# Anatomy of $\Lambda_{c}^{+}$ semileptonic decays

Chao-Qiang<EMAIL_ADDRESS>Xiang-Nan
<EMAIL_ADDRESS>and Chia-Wei<EMAIL_ADDRESS>School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute
for Advanced Study, UCAS, Hangzhou 310024, China
University of Chinese Academy of Sciences, 100190 Beijing, China

###### Abstract

We present a systematic study of $\Lambda_{c}^{+}\to{\cal
B}_{q}\ell^{+}\nu_{\ell}$ with ${\cal B}_{q}=(\Lambda,n)$ and $\ell=(e,\mu)$,
examining all the possible decay observables based on the homogeneous bag
model (HBM) and lattice QCD (LQCD). With the HBM, we find that the branching
fractions and polarization asymmetries of the daughter baryon $\Lambda$ are
${\cal B}(\Lambda_{c}^{+}\to\Lambda
e^{+}\nu_{e},\Lambda\mu^{+}\nu_{\mu},n\ell^{+}\nu_{\ell})=(3.78\pm
0.25,3.67\pm 0.23,0.40\pm 0.04)\%$ and
$\alpha_{\Lambda}(\Lambda_{c}^{+}\to\Lambda
e^{+}\nu_{e},\Lambda\mu^{+}\nu_{\mu})=(-82.6,-82.3)\%$, respectively. From the
LQCD, we obtain that $\alpha_{\Lambda}(\Lambda_{c}^{+}\to\Lambda
e^{+}\nu_{e},\mu^{+}\nu_{\mu})=(-87.4\pm 1.0,-87.2\pm 1.0)\%$. We also explore
the time-reversal asymmetries due to new physics beyond the standard model.
All our results are consistent with the current experimental data, while some
of them are accessible to the experiments at BESIII and Belle II.

##  I Introductions

Very recently, the BESIII BESIII:2022ysa  collaboration has reported the decay
branching fraction of $\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e}$ to be

$\displaystyle{\cal B}(\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e})=(3.56\pm
0.11\pm 0.07)\%\,,$ (1)

which is the most precise measurement in the heavy baryon semileptonic decays
up to date, providing an excellent opportunity to examine the standard model
(SM) and various specific quark models. On the other hand, the polarization
asymmetry of the daughter baryon $\Lambda$ ($\alpha_{\Lambda}$) was measured
nearly 20 years ago at CLEO CLEO:2004txf , given by

$\displaystyle\alpha_{\Lambda}(\Lambda_{c}^{+}\to\Lambda
e^{+}\nu_{e})=(-0.86\pm 0.03\pm 0.02)\%\,.$ (2)

It is reasonable to expect that some angular observables including
$\alpha_{\Lambda}$ can be soon measured at BESIII and Belle II. In the view of
the recent experiments of $\Lambda_{c}^{+}$ at BESIII EXP1 ; EXP2 ; EXP3 ;
EXP4 ; EXP5 ; BESIII:2016ffj , the strange charmed baryons at Belle
Belle:2021crz ; Belle:2021dgc ; Belle:2022n ; Belle:2022kqi , and the doubly
charmed baryons at LHCb LHCb:2018pcs , it is no doubt that the era of the
precision measurements in the charm baryons has begun.

On the theoretical side, the charmed baryons have also been considered with
great interests recently TH0 ; TH1 ; TH2 ; TH3 ; TH4 ; TH5 ; TH6 ; TH7 . See
Ref. REV  for a review. In particular, due to the clean theoretical
background, $\Lambda_{c}^{+}\to{\cal B}_{q}\ell^{+}\nu_{\ell}$ with ${\cal
B}_{q}=(\Lambda,n)$ and $\ell=(e,\mu)$ are widely studied in the literature
Faustov:2019ddj ; Gutsche:2015rrt , including the lattice QCD (LQCD)
Meinel:2016dqj ; Meinel:2017ggx ; LATTICE1 ; LATTICE0 , light-front quark
model (LFQM) Li:2021qod ; Zhao:2018zcb ; Geng:2020gjh , and $SU(3)$ flavor
$(SU(3)_{F})$ symmetry He:2021qnc ; Geng:2019bfz . However, there are some
tensions. For instance, the branching fractions of
$\Lambda_{c}^{+}\to\Lambda\ell^{+}\nu_{\ell}$ based on the LFQM in Refs.
Zhao:2018zcb  and Li:2021qod ; Geng:2020gjh  deviate by a factor of 2.
Clearly, a different angle in analyzing $\Lambda_{c}^{+}\to{\cal
B}_{q}\ell^{+}\nu$ may shed light on the problem.

In this paper, we concentrate on the decays of $\Lambda_{c}^{+}\to{\cal
B}_{q}\ell^{+}\nu$ with the bag model (BM) and LQCD. We compute all the
possible decay observables of $\Lambda_{c}^{+}\to{\cal B}_{q}\ell^{+}\nu$ in
the SM and discuss some of the possible effects from new physics (NP). In
particular, we discuss the time-reversal (T) asymmetries. As these asymmetries
are contaminated little by the hadronic uncertainties, they provide a reliable
way to probe NP.

This work is organized as follows. In Sec. II, we give the baryon wave
functions and helicity amplitudes from the homogeneous BM (HBM). In Sec. III,
we present the angular distributions and extract the physical observables in
$\Lambda_{c}^{+}\to{\cal B}_{q}\ell^{+}\nu$. In Sec. IV, we study the T
asymmetries from NP. Finally, we conclude this study in Sec. V.

##  II Baryon wave functions and Helicity amplitudes

The amplitudes of $\Lambda_{c}^{+}\to{\cal B}_{q}\ell^{+}\nu$ are given as

$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cq}g^{\mu\nu}\overline{v}{\gamma_{\mu}}(1-{\gamma_{5}})u\langle{\cal
B}_{q}|\overline{q}{\gamma_{\nu}}(1-{\gamma_{5}})c|\Lambda_{c}\rangle,$ (3)

where $V_{cs}=0.973$ and $V_{cd}=0.221$ are the Cabibbo-Kobayashi-Maskawa
quark mixing matrix elements pdg , $G_{F}$ is the Fermi constant, $q=(s,d)$
for ${\cal B}_{q}=(\Lambda,n)$, and $\overline{v}$ and $u$ are the Dirac
spinors of $\nu$ and $\ell^{+}$, respectively. We expand the Minkowski metric
$g^{\mu\nu}$ by

$g^{\mu\nu}=\varepsilon_{t}^{\mu}(q)\varepsilon_{t}^{\ast\nu}(q)-\sum_{\lambda=0,\pm}\varepsilon^{\mu}_{\lambda}(q)\varepsilon^{\ast\nu}_{\lambda}(q)\,,$
(4)

where $\varepsilon$ and $q$ are the polarization vector and four-momentum of
the off-shell $W$ boson ($W^{*}$), respectively, and the subscript in
$\varepsilon$ denotes the helicity of $W^{*}$. In the center of frames of
$W^{*}$ and $\Lambda_{c}^{+}$, we have Timereversal

${\varepsilon}^{\mu}_{\pm}=\frac{1}{\sqrt{2}}(0,\pm
1,i,0)^{T}\,,\quad{\varepsilon}_{0}^{\mu}=(0,0,0,-1)^{T}\,,\quad{\varepsilon}^{\mu}_{t}=(-1,0,0,0)^{T}\,,$
(5)

and

${\varepsilon}^{\mu}_{\pm}=\frac{1}{\sqrt{2}}(0,\mp
1,i,0)^{T}\,,\quad{\varepsilon}_{0}^{\mu}=\frac{1}{\sqrt{q^{2}}}(-|\vec{q}\,|,0,0,q^{0})^{T}\,,\quad{\varepsilon}^{\mu}_{t}=\frac{-1}{\sqrt{q^{2}}}q^{\mu},$
(6)

respectively. We note that in the calculations, we always choose the
3-momentum of the outgoing fermion toward the $\hat{z}$ direction.

By inserting Eq. (4) into Eq. (3), the three-body problem is reduced to a
product of the two-body ones, given by

$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cq}\left(L_{t}B_{t}-\sum_{\lambda=0,\pm}L_{\lambda}B_{\lambda}\right)$
(7)

along with

$\displaystyle
L_{\lambda_{W}}=\varepsilon_{\lambda_{W}}^{\mu}\overline{v}_{\ell}{\gamma_{\mu}}(1-{\gamma_{5}})u\,,~{}~{}~{}B_{\lambda_{W}}=\varepsilon_{\lambda_{W}}^{\ast\mu}\langle{\cal
B}_{q}|\overline{s}{\gamma_{\mu}}(1-{\gamma_{5}})c|\Lambda_{c}^{+}\rangle\,,$
(8)

describing $W^{*}\to\ell^{+}\nu$ and $\Lambda_{c}^{+}\to{\cal B}_{q}W^{*}$,
respectively.

