#  $\Lambda_{c}$ semileptonic decays

Sheng-Qi Zhang1 and Cong-Feng Qiao1111Corresponding author<EMAIL_ADDRESS>

1School of Physical Sciences, University of Chinese Academy of Sciences
YuQuan Road 19A, Beijing 100049, China

###### Abstract

Motivated by the recent experimental progress in the $\Lambda_{c}$ decay that
contains a neutron in the final state, we analyze the semileptonic decay
$\Lambda_{c}\rightarrow n\ell\nu_{\ell}$ in the framework of QCD sum rules.
The transition form factors are analytically computed using three-point
correlation functions and the Cutkosky cutting rules, which can be
extrapolated into the physical region by employing the dipole parametrization.
The branching fractions of $\Lambda_{c}\rightarrow ne^{+}\nu_{e}$ and
$\Lambda_{c}\rightarrow n\mu^{+}\nu_{\mu}$ are estimated to be $(0.280\pm
0.031)\%$ and $(0.274\pm 0.030)\%$, respectively. Furthermore, we calculate as
well the relevant decay asymmetry observables sensitive to new physics beyond
the standard model. The numerical results of semileptonic decays
$\Lambda_{c}\rightarrow\Lambda\ell\nu_{\ell}$ are also given and confronted to
the latest experimental data.

##  1 Introduction

The semileptonic decay of the lightest charmed baryon $\Lambda_{c}$ plays an
important role in exploring strong and weak interactions in charm sectors. It
can help elucidate the role of nonperturbative effects in strong interactions
and provide crucial inputs for studying heavier charmed baryons and bottom
baryons decay. Additionally, the precise measurement of the Cabibbo-Kobayashi-
Maskawa (CKM) matrix elements $|V_{cs}|$ and $|V_{cd}|$ can also provide the
significant test for the standard model and the probable evidence for new
physics beyond the standard model [1].

Recent years, there have been extensive measurements of the semileptonic decay
modes $\Lambda_{c}\rightarrow\Lambda\ell\nu_{\ell}$ [2, 3, 4, 5]. The most
precise results of branching fractions yet are
$\mathcal{B}\left(\Lambda_{c}\rightarrow\Lambda e^{+}\nu_{e}\right)=(3.56\pm
0.11\pm 0.07)\%$ [4] and
$\mathcal{B}\left(\Lambda_{c}\rightarrow\Lambda\mu^{+}\nu_{\mu}\right)=(3.48\pm
0.14\pm 0.10)\%$ [5], respectively. Comparing the former result with
$\Lambda_{c}$ inclusive semileptonic decay mode
$\mathcal{B}\left(\Lambda_{c}\rightarrow Xe^{+}\nu_{e}\right)=(3.95\pm 0.34\pm
0.09)\%$ [6], it can be inferred that there still remain some potential
exclusive semileptonic decay modes measureable. Recently, BESIII collaboration
reported the evidence of the decay modes containing excited states,
specifically $\Lambda_{c}\rightarrow\Lambda(1520)e^{+}\nu_{e}$ and
$\Lambda_{c}\rightarrow\Lambda(1405)e^{+}\nu_{e}$ [7]. These two decay modes
yield a relative small branching fractions to be $(1.02\pm 0.52\pm 0.11)\times
10^{-3}$ and $(0.42\pm 0.19\pm 0.04)\times 10^{-3}$, respectively. Moreover,
the measurement of two five-body semileptonic decay modes
$\Lambda_{c}\rightarrow\Lambda\pi^{+}\pi^{-}e^{+}\nu_{e}$ and
$\Lambda_{c}\rightarrow pK_{s}^{0}\pi^{-}e^{+}\nu_{e}$ are also performed [8],
in which the upper limits are set to be
$\mathcal{B}(\Lambda_{c}\rightarrow\Lambda\pi^{+}\pi^{-}e^{+}\nu_{e})<3.9\times
10^{-4}$ and $\mathcal{B}(\Lambda_{c}\rightarrow
pK_{s}^{0}\pi^{-}e^{+}\nu_{e})<3.3\times 10^{-4}$. In physics, besides the
$\Lambda_{c}$ semileptonic decay modes that include $\Lambda(\Lambda^{*})$
baryon in the final state, the exclusive semileptonic decay modes
$\Lambda_{c}\rightarrow n\ell\nu_{\ell}$ are also permitted by the standard
model. However, there is still a lack of experimental data in this regard.

Theoretically, $\Lambda_{c}\rightarrow n\ell\nu_{\ell}$ is dominated by the
Cabibbo-suppressed transition $c\rightarrow d\ell\nu_{\ell}$. As a result, the
decay width is anticipated to be much smaller compared to the
$\Lambda_{c}\rightarrow\Lambda\ell\nu_{\ell}$ mode, which is dominated by the
Cabibbo-favored transition $c\rightarrow s\ell\nu_{\ell}$. While
experimentally, the main challenge lies in distinguishing neutron signals from
neutral noises, which leads to the problem of direct neutron detections [9,
10]. Fortunately, with the improvement of detector performance and analysis
technique, BESIII collaboration has made notable progress in measuring
$\Lambda_{c}$ decays that involve neutron signals in the final state [9, 10,
11, 12]. It is predictable that the experimental data for the decay mode
$\Lambda_{c}\rightarrow n\ell\nu_{\ell}$ will be available in the near future,
making it beneficial to explore this process theoretically. Furthermore, the
semileptonic decay $\Lambda_{c}\rightarrow n\ell\nu_{\ell}$ is an exceptional
candidate for extracting the magnitude of CKM matrix element $|V_{cd}|$.
Currently, the determination of $|V_{cd}|$ relies primarily on the charm meson
semileptonic decay $D\rightarrow\pi\ell\nu_{\ell}$ [13, 14, 15, 16, 17].
Therefore, it is of great importance to investigate the semileptonic decay
$\Lambda_{c}\rightarrow n\ell\nu_{\ell}$ both experimentally and theoretically
since such studies are crucial for providing precise verification for
$|V_{cd}|$ in the charm baryon sector.

