{2}+|h_{\lambda_{V}+1/2,\lambda_{V};1/2}^{\text{PC}}|^{2}\Big{)},\\\
\alpha=&\frac{\sum_{\lambda_{V}}2h_{\lambda_{V}+1/2,\lambda_{V};1/2}^{\text{PV}}h_{\lambda_{V}+1/2,\lambda_{V};1/2}^{\text{PC}}}{\sum_{\lambda_{V}}(|h_{\lambda_{V}+1/2,\lambda_{V};1/2}^{\text{PV}}|^{2}+|h_{\lambda_{V}+1/2,\lambda_{V};1/2}^{\text{PC}}|^{2})}\end{split}$
(4.37)

with

$\begin{split}h_{3/2,1;1/2}^{\text{PV(PC)}}=&\mp\sqrt{2s_{\pm}}C_{1}(D_{1}),\\\
h_{-1/2,-1;1/2}^{\text{PV(PC)}}=&\mp\sqrt{\frac{2s_{\pm}}{3}}\Big{[}C_{1}(D_{1})-\frac{s_{\mp}}{M^{\prime}}C_{2}(D_{2})\Big{]},\\\
h_{1/2,0;1/2}^{\text{PV(PC)}}=&\mp\frac{\sqrt{s_{\pm}}}{2\sqrt{3}M^{\prime}m_{V}}\Big{[}2(M^{2}-M^{\prime
2}-m_{V}^{2})C_{1}(D_{1})\\\ &\pm 2s_{\mp}(M\pm
M^{\prime})C_{2}(D_{2})+s_{+}s_{-}C_{3}(D_{3})\Big{]},\end{split}$ (4.38)

for the cases associated with the pseudoscalar and the vector meson emitted
processes, respectively. The $\vec{p}_{c}$ is the three-momentum of the
daughter baryon (or meson) in the rest frame of the parent baryon, while the
$M(M^{\prime})$ is the mass of parent (daughter) baryon and $E^{\prime}$
denotes the energy of the daughter baryon.

