#  $D^{(\ast)}N$ interaction and the structure of $\Sigma_{c}(2800)$ and
$\Lambda_{c}(2940)$ in chiral effective field theory

Bo Wang1,2<EMAIL_ADDRESS>Lu Meng2<EMAIL_ADDRESS>Shi-Lin Zhu2,1
<EMAIL_ADDRESS>1Center of High Energy Physics, Peking University, Beijing
100871, China
2 School of Physics and State Key Laboratory of Nuclear Physics and
Technology, Peking University, Beijing 100871, China

###### Abstract

We study the $DN$ and $D^{\ast}N$ interactions to probe the inner structure of
$\Sigma_{c}(2800)$ and $\Lambda_{c}(2940)$ with the chiral effective field
theory to the next-to-leading order. We consider the contact term, one-pion-
exchange and two-pion-exchange contributions to characterize the short-, long-
and mid-range interactions of the $D^{(\ast)}N$ systems. The low energy
constants of the $D^{(\ast)}N$ systems are related to those of the $N\bar{N}$
interaction with quark level Lagrangian that inspired by the resonance
saturation model. The $\Delta(1232)$ degree of freedom is also included in the
loop diagrams. The attractive potential in the $[DN]_{J=1/2}^{I=1}$ channel is
too weak to form bound state, which indicates the explanation of
$\Sigma_{c}(2800)$ as the compact charmed baryon is more reasonable.
Meanwhile, the potentials of the isoscalar channels are deep enough to yield
the molecular states. We obtain the masses of the $[DN]_{J=1/2}^{I=0}$,
$[D^{\ast}N]_{J=1/2}^{I=0}$ and $[D^{\ast}N]_{J=3/2}^{I=0}$ systems to be
$2792.0$ MeV, $2943.6$ MeV and $2938.4$ MeV, respectively. The
$\Lambda_{c}(2940)$ is probably the isoscalar $D^{\ast}N$ molecule considering
its low mass puzzle. Besides, the $\Lambda_{c}(2940)$ signal might contain the
spin-$\frac{1}{2}$ and spin-$\frac{3}{2}$ two structures, which can
qualitatively explain the significant decay ratio to $D^{0}p$ and
$\Sigma_{c}\pi$. We also study the $\bar{B}^{(\ast)}N$ systems and predict the
possible molecular states in the isoscalar channels. We hope experimentalists
could hunt for the open charmed molecular pentaquarks in the
$\Lambda_{c}^{+}\pi^{+}\pi^{-}$ final state.

##  I Introduction

Hadron spectroscopy plays an important role in understanding the low energy
behaviors of QCD. Quark model is very successful in describing the hadron
spectra Tanabashi:2018oca . But it is rather difficult to assign the near-
threshold states, such as $X(3872)$ Choi:2003ue  and $D_{s0}(2317)$
Aubert:2003fg  to the quark model predictions Chen:2016qju ; Guo:2017jvc ;
Liu:2019zoy ; Lebed:2016hpi ; Esposito:2016noz ; Brambilla:2019esw . In the
charmed baryon family, a state $\Lambda_{c}(2940)$ also falls into the same
situation as the $X(3872)$ and $D_{s0}(2317)$.

In 2007, the BaBar Collaboration observed a charmed baryon $\Lambda_{c}(2940)$
in the $D^{0}p$ invariant mass spectrum Aubert:2006sp , which is an isosinglet
since no signal is observed in the $D^{+}p$ final state. It was subsequently
confirmed by the Belle experiment in the decay mode
$\Lambda_{c}(2940)\to\Sigma_{c}\pi$ Abe:2006rz . In 2017, the $J^{P}$ quantum
numbers of $\Lambda_{c}(2940)$ was constrained by the LHCb measurement, and
the most likely spin-parity assignment for $\Lambda_{c}(2940)$ is
$J^{P}=\frac{3}{2}^{-}$ Aaij:2017vbw  (The mass and width of
$\Lambda_{c}(2940)$ obtained by the BaBar, Belle and LHCb experiments are
shown in Table 1).

Table 1: The mass and width of $\Lambda_{c}(2940)$ in experiments (in units of MeV). BaBar | $2939.8\pm 1.3\pm 1.0$ | $17.5\pm 5.2\pm 5.9$
---|---|---
Belle | $2938.0\pm 1.3^{+2.0}_{-4.0}$ | $13^{+8+27}_{-5-7}$
LHCb | $2944.8^{+3.5}_{-2.5}\pm 0.4^{+0.1}_{-4.6}$ | $27.7^{+8.2}_{-6.0}\pm 0.9^{+5.2}_{-10.4}$

Up to now, there are two different interpretations of the internal structure
of $\Lambda_{c}(2940)$. One is the ordinal charmed baryon, and the other one
is the $D^{\ast}N$ molecular state. However, it is difficult to arrange
$\Lambda_{c}(2940)$ to the $2P$ state in the charmed baryon spectroscopy,
since its mass is about $60$$-$$100$ MeV smaller than the calculations of the
quark models Capstick:1986bm ; Ebert:2011kk ; Chen:2014nyo ; Lu:2016ctt .
Considering the $\Lambda_{c}(2940)$ lies about $6$ MeV below the $D^{\ast 0}p$
threshold, the molecular explanation was firstly proposed in Ref. He:2006is ,
where the $\Lambda_{c}(2940)$ as the $\frac{1}{2}^{-}$ molecular state is
preferred by analyzing its decay behaviors. In Ref. He:2010zq , He et al.
studied the $D^{\ast}N$ interaction with the one-boson-exchange model, and
their calculation supports the interpretation of the $\Lambda_{c}(2940)$ as
the $D^{\ast}N$ bound state with $I(J^{P})=0(\frac{1}{2}^{+})$ or
$0(\frac{3}{2}^{-})$. In Ref. Ortega:2012cx , Ortega et al. investigated the
$\Lambda_{c}(2940)$ as a $D^{\ast}N$ molecule in the constituent quark model,
and they obtain the binding solution in isoscalar $J^{P}=\frac{3}{2}^{-}$
channel. In Refs. Dong:2009tg ; Dong:2010xv , the strong and radiative decays
of $\Lambda_{c}(2940)$ are calculated in the molecular picture. A QCD sum rule
study in Ref. Zhang:2012jk  indicates the $\Lambda_{c}(2940)$ is not a compact
state. Some recent calculations based on the chiral quark model also support
the molecular explanation for $\Lambda_{c}(2940)$ Zhao:2016zhf ; Zhang:2019vqe
. (see Refs. Klempt:2009pi ; Crede:2013sze ; Cheng:2015iom ; Chen:2016spr ;
Kato:2018ijx  for review and Refs. Huang:2016ygf ; Wang:2015rda ;
Cheng:2015naa ; Xie:2015zga ; Romanets:2012hm ; He:2011jp ; Cheng:2006dk ;
Chen:2007xf ; Zhong:2007gp ; Chen:2015kpa  for other related works).

Another charmed baryon related with the $DN$ threshold is $\Sigma_{c}(2800)$,
which is an isotriplet and firstly observed by the Belle Collaboration in the
$\Lambda_{c}\pi$ mass spectrum Mizuk:2004yu . The neutral state
$\Sigma_{c}(2800)^{0}$ was possibly confirmed by the BaBar experiment
Aubert:2008ax , but the measured mass from BaBar is about $50$ MeV larger. The
$J^{P}$ of $\Sigma_{c}(2800)$ is still undetermined yet Tanabashi:2018oca .
Like the $\Lambda_{c}(2940)$, $\Sigma_{c}(2800)$ is interpreted as the
$P$-wave excitation of the charmed baryon in the $\lambda$ mode Ebert:2011kk ;
Cheng:2006dk ; Chen:2016iyi ; Garcilazo:2007eh ; Wang:2017kfr , and $DN$
molecule Zhang:2012jk ; Zhao:2016zhf ; Zhang:2019vqe ; Dong:2010gu ,
respectively.

Investigating the $DN$ and $D^{\ast}N$ interaction is essential to disentangle
the puzzles of $\Sigma_{c}(2800)$ and $\Lambda_{c}(2940)$. Besides,
understanding the $D^{(\ast)}N$ interaction is also crucial to probe the
$D$-mesic nuclei Tsushima:1998ru ; GarciaRecio:2010vt  and the properties of
the charmed mesons in the nuclear matter Hosaka:2016ypm ; Krein:2017usp . An
alternative approach based on the meson-exchange model Machleidt:1987hj  has
been employed to construct the $DN$ and $\bar{D}N$ interaction by the Jülich
group Haidenbauer:2007jq ; Haidenbauer:2008ff ; Haidenbauer:2010ch .

