# Revisiting semileptonic decays of $\Lambda_{b(c)}$ supported by baryon
spectroscopy

Yu-Shuai Li1,2<EMAIL_ADDRESS>Xiang Liu1,2,3<EMAIL_ADDRESS>Fu-Sheng
Yu3,4,5<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
4School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000,
China
5Center for High Energy Physics, Peking University, Beijing 100871, China

###### Abstract

The semileptonic decays of $\Lambda_{b}\to\Lambda_{c}^{(*)}\ell\nu_{\ell}$ and
$\Lambda_{c}\to\Lambda^{(*)}\ell\nu_{\ell}$ are studied in the light-front
quark model in this work. Instead of the quark-diquark approximation, we use
the three-body wave functions obtained by baryon spectroscopy. The ground
states $(1/2^{+})$, the $\lambda$-mode orbital excited states $(1/2^{-})$, and
the first radial excited states $(1/2^{+})$ of $\Lambda_{(c)}^{(*)}$ are
considered. The discussions are given for the form factors, partial widths,
branching fractions, leptonic forward-backward asymmetries, hadron
polarizations, lepton polarizations, and the lepton flavor universalities. Our
results are useful for the inputs of heavy baryon decays and understanding the
baryon structures, and helpful for experimental measurements.

##  I introduction

In recent years, a lot of progress has been made in understanding of the weak
decays of heavy-flavor baryons, such as the observation of the double-charm
baryon $\Xi_{cc}^{++}$ Aaij:2017ueg ; Aaij:2018gfl , the first measurements of
the absolute branching fractions of $\Lambda_{c}^{+}$ and $\Xi_{c}^{+,0}$
decays Zupanc:2013iki ; Ablikim:2015prg ; Ablikim:2015flg ; Ablikim:2016vqd ;
Ablikim:2018woi ; Li:2018qak , and the evidence of $CP$ violation in
$\Lambda_{b}^{0}\to p\pi^{+}\pi^{-}\pi^{-}$ Aaij:2016cla . The semileptonic
decays play an important role in the understanding of the dynamics of beauty
and charm baryon decays. The simpler dynamics of semileptonic decays compared
to the nonleptonic ones, can help us to search for the new physics beyond the
Standard Model (SM) and study the structures of excited states of baryons.

The lepton flavor universality violation (LFUV) was tested in the recent years
using the ratio of $R(D^{(*)})=\mathcal{B}(B\to
D^{(*)}\tau\nu_{\tau})/\mathcal{B}(B\to D^{(*)}e(\mu)\nu_{e(\mu)})$. The
experimental measurements deviate from the Standard Model predictions by
3.1$\sigma$ Amhis:2019ckw , indicating a hint for new physics. It was pointed
out that the $\Lambda_{b}\to\Lambda_{c}\ell\nu$ decays provide a theoretical
cleaner place to test the LFUV Bernlochner:2018kxh ; Bernlochner:2018bfn . Up
to the order of $\mathcal{O}(\Lambda_{\rm QCD}^{2}/m_{c}^{2})$ in the heavy-
quark effective theory, there are only three Isgur-Wise functions in the
$\Lambda_{b}\to\Lambda_{c}$ transitions, compared to the 10 parameters in
$B\to D^{(*)}$ transitions. Due to the large data collected by LHCb in the
recent past, some excited states involving processes of
$\Lambda_{b}\to\Lambda_{c}^{*}\ell\nu_{\ell}$ can also be measured
Bernlochner:2021vlv . Thus, the
$\Lambda_{b}\to\Lambda_{c}^{(*)}\ell\nu_{\ell}$ decays in the same theoretical
framework deserve to be systematically studied.

The comparison between the semileptonic exclusive decay of
$\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e}$ and the inclusive $\Lambda_{c}^{+}\to
e^{+}X$ decay provides an interesting result. The BESIII measurements show
that the branching fraction of the exclusive process of
$\mathcal{B}(\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e})=(3.63\pm 0.43)\%$
Ablikim:2015prg  dominates the inclusive decay of
$\mathcal{B}(\Lambda_{c}^{+}\to e^{+}X)=(3.95\pm 0.35)\%$ Ablikim:2018woi ,
with a fraction of $(91.9\pm 13.6)\%$. It implies a little room for other
semileptonic processes involving excited final states. This phenomenon is
quite different from the charmed meson decays. For example,
$\mathcal{B}(D^{+}\to\overline{K}^{0}e^{+}\nu_{e})=(8.73\pm 0.10)\%$ is much
smaller than $\mathcal{B}(D^{+}\to e^{+}X)=(16.07\pm 0.30)\%$ Zyla:2020zbs .
Therefore, it is necessary to explore the excited-state involving processes of
$\Lambda_{c}^{+}\to\Lambda^{(*)}e^{+}\nu_{e}$, to solve this problem.

The key issue in the theoretical study of semileptonic decays is to calculate
the weak transition form factors. They are also the important inputs in the
nonleptonic decays, such as in the prediction of the discovery channels of the
doubly charmed baryons Yu:2017zst ; Wang:2017mqp ; Han:2021azw ; Yu:2019lxw .
The current research status on the form factors is reviewed in the next
section. The main difficulty in the calculations is from the three-body
problem in the baryons. A lot of theoretical works are based on the
approximation of the quark-diquark scheme Guo:2005qa ; Zhao:2018zcb ;
Ke:2007tg ; Zhu:2018jet ; Chua:2018lfa ; Chua:2019yqh .

In this work, we calculate the form factors of
$\Lambda_{b}\to\Lambda_{c}^{(*)}\left(1/2^{\pm}\right)$ and
$\Lambda_{c}\to\Lambda^{(*)}\left(1/2^{\pm}\right)$ transitions in the light-
front quark model with a triquark picture. There are some important
improvements. Firstly, the triquark picture is closer to the conventional
baryons’ structures compared to the diquark approximation. Secondly, the
baryons’ wave functions are obtained by the Gaussian expansion method (GEM),
avoiding the effective parameters $\beta$’s which are the major uncertainties
in the previous works Ke:2007tg ; Wang:2017mqp ; Zhu:2018jet ; Ke:2019smy ;
Chua:2019yqh ; Geng:2020fng ; Geng:2020gjh . Finally, the parameters in the
wave functions are fixed by the baryon spectra, since a series of baryons are
involved in this work.

The organization of this paper is as follows. In Sec. II the current research
status on the theoretical calculations of form factors are given. In Sec. III,
the formalisms for the form factors of the related processes are derived in
the framework of light-front quark model. The formulas of the Godfrey-Isgure
(GI) model and the GEM, which are used to obtain the spatial wave functions of
baryons, are illustrated. In Sec. IV we give our numerical results; either the
spatial wave functions for involved baryons or the numerical results of form
factors. Besides, the relevant semileptonic differential decay rates and the
branching fractions are also obtained by using these form factors. We also
made a comparison with other theoretical predictions and experimental data.
The conclusion and discussion are given in the last section.

##  II The present research status

The semileptonic decays of $\Lambda_{Q}$ have been widely studied with various
approaches, which include flavor symmetry, various quark models, the light-
front approach, QCD sum rules, light-cone sum rules, lattice QCD (LQCD) and so
on. In this section, we briefly introduce the present status of the
semileptonic decays of $\Lambda_{Q}$.

In the early 1980s, the evidence of $\Lambda_{c}^{+}$ semileptonic decay
Ballagh:1981yh ; Vella:1982ei  was found. However, the precisions on the
branching ratios were very low in the experiments Klein:1989pu ;
Albrecht:1991bu ; Bergfeld:1994gt . The recent progress on this issue was made
by the BESIII Collaboration Ablikim:2015prg ; Ablikim:2016vqd , where the
first measurement of the absolute branching ratio for
$\Lambda_{c}^{+}\to\Lambda\ell^{+}\nu_{\ell}$ was given

$\begin{split}\mathcal{B}(\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e})&=(3.63\pm
0.38\pm 0.20)\%,\\\
\mathcal{B}(\Lambda_{c}^{+}\to\Lambda\mu^{+}\nu_{\mu})&=(3.49\pm 0.46\pm
0.27)\%.\end{split}$

Obviously, the new measurement can be applied to test the different
theoretical approaches describing the $\Lambda_{c}^{+}$ semileptonic decays.

In 1989, Marcial et al. PerezMarcial:1989yh  predicted the widths and
branching fractions of $\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e}$ by using both
the nonrelativistic quark model and the MIT bag model (MBM). Here, the
predicted branching ratio of $\mathcal{B}(\Lambda_{c}^{+}\to\Lambda
e^{+}\nu_{e})$ is $1.4\sim 4.2$ %. Later, Hussain and Korner Hussain:1990ai
studied the same topic with a relativistic spectator quark model where the
interaction between the spectator quark and the acting quark is ignored. This
treatment was widely used in the study of heavy to heavy transitions, and was
expanded to $c\to s$ sector by them. The estimated branching fraction is
$4.45\%$ Hussain:1990ai . Efimov et al. also focused on
$\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e}$ decay at the same time. Using the
quark confinement model, they predicted $\mathcal{B}(\Lambda_{c}^{+}\to\Lambda
e^{+}\nu_{e})=5.72\%$ Efimov:1991ex . Cheng and Tseng Cheng:1995fe  applied
the nonrelativistic quark model and considered the flavor suppression factor,
and obtained $\mathcal{B}(\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e})=1.44\%$. In
fact, QCDSR is also an effective approach to study the semileptonic decay of
$\Lambda_{c}$. For example, Carvalho et al. estimated
$\mathcal{B}(\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e})=(2.69\pm 0.37)\%$
MarquesdeCarvalho:1999bqs . By including the higher twist contributions in the
light cone sum rule calculation, Liu et al. Liu:2009sn  calculated
$\mathcal{B}(\Lambda_{c}^{+}\to\Lambda l^{+}\nu_{l})$, which is $3.0\%$ or
$2.0\%$ if adopting the Chernyak-Zhitnitsky-type or the Ioffe-type
interpolating current, respectively. In addition, Zhao et al. Zhao:2020mod
also presented the involved form factors and decay rates in QCDSR, where the
results are consistent with the experimental data within errors. Pervin et al.
Pervin:2005ve  performed the calculation in the framework of the consistent
quark model with both nonrelativistic and semirelativistic Hamiltonians. The
obtained decay rate from the semirelativistic Hamiltonian can describe the
experimental data well, while the one from the nonrelativistic Hamiltonian
shows apparent divergence Pervin:2005ve . Migura et al. used a
relativistically covariant constituent quark model with the Bethe-Salpeter
equation and found $\mathcal{B}(\Lambda_{c}^{+}\to\Lambda
e^{+}\nu_{e})=3.19\%$ and
$\mathcal{B}(\Lambda_{c}^{+}\to\Lambda\mu^{+}\nu_{\mu})=2.97\%$ Migura:2006en
.

After measuring the absolute branching fractions of
$\mathcal{B}(\Lambda_{c}^{+}\to\Lambda\ell^{+}\nu_{\ell})$ by BESIII
Ablikim:2015prg ; Ablikim:2016vqd , Faustov and Galkin Faustov:2020thr
investigated the semileptonic decays of $\Lambda_{c}$ within a relativistic
quark model (RQM) based on the quasipotential approach. By taking into account
the relativistic effects, their estimations for
$\mathcal{B}(\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e})$ and
$\mathcal{B}(\Lambda_{c}^{+}\to\Lambda\mu^{+}\nu_{\mu})$ are $3.25\%$ and
$3.14\%$, respectively, which are good comparison with the current
measurements. Besides, Gutsche et al. Gutsche:2015rrt  applied the covariant
confined quark model (CCQM). Apart from the decay rates, they also presented
some detailed results for other physical observables, which are also important
to the study of semileptonic decay. The light-front approach was also widely
used to analyze the semileptonic decays of $\Lambda_{c}$. Zhao Zhao:2018zcb
studied the $\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e}$ process in the light-
front approach by treating the spectator quarks as a diquark system and used
an effective parameter to simulate the baryon wave function. With this
approximation, he obtained $\mathcal{B}(\Lambda_{c}^{+}\to\Lambda
e^{+}\nu_{e})=1.63\%$, which is smaller than the BESIII data. On the contrary,
without considering diquark approximation, Geng et al. Geng:2020fng ;
Geng:2020gjh  estimated $\mathcal{B}(\Lambda_{c}^{+}\to\Lambda
e^{+}\nu_{e})=(3.36\pm 0.87)\%$ and
$\mathcal{B}(\Lambda_{c}^{+}\to\Lambda\mu^{+}\nu_{\mu})=(3.21\pm 0.85)\%$ in
the MIT bag model with the spacial wave functions determined by the baryon
spectroscopy Geng:2020fng  and $\mathcal{B}(\Lambda_{c}^{+}\to\Lambda
e^{+}\nu_{e})=(3.55\pm 1.04)\%$ and
$\mathcal{B}(\Lambda_{c}^{+}\to\Lambda\mu^{+}\nu_{\mu})=(3.40\pm 1.02)\%$ in
the light-front formalism Geng:2020gjh  , which are all consistent with the
BESIII data.