Table 6: The absolute branching ratios and up-down asymmetry parameters of the $\Lambda_{b}^{0}\to\Lambda_{c}^{+}(2625)M^{-}$ decays with $M$ denoting a pseudoscalar or vector meson. We also compare the branching ratios (in the unit of $10^{-3}$) with those given by Ref. Chua:2019yqh  in the fourth column. Mode | $\mathcal{B}\ (\times 10^{-3})$ | $\alpha$ | Ref. Chua:2019yqh
---|---|---|---
$\Lambda_{b}^{0}\to\Lambda_{c}(2625)^{+}\pi^{-}$ | $3.12\pm 0.15$ | $-0.99\pm 0.07$ | $2.40^{+4.09}_{-1.82}$
$\Lambda_{b}^{0}\to\Lambda_{c}(2625)^{+}\rho^{-}$ | $4.25\pm 0.23$ | $-0.88\pm 0.07$ | $4.38^{+6.78}_{-3.17}$
$\Lambda_{b}^{0}\to\Lambda_{c}(2625)^{+}K^{-}$ | $0.232\pm 0.012$ | $-0.99\pm 0.07$ | $0.17^{+0.30}_{-0.13}$
$\Lambda_{b}^{0}\to\Lambda_{c}(2625)^{+}K^{*-}$ | $0.212\pm 0.011$ | $-0.85\pm 0.07$ | $0.22^{+0.33}_{-0.16}$
$\Lambda_{b}^{0}\to\Lambda_{c}(2625)^{+}D^{-}$ | $0.266\pm 0.016$ | $-0.92\pm 0.07$ | $0.13^{+0.22}_{-0.10}$
$\Lambda_{b}^{0}\to\Lambda_{c}(2625)^{+}D^{*-}$ | $0.161\pm 0.007$ | $-0.45\pm 0.05$ | $0.13^{+0.17}_{-0.08}$
$\Lambda_{b}^{0}\to\Lambda_{c}(2625)^{+}D_{s}^{-}$ | $6.60\pm 0.40$ | $-0.90\pm 0.07$ | $2.88^{+4.92}_{-2.16}$
$\Lambda_{b}^{0}\to\Lambda_{c}(2625)^{+}D_{s}^{*-}$ | $3.15\pm 0.13$ | $-0.41\pm 0.04$ | $2.41^{+2.98}_{-1.52}$
Table 7: The absolute branching ratios and up-down asymmetry parameters of the $\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}(2815)M^{-}$ decays with $M$ denoting a pseudoscalar or vector meson, where the branching ratios out of or in brackets correspond to the $\Xi_{b}^{0}\to\Xi_{c}(2815)^{+}$ and $\Xi_{b}^{-}\to\Xi_{c}(2815)^{0}$ transitions, respectively. We also compare the branching ratios (in the unit of $10^{-3}$) with those given by Ref. Chua:2019yqh  in the fourth column. Mode | $\mathcal{B}\ (\times 10^{-3})$ | $\alpha$ | Ref. Chua:2019yqh
---|---|---|---
$\Xi_{b}^{0,-}\to\Xi_{c}(2815)^{+,0}\pi^{-}$ | $3.13\pm 0.17\ (3.33\pm 0.18)$ | $-0.99\pm 0.07$ | $3.32^{+6.08}_{-2.63}\ (3.53^{+6.46}_{-2.80})$
$\Xi_{b}^{0,-}\to\Xi_{c}(2815)^{+,0}\rho^{-}$ | $4.29\pm 0.25\ (4.55\pm 0.27)$ | $-0.88\pm 0.07$ | $6.10^{+9.95}_{-4.55}\ (6.49^{+10.58}_{-4.84})$
$\Xi_{b}^{0,-}\to\Xi_{c}(2815)^{+,0}K^{-}$ | $0.233\pm 0.012\ (0.248\pm 0.014)$ | $-0.99\pm 0.07$ | $0.24^{+0.44}_{-0.19}\ (0.26^{+0.47}_{-0.20})$
$\Xi_{b}^{0,-}\to\Xi_{c}(2815)^{+,0}K^{\ast-}$ | $0.214\pm 0.012\ (0.227\pm 0.013)$ | $-0.85\pm 0.07$ | $0.30^{+0.48}_{-0.22}\ (0.32^{+0.51}_{-0.24})$
$\Xi_{b}^{0,-}\to\Xi_{c}(2815)^{+,0}D^{-}$ | $0.275\pm 0.017\ (0.292\pm 0.019)$ | $-0.92\pm 0.07$ | $0.19^{+0.33}_{-0.14}\ (0.20^{+0.35}_{-0.15})$
$\Xi_{b}^{0,-}\to\Xi_{c}(2815)^{+,0}D^{\ast-}$ | $0.167\pm 0.008\ (0.177\pm 0.009)$ | $-0.45\pm 0.05$ | $0.19^{+0.24}_{-0.12}\ (0.20^{+0.26}_{-0.13})$
$\Xi_{b}^{0,-}\to\Xi_{c}(2815)^{+,0}D_{s}^{-}$ | $6.80\pm 0.40\ (7.30\pm 0.40)$ | $-0.90\pm 0.07$ | $4.34^{+7.54}_{-3.25}\ (4.65^{+8.08}_{-3.48})$
$\Xi_{b}^{0,-}\to\Xi_{c}(2815)^{+,0}D_{s}^{\ast-}$ | $3.27\pm 0.015\ (3.47\pm 0.017)$ | $-0.41\pm 0.04$ | $3.51^{+4.30}_{-2.18}\ (3.74^{+4.58}_{-2.32})$

By substituting our numerical results of the form factors and the decay
parameters into Eq. (4.36) and Eq. (4.37), the branching ratios and asymmetry
parameters can be further obtained, which are collected in Tables 6-7 for the
$\Lambda_{b}^{0}\to\Lambda_{c}(2625)^{+}M^{-}$ and
$\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}(2815)M^{-}$ decays, respectively with emitting
a pseudoscalar meson ($\pi^{-}$, $K^{-}$, $D^{-}$, and $D_{s}^{-}$) or a
vector meson ($\rho^{-}$, $K^{*-}$, $D^{*-}$, and $D_{s}^{*-}$). Our results
show the process emitted a $\pi^{-}$, $\rho^{-}$, or $D_{s}^{(*)-}$ meson has
considerable branching ratio, which is possible to be explored in the future
experiment, like LHCb. As for other processes, the branching ratios are
suppressed by an order of magnitude due to the smaller values of CKM matrix
element.

