$\Lambda_c(2910)$ and $\Lambda_c(2940)$ as the conventional baryons dressed with the $D^*N$ channel

In this work, we treat $\Lambda_c(2910)^+$ and $\Lambda_c(2940)^+$ as the conventional $udc$ cores dressed with the $D^*N$ channel. We provide a possible interpretation to both $\Lambda_c(2910)^+$ and $\Lambda_c(2940)^+$ within the same framework. In the study, we consider not only the effects between the conventional triquark core and the $D^*N$ channel but also the $D^*N$-$D^*N$ interactions. The mass of $\Lambda_c$ state with $J^P=1/2^-$ is larger than that with $J^P=3/2^-$ in this unquenched picture, which is very different from the prediction of the conventional quenched quark model. Based on the mass spectrum, the spin-parity of $\Lambda_c(2940)^+$ is more likely to be $1/2^-$ while $\Lambda_c(2910)^+$ prefers $3/2^-$. We look forward to the future experiments can test our results with more precise experimental data.

In the present work, we try to interpret Λ c (2910) + and Λ c (2940) + at the same time within a consistent framework with the coupled-channel effects between the bare triquark states and the D * N channel, as well as the D * N-D * N interactions. The physical states Λ c (2910) + and Λ c (2940) + can be produced as a competition between the bare 2P-wave udc cores and S -wave D * N components. We believe our effort can lead us to disclose the true natures of Λ c (2910) + and Λ c (2940) + . This article is organized as follows. After the introduction, we present the details of the theoretical framework in Section II, which includes the full Hamiltonian and coupledchannel equation involving the interaction of the hadronhadron channel. The detailed interactions will be given in Section III within the chiral effective field theory and quarkpair-creation model. In Section IV, we present our numerical results and discussion. A short summary follows in Section V.

II. FRAMEWORK
If we consider the mixing between the bare state |Ψ 0 and |BC, p channel, the physical state [64,[70][71][72][73][74][75][76][77][78][79][80] can be represented by Here, c 0 is the possible amplitude to discover the bare state |Ψ 0 in the physical state |Ψ , χ BC (p) denotes the relative wave function in the hadron-hadron channel |BC, p , and the normalizing condition is given by Then the full coupled-channel equation can be formally expressed as Here,Ĥ 0 only works on the bare state |Ψ 0 , i.e., where M 0 is the bare mass, which can be well determined by the traditional potential models, such as non-relativistic three quark model [46,[81][82][83][84], quark-diquark model [2,3,13], Capstick-Isgur model [4], and so on. For the most low-lying singly charmed baryons, these traditional potential models have good predictive powers in studying the mass spectra.Ĥ 0 is usually written as a sum of two parts, i.e., where E i is free energy for i-th constituent quark, and the V i j is the effective potential between two quarks where the form depends on specific potential models.
Then c 0 can be obtained On the other hand, multiplying BC, p | on each side of Eq. (2.5), we have (2.10) TheĤ BC is the Hamiltonian of the intermediate BC channel, which includes two parts, i.e., the free and inner interactions. Explicitly, the matrix element sandwiching theĤ BC can be expressed as where E BC (p ) = m B + m C + p 2 2m B + p 2 2m C is the free energy of the intermediate BC channel and V BC→BC (p, p ) is the hadronhadron interaction in the momentum space. With Eqs. (2.9)-(2.11), we can obtain the following coupled-channel equation (2.12) It is equivalent to the following expression Thus, the final χ BC -coupled-channel equation (2.13) contains the interaction between hadron B and C. By solving Eq. (2.13), we can get the physical mass that includes the contribution of both the hadron BC channel and the bare state. The two terms inside the parenthesis of Eq. (2.13) are responsible for the mass shift.
For obtaining the solution of Eq. (2.13), we use a set of complete base expansion method, where the complete basis can be chosen as the Harmonic oscillator basis, the Gaussian basis, and so on. For example, we use the Gaussian basis to replace the χ BC (p) in the Eq. (2.13) [85,86] where C il is the coefficient of the corresponding basis, and φ p ilm (p) is the Gaussian basis. In the coordinate space, φ r nlm (ν n , r) = N nl r l e −v n r 2 Y lm (r), (2.15) where N nl is normalization constant. By the Fourier transform, the Gaussian basis in momentum space can be written as In Eqs. (2.15) and (2.16), v n is the Gaussian ranges, i.e., v n = 1/r 2 n , r n = r 1 a n−1 (n = 1, 2 . . . N max ). (2.17) With the above Gaussian basis, all the Hamilton matrix elements can be expressed in simple forms. We define , With the above matrix elements, Eq. (2.13) is equivalent to following general eigenvalue equation Here the coefficient C il can be solved by the Rayleigh-Ritz variational principle. Because both sides of Eq. If the interaction V BC→BC (p, p ) of the direct hadron-hadron channel is neglected, we can extract the approximate mass M and obtain approximate wave function χ BC (p)