In past, theoretical investigations for $\Lambda_{c}\rightarrow
n\ell\nu_{\ell}$ semileptonic decay have been performed in depth in a variety
of methods, such as the light-cone sum rules (LCSR) [18, 19, 20], the light
front approach (LF) [21], the covariant confined quark model (CCQM) [22], the
constituent quark model (CQM) [23], the relativistic quark model (RQM) [24],
the $SU(3)$ flavor symmetry [25, 26], the MIT bag model (MBM) [27], and the
lattice QCD (LQCD) [28]. Additionally, QCD sum rules (QCDSR) has also been
widely utilized to deal with the baryonic decay mode [29, 30, 31, 32, 33, 34,
35, 36, 37]. Rather than a phenomenological model, QCDSR is a QCD-based
theoretical framework that systematically incorporates nonperturbative effects
at each dimension. To evaluate the form factors in the weak transitions, the
three-point correlation functions are constructed with appropriate
interpolating currents. The QCDSR will be formally established by equating two
representations of the three-point correlation functions, namely the QCD
representation and the phenomenological representation, which enables the
determination of the transition form factors. In this work, we will apply
QCDSR to calculate the form factors of the $\Lambda_{c}\rightarrow
n\ell\nu_{\ell}$ semileptonic decay mode, after which the branching fractions
as well as some other relevant decay asymmetry observables are also obtained.
Besides, the numerical results of
$\Lambda_{c}\rightarrow\Lambda\ell\nu_{\ell}$ semileptonic decay are also
given and compared with latest experimental results.

The rest of the paper is structured as follows: in Sec. 2 we interpret the
basic idea of QCDSR for the three-point correlation functions. The numerical
results and analysis are presented in Sec. 3. The conclusions and discussions
are given in the last section.

##  2 Formalism

The $\Lambda_{c}\rightarrow n\ell\nu_{\ell}$ decay is dominated by the
Cabibbo-suppressed transition $c\rightarrow d\ell\nu_{\ell}$ at quark level.
The effective Hamiltonian depicting this transition is written as

$\displaystyle\mathcal{H}_{eff}=\frac{G_{F}}{\sqrt{2}}V_{cd}[\bar{\ell}\gamma_{\mu}\left(1-\gamma_{5}\right)v_{\ell}][\bar{d}\gamma^{\mu}\left(1-\gamma_{5}\right)c],$
(1)

where $G_{F}$ denotes the Fermi constant and $V_{cd}$ is the CKM matrix
element. The Feynman diagram of $\Lambda_{c}\rightarrow n\ell\nu_{\ell}$ is
shown in Fig. 1. The leptonic part of this decay mode can be obtained through
electro-weak perturbation theory, while the hadronic part can not be
calculated perturbatively due to its involvement in the low-energy aspects of
QCD. In general, the weak transition matrix element of hadronic part can be
parametrized in terms of transition form factors

$\displaystyle\bra{\Lambda_{c}(q_{1})}j_{\mu}\ket{n(q_{2})}$
$\displaystyle=\bar{u}_{\Lambda_{c}}(q_{1})\bigg{[}f_{1}(q^{2})\gamma_{\mu}+if_{2}(q^{2})\sigma^{\mu\nu}\frac{q_{\nu}}{M_{\Lambda_{c}}}+f_{3}(q^{2})\frac{q_{\mu}}{M_{\Lambda_{c}}}\bigg{]}u_{n}(q_{2})$
$\displaystyle-\bar{u}_{\Lambda_{c}}(q_{1})\bigg{[}g_{1}(q^{2})\gamma_{\mu}+ig_{2}(q^{2})\sigma^{\mu\nu}\frac{q_{\nu}}{M_{\Lambda_{c}}}+g_{3}(q^{2})\frac{q_{\mu}}{M_{\Lambda_{c}}}\bigg{]}\gamma_{5}u_{n}(q_{2}),$
(2)

where $q_{1}$ and $q_{2}$ are the four-vector momentum of initial state
$\Lambda_{c}$ and final state neutron, respectively. The momentum transfer $q$
is defined as $q=q_{1}-q_{2}$.

Figure 1: Feynman diagram for $\Lambda_{c}\rightarrow n\ell\nu_{\ell}$
semileptonic decay.

To calculate the transition form factors by QCDSR, the three-point correlation
functions can be formally constructed as

$\Pi_{\mu}(q_{1}^{2},q_{2}^{2},q^{2})=i^{2}\int
d^{4}x\;d^{4}y\;e^{i(q_{1}x-q_{2}y)}<0|T\\{j_{\Lambda_{c}}(x)j_{\mu}(0)j^{\dagger}_{n}(y)\\}|0>.$
(3)

The weak transition current $j_{\mu}$ is defined as
$j_{\mu}=\bar{c}\gamma_{\mu}\left(1-\gamma_{5}\right)d$ and the interpolating
currents of $\Lambda_{c}$ and neutron take the following quark structure [35,
38],

$\displaystyle j_{\Lambda_{c}}$
$\displaystyle=\epsilon_{ijk}\left(u_{i}^{T}C\gamma_{5}d_{j}\right)c_{k},$ (4)
$\displaystyle j_{n}$
$\displaystyle=\epsilon_{ijk}\left(u_{i}^{T}C\gamma_{5}d_{j}\right)d_{k},$ (5)

where the subscripts $i$, $j$, and $k$ represent the color indices and $C$ is
the charge conjugation matrix.

On the phenomenological side, after inserting a complete set of intermediate
hadronic states and exploiting double dispersion relations, the three-point
correlation functions in Eq. (3) can be described as,

$\displaystyle\Pi_{\mu}^{\text{phe}}(q_{1}^{2},q_{2}^{2},q^{2})=$
$\displaystyle\sum_{\text{spins}}\frac{\bra{0}j_{\Lambda_{c}}\ket{\Lambda_{c}(q_{1})}\bra{\Lambda_{c}(q_{1})}j_{\mu}\ket{n(q_{2})}\bra{n(q_{2})}j_{n}\ket{0}}{(q_{1}^{2}-M_{\Lambda_{c}}^{2})(q_{2}^{2}-M_{n}^{2})}$
$\displaystyle+$ $\displaystyle\text{higher resonances and continuum states},$
(6)

where $M_{\Lambda_{c}}$ and $M_{n}$ denote the mass of $\Lambda_{c}$ and
neutron, respectively. The vacuum-to-baryon transition amplitudes can be
parametrized by defining the decay constants,