In experiment, the LHCb Collaboration measured ParticleDataGroup:2022pth ;
LHCb:2011poy

$\begin{split}\mathcal{B}&(\Lambda_{b}\to\Lambda_{c}(2625)\pi^{-},\Lambda_{c}(2625)\to\Lambda_{c}\pi^{+}\pi^{-})\\\
&=(3.3\pm 1.3)\times 10^{-4}.\end{split}$

Based on the narrow-width approximation and
$\mathcal{B}(\Lambda_{c}(2625)\to\Lambda\pi^{+}\pi^{-})\approx 67\%$
ParticleDataGroup:2022pth , we have
$\mathcal{B}(\Lambda_{b}\to\Lambda_{c}(2625)\pi^{-})=(4.9\pm 1.9)\times
10^{-4}$, which is apparently smaller than our result. It should be clarified
by more precise measurement in future.

In addition, we also compare our results with that in Ref. Chua:2019yqh  as
shown in the fourth column of Table 6 and Table 7 for the
$\mathcal{B}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}(2625)M^{-})$ and
$\mathcal{B}(\Xi_{b}^{0,-}\to\Xi_{c}^{+,0}(2815)M^{-})$ decays, respectively.
Our results are consistent with the results in Ref. Chua:2019yqh , but have
smaller uncertainties. It benefits from our improved treatment of the baryon
wave function. By hypothesizing the $\Lambda_{c}(2625)$ and $\Xi_{c}(2815)$ as
the dynamically generated resonances from the vector meson-baryon
interactions, the authors of Refs. Liang:2016ydj ; Pavao:2017cpt  calculated
the $\Lambda_{b}\to\Lambda_{c}(2625)D_{s}^{-}$ and
$\Xi_{b}\to\Xi_{c}(2815)\pi^{-}$ channels. Their results show apparently
smaller widths compared with the results from the present work and Ref.
Chua:2019yqh  based on the $udc$-scheme of the $\Lambda_{c}(2625)$ and
$\Xi_{c}(2815)$ states. So we also expect the LHCb Collaboration to measure
the corresponding $\pi^{-}$ and $D_{s}^{-}$ channels, which not only is useful
to reveal the inner structures of the $\Lambda_{c}(2625)$ and $\Xi_{c}(2815)$,
but also can enrich the observed modes of $b$-decay.

##  V Discussion and conclusion

With the update of High Luminosity Large Hadron Collider and the accumulation
of experimental data, the exploration of the bottom baryon decays into the
$P$-wave excited charmed baryon becomes highlight. In this work, we study the
form factors of the $\Lambda_{b}\to\Lambda_{c}(2625)$ and
$\Xi_{b}\to\Xi_{c}(2815)$ transitions, and further discuss the corresponding
semileptonic decays and color-allowed two-body nonleptonic decays.

As the first step, the weak transition form factors are obtained via three-
body LFQM, where the important inputs, the spatial wave functions of these
concerned baryons, are extracted by solving the Schrödinger equation with the
support of GEM Hiyama:2003cu ; Hiyama:2012sma ; Yoshida:2015tia  and by
adopting a semirelativistic three-body potential model Capstick:1985xss ;
Li:2021qod ; Li:2021kfb ; Li:2022nim . By fitting the mass spectrum of the
single bottom and single charmed baryons, the parameters in semirelativistic
potential model can be fixed. This treatment is different from taking a simple
harmonic oscillator wave function with a phenomenal parameter $\beta$. Thus,
we can avoid the $\beta$ dependence of the result, where the present work is
supported by the baryon spectroscopy. Additionally, these calculated form
factors in this work are comparable with the result from LQCD and consistent
with the expectation from HQET.