Instead of the boson-exchange model, the modern theory of nuclear force is
built upon the pioneer work of Weinberg Weinberg:1990rz ; Weinberg:1991um  and
largely developed in the framework of effective field theory. The chiral
effective field theory was extensively exploited to study the $NN$ interaction
with great success Bernard:1995dp ; Epelbaum:2008ga ; Machleidt:2011zz ;
Meissner:2015wva ; Hammer:2019poc ; Machleidt:2020vzm . The chiral effective
field theory was also utilized to study the systems with heavy flavors in
Refs. Liu:2012vd ; Xu:2017tsr ; Wang:2018atz ; Meng:2019ilv ; Wang:2019ato ;
Meng:2019nzy ; Wang:2019nvm , which is a powerful tool in predicting the
$BB^{\ast}$ and $B^{\ast}B^{\ast}$ bound states Wang:2018atz , reproducing the
newly observed pentaquarks Wang:2019ato , extrapolating the $\Sigma_{c}N$
potential from lattice QCD result to the physical pion mass Meng:2019nzy , and
so on. As a natural extension of the $NN$ interaction, in this work, we use
the chiral effective field theory to study the $D^{(\ast)}N$ interaction up to
the next-to-leading order. We simultaneously consider the long-, mid- and
short-range interactions, and include the contribution of $\Delta(1232)$ in
the loops as an intermediate state. With the chiral effective field theory, we
calculate the $D^{(\ast)}N$ effective potentials and search for the possible
bound states. The numerical results can be compared with the experimental data
of $\Lambda_{c}(2940)$ and $\Sigma_{c}(2800)$ to see whether they are the
genuine charmed baryons or the molecular nature.

This paper is organized as follows. In Sec. II, we give the Lagrangians and
effective potentials of the $D^{(\ast)}N$ systems. In Sec. III, we illustrate
our numerical results and discussions. In Sec. IV, we conclude with a short
summary. In Appendix A, we relate the low energy constants to those of the
$N\bar{N}$ system with quark model.

##  II Lagrangians and effective potentials

###  II.1 Effective chiral Lagrangians

We first show the leading order Lagrangian of the nucleon and pion interaction
under the heavy baryon reduction Scherer:2002tk , which reads

$\displaystyle\mathcal{L}_{\mathcal{N}\varphi}=\bar{\mathcal{N}}(iv\cdot\mathcal{D}+2g_{a}\mathcal{S}\cdot
u)\mathcal{N},$ (1)

where $\mathcal{N}=(p,n)^{T}$ denotes the the large component of the nucleon
field under the nonrelativistic reduction. $v=(1,\bm{0})$ is the 4-velocity of
the nucleon and $\mathcal{D}_{\mu}=\partial_{\mu}+\Gamma_{\mu}$. $g_{a}\simeq
1.29$ is the axial-vector coupling constant.
$\mathcal{S}^{\mu}=\frac{i}{2}\gamma_{5}\sigma^{\mu\nu}v_{\nu}$ stands for the
spin operator of the nucleon. $\Gamma_{\mu}$ and $u_{\mu}$ are the chiral
connection and axial-vector current, respectively. Their expressions read

$\displaystyle\Gamma_{\mu}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\left[\xi^{\dagger},\partial_{\mu}\xi\right]\equiv\tau^{i}\Gamma_{\mu}^{i},u_{\mu}\equiv\frac{i}{2}\left\\{\xi^{\dagger},\partial_{\mu}\xi\right\\}\equiv\tau^{i}\omega_{\mu}^{i},$
(2)

where $\tau^{i}$ is the Pauli matrix,

$\displaystyle\xi^{2}=U=\exp\left(\frac{i\varphi}{f_{\pi}}\right),\qquad\varphi=\left[\begin{array}[]{cc}\pi^{0}&\sqrt{2}\pi^{+}\\\
\sqrt{2}\pi^{-}&-\pi^{0}\end{array}\right],$ (5)

and $f_{\pi}=92.4$ MeV is the pion decay constant.

Considering the importance of $\Delta(1232)$ in the $NN$ interaction
Machleidt:1987hj ; Holinde:1977rh ; Krebs:2007rh ; Epelbaum:2007sq ;
Kaiser:1998wa , we adopt the small scale expansion method Hemmert:1997ye  to
explicitly include the $\Delta(1232)$ in the Lagrangians. The Lagrangian that
delineates the $\Delta$-$N$-$\pi$ coupling is given as

$\displaystyle\mathcal{L}_{\Delta\varphi}$ $\displaystyle=$
$\displaystyle-\bar{\mathcal{T}}_{i}^{\mu}(iv\cdot\mathcal{D}^{ij}-\delta^{ij}\delta_{a}+2g_{1}\mathcal{S}\cdot
u^{ij})g_{\mu\nu}\mathcal{T}_{j}^{\nu},$ (6)
$\displaystyle\mathcal{L}_{\Delta\mathcal{N}\varphi}$ $\displaystyle=$
$\displaystyle
2g_{\delta}(\bar{\mathcal{T}}_{i}^{\mu}g_{\mu\alpha}\omega_{i}^{\alpha}\mathcal{N}+\bar{\mathcal{N}}\omega_{i}^{\alpha\dagger}g_{\alpha\mu}\mathcal{T}_{i}^{\mu}),$
(7)

where $\delta_{a}=m_{\Delta}-m_{N}$. $g_{1}=\frac{9}{5}g_{a}$ is estimated
with the quark model Hemmert:1997ye . $g_{\delta}\simeq 1.05$ is the coupling
constant for $\Delta N\pi$ vertex. $\mathcal{T}_{i}^{\mu}$ denotes the
spin-$\frac{3}{2}$ and isospin-$\frac{3}{2}$ field $\Delta(1232)$ after
performing the nonrelativistic reduction. Its matrix form reads

$\displaystyle\mathcal{T}_{\mu}^{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[\begin{array}[]{c}\Delta^{++}-\frac{1}{\sqrt{3}}\Delta^{0}\\\
\frac{1}{\sqrt{3}}\Delta^{+}-\Delta^{-}\end{array}\right]_{\mu},$ (10)
$\displaystyle\mathcal{T}_{\mu}^{2}$ $\displaystyle=$
$\displaystyle\frac{i}{\sqrt{2}}\left[\begin{array}[]{c}\Delta^{++}+\frac{1}{\sqrt{3}}\Delta^{0}\\\
\frac{1}{\sqrt{3}}\Delta^{+}+\Delta^{-}\end{array}\right]_{\mu},\mathcal{T}_{\mu}^{3}=-\sqrt{\frac{2}{3}}\left[\begin{array}[]{c}\Delta^{+}\\\
\Delta^{0}\end{array}\right]_{\mu}.$ (15)

The leading order Lagrangian that depicts the interaction between the charmed
mesons and light Goldstones reads Wise:1992hn ; Manohar:2000dt

$\displaystyle\mathcal{L}_{\mathcal{H}\varphi}=i\langle\mathcal{H}v\cdot\mathcal{D}\bar{\mathcal{H}}\rangle-\frac{1}{8}\delta_{b}\langle\mathcal{H}\sigma^{\mu\nu}\bar{\mathcal{H}}\sigma_{\mu\nu}\rangle+g\langle\mathcal{H}\not{u}\gamma_{5}\bar{\mathcal{H}}\rangle,$
(16)

where $\langle\cdots\rangle$ represents the trace in spinor space.
$\delta_{b}$ is defined as $\delta_{b}=m_{D^{\ast}}-m_{D}$. $g\simeq-0.59$
stands for the axial coupling, whose sign is determined with the help of quark
model. The $\mathcal{H}$ is the super-field for the charmed mesons, which
reads

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\frac{1+\not{v}}{2}\left(P_{\mu}^{\ast}\gamma^{\mu}+iP\gamma_{5}\right),$
$\displaystyle\bar{\mathcal{H}}$ $\displaystyle=$
$\displaystyle\gamma^{0}H^{\dagger}\gamma^{0}=\left(P_{\mu}^{\ast{\dagger}}\gamma^{\mu}+iP^{\dagger}\gamma_{5}\right)\frac{1+\not{v}}{2}.$
(17)

with $P=(D^{0},D^{+})^{T}$ and $P^{\ast}=(D^{\ast 0},D^{\ast+})^{T}$,
respectively. We construct the leading order contact Lagrangian to describe
the short distance interaction between the nucleon and charmed meson,