III. DETAILED INTERACTIONS
In this section, we provide the D * N interaction in Sec. III A and the coupling between the bare state and the D * N in Sec. III B. These determine the coupled-channel effects of Λ c (2940) + and Λ c (2910) + .
We will investigate four channels in the D * N system, i.e., isospin I = 0, 1 and spin J = 1/2, 3/2. Therefore, we consider the following flavor wave functions In the following, we first give the chiral Lagrangians and Feynman diagrams including tree and one-loop diagrams of the D * N system. Then, we provide the D * N effective potentials in momentum space at the next-to-leading order O( 2 ). They include the contact term, one-pion-exchange, and twopion-exchange contributions, which approximatively corresponds to the short-range, long-range, and middle-range interactions, respectively. In addition, we also consider the ∆(1232) contribution. Through the Fourier transformation, we also obtain the effective potentials in coordinate space. The leading order πN Lagrangian [106][107][108] is given by 22) where N = (p, n) T denotes the the large component of the nucleon field under the nonrelativistic reduction, v = (1, 0, 0, 0) stands for the four-velocity of the nucleon, D µ = ∂ µ +Γ µ is covariant derivative, g a is the axial-vector coupling constant, and S µ = i 2 γ 5 σ µν v ν denotes the Pauli-Lubanski spin vector. The chiral connection Γ µ and axial-vector current u µ are expressed as where τ i is a 2-component Pauli matrix in the isospin space, i.e., and f π is the pion decay constant. The D ( * ) π Lagrangian at the leading order is given by [109][110][111][112] where · · · denotes the trace in the spinor space, δ b is mass shift between D * and D, i.e., δ b = m D * − m D , and it does not disappear in the chiral limit. g stands for the axial coupling constant. The H represents the super-field for the charmed mesons, which reads The leading order contact Lagrangian describes the short distance interaction between the nucleon and charmed meson [44], which can be written as where D a , D b , E a and E b are four low energy constants (LECs).
In the framework of the heavy hadron chiral perturbation theory, the scattering amplitudes of the D * N system can be expanded order by order in powers of a small quantity ε = q/Λ χ ∼ 1 GeV, where q is either the momentum of Goldstone bosons or the residual momentum of heavy flavor hadrons, and Λ χ represents either the chiral breaking scale or the mass of a heavy hadron. The expansion respects the power counting rule [106,107]. The Feynman diagrams of O( 0 ) contact term, O( 0 ) one-pion-exchange, and O( 2 ) two-pion-exchange are illustrated in Fig. 2.
With the chiral Lagrangians and these Feynman diagrams, we can obtain their Feynman amplitudes M. respectively. These effective potentials of the Feynman diagrams in Fig. 2 consist of following parts where V LO contact , V LO 1−π , and V NLO 2−π denote O( 0 ) contact term, O( 0 ) one-pion-exchange, and O( 2 ) two-pion-exchange potentials, respectively.
The V NLO 2−π is the sum of the football diagram, triangle diagram, box diagram, and crossed box diagram potentials in Fig. 2 These two-pion-exchange diagrams need some loop functions J F i j , J T i j , J B i j , and J R i j , which can be found in Refs. [96-98, 102, 105]. In order to obtain the effective potentials, the two-particle-reducible (2PR) contribution should be subtracted from these crossed box diagrams by the principal value integral method in the Appendix B of Ref. [102]. Additionally, the divergent parts in two-pion-exchange diagrams can be absorbed by unrenormalized LECs in Ref. [44]. In this work, all the parameters for the D * N interaction are from Ref. [44].
In Fig. 3, we give the D * N effective potential in momentum space, which can help us to understand the D * N interaction clearer. For the [D * N] I=0 J=1/2 channel in Fig. 3 (a), the O( 0 ) one-pion-exchange potential and O( 2 ) two-pionexchange potential are both repulsive. But their repulsive interaction is rather weak. The attractive interaction is dominantly provided by the O( 0 ) contact interaction. In Fig. 3 (b), the O( 0 ) one-pion-exchange potential is attractive and the O( 2 ) two-pion-exchange potential is repulsive, but both of them are rather weak. However, the potential of O( 0 ) contact term has strong attractive interaction. For the [D * N] I=1 J=1/2 channel in Fig. 3 (c), the potential of O( 0 ) contact term and O( 2 ) two-pion-exchange are both repulsive, but the O( 0 ) one-pion-exchange is attractive. The total potential has rather weak attraction. From Fig. 3 (d), the total potential are not attractive enough. Therefore, the two isoscalar channels I(J P ) = 0(1/2 − ) and I(J P ) = 0(3/2 − ) provide the stronger attractive interaction in momentum space, and we identify the attraction in two I = 1 channels is weak. In the following parts, we mainly consider the two isospin I = 0 channels. Furthermore, the ∆(1232) plays a particular role in the D * N interaction because the coupling between ∆(1232) and πN sys-tem is very strong. The total potential is sensitive to ∆(1232), which determines its importance in the D * N interaction.
Based on the obtained effective potential in momentum space V(q), the effective potential in coordinate space V(r) can be obtained by the following Fourier transformation Here, F (q) = e −q 2n /Λ 2n is the form factor with Gauss form to suppress the high momentum and renormalize the potential [44,97]. When we employ the chiral effective field theory and use some approximation, such as the transferred energy q 0 between D * and N, and the residual energies of N and D * are all set to zero, then the effective potential of D * N will have the simple form of V BC→BC (p − p). Thus, the V BC→BC (p , p) in Eq. (2.11) can be defined as V BC→BC (p , p) = 1 (2π) 3 V BC→BC (r)e i(p −p)·r d 3 r.
In this scheme, the matrix element V f i in Eq. (2.18) can be conveniently calculated by For the interaction HamiltonianĤ I between the bare state and D * N in Eq. (2.3), we employ the quark-pair-creation model [119,120], which has the expression (3.34) Here, g = 2m q γ, m q is the mass of the creation quark, and the dimensionless parameter γ describes the strength of the quark and antiquark pair creation from the vacuum, which can be determined phenomenologically by the Okubo-Zweig-Iizuka (OZI) allowed decay widths of charmonia. In the non-relativistic limit,Ĥ I is equivalent to [13] Here, ω, φ, χ, and Y m 1 describe the color, flavor, spin, and orbital angular momentum functions of the quark pair, respectively. b † µ and d † ν are quark and antiquark creation operators, respectively. In this work, we fit the dimensionless parameter γ as 9.45 from the total decay width of Σ c (2520) [1].
The masses of bare Λ c (2P, 1/2 − ) and Λ c (2P, 3/2 − ) can be obtained by the traditional quark potential models combined with the Gaussian expansion method by solving Eqs. (2.6) and (2.7). In this work, we use different bare masses as input and collect them in Table I

