#  Semileptonic weak decays of anti-triplet charmed baryons in the light-front
formalism

C.Q. Geng1,2,3, Chia-Wei Liu1,2,3 and Tien-Hsueh Tsai3 1School of Fundamental
Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study,
UCAS, Hangzhou 310024, China
2International Centre for Theoretical Physics Asia-Pacific, Beijing/Hangzhou,
China
3Department of Physics, National Tsing Hua University, Hsinchu 300, Taiwan

###### Abstract

We systematically study the semileptonic decays of ${\bf B_{c}}\to{\bf
B_{n}}\ell^{+}\nu_{\ell}$ in the light-front constituent quark model, where
${\bf B_{c}}$ represent the anti-triplet charmed baryons of
$(\Xi_{c}^{0},\Xi_{c}^{+},\Lambda_{c}^{+})$ and ${\bf B_{n}}$ correspond to
the octet ones. We determine the spin-flavor structures of the constituents in
the baryons with the Fermi statistics and calculate the decay branching ratios
(${\cal B}$s) and averaged asymmetry parameters ($\alpha$s) with the helicity
formalism. In particular, we find that ${\cal B}(\Lambda_{c}^{+}\to\Lambda
e^{+}\nu_{e},ne^{+}\nu_{e})=(3.55\pm 1.04,0.36\pm 0.15)\%$, ${\cal
B}(\Xi_{c}^{+}\to\Xi^{0}e^{+}\nu_{e},\Sigma^{0}e^{+}\nu_{e},\Lambda
e^{+}\nu_{e})=(11.3\pm 3.35),0.33\pm 0.09,0.12\pm 0.04\%$ and ${\cal
B}(\Xi_{c}^{0}\to\Xi^{-}e^{+}\nu_{e},\Sigma^{-}e^{+}\nu_{e})=(3.49\pm
0.95,0.22\pm 0.06)\%$. Our results agree with the current experimental data.
Our prediction for ${\cal B}(\Lambda_{c}^{+}\to ne^{+}\nu_{e})$ is consistent
with those in the literature, which can be measured by the charm-facilities,
such as BESIII and BELLE. Some of our results for the $\Xi_{c}^{+(0)}$
semileptonic channels can be tested by the experiments at BELLE as well as the
ongoing ones at LHCb and BELLEII.

##  I introduction

In the recent few years, there have been two important experiments in charmed
baryon physics. One is the measurement of the absolute branching ratios of
${\cal B}(\Xi_{c}^{0}\to\Xi^{-}\pi^{+})=(1.8\pm 0.5)\%$ Li:2018qak  and ${\cal
B}(\Xi_{c}^{+}\to\Xi^{-}\pi^{+}\pi^{+})=(2.86\pm 1.21\pm 0.38)\%$ Li:2019atu
from the Belle Collaboration, and the other is the most precise measurement of
the $\Xi_{c}^{0}$’s lifetime of $\tau_{\Xi_{c}^{0}}=154.5\pm 1.7\pm 1.6\pm
1.0\text{ fs}$ from the LHCb Collaboration Aaij:2019lwg , which significantly
deviates from the past world averaged value of
$\tau_{\Xi_{c}^{0}}=112^{+13}_{-10}~{}\text{fs}$ in PDG Zyla:2020zbs . Both of
them bring us new hints as well as new problems in charm physics. We are now
in a precision-era of charm physics. It is expected that as more high quality
data will be accumulated in the future, stronger constraints on various
baryonic QCD models as well as physics beyond the standard model can be given.

Recently, the anti-triplet charm baryon decays have been extensively discussed
in the literature. However, due to the large non-perturbative effects from the
quantum chromodynamics (QCD), the decay amplitudes and observables of hadrons
are very hard to obtain from the QCD first principle. To avoid the
difficulties, various charmed baryon decay processes have been studied based
on the flavor symmetry of $SU(3)_{f}$, such as semi-leptonic, two-body and
three-body non-leptonic decays, to get reliable results Geng:2017mxn ;
Savage:1989qr ; Savage:1991wu ; zero ; He:2018joe ; He:2018php ; Wang:2017azm
; Geng:2018plk ; Geng:2018rse ; Geng:2018upx ; Geng:2019awr ; Geng:2019bfz ;
Grossman:2019xcj ; Lu:2016ogy ; Wang:2017gxe ; Cen:2019ims ; Hsiao:2019yur ;
Roy:2019cky ; Jia:2019zxi ; Geng:2020fng . It is known that $SU(3)_{f}$ is an
approximate symmetry, resulting in about $10\%$ of uncertainties for the
predictions inherently. Moreover, the $SU(3)_{f}$ symmetry itself does not
directly reveal any clue about the QCD dynamics. In fact, its applications
heavily rely on the experimental data as inputs. In order to have precise
calculations, we need a specific dynamical QCD model to understand each decay
process. For simplicity, we only discuss the semi-leptonic processes of the
anti-triplet charmed baryons in this study, which involve purely factorizable
contributions. There are several theoretical calculations on these decay
processes with different QCD frameworks in the literature Cheng:1995fe ;
Zhao:2018zcb ; PerezMarcial:1989yh ; Faustov:2019ddj ; Faustov:2016yza ;
Gutsche:2015rrt ; Meinel:2016dqj ; Meinel:2017ggx ; Azizi:2009wn ; Liu:2009sn
.

The light front (LF) QCD formalism in the quark model is a consistent
relativistic approach, which has been tested successfully in the mesonic and
light quark sectors in early times Schlumpf:1992ce ; Zhang:1994ti . Because of
these successes, it has been used in other generalized systems, such as those
containing the heavy mesons, pentaquarks and so on Cheng:2004cc ;
Chang:2019obq ; Zhao:2018mrg ; Xing:2018lre ; Cheng:2017pcq ; Geng:2000fs ;
Geng:2001vy ; Geng:2016pyr . Apart from the charm system, the bottom to
charmed baryon nonleptonic decays have been recently analyzed in the LF
formalism Chua:2018lfa . For a review on the comprehensive introduction of the
LF QCD and its vacuum structure, one can refer to Ref. Zhang:1994ti . For the
LF constituent quark model (LFCQM), we recommend Ref. Schlumpf:1992ce  and
references therein. The advantage of LFCQM is that we can boost the reference
frame without changing the equation of motion because of the commutativity of
the LF Hamiltonian and boost generators. It provides us with a great
convenience to calculate the wave-function in different inertial frames
because of the recoil effects in the transition form factors.

This paper is organized as follows. We first present our formal calculations
of the branching ratios and averaged decay asymmetries in terms of the
helicity amplitudes, the baryonic states in LFCQM, and the baryonic transition
form factors in Sec. II. In Sec. III, we show our numerical results and
compare them with those in the literature. In Sec. IV, we give our
conclusions.

##  II Formalism

###  II.1 Helicity Amplitudes and Observables

The effective Hamiltonian for the anti-triplet charmed baryon semi-leptonic
weak decays can be written as

$\displaystyle{\cal
H}_{eff}=\frac{G_{F}}{\sqrt{2}}V_{cq}(\bar{\nu_{\ell}}\ell)_{V-A}(\bar{q}c)_{V-A}$
(1)

where $G_{F}$ is the Fermi constant, $V_{cq}$ is the CKM matrix element, and
$(\bar{\ell}\nu_{\ell})_{V-A}$ and $(\bar{q}c)_{V-A}$ denote as the usual
$V-A$ currents $\bar{\ell}\gamma^{\mu}(1-\gamma_{5})\nu_{\ell}$ and
$\bar{q}\gamma_{\mu}(1-\gamma_{5})c$ with $q=d,s$, respectively. The weak
transition amplitudes of the anti-triplet charmed baryons are given by