In the following, we continue to introduce the research status of the
semileptonic decays of $\Lambda_{b}$. Firstly, we should show the current
experimental data in the Particle Data Group (PDG) Zyla:2020zbs

$\begin{split}\mathcal{B}(\Lambda_{b}\to\Lambda_{c}^{+}\ell^{-}\nu_{\ell})&=(6.4^{+1.4}_{-1.3})\%,\\\
\mathcal{B}(\Lambda_{b}\to\Lambda_{c}(2595)^{+}\ell^{-}\nu_{\ell})&=(0.79^{+0.40}_{-0.35})\%\end{split}$
(1)

with $\ell^{-}=e^{-}$ or $\mu^{-}$. Obviously, there still needs to be some
improvement in these measurements.

The $\Lambda_{b}\to\Lambda_{c}^{+}\ell^{-}\nu_{\ell}$ semileptonic decays have
been studied for a long time Mannel:1991ii ; Korner:1991ph ; Mannel:1991bs ;
Singleton:1990ye ; Dai:1996xv ; Lee:1998bj ; Ivanov:1998ya ; Cardarelli:1998tq
; MarquesdeCarvalho:1999bqs ; Guo:1999ss . Different from the
$\Lambda_{c}^{+}$ semileptonic decays, the $\tau^{-}$-mode of the
$\Lambda_{b}\to\Lambda_{c}^{+}\ell^{-}\nu_{\ell}$ semi-leptonic decay is
allowed kinematically. Similar to the discussion of the $B\to D^{(*)}$
semileptonic decays Lees:2012xj ; Belle:2019rba , investigating the ratio
$R(\Lambda_{c})=\mathcal{B}(\Lambda_{b}\to\Lambda_{c}^{+}\ell^{-}\nu_{\ell})/\mathcal{B}(\Lambda_{b}\to\Lambda_{c}^{+}\tau^{-}\nu_{\tau})$
with $\ell^{-}=e^{-}$ or $\mu^{-}$ is also an interesting research topic.
Thus, in this work, we will focus on this issue. There were some theoretical
investigations of the $\Lambda_{b}$ semileptonic decays. Ke et al. Ke:2007tg ;
Zhu:2018jet ; Ke:2019smy  calculated the branching fractions in standard and
covariant light-front quark models within the diquark picture. Their results
show that the quark-diquark picture works well for heavy baryons.
Additionally, they also reinvestigated the same topic by the same approach
without introducing diquark approximation. A RQM Ebert:2006rp ;
Faustov:2016pal  with quark-diquark approximation was used by Ebert, Faustov
and Galkin. The calculated branching fraction is consistent with the
experimental data; other physical observables were also given. Gutsche et al.
Gutsche:2015mxa ; Gutsche:2018nks  calculated the
$\Lambda_{b}\to\Lambda_{c}^{+}\ell^{-}\nu_{\ell}$$(\ell^{-}=e^{-},\mu^{-},\tau^{-})$
observables by using the covariant confined quark model. Ranmani et al.
Rahmani:2020kjd  calculated them in a potential model with a modified QCD
Cornell interaction. Thakkar Thakkar:2020vpv  studied the semileptonic decays
by using the hypercentral constituent quark model, where the six-dimensional
hypercentral Schrödinger equation was solved for extracting the wave functions
of heavy baryons. Of course, the QCDSR was also applied to the study of the
$\Lambda_{b}$ semileptonic decays Wang:2003it ; Huang:2005mea ; Wang:2009yma ;
Azizi:2018axf .

Until now, theorists have paid more attentions to the semileptonic decays of
$\Lambda_{Q}$ to a ground state of $\Lambda$ or $\Lambda_{c}$. Since 2005, the
semileptonic decays of $\Lambda_{Q}$ to an excited state of $\Lambda$ or
$\Lambda_{c}$ have been studied. Pervin et al. Pervin:2005ve  studied the
semileptonic decays of $\Lambda_{Q}$ to
$J^{P}=\frac{1}{2}^{\pm},\frac{3}{2}^{-}$ final states in a constituent quark
model with both nonrelativistic and semirelativistic Hamiltonians. Gussain and
Roberts Hussain:2017lir  studied the semileptonic decays of $\Lambda_{c}^{+}$
to excited $\Lambda$ states with a nonrelativistic quark model. Gutsche et al.
Gutsche:2018nks  analyzed
$\Lambda_{b}\to\Lambda_{c}\left(\frac{1}{2}^{\pm},\frac{3}{2}^{-}\right)\ell^{-}\nu_{\ell}$
with CCQM. Nieves et al. Nieves:2019kdh  studied the
$\Lambda_{b}\to\Lambda_{c}(2595,2625)\ell^{-}\nu_{\ell}$ with heavy quark spin
symmetry. Bečirević et al. Becirevic:2020nmb  studied
$\Lambda_{b}\to\Lambda_{c}\left(\frac{1}{2}^{\pm}\right)\ell^{-}\nu_{\ell}$ by
using the Bakamjian-Thomas approach with a spectroscopic model. Recently, the
BESIII Collaboration Ablikim:2019hff  released a white paper of future plans,
where measuring $\Lambda_{c}\to\Lambda^{*}\ell^{+}\nu_{\ell}$ was mentioned.
To enlarge our knowledge on the semileptonic decays of $\Lambda_{Q}$ to
excited $\Lambda$ or $\Lambda_{c}$ state, joint effort from experimentalists
and theorists are needed.

In fact, the LQCD Bowler:1997ej ; Gottlieb:2003yb ; Detmold:2015aaa ;
Meinel:2016dqj ; Meinel:2021rbm  is an effective approach to study the
$\Lambda_{Q}$ semileptonic decays. Meinal et al. Detmold:2015aaa ;
Meinel:2016dqj  calculated the form factors and decay rates of
$\Lambda_{b}\to\Lambda_{c}\ell^{-}\nu_{\ell}$ and
$\Lambda_{c}\to\Lambda\ell^{+}\nu_{\ell}$ processes with all possible leptonic
channels. Very recently, they presented the first lattice QCD calculation of
the form factors describing the
$\Lambda_{b}\to\Lambda_{c}\left(\frac{1}{2}^{-},\frac{3}{2}^{-}\right)\ell^{-}\nu_{\ell}$
decays Meinel:2021rbm . We are looking forward to more progress on the
$\Lambda_{Q}$ semileptonic decays by LQCD, which may provide valuable
information to theoretical studies.

The numerical results from literature are summarized in the table in Sec.V,
with a comparison to our predictions.

##  III semileptonic decays of $\Lambda_{Q}$

###  III.1 The form factors relevant to semileptonic decays of $\Lambda_{Q}$

In this subsection, we briefly present how to calculate the form factors
involved in these semi-leptonic decays of $\Lambda_{b}$ and $\Lambda_{c}$. For
illustration, we take $\Lambda_{b}\to\Lambda_{c}^{(*)}\left(1/2^{\pm}\right)$
as an example, where $\Lambda_{c}\left(1/2^{+}\right)$ and
$\Lambda_{c}\left(1/2^{-}\right)$ denote $\Lambda_{c}\left(2286\right)$ and
$\Lambda_{c}\left(2595\right)$, respectively. In addition, we use subscript
$*$ to mark the radial excitation.

The Hamiltonian depicted corresponding semileptonic decays is written as

$\mathcal{H}_{eff}=\frac{G_{F}}{\sqrt{2}}V_{cb}\bar{\ell}\gamma_{\mu}(1-\gamma_{5})\nu_{\ell}\bar{c}\gamma^{\mu}(1-\gamma_{5})b,$
(2)

where $G_{F}$ is the Fermi constant and $V_{cb}$ is the CKM matrix element. In
Fig. 1, the Feynman diagram for the weak transition is given.

Given that the quarks are confined in hadron, the weak transition matrix
element cannot be calculated by QCD since these weak transitions are involved
in low-energy aspects of QCD. Generally, the matrix elements of hadron section
$\langle\Lambda_{c}^{(*)}(1/2^{\pm})|\bar{c}\gamma^{\mu}(1-\gamma_{5})b|\Lambda_{b}(1/2^{+})\rangle$
can be parametrized in terms of several dimensionless form factors as

Figure 1: The diagram for depicting the
$\Lambda_{b}\to\Lambda_{c}^{(*)}\ell^{-}\nu_{\ell}$ decays in tree level.

$\begin{split}\langle&\Lambda_{c}^{(*)}(1/2^{+})(P^{\prime},J^{\prime}_{z})|\bar{c}\gamma^{\mu}(1-\gamma_{5})b|\Lambda_{b}(P,J_{z})\rangle\\\
&=\bar{u}(P^{\prime},J^{\prime}_{z})\left[f^{V}_{1}(q^{2})\gamma^{\mu}+i\frac{f^{V}_{2}(q^{2})}{M}\sigma^{\mu\nu}q_{\nu}+\frac{f^{V}_{3}(q^{2})}{M}q^{\mu}\right.\\\
&\quad\left.-\left(g^{A}_{1}(q^{2})\gamma^{\mu}+i\frac{g^{A}_{2}(q^{2})}{M}\sigma^{\mu\nu}q_{\nu}+\frac{g^{A}_{3}(q^{2})}{M}q^{\mu}\right)\gamma_{5}\right]u(P,J_{z}),\end{split}$
(3)
$\begin{split}\langle&\Lambda_{c}(1/2^{-})(P^{\prime},J^{\prime}_{z})|\bar{c}\gamma^{\mu}(1-\gamma_{5})b|\Lambda_{b}(P,J_{z})\rangle\\\
&=\bar{u}(P^{\prime},J^{\prime}_{z})\left[\left(g^{V}_{1}(q^{2})\gamma^{\mu}+i\frac{g^{V}_{2}(q^{2})}{M}\sigma^{\mu\nu}q_{\nu}+\frac{g^{V}_{3}(q^{2})}{M}q^{\mu}\right)\gamma_{5}\right.\\\
&\quad\left.-\left(f^{A}_{1}(q^{2})\gamma^{\mu}+i\frac{f^{A}_{2}(q^{2})}{M}\sigma^{\mu\nu}q_{\nu}+\frac{f^{A}_{3}(q^{2})}{M}q^{\mu}\right)\right]u(P,J_{z}),\end{split}$
(4)

for $1/2^{+}\to 1/2^{+}$ and $1/2^{+}\to 1/2^{-}$ transitions, respectively.
Here, $M(P)$, $M^{\prime}(P^{\prime})$ denote the masses (four momentums) of
the initial and final hadrons respectively, and $q=P-P^{\prime}$.

Following the standard procedure, the angular distribution for the decay
$\Lambda_{b}\to\Lambda_{c}^{(*)}W^{-}(\to\ell^{-}\bar{\nu}_{\ell})$ reads as

$\begin{split}\frac{d\Gamma}{dq^{2}d\cos\theta_{l}}=&\frac{G_{F}^{2}}{(2\pi)^{3}}|V_{cb}|^{2}\frac{\sqrt{\lambda(M^{2},M^{\prime
2},q^{2})}(q^{2}-m_{l}^{2})^{2}}{128M^{3}q^{2}}\\\
&\times\left[A_{1}+\frac{m_{l}^{2}}{q^{2}}A_{2}\right],\end{split}$ (5)

where $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2xz-2yz$ is the kinematical
triangle Kallen function and

$\begin{split}A_{1}=&2\sin^{2}\theta_{l}(H^{2}_{1/2,0}+H^{2}_{-1/2,0})+(1-\cos\theta_{l})^{2}H^{2}_{1/2,1}\\\
&+(1+\cos\theta_{l})^{2}H^{2}_{-1/2,-1},\\\
A_{2}=&2\cos^{2}\theta_{l}(H^{2}_{1/2,0}+H^{2}_{-1/2,0})+\sin^{2}\theta_{l}(H^{2}_{1/2,1}+H^{2}_{-1/2,-1})\\\
&-4\cos\theta_{l}(H_{1/2,t}H_{1/2,0}+H_{-1/2,t}H_{-1/2,0})\\\
&+2(H^{2}_{1/2,t}+H^{2}_{-1/2,t}).\end{split}$ (6)

$\theta_{l}$ is the angle between momenta of the final hadron and the lepton
in the $q^{2}$ rest frame. The helicity amplitudes in Eq. (6),
$H^{V,A}_{\lambda_{\Lambda_{c}^{(*)}},\lambda_{W^{-}}}$, can be expressed as a
series functions of the form factors defined in Eqs. (3) and (4),