$\displaystyle\mathcal{L}_{\mathcal{NH}}$ $\displaystyle=$ $\displaystyle
D_{a}\bar{\mathcal{N}}\mathcal{N}\langle\bar{\mathcal{H}}\mathcal{H}\rangle+D_{b}\bar{\mathcal{N}}\gamma_{\mu}\gamma_{5}\mathcal{N}\langle\bar{\mathcal{H}}\gamma^{\mu}\gamma_{5}\mathcal{H}\rangle$
$\displaystyle+E_{a}\bar{\mathcal{N}}\tau_{i}\mathcal{N}\langle\bar{\mathcal{H}}\tau_{i}\mathcal{H}\rangle+E_{b}\bar{\mathcal{N}}\gamma_{\mu}\gamma_{5}\tau_{i}\mathcal{N}\langle\bar{\mathcal{H}}\gamma^{\mu}\gamma_{5}\tau_{i}\mathcal{H}\rangle,$

where $D_{a}$, $D_{b}$, $E_{a}$ and $E_{b}$ are four low energy constants
(LECs). $D_{a}$ and $D_{b}$ contribute to the central potential and spin-spin
interaction, respectively. $E_{a}$ and $E_{b}$ are related with the isospin-
isospin interaction and contribute to the central and spin-spin interaction in
spin space, respectively. With the quark model, we fix their values with the
$N\bar{N}$ interaction as inputs, which is given in the Appendix A.

###  II.2 Expressions of the effective potentials

In the framework of heavy hadron chiral perturbation theory, the scattering
amplitudes of the $D^{(\ast)}N$ systems can be expanded order by order in
powers of a small parameter $\varepsilon=q/\Lambda_{\chi}$, where $q$ is
either the momentum of Goldstone bosons or the residual momentum of heavy
hadrons, and $\Lambda_{\chi}$ represents either the chiral breaking scale or
the mass of a heavy hadron111The mass splitting $\delta_{b}\sim m_{\pi}$, so
$\delta_{b}/\Lambda_{\chi}$ can be safely treated as the small parameter in
chiral expansions. However, the mass difference $\delta_{a}\sim 2m_{\pi}\ll
m_{N}$, so strictly speaking, the ‘small parameter’
$\delta_{a}/\Lambda_{\chi}$ therefore has to be regarded as a phenomenological
one. With the current accuracy, expanding to $(\delta_{a}/\Lambda_{\chi})^{2}$
at the one-loop level can fulfill the convergence of the chiral expansions..
The expansion is organized by the power counting rule in Refs. Weinberg:1990rz
; Weinberg:1991um . The $\mathcal{O}(\varepsilon^{0})$ Feynman diagrams for
the $DN$ and $D^{\ast}N$ systems are shown in Fig. 1, which contain the
contact and one-pion-exchange diagrams. The one-pion-exchange diagram for the
$DN$ system vanishes since the $DD\pi$ vertex is forbidden. The corresponding
momentum-space potentials of graphs in Fig. 1 read

Figure 1: The leading order Feynman diagrams that account for the
$\mathcal{O}(\varepsilon^{0})$ effective potentials of the $ND$ ($X_{1.1}$),
and $ND^{\ast}$ ($X_{2.1},H_{2.1}$) systems. We use the thick, thin, double-
thin and dashed lines to denote the $N$, $D$, $D^{\ast}$ and pion fields,
respectively.

$\displaystyle\mathcal{V}_{DN}^{X_{1.1}}$ $\displaystyle=$ $\displaystyle
D_{a}-4E_{a}(\mathbf{I}_{1}\cdot\mathbf{I}_{2}),$ (19)
$\displaystyle\mathcal{V}_{D^{\ast}N}^{X_{2.1}}=D_{a}+D_{b}\bm{\sigma}\cdot\mathbf{T}-4(E_{a}+E_{b}\bm{\sigma}\cdot\mathbf{T})(\mathbf{I}_{1}\cdot\mathbf{I}_{2}),$
$\displaystyle\mathcal{V}_{D^{\ast}N}^{H_{2.1}}$ $\displaystyle=$
$\displaystyle(\mathbf{I}_{1}\cdot\mathbf{I}_{2})\frac{gg_{a}}{f_{\pi}^{2}}\frac{(\bm{q}\cdot\bm{\sigma})(\bm{q}\cdot\mathbf{T})}{\bm{q}^{2}+m_{\pi}^{2}},$
(21)

where $\mathbf{I}_{1}$ and $\mathbf{I}_{2}$ are the isospin operators of $D$
and $N$, respectively. The operators $\bm{\sigma}$ and $\mathbf{T}$ are
related to the spin operators of the spin-$1\over{2}$ baryon, spin-$1$ meson
as $\frac{1}{2}\bm{\sigma}$ and $-\mathbf{T}$, respectively (see Ref.
Wang:2019ato  for details). The Breit approximation

$\displaystyle\mathcal{V}(\bm{q})=-\frac{\mathcal{M}(\bm{q})}{\sqrt{2\Pi_{i}m_{i}2\Pi_{f}m_{f}}}$
(22)

is used to relate the scattering amplitude $\mathcal{M}(\bm{q})$ to the
effective potential $\mathcal{V}(\bm{q})$ in momentum space ($m_{i}$ and
$m_{f}$ are the masses of the initial and final states, respectively).

The next-to-leading order two-pion-exchange diagrams for the $DN$ system are
illustrated in Fig. 2. The effective potentials from these graphs read

Figure 2: The two-pion-exchange diagrams of the $DN$ system at
$\mathcal{O}(\varepsilon^{2})$. These diagrams are classified as the football
diagram ($F_{1.1}$), triangle diagrams ($T_{1.i}$), box diagrams ($B_{1.i}$)
and crossed box diagrams ($R_{1.i}$). We use the heavy-thick line to denote
the $\Delta(1232)$ in the loops. Other notations are the same as those in Fig.
1.

$\displaystyle\mathcal{V}_{DN}^{F_{1.1}}$ $\displaystyle=$
$\displaystyle(\mathbf{I}_{1}\cdot\mathbf{I}_{2})\frac{1}{f_{\pi}^{4}}J_{22}^{F}(m_{\pi},q),$
(23)

$\displaystyle\mathcal{V}_{DN}^{T_{1.1}}$ $\displaystyle=$
$\displaystyle(\mathbf{I}_{1}\cdot\mathbf{I}_{2})\frac{g^{2}}{f_{\pi}^{4}}\Big{[}(d-1)J_{34}^{T}$
(24)
$\displaystyle-\bm{q}^{2}\left(J_{24}^{T}+J_{33}^{T}\right)\Big{]}(m_{\pi},\mathcal{E}-\delta_{b},q),$
$\displaystyle\mathcal{V}_{DN}^{T_{1.2}}$ $\displaystyle=$
$\displaystyle(\mathbf{I}_{1}\cdot\mathbf{I}_{2})\frac{4g_{\delta}^{2}}{3f_{\pi}^{4}}\Big{[}(2-d)J_{34}^{T}$
(25)
$\displaystyle-\bm{q}^{2}\frac{2-d}{d-1}\left(J_{24}^{T}+J_{33}^{T}\right)\Big{]}(m_{\pi},\mathcal{E}-\delta_{a},q),$
$\displaystyle\mathcal{V}_{DN}^{T_{1.3}}$ $\displaystyle=$
$\displaystyle(\mathbf{I}_{1}\cdot\mathbf{I}_{2})\frac{g_{a}^{2}}{f_{\pi}^{4}}\Big{[}(d-1)J_{34}^{T}$
(26)
$\displaystyle-\bm{q}^{2}\left(J_{24}^{T}+J_{33}^{T}\right)\Big{]}(m_{\pi},\mathcal{E},q),$
$\displaystyle\mathcal{V}_{DN}^{B_{1.1}}$ $\displaystyle=$
$\displaystyle\left[\frac{1}{8}-\frac{1}{3}\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right]\frac{3g^{2}g_{a}^{2}}{2f_{\pi}^{4}}\Big{[}\bm{q}^{4}\left(J_{22}^{B}+2J_{32}^{B}+J_{43}^{B}\right)$
(27)
$\displaystyle+(d^{2}-1)J_{41}^{B}-2\bm{q}^{2}(d+1)\left(J_{31}^{B}+J_{42}^{B}\right)$
$\displaystyle-\bm{q}^{2}J_{21}^{B}\Big{]}(m_{\pi},\mathcal{E},\mathcal{E}-\delta_{b},q),$
$\displaystyle\mathcal{V}_{DN}^{B_{1.2}}$ $\displaystyle=$
$\displaystyle\left[\frac{1}{2}+\frac{2}{3}\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right]\frac{g^{2}g_{\delta}^{2}}{f_{\pi}^{4}}\Big{[}\bm{q}^{4}\frac{d-2}{d-1}\left(J_{22}^{B}+2J_{32}^{B}+J_{43}^{B}\right)$
(28)
$\displaystyle+(d^{2}-d-2)J_{41}^{B}-2\bm{q}^{2}\frac{d^{2}-d-2}{d-1}\left(J_{31}^{B}+J_{42}^{B}\right)$
$\displaystyle-\bm{q}^{2}\frac{d-2}{d-1}J_{21}^{B}\Big{]}(m_{\pi},\mathcal{E}-\delta_{a},\mathcal{E}-\delta_{b},q),$