$\displaystyle{\cal A}({\bf B}_{c}\to{\bf
B}_{n}\ell^{+}\nu_{\ell})=\frac{G_{F}}{\sqrt{2}}V_{cq}\bar{\ell}\gamma^{\mu}(1-\gamma_{5})\nu_{\ell}\langle{\bf
B}_{n}|\bar{q}\gamma_{\mu}(1-\gamma_{5})c|{\bf B}_{c}\rangle$ (2)

where the baryon transition matrix elements are parameterized by

$\displaystyle\langle{\bf
B}_{n},p_{f},S_{z}^{\prime}|\bar{q}\gamma_{\mu}(1-\gamma_{5})c|{\bf
B}_{c},p_{i},S_{z}\rangle$ $\displaystyle=\bar{u}_{{\bf
B}_{n}}(p_{f},S_{z}^{\prime})\left[\gamma_{\mu}f_{1}(k^{2})-i\sigma_{\mu\nu}\frac{k^{\nu}}{M_{{\bf
B}_{c}}}f_{2}(k^{2})+f_{3}(k^{2})\frac{k_{\mu}}{M_{{\bf B}_{c}}}\right]u_{{\bf
B}_{c}}(p_{i},S_{z})$ $\displaystyle-\bar{u}_{{\bf
B}_{n}}(p_{f},S_{z}^{\prime})\left[\gamma_{\mu}g_{1}(k^{2})-i\sigma_{\mu\nu}\frac{k^{\nu}}{M_{{\bf
B}_{c}}}g_{2}(k^{2})+g_{3}(k^{2})\frac{k_{\mu}}{M_{{\bf
B}_{c}}}\right]\gamma_{5}u_{{\bf B}_{c}}(p_{i},S_{z})$ (3)

with $k^{\mu}=p_{i}^{\mu}-p_{f}^{\mu}$,
$\sigma_{\mu\nu}=\frac{i}{2}\left[\gamma_{\mu},\gamma_{\nu}\right]$, and
$f_{i}(k^{2})$ and $g_{i}(k^{2})$ being the form factors describing the non-
perturbative QCD effect in the time-like range of $m_{\ell}^{2}<k^{2}<(M_{\bf
B_{c}}-M_{\bf B_{n}})^{2}$. We introduce a set of helicity amplitudes
$H^{V(A)}_{\lambda_{2}\lambda_{W}}$ to calculate the decay branching ratios
and other physical quantities, where $\lambda_{2}$ and $\lambda_{W}$ represent
the helicity quantum numbers of the daughter baryon and off-shell $W^{+}$
boson in the decay processes, respectively. These amplitudes give more
intuitive physical pictures about the helicity structures of the decay
processes. Furthermore, when we evaluate the asymmetries of these processes,
such as the integrated (averaged) decay asymmetry, also known as the
longitudinal polarization of the daughter baryon, these amplitudes result in
much simpler expressions than the traditional ones. Relations between the
helicity amplitudes and form factors are given byKadeer:2005aq ; Zhao:2018zcb

$\displaystyle H^{V}_{\frac{1}{2}1}$ $\displaystyle=$
$\displaystyle\sqrt{2K_{-}}\left(-f_{1}(k^{2})-\frac{M_{\bf B_{c}}+M_{\bf
B_{n}}}{M_{\bf B_{c}}}f_{2}(k^{2})\right)\,,$ $\displaystyle
H^{V}_{\frac{1}{2}0}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{K_{-}}}{\sqrt{k^{2}}}\left((M_{\bf B_{c}}+M_{\bf
B_{n}})f_{1}(k^{2})+\frac{k^{2}}{M_{\bf B_{c}}}f_{2}(k^{2})\right)\,,$
$\displaystyle H^{V}_{\frac{1}{2}t}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{K_{+}}}{\sqrt{k^{2}}}\left((M_{\bf B_{c}}+M_{\bf
B_{n}})f_{1}(k^{2})+\frac{k^{2}}{M_{\bf B_{c}}}f_{3}(k^{2})\right)\,,$
$\displaystyle H^{A}_{\frac{1}{2}1}$ $\displaystyle=$
$\displaystyle\sqrt{2K_{+}}\left(-g_{1}(k^{2})-\frac{M_{\bf B_{c}}+M_{\bf
B_{n}}}{M_{\bf B_{c}}}g_{2}(k^{2})\right)\,,$ $\displaystyle
H^{A}_{\frac{1}{2}0}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{K_{+}}}{\sqrt{k^{2}}}\left((M_{\bf B_{c}}-M_{\bf
B_{n}})g_{1}(k^{2})-\frac{k^{2}}{M_{\bf B_{c}}}g_{2}(k^{2})\right)\,,$
$\displaystyle H^{A}_{\frac{1}{2}t}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{K_{-}}}{\sqrt{k^{2}}}\left((M_{\bf B_{c}}-M_{\bf
B_{n}})g_{1}(k^{2})-\frac{k^{2}}{M_{\bf B_{c}}}g_{3}(k^{2})\right)\,,$ (4)

where $K_{\pm}=(M_{\bf B_{c}}\pm M_{\bf B_{n}})^{2}-k^{2}$.

The differential decay widths and asymmetries can be expressed in the
following analytic forms in terms of the helicity amplitudes with the non-
vanishing lepton masses of $m_{\ell}$ Geng:2019bfz ,

$\displaystyle\frac{d\Gamma}{dk^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\frac{G_{F}^{2}}{(2\pi)^{3}}\left|V_{qc}\right|^{2}\frac{\left(k^{2}-m_{\ell}^{2}\right)^{2}p}{8M_{\bf{B}_{c}}^{2}k^{2}}\left[\left(1+\frac{m_{\ell}^{2}}{2k^{2}}\right)\right.$
(5)
$\displaystyle\left.\left(\left|H_{\frac{1}{2}1}\right|^{2}+\left|H_{-\frac{1}{2}-1}\right|^{2}+\left|H_{\frac{1}{2}0}\right|^{2}+\left|H_{-\frac{1}{2}0}\right|^{2}\right)+\frac{3m_{\ell}^{2}}{2k^{2}}\left(\left|H_{\frac{1}{2}t}\right|^{2}+\left|H_{-\frac{1}{2}t}\right|^{2}\right)\right]\,,$
$\displaystyle\langle\alpha(k^{2})\rangle=\frac{\int
dk^{2}\frac{\left(k^{2}-m_{\ell}^{2}\right)^{2}p}{8M_{\bf{B}_{c}}^{2}k^{2}}\left[\left(1+\frac{m_{\ell}^{2}}{2k^{2}}\right)\right.\left.\left(\left|H_{\frac{1}{2}1}\right|^{2}-\left|H_{-\frac{1}{2}-1}\right|^{2}+\left|H_{\frac{1}{2}0}\right|^{2}-\left|H_{-\frac{1}{2}0}\right|^{2}\right)+\frac{3m_{\ell}^{2}}{2k^{2}}\left(\left|H_{\frac{1}{2}t}\right|^{2}-\left|H_{-\frac{1}{2}t}\right|^{2}\right)\right]}{\int
dk^{2}\frac{\left(k^{2}-m_{\ell}^{2}\right)^{2}p}{8M_{\bf{B}_{c}}^{2}k^{2}}\left[\left(1+\frac{m_{\ell}^{2}}{2k^{2}}\right)\right.\left.\left(\left|H_{\frac{1}{2}1}\right|^{2}+\left|H_{-\frac{1}{2}-1}\right|^{2}+\left|H_{\frac{1}{2}0}\right|^{2}+\left|H_{-\frac{1}{2}0}\right|^{2}\right)+\frac{3m_{\ell}^{2}}{2k^{2}}\left(\left|H_{\frac{1}{2}t}\right|^{2}+\left|H_{-\frac{1}{2}t}\right|^{2}\right)\right]}\,,$
(6)

where $p=\sqrt{K_{+}K_{-}}/{2M_{\bf B_{c}}}$ and
$H_{\lambda_{2}\lambda_{W}}=H_{\lambda_{2}\lambda_{W}}^{V}-H_{\lambda_{2}\lambda_{W}}^{A}$.