$\begin{split}&H^{V}_{\frac{1}{2},0}(1/2^{+}\to
1/2^{\pm})=\frac{\sqrt{Q_{\mp}}}{\sqrt{q^{2}}}\left(M_{\pm}f(g)^{V}_{1}(q^{2})\mp\frac{q^{2}}{M}f(g)^{V}_{2}(q^{2})\right),\\\
&H^{A}_{\frac{1}{2},0}(1/2^{+}\to
1/2^{\pm})=\frac{\sqrt{Q_{\pm}}}{\sqrt{q^{2}}}\left(M_{\mp}g(f)^{A}_{1}(q^{2})\pm\frac{q^{2}}{M}g(f)^{A}_{2}(q^{2})\right),\\\
&H^{V}_{\frac{1}{2},1}(1/2^{+}\to
1/2^{\pm})=\sqrt{2Q_{\mp}}\left(f(g)^{V}_{1}(q^{2})\mp\frac{M_{\pm}}{M}f(g)^{V}_{2}(q^{2})\right),\\\
&H^{A}_{\frac{1}{2},1}(1/2^{+}\to
1/2^{\pm})=\sqrt{2Q_{\pm}}\left(g(f)^{A}_{1}(q^{2})\pm\frac{M_{\mp}}{M}g(f)^{A}_{2}(q^{2})\right),\\\
&H^{V}_{\frac{1}{2},t}(1/2^{+}\to
1/2^{\pm})=\frac{\sqrt{Q_{\pm}}}{\sqrt{q^{2}}}\left(M_{\mp}f(g)^{V}_{1}(q^{2})\pm\frac{q^{2}}{M}f(g)^{V}_{3}(q^{2})\right),\\\
&H^{A}_{\frac{1}{2},t}(1/2^{+}\to
1/2^{\pm})=\frac{\sqrt{Q_{\mp}}}{\sqrt{q^{2}}}\left(M_{\pm}g(f)^{A}_{1}(q^{2})\mp\frac{q^{2}}{M}g(f)^{A}_{3}(q^{2})\right),\\\
\end{split}$ (7)

where $Q_{\pm}$ is defined as $Q_{\pm}=(M\pm M^{\prime})^{2}-q^{2}$ and
$M_{\pm}=M\pm M^{\prime}$. The negative helicities for transitions involved
final hadrons with $J^{P}=$ $1/2^{+}$ and $1/2^{-}$ can be obtained by

$H^{V}_{-\lambda^{\prime},-\lambda_{W}}=\mathcal{P}^{V}H^{V}_{\lambda^{\prime},\lambda_{W}},~{}H^{A}_{-\lambda^{\prime},-\lambda_{W}}=\mathcal{P}^{A}H^{A}_{\lambda^{\prime},\lambda_{W}}$
(8)

with $(\mathcal{P}^{V},\mathcal{P}^{A})=$ $(+,-)$ and $(-,+)$, respectively.
The total helicity amplitudes can be obtained by

$H_{\lambda^{\prime},\lambda_{W}}=H^{V}_{\lambda^{\prime},\lambda_{W}}-H^{A}_{\lambda^{\prime},\lambda_{W}}.$
(9)

It is obvious that the form factors play important roles in the calculation of
the semileptonic decays, which can be calculated by concrete models. In this
work, we adopt the light-front quark model, where the general expressions of
$\Lambda_{b}\to\Lambda_{c}^{(*)}\left(1/2^{\pm}\right)$ weak transitions are
written as Ke:2019smy

$\begin{split}\langle\Lambda_{c}&{}^{(*)}\left(1/2^{+}\right)(\bar{P}^{\prime},J^{\prime}_{z})|\bar{c}\gamma^{\mu}(1-\gamma_{5})b|\Lambda_{b}\left(1/2^{+}\right)(\bar{P},J_{z})\rangle\\\
=&\int\left(\frac{dx_{1}d^{2}\vec{k}_{1\bot}}{2(2\pi)^{3}}\right)\left(\frac{dx_{2}d^{2}\vec{k}_{2\bot}}{{2(2\pi)^{3}}}\right)\frac{\phi_{\Lambda_{b}}(x_{i},\vec{k}_{i\bot})\phi^{*}_{\Lambda_{c}^{(*)}}(x_{i}^{\prime},\vec{k}_{i\bot}^{\prime})}{16\sqrt{x_{3}x_{3}^{\prime}{M_{0}}^{3}{M_{0}^{\prime}}^{3}}}\frac{\text{Tr}[(\not{\bar{P}}^{\prime}-M_{0}^{\prime})\gamma_{5}(\not{p}_{1}+m_{1})(\not{\bar{P}}+M_{0})\gamma_{5}(\not{p}_{2}-m_{2})]}{\sqrt{(e_{1}+m_{1})(e_{2}+m_{2})(e_{3}+m_{3})(e_{1}^{\prime}+m_{1}^{\prime})(e_{2}^{\prime}+m_{2}^{\prime})(e_{3}^{\prime}+m_{3}^{\prime})}}\\\
&\times\bar{u}(\bar{P}^{\prime},J_{z}^{\prime})(\not{p}_{3}^{\prime}+m_{3}^{\prime})\gamma^{\mu}(1-\gamma_{5})(\not{p}_{3}+m_{3})u(\bar{P},J_{z}),\end{split}$
(10)
$\begin{split}\langle\Lambda_{c}&{}^{*}\left(1/2^{-}\right)(\bar{P}^{\prime},J^{\prime}_{z})|\bar{c}\gamma^{\mu}(1-\gamma_{5})b|\Lambda_{b}\left(1/2^{+}\right)(\bar{P},J_{z})\rangle\\\
=&\int\left(\frac{dx_{1}d^{2}\vec{k}_{1\bot}}{2(2\pi)^{3}}\right)\left(\frac{dx_{2}d^{2}\vec{k}_{2\bot}}{{2(2\pi)^{3}}}\right)\frac{\phi_{\Lambda_{b}}(x_{i},\vec{k}_{i\bot})\phi^{*}_{\Lambda_{c}^{*}}(x_{i}^{\prime},\vec{k}_{i\bot}^{\prime})}{16\sqrt{3x_{3}x_{3}^{\prime}{M_{0}}^{3}{M_{0}^{\prime}}^{3}}}\frac{\text{Tr}[(\not{\bar{P}}^{\prime}-M_{0}^{\prime})\gamma_{5}(\not{p}_{1}+m_{1})(\not{\bar{P}}+M_{0})\gamma_{5}(\not{p}_{2}-m_{2})]}{\sqrt{(e_{1}+m_{1})(e_{2}+m_{2})(e_{3}+m_{3})(e_{1}^{\prime}+m_{1}^{\prime})(e_{2}^{\prime}+m_{2}^{\prime})(e_{3}^{\prime}-m_{3}^{\prime})}}\\\
&\times\frac{\sum_{i}m_{i}^{\prime}/m_{3}^{\prime}}{\sum_{i}m_{i}^{\prime}+M_{0}^{\prime}}\bar{u}(\bar{P}^{\prime},J_{z}^{\prime})\left(\not{K}^{\prime}-\frac{m_{3}^{\prime
2}-\mu^{\prime
2}}{2M_{0}^{\prime}}\right)\gamma_{5}(\not{p}_{3}^{\prime}+m_{3}^{\prime})\gamma^{\mu}(1-\gamma_{5})(\not{p}_{3}+m_{3})u(\bar{P},J_{z}),\end{split}$
(11)

where
$K^{\prime}=\frac{(m_{1}^{\prime}+m_{2}^{\prime})p_{3}^{\prime}-m_{3}^{\prime}(p_{1}+p_{2})}{m_{1}^{\prime}+m_{2}^{\prime}+m_{3}^{\prime}}$
is the $\lambda-$mode momentum of the final hadron. And,
$\mu^{\prime}=m_{1}^{\prime}m_{2}^{\prime}/m_{1}^{\prime}+m_{2}^{\prime}$;
$\bar{P}=p_{1}+p_{2}+p_{3}$ and $\bar{P}^{\prime}=p_{1}+p_{2}+p_{3}^{\prime}$
are the light-front momentum for initial or final hadrons respectively,
considering $p_{1}=p_{1}^{\prime}$ and $p_{2}=p_{2}^{\prime}$ in the spectator
model. $\phi(x_{i},\vec{k}_{i\bot})$ and
$\phi^{*}(x_{i}^{\prime},\vec{k}_{i\bot}^{\prime})$ represent the wave
functions for initial or final hadrons, respectively.

We follow Ref. Chua:2019yqh , where they chose a special gauge $q^{+}=0$, to
get the form factors with $V^{+}$, $A^{+}$, $\vec{q}_{\bot}\cdot\vec{V}$,
$\vec{q}_{\bot}\cdot\vec{A}$, $\vec{n}_{\bot}\cdot\vec{V}$, and
$\vec{n}_{\bot}\cdot\vec{A}$ in the way shown below. Firstly, multiply
$\sum_{J_{z},J_{z}^{\prime}}\delta_{J_{z},J_{z}^{\prime}}$,
$\sum_{J_{z},J_{z}^{\prime}}(\sigma^{3})_{J_{z},J_{z}^{\prime}}$,
$\sum_{J_{z},J_{z}^{\prime}}(\vec{\sigma}\cdot\vec{q}_{\bot})_{J_{z},J_{z}^{\prime}}$,
$\sum_{J_{z},J_{z}^{\prime}}(\vec{\sigma}\cdot\vec{q}_{\bot}\sigma^{3})_{J_{z},J_{z}^{\prime}}$
to both sides of Eq. (3) and Eq. (4), and then use Eq. (11) of Ref.
Chua:2019yqh  in the right side, use Eq. (10), Eq. (11) and Eq. (10) of Ref.
Chua:2019yqh  in the left side. Following the above steps we can finally
obtain the concrete expressions for the involved form factors.

The expressions for the form factors describing
$\Lambda_{b}\to\Lambda_{c}^{(*)}(1/2^{+})$ transitions are

$\begin{split}f^{V}_{1}(q^{2})=&\int\mathcal{DS}\frac{1}{8P^{+}P^{\prime+}}\text{Tr}[(\not{\bar{P}}+M_{0})\gamma^{+}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})(\not{p}_{3}^{\prime}+m_{3}^{\prime})\gamma^{+}(\not{p}_{3}+m_{3})],\\\
f^{V}_{2}(q^{2})=&\int\mathcal{DS}\frac{iM}{8P^{+}P^{\prime+}q^{2}}\text{Tr}[(\not{\bar{P}}+M_{0})\sigma^{+\mu}q_{\mu}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})(\not{p}_{3}^{\prime}+m_{3}^{\prime})\gamma^{+}(\not{p}_{3}+m_{3})],\\\
f^{V}_{3}(q^{2})=&\frac{M}{M+M^{\prime}}\left(f_{1}^{V}(q^{2})(1-\frac{2\bar{P}\cdot
q}{q^{2}})+\int\mathcal{DS}\frac{1}{4\sqrt{P^{+}P^{\prime+}}q^{2}}\text{Tr}[(\not{\bar{P}}+M_{0})\gamma^{+}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})(\not{p}_{3}^{\prime}+m_{3}^{\prime})\not{q}(\not{p}_{3}+m_{3})]\right),\\\
g^{A}_{1}(q^{2})=&\int\mathcal{DS}\frac{1}{8P^{+}P^{\prime+}}\text{Tr}[(\not{\bar{P}}+M_{0})\gamma^{+}\gamma_{5}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})(\not{p}_{3}^{\prime}+m_{3}^{\prime})\gamma^{+}\gamma_{5}(\not{p}_{3}+m_{3})],\\\
g^{A}_{2}(q^{2})=&\int\mathcal{DS}\frac{-iM}{8P^{+}P^{\prime+}q^{2}}\text{Tr}[(\not{\bar{P}}+M_{0})\sigma^{+\mu}q_{\mu}\gamma_{5}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})(\not{p}_{3}^{\prime}+m_{3}^{\prime})\gamma^{+}\gamma_{5}(\not{p}_{3}+m_{3})],\\\
g^{A}_{3}(q^{2})=&\frac{M}{M-M^{\prime}}\left(g^{A}_{1}(q^{2})(-1+\frac{2\bar{P}\cdot
q}{q^{2}})+\int\mathcal{DS}\frac{-1}{4\sqrt{P^{+}P^{\prime+}}q^{2}}\text{Tr}[(\not{\bar{P}}+M_{0})\gamma^{+}\gamma_{5}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})(\not{p}_{3}^{\prime}+m_{3}^{\prime})~{}\not{q}~{}\gamma_{5}~{}(\not{p}_{3}+m_{3})]\right)\end{split}$
(12)

with

$\mathcal{DS}=\frac{dx_{1}d^{2}\vec{k}_{1\bot}dx_{2}d^{2}\vec{k}_{2\bot}}{2(2\pi)^{3}2(2\pi)^{3}}\frac{\phi^{*}(x_{i}^{\prime},\vec{k}_{i\bot}^{\prime})\phi(x_{i},\vec{k}_{i\bot})}{16\sqrt{x_{3}x_{3}^{\prime}M_{0}^{3}M_{0}^{\prime
3}}}\frac{\text{Tr}[(\not{\bar{P}}^{\prime}-M_{0}^{\prime})\gamma_{5}(\not{p}_{1}+m_{1})(\not{\bar{P}}+M_{0})\gamma_{5}(\not{p}_{2}-m_{2})]}{\sqrt{(e_{1}+m_{1})(e_{2}+m_{2})(e_{3}+m_{3})(e_{1}^{\prime}+m_{1}^{\prime})(e_{2}+m_{2}^{\prime})(e_{3}^{\prime}+m_{3}^{\prime})}},$
(13)

while the expressions for the form factors describing
$\Lambda_{b}\to\Lambda_{c}(1/2^{-})$ transition are