IV. RESULTS AND DISCUSSIONS
Using the formalism described in Sec. II and the detailed interaction in Sec. III, we can now study the dynamical coupling between the S -wave D * N channel and bare charmed baryon core Λ c (2P). In the quenched quark model, the charmed baryons are simply treated as the three-quark udc baryons [2-4, 13, 46]. In the unquenched picture, the physical states Λ c (2940) + and Λ c (2910) + consist of both udc core and the S -wave D * N component. In the former works, the authors of Ref. [46] considered the coupling between udc and D * N but did not involve the D * N interaction, while this work contains them both. We list the results of the three approaches in Table II.
From the left column in Table II, the masses of the Λ c (2P, 1/2 − ) and Λ c (2P, 3/2 − ) from quenched quark model are much larger than those of Λ c (2940) + and Λ c (2910) + . From the middle columns in Table II, the theoretical masses decrease due to the coupling between the udc core and D * N, but neither can reach the mass of lower state Λ c (2910) + [46]. Among them, we can see the coupled-channel effects between the Λ c (2P, 1/2 − ) and S -wave [D * N] I=0 J=1/2 channel is relatively weak, and thus the mass is only suppressed by about 20 MeV and is still above the D * N threshold. Neither Λ c (2940) + nor Λ c (2910) + can match to it. Meanwhile, the coupled effects between Λ c (2P, 3/2 − ) and S -wave [D * N] I=0 J=3/2 channel is strong, the mass decreases by about 70 MeV and locates below D * N threshold.
In Fig. 4, we present the physical mass M dependence on the bare mass M 0 in Λ c (2P, 1/2 − ) and Λ c (2P, 3/2 − ). One can notice that the physical masses increase as the bare masses. There are differences between the vertical axis and horizontal axis values of points in the curves, which shows that the bare udc core has a significant mass shift due to the effects of Swave D * N channel.
By solving Eq. (2.13), one can obtain the radial wave function of D * N channel and the probability amplitude of bare udc core, which can help us to reveal the role of D * N channel and bare udc core in physical state Λ c (2940) + and Λ c (2910) + . In Table II, we also provide the root-mean-square radius of D * N component and the probability of bare udc core in the bound state, and we can see the 3/2 − state is a little thinner and contains more bare udc core than the 1/2 − one.
If neglecting the triquark core, the pure D * N can also form bound states with masses around 2940 MeV with J P = 1/2 − and 3/2 − [44]. But such assumptions make us difficult to place the bare 2P udc states. In Fig. 5, we present the D * N radial wave functions for the three circumstances: the pure D * N molecule, the udc core dressed by D * N without considering D * N interaction, and the dressed state with D * N interaction. From Fig. 5, we can find that the existence of bare state can make the D * N bind more tightly. These wave functions can be used to analyze other properties of charmed baryons, and it can help us to distinguish which model is better in future.
As mentioned earlier, the ∆(1232) plays an important role in these two physical states. If ignoring the contribution of ∆(1232), the two-pion-exchange potential becomes smaller. We list the results in Table III if turning off ∆(1232). From the table, the masses of dressed states decrease by about 10 MeV. In two isoscalar states [D * N] I=0 J=1/2 and [D * N] I=0 J=3/2 , the total potentials become more attractive, which bring these two states thinner.
How to get rid of the cutoff dependence in nonperturbative calculations is still an outstanding problem both in hadron physics and nuclear physics. In order to investigate the cutoff dependence of our results, we list the charmed baryon masses with two different cutoffs in Table IV. One can see that the unquenched masses all become lower with the cutoff becomes larger, and the results at Λ = 0.4 GeV are more in line with the experimental data. Currently, we can only provide reasonable results in a very narrow range of cutoff due to the restriction of the validity region of the chiral effective field theory.
In our calculation, we gave the possible interpretation that the J P for Λ c (2940) is 1/2 − , which is in clear conflict with the preferred 3/2 − assignment from the LHCb experiment [19]. However, our results cannot be completely ruled out by the current experiment, and some other articles also agree with the J P = 1/2 − assignment [2,7,29,34,37,38,44]. Moreover, the LHCb Collaboration concludes that the other solutions with spins 1/2 to 7/2 cannot be excluded [19].
The D 0 p mass region in the amplitude fit is 2.8 ∼ 3.0 GeV in Ref. [19], but only Λ c (2940) was included. Two resonances Λ c (2940) and Λ c (2910) should be taken into account at the same time, and the conclusion may be changed from the new fit in experiment. The J P assignment for Λ c (2940) should further be measured by other way like the partial wave analysis. Our results should also be further checked by analyzing their other properties in theory.