$\displaystyle\bra{0}j_{\Lambda_{b}}\ket{\Lambda_{c}(q_{1})}$
$\displaystyle=\lambda_{\Lambda_{c}}u_{\Lambda_{c}}(q_{1}),$ (7)
$\displaystyle\bra{0}j_{n}\ket{n(q_{2})}$
$\displaystyle=\lambda_{n}u_{n}(q_{2}),$ (8)

where $\lambda_{\Lambda_{c}}$ and $\lambda_{n}$ represent the decay constants
of $\Lambda_{c}$ and neutron, respectively. By introducing the hadronic
transition matrix elements in Eq. (2) and utilizing the spin sum completeness
relations, $\sum
u_{\Lambda_{c}}(q_{1})\bar{u}_{\Lambda_{c}}(q_{1})=\not{q}_{1}+M_{\Lambda_{c}}$
and $\sum u_{n}(q_{2})\bar{u}_{n}(q_{2})=\not{q}_{2}+M_{n}$, we can finally
obtain the phenomenological representation of the three-point correlation
functions of Eq. (3),

$\displaystyle\Pi_{\mu}^{\text{phe}}(q_{1}^{2},q_{2}^{2},q^{2})$
$\displaystyle=\frac{\lambda_{n}\left(\not{q}_{2}+M_{n}\right)\bigg{[}f_{1}(q^{2})\gamma_{\mu}+if_{2}(q^{2})\sigma^{\mu\nu}\frac{q_{\nu}}{M_{\Lambda_{c}}}+f_{3}(q^{2})\frac{q_{\mu}}{M_{\Lambda_{c}}}\bigg{]}\lambda_{\Lambda_{c}}\left(\not{q}_{1}+M_{\Lambda_{c}}\right)}{(q_{1}^{2}-M_{\Lambda_{c}}^{2})(q_{2}^{2}-M_{n}^{2})}$
$\displaystyle-\frac{\lambda_{n}\left(\not{q}_{2}+M_{n}\right)\bigg{[}g_{1}(q^{2})\gamma_{\mu}+ig_{2}(q^{2})\sigma^{\mu\nu}\frac{q_{\nu}}{M_{\Lambda_{c}}}+g_{3}(q^{2})\frac{q_{\mu}}{M_{\Lambda_{c}}}\bigg{]}\gamma_{5}\lambda_{\Lambda_{c}}\left(\not{q}_{1}+M_{\Lambda_{c}}\right)}{(q_{1}^{2}-M_{\Lambda_{c}}^{2})(q_{2}^{2}-M_{n}^{2})}.$
(9)

It should be noted that we assume $f_{3}(q^{2})$ and $g_{3}(q^{2})$ to be
negligible in this study as they will contribute to semileptonic decays at
$\mathcal{O}\left(m_{\ell}^{2}\right)$ [39, 33].

On the QCD side, the three-point correlation functions of Eq. (3) can be
expressed by operator-product expansion (OPE) and double dispersion relations,

$\displaystyle\Pi_{\mu}^{\text{QCD}}(q_{1}^{2},q_{2}^{2},q^{2})=\int_{s_{1}^{\text{min}}}^{\infty}ds_{1}\int_{s_{2}^{\text{min}}}^{\infty}ds_{2}\frac{\rho^{\text{QCD}}_{\mu}(s_{1},s_{2},q^{2})}{(s_{1}-q_{1}^{2})(s_{2}-q_{2}^{2})},$
(10)

where $s_{1(2)}^{\text{min}}$ is the kinematic limit.
$\rho^{\text{QCD}}_{\mu}(s_{1},s_{2},q^{2})$ stands for the spectral density,
which can be obtained through the application of Cutkosky cutting rules [35,
33, 34, 36, 37, 40, 41, 42]. In this work, contributions up to dimension 6 are
considered in $\rho^{\text{QCD}}_{\mu}(s_{1},s_{2},q^{2})$, which can be
expressed as,

$\displaystyle\rho^{\text{QCD}}_{\mu}(s_{1},s_{2},q^{2})$
$\displaystyle=\rho^{\text{pert}}_{\mu}(s_{1},s_{2},q^{2})+\rho^{\langle\bar{q}q\rangle}_{\mu}(s_{1},s_{2},q^{2})+\rho^{\langle
g_{s}^{2}G^{2}\rangle}_{\mu}(s_{1},s_{2},q^{2})$ $\displaystyle+\rho^{\langle
g_{s}\bar{q}\sigma\cdot
Gq\rangle}_{\mu}(s_{1},s_{2},q^{2})+\rho^{\langle\bar{q}q\rangle^{2}}_{\mu}(s_{1},s_{2},q^{2}).$
(11)

The first term corresponds to the perturbative contribution, while
$\langle\bar{q}q\rangle$, $\langle g_{s}^{2}G^{2}\rangle$, $\langle
g_{s}\bar{q}\sigma\cdot Gq\rangle$, and $\langle\bar{q}q\rangle^{2}$ represent
condensates that describe the nonperturbative effects. The relevant Feynman
diagrams are plotted in Fig. 2.

To establish the relation between phenomenological representation and QCD
representation, the quark-hadron duality is adopted,

$\Pi_{\mu}^{\text{phe}}(q_{1}^{2},q_{2}^{2},q^{2})\simeq\int_{s_{1}^{\text{min}}}^{s_{1}^{0}}ds_{1}\int_{s_{2}^{\text{min}}}^{s_{2}^{0}}ds_{2}\frac{\rho^{\text{QCD}}_{\mu}(s_{1},s_{2},q^{2})}{(s_{1}-q_{1}^{2})(s_{2}-q_{2}^{2})}.$
(12)

Here, $s_{1}^{0}$ and $s_{2}^{0}$ denote the threshold parameters of
$\Lambda_{c}$ and neutron, respectively. After taking into account the double
Borel transform to suppress the higher excited states and continuum states
contributions, the analytic expression of $f_{i}(q^{2})$ and $g_{i}(q^{2})$
can be derived,