$\begin{split}g^{V}_{1}(q^{2})=&\int\mathcal{DP}\frac{1}{8P^{+}P^{\prime+}}\text{Tr}[(\not{\bar{P}}+M_{0})\gamma^{+}\gamma_{5}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})\left(\not{K}^{\prime}-\frac{m_{3}^{\prime
2}-\mu^{\prime
2}}{2M_{0}^{\prime}}\right)\gamma_{5}(\not{p}_{3}^{\prime}+m_{3}^{\prime})\gamma^{+}(\not{p}_{3}+m_{3})],\\\
g^{V}_{2}(q^{2})=&\int\mathcal{DP}\frac{-iM}{8P^{+}P^{\prime+}q^{2}}\text{Tr}[(\not{\bar{P}}+M_{0})\sigma^{+\mu}q_{\mu}\gamma_{5}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})\left(\not{K}^{\prime}-\frac{m_{3}^{\prime
2}-\mu^{\prime
2}}{2M_{0}^{\prime}}\right)\gamma_{5}(\not{p}_{3}^{\prime}+m_{3}^{\prime})\gamma^{+}(\not{p}_{3}+m_{3})],\\\
g^{V}_{3}(q^{2})=&\frac{M}{M+M^{\prime}}\left(f_{1}^{V}(q^{2})(1-\frac{2\bar{P}\cdot
q}{q^{2}})\right.\\\
&\left.+\int\mathcal{DP}\frac{1}{4\sqrt{P^{+}P^{\prime+}}q^{2}}\text{Tr}[(\not{\bar{P}}+M_{0})\gamma^{+}\gamma_{5}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})\left(\not{K}^{\prime}-\frac{m_{3}^{\prime
2}-\mu^{\prime
2}}{2M_{0}^{\prime}}\right)\gamma_{5}(\not{p}_{3}^{\prime}+m_{3}^{\prime})\not{q}(\not{p}_{3}+m_{3})]\right),\\\
f^{A}_{1}(q^{2})=&\int\mathcal{DP}\frac{1}{8P^{+}P^{\prime+}}\text{Tr}[(\not{\bar{P}}+M_{0})\gamma^{+}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})\left(\not{K}^{\prime}-\frac{m_{3}^{\prime
2}-\mu^{\prime
2}}{2M_{0}^{\prime}}\right)\gamma_{5}(\not{p}_{3}^{\prime}+m_{3}^{\prime})\gamma^{+}\gamma_{5}(\not{p}_{3}+m_{3})],\\\
f^{A}_{2}(q^{2})=&\int\mathcal{DP}\frac{iM}{8P^{+}P^{\prime+}q^{2}}\text{Tr}[(\not{\bar{P}}+M_{0})\sigma^{+\mu}q_{\mu}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})\left(\not{K}^{\prime}-\frac{m_{3}^{\prime
2}-\mu^{\prime
2}}{2M_{0}^{\prime}}\right)\gamma_{5}(\not{p}_{3}^{\prime}+m_{3}^{\prime})\gamma^{+}\gamma_{5}(\not{p}_{3}+m_{3})],\\\
f^{A}_{3}(q^{2})=&\frac{M}{M-M^{\prime}}\left(g^{A}_{1}(q^{2})(-1+\frac{2\bar{P}\cdot
q}{q^{2}})\right.\\\
&\left.+\int\mathcal{DP}\frac{-1}{4\sqrt{P^{+}P^{\prime+}}q^{2}}\text{Tr}[(\not{\bar{P}}+M_{0})\gamma^{+}(\not{\bar{P}}^{\prime}+M_{0}^{\prime})\left(\not{K}^{\prime}-\frac{m_{3}^{\prime
2}-\mu^{\prime
2}}{2M_{0}^{\prime}}\right)\gamma_{5}(\not{p}_{3}^{\prime}+m_{3}^{\prime})\not{q}\gamma_{5}(\not{p}_{3}+m_{3})]\right)\end{split}$
(14)

with

$\mathcal{DP}=\frac{dx_{1}d^{2}\vec{k}_{1\bot}dx_{2}d^{2}\vec{k}_{2\bot}}{2(2\pi)^{3}2(2\pi)^{3}}\frac{\phi^{*}(x_{i}^{\prime},\vec{k}_{i\bot}^{\prime})~{}\phi(x_{i},\vec{k}_{i\bot})}{16\sqrt{3x_{3}x_{3}^{\prime}M_{0}^{3}M_{0}^{\prime
3}}}\frac{(\sum_{i}m_{i}^{\prime})/m_{3}^{\prime}}{(M_{0}^{\prime}+\sum_{i}m_{i}^{\prime})}\frac{\text{Tr}[(\not{\bar{P}}^{\prime}-M_{0}^{\prime})\gamma_{5}(\not{p}_{1}+m_{1})(\not{\bar{P}}+M_{0})\gamma_{5}(\not{p}_{2}-m_{2})]}{\sqrt{(e_{1}+m_{1})(e_{2}+m_{2})(e_{3}+m_{3})(e_{1}^{\prime}+m_{1}^{\prime})(e_{2}+m_{2}^{\prime})(e_{3}^{\prime}-m_{3}^{\prime})}}.$
(15)

The ones for $\Lambda_{c}\to\Lambda^{(*)}\left(1/2^{\pm}\right)$ modes have
the same expressions. Since the light-front quark model has been widely used
for studying the weak transition form factors, we will not repeatedly
introduce the explicit forms for this model here. One can turn to Refs.
Ke:2019smy ; Chua:2019yqh ; Chua:2018lfa ; Ke:2007tg ; Cheng:2004cc ;
Wei:2009np ; Chang:2019obq ; Chang:2020wvs ; Ke:2019lcf ; Ke:2012wa ;
Wang:2017mqp  for detailed introductions.

The spatial wave functions of these baryons involved in the discussed
semileptonic decay are crucial input when calculating the form factors. In the
previous works, the spatial wave functions were generally used in simple
harmonic oscillator form with the oscillator parameter $\beta$ Ke:2019smy ;
Chua:2019yqh ; Ke:2007tg ; Chua:2018lfa . In this work, we adopt the exact
spatial wave functions for the involved baryons with the help of the GEM,
which will be introduced in the next subsection. This approach should be
supported by the study of the mass spectrum of the baryon.

###  III.2 The spatial wave function of these involved baryons

Differing from former treatments Ke:2019smy ; Chua:2019yqh ; Ke:2007tg ;
Chua:2018lfa , in this work we adopt GEM to get the wave functions of baryons.
As is known, baryons are formed by three constituent quarks in the traditional
constituent quark model. Different from mesons, one baryon bound state is the
typical three-body system, and its wave function can be extracted by solving
the three-body Schrödinger equation. In this work, we use the semirelativistic
potentials introduced by Godfrey and Isgur Godfrey:1985xj , which was applied
by Capstick and Isgur to study baryon spectroscopy Capstick:1986bm .

In the light quark sector, the relativistic effects in effective potential
should be considered. Those effects are introduced by two ways in the GI
model. At the first step, the GI model introduces a smeared function, and thus
the Hamiltonian depicted three-body bound system can be written as

$\mathcal{H}=K+\sum_{i=1<j}^{3}(S_{ij}+G_{ij}+V^{\text{so}(s)}_{ij}+V^{\text{so}(v)}_{ij}+V^{\text{tens}}_{ij}+V^{\text{con}}_{ij}),$
(16)

where $K$, $S$, $G$, $V^{\text{so}(s)}$, $V^{\text{so}(v)}$,$V^{\text{tens}}$
and $V^{\text{con}}$ are the kinetic energy, the spin-independent linear
confinement piece, the Coulomb-like potential, the scalar type-spin-orbit
interaction, the vector type-spin-orbit interaction, the tensor potential, and
the spin-dependent contact potential, respectively. They are written as
Godfrey:1985xj ; Capstick:1986bm ; Song:2015nia ; Pang:2017dlw

$\displaystyle K$ $\displaystyle=$
$\displaystyle\sum_{i=1,2,3}\sqrt{m_{i}^{2}+p_{i}^{2}},$ (17) $\displaystyle
S_{ij}$ $\displaystyle=$
$\displaystyle-\frac{3}{4}\left(br_{ij}\left[\frac{e^{-\sigma^{2}r_{ij}^{2}}}{\sqrt{\pi}\sigma
r_{ij}}+\left(1+\frac{1}{2\sigma^{2}r_{ij}^{2}}\right)\frac{2}{\sqrt{\pi}}\right.\right.$
(18) $\displaystyle\left.\left.\times\int_{0}^{\sigma
r_{ij}}e^{-x^{2}}dx\right]\right)\mathbf{F_{i}}\cdot\mathbf{F_{j}}+\frac{c}{3},$
$\displaystyle G_{ij}$ $\displaystyle=$
$\displaystyle\sum_{k}\frac{\alpha_{k}}{r_{ij}}\left[\frac{2}{\sqrt{\pi}}\int_{0}^{\tau_{k}r_{ij}}e^{-x^{2}}dx\right]\mathbf{F_{i}}\cdot\mathbf{F_{j}},$
(19)

for spin-independent terms with

$\sigma^{2}=\sigma_{0}^{2}\left[\frac{1}{2}+\frac{1}{2}\left(\frac{4m_{i}m_{j}}{(m_{i}+m_{j})^{2}}\right)^{4}+s^{2}\left(\frac{2m_{i}m_{j}}{m_{i}+m_{j}}\right)^{2}\right],$
(20)

and

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

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

At the second step, the general potential (which relies on the center-of-mass
of interacting quarks and momentum) is made up for the loss of relativistic
effects in the nonrelativistic limit Godfrey:1985xj ; Capstick:1986bm ;
Song:2015nia ; Pang:2017dlw ; Wang:2018rjg ,

$\begin{split}&G_{ij}\to\left(1+\frac{p^{2}}{E_{i}E_{j}}\right)^{1/2}G_{ij}\left(1+\frac{p^{2}}{E_{i}E_{j}}\right)^{1/2},\\\
&\frac{V^{k}_{ij}}{m_{i}m_{j}}\to\left(\frac{m_{i}m_{j}}{E_{i}E_{j}}\right)^{1/2+\epsilon_{k}}\frac{V^{k}_{ij}}{m_{i}m_{j}}\left(\frac{m_{i}m_{j}}{E_{i}E_{j}}\right)^{1/2+\epsilon_{k}}\end{split}$
(21)