With the obtained form factors, we further evaluate the weak decays. For the
semileptonic processes, our result of
$\mathcal{B}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}(2625)\mu^{-}\nu_{\mu})=(1.641\pm
0.113)\%$ is consistent with current experimental data, and the branching
ratios of electron and muon channels can reach up to the magnitude of $1\%$,
which are accessible at the LHCb experiment in future. Besides, other
important physical observables, including the leptonic forward-backward
asymmetry ($A_{FB}$), the final hadron polarization ($P_{B}$), and the lepton
polarization ($P_{\ell}$) are also investigated. As for the nonleptonic
processes, the $\pi^{-}$, $\rho^{-}$, and $D_{s}^{(*)-}$-emitted channels have
considerable widths, and they are worthy to be focused by LHCb.

In this work, our study shows the $\Lambda_{b}\to\Lambda_{c}(2625)$ and
$\Xi_{b}\to\Xi_{c}(2815)$ weak transitions have sizable branching ratios,
which can be accessible at experiment. Especially, we notice that different
theoretical groups gave different results of these discussed transitions by
different theoretical frameworks and different structure assignments to the
$\Lambda_{c}(2625)$ and $\Xi_{c}(2815)$ Pervin:2005ve ; Liang:2016ydj ;
Pavao:2017cpt ; Gutsche:2018nks ; Chua:2019yqh ; Nieves:2019kdh , which can be
tested by future experimental measurement. At present, only the
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}(2625)\mu^{-}\nu_{\mu}$ was measured
ParticleDataGroup:2022pth . Considering the high-luminosity upgrade to LHC,
the LHCb experiment will have enough interest and potential to carry our the
measurement to these discussed weak transitions in this work. Taking this
opportunity, we suggest LHCb to measure these discussed channels, where these
measurements no doubt can be useful to enrich the $b$-decay modes and can of
course be applied to distinguish different structure assignments to the
$\Lambda_{c}(2625)$ and $\Xi_{c}(2815)$ states.

## 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. 12247101, the project for top-notch innovative talents of Gansu province,
and by the Fundamental Research Funds for the Central Universities under Grant
No. lzujbky-2022-it17.