###  II.2 Light Front Constituent Quark Model

From Eqs. (5) and (6), we see that as long as the time-like form factors are
known, both branching ratios and averaged decay asymmetries can be determined.
To calculate these time-like form factors, we use LFCQM, in which a baryon is
treated as a bound state of three constituent quarks quantized in the LF
formalism and its state is denoted by the momentum $P$, canonical spin $S$ and
the z-direction projection of spin $S_{z}$, respectively. As a result, the
baryon state can be expressed by Zhang:1994ti ; Schlumpf:1992ce ; Cheng:2004cc
; Ke:2007tg ; Ke:2012wa ; Ke:2019smy

$\displaystyle|{\bf B}$
$\displaystyle,P,S,S_{z}\rangle=\int\\{{d^{3}{\tilde{p}}}\\}2(2\pi)^{3}\frac{1}{\sqrt{P^{+}}}\delta^{3}(\tilde{P}-\tilde{p}_{1}-\tilde{p}_{2}-\tilde{p}_{3})$
(7) $\displaystyle\times$
$\displaystyle\sum_{\lambda_{1},\lambda_{2},\lambda_{3}}\Psi^{SS_{z}}(\tilde{p}_{1},\tilde{p}_{2},\tilde{p}_{3},\lambda_{1},\lambda_{2},\lambda_{3})C^{\alpha\beta\gamma}F_{abc}|q_{\alpha}^{a}(\tilde{p}_{1},\lambda_{1})q_{\beta}^{b}(\tilde{p}_{2},\lambda_{2})q_{\gamma}^{c}(\tilde{p}_{3},\lambda_{3})\rangle\,,$

where
$\Psi^{SS_{z}}(\tilde{p}_{1},\tilde{p}_{2},\tilde{p}_{3},\lambda_{1},\lambda_{2},\lambda_{3})$
is the vertex function describing the overlapping between the baryon and its
constituents, which can be formally solved from the three-body Bethe-Salpeter
equations, $C^{\alpha\beta\gamma}$ ($F_{abc}$) are the color (flavor) factors,
$\lambda_{i}$ and $\tilde{p}_{i}$ with $i=1,2,3$ are the LF helicities and
3-momenta of the on-mass-shell constituent quarks, defined as

$\displaystyle\tilde{p}_{i}=(p_{i}^{+},p_{i\perp})\,,~{}p_{i\perp}=(p_{i}^{1},p_{i}^{2})\,,~{}p_{i}^{-}=\frac{m_{i}^{2}+p_{i\perp}^{2}}{p_{i}^{+}}\,,$
(8)

and $|q_{\alpha}^{a}(\tilde{p},\lambda)\rangle$ and $\\{{d^{3}\tilde{p}}\\}$
correspond to the light front constituent quark states and the integral
measure, given by

$\displaystyle|q_{\alpha}^{a}(\tilde{p},\lambda)\rangle=d^{\dagger
a}_{\alpha}(\tilde{p},\lambda)|0\rangle\,,~{}\\{{d^{3}\tilde{p}}\\}\equiv\prod_{i=1,2,3}\frac{dp_{i}^{+}d^{2}p_{i\perp}}{2(2\pi)^{3}}\,,$
(9)

respectively, with the quark field operators satisfied the following anti-
commutation relations

$\displaystyle~{}\\{d_{\alpha^{\prime}}^{a^{\prime}}(\tilde{p^{\prime}},\lambda^{\prime}),d^{\dagger
a}_{\alpha}(\tilde{p},\lambda)\\}=2(2\pi)^{3}\delta^{3}(\tilde{p^{\prime}}-\tilde{p})\delta_{\lambda^{\prime}\lambda}\delta_{\alpha^{\prime}\alpha}\delta^{a^{\prime}a}\,,\delta^{3}(\tilde{p})=\delta(p^{+})\delta^{2}(p_{\perp})\,.$
(10)

To separate the internal motion of the constituents from the bulk motion, we
use the kinematic variables of $(q_{\perp},\xi)$, $(Q_{\perp},\eta)$ and
$P_{tot}$, given by

$\displaystyle P_{tot}$ $\displaystyle=$
$\displaystyle\tilde{P}_{1}+\tilde{P}_{2}+\tilde{P}_{3},\qquad\xi=\frac{p_{1}^{+}}{p_{1}^{+}+p_{2}^{+}},\qquad\eta=\frac{p_{1}^{+}+p_{2}^{+}}{P_{tot}^{+}}\,,$
$\displaystyle q_{\perp}$ $\displaystyle=$ $\displaystyle(1-\xi)p_{1\perp}-\xi
p_{2\perp},\quad Q_{\perp}=(1-\eta)(p_{1\perp}+p_{2\perp})-\eta p_{3\perp}\,,$
(11)

where $(q_{\perp},\xi)$ and $(Q_{\perp},\eta)$ capture the relative motions
between the first and second quarks, and the third and other two quarks,
respectively. We consider the three constituent quarks in the baryon
independently with suitable spin-flavor structures satisfying the Fermi
statistics to have a correct baryon bound state system. The vertex function of
$\Psi^{SS_{z}}(\tilde{p}_{1},\tilde{p}_{2},\tilde{p}_{3},\lambda_{1},\lambda_{2},\lambda_{3})$
in Eq. (7) can be further written into two parts Lorce:2011dv ; Zhang:1994ti ;
Schlumpf:1992ce ,

$\displaystyle\Psi^{SS_{z}}(\tilde{p}_{1},\tilde{p}_{2},\tilde{p}_{3},\lambda_{1},\lambda_{2},\lambda_{3})$
$\displaystyle=$
$\displaystyle\phi(q_{\perp},\xi,Q_{\perp},\eta)\Xi^{SS_{z}}(\lambda_{1},\lambda_{2},\lambda_{3})\,,$
(12)

where $\phi(q_{\perp},\xi,Q_{\perp},\eta)$ is the momentum distribution of
constituent quarks and $\Xi^{SS_{z}}(\lambda_{1},\lambda_{2},\lambda_{3})$
represents the momentum-dependent spin wave function, given by

$\displaystyle\Xi^{SS_{z}}(\lambda_{1},\lambda_{2},\lambda_{3})$
$\displaystyle=$
$\displaystyle\sum_{s_{1},s_{2},s_{3}}\langle\lambda_{1}|R^{\dagger}_{1}|s_{1}\rangle\langle\lambda_{2}|R^{\dagger}_{2}|s_{2}\rangle\langle\lambda_{3}|R^{\dagger}_{3}|s_{3}\rangle\bigg{\langle}\frac{1}{2}s_{1},\frac{1}{2}s_{2},\frac{1}{2}s_{3}\bigg{|}SS_{z}\bigg{\rangle}\,,$
(13)

with the $SU(2)$ Clebsch-Gordan coefficients of
$\big{\langle}\frac{1}{2}s_{1},\frac{1}{2}s_{2},\frac{1}{2}s_{3}\big{|}SS_{z}\big{\rangle}$,
and $R_{i}$ is the Melosh transformation matrix Polyzou:2012ut ;
Schlumpf:1992ce , which corresponds to the $i$th constituent quark, expressed
by

$\displaystyle R_{M}(x,p_{\perp},m,M)$ $\displaystyle=$
$\displaystyle\frac{m+xM-i\vec{\sigma}\cdot(\vec{n}\times\vec{q})}{\sqrt{(m+xM)^{2}+q_{\perp}^{2}}}\,,$
(14) $\displaystyle
R_{1}=R_{M}(\eta,Q_{\perp},M_{3},M)R_{M}(\xi,q_{\perp},m_{1},M_{3})\,,$
$\displaystyle
R_{2}=R_{M}(\eta,Q_{\perp},M_{3},M)R_{M}(1-\xi,-q_{\perp},m_{2},M_{3})\,,$
$\displaystyle R_{3}=R_{M}(1-\eta,-Q_{\perp},m_{3},M)\,.$

Here, $\vec{\sigma}$ stands for the Pauli matrix, $\vec{n}=(0,0,1)$, and $M$
and $M_{3}$ are invariant masses of $(q_{\perp},\xi)$ and $(Q_{\perp},\eta)$
systems, represented by Schlumpf:1992ce

$\displaystyle
M_{3}^{2}=\frac{q_{\perp}^{2}}{\xi(1-\xi)}+\frac{m_{1}^{2}}{\xi}+\frac{m_{2}^{2}}{1-\xi}\,,$
$\displaystyle
M^{2}=\frac{Q_{\perp}^{2}}{\eta(1-\eta)}+\frac{M_{3}^{2}}{\eta}+\frac{m_{3}^{2}}{1-\eta}\,,$
(15)

respectively.