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

Table 1: The parameters used in the semirelativistic potential model. Parameters | Values | Parameters | Values
---|---|---|---
$m_{u}~{}(\text{GeV})$ | $0.220$ | $\epsilon^{\text{so}(s)}$ | $0.448$
$m_{d}~{}(\text{GeV})$ | $0.220$ | $\epsilon^{\text{so}(v)}$ | $-0.062$
$m_{s}~{}(\text{GeV})$ | $0.419$ | $\epsilon^{\text{tens}}$ | $0.379$
$m_{c}~{}(\text{GeV})$ | $1.628$ | $\epsilon^{\text{con}}$ | $-0.142$
$m_{b}~{}(\text{GeV})$ | $4.977$ | $\sigma_{0}~{}(\text{GeV})$ | $2.242$
$b~{}(\text{GeV}^{2})$ | $0.142$ | $s$ | $0.805$
$c~{}(\text{GeV})$ | $-0.302$ |  |
Table 2: The spatial wave functions of $\Lambda_{b}$, $\Lambda_{c}^{(*)}\left(1/2^{\pm}\right)$ as well as $\Lambda^{(*)}\left(1/2^{\pm}\right)$ with the GI model by the GEM. The Gaussian bases $(n_{\rho},n_{\lambda})$ listed in the third column are arranged as: $[(1,1),(1,2),\cdots,(1,n_{\lambda_{max}}),(2,1),(2,2),\cdots,(2,n_{\lambda_{max}}),\cdots,(n_{\rho_{max}},1),(n_{\rho_{max}},2),\cdots,(n_{\rho_{max}},n_{\lambda_{max}})]$. Here, we also listed the names of states, which are given in PDG, in the second column. States | Name in PDG | Eigenvectors
---|---|---
$\Lambda_{b}\left(\frac{1}{2}^{+}\right)$ | $\Lambda_{b}$ | $[-0.0177,-0.0561,-0.0930,-0.0020,-0.0022,0.0007,-0.0382,0.0018,-0.0009,-0.0024,0.0020,-0.0006,$
$0.0172,-0.3036,-0.2450,-0.0090,-0.0030,0.0010,0.0070,0.0590,-0.3754,-0.0091,-0.0024,0.0008,$
$-0.0090,-0.0043,-0.0265,-0.1238,0.0256,-0.0057,0.0026,0.0022,0.0169,0.0149,-0.0088,0.0018]$
$\Lambda_{c}\left(\frac{1}{2}^{+}\right)$ | $\Lambda_{c}^{+}$ | $[-0.0162,-0.0404,-0.1010,-0.0174,0.0003,0.0001,-0.0321,-0.0094,-0.0024,-0.0027,0.0019,-0.0005,$
$0.0163,-0.2232,-0.2985,-0.0527,0.0045,-0.0007,0.0002,0.0701,-0.3035,-0.0831,0.0127,-0.0027,$
$-0.0054,-0.0095,-0.0129,-0.1277,0.0183,-0.0042,0.0018,0.0024,0.0135,0.0186,-0.0085,0.0016]$
$\Lambda_{c}\left(\frac{1}{2}^{-}\right)$ | $\Lambda_{c}(2595)^{+}$ | $[0.0044,0.0227,0.0983,0.0423,-0.0049,0.0010,0.0149,-0.0018,0.0083,0.0016,-0.0011,0.0003,$
$-0.0065,0.1155,0.3076,0.1148,-0.0139,0.0030,0.0021,-0.0485,0.2750,0.1689,-0.0283,0.0067,$
$0.0037,0.0011,0.0136,0.1510,-0.0104,0.0026,-0.0013,0.0005,-0.0131,-0.0266,0.0080,-0.0015]$
$\Lambda_{c}^{*}\left(\frac{1}{2}^{+}\right)$ | $\Lambda_{c}(2765)^{+}$ | $[-0.0244,-0.0944,-0.127630,0.1982,-0.0127,0.0034,-0.0375,-0.0588,0.0207,0.0079,-0.0028,0.0004,$
$0.0206,-0.4044,-0.4802,0.5686,-0.0447,0.0110,0.0099,0.0386,-0.2691,0.5855,-0.0219,0.0051,$
$-0.0080,-0.0099,-0.1144,0.3734,0.0859,-0.0156,0.0014,0.0089,0.0311,-0.0673,0.0101,0.0002]$
$\Lambda\left(\frac{1}{2}^{+}\right)$ | $\Lambda$ | $[0.0171,0.0239,0.0847,0.0412,0.0035,-0.0005,0.0373,-0.0047,0.0106,0.0004,-0.0013,0.0006,$
$-0.0192,0.1969,0.2244,0.1544,-0.0024,0.0017,0.0014,-0.0693,0.2616,0.1231,0.0116,-0.0003,$
$0.0041,0.0115,-0.0091,0.1469,-0.0115,0.0038,-0.0016,-0.0021,-0.0071,-0.0248,0.0090,-0.0022]$
$\Lambda\left(\frac{1}{2}^{-}\right)$ | $\Lambda(1405)$ | $[0.0058,0.0145,0.0715,0.0639,0.0019,0.0002,0.0193,-0.0075,0.0074,0.0028,-0.0015,0.0004,$
$-0.0101,0.1087,0.2125,0.1904,0.0020,0.0013,0.0027,-0.0469,0.2260,0.2215,-0.0009,0.0029,$
$0.0022,0.0061,-0.0113,0.1742,0.0127,-0.0007,-0.0010,-0.0005,-0.0054,-0.0329,0.0063,-0.0012]$
$\Lambda^{*}\left(\frac{1}{2}^{+}\right)$ | $\Lambda(1600)$ | $[0.0310,0.0424,0.1590,-0.1262,-0.0378,0.0031,0.0429,0.0480,-0.0085,-0.0117,0.0011,0.0000,$
$-0.0119,0.2645,0.5786,-0.2742,-0.1134,0.0120,-0.0001,-0.0639,0.2197,-0.4162,-0.1299,0.0128,$
$0.0063,0.0065,0.1069,-0.3538,-0.2064,0.0247,-0.0012,-0.0062,-0.0280,0.0514,0.0050,-0.0022]$

Now we illustrate the construction of the baryon wave function. The total wave
function is composed of color, flavor, space, and spin wave functions as

$\Psi_{\mathbf{J},\mathbf{M_{J}}}=\chi^{c}\left\\{\chi^{s}_{\mathbf{S},\mathbf{M_{S}}}\psi^{p}_{\mathbf{L},\mathbf{M_{L}}}\right\\}_{\mathbf{J},\mathbf{M_{J}}}\psi^{f},$
(22)

where $\chi^{c}=\frac{1}{\sqrt{6}}(rgb-rbg+gbr-grb+brg-bgr)$ is the color wave
function and is universal. The subscripts S, L and J are the total spin
angular momentum, the total orbital angular momentum, and the total angular
momentum quantum numbers, respectively. $\psi^{p}_{\mathbf{L},\mathbf{M_{L}}}$
denotes the spatial wave function consisting of the $\rho$-mode and
$\lambda$-mode as

$\psi^{p}_{\mathbf{L},\mathbf{M_{L}}}(\vec{\rho},\vec{\lambda})=\left\\{\phi_{\boldsymbol{l_{\rho}},\boldsymbol{ml_{\rho}}}(\vec{\rho})\phi_{\boldsymbol{l_{\lambda}},\boldsymbol{ml_{\lambda}}}(\vec{\lambda})\right\\}_{\mathbf{L},\mathbf{M_{L}}},$
(23)

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

$\vec{\rho}=\vec{r}_{1}-\vec{r}_{2},~{}~{}\vec{\lambda}=\vec{r}_{3}-\frac{m_{1}\vec{r}_{1}+m_{2}\vec{r}_{2}}{m_{1}+m_{2}}.$
(24)

Note that we only use one channel here. It is suitable for singly heavy
baryons while it is an approximation for light baryons.

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

$\mathcal{H}\Psi_{\mathbf{J},\mathbf{M_{J}}}=E\Psi_{\mathbf{J},\mathbf{M_{J}}},$
(25)

and thus we can obtain the eigenvalues $E$, and the spatial wave functions.
The Gaussian bases Hiyama:2003cu ; Yoshida:2015tia ; Yang:2019lsg

$\phi_{n,l,m}^{G}(\vec{r})=\phi^{G}_{n,l}(r)Y_{l,m}(\hat{r}),$ (26)

are used to expand the spatial wave function, where $n=1,2,\cdots,n_{max}$
with a freedom parameter $n_{max}$ which should be chosen from positive
integers, and the Gaussian size parameters $\nu_{n}$ are settled as a
geometric progression as

$\nu_{n}=1/r^{2}_{n},~{}r_{n}=r_{min}~{}a^{n-1}$ (27)

with

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

Using the parameters collected in Table 1 as input, we can obtain the masses
111 For the ground states, the estimated masses are 1131 MeV, 2286 MeV and
5621 MeV for $\Lambda$, $\Lambda_{c}$ and $\Lambda_{b}$, respectively, which
are close to the experimental data $1116$ MeV, $2286.46\pm 0.14$ MeV and
$5619.60\pm 0.17$ MeV Zyla:2020zbs . additionally, the estimated masses for
$\Lambda_{c}(1/2^{-})$ and $\Lambda_{c}^{*}(1/2^{+})$ are 2615 MeV and 2765
MeV, respectively, which can also reproduce the experimental data $2592.25\pm
0.28$ and $2766.6\pm 2.4$ MeV Zyla:2020zbs . For the $(uds)$-type
$\Lambda^{*}$ resonances the estimations for $\Lambda(1/2^{-})$ and
$\Lambda^{*}(1/2^{+})$ are 1517 MeV and 1679 MeV, respectively, which is
markedly different compared to $\Lambda(1405)$’s $1405.1^{+1.3}_{-1.0}$ Mev
and $\Lambda(1600)$’s $(1570\sim 1630)$ Mev Zyla:2020zbs . It is worth
mentioning mention that there is only one channel Yoshida:2015tia ;
Yang:2019lsg  used in this work. The big differences indicate more complicated
structures for light baryons. For $\Lambda(1405)$, our estimation is roughly
consistent with the constituent quark model Pervin:2005ve ; Yoshida:2015tia
although its structure is still a mystery and needs to be understood. More
discussions on this resonance can be found in Refs. Hall:2014uca ; Liu:2016wxq
; Azizi:2017xyx ; Mai:2020ltx  and the references therein. and spatial wave
functions of baryons. Additionally, in our calculation the values of
$\rho_{min}$ and $\rho_{max}$ are chosen as $0.2$ fm and $2.0$ fm,
respectively, and $n_{\rho_{max}}=6$. For $\lambda$-mode, we also use the same
Gaussian sized parameters. In Table 2, we collect the numerical spatial wave
functions of bottom and charmed baryons, and the ground and excited states of
$\Lambda$ involved in these discussed semileptonic decays. It should be noted
that the spatial wave functions obtained in this work are based on a
systematic analysis of all the related $\Lambda_{b}$, $\Lambda_{c}^{(*)}$ and
$\Lambda^{(*)}$ baryons.

##  IV numerical results

###  IV.1 The calculated form factors

In this subsection, we present our results for the form factors of
$\Lambda_{b}\to\Lambda_{c}^{(*)}(1/2^{\pm})$ and
$\Lambda_{c}\to\Lambda^{(*)}(1/2^{\pm})$ transitions. The mass of baryons we
used in our calculation are cited from Ref. Zyla:2020zbs , and the spatial
wave functions are obtained from GEM. The form factors obtained in Sec. II are
calculated out in a spacelike region, so we need to expand them to a timelike
region. We would use the dipole parametrization to obtain our form factors in
physical region. The concrete expression reads as Chua:2019yqh ; Chua:2018lfa

$F(q^{2})=\frac{F(0)}{\left(1-q^{2}/M^{2}\right)\left(1-b_{1}q^{2}/M^{2}+b_{2}(q^{2}/M^{2})^{2}\right)},$
(28)

where $F(0)$ is the form factor at $q^{2}=0$, $b_{1}$, and $b_{2}$ are
parameters needing to be fitted. We numerically compute 24 points for each
form factor from $q^{2}=-q^{2}_{\text{max}}$ to $q^{2}=0$ and fit them with
MINUIT program. The fitting of form factor is very good, which can be seen in
Fig. 2. Since it is difficult to estimate the uncertainty from the model
itself, we roughly take an error of ten percent for the fitting points.

With the spatial wave functions for $\Lambda_{Q}^{(*)}$, we can obtain the
form factors numerically in the framework of the light-front quark model. In
this way, we avoid the effective $\beta$ parameters Ke:2019smy ; Chua:2019yqh
; Zhu:2018jet ; Ke:2007tg ; Chen:2017vgi ; Xu:2014mqa , and all the free
parameters can be fixed by fitting the mass spectra of the baryons. The
parameters for the extended form factors are collected in Table 3 and 4 for
$\Lambda_{b}\to\Lambda_{c}^{(*)}(1/2^{\pm})$ and
$\Lambda_{c}\to\Lambda^{(*)}(1/2^{\pm})$ transitions, respectively.