As $L_{\lambda_{W}}$ and $B_{\lambda_{W}}$ are Lorentz scalars, they can be
calculated in different Lorentz frames. We adopt the rest frames of $W^{*}$
and $\Lambda_{c}^{+}$ for $L_{\lambda_{W}}$ and $B_{\lambda_{W}}$,
respectively. In the SM, the helicity amplitudes are given as

$\displaystyle
h_{\pm}=\varepsilon_{-\frac{1}{2}\pm\frac{1}{2}}^{\mu}\overline{v}_{-}{\gamma_{\mu}}(1-{\gamma_{5}})u_{\pm}\,.$
$\displaystyle
H_{\lambda_{q}\lambda_{W}}=\varepsilon_{\lambda_{W}}^{\ast\mu}\left({\cal
V}_{\mu}^{\lambda_{q}\lambda_{c}}-{\cal
A}_{\mu}^{\lambda_{q}\lambda_{c}}\right)\,,$ (9)

with $\lambda_{c}=\lambda_{q}-\lambda_{W}$ and

$\displaystyle{\cal V}_{\mu}^{\lambda_{q}\lambda_{c}}$ $\displaystyle=$
$\displaystyle\langle{\cal
B}_{q},\hat{J}\cdot\hat{p}=\lambda_{q},\vec{v}=v_{0}\hat{z}|\overline{q}{\gamma_{\mu}}c|\Lambda_{c}^{+},J_{z}=\lambda_{c},\vec{v}=0\rangle$
$\displaystyle=$ $\displaystyle N_{q}\langle
duq,J^{3}_{z}=\lambda_{q},\vec{v}=v_{0}\hat{z}|\overline{q}_{3}{\gamma_{\mu}}c_{3}|duc,J^{3}_{z}=\lambda_{c},\vec{v}=0\rangle\,,$
$\displaystyle{\cal A}_{\mu}^{\lambda_{q}\lambda_{c}}$ $\displaystyle=$
$\displaystyle\langle{\cal
B}_{q},\hat{J}\cdot\hat{p}=\lambda_{q},\vec{v}=v_{0}\hat{z}|\overline{q}{\gamma_{\mu}}\gamma_{5}c|\Lambda_{c}^{+},J_{z}=\lambda_{c},\vec{v}=0\rangle$
(10) $\displaystyle=$ $\displaystyle N_{q}\langle
duq,J^{3}_{z}=\lambda_{q},\vec{v}=v_{0}\hat{z}|\overline{q}_{3}{\gamma_{\mu}}\gamma_{5}c_{3}|duc,J^{3}_{z}=\lambda_{c},\vec{v}=0\rangle\,,$

where $N_{q}=(1,\sqrt{3/2})$ for $q=(s,d)$ are the spin-flavor overlappings,
the subscript of Dirac spinors stands for helicity, $\overline{s}_{3}$ and
$c_{3}$ act only on the third quarks, $J^{3}$ is the angular momentum of the
third quark, and $\vec{v}$ represents the velocity.

The adopted convention of the Dirac spinors can be found in Appendix of Ref.
Timereversal . The helicity amplitudes $h_{\pm}$ are calculated as

$h_{+}=-2\sqrt{2(q^{2}-M_{\ell}^{2})}\,,~{}~{}~{}h_{-}=\sqrt{\delta_{\ell}}h_{+}\,,$
(11)

where $\delta_{\ell}={2M_{\ell}^{2}/q^{2}}$ and $M_{\ell}$ corresponds to the
charged lepton mass. Due to the left-handed nature of the weak interaction, we
have $h_{+}\gg h_{-}$ as $\ell^{+}$ has a positive helicity in the massless
limit.

On the other hand, ${\cal V}$ and ${\cal A}$ depend on the baryon wave
functions and vary in quark models. The relevant baryon wave functions are
given as

$\displaystyle|\Lambda_{c}^{+},\updownarrow\rangle=\int\frac{1}{\sqrt{6}}\epsilon^{\alpha\beta\gamma}d_{a\alpha}^{\dagger}(\vec{x}_{1})u_{b\beta}^{\dagger}(\vec{x}_{2})c_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi_{A_{\updownarrow}(duc)}^{abc}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
$\displaystyle|\Lambda,\updownarrow\rangle=\int\frac{1}{\sqrt{6}}\epsilon^{\alpha\beta\gamma}d_{a\alpha}^{\dagger}(\vec{x}_{1})u_{b\beta}^{\dagger}(\vec{x}_{2})s_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi_{A_{\updownarrow}(dus)}^{abc}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
$\displaystyle|n,\updownarrow\rangle=\int\frac{1}{2\sqrt{3}}\epsilon^{\alpha\beta\gamma}d_{a\alpha}^{\dagger}(\vec{x}_{1})u_{b\beta}^{\dagger}(\vec{x}_{2})d_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi_{A_{\updownarrow}(dus)}^{abc}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
(12)

where $\epsilon^{\alpha\beta\gamma}$ is the totally antisymmetric tensor, the
Latin and Greek letters stand for the spinor and color indices, and
$q^{\dagger}(\vec{x})$ is the creation operator of a quark at $\vec{x}$ and
$t=0$. Without specifying the distributions of $\Psi$, the wave functions in
Eq. (II) are general results of quark models in the instant form. In the MIT
BM, the distributions read as bag

$\displaystyle\Psi_{A_{\updownarrow}(q_{1}q_{2}q_{3})}^{abc}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})=$
$\displaystyle\frac{{\cal
N}}{\sqrt{2}}\left(\phi^{a}_{q_{1}\uparrow}(\vec{x}_{1})\phi^{b}_{q_{2}\downarrow}(\vec{x}_{2})-\phi^{a}_{q_{1}\downarrow}(\vec{x}_{1})\phi^{b}_{q_{2}\uparrow}(\vec{x}_{2})\right)\phi^{c}_{q_{3}\updownarrow}(\vec{x}_{3}),$
(13)

where ${\cal N}$ is the normalization constant, and
$\phi^{a}_{q\updownarrow}(\vec{x})$ is a bag state centering at $\vec{x}=0$

$\phi_{q\updownarrow}^{a}(\vec{x})=\left(\begin{array}[]{c}\omega_{q}^{+}j_{0}(p_{q}r)\chi_{\updownarrow}\\\
i\omega_{q}^{-}j_{1}(p_{q}r)\hat{x}\cdot\vec{\sigma}\chi_{\updownarrow}\\\
\end{array}\right)_{a}\,,$ (14)

with $\chi_{\uparrow}=(1,0)^{T}$ and $\chi_{\downarrow}=(0,1)^{T}$
representing $J_{z}=\pm 1/2$, and $j_{0,1}$ the spherical Bessel functions,
respectively. Here, the kinematic factors are defined as
$\omega_{q}^{\pm}=\sqrt{E_{q}^{k}\pm m_{q}}$ with $E_{q}^{k}$, $p_{q}$ and
$m_{q}$ the kinematic energy, 3-momentum and mass of the quark. In turn,
$p_{q}$ has to obey the boundary condition

$\tan(p_{q}R)=\frac{p_{q}R}{1-m_{q}R-E_{q}^{k}R}\,,$ (15)

with $R$ the bag radius.

Although the MIT BM successfully explains most of the low-lying baryon masses,
it is problematic when it applies to decays. It is due to that the baryon wave
functions in Eq. (13) are localized at $\vec{x}=0$. According to the
Heisenberg uncertainty principle, it can not be a momentum eigenstate, which
is a basic requirement in calculating decay widths. To resolve the problem, we
take the baryon wave functions as infinite linear superpositions of Eq. (13),
given by

$\displaystyle\Psi_{A_{\updownarrow}(q_{1}q_{2}q_{3})}^{abc}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})=$
$\displaystyle\frac{{\cal N}_{{\cal
B}}}{\sqrt{2}}\int\left(\phi^{a}_{q_{1}\uparrow}(\vec{x}_{1}-\vec{x}_{\Delta})\phi^{b}_{q_{2}\downarrow}(\vec{x}_{2}-\vec{x}_{\Delta})\right.$
(16)
$\displaystyle\left.-\phi^{a}_{q_{1}\downarrow}(\vec{x}_{1}-\vec{x}_{\Delta})\phi^{b}_{q_{2}\uparrow}(\vec{x}_{2}-\vec{x}_{\Delta})\right)\phi^{c}_{q_{3}\updownarrow}(\vec{x}_{3}-\vec{x}_{\Delta})d^{3}\vec{x}_{\Delta},$

which clearly distribute homogeneously over the space, and thus are named as
the HBM Liu:2022pdk . The overall normalization constants, depending on the
quark components, are calculated as

${\cal N}_{{\cal B}}=\frac{1}{\overline{u}_{{\cal B}}u_{{\cal B}}}\int
d^{3}\vec{x}_{\Delta}\prod_{i=1,2,3}D^{0}_{q_{i}}(\vec{x}_{\Delta})\,,$ (17)

with

$D^{v}_{q}(\vec{x}_{\Delta})=\frac{1}{\gamma}\int
d^{3}\vec{x}\phi_{q}^{\dagger}\left(\vec{x}^{+}\right)\phi_{q}\left(\vec{x}^{-}\right)e^{-2iE_{q}\vec{v}\cdot\vec{x}}\,,$
(18)

where $\vec{x}^{\pm}=\vec{x}\pm\vec{x}_{\Delta}/2$, and the dependencies on
$v$ are for the later convenience. The wave functions of the low-lying baryons
are collected in Appendix A.