## 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) 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.
  * (9) 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.
  * (10) R. Aaij et al. [LHCb], Evidence for a new structure in the $J/\psi p$ and $J/\psi\bar{p}$ systems in $B_{s}^{0}\to J/\psi p\bar{p}$ decays, Phys. Rev. Lett. 128 (2022) no.6, 062001.
  * (11) 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.
  * (12) S. A. Gottlieb and S. Tamhankar, A Lattice study of $\Lambda_{b}$ semileptonic decay, Nucl. Phys. B Proc. Suppl. 119 (2003), 644-646.
  * (13) W. Detmold, C. Lehner and S. Meinel, $\Lambda_{b}\to p\ell^{-}\bar{\nu}_{\ell}$ and $\Lambda_{b}\to\Lambda_{c}\ell^{-}\bar{\nu}_{\ell}$ form factors from lattice QCD with relativistic heavy quarks, Phys. Rev. D 92 (2015) no.3, 034503.
  * (14) M. Q. Huang, H. Y. Jin, J. G. Korner and C. Liu, Note on the slope parameter of the baryonic $\Lambda_{b}\to\Lambda_{c}$ Isgur-Wise function, Phys. Lett. B 629 (2005), 27-32.
  * (15) 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.
  * (16) K. Azizi and J. Y. Süngü, Semileptonic $\Lambda_{b}\rightarrow\Lambda_{c}{\ell}\bar{\nu}_{\ell}$ Transition in Full QCD, Phys. Rev. D 97 (2018) no.7, 074007.
  * (17) H. H. Duan, Y. L. Liu and M. Q. Huang, Light-cone sum rule analysis of semileptonic decays $\varLambda_{b}^{0}\rightarrow\varLambda_{c}^{+}\ell^{-}{\overline{\nu}}_{\ell}$, Eur. Phys. J. C 82 (2022) no.10, 951.
  * (18) Y. Miao, H. Deng, K. S. Huang, J. Gao and Y. L. Shen, $\Lambda_{b}\to\Lambda_{c}$ form factors from QCD light-cone sum rules*, Chin. Phys. C 46 (2022) no.11, 113107.
  * (19) Z. G. Wang, Analysis of the Isgur-Wise function of the $\Lambda_{b}\to\Lambda_{c}$ transition with light-cone QCD sum rules, [arXiv:0906.4206 [hep-ph]].
  * (20) M. Pervin, W. Roberts and S. Capstick, Semileptonic decays of heavy lambda baryons in a quark model, Phys. Rev. C 72 (2005), 035201.
  * (21) 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.
  * (22) H. W. Ke, X. Q. Li and Z. T. Wei, Diquarks and $\Lambda_{b}\to\Lambda_{c}$ weak decays, Phys. Rev. D 77 (2008), 014020.
  * (23) R. N. Faustov and V. O. Galkin, Semileptonic decays of $\Lambda_{b}$ baryons in the relativistic quark model, Phys. Rev. D 94 (2016) no.7, 073008.
  * (24) T. Gutsche, M. A. Ivanov, J. G. Körner, V. E. Lyubovitskij, P. Santorelli and N. Habyl, Semileptonic decay $\Lambda_{b}\to\Lambda_{c}+\tau^{-}+\bar{\nu_{\tau}}$ in the covariant confined quark model, Phys. Rev. D 91 (2015) no.7, 074001 [erratum: Phys. Rev. D 91 (2015) no.11, 119907].
  * (25) S. Rahmani, H. Hassanabadi and J. Kříž, Nonleptonic and semileptonic ${\Lambda_{b}}\rightarrow{\Lambda_{c}}$ transitions in a potential quark model, Eur. Phys. J. C 80 (2020) no.7, 636.
  * (26) C. K. Chua, Color-allowed bottom baryon to charmed baryon nonleptonic decays, Phys. Rev. D 99 (2019) no.1, 014023.
  * (27) 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.
  * (28) 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.
  * (29) T. Gutsche, M. A. Ivanov, J. G. Körner, V. E. Lyubovitskij, P. Santorelli and C. T. Tran, Analyzing lepton flavor universality in the decays $\Lambda_{b}\to\Lambda_{c}^{(\ast)}(\frac{1}{2}^{\pm},\frac{3}{2}^{-})+\ell\,\bar{\nu}_{\ell}$, Phys. Rev. D 98 (2018) no.5, 053003.