Table 3: The form factors for $\Lambda_{b}\to\Lambda_{c}^{(*)}$ in this work. We use a three-parameter form as illustrated in Eq. (28) for these form factors. | $F(0)$ | $F(q^{2}_{max})$ | $b_{1}$ | $b_{2}$
---|---|---|---|---
| $\Lambda_{b}\to\Lambda_{c}\left(\frac{1}{2}^{+}\right)$
$f^{V}_{1}$ | $0.50\pm 0.05$ | $1.01\pm 0.10$ | $0.71\pm 0.01$ | $0.14\pm 0.05$
$f^{V}_{2}$ | $-0.12\pm 0.01$ | $-0.29\pm 0.03$ | $1.14\pm 0.02$ | $0.40\pm 0.06$
$f^{V}_{3}$ | $-0.04\pm 0.00$ | $-0.11\pm 0.01$ | $1.16\pm 0.02$ | $0.32\pm 0.06$
$g^{A}_{1}$ | $0.49\pm 0.05$ | $0.98\pm 0.10$ | $0.68\pm 0.01$ | $0.14\pm 0.05$
$g^{A}_{2}$ | $-0.02\pm 0.00$ | $-0.05\pm 0.00$ | $0.98\pm 0.02$ | $0.34\pm 0.06$
$g^{A}_{3}$ | $-0.15\pm 0.02$ | $-0.37\pm 0.04$ | $1.29\pm 0.02$ | $0.56\pm 0.06$
| $\Lambda_{b}\to\Lambda_{c}\left(\frac{1}{2}^{-}\right)$
$g^{V}_{1}$ | $0.39\pm 0.04$ | $0.56\pm 0.06$ | $0.15\pm 0.01$ | $0.33\pm 0.06$
$g^{V}_{2}$ | $-0.33\pm 0.03$ | $-0.71\pm 0.07$ | $1.42\pm 0.02$ | $0.69\pm 0.07$
$g^{V}_{3}$ | $-0.55\pm 0.06$ | $-1.23\pm 0.12$ | $1.48\pm 0.02$ | $0.66\pm 0.07$
$f^{A}_{1}$ | $0.44\pm 0.04$ | $0.70\pm 0.07$ | $0.45\pm 0.01$ | $0.14\pm 0.06$
$f^{A}_{2}$ | $-0.38\pm 0.04$ | $-0.80\pm 0.08$ | $1.27\pm 0.02$ | $0.53\pm 0.07$
$f^{A}_{3}$ | $-0.45\pm 0.05$ | $-0.98\pm 0.10$ | $1.45\pm 0.02$ | $0.74\pm 0.07$
| $\Lambda_{b}\to\Lambda_{c}^{*}\left(\frac{1}{2}^{+}\right)$
$f^{V}_{1}$ | $0.20\pm 0.02$ | $0.18\pm 0.02$ | $-1.19\pm 0.02$ | $2.05\pm 0.07$
$f^{V}_{2}$ | $-0.07\pm 0.01$ | $-0.08\pm 0.01$ | $-0.33\pm 0.02$ | $1.26\pm 0.08$
$f^{V}_{3}$ | $-0.11\pm 0.01$ | $-0.20\pm 0.02$ | $1.13\pm 0.02$ | $0.40\pm 0.10$
$g^{A}_{1}$ | $0.19\pm 0.02$ | $0.17\pm 0.02$ | $-1.31\pm 0.02$ | $2.24\pm 0.07$
$g^{A}_{2}$ | $-0.08\pm 0.01$ | $-0.17\pm 0.02$ | $1.29\pm 0.01$ | $0.12\pm 0.07$
$g^{A}_{3}$ | $-0.09\pm 0.01$ | $-0.12\pm 0.01$ | $0.15\pm 0.01$ | $0.89\pm 0.06$
Table 4: The form factors for $\Lambda_{c}\to\Lambda^{(*)}$ in this work. We use a three-parameter form as illustrated in Eq. (28) for these form factors. | $F(0)$ | $F(q^{2}_{max})$ | $b_{1}$ | $b_{2}$
---|---|---|---|---
| $\Lambda_{c}\to\Lambda\left(\frac{1}{2}^{+}\right)$
$f^{V}_{1}$ | $0.71\pm 0.07$ | $1.02\pm 0.10$ | $0.28\pm 0.01$ | $0.13\pm 0.06$
$f^{V}_{2}$ | $-0.36\pm 0.04$ | $-0.58\pm 0.06$ | $0.65\pm 0.01$ | $0.23\pm 0.06$
$f^{V}_{3}$ | $-0.29\pm 0.03$ | $-0.49\pm 0.05$ | $0.86\pm 0.01$ | $0.21\pm 0.07$
$g^{A}_{1}$ | $0.62\pm 0.06$ | $0.86\pm 0.09$ | $0.10\pm 0.01$ | $0.21\pm 0.06$
$g^{A}_{2}$ | $-0.11\pm 0.01$ | $-0.19\pm 0.02$ | $0.79\pm 0.01$ | $0.05\pm 0.06$
$g^{A}_{3}$ | $-0.60\pm 0.06$ | $-1.07\pm 0.11$ | $1.08\pm 0.01$ | $0.56\pm 0.07$
| $\Lambda_{c}\to\Lambda\left(\frac{1}{2}^{-}\right)$
$g^{V}_{1}$ | $0.46\pm 0.05$ | $0.49\pm 0.05$ | $-0.56\pm 0.03$ | $1.13\pm 0.35$
$g^{V}_{2}$ | $-0.41\pm 0.04$ | $-0.53\pm 0.05$ | $0.81\pm 0.03$ | $1.54\pm 0.38$
$g^{V}_{3}$ | $-1.45\pm 0.15$ | $-1.92\pm 0.19$ | $1.04\pm 0.04$ | $1.94\pm 0.39$
$f^{A}_{1}$ | $0.63\pm 0.06$ | $0.73\pm 0.07$ | $0.05\pm 0.03$ | $0.64\pm 0.37$
$f^{A}_{2}$ | $-0.56\pm 0.06$ | $-0.71\pm 0.07$ | $0.61\pm 0.03$ | $0.75\pm 0.38$
$f^{A}_{3}$ | $-0.75\pm 0.08$ | $-0.98\pm 0.10$ | $0.86\pm 0.03$ | $1.38\pm 0.38$
| $\Lambda_{c}\to\Lambda^{*}\left(\frac{1}{2}^{+}\right)$
$f^{V}_{1}$ | $0.19\pm 0.02$ | $0.17\pm 0.02$ | $-2.81\pm 0.04$ | $6.92\pm 0.70$
$f^{V}_{2}$ | $-0.11\pm 0.01$ | $-0.09\pm 0.01$ | $-2.68\pm 0.04$ | $7.53\pm 0.71$
$f^{V}_{3}$ | $-0.38\pm 0.04$ | $-0.42\pm 0.04$ | $0.31\pm 0.04$ | $1.33\pm 0.80$
$g^{A}_{1}$ | $0.13\pm 0.01$ | $0.11\pm 0.01$ | $-3.99\pm 0.04$ | $11.42\pm 0.73$
$g^{A}_{2}$ | $-0.23\pm 0.02$ | $-0.25\pm 0.03$ | $0.35\pm 0.05$ | $1.89\pm 0.81$
$g^{A}_{3}$ | $-0.35\pm 0.04$ | $-0.36\pm 0.04$ | $-0.60\pm 0.04$ | $3.27\pm 0.77$

The form factors for $\Lambda_{b}\to\Lambda_{c}(1/2^{+},2286)$ transition can
be worked out with the help of Eq. (10). The $q^{2}$ dependence of
$f^{V}_{1,2,3}$ and $g^{A}_{1,2,3}$ for this transition are plotted in the
first panel of Fig. 2, and the fitting parameters in Eq. (28) are listed in
Table 3. In the heavy quark limit (HQL), the corresponding transition matrix
element can be rewritten as Bernlochner:2018bfn ; Bernlochner:2018kxh ;
Georgi:1990ei

$\left<\Lambda_{c}(\nu^{\prime},s^{\prime})|\bar{c}_{\nu^{\prime}}\Gamma
b_{\nu}|\Lambda_{b}(\nu,s)\right>=\zeta(\omega)\bar{u}(\nu^{\prime},s^{\prime})\Gamma
u(\nu,s),$ (29)

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

$\begin{split}&f^{V}_{1}(q^{2})=g^{A}_{1}(q^{2})=\zeta(\omega),\\\
&f^{V}_{2}=f^{V}_{3}=g^{A}_{2}=g^{A}_{3}=0,\end{split}$ (30)

where $\omega=\nu\cdot\nu^{\prime}=(M^{2}+M^{\prime 2}-q^{2})/(2MM^{\prime})$
while $\nu^{\prime}=p^{\prime}/M^{\prime}$ and $\nu=p/M$ are the four-
velocities for $\Lambda_{c}$ and $\Lambda_{b}$, respectively. $\zeta(\omega)$
is the so-called Isgur-Wise function (IWF) and usually expressed as a Taylor
series expansion as Aaij:2017svr

$\zeta(\omega)=1-\rho^{2}(\omega-1)+\frac{\sigma^{2}}{2}(\omega-1)^{2}+\cdots,$
(31)

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

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

It provides one strong restriction for our results. Either
$f^{V}_{1}(q_{max}^{2})=1.010\pm 0.100$ or $g^{A}_{1}(q_{max}^{2})=0.980\pm
0.100$ in our results satisfies the above restriction. Besides, we can easily
see that $f^{V}_{1}$ and $g^{A}_{1}$ are very close to each other, and
dominate over $f^{V}_{2,3}$ and $g^{A}_{2,3}$. These results satisfy the
predictions of heavy quark effective theory. At the mean time, we obtain the
$\rho^{2}$ and $\sigma^{2}$ in Eq. (31), by fitting $\zeta(\omega)$ from
$f^{V}_{1}(q^{2})$ and $g^{A}_{1}(q^{2})$. The comparison with other
theoretical predictions and the fitting of experimental data are listed in
Table 5. The slope parameter $\rho^{2}$, either from fitting $f^{V}_{1}$ or
$g^{A}_{1}$, is close to the results from DELPHI Abdallah:2003gn  and LHCb
Aaij:2017svr . The parameter $\sigma^{2}$ from $g^{A}_{1}(q^{2})$ is slightly
greater than the measurement of LHCb Collaboration, while the one from
$f^{V}_{1}(q^{2})$ is agrees well with the measurement.

Table 5: Our results for the IWF’s shape parameters of the $\Lambda_{b}\rightarrow\Lambda_{c}$ transition. The superscripts $[a]$ and $[b]$ in the second and third rows represent the fitting of $f^{V}_{1}$ and $g^{A}_{1}$ respectively. In Ref. Thakkar:2020vpv , they list various theoretical predictions of $\rho^{2}$ (in Table 3) and we do not list those here. | $\rho^{2}$ | $\sigma^{2}$
---|---|---
This work[a] | $1.67\pm 0.11$ | $2.45\pm 0.63$
This work[b] | $1.85\pm 0.11$ | $3.25\pm 0.61$
RQM Ebert:2006rp  | $1.51$ | $4.06$
QCDSR Huang:2005mea  | $1.35\pm 0.13$ |
LQCD Bowler:1997ej  | $1.2^{+0.8}_{-1.1}$ |
DELPHI Abdallah:2003gn  | $2.03\pm 0.46^{+0.72}_{-1.00}$ |
LHCb Aaij:2017svr  | $1.63\pm 0.07$ | $2.16\pm 0.34$
Table 6: Theoretical predictions for the form factors $f_{1,2}^{V}(0)$ and $g_{1,2}^{A}(0)$ of $\Lambda_{b}\to\Lambda_{c}$ and $\Lambda_{c}\to\Lambda$ at $q^{2}=0$ using different approaches. | $f^{V}_{1}(0)$ | $f^{V}_{2}(0)$ | $g^{A}_{1}(0)$ | $g^{A}_{2}(0)$
---|---|---|---|---
$\Lambda_{b}\to\Lambda_{c}$ |  |  |  |
This work | $0.50\pm 0.05$ | $-0.12\pm 0.01$ | $0.49\pm 0.05$ | $-0.02\pm 0.00$
LFQM Chua:2019yqh  | $0.474^{+0.069}_{-0.072}$ | $-0.109^{+0.019}_{-0.021}$ | $0.468^{+0.067}_{-0.007}$ | $0.051^{+0.008}_{-0.012}$
LFQM Zhu:2018jet  | $0.500^{+0.028}_{-0.031}$ | $-0.098^{+0.005}_{-0.004}$ | $0.509^{+0.029}_{-0.031}$ | $-0.014^{+0.001}_{-0.001}$
LFQM Ke:2019smy  | $0.488$ | $-0.180$ | $0.470$ | $-0.0479$
RQM Faustov:2016pal  | $0.526$ | $-0.137$ | $0.505$ | $0.027$
CCQM Gutsche:2015mxa  | $0.549$ | $-0.110$ | $0.542$ | $-0.018$
LQCD Detmold:2015aaa  | $0.418\pm 0.161$ | $-0.100\pm 0.055$ | $0.378\pm 0.102$ | $-0.004\pm 0.002$
$\Lambda_{c}\to\Lambda$ |  |  |  |
This work | $0.71\pm 0.07$ | $-0.36\pm 0.04$ | $0.62\pm 0.06$ | $-0.11\pm 0.01$
LFQM Zhao:2018zcb  | $0.468$ | $-0.222$ | $0.407$ | $-0.035$
LFQM Geng:2020fng ; Geng:2020gjh  | $0.67\pm 0.01$ | $-0.76\pm 0.02$ | $0.59\pm 0.01$ |  $(1.59\pm 0.05)\times 10^{-3}$ Geng:2020fng
$(3.8\pm 1.2)\times 10^{-3}$ Geng:2020gjh
MBM Geng:2020fng  | $0.54$ | $-0.22$ | $0.52$ | $0.06$
CCQM Gutsche:2015rrt  | $0.511$ | $-0.289$ | $0.466$ | $0.025$
RQM Faustov:2016yza  | $0.700$ | $-0.295$ | $0.488$ | $0.135$
LCSRLiu:2009sn  | $0.517$ | $-0.123$ | $0.517$ | $0.123$
LCSR Huang:2006ny  | $0.449$ | $-0.193$ | $0.449$ | $0.193$
LQCD Meinel:2016dqj  | $0.647\pm 0.122$ | $-0.310\pm 0.075$ | $0.577\pm 0.076$ | $0.001\pm 0.048$
---
Figure 2: The $q^{2}$ dependence of form factors of
$\Lambda_{b}\to\Lambda_{c}^{(*)}(1/2^{\pm})$ and
$\Lambda_{c}\to\Lambda^{(*)}(1/2^{\pm})$ transitions, in which the solid and
dashed lines represent the vector or pseudoscalar-types form factors
respectively, while the blue, red and purple lines (both solid and dashed)
represent the $i$th form factors denoted by the subscripts respectively for
each types. In addition, the symbols e.g., point, triangle and diamond in
$q^{2}<0$ regions denote the fitted points for each of the form factors.