where the loop functions $J_{ij}^{F}$, $J_{ij}^{T}$, $J_{ij}^{B}$ and
$J_{ij}^{R}$ are defined and given in Refs. Meng:2019ilv ; Wang:2019ato ;
Wang:2018atz . $d$ is the dimension introduced in the dimensional
regularization. $\mathcal{E}=E_{i}-m_{i}~{}[i=N,D^{(\ast)}]$ represents the
residual energies of the $N$ and $D^{(\ast)}$. $\mathcal{E}$ is set to zero in
our calculations. The expressions of the crossed box diagrams $(R_{1.i})$ can
be obtained with the relation

$\displaystyle\mathcal{V}_{DN}^{R_{1.i}}$ $\displaystyle=$
$\displaystyle\mathcal{V}_{DN}^{B_{1.i}}\Big{|}_{J_{ij}^{B}\to
J_{ij}^{R},~{}\mathbf{I}_{1}\cdot\mathbf{I}_{2}\to-\mathbf{I}_{1}\cdot\mathbf{I}_{2}}.$
(29)

In order to generate the effective potential, one needs to subtract the two
particle reducible contributions from the box diagrams. A detailed deduction
that based on the principal value integral method is given in the Appendix B
of Ref. Wang:2019ato .

The $\mathcal{O}(\varepsilon^{2})$ two-pion-exchange diagrams for the
$D^{\ast}N$ system are shown in Fig. 3. Their analytical expressions are
written as

Figure 3: The two-pion-exchange diagrams of the $D^{\ast}N$ system at
$\mathcal{O}(\varepsilon^{2})$. The notations are the same as those in Fig. 2.

$\displaystyle\mathcal{V}_{D^{\ast}N}^{F_{2.1}}$ $\displaystyle=$
$\displaystyle(\mathbf{I}_{1}\cdot\mathbf{I}_{2})\frac{1}{f_{\pi}^{4}}J_{22}^{F}(m_{\pi},q),$
(30) $\displaystyle\mathcal{V}_{D^{\ast}N}^{T_{2.1}}$ $\displaystyle=$
$\displaystyle(\mathbf{I}_{1}\cdot\mathbf{I}_{2})\frac{g^{2}}{f_{\pi}^{4}}\left[2J_{34}^{T}-\bm{q}^{2}\frac{d-2}{d-1}\left(J_{24}^{T}+J_{33}^{T}\right)\right](m_{\pi},\mathcal{E},q),$
(31) $\displaystyle\mathcal{V}_{D^{\ast}N}^{T_{2.2}}$ $\displaystyle=$
$\displaystyle(\mathbf{I}_{1}\cdot\mathbf{I}_{2})\frac{g^{2}}{f_{\pi}^{4}}\left[J_{34}^{T}-\frac{\bm{q}^{2}}{d-1}\left(J_{24}^{T}+J_{33}^{T}\right)\right](m_{\pi},\mathcal{E}+\delta_{b},q),$
(32) $\displaystyle\mathcal{V}_{D^{\ast}N}^{T_{2.3}}$ $\displaystyle=$
$\displaystyle(\mathbf{I}_{1}\cdot\mathbf{I}_{2})\frac{g_{a}^{2}}{f_{\pi}^{4}}\bigg{[}(d-1)J_{34}^{T}-\bm{q}^{2}\left(J_{24}^{T}+J_{33}^{T}\right)\bigg{]}(m_{\pi},\mathcal{E},q),$
(33) $\displaystyle\mathcal{V}_{D^{\ast}N}^{T_{2.4}}$ $\displaystyle=$
$\displaystyle(\mathbf{I}_{1}\cdot\mathbf{I}_{2})\frac{4g_{\delta}^{2}}{3f_{\pi}^{4}}\bigg{[}(2-d)J_{34}^{T}-\bm{q}^{2}\frac{2-d}{d-1}\left(J_{24}^{T}+J_{33}^{T}\right)\bigg{]}(m_{\pi},\mathcal{E}-\delta_{a},q),$
(34) $\displaystyle\mathcal{V}_{D^{\ast}N}^{B_{2.1}}$ $\displaystyle=$
$\displaystyle\left[\frac{1}{8}-\frac{1}{3}\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right]\frac{3g^{2}g_{a}^{2}}{2f_{\pi}^{4}}\bigg{[}\frac{4d^{2}-10d+6}{d-1}J_{41}^{B}-\bm{q}^{2}\frac{d^{2}+3d-8}{d-1}\left(J_{31}^{B}+J_{42}^{B}\right)$
(35)
$\displaystyle-\bm{q}^{2}\frac{d-2+\bm{\sigma}\cdot\bm{T}}{d-1}J_{21}^{B}+\bm{q}^{4}\frac{d-2}{d-1}\left(J_{22}^{B}+2J_{32}^{B}+J_{43}^{B}\right)\bigg{]}(m_{\pi},\mathcal{E},\mathcal{E},q),$
$\displaystyle\mathcal{V}_{D^{\ast}N}^{B_{2.2}}$ $\displaystyle=$
$\displaystyle\left[\frac{1}{8}-\frac{1}{3}\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right]\frac{3g^{2}g_{a}^{2}}{2f_{\pi}^{4}}\bigg{[}-2\bm{q}^{2}\frac{d+1}{d-1}\left(J_{31}^{B}+J_{42}^{B}\right)-\bm{q}^{2}\frac{1}{d-1}(1+\bm{\sigma}\cdot\bm{T})J_{21}^{B}$
(36)
$\displaystyle+(d+1)J_{41}^{B}+\bm{q}^{4}\frac{1}{d-1}\left(J_{22}^{B}+2J_{32}^{B}+J_{43}^{B}\right)\bigg{]}(m_{\pi},\mathcal{E},\mathcal{E}+\delta_{b},q),$
$\displaystyle\mathcal{V}_{D^{\ast}N}^{B_{2.3}}$ $\displaystyle=$
$\displaystyle\left[\frac{1}{2}+\frac{2}{3}\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right]\frac{g^{2}g_{\delta}^{2}}{f_{\pi}^{4}}\bigg{[}-\bm{q}^{2}\frac{(d-2)^{2}-\bm{\sigma}\cdot\bm{T}}{(d-1)^{2}}J_{21}^{B}-\bm{q}^{2}\frac{(d-2)(d^{2}+3d-8)}{(d-1)^{2}}\left(J_{31}^{B}+J_{42}^{B}\right)$
(37)
$\displaystyle+\frac{2(d^{2}-2d+2)}{d-1}J_{41}^{B}+\bm{q}^{4}\frac{(d-2)^{2}}{(d-1)^{2}}\left(J_{22}^{B}+2J_{32}^{B}+J_{43}^{B}\right)\bigg{]}(m_{\pi},\mathcal{E}-\delta_{a},\mathcal{E},q),$
$\displaystyle\mathcal{V}_{D^{\ast}N}^{B_{2.4}}$ $\displaystyle=$
$\displaystyle\left[\frac{1}{2}+\frac{2}{3}\mathbf{I}_{1}\cdot\mathbf{I}_{2}\right]\frac{g^{2}g_{\delta}^{2}}{f_{\pi}^{4}}\frac{1}{d-1}\bigg{[}-2\bm{q}^{2}\frac{(d+1)(d-2)}{d-1}\left(J_{31}^{B}+J_{42}^{B}\right)-\bm{q}^{2}\frac{d-2-\bm{\sigma}\cdot\bm{T}}{d-1}J_{21}^{B}$
(38)
$\displaystyle+\bm{q}^{4}\frac{d-2}{d-1}\left(J_{22}^{B}+2J_{32}^{B}+J_{43}^{B}\right)+(d^{2}-d-2)J_{41}^{B}\bigg{]}(m_{\pi},\mathcal{E}-\delta_{a},\mathcal{E}+\delta_{b},q).$