  * (30) 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.
  * (31) 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.
  * (32) Y. S. Li and X. Liu, Restudy of the color-allowed two-body nonleptonic decays of bottom baryons $\Xi_{b}$ and $\Omega_{b}$ supported by hadron spectroscopy, Phys. Rev. D 105 (2022) no.1, 013003.
  * (33) J. Nieves, R. Pavao and S. Sakai, $\Lambda_{b}$ decays into $\Lambda_{c}^{*}\ell\bar{\nu}_{\ell}$ and $\Lambda_{c}^{*}\pi^{-}$ $[\Lambda_{c}^{*}=\Lambda_{c}(2595)$ and $\Lambda_{c}(2625)]$ and heavy quark spin symmetry, Eur. Phys. J. C 79 (2019) no.5, 417.
  * (34) S. Meinel and G. Rendon, $\Lambda_{b}\to\Lambda_{c}^{*}(2595,2625)\ell^{-}\bar{\nu}$form factors from lattice QCD, Phys. Rev. D 103 (2021) no.9, 094516.
  * (35) S. Meinel and G. Rendon, $\Lambda_{c}\to\Lambda^{*}(1520)$ form factors from lattice QCD and improved analysis of the $\Lambda_{b}\to\Lambda^{*}(1520)$ and $\Lambda_{b}\to\Lambda_{c}^{*}(2595,2625)$ form factors, Phys. Rev. D 105 (2022) no.5, 054511.
  * (36) W. H. Liang, M. Bayar and E. Oset, $\Lambda_{b}\to\pi^{-}(D_{s}^{-})\Lambda_{c}(2595),~{}\pi^{-}(D_{s}^{-})\Lambda_{c}(2625)$ decays and $DN,~{}D^{*}N$ molecular components, Eur. Phys. J. C 77 (2017) no.1, 39.
  * (37) W. H. Liang, E. Oset and Z. S. Xie, Semileptonic $\Lambda_{b}\to\bar{\nu}_{l}l\Lambda_{c}(2595)$ and $\Lambda_{b}\to\bar{\nu}_{l}l\Lambda_{c}(2625)$ decays in the molecular picture of $\Lambda_{c}(2595)$ and $\Lambda_{c}(2625)$, Phys. Rev. D 95 (2017) no.1, 014015.
  * (38) R. P. Pavao, W. H. Liang, J. Nieves and E. Oset, Predictions for $\Xi_{b}^{-}\rightarrow\pi^{-}\left(D_{s}^{-}\right)\ \Xi_{c}^{0}(2790)\left(\Xi_{c}^{0}(2815)\right)$ and $\Xi_{b}^{-}\rightarrow\bar{\nu}_{l}l\ \Xi_{c}^{0}(2790)\left(\Xi_{c}^{0}(2815)\right)$, Eur. Phys. J. C 77 (2017) no.4, 265.
  * (39) R. Aaij et al. [LHCb], Study of the $D^{0}p$ amplitude in $\Lambda_{b}^{0}\to D^{0}p\pi^{-}$ decays, JHEP 05, 030 (2017).
  * (40) J. Yelton et al. [Belle], Study of Excited $\Xi_{c}$ States Decaying into $\Xi_{c}^{0}$ and $\Xi_{c}^{+}$ Baryons, Phys. Rev. D 94, no.5, 052011 (2016).
  * (41) B. Chen, X. Liu and A. Zhang, Newly observed $\Lambda_{c}(2860)^{+}$ at LHCb and its \emphD-wave partners $\Lambda_{c}(2880)^{+}$, $\Xi_{c}(3055)^{+}$ and $\Xi_{c}(3080)^{+}$, Phys. Rev. D 95, no.7, 074022 (2017).
  * (42) B. Chen, K. W. Wei, X. Liu and T. Matsuki, Low-lying charmed and charmed-strange baryon states, Eur. Phys. J. C 77, no.3, 154 (2017).
  * (43) S. Capstick and N. Isgur, Baryons in a Relativized Quark Model with Chromodynamics, AIP Conf. Proc. 132 (1985), 267-271.
  * (44) H. X. Chen, W. Chen, Q. Mao, A. Hosaka, X. Liu and S. L. Zhu, P-wave charmed baryons from QCD sum rules, Phys. Rev. D 91 (2015) no.5, 054034.
  * (45) J. J. Guo, P. Yang and A. Zhang, Strong decays of observed $\Lambda_{c}$ baryons in the ${}^{3}P_{0}$ model, Phys. Rev. D 100 (2019) no.1, 014001.
  * (46) G. L. Yu, Z. Y. Li, Z. G. Wang, J. Lu and M. Yan, Systematic analysis of single heavy baryons $\Lambda_{Q}$, $\Sigma_{Q}$ and $\Omega_{Q}$, [arXiv:2206.08128 [hep-ph]].