For $\Lambda_{b}\to\Lambda_{c}(1/2^{-},2595)$ transition, the $q^{2}$
dependence of $g^{V}_{1,2,3}$ and $f^{A}_{1,2,3}$ are plotted in the second
panel of Fig. 2. Similarly, the corresponding transition matrix element can be
simplified and be rewritten as Chua:2019yqh

$\left<\Lambda_{c}^{*}(1/2^{-})(\nu^{\prime})|\bar{c}_{\nu^{\prime}}\Gamma
b_{\nu}|\Lambda_{b}(\nu)\right>\\\
=\frac{\sigma(\omega)}{\sqrt{3}}\bar{u}(\nu^{\prime})\gamma_{5}(\not{\nu}+\nu\cdot\nu)\Gamma
u(\nu),$ (32)

in HQL. At the mean time, the form factors can be simplified as

$\begin{split}g_{1}^{V}(q^{2})=f_{1}^{A}(q^{2})&=\left(\omega-\frac{M^{\prime}}{M}\right)\frac{\sigma(\omega)}{\sqrt{3}},\\\
g_{2}^{V}(q^{2})=g_{3}^{V}(q^{2})=&f_{2}^{A}(q^{2})=f_{3}^{A}(q^{2})=-\frac{\sigma(\omega)}{\sqrt{3}}.\end{split}$
(33)

When comparing our results with the predictions of the heavy quark limit, we
can draw the conclusion that our results match the main requirements of the
above analysis with: 1. $g^{V}_{1}$ is close to $f^{A}_{1}$, while
$g^{V}_{2,3}$ and $f^{A}_{2,3}$ are close to each others. 2.
$f^{A}_{1}/f^{A}_{2,3}$ and $f^{A}_{1}/f^{A}_{2,3}$ roughly close to
$(M^{\prime}-M)/M$ at $q^{2}=q_{max}^{2}$.

For the $\Lambda_{b}\to\Lambda_{c}^{*}(1/2^{+},2765)$ transition, the $q^{2}$
dependence of $f^{V}_{1,2,3}$ and $g^{A}_{1,2,3}$ are plotted in the third
panel of Fig. 2. The HQL expects $f_{1}^{V}=g_{1}^{A}=0$ at
$q^{2}=q^{2}_{max}$, because the wave functions of the low-lying $\Lambda_{b}$
and the radial excited state $\Lambda_{c}^{*}(2765)$ are orthogonal
Chua:2019yqh . Evidently our results embody this character according to Fig. 2
and Table 3.

For $\Lambda_{c}\to\Lambda(1/2^{+},1116)$ form factors, The $q^{2}$ dependence
of $f^{V}_{1,2,3}$ and $g^{A}_{1,2,3}$ are plotted in the fourth panel of Fig.
2. One interesting prediction from heavy quark symmetry at $q^{2}=0$ is that
the daughter $\Lambda$ baryon is predicted to emerge $100\%$ polarized, with
the decay asymmetry parameter $\alpha_{q^{2}=0}=-1$. Our result of $-0.99$ is
very close to the prediction. Besides, the mean value of the polarization
$\langle\alpha\rangle$ has been measured as $-0.86\pm 0.03\pm 0.02$ by CLEO
with high precision Hinson:2004pj . Our result of $-0.87$ is very consistent
with that experiment.

As the form factors depict $\Lambda_{c}\to\Lambda(1/2^{-},1P)$ and
$\Lambda_{c}\to\Lambda^{*}(1/2^{+},2S)$ transitions, we also plot the $q^{2}$
dependence of $f_{1,2,3}$ and $g_{1,2,3}$ in the fifth and sixth panels of
Fig. 2. Presently, we do not have enough information to test these channels,
we expect more experiments and more theories (especially the LQCD) may bring
us the crucial breakthroughs in the future.

Lastly, we list our results for form factors of the
$\Lambda_{b}\to\Lambda_{c}$ and $\Lambda_{c}\to\Lambda$ transitions at
$q^{2}=0$ in Table 6 and compare them with other approaches. The calculations
in Refs. Zhu:2018jet ; Ke:2019smy ; Chua:2019yqh ; Geng:2020fng ; Geng:2020gjh
were based on the light-front quark model. Reference. Chua:2019yqh  worked in
the diquark picture while Ref. Ke:2019smy ; Geng:2020fng ; Geng:2020gjh
worked in the triquark picture. In addition, the MBM was also used in Ref.
Geng:2020fng , in which the inputs are based on the baryon spectroscopy. Refs.
Zhu:2018jet  used covariant light-front quark model. A covariant consistent
quark model was used in Ref. Gutsche:2015mxa  and a RQM was used in Ref.
Faustov:2016pal . (The result from LQCD Meinel:2016dqj ; Detmold:2015aaa  are
also listed here.) It can be seen from Table 6 that our results are consistent
with most of the other theoretical predictions including LQCD. These form
factors are extremely important to help us to understand the weak decays and
search for the undiscovered semileptonic channels. We expect more theoretical
and experimental data for testing these results ulteriorly.

These transition form factors would be useful as inputs not only in the
studies of semileptonic decays as to be shown in the following subsection, but
also in the nonleptonic decays of $\Lambda_{b}$ and $\Lambda_{c}$ Lu:2016ogy ;
Geng:2017esc ; Geng:2017mxn ; Wang:2017gxe ; Geng:2018plk ; Cheng:2018hwl ;
Jiang:2018iqa ; Geng:2018bow ; Geng:2018upx ; Zhao:2018mov ; Jia:2019zxi ;
Zou:2019kzq ; Niu:2020gjw ; Hu:2020nkg ; Meng:2020euv .

###  IV.2 Numerical results of the semileptonic decays observables

Table 7: Our results for the absolute branching fractions $\mathcal{B}(\Lambda_{b}\to\Lambda_{c}^{(*)}\ell^{-}\nu_{\ell})$ with $\ell^{-}=e^{-},\mu^{-},\tau^{-}$ and $\mathcal{B}(\Lambda_{c}\to\Lambda^{(*)}\ell^{+}\nu_{\ell})$ with $\ell^{+}=e^{+},\mu^{+}$. The numbers out of the brackets are our predictions, while the ones in the brackets are the referential numbers with the superscript [exp] and [LQCD] denoting the data which comes from experiments Zyla:2020zbs ; Ablikim:2015prg ; Ablikim:2016vqd ; Hinson:2004pj  or LQCD Meinel:2016dqj ; Detmold:2015aaa , respectively. All the values are given as a percent ($\%$). | $\Lambda_{b}\to\Lambda_{c}^{(*)}\ell\nu_{\ell}$ | $\Lambda_{c}\to\Lambda^{(*)}\ell\nu_{\ell}$
---|---|---
| $\Lambda_{c}\left(\frac{1}{2}^{+}\right)$ | $\Lambda_{c}\left(\frac{1}{2}^{-}\right)$ | $\Lambda_{c}^{*}\left(\frac{1}{2}^{+}\right)$ | $\Lambda\left(\frac{1}{2}^{+}\right)$ | $\Lambda\left(\frac{1}{2}^{-}\right)$ | $\Lambda^{*}\left(\frac{1}{2}^{+}\right)$
$\ell=e$ |  $6.47\pm 0.96$ (${6.2^{+1.4}_{-1.3}}^{\text{[exp]}},$ | $1.73\pm 0.59$ | $0.21\pm 0.05$ |  $4.04\pm 0.75$ (${3.63\pm 0.43}^{\text{[exp]}},$ | $0.31\pm 0.08$ | $0.007\pm 0.002$
${5.49\pm 0.34}^{\text{[LQCD]}}$) |  |  |  $3.80\pm 0.22^{\text{[LQCD]}}$) |  |
$\ell=\mu$ |  $6.45\pm 0.95$ (${6.2^{+1.4}_{-1.3}}^{\text{[exp]}},$ | $1.72\pm 0.58$ | $0.21\pm 0.05$ |  $3.90\pm 0.73$ (${3.49\pm 0.46}^{\text{[exp]}},$ | $0.28\pm 0.07$ | $0.005\pm 0.002$
${5.49\pm 0.34}^{\text{[LQCD]}}$) | (${0.79^{+0.40}_{-0.35}}^{\text{[exp]}}$) |  |  $3.69\pm 0.22^{\text{[LQCD]}}$) |  |
$\ell=\tau$ |  $1.97\pm 0.29$ (${1.57\pm 0.08}^{\text{[LQCD]}}$) | $0.24\pm 0.11$ | $0.02\pm 0.005$ | - | - | -
---
Figure 3: The $q^{2}$ dependence of differential decay widths for $\Lambda_{b}\to\Lambda_{c}^{(*)}\ell^{-}\nu_{\ell}$ semileptonic decays with $\ell^{-}=e^{-}$ (red solid line), $\mu^{-}$ (blue dashed line), $\tau^{-}$ (purple dot dashed line) and for $\Lambda_{c}\to\Lambda^{(*)}\ell^{+}\nu_{\ell}$ semileptonic decays with $\ell^{+}=e^{+}$ (red solid line), $\mu^{+}$ (blue dashed line). Table 8: The predictions for averaged leptonic forward-backward asymmetry $\langle A_{FB}\rangle$, the averaged final hadron polarization $\langle P_{B}\rangle$ and the averaged lepton polarization $\langle P_{\ell}\rangle$. The values in the same column for $\Lambda_{b}\to\Lambda_{c}^{(*)}\ell^{-}\nu_{\ell}$ processes are the results for $\ell^{-}=e^{-},\mu^{-},\tau^{-}$ (top to bottom), while the ones for $\Lambda_{c}\to\Lambda^{(*)}\ell^{+}\nu_{\ell}$ processes are the results for $\ell^{+}=e^{+},\mu^{+}$ (top to bottom). Channels | $\ell=$ | $\langle A_{FB}\rangle$ | $\langle P_{B}\rangle$ | $\langle P_{\ell}\rangle$
---|---|---|---|---
$\Lambda_{b}\to\Lambda_{c}(1/2^{+})\ell^{-}\nu_{\ell}$ | $e$ | $0.18\pm 0.05$ | $-0.81\pm 0.12$ | $-1.00\pm 0.00$
$\mu$ | $0.17\pm 0.05$ | $-0.81\pm 0.12$ | $-0.98\pm 0.00$
$\tau$ | $-0.08\pm 0.03$ | $-0.77\pm 0.11$ | $-0.24\pm 0.03$
$\Lambda_{b}\to\Lambda_{c}(1/2^{-})\ell^{-}\nu_{\ell}$ | $e$ | $0.23\pm 0.04$ | $-0.95\pm 0.08$ | $-1.00\pm 0.00$
$\mu$ | $0.22\pm 0.04$ | $-0.95\pm 0.08$ | $-0.98\pm 0.00$
$\tau$ | $-0.07\pm 0.07$ | $-0.95\pm 0.06$ | $-0.18\pm 0.04$
$\Lambda_{b}\to\Lambda_{c}^{*}(1/2^{+})\ell^{-}\nu_{\ell}$ | $e$ | $0.18\pm 0.07$ | $-0.91\pm 0.12$ | $-1.00\pm 0.00$
$\mu$ | $0.17\pm 0.07$ | $-0.91\pm 0.12$ | $-0.97\pm 0.00$
$\tau$ | $-0.13\pm 0.05$ | $-0.88\pm 0.13$ | $-0.13\pm 0.04$
$\Lambda_{c}\to\Lambda(1/2^{+})\ell^{+}\nu_{\ell}$ | $e$ | $0.20\pm 0.05$ | $-0.87\pm 0.09$ | $-1.00\pm 0.00$
$\mu$ | $0.16\pm 0.04$ | $-0.87\pm 0.09$ | $-0.89\pm 0.04$
$\Lambda_{c}\to\Lambda(1/2^{-})\ell^{+}\nu_{\ell}$ | $e$ | $0.18\pm 0.22$ | $-0.91\pm 0.05$ | $-1.00\pm 0.00$
$\mu$ | $0.10\pm 0.13$ | $-0.91\pm 0.05$ | $-0.80\pm 0.07$
$\Lambda_{c}\to\Lambda^{*}(1/2^{+})\ell^{+}\nu_{\ell}$ | $e$ | $0.11\pm 0.002$ | $-0.90\pm 0.06$ | $-1.00\pm 0.00$
$\mu$ | $-0.001\pm 0.001$ | $-0.90\pm 0.06$ | $-0.71\pm 0.09$
---
Figure 4: The $q^{2}$ dependence of lepton forward-backward asymmetry
($A_{FB}$) for $\Lambda_{b}\to\Lambda_{c}^{(*)}\ell^{-}\nu_{\ell}$
semileptonic decays with $\ell^{-}=e^{-}$ (red solid line), $\mu^{-}$ (blue
dashed line), $\tau^{-}$ (purple dot dashed line) and for
$\Lambda_{c}\to\Lambda^{(*)}\ell^{+}\nu_{\ell}$ semileptonic decays with
$\ell^{+}=e^{+}$ (red solid line), $\mu^{+}$ (blue dashed line).
---
Figure 5: The $q^{2}$ dependence of final hadron longitudinal polarization
($P_{B}$)for $\Lambda_{b}\to\Lambda_{c}^{(*)}\ell^{-}\nu_{\ell}$ semileptonic
decays with $\ell^{-}=e^{-}$ (red solid line), $\mu^{-}$ (blue dashed line),
$\tau^{-}$ (purple dot dashed line) and for
$\Lambda_{c}\to\Lambda^{(*)}\ell^{+}\nu_{\ell}$ semileptonic decays with
$\ell^{+}=e^{+}$ (red solid line), $\mu^{+}$ (blue dashed line).
---
Figure 6: The $q^{2}$ dependence of final lepton longitudinal polarization
($P_{l}$) for $\Lambda_{b}\to\Lambda_{c}^{(*)}\ell^{-}\nu_{\ell}$ semileptonic
decays with $\ell^{-}=e^{-}$ (red solid line), $\mu^{-}$ (blue dashed line),
$\tau^{-}$ (purple dot dashed line) and for
$\Lambda_{c}\to\Lambda^{(*)}\ell^{+}\nu_{\ell}$ semileptonic decays with
$\ell^{+}=e^{+}$ (red solid line), $\mu^{+}$ (blue dashed line).

