Doubly charmed multibaryon systems

We study two- and three-baryon systems with two units of charm looking for possible bound states or resonances. All two-baryon interactions are consistently derived from a constituent quark model tuned in the light-flavor hadron phenomenology: spectra and interactions. The presence of the heavy quarks makes the two-body interactions simpler than in the light-flavor sector. Our results show a narrow two-body resonance with quantum numbers $(I,J^P)=(0,0^+)$. It is located 6.2 MeV below the $\Sigma_c\Sigma_c$ threshold and has a width of 4.7 MeV. The foregoing two-body state contributes to generate a $N \Sigma_c\Sigma_c$ resonance with quantum numbers $(I,J^P)=(1/2,1/2^+)$ and a separation energy of 0.2 MeV.


I. INTRODUCTION
The existence of molecules made of heavy baryons is a hot topic in nowadays hadronic physics [1][2][3][4][5][6][7]. The observation in 2017 by the LHCb Collaboration of a doubly charmed baryon [8] increased the interest in exotic states containing pairs of charmed quarks. Right now, the LHCb Collaboration has reported two structures matching the lineshape of a resonance just above twice the J/Ψ mass, that could originate from a hadron containing two charm quarks [9]. Although the existence of exotic structures containing pairs of heavy quarks is a long-term challenge [10], it has recently been noticed, for example in Refs. [11][12][13], that doubly charmed tetraquarks are the first to be at the edge of binding.
On general grounds, the main motivation to wonder about the existence of heavy-baryon molecules is rooted in the reduction of the kinetic energy due to larger reduced masses.
However, such molecular states could be the concatenation of several effects and not just a fairly attractive interaction. The coupling between nearby channels, conflicts between different terms of the interaction, and non-central forces often play a significant role. Some of these contributions may be reinforced by the presence of heavy quarks while others may become weaker [14,15].
Behind all this there is the understanding of the hadron-hadron interaction governed by the dynamics of quarks and gluons, which is a topical issue. To encourage new experiments and analysis of existing data it is essential to have detailed theoretical investigations. Despite some uncertainty in contemporary interaction models, the possible existence of bound states or resonances is a key element, because their signs might be clear enough to be identified in the experimental data [9]. Thus, it is the purpose of this work to study the possible existence of hadronic molecules or resonances in two-and three-baryon systems with two units of charm, in particular, Λ c Λ c , Σ c Σ c , NΛ c Λ c and NΣ c Σ c states. When tackling this problem, one has to manage with an important difficulty, namely the complete lack of experimental data. Therefore, the generalization of models describing two-hadron interactions in the light-flavor sector could offer insight about the unknown interaction of hadrons with heavy flavors.
Following these ideas, we will make use of a constituent quark model (CQM) tuned on the description of the NN interaction [16] as well as the meson [17] and baryon [18,19] spectra in all flavor sectors, to obtain parameter-free predictions that will hopefully be testable in future experiments. Let us note that the study of the interaction between charmed baryons has become an interesting subject in several contexts [9,[20][21][22][23] and it may shed light on the possible existence of exotic nuclei with heavy flavors [24][25][26][27][28][29].
The paper is organized as follows. In Sec. II we describe and analyze particular aspects of the S wave two-body subsystems: NΛ c , NΣ c , Λ c Λ c and Σ c Σ c . Section III is devoted to the study of the lightest NΛ c Λ c and NΣ c Σ c three-body systems. Finally, in Sec. IV we summarize our main conclusions.