The spin-flavor structures of ${\bf B_{c}}$ and ${\bf B_{n}}$ are given by

$\displaystyle|{\bf B_{c}}\rangle=\frac{1}{\sqrt{6}}[\phi_{3}\chi^{\rho
3}(|q_{1}q_{2}c\rangle-|q_{2}q_{1}c\rangle)+\phi_{2}\chi^{\rho
2}(|q_{1}cq_{2}\rangle-|q_{2}cq_{1}\rangle)+\phi_{1}\chi^{\rho
1}(|cq_{1}q_{2}\rangle-|cq_{2}q_{1}\rangle)]\,,$
$\displaystyle|\Lambda^{0}\rangle=\frac{1}{\sqrt{6}}\phi[\chi^{\rho
3}(|dus\rangle-|uds\rangle)+\chi^{\rho 2}(|dsu\rangle-|usd\rangle)+\chi^{\rho
1}(|sdu\rangle-|sud\rangle)]\,,$
$\displaystyle|\Sigma^{0}\rangle=\frac{1}{\sqrt{6}}\phi[\chi^{\lambda
3}(|dus\rangle+|uds\rangle)+\chi^{\lambda
2}(|dsu\rangle+|usd\rangle)+\chi^{\lambda 1}(|sdu\rangle+|sud\rangle)]\,,$
$\displaystyle|B_{n}\rangle=\frac{1}{\sqrt{3}}\phi[\chi^{\lambda_{3}}|q_{1}q_{1}q_{2}\rangle+\chi^{\lambda_{2}}|q_{1}q_{2}q_{1}\rangle+\chi^{\lambda_{1}}|q_{2}q_{1}q_{1}\rangle]\,,$
(16)

where $q_{1(2)}=u,d,s$, $B_{n}=(N,\Xi,\Sigma^{+},\Sigma^{-})$, $\phi_{(i)}$ is
the momentum distribution of the constituents with the corresponding spin-
flavor configuration, and

$\displaystyle\chi^{\rho 3}_{\uparrow}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|\uparrow\downarrow\uparrow\rangle-|\downarrow\uparrow\uparrow\rangle)\,,\quad\chi^{\lambda
3}_{\uparrow}=\frac{1}{\sqrt{6}}(|\uparrow\downarrow\uparrow\rangle+|\downarrow\uparrow\uparrow\rangle-2|\uparrow\uparrow\downarrow\rangle)\,,$
(17)

In principle, one could solve $\phi_{(i)}$ from the Bethe-Salpeter equation
with an explicit QCD-inspired potential but it is beyond the scope of this
paper. Nonetheless, we use a Gaussian type distribution with the
phenomenological shape parameters $\beta_{Q}$ and $\beta_{q}$ to describe the
relative motions of constituents. Consequently, we represent the LF kinematic
variables $(\xi,q_{\perp})$ and $(\eta,Q_{\perp})$ in the forms of ordinary
3-momenta ${\bf q}=(q_{\perp},q_{z})$ and ${\bf Q}=(Q_{\perp},Q_{z})$:

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle{\cal
N}\sqrt{\frac{\partial q_{z}}{\partial\xi}\frac{\partial
Q_{z}}{\partial\eta}}e^{-\frac{{\bf Q}^{2}}{2\beta_{Q}^{2}}-\frac{{\bf
q}^{2}}{2\beta_{q}^{2}}}\,,$ $\displaystyle q_{z}$ $\displaystyle=$
$\displaystyle\frac{\xi
M_{3}}{2}-\frac{m_{1}^{2}+q_{\perp}^{2}}{2M_{3}\xi}\,,\quad Q_{z}=\frac{\eta
M}{2}-\frac{M_{3}^{2}+Q_{\perp}^{2}}{2M\eta}\,,$ (18)

where $\cal N$ is the normalization constant. Since the one-particle baryonic
state is normalized as

$\displaystyle\langle{\bf B}$
$\displaystyle,P^{\prime},S^{\prime},S^{\prime}_{z}|{\bf B}$
$\displaystyle,P,S,S_{z}\rangle=2(2\pi)^{3}P^{+}\delta^{3}(\tilde{P^{\prime}}-\tilde{P})\delta_{S_{z}^{\prime}S_{z}}\,,$
(19)

the normalization condition of the momentum wave function is given by

$\displaystyle\frac{1}{2^{2}(2\pi)^{6}}\int d\xi d\eta
d^{2}q_{\perp}d^{2}Q_{\perp}|\phi_{(3)}|^{2}=1\,.$ (20)

In this paper, we take different shape parameters of $\beta_{q}$ and
$\beta_{Q}$ in the momentum wave functions $\phi_{i}$ to describe the scalar
diquark effects in ${\bf B_{c}}$. On the other hand, we assume the momentum-
distribution function $\phi$ of octet baryons ${\bf B_{n}}$ is flavor
symmetric for all constituents. In other words, the $SU(3)_{f}$ flavor
symmetry is hold in the momentum wave function of ${\bf B_{n}}$. As a result,
the shape parameters of $\phi$ are equal, $i.e.$ $\beta_{Q{\bf
B_{n}}}=\beta_{q{\bf B_{n}}}=\beta_{{\bf B_{n}}}$. Note that there is no
$SU(6)$ spin-flavor symmetry in ${\bf B_{c}}$ and ${\bf B_{n}}$ because of the
momentum-dependent Melosh transformation even though the forms of these states
are similar to those with the $SU(6)$ spin-flavor wave functions.

###  II.3 Transition form factors

We pick the $\bar{q}\gamma^{+}(1-\gamma_{5})c$ current or so-called “good
component” of the baryon transition amplitudes

$\displaystyle\langle{\bf
B}_{n},p_{f},S_{z}^{\prime}|\bar{q}\gamma^{+}(1-\gamma_{5})c|{\bf
B}_{c},p_{i},S_{z}\rangle$ $\displaystyle=\bar{u}_{{\bf
B}_{n}}(p_{f},S_{z}^{\prime})\left[\gamma^{+}f_{1}(k^{2})-i\sigma^{+\nu}\frac{k_{\nu}}{M_{{\bf
B}_{c}}}f_{2}(k^{2})+f_{3}(k^{2})\frac{k^{+}}{M_{{\bf B}_{c}}}\right]u_{{\bf
B}_{c}}(p_{i},S_{z})$ $\displaystyle-\bar{u}_{{\bf
B}_{n}}(p_{f},S_{z}^{\prime})\left[\gamma^{+}g_{1}(k^{2})-i\sigma^{+\nu}\frac{k_{\nu}}{M_{{\bf
B}_{c}}}g_{2}(k^{2})+g_{3}(k^{2})\frac{k^{+}}{M_{{\bf
B}_{c}}}\right]\gamma_{5}u_{{\bf B}_{c}}(p_{i},S_{z})$ (21)

and choose the frame such that $p^{+}_{i(f)}$ is conserved
($k^{+}=0,k^{2}=-k_{\perp}^{2}$) to calculate the form factors to avoid zero-
mode contributions and other $x^{+}$-ordered diagrams in the LF formalism
Schlumpf:1992ce ; Zhang:1994ti . The Matrix elements of the vector and axial-
vector currents at quark level correspond to three different lowest-order
Feynman diagrams as shown in Fig. 1.

(a)

(b) .

(c)

Figure 1: Feynman diagrams for the baryonic weak transitions at the lowest
order, where the sign of “$\bullet$” denotes the V-A current vertex, where (a)
$p^{\prime}_{1}-p_{1}=k$, (b) $p^{\prime}_{2}-p_{2}=k$ and (c)
$p^{\prime}_{3}-p_{3}=k$.

Since the spin-flavor-momentum wave functions of baryons are totally symmetric
under the permutation of constituents, we have that
$(a)+(b)+(c)=3(a)=3(b)=3(c)$ Schlumpf:1992ce . We only present the calculation
for the diagram (c), which contains simpler and cleaner forms with our
notation $(q_{\perp},Q_{\perp},\xi,\eta)$ as a demonstration. We can extract
the form factors from the matrix elements through the relations

$\displaystyle f_{1}(k^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2p^{+}_{i}}\langle{\bf
B_{n}},p_{f},\uparrow|\bar{q}\gamma^{+}c|{\bf B_{c}},p_{i},\uparrow\rangle\,,$
$\displaystyle f_{2}(k^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2p^{+}_{i}}\frac{M_{{\bf B_{c}}}}{k_{\perp}}\langle{\bf
B_{n}},p_{f},\uparrow|\bar{q}\gamma^{+}c|{\bf
B_{c}},p_{i},\downarrow\rangle\,,$ $\displaystyle g_{1}(k^{2})$
$\displaystyle=$ $\displaystyle\frac{1}{2p^{+}_{i}}\langle{\bf
B_{n}},p_{f},\uparrow|\bar{q}\gamma^{+}\gamma_{5}c|{\bf
B_{c}},p_{i},\uparrow\rangle\,,$ $\displaystyle g_{2}(k^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2p^{+}_{i}}\frac{M_{{\bf B_{c}}}}{k_{\perp}}\langle{\bf
B_{n}},p_{f},\uparrow|\bar{q}\gamma^{+}\gamma_{5}|{\bf
B_{c}},p_{i},\downarrow\rangle\,.$ (22)