Conventionally, though not necessarily, matrix elements are evaluated at the
Briet frame, of which the initial and final baryons have opposite velocities
$-\vec{v}$ and $\vec{v}$, respectively. They are related to Eq. (II) as

$\displaystyle{\cal
V}_{\mu}^{\prime\lambda_{q}\lambda_{c}}=\Lambda_{\mu}\,^{\nu}{\cal
V}_{\nu}^{\lambda_{q}\lambda_{c}}\,,~{}~{}~{}{\cal
A}_{\mu}^{\prime\lambda_{q}\lambda_{c}}=\Lambda_{\mu}\,^{\nu}{\cal
A}_{\nu}^{\lambda_{q}\lambda_{c}}\,,$ (19)

where $\Lambda^{\mu}\,_{\nu}$ is a Lorentz boost toward the $-\hat{z}$
direction, defined as $\Lambda^{3}\,_{0}=-\gamma v$ with
$\gamma=\sqrt{1/(1-v^{2})}$. From the baryon wave functions given in Eq. (16),
we obtain

$\displaystyle{\cal V}_{\mu}^{\prime\lambda_{q}\lambda_{c}}={\cal N}_{{\cal
B}_{c}}{\cal N}_{{\cal B}_{q}}\int
d^{3}\vec{x}_{\Delta}\Upsilon^{\lambda_{q}\lambda_{c}}_{\mu}(\vec{x}_{\Delta})\prod_{l=d,u}D^{v}_{q}(\vec{x}_{\Delta})\,,$
(20) $\displaystyle{\cal A}_{\mu}^{\prime\lambda_{q}\lambda_{c}}={\cal
N}_{{\cal B}_{c}}{\cal N}_{{\cal B}_{q}}\int
d^{3}\vec{x}_{\Delta}\Upsilon^{\lambda_{q}\lambda_{c}}_{5\mu}(\vec{x}_{\Delta})\prod_{l=u,d}D^{v}_{q}(\vec{x}_{\Delta})\,,$

with

$\displaystyle\Upsilon_{\mu}^{\lambda_{q}\lambda_{c}}(\vec{x}_{\Delta})$
$\displaystyle=$ $\displaystyle\int
d^{3}\vec{x}\phi_{q{\lambda_{1}}}^{\dagger}\left(\vec{x}^{+}\right)S_{v}\gamma_{0}\gamma_{\mu}S_{-v}\phi_{c{\lambda_{2}}}\left(\vec{x}^{-}\right)e^{2i(E_{u}+E_{d})\vec{v}\cdot\vec{x}}\,,$
$\displaystyle\Upsilon_{5\mu}^{\lambda_{q}\lambda_{c}}(\vec{x}_{\Delta})$
$\displaystyle=$ $\displaystyle\int
d^{3}\vec{x}\phi_{q{\lambda_{q}}}^{\dagger}\left(\vec{x}^{+}\right)S_{v}\gamma_{0}\gamma_{\mu}\gamma_{5}S_{-v}\phi_{c{\lambda_{c}}}\left(\vec{x}^{-}\right)e^{2i(E_{u}+E_{d})\vec{v}\cdot\vec{x}}\,,$
(21)

where $S_{\pm v}=a_{+}\pm a_{-}\gamma^{0}\gamma^{3}$ are the Lorentz boost
matrix for Dirac spinors toward the $\pm\hat{z}$ direction, and
$a_{\pm}=\sqrt{(\gamma\pm 1)/2}$. In Eq. (20), $\Upsilon_{(5)}$ describes the
quark transition of the (axial) current operator, and $D_{q}^{v}$ are the
overlappings of the spectator quarks.

The main uncertainties of the model calculations come from the quark energies.
In this work, we take the values Liu:2022pdk

$\displaystyle M_{p}/3~{}\text{GeV}<E_{u}<0.368~{}\text{GeV}\,,$ (22)

with $M_{p}$ the proton mass. We use the capital $M$ to represent hadron
masses and the lower case $m$ for quark masses. The adopted bag radius and
quark masses are Liu:2022pdk ; bag

$R=4.9~{}\text{GeV}^{-1}\,,~{}~{}~{}m_{u,d}=0\,,~{}~{}~{}m_{s}=(0.19\pm
0.09)~{}\text{GeV}~{}~{}~{}m_{c}=1.655~{}\text{GeV}\,.$ (23)

By collecting Eqs. (19), (20) and (II), one shall be able to evaluate Eq.
(II), which completes the evaluations. See Ref. Liu:2022pdk  for calculation
details.

Before we end this section, we note that the baryon transition matrix elements
are conventionally parameterized as follows:

$\displaystyle\langle{\cal
B}_{q}|\overline{s}{\gamma_{\nu}}(1-{\gamma_{5}})c|\Lambda^{+}_{c}\rangle=\overline{u}_{q}\left[\left(f_{1}(q^{2}){\gamma_{\mu}}-if_{2}(q^{2})\frac{\sigma_{\mu\nu}}{M_{c}}q^{\nu}+f_{3}(q^{2})\frac{q_{\mu}}{M_{c}}\right)\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\left.\left(g_{1}(q^{2}){\gamma_{\mu}}-ig_{2}(q^{2})\frac{\sigma_{\mu\nu}}{M_{c}}q^{\nu}+g_{3}(q^{2})\frac{q_{\mu}}{M_{c}}\right){\gamma_{5}}\right]u_{c},$
(24)

where $\overline{u}_{q}$ and $u_{c}$ are the Dirac spinors of ${\cal B}_{q}$
and $\Lambda_{c}^{+}$,
$\sigma_{\mu\nu}=i(\gamma_{\mu}\gamma_{\nu}-\gamma_{\nu}\gamma_{\mu})/2\,,$
and $M_{q}$ and $M_{c}$ are the masses of ${\cal B}_{q}$ and
$\Lambda_{c}^{+}$, respectively. The form factors in Eq. (II) can be
numerically extracted by matching Eqs. (II) and (II) once ${\cal V}$ and
${\cal A}$ are computed. The relations between the helicity amplitudes and
form factors are

$\displaystyle H_{\pm\frac{1}{2}\pm
1}=\sqrt{2Q_{-}}\left(f_{1}+\frac{M_{+}}{M_{c}}f_{2}\right)\pm\sqrt{2Q_{+}}\left(-g_{1}+\frac{M_{-}}{M_{c}}g_{2}\right)\,,$
$\displaystyle
H_{\pm\frac{1}{2}0}=-\sqrt{\frac{Q_{-}}{q^{2}}}\left(M_{+}f_{1}+\frac{q^{2}}{M_{c}}f_{2}\right)\pm\sqrt{\frac{Q_{+}}{q^{2}}}\left(M_{-}g_{1}-\frac{q^{2}}{M_{c}}g_{2}\right)\,,$
$\displaystyle
H_{\pm\frac{1}{2}t}=-\sqrt{\frac{Q_{+}}{q^{2}}}\left(M_{-}f_{1}+\frac{q^{2}}{M_{c}}f_{3}\right)\pm\sqrt{\frac{Q_{-}}{q^{2}}}\left(M_{+}g_{1}-\frac{q^{2}}{M_{c}}g_{3}\right)\,,$
(25)

with $M_{\pm}=M_{c}\pm M_{q}$ and $Q_{\pm}=M_{\pm}^{2}-q^{2}$.

##  III Decay observables

In this section, we study physical observables by the means of the angular
distributions. For convenience, we take the following abbreviations:

$a_{\pm}=H_{\pm\frac{1}{2}0}\,,~{}~{}~{}b_{\pm}=H_{\mp\frac{1}{2}\mp
1}\,,~{}~{}~{}t_{\pm}=H_{\pm\frac{1}{2}t}\,,~{}~{}~{}|\xi^{2}|=|\xi_{+}^{2}|+|\xi_{-}^{2}|\,,~{}~{}~{}|\xi_{\Delta}^{2}|=|\xi_{+}^{2}|-|\xi_{-}^{2}|\,,$
(26)

with $\xi=a,b$ and $t$. Here, we do not explicitly write down the $q^{2}$
dependencies of the helicity amplitudes.

###  III.1 Observables in $\Lambda_{c}^{+}\to\Lambda(\to
p\pi^{-})\ell^{+}\nu_{\ell}$

We start with the partial decay width, given as

$\displaystyle\frac{\partial\Gamma^{\ell}}{\partial
q^{2}}=\zeta\big{[}\left(1+\delta_{\ell}\right)\left(|a^{2}|+|b^{2}|\right)+3\delta_{\ell}|t^{2}|\big{]}$
$\displaystyle\zeta=\frac{G_{F}^{2}V_{cs}^{2}}{192\pi^{3}}\frac{(q^{2}-M_{\ell}^{2})^{2}}{M_{c}^{2}q^{2}}|\vec{p}_{s}|\,,$
(27)

where $\vec{p}_{s}$ is the three-momentum of $\Lambda$ in the rest frame of
$\Lambda_{c}^{+}$. The angular momentum of a timelike $W^{*}$ in its rest
frame is essentially zero, resulting in that $\ell^{+}$ and $\nu_{\ell}$ have
opposite helicities. As $t_{\pm}$ attribute to the timelike $W^{*}$ solely,
they are always followed by $\delta_{\ell}$ as shown in Eq. (III.1).
Numerically, $\delta_{\ell}$ can be taken as zero.

The branching fractions along with those from the literature and experiments
are collected in Table. 1, where Ref. Gutsche:2015rrt  employs the covariant
quark model (CQM), Ref. Faustov:2019ddj  considers the relativistic quark
model (RQM), Ref. Meinel:2016dqj  calculates the form factors by the LQCD,
Refs. Zhao:2018zcb ; Geng:2020gjh ; Li:2021qod  adopt the LFQM, and Refs.
He:2021qnc ; Geng:2019bfz  utilize the $SU(3)_{F}$ symmetry. All the results
in the literature show that ${\cal B}(\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e})$
is larger than ${\cal B}(\Lambda_{c}^{+}\to\Lambda\mu^{+}\nu_{\mu})$.
Therefore, an opposite behavior in the experimental measurements shall be a
clear evidence of NP. The branching fractions in the literature are all in the
same order as the experimental ones. The results of $SU(3)_{F}$ are well
consistent with those from the experiments. On the other hand, the branching
fraction of $\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e}$ in Ref. Zhao:2018zcb
from the LFQM is twice smaller than the experimental one. By adopting a
different set of parameter input, Refs. Geng:2020gjh ; Li:2021qod  show that
the LFQM is capable of explaining the experimental data, but the predictive
power is questionable in turn. A parameter-independent study of the LFQM is
clearly required. One of the great advantage of the BM is that the parameters
are fitted from the mass spectra. In particular, the uncertainties are
considerably smaller than other quark models. Our central values of the
branching fractions are slightly larger than the ones from the experiments but
smaller than those of the LQCD.