In order to calculate the physical observables in semileptonic precess, we
adopt the helicity formalism. The baryon and lepton masses used in this paper
come from from the PDG Zyla:2020zbs , as well as
$\tau_{\Lambda_{b}}=1.470~{}\rm{ps}$, and
$\tau_{\Lambda_{c}^{+}}=203.5~{}\rm{fs}$ Aaij:2019lwg . The CKM matrix
elements are chosen as $V_{cs}=0.987$ and $V_{cb}=0.041$ Zyla:2020zbs .

The $q^{2}$ dependence of the differential decay widths of
$\Lambda_{b}\to\Lambda_{c}^{(*)}(1/2^{\pm})\ell^{-}\nu_{\ell}$ and
$\Lambda_{c}\to\Lambda^{(*)}(1/2^{\pm})\ell^{-}\nu_{\ell}$ processes, which
can be obtained by integrating $\theta_{\ell}$, are shown in Fig. 3.In the
meantime we list the branching fractions in Table 7, and compare our results
with the data from experiments or LQCD.

It is obvious that our predictions for
$\mathcal{B}(\Lambda_{b}\to\Lambda_{c}e^{-}\nu_{e}(\mu^{-}\nu_{\mu}))$ are
consistent with the experimental data although slightly greater than the
LQCDs’ results. As for the $\tau$ one, it is slightly over the upper limit of
LQCD but tallies with Ref. Gutsche:2015mxa  and Ref. Faustov:2016pal .
$\mathcal{B}(\Lambda_{c}\to\Lambda e^{+}\nu_{e}(\mu^{+}\nu_{\mu}))$ is
consistent with either the LQCD’s or the BESIII’s results. In 2009, the CDF
Collaboration reported their estimation for
$\mathcal{B}(\Lambda_{b}\to\Lambda_{c}(2595)\mu^{-}\nu_{\mu})=0.9\%$ with
almost $50\%$ uncertainty. This is the only experimental data for excited
channels. Our prediction for this one is near $1.8\%$, which is about two
times better than the limit for the current data. This leads us to search for
more information, in particular the form factors and the absolute branching
rates (by further experiment), on these channels. While for the
$\Lambda_{c}^{*}(1/2^{+})$ and $\Lambda(1/2^{-})$ channels, considering that
our predictions for the $e$ and $\mu$-modes are up to a few thousand of
magnitude-we would like to take a positive attitude to discovering them in
future experiments. These decays can provide a unique platform for studying
the nature of the involved excited hadron states ulteriorly.

We briefly estimate the contributions from ground states, and obtain

$\begin{split}\frac{\mathcal{B}(\Lambda_{b}\to\Lambda_{c}\ell^{-}\nu_{\ell})}{\sum\mathcal{B}(\Lambda_{b}\to\Lambda_{c}^{(*)}\ell^{-}\nu_{\ell})}=&\
0.77\pm 0.21,\\\
\frac{\mathcal{B}(\Lambda_{c}\to\Lambda\ell^{+}\nu_{\ell})}{\sum\mathcal{B}(\Lambda_{c}\to\Lambda^{(*)}\ell^{+}\nu_{\ell})}=&\
0.93\pm 0.34,\end{split}$ (34)

with $\ell=e$ or $\mu$. It indicates the decays to ground state account for a
large proportion of the decays in the corresponding $\Lambda_{b}$ and
$\Lambda_{c}$ semileptonic channels, which is consistent with the BESIII
measurements of $\mathcal{B}(\Lambda_{c}\to\Lambda
e^{+}\nu_{e})/\mathcal{B}(\Lambda_{c}\to e^{+}X)=(91.9\pm 13.6)\%$.

Additionally, other important physical observables, e.g., the leptonic
forward-backward asymmetry ($A_{FB}$), the final hadron polarization
($P_{B}$), and the lepton polarization ($P_{l}$) are also explored. Their
characters dependant on $q^{2}$ are shown in Figs. 4, 5, and 6, respectively.
Their mean values are also listed in Table 8. These parameters are defined as
Azizi:2019tcn

$A_{FB}(q^{2})=\frac{\int_{0}^{1}\frac{d\Gamma}{dq^{2}d\cos\theta_{\ell}}d\cos\theta_{\ell}-\int_{-1}^{0}\frac{d\Gamma}{dq^{2}d\cos\theta_{\ell}}d\cos\theta_{\ell}}{\int_{0}^{1}\frac{d\Gamma}{dq^{2}d\cos\theta_{\ell}}d\cos\theta_{\ell}+\int_{-1}^{0}\frac{d\Gamma}{dq^{2}d\cos\theta_{\ell}}d\cos\theta_{\ell}},$
(35)
$P_{B}(q^{2})=\frac{d\Gamma^{\lambda_{2}=1/2}/dq^{2}-d\Gamma^{\lambda_{2}=-1/2}/dq^{2}}{d\Gamma/dq^{2}},$
(36)
$P_{\ell}(q^{2})=\frac{d\Gamma^{\lambda_{\ell}=1/2}/dq^{2}-d\Gamma^{\lambda_{\ell}=-1/2}/dq^{2}}{d\Gamma/dq^{2}},$
(37)

respectively, where

$\displaystyle\frac{d\Gamma^{\lambda_{2}=1/2}}{dq^{2}}=$
$\displaystyle\frac{4m_{l}^{2}}{3q^{2}}\left(H^{2}_{1/2,1}+H^{2}_{1/2,0}+3H^{2}_{1/2,t}\right)$
$\displaystyle+\frac{8}{3}\left(H^{2}_{1/2,0}+H^{2}_{1/2,1}\right),$ (38)
$\displaystyle\frac{d\Gamma^{\lambda_{2}=-1/2}}{dq^{2}}=$
$\displaystyle\frac{4m_{l}^{2}}{3q^{2}}\left(H^{2}_{-1/2,-1}+H^{2}_{-1/2,0}+3H^{2}_{-1/2,t}\right)$
$\displaystyle+\frac{8}{3}\left(H^{2}_{-1/2,-1}+H^{2}_{-1/2,0}\right),$ (39)
$\displaystyle\frac{d\Gamma^{\lambda_{\ell}=1/2}}{dq^{2}}=$
$\displaystyle\frac{m_{l}^{2}}{q^{2}}\left[\frac{4}{3}\left(H^{2}_{1/2,1}+H^{2}_{1/2,0}+H^{2}_{-1/2,-1}+H^{2}_{-1/2,0}\right)\right.$
$\displaystyle\left.+4\left(H^{2}_{1/2,t}+H^{2}_{-1/2,t}\right)\right],$ (40)
$\displaystyle\frac{d\Gamma^{\lambda_{\ell}=-1/2}}{dq^{2}}=$
$\displaystyle\frac{8}{3}\left(H^{2}_{1/2,1}+H^{2}_{1/2,0}+H^{2}_{-1/2,-1}+H^{2}_{-1/2,0}\right).$
(41)

In Fig. 4, the leptonic forward-backward asymmetry $F_{FB}$ always changes its
sign at a $q^{2}$ value which is associated with the mass of a lepton. This
character is consistent with Refs. Faustov:2016pal ; Gutsche:2015rrt ;
Becirevic:2020nmb . In Fig. 5, the hadron polarization always varies from
$P_{B}=-1$ to $P_{B}=0$ as the $q^{2}$ increases from zero to $q^{2}_{\rm
max}$. The nonmonotone behavior of
$\Lambda_{b}\to\Lambda_{c}(1/2^{-})\ell^{-}\nu_{\ell}$ is consistent with
Becirevic:2020nmb . The dependence on the lepton mass in the hadron
polarization is less than the other physical observables. Our estimation of
$\langle P_{B}\rangle_{\Lambda_{c}\to\Lambda e^{+}\nu_{e}}=-0.87\pm 0.09$
agrees with $-0.86\pm 0.03\pm 0.02$ Hinson:2004pj  reported by CLEO
Collaboration. In Fig. 6, the lepton polarization is close to $P_{\ell}=-1$,
especially for electron modes. It stems from the weak interaction which is
purely left handed. With the negligible mass, the electrons are approximately
absolutely polarized. The line shapes of $\mu$ and $\tau$ modes can be
understood by their masses which change the chirality. As the $q^{2}$
increases, the leptons are highly boosted, so that their helicity are more
left-handed with the polarization approaching $-1$. The future experiments on
the values of these observables and their comparison with the predictions of
the present study would help us understand the corresponding channels and the
internal structures of the baryons. These observables are also important to
investigate the new physics effects beyond the SM Azizi:2019tcn .

In conclusion, we would like to focus on the ratios of branching fractions
which reflect the leptonic flavor universality

$R(\Lambda_{c}^{(*)})=\frac{Br(\Lambda_{b}\to\Lambda_{c}^{(*)}\tau^{-}\nu_{\tau})}{Br(\Lambda_{b}\to\Lambda_{c}^{(*)}\ell^{-}\nu_{\ell})},$
(42)

with $\ell^{-}=e^{-}$ or $\mu^{-}$. These ratios, obtained by our and other
approaches, e.g. HQET Bernlochner:2018kxh ; Bernlochner:2018bfn ; Han:2020sag
, QCDSR Azizi:2018axf , various quark models Pervin:2005ve ; Faustov:2016pal ;
Faustov:2016yza ; Gutsche:2018nks , the SU(3) flavor symmetry Geng:2019bfz ,
as well as LQCD Bernlochner:2018bfn ; Mu:2019bin , are displayed in Table 9.
Our estimation for $R(\Lambda_{c})$ is consistent with other models. The
$\tau$-channel for $\Lambda_{b}\to\Lambda_{c}^{*}$ decays is smaller than the
$\Lambda_{b}\to\Lambda_{c}$ process, due to the smaller phase spaces.

Table 9: Our predictions for the ratios of branching fractions $R(\Lambda_{c}^{(*)})$. | This work | Others
---|---|---
$R(\Lambda_{c})$ | $0.30\pm 0.09$ | about 0.3 Azizi:2018axf ; Bernlochner:2018bfn ; Faustov:2016pal ; Gutsche:2018nks ; Mu:2019bin ; Han:2020sag
$R\left(\Lambda_{c}\left(1/2^{-}\right)\right)$ | $0.14\pm 0.01$ | 0.13 Gutsche:2018nks , 0.21-0.31 Pervin:2005ve
$R\left(\Lambda_{c}^{*}\left(1/2^{+}\right)\right)$ | $0.10\pm 0.05$ |

##  V conclusion and discussion

In this work, we study the weak transition form factors for
$\Lambda_{b}\to\Lambda_{c}^{(*)}\left(1/2^{\pm}\right)$ and
$\Lambda_{c}\to\Lambda^{(*)}\left(1/2^{\pm}\right)$ in the framework of light-
front quark model, and investigate the involved semileptonic decays. A
semirelativistic three-body potential model and GEM are used to extract the
baryon wave functions, which are the input of our calculation. This treatment
of baryon wave functions is different from former ways of taking a simple
harmonic oscillator wave function with a $\beta$ value in the calculation. To
some extent, we can avoid the $\beta$ parameter dependence of the results
based on the support from baryon spectroscopy.

Our results of form factors for the ground-state modes are comparable to other
theoretical studies, especially to the LQCD or HQL. The ones for the excited
states involving modes need to be tested by more theories. These form factors
will be useful for studying the weak decays of $\Lambda_{b}$ and
$\Lambda_{b}$.

With the obtained form factors, we predict the branching fractions of the
considered channels. Our results are consistent with the experimental data and
the LQCD calculations. The excited-state modes have much smaller branching
fractions compared to the ground-state modes, which can explain the measured
result of $\mathcal{B}(\Lambda_{c}\to\Lambda
e^{+}\nu_{e})/\mathcal{B}(\Lambda_{c}\to e^{+}X)=(91.9\pm 13.6)\%$. The ratios
of
$R(\Lambda_{c}^{(*)})=\mathcal{B}(\Lambda_{b}\to\Lambda_{c}^{(*)}\tau^{-}\nu_{\tau})/\mathcal{B}(\Lambda_{b}\to\Lambda_{c}^{(*)}\ell^{-}\nu_{\ell})$
are predicted to be tested for the lepton flavor universality. Moreover, the
leptonic forward-backward asymmetry ($A_{FB}$), the final hadron polarization
($P_{B}$) and the lepton polarization ($P_{\ell}$) are also investigated. The
future BESIII, LHCb and Belle II experiments can measure these observables and
test our results.

## ACKNOWLEDGMENTS

We would like to thank Chun-Khiang Chua, Si-Qiang Luo and Jian-Peng Wang for
useful discussions. 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, and the National Natural Science Foundation of
China under Grant No. 11975112 and 12047501.

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