Note that $f_{3}$ and $g_{3}$ are not available when $k^{+}=0$, but they are
negligible because of the suppressions of $k^{2}/M_{\bf B_{c}}^{2}$. In fact,
$f_{3}$ and $g_{3}$ are only associated with $H_{\frac{1}{2}t}^{V(A)}$ which
do not contribute to the semileptonic decays in the massless lepton limit
Geng:2019bfz . As a result, we can safely set both $f_{3}$ and $g_{3}$ to be 0
in this study. With the help of the momentum distribution functions and Melosh
transformation matrix in Eq. (7), the transition matrix elements can be
written as

$\displaystyle\langle{\bf
B_{n}},p_{f},S^{\prime}_{z}|\bar{q}\gamma^{+}(\gamma^{5})c|{\bf
B_{c}},p_{i},S_{z}\rangle$ $\displaystyle=\frac{1}{2^{2}(2\pi)^{6}}\int d\xi
d\eta
d^{2}q_{\perp}d^{2}Q_{\perp}\phi(q^{\prime}_{\perp},\xi,Q^{\prime}_{\perp},\eta)\phi_{3}(q_{\perp},\xi,Q_{\perp},\eta)F^{lmn}F_{ijk}\delta_{l}^{i}\delta_{m}^{j}$
$\displaystyle\times\sum_{s_{1},s_{2},s_{3}}\sum_{s^{\prime}_{1},s^{\prime}_{2},s^{\prime}_{3}}\langle
S^{\prime},S^{\prime}_{z}|s^{\prime}_{1},s^{\prime}_{2},s^{\prime}_{3}\rangle\langle
s_{1},s_{2},s_{3}|S,S_{z}\rangle\langle
s^{\prime}_{1}|R^{\prime}_{1}R^{\dagger}_{1}|s_{1}\rangle\langle
s^{\prime}_{2}|R^{\prime}_{2}R^{\dagger}_{2}|s_{2}\rangle$
$\displaystyle\times 2P^{+}(3\delta_{n}^{q}\delta^{k}_{c})\langle
s^{\prime}_{3}|R^{\prime}_{3}\sum_{\lambda^{\prime}_{3}\lambda_{3}}\left(\delta^{\lambda_{3}}_{\lambda^{\prime}_{3}}([\sigma_{z}]^{\lambda_{3}}_{\lambda^{\prime}_{3}})|\lambda^{\prime}_{3}\rangle\langle\lambda_{3}|\right)R_{3}^{\dagger}|s_{3}\rangle\,,$
(23)

where the indices of $q$ and $c$ in the delta symbols correspond to the quark
flavors in the $\bar{q}\gamma^{+}(\gamma^{5})c$ current,
$q_{\perp}^{\prime}=q_{\perp}$ and $Q_{\perp}^{\prime}=Q_{\perp}+k_{\perp}$.
Using Eqs. (22) and (23), we find that

$\displaystyle f_{1}(k^{2})=\frac{3}{2^{2}(2\pi)^{6}}\int d\xi d\eta
d^{2}q_{\perp}d^{2}Q_{\perp}\phi(q^{\prime}_{\perp},\xi,Q^{\prime}_{\perp},\eta)\phi_{3}(q_{\perp},\xi,Q_{\perp},\eta)(F^{lmn}F_{ijk}\delta_{n}^{q}\delta^{k}_{c}\delta_{l}^{i}\delta_{m}^{j})$
$\displaystyle\times\sum_{s_{1},s_{2},s_{3}}\sum_{s^{\prime}_{1},s^{\prime}_{2},s^{\prime}_{3}}\langle
S^{\prime},\uparrow|s^{\prime}_{1},s^{\prime}_{2},s^{\prime}_{3}\rangle\langle
s_{1},s_{2},s_{3}|S,\uparrow\rangle\prod_{i=1,2,3}\langle
s^{\prime}_{i}|R^{\prime}_{i}R^{\dagger}_{i}|s_{i}\rangle\,,$ (24)
$\displaystyle g_{1}(k^{2})=\frac{3}{2^{2}(2\pi)^{6}}\int d\xi d\eta
d^{2}q_{\perp}d^{2}Q_{\perp}\phi(q^{\prime}_{\perp},\xi,Q^{\prime}_{\perp},\eta)\phi_{3}(q_{\perp},\xi,Q_{\perp},\eta)(F^{lmn}F_{ijk}\delta_{n}^{q}\delta^{k}_{c}\delta_{l}^{i}\delta_{m}^{j})$
$\displaystyle\times\sum_{s_{1},s_{2},s_{3}}\sum_{s^{\prime}_{1},s^{\prime}_{2},s^{\prime}_{3}}\langle
S^{\prime},\uparrow|s^{\prime}_{1},s^{\prime}_{2},s^{\prime}_{3}\rangle\langle
s_{1},s_{2},s_{3}|S,\uparrow\rangle\prod_{i=1,2}\langle
s^{\prime}_{i}|R^{\prime}_{i}R^{\dagger}_{i}|s_{i}\rangle\langle
s^{\prime}_{3}|R^{\prime}_{3}\sigma_{z}R^{\dagger}_{3}|s_{3}\rangle\,,$ (25)
$\displaystyle f_{2}(k^{2})=\frac{3}{2^{2}(2\pi)^{6}}\frac{M_{{\bf
B_{c}}}}{k_{\perp}}\int d\xi d\eta
d^{2}q_{\perp}d^{2}Q_{\perp}\phi(q^{\prime}_{\perp},\xi,Q^{\prime}_{\perp},\eta)\phi_{3}(q_{\perp},\xi,Q_{\perp},\eta)(F^{lmn}F_{ijk}\delta_{n}^{q}\delta^{k}_{c}\delta_{l}^{i}\delta_{m}^{j})$
(26)
$\displaystyle\times\sum_{s_{1},s_{2},s_{3}}\sum_{s^{\prime}_{1},s^{\prime}_{2},s^{\prime}_{3}}\langle
S^{\prime},\uparrow|s^{\prime}_{1},s^{\prime}_{2},s^{\prime}_{3}\rangle\langle
s_{1},s_{2},s_{3}|S,\downarrow\rangle\prod_{i=1,2,3}\langle
s^{\prime}_{i}|R^{\prime}_{i}R^{\dagger}_{i}|s_{i}\rangle\,,$ $\displaystyle
g_{2}(k^{2})=\frac{3}{2^{2}(2\pi)^{6}}\frac{M_{{\bf B_{c}}}}{k_{\perp}}\int
d\xi d\eta
d^{2}q_{\perp}d^{2}Q_{\perp}\phi(q^{\prime}_{\perp},\xi,Q^{\prime}_{\perp},\eta)\phi_{3}(q_{\perp},\xi,Q_{\perp},\eta)(F^{lmn}F_{ijk}\delta_{n}^{q}\delta^{k}_{c}\delta_{l}^{i}\delta_{m}^{j})$
$\displaystyle\times\sum_{s_{1},s_{2},s_{3}}\sum_{s^{\prime}_{1},s^{\prime}_{2},s^{\prime}_{3}}\langle
S^{\prime},\uparrow|s^{\prime}_{1},s^{\prime}_{2},s^{\prime}_{3}\rangle\langle
s_{1},s_{2},s_{3}|S,\downarrow\rangle\prod_{i=1,2}\langle
s^{\prime}_{i}|R^{\prime}_{i}R^{\dagger}_{i}|s_{i}\rangle\langle
s^{\prime}_{3}|R^{\prime}_{3}\sigma_{z}R^{\dagger}_{3}|s_{3}\rangle\,.$ (27)

##  III Numerical Results

To find out the decay branching ratios and averaged asymmetries in the
helicity formalism, we first calculate the transition form factors with LFCQM.
By imposing the condition $k^{+}=0$, the form factors can be evaluated only in
the space-like region ($k^{2}=-k^{2}_{\perp}$) instead of the time-like one.
Nonetheless, we still can extract the time-like information of the form
factors via analytically continuations Ke:2007tg ; Cheng:2004cc ; Jaus:1991cy
; Jaus:1996np .

We fit $f_{1(2)}(k^{2})$ and $g_{1(2)}(k^{2})$ with the analytic functions in
the space-like region with the following form

$\displaystyle F(k^{2})=\frac{F(0)}{1-q_{1}k^{2}+q_{2}k^{4}}\,.$ (28)

We employ the numerical values of the constituent quark masses and shape
parameters in Table. I.