$\displaystyle f_{1}(t)=g_{1}(t)=$
$\displaystyle\frac{e^{M_{\Lambda_{c}}^{2}/M_{B_{1}}^{2}}e^{M_{n}^{2}/M_{B_{2}}^{2}}}{\lambda_{\Lambda_{c}}\lambda_{n}M_{\Lambda_{c}}}\bigg{[}\int_{s_{1}^{\text{min}}}^{s_{1}^{0}}ds_{1}\int_{s_{2}^{\text{min}}}^{s_{2}^{0}}ds_{2}\int
d\xi\frac{3\,m_{c}\,\xi}{64\,\pi^{4}\lambda(s_{1},s_{2},t)^{3/2}}\times$
$\displaystyle\big{(}m_{c}^{2}(s_{1}-t-s_{2})-t\,(s_{1}-t+s_{2}-2\,\xi)\big{)}\times
e^{-s_{1}/M_{B_{1}}^{2}}e^{-s_{2}/M_{B_{2}}^{2}}$
$\displaystyle+\frac{m_{c}\langle\bar{q}q\rangle^{2}}{6}e^{-m_{c}^{2}/M_{B_{1}}^{2}}e^{-m_{d}^{2}/M_{B_{2}}^{2}}\bigg{]},$
(13) $\displaystyle f_{2}(t)=g_{2}(t)=$
$\displaystyle\frac{e^{M_{\Lambda_{c}}^{2}/M_{B_{1}}^{2}}e^{M_{n}^{2}/M_{B_{2}}^{2}}}{\lambda_{\Lambda_{c}}\lambda_{n}}\int_{s_{1}^{\text{min}}}^{s_{1}^{0}}ds_{1}\int_{s_{2}^{\text{min}}}^{s_{2}^{0}}ds_{2}\int
d\xi\frac{3\,\xi}{64\,\pi^{4}\lambda(s_{1},s_{2},t)^{5/2}}\Big{[}$
$\displaystyle
m_{c}^{4}\big{(}s_{2}(s_{1}+t)+(s_{1}-t)^{2}-2\,s_{2}^{2}\big{)}-$
$\displaystyle
m_{c}^{2}\big{(}s_{1}^{3}-s_{1}^{2}(t+s_{2}+2\,\xi)-s_{1}(t^{2}+2\,t\,(\xi-3\,s_{2})+s_{2}(s_{2}-4\,\xi))$
$\displaystyle+(t-s_{2})(t^{2}+4\,t\,\xi-s_{2}(s_{2}-2\,\xi))\big{)}-$
$\displaystyle
t\big{(}-2\xi(-2\,s_{1}^{2}+s_{1}(t+s_{2})+(t-s_{2})^{2})-3\,\xi^{2}(s_{1}+t-s_{2})$
$\displaystyle-
s_{1}s_{2}(s_{1}+t)-s_{1}(s_{1}-t)^{2}+2\,s_{1}s_{2}^{2}\big{)}\Big{]}\times
e^{-s_{1}/M_{B_{1}}^{2}}e^{-s_{2}/M_{B_{2}}^{2}},$ (14)

where we define $t=q^{2}$ and
$\lambda(s_{1},s_{2},t)=s_{1}^{2}+s_{2}^{2}+t^{2}-2s_{1}s_{2}-2s_{1}t-2s_{2}t$.
$M_{B_{1}}^{2}$ and $M_{B_{2}}^{2}$ represent the Borel parameters which will
appear after double Borel transform. The variable $\xi$ is introduced in the
integral through the phase space integration by utilizing Cutkosky cutting
rules. It can be observed from Eq. (2) and (2) that the quark condensate
$\langle\bar{q}q\rangle$ and the mixed quark-gluon condensate $\langle
g_{s}\bar{q}\sigma\cdot Gq\rangle$ do not contribute to the transition form
factors, while we find the contribution from gluon condensate $\langle
g_{s}^{2}G^{2}\rangle$ is negligible and can be ignored. Thus, only the four-
quark condensate $\langle\bar{q}q\rangle^{2}$ determines the primary
nonperturbative contribution to $f_{1}$, which is in agreement with previous
theoretical studies of heavy to light transitions [33, 31, 32].

Figure 2: The Feynman diagrams for the computation of
$\rho^{\text{QCD}}_{\mu}(s_{1},s_{2},q^{2})$. Double solid lines represent
charm quark, and ordinary solid lines denote light quark.

##  3 Numerical results and discussions

In our numerical calculation, the following input parameters are adopted [43,
44, 42, 41, 45, 20, 46, 47],

$\displaystyle\langle\bar{q}q\rangle$ $\displaystyle=-(0.24\pm
0.01)^{3}\text{GeV}^{3},$ $\displaystyle s_{1}^{0}$ $\displaystyle=(9.5\sim
10.5)\,\text{GeV}^{2},\quad s_{2}^{0}=(2.4\sim 3.0)\,\text{GeV}^{2}$
$\displaystyle m_{c}$ $\displaystyle=1.27\pm 0.02\,\text{GeV},\quad
m_{d}=4.67^{+0.48}_{-0.17}\,\text{MeV},$ $\displaystyle\lambda_{\Lambda_{c}}$
$\displaystyle=0.0119\,\text{GeV}^{3},\quad\lambda_{n}=0.02\,\text{GeV}^{3},$
$\displaystyle M_{\Lambda_{c}}$ $\displaystyle=2.286\,\text{GeV},\quad
M_{n}=0.938\,\text{GeV}.$ (15)

Here, the standard value of quark condensate $\langle\bar{q}q\rangle$ is taken
at the renormalization point $\mu=1\,\text{GeV}$. The decay constants and the
threshold parameters are determined using the two-point sum rules [20, 46],
employing the same interpolating currents of Eq. (4) and Eq. (5).