The expressions of the diagrams $(R_{2.i})$ can be obtained with

$\displaystyle\mathcal{V}_{D^{\ast}N}^{R_{2.i}}=\mathcal{V}_{D^{\ast}N}^{B_{2.i}}\Big{|}_{J_{x}^{B}\to
J_{x}^{R},~{}\mathbf{I}_{1}\cdot\mathbf{I}_{2}\to-\mathbf{I}_{1}\cdot\mathbf{I}_{2},~{}\bm{\sigma}\cdot\bm{T}\to-\bm{\sigma}\cdot\bm{T}}.$
(39)

In above equations, the spin operator and transferred momentum are defined in
$d$ dimensions, such as $\bm{S}^{2}=(d-1)/4$ Scherer:2002tk  and
$q_{i}q_{j}=1/(d-1)\bm{q}^{2}\delta_{ij}$ for $S$-wave. The relation between
the $\bm{T}$ operator and the polarization vector of $D^{\ast}$ meson can be
found in the Appendix C of Ref. Wang:2019ato .

Notably, because of $\delta_{b}>m_{\pi}$, some diagrams in Fig. 3, such as
$(T_{2.2})$ and $(B_{2.2})$, would also generate imaginary parts, which
contribute to the width of the corresponding bound state. If solving the
Lippmann-Schwinger equation, the imaginary part would shift the pole position
in the Riemann sheet. In this work, we solve the Schrödinger equation to only
focus on the binding energies, thus the imaginary parts are ignored in our
calculations.

At the next-to-leading order, besides the two-pion-exchange potentials
illustrated above, there also have the one-loop corrections to the one-pion-
exchange and contact terms. These corrections can be replaced by using the
physical values of the couplings, decay constant and masses of pion, $N$ and
$D$ mesons, etc.. In addition, the subleading contact Lagrangians are also
necessary. On the one hand, they are responsible for the renormalizations of
two-pion-exchange loop diagrams. Because these diagrams are usually
ultraviolet divergent in $d=4$ dimensions, we need the unrenormalized
$\mathcal{O}(\varepsilon^{2})$ LECs to absorb the divergent parts of the loop
functions in Eqs. (23)-(38). On the other hand, these
$\mathcal{O}(\varepsilon^{2})$ Lagrangians may generate contact interactions
with two powers of momenta in the following form Machleidt:2011zz ,

$\displaystyle\mathscr{V}$ $\displaystyle=$ $\displaystyle
C_{1}\bm{q}^{2}+C_{2}\bm{k}^{2}+(C_{3}\bm{q}^{2}+C_{4}\bm{k}^{2})\bm{\sigma}\cdot\bm{T}$
(40)
$\displaystyle+C_{5}[i\bm{S}\cdot(\bm{q}\times\bm{k})]+C_{6}(\bm{\sigma}\cdot\bm{q})(\bm{T}\cdot\bm{q})$
$\displaystyle+C_{7}(\bm{\sigma}\cdot\bm{k})(\bm{T}\cdot\bm{k}),$

where $\bm{k}=(\bm{p}_{N}+\bm{p}_{D})/2$ is the average momentum and
$\bm{S}=(\bm{S}_{N}+\bm{S}_{D^{\ast}})/2$ is the total spin (Note that for
$DN$ system, only the $C_{1}$, $C_{2}$ and $C_{5}$ terms survive). Unlike the
$NN$ case, these $\mathcal{O}(\varepsilon^{2})$ LECs cannot be fitted at
present since the $D^{(\ast)}N$ scattering data is still unavailable. So we
ignore their contributions and only consider the leading order contact terms.
Once the $D^{(\ast)}N$ scattering phase shifts in lattice QCD simulations are
available, it would be an intriguing topic to restudy this system with
considering higher order contributions.

##  III Numerical results and discussions

With the momentum-space potentials $\mathcal{V}(\bm{q})$ obtained in Sec.
II.2, we make the following Fourier transformation to get the effective
potential $V(r)$ in the coordinate space,

$\displaystyle V(r)=\int\frac{d^{3}\bm{q}}{(2\pi)^{3}}e^{i\bm{q\cdot
r}}\mathcal{V}(\bm{q})\mathcal{F}(\bm{q}).$ (41)

We need to introduce a regulator $\mathcal{F}(\bm{q})$ to suppress the high
momentum contribution. We choose the Gauss form
$\mathcal{F}(\bm{q})=\exp(-\bm{q}^{2n}/\Lambda^{2n})$ as used in the $NN$ and
$N\bar{N}$ systems Epelbaum:2004fk ; Epelbaum:2003xx . The Taylor expansion of
the regulator function gives
$\mathcal{F}(\bm{q})=1-\bm{q}^{2n}/\Lambda^{2n}+\dots$. The power $n$ is
chosen to be sufficiently large in order that the cutoff induced contributions
$\mathcal{V}(\bm{q})\mathcal{O}(\bm{q}^{2n}/\Lambda^{2n})$ are beyond the
chiral order one is working at. In our calculation, we only consider the
leading and subleading contributions, thus $n=1$ is already enough in our
case. However, we use the $N\bar{N}$ LECs fitted in Ref. Kang:2013uia  to
estimate the LECs of $D^{(\ast)}N$ systems, where the $n=3$ is adopted in Ref.
Kang:2013uia , so we also choose $n=3$ for consistency. The power $n=3$ and
cutoff $\Lambda\simeq 0.5\pm 0.1$ GeV are always adopted to fit the
experimental data and make predictions Kang:2013uia ; Epelbaum:2014efa ;
Epelbaum:2014efa ; Entem:2003ft ; Dai:2017ont .

###  III.1 Numerical results

In order to get the numerical results, we also need to know the values of the
four LECs in Eq. (II.1). Generally, these LECs should be determined by fitting
the $D^{(\ast)}N$ scattering data in experiments or in lattice QCD
simulations. However, the data in this area are scarce, thus we have to resort
to other alternative ways. As proposed in Refs. Meng:2019nzy ; Wang:2019nvm ,
we estimate the LECs by constructing the contact Lagrangian at the quark
level, and then extract the couplings from the $N\bar{N}$ interaction, which
is demonstrated in Appendix A.

Figure 4: The effective potentials of the $D^{(\ast)}N$ systems. Their
$I(J^{P})$ are marked in each subfigure. The potentials with the solid lines
are obtained with the cutoff parameter $\Lambda=0.4$ GeV and the LECs
estimated in Appendix A. The corresponding total potentials with the dashed
lines are obtained with the cutoff parameter $\Lambda=0.6$ GeV.

We show the effective potentials of each possible $I(J^{P})$ configurations in
Fig. 4. In the following, we analyze the behaviors of effective potentials for
each system.

$DN$ system: The result in Fig. 4($a$) shows that the
$\mathcal{O}(\varepsilon^{0})$ contact and $\mathcal{O}(\varepsilon^{2})$ two-
pion-exchange potentials of the $[DN]_{J=1/2}^{I=0}$ system are both
attractive. But the attraction of two-pion-exchange potential is rather weak.
The attractive potential is dominantly provided by the contact interaction. We
find a bound state in this channel. The binding energy and mass of this state
are predicted, respectively,

$\displaystyle\Delta E_{[DN]_{J=1/2}^{I=0}}$ $\displaystyle\simeq$
$\displaystyle-11.1~{}\textrm{MeV},$ $\displaystyle M_{[DN]_{J=1/2}^{I=0}}$
$\displaystyle\simeq$ $\displaystyle 2792.0~{}\textrm{MeV}.$ (42)

For the $[DN]_{J=1/2}^{I=1}$ system in Fig. 4($b$), the
$\mathcal{O}(\varepsilon^{0})$ contact interaction vanishes in our
calculation, and the total potential arises from the two-pion-exchange
contribution. We notice the potential in this channel is much shallower than
that of the $[DN]_{J=1/2}^{I=0}$ channel, i.e., the attraction is too feeble
to form the bound state. Thus the binding solution does not exist in this
channel.