II. TWO-BARYON SYSTEMS
The two-body interactions that are necessary to study the charm +2 two-and threebaryon systems have been discussed at length in the literature [30,31]. They are derived from the CQM [16][17][18][19]. The capability of the model is endorsed by the nice description of the NN phase shifts, as can be seen in Figs. 2, 3 and 4 of Ref. [32]. The NΛ c and NΣ c interactions have been presented and discussed in detail in Ref. [30], in comparison with the other approaches available in the literature, in particular recent lattice QCD studies [29].
The Λ c Λ c and Σ c Σ c interactions have been consistently derived within the CQM in Ref. [31], also in comparison with the alternative approaches available in the literature. We refer the reader to Refs. [30,31] for a thorough description of the derivation and analysis of the twobody interactions. As can be seen in Table 1 of Ref. [30] and Table II of Ref. [31] all two-body interactions are consistently derived with the same set of parameters. In the following we highlight some peculiarities of the two-body interactions that are relevant to the purpose of the present work.
We summarize in Table I the low-energy parameters of the two-body systems in the charm +1 and +2 sectors. The scattering length becomes complex for those two-body channels with open lower mass two-body states. The two-body interactions are in general attractive but not sufficient for having bound states, in agreement with lattice QCD estimations [29].
The singlet isospin 1/2 and triplet isospin 3/2 Σ c N interactions are the only repulsive ones.
The last line of Table I presents the results for the uncoupled Σ c Σ c isosinglet system 1 . It can be seen how the scattering length is positive and larger than the range of the interaction, I: CQM results for the 1 S 0 and 3 S 1 scattering lengths (a s and a t ) and effective range parameters (r s and r t ) in fm for the different S wave Y c N and Y c Y c systems (Y c = Λ c or Σ c ). The results shown in the last line, marked by a †, correspond to the uncoupled Σ c Σ c system. Fig. 1, pointing to existence of a bound state that will be discussed further below.
Of particular interest are the results for the lightest charm +2 channel, with quantum numbers (I, J P ) = (0, 0 + ). We show in Fig. 1 the two-body potentials involved in this channel. The Λ c Λ c interaction is slightly attractive at intermediate distances but, however, repulsive at short range. It is decoupled from the closest two-baryon threshold, the NΞ cc state [31], which is relevant for the possible existence of a resonance in the strange sector [33,34]. There is a general agreement on the overall attractive character of the Σ c Σ c   interaction [1,2,5]. Finally, the CQM coupling between the Λ c Λ c and Σ c Σ c channels is a bit stronger than in hadronic theories, based solely on a one-pion exchange potential [2], due to quark-exchange effects [31]. All of this fits the scenario of the strange sector, as can be seen by comparing with Fig. 1(b) of Ref. [35], but the absence of the one-kaon exchange potential gives rise to a less attractive interaction.
In Fig. 2 we present the Fredholm determinant [36] for the two-body (I, J P ) = (0, 0 + ) charm +2 channel in two different cases. The dashed line corresponds to the result considering the full coupling between the Λ c Λ c and Σ c Σ c states, whereas the solid line considers only the Σ c Σ c channel. The coupled channel calculation shows an attractive character but not sufficient to generate a bound state, the Fredholm determinant does not become negative for energies below threshold. This result is in agreement with other estimations in the literature [1,2,5] in which in spite of the attractive character of the Λ c Λ c interaction, the central potential alone is not enough to generate a bound state. The coupling to larger mass channels could be important for the existence of a bound state or a resonance. However, due to the large mass difference between the two coupled channels in the (I, J P ) = (0, 0 + ) partial wave, 338 MeV, the coupled channel effect is weakened. Let us just note that, for example, in the strange sector the coupling to the NΣ state is relevant for the NΛ system [37] due to a smaller mass difference, M(Σ) − M(Λ) = 77 MeV. Heavier mass channels play a minor role, such as the ∆∆ channel (584 MeV above the NN threshold) in the NN system [38].
Thus, one does not expect higher channels, as it could be Σ * c Σ * c (468 MeV above the Λ c Λ c threshold) to play a relevant role, as it has been explicitly checked in the literature [5].
Due to the large mass difference between the Λ c Λ c and Σ c Σ c channels, we have studied the uncoupled Σ c Σ c system. The dynamics could be dominated by the attraction in the Σ c Σ c channel in a way that the Λ c Λ c channel would be mainly a tool for detection. This mechanism is somewhat related to the 'synchronization of resonances' proposed by D. Bugg [39]. A similar situation could be the case of the d * (2380) resonance in the ∆∆ system, see Ref. [40] for a recent review. The result is depicted in Fig. 2 [41]. It was demonstrated how the width of the resonance does not come only determined by the available phase space for its decay to the detection channel, but it greatly depends on the relative position of the mass of the resonance with respect to the masses of the coupled-channels generating the state. 2 Thus, making use of the interactions given in Fig. 1, we have studied the width of the resonance produced in between the two thresholds, Λ c Λ c and Σ c Σ c . The Lippmann-Schwinger equation in the case of S-wave interactions is written as, where µ 1 = m Λc /2, µ 2 = m Σc /2, and ∆E = 2m Σc − 2m Λc . The interactions in momentum 2 The equivalence of the results obtained using a two-cluster interaction or a variational approach for the multiquark problem has been recently shown, see for example in Ref. [42]. Dealing with resonances, the two-cluster interaction allows to look for the poles of the propagator without resorting to numerical extensions of the variational approach, like the complex scaling method, that would just give an indication about the possible existence of a resonance.
space are given by, where V ij (r) are the two-body potentials in Fig. 1. The resonance exists at an energy E = E R such that the phase shift δ(E R ) = 90 • , for energies between the Λ c Λ c and Σ c Σ c thresholds, i.e., 0 < E R < ∆E. The mass of the resonance is given by The width of the resonance is calculated using the Breit-Wigner formula as [43][44][45], Although the Breit-Wigner formula is not very accurate close to threshold, however, we have explicitly checked by analytic continuation of the S matrix on the second Riemann sheet that at low energy the width follows the expected Γ ∼ E 1/2 behavior.
Using the formalism described above we have calculated the width of the Σ c Σ c state. We