Table 1:  The branching fractions of $\Lambda_{c}^{+}\to\Lambda\ell^{+}\nu_{\ell}$ in units of $\%$, where the numbers in the parentheses are the uncertainties counting backward in digits, i.e. $3.75(25)=3.75\pm 0.25$.  Model | ${\cal B}(\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e})$ | ${\cal B}(\Lambda_{c}^{+}\to\Lambda{\mu}^{+}\nu_{\mu})$
---|---|---
HBM | $3.78(25)$ | $3.67(23)$
Data BESIII:2022ysa ; BESIII:2016ffj  | $3.56(13)$ | $3.49(53)$
CQM Gutsche:2015rrt  | $2.78$ | $2.69$
RQM Faustov:2019ddj  | $3.25$ | $3.14$
LQCD Meinel:2016dqj  | $3.80(22)$ | $3.69(22)$
LFQM Zhao:2018zcb  | $1.63$ | $-$
LFQM Geng:2020gjh  | $3.55(104)$ | $3.40(102)$
LFQM Li:2021qod  | $4.04(75)$ | $3.90(73)$
$SU(3)_{F}$ Geng:2019bfz  | $3.6(4)$ | $3.5(4)$
$SU(3)_{F}$ He:2021qnc  | $3.62(32)$ | $3.45(30)$

To examine the decays further, we study the angular distributions of
$\Lambda_{c}^{+}\to\Lambda(\to p\pi^{-})\ell^{+}\nu_{\ell}$. The fivefold
angular distributions are obtained by piling up the Wigner-$d$ matrices of
$d^{J}$, given as

$\displaystyle{\cal
D}(q^{2},\vec{\Omega})\equiv\frac{\partial^{6}\Gamma^{\ell}}{\partial
q^{2}\partial\partial\vec{\Omega}}={\cal B}(\Lambda\to
p\pi^{+})\frac{\zeta}{32\pi^{2}}\sum_{\lambda_{\ell}\,,\lambda_{p}\,,\lambda_{c}}\rho_{\lambda_{c}\lambda_{c}}\left|A_{\lambda_{p}}h_{\lambda_{\ell}}\right|^{2}$
$\displaystyle\left|\sum_{\lambda_{s}\,,\lambda_{W}}(-1)^{J_{W}}H_{\lambda_{q}\lambda_{W}}d^{\frac{1}{2}}(\theta_{c})^{\lambda_{c}}\,_{\lambda_{s}-\lambda_{W}}d^{\frac{1}{2}}(\theta_{s})^{\lambda_{s}}\,_{\lambda_{p}}d^{J_{W}}(\theta_{\ell})^{\lambda_{W}}\,_{\frac{1}{2}-\lambda_{\ell}}e^{i(\lambda_{s}\phi_{c}+\lambda_{\ell}\phi_{\ell})}\right|^{2},$
(28)

where
$\vec{\Omega}=(\cos\theta_{c},\cos\theta_{s},\cos\theta_{\ell},\phi_{c},\phi_{\ell})$,
$\rho_{\pm\pm}=(1\pm P_{b})/2$ with $P_{b}$ the polarization fraction of
$\Lambda_{c}^{+}$, $\lambda_{c,s,\ell,p}=\pm 1/2$ ,
$|\vec{p}_{c}|=\sqrt{Q_{+}Q_{-}}/2M_{c}$, $A_{\pm}^{2}=(1\pm\alpha)/2$ with
$\alpha$ the up-down asymmetry of $\Lambda\to p\pi^{+}$, and $J_{W}$ is the
angular momentum of $W^{*}$. The derivation can be found in Appendix of Ref.
Timereversal , and the angles $\vec{\Omega}$ are defined in FIG. 1.

Figure 1:  Definitions of angles, where the spin of $\Lambda_{c}^{+}$ points
toward the direction of $\theta_{c}=0$.

By expanding Eq. (III.1), we arrive at

$\displaystyle{\cal D}(q^{2},\vec{\Omega})\propto 1+{\cal X}_{2}P_{2}+{\cal
X}_{3}\cos\theta_{\ell}+P_{b}\Big{(}{\cal X}_{4}\cos\theta_{c}+{\cal
X}_{5}\cos\theta_{c}P_{2}+{\cal X}_{6}\cos\theta_{c}\cos\theta_{\ell}$
$\displaystyle+\text{Re}({\cal
X}_{7}e^{i\phi_{\ell}})\sin\theta_{c}\sin\theta_{\ell}+\text{Re}({\cal
X}_{8}e^{i\phi_{\ell}})\sin\theta_{c}\sin\theta_{\ell}\cos\theta_{\ell}\Big{)}+\alpha\Big{(}{\cal
X}_{9}\cos\theta_{s}+{\cal X}_{10}\cos\theta_{s}P_{2}$ $\displaystyle+{\cal
X}_{11}\cos\theta_{s}\cos\theta_{\ell}+\text{Re}({\cal
X}_{12}e^{i\Phi})\sin\theta_{s}\cos\theta_{s}+\text{Re}({\cal
X}_{13}e^{i\Phi})\sin\theta_{s}\cos\theta_{s}\cos\theta_{\ell}\Big{)}$
$\displaystyle+P_{b}\alpha\Big{(}{\cal
X}_{14}\sin\theta_{c}\sin\theta_{s}P_{2}+{\cal
X}_{15}\cos\theta_{c}\cos\theta_{s}P_{2}+{\cal
X}_{16}\cos\theta_{c}\cos\theta_{s}+{\cal
X}_{17}\cos\theta_{c}\cos\theta_{s}\cos\theta_{\ell}$
$\displaystyle+\text{Re}({\cal
X}_{18}e^{-i\phi_{c}})\sin\theta_{c}\sin\theta_{s}P_{2}+\text{Re}({\cal
X}_{19}e^{i(\phi_{c}+2\phi_{\ell})})\sin\theta_{c}\sin\theta_{s}P_{2}+\text{Re}({\cal
X}_{20}e^{i\phi_{c}})\sin\theta_{c}\sin\theta_{s}$
$\displaystyle+\text{Re}({\cal
X}_{21}e^{i(\phi_{c}+2\phi_{\ell})})\sin\theta_{c}\sin\theta_{s}+\text{Re}({\cal
X}_{22}(e^{i\phi_{c}})\sin\theta_{c}\sin\theta_{s}\cos\theta_{\ell}$
$\displaystyle+\text{Re}({\cal
X}_{23}e^{i\phi_{\ell}})\sin\theta_{c}\sin\theta_{\ell}\cos\theta_{s}+\text{Re}({\cal
X}_{24}e^{i\Phi})\sin\theta_{s}\sin\theta_{\ell}\cos\theta_{c}$
$\displaystyle+\text{Re}({\cal
X}_{25}e^{i\phi_{\ell}})\sin\theta_{c}\sin\theta_{\ell}\cos\theta_{s}\cos\theta_{\ell}+\text{Re}({\cal
X}_{26}e^{i\Phi})\sin\theta_{s}\sin\theta_{\ell}\cos\theta_{c}\cos\theta_{\ell}\Big{)}\,.$
(29)

where $P_{2}=(3\cos\theta_{\ell}-1)/2$, $\Phi=\phi_{c}+\phi_{\ell}$, and the
observables ${\cal X}_{2\text{-}26}$ are defined in TABLE 2. Note that within
the SM, $\xi_{\pm}$ are real, leading to Re$({\cal X})={\cal X}$. To obtain
the distributions of the charge conjugate processes, one can take the
transformation of $\theta_{\ell}\to\pi-\theta_{\ell}$ and
$\alpha\to\overline{\alpha}$ so that ${\cal X}_{i}=\overline{{\cal X}}_{i}$ in
the absence of NP, where the overline denotes the charge conjugation.

To test the results with the experiments, we define the $q^{2}$ averages of
the decay observables in TABLE 2, as

$\langle{\cal
X}_{i}(\ell)\rangle=\frac{1}{\Gamma^{\ell}}\int_{M_{\ell}^{2}}^{M_{-}^{2}}\zeta{\cal
X}_{i}dq^{2}.$ (30)

The form factors calculated in the LQCD can be found in Refs. Meinel:2016dqj ;
Meinel:2017ggx , where ${\cal B}$ are also provided. In this work, we
calculate the angular observables from both the HBM and LQCD, listed in TABLE
2. The central values and uncertainties of the LQCD come from the nominal and
higher order fits, respectively Meinel:2016dqj ; Meinel:2017ggx . Large parts
of the uncertainties are canceled due to the correlations between ${\cal
X}_{i}$ and $\Gamma$. It is worth to mention that the HBM results agree well
with the LQCD ones except for $\langle{\cal X}_{14,18,20}(\ell)\rangle$,
attributed by $a_{+}a_{-}^{*}$.