  * (47) Z. Y. Li, G. L. Yu, Z. G. Wang, J. Lu and J. Z. Gu, Systematic analysis of strange single heavy baryons, [arXiv:2207.04167 [hep-ph]].
  * (48) R. Aaij et al. [LHCb], Measurement of the mass and production rate of $\Xi_{b}^{-}$ baryons, Phys. Rev. D 99 (2019) no.5, 052006.
  * (49) W. Wang, F. S. Yu and Z. X. Zhao, Weak decays of doubly heavy baryons: the $1/2\rightarrow 1/2$ case, Eur. Phys. J. C 77 (2017) no.11, 781.
  * (50) Z. X. Zhao, Weak decays of heavy baryons in the light-front approach, Chin. Phys. C 42 (2018) no.9, 093101.
  * (51) Z. X. Zhao, Weak decays of doubly heavy baryons: the $1/2\rightarrow 3/2$ case, Eur. Phys. J. C 78 (2018) no.9, 756.
  * (52) H. W. Ke, N. Hao and X. Q. Li, $\Sigma_{b}\to\Sigma_{c}^{*}$ weak decays in the light-front quark model with two schemes to deal with the polarization of diquark, J. Phys. G 46 (2019) no.11, 115003.
  * (53) 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.
  * (54) W. Wang and Z. P. Xing, Weak decays of triply heavy baryons in light front approach, Phys. Lett. B 834 (2022), 137402.
  * (55) Z. X. Zhao, Weak decays of triply heavy baryons: the $3/2\to 1/2$ case, [arXiv:2204.00759 [hep-ph]].
  * (56) M. V. Terentev, On the Structure of Wave Functions of Mesons as Bound States of Relativistic Quarks, Sov. J. Nucl. Phys. 24 (1976), 106 ITEP-5-1976.
  * (57) V. B. Berestetsky and M. V. Terentev, Nucleon Form-Factors and Dynamics of the Light Front, Sov. J. Nucl. Phys. 25 (1977), 347-354.
  * (58) Q. Chang, L. T. Wang and X. N. Li, Form factors of $V^{\prime}\to V^{\prime\prime}$ transition within the light-front quark models, JHEP 12 (2019), 102.
  * (59) H. W. Ke, F. Lu, X. H. Liu and X. Q. Li, Study on $\Xi_{cc}\to\Xi_{c}$ and $\Xi_{cc}\to\Xi^{\prime}_{c}$ weak decays in the light-front quark model, Eur. Phys. J. C 80 (2020) no.2, 140.
  * (60) H. W. Ke, Q. Q. Kang, X. H. Liu and X. Q. Li, Weak decays of $\Xi_{c}^{(\prime)}\to\Xi$ in the light-front quark model, Chin. Phys. C 45 (2021) no.11, 113103.
  * (61) H. Y. Cheng, C. K. Chua and C. W. Hwang, Light front approach for heavy pentaquark transitions, Phys. Rev. D 70 (2004), 034007.
  * (62) H. Y. Cheng, C. Y. Cheung and C. W. Hwang, Mesonic form-factors and the Isgur-Wise function on the light front, Phys. Rev. D 55 (1997), 1559-1577.
  * (63) C. Y. Cheung, W. M. Zhang and G. L. Lin, Light front heavy quark effective theory and heavy meson bound states, Phys. Rev. D 52 (1995), 2915-2925.
  * (64) C. Q. Geng, C. C. Lih and W. M. Zhang, Radiative leptonic B decays in the light front model, Phys. Rev. D 57 (1998), 5697-5702.
  * (65) C. Q. Geng, C. W. Liu, Z. Y. Wei and J. Zhang, Weak radiative decays of antitriplet bottomed baryons in light-front quark model, Phys. Rev. D 105 (2022) no.7, 7.
  * (66) J. G. Korner, M. Kramer and D. Pirjol, Heavy baryons, Prog. Part. Nucl. Phys. 33 (1994), 787-868.
  * (67) F. Hussain, J. G. Korner, J. Landgraf and S. Tawfiq, $SU(2N_{f})\otimes O(3)$ light diquark symmetry and current induced heavy baryon transition form-factors, Z. Phys. C 69 (1996), 655-662.