III. THE THREE-BARYON SYSTEM
The Λ c Λ c − Σ c Σ c system in a pure S wave configuration has quantum numbers (I, J P ) = (0, 0 + ) so that adding one more nucleon, the NΛ c Λ c system has necessarily quantum numbers (I, J P ) = (1/2, 1/2 + ). It is coupled to the NΣ c Σ c channel. A detailed description of the Faddeev equations of the three-body system can be found in Ref. [46]. It has been explained how to deal with coupled channels containing identical particles of various types in the upper and lower channels. We show in Table II the different two-body channels that contribute to the NΛ c Λ c − NΣ c Σ c (I, J P ) = ( 1 2 , 1 2 + ) three-body system. Notice that the charm +2 S wave channels Λ c Σ c and Σ c Σ c with isospin 1 are not considered since they do not couple to the isosinglet Λ c Λ c two-body subsystem. Therefore, the Faddeev equations of the (I, J P ) = (1/2, 1/2 + ) three-body system are of the form, where t k ij are the two-body t-matrices that already contain the coupling among all two-body channels contributing to a given three-body state, see Table II. G 0 is the propagator of three free particles. The superscript of the amplitudes indicates the spectator particle and the subscript the interacting pair.
The results are presented in Figure 3. We have performed three different calculations.
First, we have included the three-body amplitudes of the first two lines of Table II that do not contain the Σ c baryon, the result being depicted by the dotted line. As it could have been expected, there exists attraction due to the attractive character of the NΛ c and Λ c Λ c interactions discussed in Sect. II. However, the attraction is not sufficient for having a bound state. Then, we have included the amplitudes containing the Σ c Σ c two-body subsystem, third line in Table II, and all isospin 1/2 three-body amplitudes containing a Σ c baryon either as spectator or as a member of the interacting pair, lines 4 to 6 of Table II. The result Guided by the resonance found in the Σ c Σ c system, we have finally studied the NΣ c Σ c system without coupling to NΛ c Λ c , looking for a three-body resonance. The results are promising if the triplet isospin 3/2 amplitude is not considered. Thus, considering only the isospin 1/2 amplitudes we obtain a bound state with a separation energy of 0.6 MeV. If the singlet isospin 3/2 amplitude is included, the separation energy increases to 0.7 MeV. If the repulsive triplet isospin 3/2 (NΣ c )Σ c amplitude is considered, the signal of the resonance is lost. However, adopting the best suited value for the charm sector, b c = 0.2 fm, the NΣ c Σ c three-body resonance is still there with a separation energy of 0.2 MeV.

IV. SUMMARY
In short, we have studied two-and three-baryon systems with two units of charm looking for possible bound states or resonances. All two-baryon interactions are consistently derived from a constituent quark model tuned in the light-flavor hadron phenomenology. Our results show a narrow two-body resonance with quantum numbers (I, J P ) = (0, 0 + ). It is located 6.2 MeV below the Σ c Σ c threshold and has a width of 4.7 MeV. A detailed study of the coupled NΛ c Λ c − NΣ c Σ c three-body system as well as the uncoupled NΣ c Σ c system shows that the foregoing two-body state contributes to generate a NΣ c Σ c resonance with quantum numbers (I, J P ) = (1/2, 1/2 + ) and a separation energy of 0.2 MeV.
Weakly bound states and resonances are usually very sensitive to potential details and therefore theoretical investigations with different phenomenological models are highly desirable. The existence of these states could be scrutinized in the future at the LHC, J-PARC and RHIC providing a great opportunity for extending our knowledge to some unreached part in our matter world.