$D^{(\ast)}N$ interaction and the structure of $\Sigma_c(2800)$ and $\Lambda_c(2940)$ in chiral effective field theory

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 $D^0p$ 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 [1]. But it is rather difficult to assign the near-threshold states, such as X(3872) [2] and D s0 (2317) [3] to the quark model predictions [4][5][6][7][8][9]. In the charmed baryon family, a state Λ 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 Λ c (2940) in the D 0 p invariant mass spectrum [10], 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 Λ c (2940) → Σ c π [11]. In 2017, the J P quantum numbers of Λ c (2940) was constrained by the LHCb measurement, and the most likely spin-parity assignment for Λ c (2940) is J P = 3 2 − [12] (The mass and width of Λ c (2940) obtained by the BaBar, Belle and LHCb experiments are shown in Table I). Up to now, there are two different interpretations of the internal structure of Λ c (2940). One is the ordinal charmed baryon, and the other one is the D * N molecular state. However, it is difficult to arrange Λ c (2940) to the 2P state in the of nuclear force is built upon the pioneer work of Weinberg [54,55] and largely developed in the framework of effective field theory. The chiral effective field theory was extensively exploited to study the N N interaction with great success [56][57][58][59][60][61]. The chiral effective field theory was also utilized to study the systems with heavy flavors in Refs. [62][63][64][65][66][67][68], which is a powerful tool in predicting the BB * and B * B * bound states [64], reproducing the newly observed pentaquarks [66], extrapolating the Σ c N potential from lattice QCD result to the physical pion mass [67], and so on. As a natural extension of the N N interaction, in this work, we use the chiral effective field theory to study the D ( * ) N interaction up to the next-to-leading order. We simultaneously consider the long-, mid-and short-range interactions, and include the contribution of ∆(1232) in the loops as an intermediate state.
With the chiral effective field theory, we calculate the D ( * ) N effective potentials and search for the possible bound states. The numerical results can be compared with the experimental data of Λ c (2940) and Σ 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 ( * ) 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 NN system with quark model.

A. Effective chiral Lagrangians
We first show the leading order Lagrangian of the nucleon and pion interaction under the heavy baryon reduction [69], which reads where N = (p, n) T denotes the the large component of the nucleon field under the nonrelativistic reduction. v = (1, 0) is the 4-velocity of the nucleon and D µ = ∂ µ + Γ µ . g a 1.29 is the axial-vector coupling constant. S µ = i 2 γ 5 σ µν v ν stands for the spin operator of the nucleon. Γ µ and u µ are the chiral connection and axial-vector current, respectively. Their expressions read where τ i is the Pauli matrix, and f π = 92.4 MeV is the pion decay constant. Considering the importance of ∆(1232) in the N N interaction [50,[70][71][72], we adopt the small scale expansion method [73] to explicitly include the ∆(1232) in the Lagrangians. The Lagrangian that delineates the ∆-N -π cou-pling is given as where δ a = m ∆ − m N . g 1 = 9 5 g a is estimated with the quark model [73]. g δ 1.05 is the coupling constant for ∆N π vertex. T µ i denotes the spin-3 2 and isospin-3 2 field ∆(1232) after performing the nonrelativistic reduction. Its matrix form reads The leading order Lagrangian that depicts the interaction between the charmed mesons and light Goldstones reads [74,75] L Hϕ = i Hv · DH − 1 8 where · · · represents the trace in spinor space. δ b is defined 59 stands for the axial coupling, whose sign is determined with the help of quark model. The H is the super-field for the charmed mesons, which reads with P = (D 0 , D + ) T and P * = (D * 0 , D * + ) T , respectively. We construct the leading order contact Lagrangian to describe the short distance interaction between the nucleon and charmed meson, 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 NN interaction as inputs, which is given in the Appendix A.

B. Expressions of the effective potentials
In the framework of heavy hadron chiral perturbation theory, the scattering amplitudes of the D ( * ) N systems can be expanded order by order in powers of a small parameter ε = q/Λ χ , where q is either the momentum of Goldstone bosons or the residual momentum of heavy hadrons, and Λ χ represents either the chiral breaking scale or the mass of a heavy hadron. The expansion is organized by the power counting rule in Refs. [54,55]. The O(ε 0 ) Feynman diagrams for the DN and D * 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 DDπ vertex is forbidden. The corresponding momentum-space potentials of graphs in Fig. 1 where I 1 and I 2 are the isospin operators of D and N , respectively. The operators σ and T are related to the spin operators of the spin-1 2 baryon, spin-1 meson as 1 2 σ and −T, respectively (see Ref. [66] for details). The Breit approximation is used to relate the scattering amplitude M(q) to the effective potential V(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 where the loop functions J F ij , J T ij , J B ij and J R ij are defined and given in Refs. [64][65][66]. d is the dimension introduced in the dimensional regularization.
represents the residual energies of the N and D ( * ) . E is set to zero in our calculations. The expressions of the crossed box diagrams (R 1.i ) can be obtained with the relation The O(ε 2 ) two-pion-exchange diagrams for the D * N system are shown in Fig. 3. Their analytical expressions are written as The two-pion-exchange diagrams of the DN system at O(ε 2 ). These diagrams are classified as the football diagram (F1.1), triangle diagrams (T1.i), box diagrams (B1.i) and crossed box diagrams (R1.i). We use the heavy-thick line to denote the ∆(1232) in the loops. Other notations are the same as those in Fig. 1.
The notations are the same as those in Fig. 2.
The expressions of the diagrams (R 2.i ) can be obtained with V R2.i