Table 2:  Definitions of ${\cal X}_{i}$ and their average values in units of $\%$.  $i$ | ${\cal X}_{i}(\Lambda_{c}^{+}\to\Lambda\ell^{+}\nu)$ | HBM | LQCD |
---|---|---|---|---
$\langle{\cal X}_{i}(e)\rangle$ | $\langle{\cal X}_{i}(\mu)\rangle$ | $\langle{\cal X}_{i}(e)\rangle$ | $\langle{\cal X}_{i}(\mu)\rangle$ |
$2~{}$ | $(2\delta_{\ell}-1)(|a|^{2}-\frac{1}{2}|b|^{2})$ | $-43.1(9)$ | $-37.1(7)$ | $-43.8(13)$ | $-37.6(11)$ |
$3$ | $-6\delta_{\ell}\left(\text{Re}\left(a_{+}t_{+}^{*}\right)+\text{Re}\left(a_{-}t_{-}^{*}\right)\right)+\frac{3}{2}|b_{\Delta}|^{2}$ | $35.2(10)$ | $28.6(12)$ | $40.2(12)$ | $33.7(13)$ |
$4$ | $(\delta_{\ell}+1)(|a_{\Delta}|^{2}+|b_{\Delta}|^{2})+3\delta_{\ell}|t_{\Delta}|^{2}$ | $-35.7(13)$ | $-35.4(13)$ | $-33.7(15)$ | $-33.3(15)$ |
$5$ | $(2\delta_{\ell}-1)(|a_{\Delta}|^{2}-\frac{1}{2}|b_{\Delta}|^{2})$ | $70.8(3)$ | $64.0(1)$ | $73.9(9)$ | $67.0(8)$ |
$6$ | $-6\delta_{\ell}\left[\text{Re}(a_{+}t_{+}^{*})-\text{Re}(a_{-}t_{-}^{*})\right]-\frac{3}{2}|b_{\Delta}|^{2}$ | $-35.2(10)$ | $-28.5(12)$ | $-40.2(12)$ | $-33.7(13)$ |
$7$ | $\frac{3}{\sqrt{2}}\left[2\delta_{\ell}(t_{-}b_{+}^{*}-b_{-}t_{+}^{*})+(a_{-}b_{+}^{*}+b_{-}a_{+}^{*})\right]$ | $-91.1(7)$ | $-91.3(6)$ | $-91.5(8)$ | $-91.7(7)$ |
$8$ | $\frac{3}{\sqrt{2}}(2\delta_{\ell}-1)(b_{-}a_{+}^{*}-a_{-}b_{+}^{*})$ | $-78.0(5)$ | $-73.9(6)$ | $-83.1(10)$ | $-79.0(9)$ |
$9$ | $(\delta_{\ell}+1)(|a_{\Delta}|^{2}-|b_{\Delta}|^{2})+3\delta_{\ell}|t_{\Delta}|^{2}$ | $-82.6(0)$ | $-82.3(0)$ | $-87.4(10)$ | $-87.2(10)$ |
$10$ | $(2\delta_{\ell}-1)(|a_{\Delta}|^{2}+\frac{1}{2}|b_{\Delta}|^{2})$ | $47.4(10)$ | $41.3(8)$ | $47.1(12)$ | $40.9(11)$ |
$11$ | $6\delta_{\ell}\left(\text{Re}\left(a_{-}t_{-}^{*}\right)-\text{Re}\left(a_{+}t_{+}^{*}\right)\right)-\frac{3}{2}|b|^{2}$ | $-56.9(9)$ | $-50.2(11)$ | $-56.2(13)$ | $-49.8(14)$ |
$12$ | $\frac{3}{\sqrt{2}}\left[2\delta_{\ell}(b_{-}t_{-}^{*}-t_{+}b_{+}^{*})-(a_{+}b_{+}^{*}+b_{-}a_{-}^{*})\right]$ | $-49.0(1)$ | $-47.5(0)$ | $-38.8(25)$ | $-38.0(24)$ |
$13$ | $\frac{3}{\sqrt{2}}\left[2\delta_{\ell}(a_{+}b_{+}^{*}-b_{-}a_{-}^{*})+(b_{-}a_{-}^{*}-a_{+}b_{+}^{*})\right]$ | $25.1(6)$ | $23.8(5)$ | $22.9(15)$ | $21.8(14)$ |
$14$ | $(2\delta_{\ell}+1)a_{+}a_{-}^{*}$ | $-5.8(1)$ | $-6.0(1)$ | $-3.0(9)$ | $-3.3(9)$ |
$15$ | $2\delta_{\ell}(|a|^{2}+|b|^{2})-(|a|^{2}+1/2|b|^{2})$ | $-81.0(3)$ | $-73.8(1)$ | $-81.3(4)$ | $-74.0(3)$ |
$16$ | $\delta_{\ell}(|a|^{2}-|b|^{2}+3|t|^{2})+(|a|^{2}-|b|^{2})$ | $24.2(12)$ | $24.0(12)$ | $25.0(17)$ | $24.5(17)$ |
$17$ | $-6\delta_{\ell}\left(\text{Re}(a_{+}t_{+}^{*})+\text{Re}(a_{-}t_{-}^{*})\right)-\frac{3}{2}|b_{\Delta}|^{2}$ | $-35.2(10)$ | $-41.0(8)$ | $-40.3(12)$ | $-46.3(12)$ |
$18$ | $(2-4\delta_{\ell})a_{-}a_{+}^{*}$ | $-11.6(1)$ | $-11.7(1)$ | $-6.1(19)$ | $-6.7(16)$ |
$19$ | $(1-2\delta_{\ell})b_{-}b_{+}^{*}$ | $-14.5(1)$ | $-14.2(1)$ | $-12.5(6)$ | $-12.3(6)$ |
$20$ | $-2\delta_{\ell}(a_{+}a_{-}^{*}+3t_{+}t_{-}^{*})-2a_{+}a_{-}^{*}$ | $11.6(1)$ | $11.6(1)$ | $6.1(19)$ | $6.4(17)$ |
$21$ | $(2\delta_{\ell}-1)b_{-}b_{+}^{*}$ | $14.5(1)$ | $14.2(1)$ | $12.5(6)$ | $12.3(6)$ |
$22$ | $6\delta_{\ell}(a_{+}t_{-}^{*}+t_{+}a_{-}^{*})$ | $0.0(0)$ | $0.4(0)$ | $0.0(0)$ | $0.5(2)$ |
$23$ | $\frac{3}{\sqrt{2}}\left[(b_{-}a_{+}^{*}-a_{-}b_{+}^{*})-2\delta_{\ell}(b_{-}t_{+}^{*}+t_{-}b_{+}^{*})\right]$ | $78.0(5)$ | $77.8(5)$ | $83.1(10)$ | $83.1(10)$ |
$24$ | $\frac{3}{\sqrt{2}}\left[(b_{-}a_{-}^{*}-a_{+}b_{+}^{*})-2\delta_{\ell}(b_{-}t_{-}^{*}+t_{+}b_{+}^{*})\right]$ | $25.1(6)$ | $23.9(5)$ | $22.9(15)$ | $22.0(15)$ |
$25$ | $\frac{3}{\sqrt{2}}\left[2\delta_{\ell}(a_{-}b_{+}^{*}+b_{-}a_{+}^{*})-(a_{-}b_{+}^{*}+b_{-}a_{+}^{*})\right]~{}~{}$ | $-49.0(1)$ | $-47.7(0)$ | $-38.8(25)$ | $-38.1(24)$ |
$26$ | $\frac{3}{\sqrt{2}}\left[2\delta_{\ell}(b_{-}a_{-}^{*}-a_{+}b_{+}^{*})-(a_{+}b_{+}^{*}+b_{-}a_{-}^{*})\right]$ | $-49.0(1)$ | $-48.0(0)$ | $-38.8(25)$ | $-38.2(24)$ |

The reason of the deviations can be traced back to the form factors in the
HBM, plotted in FIG. 2 and TABLE 3. By taking reasonable approximations of
$f_{1}=g_{1}$ and $g_{2}=0$, we have $a_{+}\propto f_{2}$ from Eq. (II). Our
approach based on the BM underestimates $f_{2}$ by a factor of two third,
leading to a smaller $a_{+}$. This factor affects little on the branching
fractions, but plays important roles in some of the decay observables. We
believe that it is somewhat a universal factor in the BM. For instance, the
magnetic dipole moments of the octet baryons, proportional to $f_{2}$, are
also systematically underestimated by a factor of two third in the MIT BM bag
.

Table 3:  The form factors of $\Lambda_{c}^{+}\to\Lambda$ at $q^{2}=0$. | $f_{1}(0)$ | $f_{2}(0)$ | $g_{1}(0)$ | $g_{2}(0)$
---|---|---|---|---
HBM | $0.604(39)$ | $0.209(5)$ | $0.566(34)$ | $0.012(2)$
LQCD Meinel:2016dqj  | $0.643(23)$ | $0.308(36)$ | $0.572(15)$ | $0.001(45)$

Figure 2:  The $q^{2}$ dependencies of the form factors in
$\Lambda_{c}^{+}\to\Lambda$ from the HBM, where the bands represent the
uncertainties from the model calculations.

To compare our results with the experiments, we calculate the form factors
$f\in\\{g_{\perp},g_{+},f_{\perp},f_{+}\\}$ adopted by the experiments. The
definitions of $f$ can be found explicitly in Ref. Meinel:2017ggx , while
their $q^{2}$ dependencies are governed by

$f\left(q^{2}\right)=\frac{a_{0}^{f}}{1-q^{2}/\left(m_{\text{pole
}}^{f}\right)^{2}}\left[1+\alpha_{1}^{f}\times z\left(q^{2}\right)\right]\,,$
(31)

where $m_{\text{pole }}^{f}$ is the pole mass,
$z=(\sqrt{t_{+}-q^{2}}-\sqrt{t_{+}-t_{0}})/(\sqrt{t_{+}-q^{2}}+\sqrt{t_{+}-t_{0}})$
with $t_{0}=(M_{c}-M_{s})^{2}$, and $t_{+}=(M_{D}+M_{K})^{2}$. The adopted
masses are
$(M_{D},M_{K},m_{\text{pole}}^{f_{\perp},f_{+}},m_{\text{pole}}^{g_{\perp},g_{+}})=(1.87,0.494,2.112,2.46)$
GeV.

The results of HBM, LQCD and BESIII are given in TABLE 4 with
$r_{f}=a^{f}_{0}/a^{g_{\perp}}_{0}$, while the form factors from HBM and
BESIII are drawn explicitly in FIG. 3. The HBM results of
$a_{0}^{g_{\perp}},\alpha_{1}^{g_{\perp}},\alpha_{1}^{g_{+}}$ lie between the
ones of LQCD and BESIII, whereas the signs of $\alpha_{1}^{f_{\perp},f_{+}}$
are barely determined in the HBM. Notice that $r_{g_{+}}=1$ holds
automatically from the definition of the form factors, which is also found
explicitly in the experiments. FIG. 3 shows that the main deviations between
the results of HBM and BESIII occur in $f_{\perp}$ at the high $q^{2}$ region.