V. SUMMARY
With the accumulation of experimental data, a series of charmed baryons have been reported in the past decades. However, some cannot be fitted into the charmed baryon family very well. Inspired by the reported Λ c (2910) + and Λ c (2940) + [1, [16][17][18][19], we use an unquenched picture to study them by considering S -wave D * N channel coupled with the bare udc core (Λ c (2P)), which gives a unified description of Λ c (2910) + and Λ c (2940) + .  [2-4, 13, 46], unquenched picture without D * N interaction [46], and unquenched picture with D * N interaction in this work. Here, r RMS refers to the root-mean-square radius of the D * N component, and P(udc) = c 2 0 represents the probability of bare udc core in the bound state.

Cases
Quenched picture Unquenched picture without D * N interaction Unquenched picture with D * N interaction   We take into account of two important factors in this work, i.e., the D * N interaction and the triquark-D * N coupling. In our unquenched picture, we reproduce the masses of Λ c (2910) + and Λ c (2940) + and assign Λ c (2910) + to J P = 3/2 − and Λ c (2940) + to 1/2 − . The results show the unquenched effects lead to the mass inversion phenomenon in the two states,  In the present work, we find the D * N interaction is crucial in forming the physical states. Let us look at Fig. 1 again. There are large gaps between the quenched quark model results and the experimental masses. The 3/2 − mass is pulled Here, the cutoff parameter Λ = 0.4 GeV and the bare masses are adopted from Ref. [46]. down more than the 1/2 − one by the coupling between the triquark core and the D * N, which causes the mass inversion phenomenon. The mass spectrum moves down further after considering the attractive D * N interaction, and eventually consistent with the experiments.
From our obtained results, we see Λ c (2910) + and Λ c (2940) + contain a significant D * N component, and the bare state can cause the D * N binding more compactly. In addition, we also study the influence of ∆(1232) in the D * N interaction. If neglecting ∆(1232), the dressed masses would be about 10 MeV smaller than before.
Revealing the mixed structure of Λ c (2910) + and Λ c (2940) + in the unquenched picture, we expect it can be further verified by other theoretical approaches like lattice QCD simulations. In addition to mass, other properties of these two states should also be analyzed within this picture in future. More importantly, we strongly suggest experiment to give a further study in future, which can provide more hints to uncover the nature of Λ c (2910) + and Λ c (2940) + .