Moreover, two additional free parameters, namely the Borel parameters
$M_{B_{1}}^{2}$ and $M_{B_{2}}^{2}$, are introduced in the framework of QCDSR.
For simplicity, we adopt the following relation of Borel parameters [34, 33,
48],

$\displaystyle\frac{M_{B_{1}}^{2}}{M_{B_{2}}^{2}}=\frac{M_{\Lambda_{c}}^{2}-m_{c}^{2}}{M_{n}^{2}-m_{d}^{2}}.$
(16)

In general, two criteria are employed to determine the values of Borel
parameters. First, the pole contribution. In order to investigate the
contribution of ground-state hadrons, the pole contribution has to dominate
the spectrum. Thus, the pole contribution can be selected larger than $40\%$
for the transition form factors, which can be formulated as follows,

$\displaystyle
R^{\text{PC}}_{\Lambda_{c}}=\frac{\int_{s_{1}^{\text{min}}}^{s_{1}^{0}}ds_{1}\int_{s_{2}^{\text{min}}}^{s_{2}^{0}}ds_{2}}{\int_{s_{1}^{\text{min}}}^{\infty}ds_{1}\int_{s_{2}^{\text{min}}}^{s_{2}^{0}}ds_{2}},$
(17) $\displaystyle
R^{\text{PC}}_{n}=\frac{\int_{s_{1}^{\text{min}}}^{s_{1}^{0}}ds_{1}\int_{s_{2}^{\text{min}}}^{s_{2}^{0}}ds_{2}}{\int_{s_{1}^{\text{min}}}^{s_{1}^{0}}ds_{1}\int_{s_{2}^{\text{min}}}^{\infty}ds_{2}}.$
(18)

These two ratios can be regarded as the pole contribution from the
$\Lambda_{c}$ chanel and neutron chanel, respectively.

The second criterion is the convergence of OPE, which ensures that the
neglected power corrections in the condensate term remain small and the
truncated OPE remains effective. In our calculation, only the four-quark
condensate $\langle\bar{q}q\rangle^{2}$ in Eq. (2) contribute to the expansion
of OPE, which means the relative contribution from the condensate
$\langle\bar{q}q\rangle^{2}$ needs to be less than $30\%$. Additionally, since
the Borel parameters $M_{B_{1}}^{2}$ and $M_{B_{2}}^{2}$ are not physical
parameters, it is necessary to find an optimal window in which the transition
form factors exhibit minimal dependence of $M_{B_{1}}^{2}$ and
$M_{B_{2}}^{2}$.

Through above preparation, the transition form factors of the semileptonic
decay $\Lambda_{c}\rightarrow n\ell\nu_{\ell}$ can be numerically calculated.
The dependence of the form factors at the maximum recoil point $q^{2}=0$ with
the required range of Borel parameter $M_{B_{2}}^{2}$ is shown in Fig. 3. In
Fig. 3, it can be observed that the variation of $s_{1}^{0}$ has a negligible
effect on $f_{1}(0)$ and $f_{2}(0)$, whereas the variation of $s_{2}^{0}$ has
a more significant impact. For comparison, we show our results and previous
theoretical predictions of transition form factors at maximum recoil point
$q^{2}=0$ in Table 1. The errors are mainly determined by the uncertainties of
the Borel parameters $M_{B_{1}}^{2}$ and $M_{B_{2}}^{2}$ and other input
parameters listed in Eq. (3). In Table. 1, our results for $f_{1}(0)$,
$f_{2}(0)$, and $g_{1}(0)$ are comparable to other predictions, while there is
significant variation for $g_{2}(0)$ obtained from different theoretical
methods. In this work, the sign of $g_{2}(0)$ aligns with the results from
LCSR [20] and LF approach [21], but differs from those derived by other
theoretical methods. Further investigations are needed to resolve this
discrepancy. Moreover, it is worth mentioning that the results from LCSR [20]
are derived using the same interpolating current as in Eq. (4), where the
transition form factors at $q^{2}=0$ show a high level of consistency with
QCDSR.

Figure 3: The transition form factors $f_{1}(0)$ and $f_{2}(0)$ as functions of Borel parameter $M_{B_{2}}^{2}$ for different values of $s_{1}^{0}$ and $s_{2}^{0}$. Table 1: Theoretical predictions for the form factors of the semileptonic decay $\Lambda_{c}\rightarrow n\ell\nu_{\ell}$ at the maximum recoil point $q^{2}=0$ with different approaches. method | $f_{1}(0)$ |  | $f_{2}(0)$ |  | $g_{1}(0)$ |  | $g_{2}(0)$
---|---|---|---|---|---|---|---
QCDSR | $0.53\pm 0.04$ |  | $-0.25\pm 0.03$ |  | $0.53\pm 0.04$ |  | $-0.25\pm 0.03$
LCSR [20] | $0.59^{+0.15}_{-0.16}$ |  | $-0.43^{+0.13}_{-0.12}$ |  | $0.55^{+0.14}_{-0.15}$ |  | $-0.16^{+0.08}_{-0.05}$
LF [21] | 0.513 |  | -0.266 |  | 0.443 |  | -0.034
CCQM [22] | 0.47 |  | -0.246 |  | 0.414 |  | 0.073
RQM [24] | 0.627 |  | -0.259 |  | 0.433 |  | 0.118
MBM [27] | 0.40 |  | -0.22 |  | 0.43 |  | 0.07
LQCD [28] | $0.672\pm 0.039$ |  | $-0.321\pm 0.038$ |  | $0.602\pm 0.031$ |  | $0.003\pm 0.052$

Considering that the QCDSR method is applicable only in the small $q^{2}$
region, and the physical region for $q^{2}$ in the $\Lambda_{c}\rightarrow
n\ell\nu_{\ell}$ decay extends from $m_{\ell}^{2}$ to
$(M_{\Lambda_{c}}-M_{n})^{2}$, we perform our calculations within a limited
range of $q^{2}\in[-0.4,0.4]\,\text{GeV}^{2}$ and employ a dipole
parametrization to extrapolate the obtained values to the entire physical
region. The expression of dipole parametrization is

$\displaystyle
f_{i}(q^{2})=\frac{f_{i}(0)}{\left(1-q^{2}/M_{\Lambda_{c}}^{2}\right)\left(1-a_{1}q^{2}/M_{\Lambda_{c}}^{2}+a_{2}(q^{2}/M_{\Lambda_{c}}^{2})^{2}\right)}.$
(19)

Here, $a_{1}$ and $a_{2}$ are fitting parameters used in dipole
parametrization. $f_{i}(0)$ represents the value of form factors at $q^{2}=0$,
which is also treated as a fitting parameter here. In order to obtain reliable
fitting parameters, the following fitting procedure is adopted. For a given
set of values of Borel parameters $M_{B_{2}}^{2}$ and threshold parameters
$s_{1}^{0}$ and $s_{2}^{0}$, the dipole parametrization in Eq. (19) is
employed to get a set of fitting parameters. Multiple estimations of the
fitting parameters are obtained by varying $M_{B_{2}}^{2}$, $s_{1}^{0}$, and
$s_{2}^{0}$. The central values are determined by averaging the different
fitting results, and the errors stem from the uncertainties of the inputs
$M_{B_{2}}^{2}$, $s_{1}^{0}$, and $s_{2}^{0}$. The fitting results are
presented in Table. 2, where the values of $f_{i}(0)$ obtained from the
fitting are consistent with our directly calculated results given in Table. 1.
The $q^{2}$ dependence of form factors is shown in Fig. 4, where the fitted
points exhibit a clear dipole behavior.