Table 4: Form factors compared with LQCD and experiments.  | $a_{0}^{g\perp}$ | $\alpha_{1}^{g_{\perp}}$ | $\alpha_{1}^{g_{+}}$ | $\alpha_{1}^{f_{\perp}}$ | $\alpha_{1}^{f_{+}}$ | $r_{f_{+}}$ | $r_{f_{\perp}}$ | $r_{g_{+}}$
---|---|---|---|---|---|---|---|---
LQCD | $0.68(2)$ | $-2.82(49)$ | $-3.61(31)$ | $-2.52(84)$ | $-3.58(55)$ | $1.19(3)$ | $1.92(7)$ | 1
HBM | $0.63(1)$ | $-0.91(60)$ | $-1.05(58)$ | $0.16(66)$ | $-0.29(72)$ | $1.03(1)$ | $1.51(5)$ | 1
Exp | $0.54(4)$ |  $1.43(209)$ |  $-8.15(158)$ | $1.75(32)$ | $3.62(65)$ | $1.13(13)$

Figure 3:  The form factors obtained from HBM and the BESIII measurements.

Some of the decay observables ${\cal X}_{i}$ have already been studied in the
literature, bearing different names. The polarization asymmetries are defined
as

$\displaystyle\alpha_{\Lambda}=\frac{\Gamma^{\ell}(\lambda_{q}=1/2)-\Gamma^{\ell}(\lambda_{q}=-1/2)}{\Gamma^{\ell}(\lambda_{q}=1/2)+\Gamma^{\ell}(\lambda_{q}=-1/2)}\,,$
(32)

which are extracted by the cascade decays of $\Lambda$ in the experiments,
given as

$\alpha_{\Lambda}=\frac{2}{\alpha}\left(\int^{1}_{0}-\int^{0}_{-1}\right)\Gamma^{\ell}_{\cos\theta_{s}}d\cos\theta_{s}=\langle{\cal
X}_{9}(\ell)\rangle\,.$ (33)

Likewise, the forward-backward asymmetries are

$\displaystyle\alpha_{\ell}=\left(\int^{1}_{0}-\int^{0}_{-1}\right)\Gamma^{\ell}_{\cos\theta_{\ell}}d\cos\theta_{\ell}=\frac{1}{2}\langle{\cal
X}_{3}(\ell)\rangle\,,$ (34)

describing the distributions in $W^{+*}\to\ell^{+}\nu$, while the up down
asymmetries are

$\displaystyle\alpha_{c}=\frac{2}{P_{b}}\left(\int^{1}_{0}-\int^{0}_{-1}\right)\Gamma^{\ell}_{\cos\theta_{c}}d\cos\theta_{c}=\langle{\cal
X}_{4}(\ell)\rangle\,.$ (35)

Note that we have adopted the shorthand notations of
$\Gamma^{\ell}_{\Omega}=\partial\Gamma^{\ell}/(\Gamma^{\ell}\partial\vec{\Omega})\,.$

Table 5:  Results of $\alpha_{\Lambda}(\Lambda_{c}^{+}\to\Lambda\ell^{+}\nu_{\ell})$ in units of $\%$. | $\alpha_{\Lambda}(\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e})$ | $\alpha_{\Lambda}(\Lambda_{c}^{+}\to\Lambda{\mu}^{+}\nu_{\mu})$
---|---|---
HBM | $-82.6(0)$ | $-82.3(0)$
LQCD Meinel:2016dqj  | $-87.4(10)$ | $-87.2(10)$
Data BESIII:2022ysa  | $-86(4)$ | $-$
RQM Faustov:2019ddj  | $-86$ | $-$
LFQM Geng:2020gjh  | $-97(3)$ | $-98(2)$
LFQM Li:2021qod  | $-87(9)$ | $-87(9)$
SU(3) Geng:2019bfz  | $-86(4)$ | $-$

We compare $\alpha_{\Lambda}$ with those in the literature in TABLE 5. In all
the quark models, $\alpha_{\Lambda}$ depend little on the lepton flavors.
Except for those in Ref. Geng:2020gjh , $\alpha_{\Lambda}$ are consistent with
the experimental values. In contrast to other approaches, the predicted values
of $\alpha_{\Lambda}$ in the HBM have little uncertainties due to the
correlations. However, they are smaller than those in the LQCD. The future
experiments on $\alpha_{\Lambda}$ shall be able to clarify the issue.

###  III.2 Observables in $\Lambda_{c}^{+}\to n\ell^{+}\nu_{\ell}$

As the cascade decays of the neutron can not be observed in the experiments,
there are only eight possible decay observables in $\Lambda_{c}^{+}\to
ne^{+}\nu$ , in which five of them require $P_{b}\neq 0$ for the measurements,
given as

$\displaystyle{\cal D}(q^{2},\vec{\Omega})\propto 1+{\cal X}_{2}P_{2}+{\cal
X}_{3}\cos\theta_{\ell}+P_{b}\Big{(}{\cal X}_{4}\cos\theta_{c}+{\cal
X}_{5}\cos\theta_{c}P_{2}+{\cal X}_{6}\cos\theta_{c}\cos\theta_{\ell}$
$\displaystyle+\text{Re}({\cal
X}_{7}e^{i\phi_{\ell}})\sin\theta_{c}\sin\theta_{\ell}+\text{Re}({\cal
X}_{8}e^{i\phi_{\ell}})\sin\theta_{c}\sin\theta_{\ell}\cos\theta_{\ell}\Big{)}\,.$
(36)

The discussions are parallel to $\Lambda_{c}^{+}\to\Lambda\ell^{+}\nu_{\ell}$.
We list out the branching fractions and decay observables in TABLE. 6 and
TABLE 7, respectively. The branching fractions in the HBM are compatible with
those in the LQCD Meinel:2017ggx  and Ref. Geng:2020gjh , but twice larger
than the results in Ref. Zhao:2018zcb  . On the other hand, the $SU(3)_{F}$
symmetry predicts relatively large branching fractions comparing to the
others.

Table 6:  The branching fractions of $\Lambda_{c}^{+}\to n\ell^{+}\nu_{\ell}$ in units of $\%$.  | ${\cal B}(\Lambda_{c}^{+}\to ne^{+}\nu_{e})$ | ${\cal B}(\Lambda_{c}^{+}\to n{\mu}^{+}\nu_{\mu})$
---|---|---
HBM | $0.40(4)$ | $0.40(4)$
RQM Faustov:2019ddj  | $0.268$ | $0.262$
LQCD Meinel:2017ggx  | $0.410(29)$ | $0.400(29)$
LFQM Zhao:2018zcb  | $0.201$ | $-$
LFQM Geng:2020gjh  | $0.36(15)$ | $0.34(15)$
$SU(3)_{F}$ He:2021qnc  | $0.520(46)$ | $0.506(45)$
Table 7:  The integrated observables in units of $\%$.  $i$ | ${\cal X}_{i}(\Lambda_{c}^{+}\to n\ell^{+}\nu)$ | HBM | LQCD |
---|---|---|---|---
$\langle{\cal X}_{i}(e)\rangle$ | $\langle{\cal X}_{i}(\mu)\rangle$ | $\langle{\cal X}_{i}(e)\rangle$ | $\langle{\cal X}_{i}(\mu)\rangle$ |
$2~{}$ | $(2\delta_{\ell}-1)(|a|^{2}-\frac{1}{2}|b|^{2})$ | $-42.3(13)$ | $-36.5(10)$ | $-43.4(16)$ | $-37.2(15)$ |
$3$ | $-6\delta_{\ell}\left(\text{Re}\left(a_{+}t_{+}^{*}\right)+\text{Re}\left(a_{-}t_{-}^{*}\right)\right)+\frac{3}{2}|b_{\Delta}|^{2}$ | $36.4(14)$ | $30.0(17)$ | $41.1(18)$ | $34.4(18)$ |
$4$ | $(\delta_{\ell}+1)(|a_{\Delta}|^{2}+|b_{\Delta}|^{2})+3\delta_{\ell}|t_{\Delta}|^{2}$ | $-34.3(19)$ | $-34.1(18)$ | $-32.8(21)$ | $-32.5(21)$ |
$5$ | $(2\delta_{\ell}-1)(|a_{\Delta}|^{2}-\frac{1}{2}|b_{\Delta}|^{2})$ | $70.7(4)$ | $64.0(2)$ | $73.9(13)$ | $66.9(12)$ |
$6$ | $-6\delta_{\ell}\left[\text{Re}(a_{+}t_{+}^{*})-\text{Re}(a_{-}t_{-}^{*})\right]-\frac{3}{2}|b_{\Delta}|^{2}$ | $-36.4(14)$ | $-29.8(17)$ | $-41.1(18)$ | $-34.4(18)$ |
$7$ | $\frac{3}{\sqrt{2}}\left[2\delta_{\ell}(t_{-}b_{+}^{*}-b_{-}t_{+}^{*})+(a_{-}b_{+}^{*}+b_{-}a_{+}^{*})\right]$ | $-91.7(9)$ | $-91.9(9)$ | $-92.1(7)$ | $-92.3(7)$ |
$8$ | $\frac{3}{\sqrt{2}}(2\delta_{\ell}-1)(b_{-}a_{+}^{*}-a_{-}b_{+}^{*})$ | $-78.7(7)$ | $-74.7(8)$ | $-83.2(15)$ | $-79.0(14)$ |

The calculated values of $\langle{\cal X}_{i}(\ell)\rangle$ show consistencies
between the HBM and LQCD. Nevertheless, ${\cal X}_{4\text{-}8}$ require
$\Lambda_{c}^{+}$ to be polarized for measurements, imposing difficulties in
the experiments. The form factors of $\Lambda_{c}^{+}\to n$ are given in
TABLE. 8 and FIG. 4. We see that once again the HBM underestimates $f_{2}$ by
a factors of two third, whereas the others are compatible with the LQCD.