$D^{\ast}N$ system: The contact potential of the $[D^{\ast}N]_{J=1/2}^{I=0}$
system in Fig. 4($c$) is attractive, while the one-pion-exchange and two-pion-
exchange interaction are both repulsive. Therefore, the total potential is
shallower than that of the $[DN]_{J=1/2}^{I=0}$ channel. However, we still
obtain a binding solution in the $[D^{\ast}N]_{J=1/2}^{I=0}$ system. The
binding energy and mass of this state are

$\displaystyle\Delta E_{[D^{\ast}N]_{J=1/2}^{I=0}}$ $\displaystyle\simeq$
$\displaystyle-1.5~{}\textrm{MeV},$ $\displaystyle
M_{[D^{\ast}N]_{J=1/2}^{I=0}}$ $\displaystyle\simeq$ $\displaystyle
2943.6~{}\textrm{MeV}.$ (43)

For the $[D^{\ast}N]_{J=1/2}^{I=1}$ system in Fig. 4($d$), the one-pion-
exchange potential is weakly attractive, but the contact and two-pion-exchange
potentials are all repulsive. Thus, the total attractive potential is not
strong enough to form molecular states in this channel.

For the channel $[D^{\ast}N]_{J=3/2}^{I=0}$ in Fig. 4($e$), the behavior of
its potentials is very interesting. We notice the one-pion- and two-pion-
exchange contributions almost cancel each other. Thus the total potential is
mainly provided by the contact term, which can reach up to $-80$ MeV at the
deepest position. By solving the Schrödinger equation, we find the binding
solution in the $[D^{\ast}N]_{J=3/2}^{I=0}$ system, and the binding energy is

$\displaystyle\Delta E_{[D^{\ast}N]_{J=3/2}^{I=0}}\simeq-6.7~{}\textrm{MeV}.$
(44)

The corresponding mass of this bound state is

$\displaystyle M_{[D^{\ast}N]_{J=3/2}^{I=0}}\simeq 2938.4~{}\textrm{MeV},$
(45)

which is in good agreement with the mass of $\Lambda_{c}(2940)$ measured by
BaBar, Belle and LHCb (e.g., see Table 1).

For the last channel $[D^{\ast}N]_{J=3/2}^{I=1}$ in Fig. 4($f$), the one-pion-
and two-pion-exchange potentials almost cancel each other, and the contact
contribution is very weakly attractive. Thus no bound state can be found in
this channel.

Role of the $\Delta(1232)$: Considering the strong coupling between
$\Delta(1232)$ and $N\pi$, we include the contribution of $\Delta(1232)$ in
the loop diagrams (e.g., see Figs. 2 and 3). Here, we discuss the role of
$\Delta(1232)$ in the the effective potentials of the $DN$ and $D^{\ast}N$
systems. We try to ignore the effect of $\Delta(1232)$, and notice that the
lineshape of the two-pion-exchange potentials changes drastically. Except for
the $[DN]_{J=1/2}^{I=1}$, the whole behavior of the other channels is totally
reversed. For example, the two-pion-exchange potential of the
$[DN]_{J=1/2}^{I=0}$ channel becomes repulsive, which renders the total
potential of this channel shallower. But for the $[D^{\ast}N]_{J=3/2}^{I=0}$
channel, the two-pion-exchange potential becomes attractive. The variation is
about $-30$ MeV, which gives rise to a deeper attractive potential, and the
binding energy is $-16$ MeV.

In general, the conclusion that there exists the bound state in isoscalar
$[D^{(\ast)}N]_{J}$ channel and no binding solution in isovector channel is
robust, no matter we consider the $\Delta(1232)$ or not. However, the
$\Delta(1232)$ plays an important role in determining the physical masses of
$\Lambda_{c}(2940)$ and other bound states, since the molecular states are
very sensitive to the subtle changes of their internal effective potentials.

It is also interesting to see the dependence of the potentials on cutoff
$\Lambda$. The results for $\Lambda=0.6$ GeV are shown as the dashed lines in
Fig. 4. We notice the change is dramatic, the total potentials are sensitive
to the cutoff, and they all become much deeper when the cutoff is varied to
$0.6$ GeV. The binding energies for $[DN]_{J=1/2}^{I=0}$,
$[D^{\ast}N]_{J=1/2}^{I=0}$ and $[D^{\ast}N]_{J=3/2}^{I=0}$ channels in this
case are $-76.7$, $-11.6$ and $-55.1$ MeV, respectively. The isovector $DN$
system also starts to form bound state with binding enery $-1.0$ MeV, but we
still cannot get binding solutions in the isovector $D^{\ast}N$ system. Of
course, this behavior is foreseeable, because we introduced a Gaussian
regulator when making the Fourier transformation [see Eq. (41)]. Even in the
$NN$ interactions, the LECs show mild dependence on the cutoff over a narrow
range only when the higher order contributions are included Marji:2013uia ;
Entem:2017gor . Generally, the LECs are used to absorb the scale dependence of
observables when making the nonperturbative renormalizations, i.e., they are
usually the functions of cutoff $\Lambda$. Therefore, varying the cutoff with
the fixed LECs might be unreasonable. In the Appendix A, we illustrate the
results with the LECs fixed at $\Lambda=0.6$ GeV from Ref. Kang:2013uia .

Inspecting the potentials in Fig. 4 one can find two defects, one is the long-
range behavior of the one-pion-exchange potential is missing, another one is a
bump appeared in the total potentials. Both are caused by the artificial
effect from the Gaussian regulator. As indicated in Refs. Epelbaum:2014efa ;
Reinert:2017usi , such a form regulator would distort the partial waves and
affect the long-range parts of the interactions. If the cutoff is sufficiently
large, the induced artefacts are expected to go beyond the accuracy at the
order we are conducting. However, they may become an issue when the cutoff
resides in the low scale. Additionally, the $\mathcal{F}(\bm{q})$ in Eq. (41)
has the risk of mixing different partial waves. But for the $S$-wave, the
mixing effect is insufficient to impact the main feature of the potentials.

###  III.2 Discussions

No binding solution in the isovector channel indicates that the
$[DN]_{J=1/2}^{I=1}$ molecular explanation of $\Sigma_{c}(2800)$ is not
favored. Although the $\Sigma_{c}(2800)$ is near the $DN$ threshold, its mass
is also consistent with the quark model predictions Ebert:2011kk ;
Chen:2016iyi ; Garcilazo:2007eh ; Yoshida:2015tia ; Ebert:2007nw . Thus
interpreting the $\Sigma_{c}(2800)$ as the $1P$ charmed baryon seems to be
more reasonable.

The situation of $\Lambda_{c}(2940)$ is very similar to the $\Lambda(1405)$,
$D_{s0}(2317)$ and $X(3872)$, i.e., there is large gap between the physical
states and quark model predictions222An unquenched study with the channel
coupling in Ref. Luo:2019qkm  declares the mass of $\Lambda_{c}(2P,3/2^{-})$
state can be lowered down to match the experimental data of
$\Lambda_{c}(2940)$.. Generally, one possible reason is these states per se
may be exotic rather than the conventional ones.

A recent analysis from LHCb gives weak constraints on the $J^{P}$ quantum
numbers of $\Lambda_{c}(2940)$, where $J^{P}=\frac{3}{2}^{-}$ is favored
Aaij:2017vbw . This is consistent with our calculations. Actually, one can
notice two peaks in the $D^{0}p$ invariant mass spectrum from $2.92$ GeV to
$2.99$ GeV in the results of LHCb (see Fig. 13(a) in Ref. Aaij:2017vbw ). The
one at $2.94$ GeV is just the reported $\Lambda_{c}(2940)$. The other peak at
$2.98$ GeV may correspond to the true $\Lambda_{c}(2P)$ baryon, since its mass
is close to the quark model prediction Capstick:1986bm ; Ebert:2011kk ;
Chen:2014nyo ; Lu:2016ctt . Our calculation indicates the $\Lambda_{c}(2940)$
is probably the $S$-wave $D^{\ast}N$ molecular state.