Table 1: Values of the constituent quark masses ($m_{i}$) and shape parameters ($\beta_{q{\bf B_{c}}}$, $\beta_{{Q\bf B_{c}}},\beta_{{\bf B_{n}}}$) in units of GeV. $m_{c}$ | $m_{s}$ | $m_{d}$ | $m_{c}$ | $\beta_{q\Lambda_{c}}$ | $\beta_{Q\Lambda_{c}}$
---|---|---|---|---|---
1.3 | 0.4 | 0.26 | 0.26 | 0.44 | 0.53
$\beta_{q\Xi_{c}}$ | $\beta_{Q\Xi_{c}}$ | $\beta_{N}$ | $\beta_{\Lambda}$ | $\beta_{\Sigma}$ | $\beta_{\Xi}$
0.49 | 0.53 | 0.32 | 0.34 | 0.34 | 0.37

The values of the shape parameters can be determined approximately by the
calculations in the mesonic cases Chang:2018zjq ; Ke:2019smy . Because the
strength of the quark-quark pairs is a half of the quark-anti-quark one
Ke:2019smy , we will get the shape parameters of the quark pairs, which are
approximately $\sqrt{2}$ smaller than those in the mesonic cases.

We adopt $\beta_{q\Lambda_{c}}\simeq 2(\beta_{u\bar{d}}/\sqrt{2})$ and
$\beta_{q\Xi_{c}}\simeq 2(\beta_{s\bar{u}(\bar{d})}/\sqrt{2})$, where the
factor of 2 comes from the effects of the diquark clustering, making the light
quark pairs to be more compact in ${\bf B_{c}}$ baryons. For the octet baryons
of ${\bf B_{n}}$, we assume that the $SU(3)_{f}$ flavor symmetry is hold, and
therefore, the shape parameters is flavor symmetric for each constituent,
$i.e.$ $\beta_{Q}=\beta_{q}$. As a result, we approximate the shape parameters
of the octet baryons equal to the mesonic ones by effectively treating any
pair of two constituents as a heavier anti-quark. By using Eqs. (II.3)-(27),
we numerically compute 32 points for all form factors from $k^{2}=0$ to
$k^{2}=-9.7\text{ GeV}^{2}$ and fit them with the MATLAB curve fitting
toolbox. We present our fitting results of the form factors in Tables 2, 3 and
4, and our predictions of the branching ratios and asymmetry parameters in
Table 5. The comparisons with different theoretical models are presented in
Tables 6, 7 and 8, respectively.

Table 2: Fitting results of the $\Lambda_{c}^{+}\to{\bf B_{n}}$ form factors
in LFCQM $\Lambda_{c}^{+}\to\Lambda$
---
| $f_{1}$ | $f_{2}$ | $g_{1}$ | $g_{2}$
$F(0)$ | $0.67\pm 0.01$ | $0.76\pm 0.02$ | $0.59\pm 0.01$ | $(3.8\pm 1.2)\times 10^{-3}$
$q_{1}$ | $1.48\pm 0.31$ | $1.44\pm 0.30$ | $1.22\pm 0.28$ | $4.99\pm 17.6$
$q_{2}$ | $2.29\pm 0.49$ | $2.23\pm 0.48$ | $1.82\pm 0.39$ | $24.8\pm 95.8$
$\Lambda_{c}^{+}\to n$
| $f_{1}$ | $f_{2}$ | $g_{1}$ | $g_{2}$
$F(0)$ | $0.83\pm 0.01$ | $1.05\pm 0.02$ | $0.71\pm 0.01$ | $0.27\pm 0.01$
$q_{1}$ | $1.25\pm 0.36$ | $1.20\pm 0.30$ | $0.94\pm 0.28$ | $1.37\pm 0.40$
$q_{2}$ | $1.85\pm 0.68$ | $1.78\pm 0.48$ | $1.36\pm 0.39$ | $2.08\pm 0.83$
Table 3: Fitting results of the $\Xi_{c}^{+}\to{\bf B_{n}}$ form factors in
LFCQM,  $\Xi_{c}^{+}\to\Xi^{0}$
---
| $f_{1}$ | $f_{2}$ | $g_{1}$ | $g_{2}$
$F(0)$ | $0.77\pm 0.02$ | $0.96\pm 0.02$ | $0.69\pm 0.01$ | $(6.8\pm 0.3)\times 10^{-3}$
$q_{1}$ | $1.50\pm 0.31$ | $1.45\pm 0.31$ | $1.25\pm 0.28$ | $2.00\pm 0.76$
$q_{2}$ | $2.30\pm 0.49$ | $2.26\pm 0.49$ | $1.85\pm 0.39$ | $3.00\pm 1.41$
$\Xi_{c}^{+}\to\Sigma^{0}$
| $f_{1}$ | $f_{2}$ | $g_{1}$ | $g_{2}$
$F(0)$ | $0.52\pm 0.01$ | $0.70\pm 0.02$ | $0.45\pm 0.01$ | $0.08\pm 0.01$
$q_{1}$ | $1.49\pm 0.32$ | $1.43\pm 0.33$ | $1.18\pm 0.28$ | $1.88\pm 0.39$
$q_{2}$ | $2.35\pm 0.51$ | $2.38\pm 0.53$ | $1.79\pm 0.38$ | $2.88\pm 0.67$
$\Xi_{c}^{+}\to\Lambda$
| $f_{1}$ | $f_{2}$ | $g_{1}$ | $g_{2}$
$F(0)$ | $0.28\pm 0.01$ | $0.38\pm 0.01$ | $0.25\pm 0.01$ | $0.04\pm 0.01$
$q_{1}$ | $1.50\pm 0.31$ | $1.35\pm 0.51$ | $1.18\pm 0.28$ | $1.71\pm 0.38$
$q_{2}$ | $2.32\pm 0.50$ | $2.30\pm 0.81$ | $1.77\pm 0.38$ | $2.78\pm 0.66$
Table 4: Fitting results of the $\Xi_{c}^{0}\to{\bf B_{n}}$ form factors in
LFCQM,  $\Xi_{c}^{0}\to\Xi^{-}$
---
| $f_{1}$ | $f_{2}$ | $g_{1}$ | $g_{2}$
$F(0)$ | $0.74\pm 0.02$ | $0.96\pm 0.02$ | $0.69\pm 0.01$ | $(6.8\pm 0.3)\times 10^{-3}$
$q_{1}$ | $1.50\pm 0.31$ | $1.47\pm 0.31$ | $1.21\pm 0.28$ | $2.00\pm 0.76$
$q_{2}$ | $2.30\pm 0.50$ | $2.25\pm 0.48$ | $1.98\pm 0.39$ | $3.00\pm 1.41$
$\Xi_{c}^{0}\to\Sigma^{-}$
| $f_{1}$ | $f_{2}$ | $g_{1}$ | $g_{2}$
$F(0)$ | $0.73\pm 0.01$ | $0.99\pm 0.02$ | $0.63\pm 0.01$ | $0.11\pm 0.01$
$q_{1}$ | $1.49\pm 0.32$ | $1.43\pm 0.33$ | $1.18\pm 0.28$ | $1.88\pm 0.39$
$q_{2}$ | $2.35\pm 0.51$ | $2.38\pm 0.53$ | $1.79\pm 0.38$ | $2.88\pm 0.70$
Table 5: Predictions of the decay branching ratios and asymmetry parameters | $\ell^{+}=e^{+}$ | $\ell^{+}=\mu^{+}$
---|---|---
| ${\cal B}(\%)$ | $\alpha$ | ${\cal B}(\%)$ | $\alpha$
$\Lambda_{c}^{+}\to\Lambda\ell^{+}\nu_{\ell}$ | $3.55\pm 1.04$ | $-0.97\pm 0.03$ | $3.40\pm 1.02$ | $-0.98\pm.0.02$
$\Lambda_{c}^{+}\to n\ell^{+}\nu_{\ell}$ | $0.36\pm 0.15$ | $-0.96\pm 0.04$ | $0.34\pm 0.15$ | $-0.96\pm 0.04$
$\Xi_{c}^{+}\to\Xi^{0}\ell^{+}\nu_{\ell}$ | $11.3\pm 3.35$ | $-0.97\pm 0.03$ | $10.8\pm 3.3$ | $-0.97\pm 0.03$
$\Xi_{c}^{+}\to\Sigma^{0}\ell^{+}\nu_{\ell}$ | $0.33\pm 0.09$ | $-0.98\pm 0.01$ | $0.31\pm 0.09$ | $-0.98\pm 0.02$
$\Xi_{c}^{+}\to\Lambda\ell^{+}\nu_{\ell}$ | $0.12\pm 0.04$ | $-0.98\pm 0.02$ | $0.11\pm 0.05$ | $-0.98\pm 0.02$
$\Xi_{c}^{0}\to\Xi^{-}\ell^{+}\nu_{\ell}$ | $3.49\pm 0.95$ | $-0.98\pm 0.02$ | $3.34\pm 0.94$ | $-0.98\pm 0.02$
$\Xi_{c}^{0}\to\Sigma^{-}\ell^{+}\nu_{\ell}$ | $0.22\pm 0.06$ | $-0.98\pm 0.02$ | $0.21\pm 0.06$ | $-0.98\pm 0.02$