Table 8:  The form factors of $\Lambda_{c}^{+}\to n$ at $q^{2}=0$. | $f_{1}(0)$ | $f_{2}(0)$ | $g_{1}(0)$ | $g_{2}(0)$
---|---|---|---|---
HBM | $0.570(56)$ | $0.210(1)$ | $0.526(50)$ | $0.015(2)$
LQCD Meinel:2017ggx  | $0.672(39)$ | $0.321(38)$ | $0.602(31)$ | $-0.003(52)$

Figure 4:  The $q^{2}$ dependencies of the form factors in $\Lambda_{c}^{+}\to
n$ from HBM, where the bands represent the uncertainties from the model
calculations.

##  IV TIME REVERSAL ASYMMETRIES AND NEW PHYSICS

In general, contaminated by the hadron uncertainties, it is often difficult to
draw a sharp conclusion on whether NP is needed to explain the experimental
data. Nevertheless, NP can generate clear signals of T asymmetries, which
vanish in the SM and are equivalent to the CP asymmetries based on the CPT
theorem. A great advantage of T violation is that a strong phase is not
necessary. The simplest T violating observables are studied in Ref.
Timereversal , and we highlight some the results here.

Two of the simplest T asymmetries in $\Lambda_{c}^{+}\to\Lambda(\to
p\pi^{-})\ell^{+}\nu_{\ell}$ are defined as

$\displaystyle{\cal T}_{\ell}$ $\displaystyle=$
$\displaystyle\frac{1}{P_{b}}\left(\int^{\pi}_{0}-\int_{\pi}^{2\pi}\right)\Gamma^{\ell}_{\phi_{\ell}}d\phi_{\ell}\quad=-\frac{\pi^{2}}{8\Gamma^{\ell}}\int_{M_{\ell}^{2}}^{M_{-}^{2}}\zeta{\bf\text{Im}}({\cal
X}_{7})dq^{2}\,,$ $\displaystyle{\cal T}_{s}$ $\displaystyle=$
$\displaystyle\frac{1}{\alpha}\left[\left(\int^{\pi}_{0}-\int_{\pi}^{2\pi}\right)d\Phi\right]\left[\left(\int_{0}^{1}-\int^{0}_{-1}\right)d\cos\theta_{s}\right]\Gamma^{\ell}_{\phi_{\ell},\cos\theta_{s}}$
(37) $\displaystyle=$
$\displaystyle-\frac{2}{3\pi\Gamma^{\ell}}\int_{M_{\ell}^{2}}^{M_{-}^{2}}\zeta{\bf\text{Im}}({\cal
X}_{12})dq^{2}\,,$

which are closely related to the azimuthal angles. The reason is that they
correspond to the triple product asymmetries of the three-momenta in the final
states. Keeping in mind $\xi_{\pm}$ are real in the SM, it is easy to see that
nonvanishing ${\cal T}_{\ell,s}$ require NP beyond the SM.

As an illustration, we explore NP with the effective Hamiltonian as

${\cal
H}^{\text{NP}}_{eff}=\frac{G_{F}}{\sqrt{2}}V_{cs}\left[C_{L}\overline{s}\gamma_{\mu}(1-\gamma_{5})c+C_{R}\overline{s}\gamma_{\mu}(1+\gamma_{5})c\right]\overline{v}\gamma_{\mu}(1-\gamma_{5})u\,.$
(38)

To the first order, we find that Timereversal

$\displaystyle\mathcal{T}_{\ell}=-\frac{3\pi^{2}}{8\sqrt{2}}\operatorname{Im}\left(C_{R}\right){\cal
Y}_{\ell}\,,~{}~{}~{}\mathcal{T}_{s}=-\frac{\sqrt{2}}{\pi}\operatorname{Im}\left(C_{R}\right){\cal
Y}_{s}\,,$ (39)

with

$\displaystyle{\cal Y}_{\ell}$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma^{\ell}}\int_{M_{\ell}^{2}}^{M_{-}^{2}}2\zeta\left(a_{+}b_{+}-a_{-}b_{-}\right)dq^{2}\,,$
$\displaystyle{\cal Y}_{s}$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma^{\ell}}\int_{M_{\ell}^{2}}^{M_{-}^{2}}2\zeta\left(a_{+}b_{-}-a_{-}b_{+}\right)dq^{2}\,,$
(40)

where $C_{L,R}$ are the Wilson coefficients from NP, and $\xi_{\pm}$ in Eq.
(IV) are calculated within the SM. In practice, $C_{L}$ can be absorbed by
redefining $V_{cs}$, so their effects vanish in the first order. We only
considered NP with the left-handed neutrinos and leptons, as the right-handed
ones are suppressed by the lepton masses Timereversal .

The values of ${\cal Y}_{\ell,s}$ can be viewed as the sensitivity
coefficients of NP. We take Im$(C_{R})=(0.1,0.2)$ as examples in TABLE 9. We
see that ${\cal T}_{\ell}$ can be as large as $10\%$, but they require
$\Lambda_{c}^{+}$ to be polarized for an experimental measurement. On the
other hand, though ${\cal T}_{c}$ can be probed with unpolarized
$\Lambda_{c}^{+}$, their values are twice smaller than ${\cal T}_{\ell}$,
making them hard to be observed. Finally, we emphasize that the T asymmetry
does not require a comparison with the charge conjugate as the strong phase is
irrelevant.

Table 9: The time-reversal asymmetries versus Im$(C_{R})$ from NP.  | Im$(C_{R})=0.1$ | Im$(C_{R})=0.2$
---|---|---
| $\ell^{+}=e^{+}$ | $\ell^{+}=\mu^{+}$ | $\ell^{+}=e^{+}$ | $\ell^{+}=\mu^{+}$
${\cal T}_{\ell}$ | $0.062(2)$ | $0.060(2)$ | $0.124(4)$ | $0.120(4)$
${\cal T}_{s}$ | $0.033(3)$ | $0.032(3)$ | $0.066(1)$ | $0.064(1)$

##  V Summary

We have given a systematical study on all the possible observables in
$\Lambda_{c}^{+}$ semileptonic decays, including the effects of NP. The model
independent angular distributions with polarized $\Lambda_{c}^{+}$ have been
presented. We have found that the BM underestimate $f_{2}$ by a factor of two
third, where the same underestimations were also found in the magnetic dipole
moments of the octet baryons. The branching fractions and polarization
asymmetries have been found to be ${\cal B}(\Lambda_{c}^{+}\to\Lambda
e^{+}\nu_{e},\Lambda\mu^{+}\nu_{\mu},n\ell^{+}\nu_{\ell})=(3.78\pm
0.25,3.67\pm 0.23,0.40\pm 0.04)\%$,and
$\alpha_{\Lambda}(\Lambda_{c}^{+}\to\Lambda
e^{+}\nu_{e},\Lambda\mu^{+}\nu_{\mu})=(-82.6,-82.3)\%$ from the HBM, and
$\alpha_{\Lambda}(\Lambda_{c}^{+}\to\Lambda
e^{+}\nu_{e},\Lambda\mu^{+}\nu_{\mu})=(-87.4\pm 1.0,-87.2\pm 1.0)\%$ from the
LQCD, which can be tested in the near future at BESIII and Belle II. In
particular, ${\cal B}(\Lambda_{c}^{+}\to\Lambda
e^{+}\nu_{e},\Lambda\mu^{+}\nu_{\mu})$ and
$\alpha_{\Lambda}(\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e})$ are in good
agreement with the current experimental data BESIII:2016ffj ; BESIII:2022ysa .
All the angular observables in the SM have been computed by both the HBM and
LQCD. Most of the results from the two approaches have been shown to be
consistent. The effects of NP on the T asymmetries have been explored, found
to be ${\cal O}(10\%)$ for Im$(C_{R})=0.2$. We stress that nonzero values of T
asymmetries in the experiments will be a smoking gun of NP.

##  Appendix A Baryon wave functions

For completeness, we display the wave functions of the low-lying baryons in
the HBM. The wave functions of the octet baryons associated with the isospin
of $I=1/2$ are given as

$\displaystyle|n,\updownarrow\rangle=\int\frac{1}{2\sqrt{3}}\epsilon^{\alpha\beta\gamma}d_{a\alpha}^{\dagger}(\vec{x}_{1})u_{b\beta}^{\dagger}(\vec{x}_{2})d_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi_{A_{\updownarrow}(dus)}^{abc}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
$\displaystyle|p,\updownarrow\rangle=\int\frac{1}{2\sqrt{3}}\epsilon^{\alpha\beta\gamma}d_{a\alpha}^{\dagger}(\vec{x}_{1})u_{b\beta}^{\dagger}(\vec{x}_{2})u_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi_{A_{\updownarrow}(dus)}^{abc}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
$\displaystyle|\Lambda,\updownarrow\rangle=\int\frac{1}{\sqrt{6}}\epsilon^{\alpha\beta\gamma}d_{a\alpha}^{\dagger}(\vec{x}_{1})u_{b\beta}^{\dagger}(\vec{x}_{2})s_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi_{A_{\updownarrow}(dus)}^{abc}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
$\displaystyle|\Xi^{-},\updownarrow\rangle=\int\frac{1}{2\sqrt{3}}\epsilon^{\alpha\beta\gamma}d_{a\alpha}^{\dagger}(\vec{x}_{1})s_{b\beta}^{\dagger}(\vec{x}_{2})s_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi_{A_{\updownarrow}(dus)}^{abc}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
$\displaystyle|\Xi^{0},\updownarrow\rangle=\int\frac{1}{2\sqrt{3}}\epsilon^{\alpha\beta\gamma}u_{a\alpha}^{\dagger}(\vec{x}_{1})s_{b\beta}^{\dagger}(\vec{x}_{2})s_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi_{A_{\updownarrow}(dus)}^{abc}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
(41)

and the ones with $I=1$ read as

$\displaystyle|\Sigma^{+},\updownarrow\rangle=\int\frac{1}{2\sqrt{3}}\epsilon^{\alpha\beta\gamma}u_{a\alpha}^{\dagger}(\vec{x}_{1})u_{b\beta}^{\dagger}(\vec{x}_{2})s_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi^{abc}_{S_{\updownarrow}(uuc)}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
$\displaystyle|\Sigma^{0},\updownarrow\rangle=\int\frac{1}{\sqrt{6}}\epsilon^{\alpha\beta\gamma}d_{a\alpha}^{\dagger}(\vec{x}_{1})u_{b\beta}^{\dagger}(\vec{x}_{2})s_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi^{abc}_{S_{\updownarrow}(uuc)}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
$\displaystyle|\Sigma^{-},\updownarrow\rangle=\int\frac{1}{2\sqrt{3}}\epsilon^{\alpha\beta\gamma}d_{a\alpha}^{\dagger}(\vec{x}_{1})d_{b\beta}^{\dagger}(\vec{x}_{2})s_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi^{abc}_{S_{\updownarrow}(uuc)}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,.$
(42)