We report three bound states in the $[DN]_{J=1/2}^{I=0}$,
$[D^{\ast}N]_{J=1/2}^{I=0}$ and $[D^{\ast}N]_{J=3/2}^{I=0}$ systems. They are
very similar to the newly observed $P_{c}(4312)$, $P_{c}(4440)$ and
$P_{c}(4457)$ at LHCb Aaij:2019vzc , which are interpreted as the
$[\bar{D}\Sigma_{c}]_{J=1/2}^{I=1/2}$,
$[\bar{D}^{\ast}\Sigma_{c}]_{J=1/2}^{I=1/2}$ and
$[\bar{D}^{\ast}\Sigma_{c}]_{J=3/2}^{I=1/2}$ molecular states Meng:2019ilv ;
Wang:2019ato , respectively. If $\Lambda_{c}(2940)$ is indeed the $D^{\ast}N$
molecular states, then it should contain two structures, i.e.,
$[D^{\ast}N]_{J=1/2}^{I=0}$ and $[D^{\ast}N]_{J=3/2}^{I=0}$. Because the mass
splitting between the spin-$\frac{1}{2}$ and spin-$\frac{3}{2}$ states is only
about $5$ MeV, it is very difficult to disassemble these two structures with
current accuracy. Similar situation has happened to the $P_{c}$ states. The
previously reported $P_{c}(4450)$ Aaij:2015tga  contains two structures,
$P_{c}(4440)$ and $P_{c}(4457)$, after increasing the data sample. More
interesting, we find the mass of the spin-$\frac{1}{2}$ state is larger than
that of the spin-$\frac{3}{2}$ one.

The signal of $\Lambda_{c}(2940)$ has been observed in the $D^{0}p$ and
$\Sigma_{c}\pi$ final states Aubert:2006sp ; Abe:2006rz ; Aaij:2017vbw .
However, if the $J^{P}$ of $\Lambda_{c}(2940)$ is $\frac{3}{2}^{-}$ as weakly
constrained by the LHCb, then it decays into the $D^{0}p$ and $\Sigma_{c}\pi$
through the $D$-wave, which is strongly suppressed333Based on the
${}^{3}P_{0}$ model calculation, the large decay width of
$\Lambda_{c}(2940)\to D^{0}p$ is reported in Refs. Lu:2018utx ; Lu:2019rtg  by
treating $\Lambda_{c}(2940)$ as a $2P$ state in the $\Lambda_{c}$ family, but
the low mass puzzle still exist.. Therefore, as mentioned above, one promising
explanation is that the $\Lambda_{c}(2940)$ signal actually contains two
structures. The spin-$\frac{1}{2}$ structure can easily decay into $D^{0}p$
and $\Sigma_{c}\pi$ via the $S$-wave.

Borrowing experiences from the discovery of $P_{c}$ states, we urge the
experimenters to reanalyze the $\Lambda_{c}^{+}\pi^{+}\pi^{-}$ invariant mass
spectrum with the accumulated data, since the $[DN]_{J=1/2}^{I=0}$,
$[D^{\ast}N]_{J=1/2}^{I=0}$ and $[D^{\ast}N]_{J=3/2}^{I=0}$ bound states can
all decay into $\Sigma_{c}^{0}\pi^{+}$.

In addition to the mass spectrum, the decay pattern can also give us some
important criteria to identify the inner structure of $\Lambda_{c}(2940)$. In
the molecular scenario, the $D^{\ast}N$ system can easily decay into the $DN$
channel via the pion exchange, while the $\Sigma_{c}\pi$ decay mode requires
the exchange of a nucleon or a $D$ meson. Thus, the decay amplitude of the
$D^{0}p$ mode should be much larger than that of the $\Sigma_{c}\pi$, because
the heavy hadron exchange is generally suppressed. However, the phase space of
the $\Sigma_{c}\pi$ mode is larger.

The three body decay mode is also very interesting. We take the decay modes of
the $X(3872)$ and other higher charmonia as an example. The branching fraction
of $X(3872)\to D^{0}\bar{D}^{0}\pi^{0}$ can reach up to $40\%$
Tanabashi:2018oca . In contrast, the open charm three body decays of the
higher charmonia is only a few percents Weng:2018ebv . Analogously, the
branching fraction of $\Lambda_{c}(2940)\to D^{0}\pi^{0}(\gamma)p$ should also
be conspicuous in the molecular picture.

Besides, our study can be easily extended to the $\bar{B}^{(\ast)}N$ systems.
The axial coupling $g$ and mass splitting $\delta_{b}$ in Eq. (16) should be
replaced by the bottomed ones, where we adopt $g=-0.52$ Ohki:2008py ;
Detmold:2012ge  and $\delta_{b}=45$ MeV Tanabashi:2018oca . The predicted
results are listed in Table. 2. There also exist bound states in the isoscalar
$[\bar{B}^{(\ast)}N]_{J}$ systems. These states might be reconstructed at the
$\Lambda_{b}^{0}\pi^{+}\pi^{-}$ final states, and the $[\bar{B}^{\ast}N]_{J}$
states could also be detected in the $B^{-}p$ mass spectrum.

Table 2: The predicted binding energies and masses for the isoscalar $[D^{(\ast)}N]_{J}$ and $[\bar{B}^{(\ast)}N]_{J}$ systems (in units of MeV). System | $[DN]_{\frac{1}{2}}$ | $[D^{\ast}N]_{\frac{1}{2}}$ | $[D^{\ast}N]_{\frac{3}{2}}$ | $[\bar{B}N]_{\frac{1}{2}}$ | $[\bar{B}^{\ast}N]_{\frac{1}{2}}$ | $[\bar{B}^{\ast}N]_{\frac{3}{2}}$
---|---|---|---|---|---|---
$\Delta E$ | $-11.1$ | $-1.5$ | $-6.7$ | $-8.8$ | $-3.5$ | $-8.4$
Mass | $2792.0$ | $2943.6$ | $2938.4$ | $6208.8$ | $6259.4$ | $6254.5$

##  IV Summary

A sophisticated investigation on the $DN$ and $D^{\ast}N$ interactions is
crucial to clarify the nature of the charmed baryons $\Sigma_{c}(2800)$ and
$\Lambda_{c}(2940)$. In this work, we systematically study the effective
potentials of the $DN$ and $D^{\ast}N$ systems with the chiral effective field
theory up to the next-to-leading order. We simultaneously consider the
contributions of the long-range one-pion-exchange, mid-range two-pion-exchange
and short-range contact term. We also include the $\Delta(1232)$ as an
intermediate state in the loop diagrams. The LECs are estimated from the
$N\bar{N}$ interaction with the help of quark model.

For the $DN$ system, our calculation shows the effective potentials of the
$[DN]_{J=1/2}^{I=0}$ and $[DN]_{J=1/2}^{I=1}$ channels are both attractive. We
find a bound state in the $[DN]_{J=1/2}^{I=0}$ channel, but the attraction in
the $[DN]_{J=1/2}^{I=1}$ channel is too weak to form a bound state. Thus the
explanation of $\Sigma_{c}(2800)$ as the $DN$ molecular state is disfavored in
our calculations. The $\Sigma_{c}(2800)$ is more likely to be the conventional
$1P$ charmed baryon, since its mass is well consistent with the quark model
prediction.

There are four channels in the $[D^{\ast}N]_{J}^{I}$ system. We find only the
isoscalar $[D^{\ast}N]_{J}$ potential is deep enough to form the molecular
state. We obtain the masses of the bound states in the
$[D^{\ast}N]_{J=1/2}^{I=0}$ and $[D^{\ast}N]_{J=3/2}^{I=0}$ channels to be
$2943.6$ and $2938.4$ MeV, respectively, which well accord with the BaBar,
Belle and LHCb measurements for $\Lambda_{c}(2940)$. Considering the small
mass splitting between the spin-$\frac{1}{2}$ and spin-$\frac{3}{2}$ states,
we conjecture the $\Lambda_{c}(2940)$ signal contains two structures.

It is not so easy to squeeze the $\Lambda_{c}(2940)$ into the conventional
charmed baryon spectrum, since the $60$$-$$100$ MeV gap between the physical
mass and quark model prediction cannot be readily remedied. However, this
problem can be easily reconciled in the molecular picture, i.e., the
$\Lambda_{c}(2940)$ is probably the isoscalar $D^{\ast}N$ molecule rather than
the $2P$ charmed baryon.

We also investigate the influence of $\Delta(1232)$ in the loop diagrams. The
binding solutions always exist in the isoscalar $[D^{\ast}N]_{J}$ channels no
matter we include the $\Delta(1232)$ or not. There still do not exist bound
states in the isovector channels even we ignore the $\Delta(1232)$. However,
the $\Delta(1232)$ is important in yielding the shallowly bound isoscalar
$[D^{(\ast)}N]_{J}$ states.

We hope experimentalist could seek for the pentaquark candidates in the open
charmed channels, where the $D^{(\ast)}N$ molecular pentaquarks in the
isoscalar systems might be reconstructed at the
$\Lambda_{c}^{+}\pi^{+}\pi^{-}$ final state.