III. NUMERICAL RESULTS AND DISCUSSIONS
With the momentum-space potentials V(q) obtained in Sec. II B, we make the following Fourier transformation to get the effective potential V (r) in the coordinate space, We need to introduce a regulator F(q) to suppress the high momentum contribution. We choose the Gauss form F(q) = exp(−q 2n /Λ 2n ) as used in the N N and NN systems [76,77]. The power n = 3 and cutoff Λ 0.5 ± 0.1 GeV are always adopted to fit the experimental data and make predictions [78][79][80][81].

A. Numerical results
In order to get the numerical results, we also need to know the values of the four LECs in Eq. (9). Generally, these LECs should be determined by fitting the D ( * ) 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. [67,68], we estimate the LECs by constructing the contact Lagrangian at the quark level, and then extract the couplings from the NN interaction, which is demonstrated in Appendix A.
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 O(ε 0 ) contact and O(ε 2 ) two-pion-exchange potentials of the [DN ] I=0 J=1/2 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, For the [DN ] I=1 J=1/2 system in Fig. 4(b), the O(ε 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 ] I=0 J=1/2 channel, i.e., the attraction is too feeble to form the bound state. Thus the binding solution does not exist in this channel. D * N system: The contact potential of the [D * N ] I=0 J=1/2 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 For the [D * N ] I=1 J=1/2 system in Fig. 4(d), the one-pionexchange 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 * N ] I=0 J=3/2 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 * N ] I=0 J=3/2 system, and the binding energy is The corresponding mass of this bound state is which is in good agreement with the mass of Λ c (2940) measured by BaBar, Belle and LHCb (e.g., see Table I).
For the last channel [D * N ] I=1 J=3/2 in Fig. 4(f ), the one-pionand 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 ∆(1232): Considering the strong coupling between ∆(1232) and N π, we include the contribution of ∆(1232) in the loop diagrams (e.g., see Figs. 2 and 3). Here, we discuss the role of ∆(1232) in the the effective potentials of the DN and D * N systems. We try to ignore the effect of ∆(1232), and notice that the lineshape of the twopion-exchange potentials changes drastically. Except for the [DN ] I=1 J=1/2 , the whole behavior of the other channels is totally reversed. For example, the two-pion-exchange potential of the [DN ] I=0 J=1/2 channel becomes repulsive, which renders the total potential of this channel shallower. But for the [D * N ] I=0 J=3/2 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 ( * ) N ] J channel and no binding solution in isovector channel is robust, no matter we consider the ∆(1232) or not. However, the ∆(1232) plays an important role in determining the physical masses of Λ c (2940) and other bound states, since the molecular states are very sensitive to the subtle changes of their internal effective potentials.

B. Discussions
No binding solution in the isovector channel indicates that the [DN ] I=1 J=1/2 molecular explanation of Σ c (2800) is not favored. Although the Σ c (2800) is near the DN threshold, its mass is also consistent with the quark model predictions [14,42,43,82,83]. Thus interpreting the Σ c (2800) as the 1P charmed baryon seems to be more reasonable.
The situation of Λ c (2940) is very similar to the Λ(1405), D s0 (2317) and X(3872), i.e., there is large gap between the physical states and quark model predictions 1 . 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 Λ c (2940), where J P = 3 2 − is favored [12]. 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. [12]). The one at 2.94 GeV is just the reported Λ c (2940). The other peak at 2.98 GeV may correspond to the true Λ c (2P ) baryon, since its mass is close to the quark model prediction [13][14][15][16]. Our calculation indicates the Λ c (2940) is probably the S-wave D * N molecular state.
We report three bound states in the and spin- 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) [86] contains two structures, P c (4440) and P c (4457), after increasing the data sample. More interesting, we find the mass of the spin-1 2 state is larger than that of the spin-3 2 one. The signal of Λ c (2940) has been observed in the D 0 p and Σ c π final states [10][11][12]. However, if the J P of Λ c (2940) is 3 2 − as weakly constrained by the LHCb, then it decays into the D 0 p and Σ c π through the D-wave, which is strongly suppressed 2 . Therefore, as mentioned above, one promising explanation is that the Λ c (2940) signal actually contains two structures. The spin-1 2 structure can easily decay into D 0 p and Σ c π via the S-wave.
Borrowing experiences from the discovery of P c states, we urge the experimenters to reanalyze the Λ + c π + π − invariant mass spectrum with the accumulated data, since the In addition to the mass spectrum, the decay pattern can also give us some important criteria to identify the inner structure of Λ c (2940). In the molecular scenario, the D * N system can easily decay into the DN channel via the pion exchange, while the Σ c π 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 Σ c π, because the heavy hadron exchange is generally suppressed. However, the phase space of the Σ c π 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) → D 0D0 π 0 can reach up to 40% [1]. In contrast, the open charm three body decays of the higher charmonia is only a few percents [89]. Analogously, the branching fraction of Λ c (2940) → D 0 π 0 (γ)p should also be conspicuous in the molecular picture.
Besides, our study can be easily extended to theB ( * ) N systems. The axial coupling g and mass splitting δ b in Eq. (7) should be replaced by the bottomed ones, where we adopt g = −0.52 [90,91] and δ b = 45 MeV [1]. The predicted results are listed in Table. II. There also exist bound states in the isoscalar [B ( * ) N ] J systems. These states might be reconstructed at the Λ 0 b π + π − final states, and the [B * N ] J states could also be detected in the B − p mass spectrum.