Table 2: Fitting results for $f_{1}(q^{2})$ and $f_{2}(q^{2})$ used dipole parametrization in Eq. (19) | $f_{i}(0)$ |  | $a_{1}$ |  | $a_{2}$
---|---|---|---|---|---
$f_{1}$ | $0.55\pm 0.03$ |  | $0.69\pm 0.09$ |  | $-0.76\pm 0.09$
$f_{2}$ | $-0.24\pm 0.02$ |  | $1.88\pm 0.11$ |  | $0.34\pm 0.05$

Figure 4: The $q^{2}$ dependence of form factors, in which the solid lines
represent the central value of fitting parameters listed in Table. 2, while
the dashed lines and dot-dashed lines represent the uncertainties allowed by
the fitting parameters, respectively. Besides, the symbol point denote the
fitted points for each of the form factors.

After deriving the $q^{2}$ dependence of transition form factors, the
branching fractions and the relevant decay asymmetry observables of
semileptonic decay $\Lambda_{c}\rightarrow n\ell\nu_{\ell}$ can be analyzed.
To facilitate this analysis, it is convenient to introduce the helicity
amplitudes, which provide a more intuitive understanding of the physical
pictures and simplify the expressions when discussing the asymmetries of the
decay processes. The relations between helicity amplitudes and the form
factors are as follows [49, 50, 34]:

$\displaystyle
H_{\frac{1}{2},0}^{V}=\frac{\sqrt{Q_{-}}}{\sqrt{q^{2}}}\big{(}M_{+}f_{1}(q^{2})-\frac{q^{2}}{M_{\Lambda_{c}}}f_{2}(q^{2})\big{)},\quad
H_{\frac{1}{2},0}^{A}=\frac{\sqrt{Q_{+}}}{\sqrt{q^{2}}}\big{(}M_{-}g_{1}(q^{2})+\frac{q^{2}}{M_{\Lambda_{c}}}g_{2}(q^{2})\big{)},$
$\displaystyle
H_{\frac{1}{2},1}^{V}=\sqrt{2Q_{-}}\big{(}-f_{1}(q^{2})+\frac{M_{+}}{M_{\Lambda_{c}}}f_{2}(q^{2})\big{)},\quad
H_{\frac{1}{2},1}^{A}=\sqrt{2Q_{+}}\big{(}-g_{1}(q^{2})-\frac{M_{-}}{M_{\Lambda_{c}}}g_{2}(q^{2})\big{)},$
$\displaystyle
H_{\frac{1}{2},t}^{V}=\frac{\sqrt{Q_{+}}}{\sqrt{q^{2}}}\big{(}M_{-}f_{1}(q^{2})+\frac{q^{2}}{M_{\Lambda_{c}}}f_{3}(q^{2})\big{)},\quad
H_{\frac{1}{2},t}^{A}=\frac{\sqrt{Q_{-}}}{\sqrt{q^{2}}}\big{(}M_{+}g_{1}(q^{2})-\frac{q^{2}}{M_{\Lambda_{c}}}g_{3}(q^{2})\big{)}.$
(20)

Here, $H_{\lambda^{\prime},\lambda_{W}}^{V(A)}$ is the helicity amplitudes for
weak transitions induced by vector and axial-vector currents, where
$\lambda^{\prime}$ and $\lambda_{W}$ represent the helicity of the neutron and
the $W$ boson, respectively. $Q_{\pm}$ is defined as
$Q_{\pm}=M_{\pm}^{2}-q^{2}$ and $M_{\pm}=M_{\Lambda_{c}}\pm M_{n}$. The
negative helicity amplitudes can be derived using following relations,

$\displaystyle
H_{-\lambda^{\prime},-\lambda_{W}}^{V}=H_{\lambda^{\prime},\lambda_{W}}^{V},\quad\quad
H_{-\lambda^{\prime},-\lambda_{W}}^{A}=-H_{\lambda^{\prime},\lambda_{W}}^{A}.$
(21)

Then the total helicity amplitudes can be obtained,

$\displaystyle
H_{\lambda^{\prime},\lambda_{W}}=H_{\lambda^{\prime},\lambda_{W}}^{V}-H_{\lambda^{\prime},\lambda_{W}}^{A}.$
(22)

With the above helicity amplitudes, the differential distribution of
$\Lambda_{c}\rightarrow n\ell\nu_{\ell}$ can be expressed as [24, 34]

$\displaystyle\frac{d\Gamma\left(\Lambda_{c}\rightarrow
n\ell\nu_{\ell}\right)}{dq^{2}}=\frac{G_{F}^{2}\left|V_{cd}\right|^{2}q^{2}\sqrt{Q_{+}Q_{-}}}{384\,\pi^{3}\,M_{\Lambda_{c}}^{3}}(1-\frac{m_{\ell}^{2}}{q^{2}})^{2}H_{\text{tot}},$
(23)

where $m_{\ell}$ denotes the lepton mass ($\ell=e,\mu$) and $H_{\text{tot}}$
is defined as

$\displaystyle H_{\text{tot}}$
$\displaystyle=\big{(}1+\frac{m_{\ell}^{2}}{2q^{2}}\big{)}\big{(}H_{\frac{1}{2},1}^{2}+H_{-\frac{1}{2},-1}^{2}+H_{\frac{1}{2},0}^{2}+H_{-\frac{1}{2},0}^{2}\big{)}$
$\displaystyle+\frac{3\,m_{\ell}^{2}}{2q^{2}}\big{(}H_{\frac{1}{2},t}^{2}+H_{-\frac{1}{2},t}^{2}\big{)}.$
(24)