The uncertainties of our results mainly arise from the extrapolations by the
analytical continuation method. The magnitudes of average asymmetry parameters
predicted by LFCQM almost reach their extremums in all channels. As shown in
Tables 6-8, we obtain that ${\cal B}(\Lambda_{c}^{+}\to\Lambda
e^{+}\nu_{e})=(3.55\pm 1.04)\%$, ${\cal
B}(\Xi_{c}^{+}\to\Xi^{0}e^{+}\nu_{e})=(11.3\pm 3.4)\%$ and ${\cal
B}(\Xi_{c}^{0}\to\Xi^{-}e^{+}\nu_{e})=(3.49\pm 0.95)\%$, which are all
consistent with the current data from PDG Zyla:2020zbs  and the BELLE
experiments Li:2018qak ; Li:2019atu . Our branching ratios are also consistent
with the predictions of the relativistic quark model (RQM) Faustov:2016yza ;
Faustov:2019ddj , the covariant constituent quark model (CCQM) Gutsche:2015rrt
and the $SU(3)_{F}$ approach Geng:2019bfz . The results given by the LF
formalism Zhao:2018zcb  and heavy quark effective theory (HQET) Cheng:1995fe
are half of ours because they choose the spin-flavor wave function of ${\bf
B_{c}}$ to be $c(q_{1}q_{2}-q_{2}q_{1})\chi_{s_{z}}^{\rho_{3}}$ instead of the
totally symmetric one. The averaged asymmetry parameter predicted by LFCQM in
$\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e}$ is $10\%$ lower than the data. Since
we do not consider any parameters about spin interactions in our
phenomenological wave functions, the helicity-structure-related results, such
as the decay asymmetries, are clearly not precise enough to explain the
experimental data. On the other hand, the prediction from RQM and the
$SU(3)_{f}$ approach are almost the same as the data. Since the authors for
RQM in Refs. Faustov:2016yza ; Faustov:2019ddj  have considered a
comprehensive QCD-inspired potential including the chromomagnetic effect,
their results are more close to the experimental values than ours. Meanwhile,
the $SU(3)_{f}$ approach is a model independent way to analyze ${\bf
B_{c}}\to{\bf B_{n}}\ell^{+}\nu_{\ell}$ decays, which automatically includes
the information related to the spin interaction when the asymmetry parameters
are used as the fitting inputs. Clearly, our results of the asymmetry
parameters could be improved by considering the full QCD potential and its
solutions.

Table 6:  Our results of $\Lambda_{c}^{+}\to{\bf B_{n}}e^{+}\nu_{e}$ decays in comparisons with the experimental data and those in various calculations in the literature. | $\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e}$ | $\Lambda_{c}^{+}\to ne^{+}\nu_{e}$
---|---|---
| ${\cal B}(\%)$ | $\alpha$ | ${\cal B}(\%)$ | $\alpha$
LFCQM | $3.55\pm 1.04$ | $-0.97\pm 0.03$ | $0.36\pm 0.15$ | $-0.96\pm 0.04$
Data Zyla:2020zbs  | $3.6\pm 0.4$ | $-0.86\pm 0.04$ | - | -
$SU(3)$ Geng:2019bfz  | $3.2\pm 0.3$ | $-0.86\pm 0.04$ | $0.51\pm 0.04$ | $-0.89\pm 0.04$
RQM Faustov:2016yza ; Faustov:2019ddj  | 3.25 | -0.86 | 0.268 | -0.91
HQET Cheng:1995fe  | 1.42 | - | - | -
LF Zhao:2018zcb  | 1.63 | - | 0.201 | -
MBM (NRQM) PerezMarcial:1989yh  | 2.6 (3.2) | - | 0.20 (0.30) | -
LQCD Meinel:2016dqj ; Meinel:2017ggx  | $3.80\pm 0.22$ | - | $0.41\pm 0.03$ | -
CCQM Gutsche:2015rrt  | 2.78 | -0.87 | 0.20 | -
LCSR Liu:2009sn ; Azizi:2009wn  | $3.0\pm 0.3$ | - | $8.69\pm 2.89$ | -
Table 7:  Our results of $\Xi_{c}^{+}\to{\bf B_{n}}e^{+}\nu_{e}$ decays in comparisons with the experimental data and those in various calculations in the literature. | $\Xi_{c}^{+}\to\Xi^{0}e^{+}\nu_{e}$ |  $\Xi_{c}^{+}\to\Sigma^{0}e^{+}\nu_{e}$
---|---|---
| $\Xi_{c}^{+}\to\Lambda e^{+}\nu_{e}$
| ${\cal B}(\%)$ | $\alpha$ | ${\cal B}(\%)$ | $\alpha$
LFCQM | $11.3\pm 3.35$ | $-0.97\pm 0.03$ | $0.33\pm 0.09$ | $-0.98\pm 0.01$
$0.12\pm 0.04$ | $-0.98\pm 0.02$
Data Zyla:2020zbs ; Li:2019atu  | $6.6^{+3.7}_{-3.5}$ | - | - | -
- | -
$SU(3)$ Geng:2019bfz  | $10.7\pm 0.9$ | $-0.83\pm 0.04$ | $0.46\pm 0.04$ | $-0.85\pm 0.04$
$0.22\pm 0.02$ | $-0.86\pm 0.04$
RQM Faustov:2019ddj  | 9.40 | -0.80 | - | -
0.13 | -0.84
LF Zhao:2018zcb  | 5.39 | - | 0.19 | -
0.08 | -
MBM (NRQM) PerezMarcial:1989yh  | 11.1 (13.3) | - | 0.28 (0.41) | -
0.09 (0.14) | -
Table 8:  Our results of $\Xi_{c}^{0}\to{\bf B_{n}}e^{+}\nu_{e}$ decays in comparisons with the experimental data and those in various calculations in the literature. | $\Xi_{c}^{0}\to\Xi^{-}e^{+}\nu_{e}$ | $\Xi_{c}^{0}\to\Sigma^{-}e^{+}\nu_{e}$
---|---|---
| ${\cal B}(\%)$ | $\alpha$ | ${\cal B}(\%)$ | $\alpha$
LFCQM | $3.49\pm 0.95$ | $-0.98\pm 0.03$ | $0.22\pm 0.06$ | $-0.98\pm 0.02$
Data Zyla:2020zbs ; Li:2018qak  | $1.8\pm 1.2$ | - | - | -
$SU(3)$ Geng:2019bfz  | $3.7\pm 0.3$ | $-0.83\pm 0.04$ | $0.33\pm 0.03$ | $-0.85\pm 0.04$
RQM Faustov:2019ddj  | 2.38 | -0.80 | - | –
LF Zhao:2018zcb  | 1.35 | - | 0.10 | -
MBM (NRQM) PerezMarcial:1989yh  | 3.55 (4.47) | - | 0.19 (0.26) | -