IV. SUMMARY
A sophisticated investigation on the DN and D * N interactions is crucial to clarify the nature of the charmed baryons Σ c (2800) and Λ c (2940). In this work, we systematically study the effective potentials of the DN and D * N systems with the chiral effective field theory up to the nextto-leading order. We simultaneously consider the contributions of the long-range one-pion-exchange, mid-range twopion-exchange and short-range contact term. We also include the ∆ (1232)  J=1/2 channel is too weak to form a bound state. Thus the explanation of Σ c (2800) as the DN molecular state is disfavored in our calculations. The Σ 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 * N ] I J system. We find only the isoscalar [D * N ] J potential is deep enough to form the molecular state. We obtain the masses of the bound states in the [D * N ] I=0 J=1/2 and [D * N ] I=0 J=3/2 channels to be 2943.6 and 2938.4 MeV, respectively, which well accord with the BaBar, Belle and LHCb measurements for Λ c (2940). Considering the small mass splitting between the spin-1 2 and spin-3 2 states, we conjecture the Λ c (2940) signal contains two structures. It is not so easy to squeeze the Λ 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 Λ c (2940) is probably the isoscalar D * N molecule rather than the 2P charmed baryon.
We also investigate the influence of ∆(1232) in the loop diagrams. The binding solutions always exist in the isoscalar [D * N ] J channels no matter we include the ∆(1232) or not. There still do not exist bound states in the isovector channels even we ignore the ∆(1232). However, the ∆(1232) is important in yielding the shallowly bound isoscalar [D ( * ) N ] J states.
We hope experimentalist could seek for the pentaquark candidates in the open charmed channels, where the D ( * ) N molecular pentaquarks in the isoscalar systems might be reconstructed at the Λ + c π + π − 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 NN interaction One needs to know the values of the LECs in Eq. (9) to study the strength of the short-range interaction. As proposed in Refs. [67,68] (more details can be found in the appendix of these two references), the LECs of D ( * ) N systems can be bridged to those of the NN interaction with the help of quark model. The way is analogous to the resonance saturation model [92], but we build the quark level Lagrangian. We assume the contact interaction stems from the heavy meson exchanging. We introduce S and A µ to produce the central potential and spin-spin interaction, respectively. The matrix form of S and A µ can be expressed as where S 3 (A µ 3 ) and S 1 (A 1 ) denote the isospin triplet and isospin singlet, respectively. The coefficient 1 3 is introduced to satisfy the SU(3) flavor symmetry.
The qq contact potential can be written as where c s and c t are the coupling constants. The minus sign in Eq. (A3) arises since the isospin triplet and the isospin singlet have the different G-parities.
With the qq contact potential V qq in Eq. (A3) and the relevant matrix element in Table III, we obtain the NN contact potential as follows, Similarly, the D * N contact potential can be easily worked out, V D * N = D * N |V qq |D * N = 3c s − 12c s I 1 · I 2 −c t σ · T + 20c t (I 1 · I 2 )(σ · T).
Matching Eq. (11) and Eq. (A5) one can get the LECs in Eq. (9), which read D a = 3c s , D b = −c t , E a = 3c s , E b = −5c t . (A6) Therefore, once we know the values of c s and c t , we can capture the short range interaction of the D ( * ) N systems. The c s and c t can be extracted from the NN interaction, and the NN scattering phase shift has been fitted in the framework of chiral effective field theory to the next-to-next-to-leading order in Ref. [78]. Using the values ofC3 S1 in the I = 0 and I = 1 channels fitted at (Λ,Λ) = (450, 500) MeV as inputs, we obtain We notice |c s |/|c t | 12.5, i.e., the spin-spin interaction only serves as a perturbation to give mass splittings between spin multiplets. Oij 1ij τi · τj σi · σj (τi · τj)(σi · σj) [NN ] I=1