## Acknowledgments

B. Wang is very grateful to X. Z. Weng for discussions on the charmed baryon
spectroscopy. This project is supported by the National Natural Science
Foundation of China under Grant 11975033.

##  Appendix A Determining the LECs from $N\bar{N}$ interaction

One needs to know the values of the LECs in Eq. (II.1) to study the strength
of the short-range interaction. As proposed in Refs. Meng:2019nzy ;
Wang:2019nvm  (more details can be found in the appendix of these two
references), the LECs of $D^{(\ast)}N$ systems can be bridged to those of the
$N\bar{N}$ interaction with the help of quark model. The way is analogous to
the resonance saturation model Epelbaum:2001fm , but we build the quark level
Lagrangian. We assume the contact interaction stems from the heavy meson
exchanging. We introduce $\mathscr{S}$ and $\mathscr{A}^{\mu}$ to produce the
central potential and spin-spin interaction, respectively. The matrix form of
$\mathscr{S}$ and $\mathscr{A}^{\mu}$ can be expressed as

$\displaystyle\mathscr{S}$ $\displaystyle=$
$\displaystyle\mathscr{S}_{3}^{i}\tau^{i}+\sqrt{\frac{1}{3}}\mathscr{S}_{1},$
(46) $\displaystyle\mathscr{A}^{\mu}$ $\displaystyle=$
$\displaystyle\mathscr{A}_{3}^{\mu
i}\tau^{i}+\sqrt{\frac{1}{3}}\mathscr{A}_{1}^{\mu},$ (47)

where $\mathscr{S}_{3}$ ($\mathscr{A}_{3}^{\mu}$) and $\mathscr{S}_{1}$
($\mathscr{A}_{1}$) denote the isospin triplet and isospin singlet,
respectively. The coefficient $\sqrt{1\over 3}$ is introduced to satisfy the
SU(3) flavor symmetry.

The $q\bar{q}$ contact potential can be written as

$\displaystyle V_{q\bar{q}}$ $\displaystyle=$ $\displaystyle
c_{s}(1-3\bm{\tau}_{1}\cdot\bm{\tau}_{2})+c_{t}(1-3\bm{\tau}_{1}\cdot\bm{\tau}_{2})\bm{\sigma}_{1}\cdot\bm{\sigma}_{2},$
(48)

where $c_{s}$ and $c_{t}$ are the coupling constants. The minus sign in Eq.
(48) arises since the isospin triplet and the isospin singlet have the
different $G$-parities.

With the $q\bar{q}$ contact potential $V_{q\bar{q}}$ in Eq. (48) and the
relevant matrix element in Table 3, we obtain the $N\bar{N}$ contact potential
as follows,

$\displaystyle V_{N\bar{N}}$ $\displaystyle=$ $\displaystyle\langle
N\bar{N}|V_{q\bar{q}}|N\bar{N}\rangle=9c_{s}-3c_{s}\bm{\tau}_{1}\cdot\bm{\tau}_{2}$
(49)
$\displaystyle+c_{t}\bm{\sigma}_{1}\cdot\bm{\sigma}_{2}-\frac{25}{3}c_{t}(\bm{\tau}_{1}\cdot\bm{\tau}_{2})(\bm{\sigma}_{1}\cdot\bm{\sigma}_{2}).$

Similarly, the $D^{\ast}N$ contact potential can be easily worked out,

$\displaystyle V_{D^{\ast}N}$ $\displaystyle=$ $\displaystyle\langle
D^{\ast}N|V_{q\bar{q}}|D^{\ast}N\rangle=3c_{s}-12c_{s}\mathbf{I}_{1}\cdot\mathbf{I}_{2}$
(50) $\displaystyle-
c_{t}\bm{\sigma}\cdot\mathbf{T}+20c_{t}(\mathbf{I}_{1}\cdot\mathbf{I}_{2})(\bm{\sigma}\cdot\mathbf{T}).$

Matching Eq. (II.2) and Eq. (50) one can get the LECs in Eq. (II.1), which
read

$\displaystyle
D_{a}=3c_{s},~{}~{}D_{b}=-c_{t},~{}~{}E_{a}=3c_{s},~{}~{}E_{b}=-5c_{t}.$ (51)

Therefore, once we know the values of $c_{s}$ and $c_{t}$, we can capture the
short range interaction of the $D^{(\ast)}N$ systems. The $c_{s}$ and $c_{t}$
can be extracted from the $N\bar{N}$ interaction, and the $N\bar{N}$
scattering phase shift has been fitted in the framework of chiral effective
field theory to the next-to-next-to-leading order in Ref. Kang:2013uia . Using
the values of $\tilde{C}_{{}^{3}S_{1}}$ in the $I=0$ and $I=1$ channels fitted
at $(\Lambda,\tilde{\Lambda})=(450,500)$ MeV as inputs, we obtain

$\displaystyle c_{s}=-8.1~{}\mathrm{GeV}^{-2},\qquad
c_{t}=0.65~{}\mathrm{GeV}^{-2}.$ (52)

We notice $|c_{s}|/|c_{t}|\simeq 12.5$, i.e., the spin-spin interaction only
serves as a perturbation to give mass splittings between spin multiplets.

Table 3: The matrix elements of the operator $\sum_{i\in h_{a},j\in h_{b}}\mathscr{O}_{ij}$, where $h_{a}$ and $h_{b}$ are two hadrons. $\mathscr{O}_{ij}$ is the two-body interaction operator between quarks. $\mathscr{O}_{ij}$ | $\bm{1}_{ij}$ | $\bm{\tau}_{i}\cdot\bm{\tau}_{j}$ | $\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}$ | $(\bm{\tau}_{i}\cdot\bm{\tau}_{j})(\bm{\sigma}_{i}\cdot\bm{\sigma}_{j})$
---|---|---|---|---
$[N\bar{N}]^{I=1}_{J=0}$ | $9$ | $1$ | $-3$ | $-\frac{25}{3}$
$[D^{\ast}N]^{I=1}_{J=3/2}$ | $3$ | $1$ | $1$ | $\frac{5}{3}$

In addition, the Ref. Kang:2013uia  also gives the fitted results with cutoff
combination $(\Lambda,\tilde{\Lambda})=(650,700)$ MeV. With the LECs of
$\tilde{C}_{{}^{3}S_{1}}$ in the $I=0$ and $I=1$ channels fitted at
$(\Lambda,\tilde{\Lambda})=(650,700)$ MeV as inputs we obtain

$\displaystyle c_{s}=-11.1~{}\mathrm{GeV}^{-2},\qquad
c_{t}=4.5~{}\mathrm{GeV}^{-2}.$ (53)

We see the $c_{s}$ value is similar to the one in Eq. (52), while $c_{t}$
becomes much larger (We find that using other channel combinations in Ref.
Kang:2013uia  as inputs always obtains the $c_{s}$ value with the similar size
and the same sign, but $c_{t}$ varies a lot). With the LECs in Eq. (53) as
input we get the binding energies of the isoscalar and isovector $D^{(\ast)}N$
system as listed in Table 4. One can notice that the binding energies of
$[DN]_{J=1/2}^{I=0}$ and $[D^{\ast}N]_{J=1/2}^{I=0}$ channels are quite large,
which can reach up to hundreds of MeV. This is highly unnatural in our
framework, so we intend to use the moderate LECs in Eq. (52) rather than the
ones in Eq. (53) to give predictions. However, we still can capture a gross
indication that the isoscalar channel is always bound, while the isovector
channel is not (except for the $[DN]_{J=1/2}^{I=1}$ channel).

Table 4: The predicted binding energies for the isoscalar and isovector $[D^{(\ast)}N]_{J}$ system (in units of MeV) in the case of $c_{s}=-11.1$ GeV-2, $c_{t}=4.5$ GeV-2 and $\Lambda=0.6$ GeV. The ``$\times$" denotes no binding solution. System | $I=0$ | $I=1$
---|---|---
$[DN]_{\frac{1}{2}}$ | $[D^{\ast}N]_{\frac{1}{2}}$ | $[D^{\ast}N]_{\frac{3}{2}}$ | $[DN]_{\frac{1}{2}}$ | $[D^{\ast}N]_{\frac{1}{2}}$ | $[D^{\ast}N]_{\frac{3}{2}}$
$\Delta E$ | $-142.3$ | $-294.8$ | $-19.2$ | $-1.0$ | $\times$ | $\times$

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