##  IV Conclusions

We have systematically studied the semi-leptonic decays of ${\bf B_{c}}\to{\bf
B_{n}}\ell^{+}\nu_{\ell}$ in LFCQM. By requiring the constituents in the
baryonic states to obey the Fermi statistics, we are able to determine the
overall spin-flavor-momentum structures of the baryons. We assume that the
momentum distribution of ${\bf B_{n}}$ is symmetric in flavor indices and any
pair of two quarks can be effectively treated as a heavier anti-quark. We have
found that ${\cal B}(\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e})=(3.55\pm
1.04)\%$, ${\cal B}(\Xi_{c}^{+}\to\Xi^{0}e^{+}\nu_{e})=(11.3\pm 3.35)\%$ and
${\cal B}(\Xi_{c}^{0}\to\Xi^{-}e^{+}\nu_{e})=(3.49\pm 0.95)\%$ in LFCQM, which
are consistent with the experimental data of $(3.6\pm 0.4)\times 10^{-2}$
Zyla:2020zbs , $(6.6^{+3.7}_{-3.5})\times 10^{-2}$ Zyla:2020zbs ; Li:2019atu
and $(1.8\pm 1.2)\times 10^{-2}$ Zyla:2020zbs ; Li:2018qak  as well as the
values predicted by $SU(3)_{F}$ Geng:2019bfz , LQCD Meinel:2016dqj ;
Meinel:2017ggx , RQM Faustov:2016yza , and CCQM Gutsche:2015rrt , but twice
larger than those in HQET Cheng:1995fe  and LF Zhao:2018zcb . We have also
obtained that ${\alpha}(\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e})=(-0.97\pm
0.03)$ in LFCQM, which is $10\%$ lower than the experimental data of $-0.86\pm
0.04$ Zyla:2020zbs . The reason of this deviation may arise from the QCD spin-
spin interacting effects, which are not included in our phenomenological wave
functions. Our results of the averaged decay asymmetries could be improved if
we consider the wave function solved from the full QCD potential. It is clear
that our predicted values for the decay branching ratios and asymmetries in
$\Xi_{c}^{+(0)}\to{\bf B_{n}}e^{+}\nu_{e}$ could be tested in the ongoing
experiments at LHCb and BELLEII.

## ACKNOWLEDGMENTS

We would like to thank Prof. Chengping Shen for discussions. This work was
supported in part by National Center for Theoretical Sciences and MoST
(MoST-107-2119-M-007-013-MY3).

## References

  * (1) Y. B. Li et al. [Belle Collaboration], Phys. Rev. Lett. 122, 082001 (2019).
  * (2) Y. B. Li et al. [Belle Collaboration], Phys. Rev. D 100, 031101 (2019).
  * (3) R. Aaij et al. [LHCb Collaboration], Phys. Rev. D 100, 032001 (2019)
  * (4) P. A. Zyla et al. [Particle Data Group], PTEP 2020, 083C01 (2020).
  * (5) M. J. Savage and R. P. Springer, Phys. Rev. D 42, 1527 (1990).
  * (6) M. J. Savage, Phys. Lett. B 257, 414 (1991).
  * (7) C. D. Lu, W. Wang and F. S. Yu, Phys. Rev. D 93, 056008 (2016).
  * (8) C. Q. Geng, Y. K. Hsiao, C. W. Liu and T. H. Tsai, JHEP 1711, 147 (2017).
  * (9) W. Wang, Z. P. Xing and J. Xu, Eur. Phys. J. C 77, 800 (2017).
  * (10) C. Q. Geng, Y. K. Hsiao, Y. H. Lin and L. L. Liu, Phys. Lett. B 776, 265 (2018).
  * (11) C. Q. Geng, Y. K. Hsiao, C. W. Liu and T. H. Tsai, Phys. Rev. D 97, 073006 (2018); Eur. Phys. J. C 78, 593 (2018).
  * (12) D. Wang, P. F. Guo, W. H. Long and F. S. Yu, JHEP 1803, 066 (2018).
  * (13) X. G. He and W. Wang, Chin. Phys. C 42, 103108 (2018).
  * (14) C. Q. Geng, Y. K. Hsiao, C. W. Liu and T. H. Tsai, Phys. Rev. D 99, 073003 (2019).
  * (15) C. Q. Geng, C. W. Liu and T. H. Tsai, Phys. Lett. B 790, 225 (2019); Phys. Lett. B 794, 19 (2019).
  * (16) C. Q. Geng, C. W. Liu, T. H. Tsai and Y. Yu, Phys. Rev. D 99, 114022 (2019).
  * (17) J. Y. Cen, C. Q. Geng, C. W. Liu and T. H. Tsai, Eur. Phys. J. C 79, 946 (2019).
  * (18) Y. K. Hsiao, Y. Yao and H. J. Zhao, Phys. Lett. B 792, 35 (2019).
  * (19) C. Q. Geng, C. W. Liu, T. H. Tsai and S. W. Yeh, Phys. Lett. B 792, 214 (2019).
  * (20) Y. Grossman and S. Schacht, JHEP 1907, 020 (2019).
  * (21) X. G. He, Y. J. Shi and W. Wang, Eur. Phys. J. C 80, 359 (2020).
  * (22) S. Roy, R. Sinha and N. G. Deshpande, Phys. Rev. D 101, 036018 (2020).
  * (23) C. Q. Geng, C. C. Lih, C. W. Liu and T. H. Tsai, Phys. Rev. D 101, 094017 (2020).
  * (24) C. P. Jia, D. Wang and F. S. Yu, Nucl. Phys. B 956, 115048 (2020).
  * (25) H. Y. Cheng and B. Tseng, Phys. Rev. D53, 1457 (1996).
  * (26) S. Meinel, Phys. Rev. Lett. 118, 082001 (2017).
  * (27) S. Meinel, Phys. Rev. D97, 034511 (2018).
  * (28) R. N. Faustov and V. O. Galkin, Eur. Phys. J. C 76, 628 (2016).
  * (29) R. N. Faustov and V. O. Galkin, Eur. Phys. J. C 79, 695 (2019); Particles 3, 208-222 (2020)
  * (30) T. Gutsche, M. A. Ivanov, J. G. Korner, V. E. Lyubovitskij and P. Santorelli, Phys. Rev. D 93, 034008 (2016).
  * (31) Y. L. Liu, M. Q. Huang and D. W. Wang, Phys. Rev. D 80, 074011 (2009)
  * (32) K. Azizi, M. Bayar, Y. Sarac and H. Sundu, Phys. Rev. D 80, 096007 (2009)
  * (33) Z. X. Zhao, Chin. Phys. C 42, 093101 (2018).
  * (34) R. Perez-Marcial, R. Huerta, A. Garcia and M. Avila-Aoki, Phys. Rev. D 40, 2955 (1989) Erratum: [Phys. Rev. D 44, 2203 (1991)].
  * (35) W. M. Zhang, Chin. J. Phys. 32, 717 (1994).
  * (36) F. Schlumpf, hep-ph/9211255.
  * (37) C. Q. Geng, C. C. Lih and W. M. Zhang, Phys. Rev. D 62, 074017 (2000).
  * (38) C. Q. Geng, C. W. Hwang and C. C. Liu, Phys. Rev. D 65, 094037 (2002).
  * (39) H. Y. Cheng, C. K. Chua and C. W. Hwang, Phys. Rev. D 70, 034007 (2004).
  * (40) C. Q. Geng, C. C. Lih and C. Xia, Eur. Phys. J. C 76, 313 (2016).
  * (41) H. Y. Cheng and X. W. Kang, Eur. Phys. J. C 77, 587 (2017) Erratum: [Eur. Phys. J. C 77, 863 (2017)].
  * (42) Z. X. Zhao, Eur. Phys. J. C 78, 756 (2018).
  * (43) Z. P. Xing and Z. X. Zhao, Phys. Rev. D 98, 056002 (2018).
  * (44) Q. Chang, L. T. Wang and X. N. Li, JHEP 1912, 102 (2019).
  * (45) C. K. Chua, Phys. Rev. D 99, 014023 (2019); Phys. Rev. D 100, 034025 (2019).
  * (46) A. Kadeer, J. G. Korner and U. Moosbrugger, Eur. Phys. J. C 59, 27-47 (2009)
  * (47) H. W. Ke, N. Hao and X. Q. Li, Eur. Phys. J. C 79, 540 (2019).
  * (48) H. W. Ke, X. Q. Li and Z. T. Wei, Phys. Rev. D 77, 014020 (2008).
  * (49) H. W. Ke, X. H. Yuan, X. Q. Li, Z. T. Wei and Y. X. Zhang, Phys. Rev. D 86, 114005 (2012).
  * (50) C. Lorce, B. Pasquini and M. Vanderhaeghen, JHEP 1105, 041 (2011).
  * (51) W. N. Polyzou, W. Glockle and H. Witala, Few Body Syst. 54, 1667 (2013).
  * (52) W. Jaus, Phys. Rev. D 44, 2851 (1991).
  * (53) W. Jaus, Phys. Rev. D 53, 1349 (1996) Erratum: [Phys. Rev. D 54, 5904 (1996)].
  * (54) Q. Chang, X. N. Li, X. Q. Li, F. Su and Y. D. Yang, Phys. Rev. D 98, 114018 (2018).