The spin-flavor-antisymmetric spatial distribution $\Psi_{A}$ is defined in
Eq. (16), and the symmetric one is given as

$\displaystyle\Psi^{abc}_{S_{\uparrow}(q_{1}q_{2}q_{3})}$
$\displaystyle(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})=\frac{{\cal N}_{{\cal
B}}}{\sqrt{6}}\int\left(2\phi^{a}_{q_{1}\uparrow}(\vec{x_{1}}^{\prime})\phi^{b}_{q_{2}\uparrow}(\vec{x_{2}}^{\prime})\phi^{c}_{q_{3}\downarrow}(\vec{x_{3}}^{\prime})\right.$
(43)
$\displaystyle\left.-\phi^{a}_{q_{1}\uparrow}(\vec{x_{1}}^{\prime})\phi^{b}_{q_{2}\downarrow}(\vec{x_{2}}^{\prime})\phi^{c}_{q_{3}\uparrow}(\vec{x_{3}}^{\prime})-\phi^{a}_{q_{1}\downarrow}(\vec{x_{1}}^{\prime})\phi^{b}_{q_{2}\uparrow}(\vec{x_{2}}^{\prime})\phi^{c}_{q_{3}\uparrow}(\vec{x_{3}}^{\prime})\right)d^{3}\vec{x}_{\Delta}\,,$
$\displaystyle\Psi^{abc}_{S_{\downarrow}(q_{1}q_{2}q_{3})}$
$\displaystyle(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})=\frac{{\cal N}_{{\cal
B}}}{\sqrt{6}}\int\left(-2\phi^{a}_{q_{1}\downarrow}(\vec{x_{1}}^{\prime})\phi^{b}_{q_{2}\downarrow}(\vec{x_{2}}^{\prime})\phi^{c}_{q_{3}\uparrow}(\vec{x_{3}}^{\prime})\right.$
$\displaystyle\left.+\phi^{a}_{q_{1}\downarrow}(\vec{x_{1}}^{\prime})\phi^{b}_{q_{2}\uparrow}(\vec{x_{2}}^{\prime})\phi^{c}_{q_{3}\downarrow}(\vec{x_{3}}^{\prime})+\phi^{a}_{q_{1}\uparrow}(\vec{x_{1}}^{\prime})\phi^{b}_{q_{2}\downarrow}(\vec{x_{2}}^{\prime})\phi^{c}_{q_{3}\downarrow}(\vec{x_{3}}^{\prime})\right)d^{3}\vec{x}_{\Delta}\,,$

where $\vec{x}^{\prime}_{i}=\vec{x}_{i}-\vec{x}_{\Delta}$.

Similarly, the heavy baryons with a single heavy quark are given as

$\displaystyle|\Lambda_{c}^{+},\updownarrow\rangle=\int\frac{1}{\sqrt{6}}\epsilon^{\alpha\beta\gamma}d_{a\alpha}^{\dagger}(\vec{x}_{1})u_{b\beta}^{\dagger}(\vec{x}_{2})c_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi_{A_{\updownarrow}(duc)}^{abc}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
$\displaystyle|\Xi_{c}^{+},\updownarrow\rangle=\int\frac{1}{\sqrt{6}}\epsilon^{\alpha\beta\gamma}u_{a\alpha}^{\dagger}(\vec{x}_{1})s_{b\beta}^{\dagger}(\vec{x}_{2})c_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi_{A_{\updownarrow}(usc)}^{abc}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
$\displaystyle|\Xi_{c}^{0},\updownarrow\rangle=\int\frac{1}{\sqrt{6}}\epsilon^{\alpha\beta\gamma}d_{a\alpha}^{\dagger}(\vec{x}_{1})s_{b\beta}^{\dagger}(\vec{x}_{2})c_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi_{A_{\updownarrow}(dsc)}^{abc}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
(44)

for the antitriplet baryons, and

$\displaystyle|\Sigma_{c}^{++},\updownarrow\rangle=\int\frac{1}{2\sqrt{3}}\epsilon^{\alpha\beta\gamma}u_{a\alpha}^{\dagger}(\vec{x}_{1})u_{b\beta}^{\dagger}(\vec{x}_{2})c_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi^{abc}_{S_{\updownarrow}(uuc)}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
$\displaystyle|\Sigma_{c}^{+},\updownarrow\rangle=\int\frac{1}{\sqrt{6}}\epsilon^{\alpha\beta\gamma}d_{a\alpha}^{\dagger}(\vec{x}_{1})u_{b\beta}^{\dagger}(\vec{x}_{2})c_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi^{abc}_{S_{\updownarrow}(duc)}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
$\displaystyle|\Sigma_{c}^{0},\updownarrow\rangle=\int\frac{1}{2\sqrt{3}}\epsilon^{\alpha\beta\gamma}d_{a\alpha}^{\dagger}(\vec{x}_{1})d_{b\beta}^{\dagger}(\vec{x}_{2})c_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi^{abc}_{S_{\updownarrow}(ddc)}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
$\displaystyle|\Xi_{c}^{\prime+},\updownarrow\rangle=\int\frac{1}{\sqrt{6}}\epsilon^{\alpha\beta\gamma}u_{a\alpha}^{\dagger}(\vec{x}_{1})s_{b\beta}^{\dagger}(\vec{x}_{2})c_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi^{abc}_{S_{\updownarrow}(usc)}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
$\displaystyle|\Xi_{c}^{\prime
0},\updownarrow\rangle=\int\frac{1}{\sqrt{6}}\epsilon^{\alpha\beta\gamma}d_{a\alpha}^{\dagger}(\vec{x}_{1})s_{b\beta}^{\dagger}(\vec{x}_{2})c_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi^{abc}_{S_{\updownarrow}(dsc)}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
$\displaystyle|\Omega_{c}^{0},\updownarrow\rangle=\int\frac{1}{2\sqrt{3}}\epsilon^{\alpha\beta\gamma}s_{a\alpha}^{\dagger}(\vec{x}_{1})s_{b\beta}^{\dagger}(\vec{x}_{2})c_{c\gamma}^{\dagger}(\vec{x}_{3})\Psi^{abc}_{S_{\updownarrow}(ssc)}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
(45)

for the sextet baryons. The bottom baryons are obtained directly by
substituting bottom quarks for the charmed quarks.

On the other hand, the low-lying spin $3/2$ baryons are constructed by

$|{\cal B},J=\frac{3}{2},J_{z}\rangle=\int\frac{1}{\sqrt{6S_{\cal
B}!}}\epsilon^{\alpha\beta\gamma}q_{1a\alpha}^{\dagger}(\vec{x}_{1})q_{2b\beta}^{\dagger}(\vec{x}_{2})q_{3c\gamma}^{\dagger}(\vec{x}_{3})\Psi^{abc}_{TJ_{z}(q_{1}q_{2}q_{3})}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})[d^{3}\vec{x}]|0\rangle\,,$
(46)

where $S_{\cal B}$ is the number of the identical quark in the baryon, $J_{z}$
the angular momentum in $\hat{z}$ direction, and

$\displaystyle\Psi^{abc}_{T\frac{3}{2}(q_{1}q_{2}q_{3})}$
$\displaystyle(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})={\cal N}_{{\cal
B}}\int\phi^{a}_{q_{1}\uparrow}(\vec{x_{1}}^{\prime})\phi^{b}_{q_{2}\uparrow}(\vec{x_{2}}^{\prime})\phi^{c}_{q_{3}\uparrow}(\vec{x_{3}}^{\prime})d^{3}\vec{x}_{\Delta}\,,$
(47) $\displaystyle\Psi^{abc}_{T\frac{1}{2}(q_{1}q_{2}q_{3})}$
$\displaystyle(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})=\frac{{\cal N}_{{\cal
B}}}{\sqrt{3}}\int\big{(}\phi^{a}_{q_{1}\downarrow}(\vec{x_{1}}^{\prime})\phi^{b}_{q_{2}\uparrow}(\vec{x_{2}}^{\prime})\phi^{c}_{q_{3}\uparrow}(\vec{x_{3}}^{\prime})d^{3}\vec{x}_{\Delta}$
$\displaystyle+\phi^{a}_{q_{1}\uparrow}(\vec{x_{1}}^{\prime})\phi^{b}_{q_{2}\downarrow}(\vec{x_{2}}^{\prime})\phi^{c}_{q_{3}\uparrow}(\vec{x_{3}}^{\prime})+\phi^{a}_{q_{1}\uparrow}(\vec{x_{1}}^{\prime})\phi^{b}_{q_{2}\uparrow}(\vec{x_{2}}^{\prime})\phi^{c}_{q_{3}\downarrow}(\vec{x_{3}}^{\prime})\big{)}d^{3}\vec{x}_{\Delta}\,.$

The baryon wave functions with negative angular momenta can be obtained by
flipping the spin directions in both sides of Eq. (47).

###### Acknowledgements.

The authors are indebted to Stefan Meinel for the help on the LQCD form
factors. This work is supported in part by the National Key Research and
Development Program of China under Grant No. 2020YFC2201501 and the National
Natural Science Foundation of China (NSFC) under Grant No. 12147103.

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