  * (68) 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.
  * (69) R. Mertig, M. Bohm and A. Denner, FEYN CALC: Computer algebraic calculation of Feynman amplitudes, Comput. Phys. Commun. 64 (1991), 345-359.
  * (70) V. Shtabovenko, R. Mertig and F. Orellana, New Developments in FeynCalc 9.0, Comput. Phys. Commun. 207 (2016), 432-444.
  * (71) V. Shtabovenko, R. Mertig and F. Orellana, FeynCalc 9.3: New features and improvements, Comput. Phys. Commun. 256 (2020), 107478.
  * (72) S. Godfrey and N. Isgur, Mesons in a Relativized Quark Model with Chromodynamics, Phys. Rev. D 32 (1985), 189-231.
  * (73) Y. S. Li, S. P. Jin, J. Gao and X. Liu, The angular analysis of $\Lambda_{b}\to\Lambda(1520)(\to N\bar{K})\ell^{+}\ell^{-}$ decay, [arXiv:2210.04640 [hep-ph]].
  * (74) E. Hiyama, Y. Kino and M. Kamimura, Gaussian expansion method for few-body systems, Prog. Part. Nucl. Phys. 51 (2003), 223-307.
  * (75) E. Hiyama, Gaussian expansion method for few-body systems and its applications to atomic and nuclear physics, PTEP 2012 (2012), 01A204.
  * (76) 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.
  * (77) S. Q. Luo, L. S. Geng and X. Liu, Double-charm heptaquark states composed of two charmed mesons and one nucleon, Phys. Rev. D 106 (2022), 014017.
  * (78) R. L. Workman et al. [Particle Data Group], Review of Particle Physics, PTEP 2022 (2022), 083C01.
  * (79) L. Lellouch, Lattice constrained unitarity bounds for anti-$\bar{B}^{0}\to\pi^{+}\ell^{-}\bar{\nu}_{\ell}$ decays, Nucl. Phys. B 479 (1996), 353-391.
  * (80) C. Bourrely and I. Caprini, Bounds on the slope and the curvature of the scalar $K\pi$ form-factor at zero momentum transfer, Nucl. Phys. B 722 (2005), 149-165.
  * (81) C. Bourrely, I. Caprini and L. Lellouch, Model-independent description of $B\to\pi\ell\nu$ decays and a determination of $|V_{cb}|$, Phys. Rev. D 79 (2009), 013008 [erratum: Phys. Rev. D 82 (2010), 099902].
  * (82) A. Bharucha, D. M. Straub and R. Zwicky, $B\to V\ell^{+}\ell^{-}$ in the Standard Model from light-cone sum rules, JHEP 08 (2016), 098.
  * (83) K. S. Huang, W. Liu, Y. L. Shen and F. S. Yu, $\Lambda_{b}\to p,N^{\ast}(1535)$ Form Factors from QCD Light-Cone Sum Rules, [arXiv:2205.06095 [hep-ph]].
  * (84) T. M. Aliev, S. Bilmis and M. Savci, Charmed baryon $\Omega_{c}^{0}\to\Omega^{-}\ell\nu_{\ell}$ and $\Omega_{c}^{0}\to\Omega^{-}\pi^{+}(\rho^{+})$ decays in light cone sum rules, Phys. Rev. D 106 (2022) no.7, 074022.
  * (85) S. Godfrey, Spectroscopy of $B_{c}$ mesons in the relativized quark model, Phys. Rev. D 70 (2004), 054017.
  * (86) T. Gutsche, M. A. Ivanov, J. G. Körner, V. E. Lyubovitskij, V. V. Lyubushkin and P. Santorelli, Theoretical description of the decays $\Lambda_{b}\to\Lambda^{(\ast)}(\frac{1}{2}^{\pm},\frac{3}{2}^{\pm})+J/\psi$, Phys. Rev. D 96 (2017) no.1, 013003.
  * (87) 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.
  * (88) 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.
  * (89) R. Aaij et al. [LHCb], Measurements of the Branching fractions for $B_{(s)}\to D_{(s)}\pi\pi\pi$ and $\Lambda_{b}^{0}\to\Lambda_{c}^{+}\pi\pi\pi$, Phys. Rev. D 84 (2011), 092001 [erratum: Phys. Rev. D 85 (2012), 039904].