According to the definition of $H_{\text{tot}}$, the contribution to the
differential decay width from $f_{3}(q^{2})$ and $g_{3}(q^{2})$ can be found
in the term $H_{\frac{1}{2},t}^{2}$ and $H_{-\frac{1}{2},t}^{2}$, which is
clearly suppressed by the factor $m_{\ell}^{2}$. Hence, we neglect the effect
of $f_{3}(q^{2})$ and $g_{3}(q^{2})$ in Eq. (2). In order to obtain the
numerical results of differential decay width, the following input parameters
related to the decay analysis are taken from Particle Data Group [45], where

$\displaystyle G_{F}=1.166\times
10^{-5}\,\text{GeV}^{-2},\quad|V_{cd}|=0.221\pm 0.004,$ $\displaystyle
m_{e}=0.511\,\text{MeV},\quad
m_{\mu}=0.106\,\text{GeV},\quad\tau_{\Lambda_{c}}=(201.5\pm 2.7)\times
10^{-15}\,\text{s}.$ (25)

Here, the mean lifetime of $\Lambda_{c}$, noted as $\tau_{\Lambda_{c}}$, is
introduced to calculate the branching fractions. We plot the $q^{2}$
dependence of differential decay width for $\Lambda_{c}\rightarrow
n\ell\nu_{\ell}$ semileptonic decay in Fig. 5(a) and list the numerical
results of branching fractions in Table. 3. It can be found that the branching
fractions for $\Lambda_{c}\rightarrow ne^{+}\nu_{e}$ semileptonic decay
obtained using QCDSR are very close to the results derived by CQM [23], RQM
[24], $SU(3)$ flavor symmetry [25], and MBM [27]. As for the
$\Lambda_{c}\rightarrow n\mu^{+}\nu_{\mu}$ decay mode, we find our results are
consistent with RQM [24] and relatively smaller than the Lattice QCD
predictions [28].

Figure 5: The $q^{2}$ dependence of the differential decay width and the relevant decay observables for $\Lambda_{c}\rightarrow n\ell\nu_{\ell}$ semileptonic decay. The red solid line denotes $\ell=e^{+}$, while the blue dashed line denotes $\ell=\mu^{+}$. Table 3: Theoretical predictions of branching fractions, the forward-backward asymmetry, and the asymmetry parameter for $\Lambda_{c}\rightarrow n\ell\nu_{\ell}$ and $\Lambda_{c}\rightarrow\Lambda\ell\nu_{\ell}$ semileptonic decay with different methods. channel | method |  | $\mathcal{B}(\%)$ | $\langle A_{FB}\rangle$ | $\langle\alpha_{\Lambda_{c}}\rangle$
---|---|---|---|---|---
$\Lambda_{c}\rightarrow ne^{+}\nu_{e}$ | QCDSR |  | $0.280\pm 0.031$ | $-0.23\pm 0.01$ | $-0.94\pm 0.03$
| LF [21] |  | $0.201$ |  |
| CCQM [22] |  | $0.207$ | $-0.236$ |
| CQM [23] |  | $0.270$ |  |
| RQM [24] |  | $0.268$ | $-0.251$ | $-0.91$
|  $SU(3)$ [25] |  | $0.289\pm 0.035$ |  |
|  $SU(3)$ [26] |  | $0.51\pm 0.04$ |  | $-0.89\pm 0.04$
| MBM [27] |  | $0.279$ |  | $-0.87$
| LFCQM [27] |  | $0.36\pm 0.15$ |  | $-0.96\pm 0.04$
| LQCD [28] |  | $0.410\pm 0.026$ |  |
$\Lambda_{c}\rightarrow n\mu^{+}\nu_{\mu}$ | QCDSR |  | $0.274\pm 0.030$ | $-0.26\pm 0.01$ | $-0.94\pm 0.03$
| CCQM [22] |  | $0.202$ | $-0.260$ |
| RQM [24] |  | $0.262$ | $-0.276$ | $-0.90$
| LQCD [28] |  | $0.400\pm 0.026$ |  |
$\Lambda_{c}\rightarrow\Lambda e^{+}\nu_{e}$ | QCDSR |  | $3.37\pm 0.36$ | $-0.20\pm 0.01$ | $-0.88\pm 0.02$
| LQCD [51] |  | $3.80\pm 0.22$ | $-0.20\pm 0.06$ | $-0.87\pm 0.10$
| Exp [4, 5] |  | $3.56\pm 0.11\pm 0.07$ | $-0.24\pm 0.03$ | $-0.86\pm 0.04$
$\Lambda_{c}\rightarrow\Lambda\mu^{+}\nu_{\mu}$ | QCDSR |  | $3.26\pm 0.35$ | $-0.23\pm 0.01$ | $-0.88\pm 0.02$
| LQCD [51] |  | $3.69\pm 0.22$ | $-0.17\pm 0.07$ | $-0.87\pm 0.10$
| Exp [4, 5] |  | $3.48\pm 0.17$ | $-0.22\pm 0.04$ | $-0.94\pm 0.08$

In addition, two relevant decay asymmetry observables, e.g., the leptonic
forward-backward asymmetry ($A_{FB}$) and the asymmetry parameter
($\alpha_{\Lambda_{c}}$) are defined as [24, 52, 5],

$\displaystyle A_{FB}(q^{2})$
$\displaystyle=\frac{\frac{d\Gamma}{dq^{2}}(\text{forward})-\frac{d\Gamma}{dq^{2}}(\text{backward})}{\frac{d\Gamma}{dq^{2}}}$
$\displaystyle=\frac{3}{4}\frac{H_{\frac{1}{2},1}^{2}-H_{-\frac{1}{2},-1}^{2}-2\frac{m_{\ell}^{2}}{q^{2}}(H_{\frac{1}{2},0}H_{\frac{1}{2},t}+H_{-\frac{1}{2},0}H_{-\frac{1}{2},t})}{H_{\text{tot}}},$
(26) $\displaystyle\alpha_{\Lambda_{c}}(q^{2})$
$\displaystyle=\frac{d\Gamma^{\lambda^{\prime}=\frac{1}{2}}/dq^{2}-d\Gamma^{\lambda^{\prime}=-\frac{1}{2}}/dq^{2}}{d\Gamma^{\lambda^{\prime}=\frac{1}{2}}/dq^{2}+d\Gamma^{\lambda^{\prime}=-\frac{1}{2}}/dq^{2}},$
(27)

where

$\displaystyle\frac{d\Gamma^{\lambda^{\prime}=\frac{1}{2}}}{dq^{2}}=$
$\displaystyle\,\frac{4m_{l}^{2}}{3q^{2}}\big{(}H_{\frac{1}{2},1}^{2}+H_{\frac{1}{2},0}^{2}+3H_{\frac{1}{2},t}^{2}\big{)}+\frac{8}{3}\big{(}H_{\frac{1}{2},1}^{2}+H_{\frac{1}{2},0}^{2}\big{)},$
(28) $\displaystyle\frac{d\Gamma^{\lambda^{\prime}=-\frac{1}{2}}}{dq^{2}}=$
$\displaystyle\,\frac{4m_{l}^{2}}{3q^{2}}\big{(}H_{-\frac{1}{2},-1}^{2}+H_{-\frac{1}{2},0}^{2}+3H_{-\frac{1}{2},t}^{2}\big{)}+\frac{8}{3}\big{(}H_{-\frac{1}{2},-1}^{2}+H_{-\frac{1}{2},0}^{2}\big{)}.$
(29)

The $q^{2}$ dependence of the decay asymmetry observables are plotted in Fig.
5(b) and (c). In Fig. 5, we can see the dependence of the differential decay
width and the decay asymmetry observables on the lepton mass is consistent
near the zero recoil region $q^{2}=(M_{\Lambda_{c}}-M_{n})^{2}$. However, near
the maximum recoil point $q^{2}=m_{\ell}^{2}$, the behavior of the
differential decay width and the leptonic forward-backward asymmetry is
significantly different. The leptonic forward-backward asymmetry is going to
$0$ for $\Lambda_{c}\rightarrow ne^{+}\nu_{e}$ decay mode and to $-0.5$ for
$\Lambda_{c}\rightarrow n\mu^{+}\nu_{\mu}$ decay mode at $q^{2}=m_{\ell}^{2}$.
This character agrees with Ref. [24]. As for the asymmetry parameter, it
varies from $\alpha_{\Lambda_{c}}=-1$ to $\alpha_{\Lambda_{c}}=0$ as the
$q^{2}$ increases from zero to $q^{2}_{max}$. Besides, it is almost
indistinguishable throughout the entire physical region, which is also
consistent with the findings in Ref. [24]. We also present the mean values of
the relevant decay asymmetry observables in Table 3, which are obtained by
separately integrating the numerators and denominators in Eq.(3) - Eq.(27)
over the physical region of $q^{2}$. From Table. 3, it can be observed that
our results for $\langle A_{FB}\rangle$ and
$\langle\alpha_{\Lambda_{c}}\rangle$ agree with the previous theoretical
predictions. Future experiments measuring these observables and comparing them
with the predictions of the present study will contribute to our understanding
of the relevant decay channels and the internal structures of baryons. In
addition, the possibility of new physics effects beyond the standard model can
be explored through these observables [49, 53].

By replacing the second $d$ quark in the current (5) with a strange quark and
applying the same analysis procedure, we also investigate the semileptonic
decay mode $\Lambda_{c}\rightarrow\Lambda\ell\nu_{\ell}$. The relevant input
parameters are $m_{s}=93.4^{+8.6}_{-3.4}\,\text{MeV}$ [45],
$\lambda_{\Lambda}=0.0208\,\text{GeV}^{3}$ [32],
$M_{\Lambda}=1.116\,\text{GeV}$ [45] and $|V_{cs}|=0.975\pm 0.006$ [45]. As
this decay mode has been thoroughly investigated both theoretically and
experimentally, we solely provide the numerical results of branching fractions
and decay asymmetry observables in Table. 3 as a validation of the QCDSR
method. It is evident that the branching fractions, the forward-backward
asymmetry, and the asymmetry parameter obtained through QCDSR for the
semileptonic decay $\Lambda_{c}\rightarrow\Lambda\ell\nu_{\ell}$ are in
excellent agreement with Lattice QCD results [51] and experimental data [4,
5].

##  4 Conclusions

In this work, we calculate the weak transition form factors of
$\Lambda_{c}\rightarrow n\ell\nu_{\ell}$ semileptonic decay in the framework
of QCD sum rules. The analytic results of the transition form factors are
obtained through the analysis of the three-point correlation functions and the
application of Cutkosky cutting rules. The numerical results for the form
factors at the maximum recoil region point $q^{2}=0$ are computed and compared
with other methods. In order to extend the form factors to the full physical
region, we utilize a dipole parametrization that adequately captures the
$q^{2}$ dependence of the form factors, ensuring a smooth extrapolation.

Based on the obtained form factors, we predict the branching fractions to be
$\mathcal{B}(\Lambda_{c}\rightarrow ne^{+}\nu_{e})=(0.280\pm 0.031)\%$ and
$\mathcal{B}(\Lambda_{c}\rightarrow n\mu^{+}\nu_{\mu})=(0.274\pm 0.030)\%$,
which will provide important information to determine the value of the CKM
matrix element $|V_{cd}|$. Moreover, the mean values of the leptonic forward-
backward asymmetry $\langle A_{FB}\rangle$ and the asymmetry parameter
$\langle\alpha_{\Lambda_{c}}\rangle$ are also given, which will play a crucial
role in probing potential new physics effects beyond the standard model.
Although there is still no experimental data for $\Lambda_{c}\rightarrow
n\ell\nu_{\ell}$ semileptonic decay to date, considering the recent
experimental progress of $\Lambda_{c}$ decay modes involving neutron final
state, we believe our predicted results can be tested by the future experiment
at BESIII, BELLEII, and LHCb.

Finally, we analyze the semileptonic decay mode
$\Lambda_{c}\rightarrow\Lambda\ell\nu_{\ell}$. Our results exhibit a strong
agreement with the experimental data, indicating that the QCDSR calculation
can be applied to other charmed baryons, such as $\Xi_{c}^{+(0)}$.
Furthermore, there is still potential for further improvement in this method.
The relatively large errors in the branching fractions compared to the
experimental data suggest the necessity for further refinement. One possible
approach to address this issue is to calculate the contributions from
radiation corrections, although this presents a significant challenge in the
application of QCD sum rules.

Acknowledgments

We thank K.S. Huang, L. Tang, and B.D. Wan for their meaningful discussions.
This work was supported in part by National Natural Science Foundation of
China(NSFC) under the Grants 11975236 and 12235008, and University of Chinese
Academy of Sciences.

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