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Hidden charm N and Delta resonances with heavy-quark symmetry

We develop a model to describe odd parity baryon resonances generated dynamically through a unitary baryon-meson coupled-channels approach. The scheme applies to channels with light and/or heavy quark content. Distinct features of the model are that, i) the interaction is an S-wave contact one, ii) it reduces to the SU(3) Weinberg-Tomozawa Hamiltonian when light pseudoscalar mesons are involved, thus, respecting chiral symmetry, iii) spin-flavor in preserved in the light quark sector, and iv) heavy quark spin symmetry is fulfilled in the heavy quark sector. In particular, baryon-meson states with different content in c or in c-bar do not mix. The model is a minimal one and it contains no free parameters. In this work, we focus on baryon resonances with hidden-charm (at least one c-bar and one c quarks). We analyze several possible sectors and, for the sector with zero net charm, we write down the most general Lagrangian consistent with SU(3) and heavy quark spin symmetry. We explicitly study the N and Delta states, which are produced from the S-wave interaction of pseudoscalar and vector mesons with 1/2+ and 3/2+ baryons within the charmless and strangeless hidden charm sector. We predict seven odd parity N-like and five Delta-like states with masses around 4 GeV, most of them as bound states. These states form heavy-quark spin multiplets, which are almost degenerate in mass. The predicted new resonances definitely cannot be accommodated by quark models with three constituent quarks and they might be looked for in the forthcoming PANDA experiment at the future FAIR facility.

The possible observation of new states with the charm degree of freedom has attracted a lot of attention over the past years in connection with many experiments such as CLEO, Belle, BaBar, and others . Moreover, the futurē PANDA and CBM experiments at FAIR facility of GSI [37,38] will aim at obtaining data in the heavy-flavor sector, thus opening the possibility for observation of new exotic states with quantum numbers of charm and strangeness. In this regard, a clear goal would be to understand the nature of these states, and in particular, whether they can be described with the usual three-quark baryon or quark-antiquark meson interpretation, or qualify better as hadron molecules within a baryon-meson coupled-channels description.
Nevertheless, these models do not explicitly incorporate heavy-quark spin symmetry (HQSS) [78][79][80], and therefore, it is unclear whether this symmetry is respected. HQSS is a QCD symmetry that appears when the quark masses, such as the charm mass, become larger than the typical confinement scale. HQSS predicts that all types of spin interactions involving heavy quarks vanish for infinitely massive quarks. Thus, HQSS connects vector and pseudoscalar mesons containing charmed quarks. On the other hand, chiral symmetry fixes the lowest order interaction between Goldstone bosons and other hadrons in a model independent way; this is the Weinberg-Tomozawa (WT) interaction. Thus, it is appealing to have a predictive model for four flavors including all basic hadrons (pseudoscalar and vector mesons, and 1/2 + and 3/2 + baryons) which reduces to the WT interaction in the sector where Goldstone bosons are involved and which incorporates HQSS in the sector where charm quarks participate. This model was developed in Ref. [81][82][83], following the steps of the SU(6) approach in the light sector of Refs. [84][85][86][87]. In these works, several resonances have been analyzed and compared to experimental states, such as the S-wave states with charm C = 1, 2, 3 [81,83] together with C = −1 states [82]. Also this scheme has been recently extended to incorporate the bottom degree of freedom [88] in order to study the nature of the newly discovered Λ b (5912) and Λ * b (5920) resonances [89] as possible molecular states.
In this paper we aim at continuing those studies on dynamically-generated baryon resonances using HQSS constraints. We will discuss extensively the details of the model and how heavy-quark spin symmetry is implemented. The model respects spin-flavor symmetry in the light sector and HQSS in the heavy sector, and it reduces to SU (3) WT in the light sector respecting chiral symmetry. Moreover, we will focus on the dynamical generation of hidden charmed states. The coupled-channels in the hidden charm sectors are characterized by containing equal number of c-andc-quarks. As we shall discuss, HQSS does not mix sectors with different number of c-orc-quarks. Thus, this model has a symmetry SU(6) × HQSS, with HQSS = SU c (2) × SUc(2) × U c (1) × Uc(1). We will pay a special attention to the charmless (C = 0) and strangeless (S = 0) sector. Recent works [74][75][76][77] predict the existence of a few nucleon-like states with masses around 4 GeV which result from the baryon-meson scattering in this hidden charm sector. In this paper we will analyze these results within our model, and predict the existence of several odd parity ∆-and N -like bound states with various spins. These resonances can be organized in heavy-quark spin multiplets, which are almost degenerate in mass and that can be subject to experimental detection.
The predicted new resonances might be subject to experimental detection in the forthcomingPANDA/FAIR experiment. If confirmed, they definitely cannot be accommodated by quark models with three constituent quarks.
The paper is organized as follows. In Sec. II we present the WT interaction implementing heavy-quark spin symmetry, and analyze the different hidden charm sectors, classified according to their charmed content. In Sec. III we introduce the unitarized coupled-channels approach used throughout this work. Sec. IV is devoted to present our results and in Sec. V we summarize the conclusions. In Appendix A we give details for the construction of the meson and baryon tensors and the computation of the different matrix elements of the interaction. The tables of the interaction matrices for the different baryon-meson channels are collected in Appendix B.

II. THE MODEL
A. Weinberg-Tomozawa interaction The theoretical model we use has been developed in previous works for baryon-meson sectors involving light and/or heavy quarks, but not for those with hidden charm, so we devote this section to fully specify the model.
The guiding principle is to blend several well established hadronic symmetries in a model as simple as possible. Specifically, to comply with chiral symmetry, SU(N F ) L × SU(N F ) R for N F flavors, we require the interaction to reproduce the Weinberg-Tomozawa (WT) Hamiltonian [90][91][92], a contact S-wave interaction, when light pseudoscalar mesons are involved. The low energy interaction of soft pseudo-Nambu-Goldstone bosons of the spontaneously broken chiral symmetry off any (flavored) target takes the WT universal form 1 where f is the decay constant of the pseudoscalar meson (∼ 93 MeV), and J i P , J i T are the SU(N F ) group generators (with the standard normalization f ikl f jkl = N F δ ij , where f ijk are the structure constants) for pseudoscalar meson and target, respectively. Further, K(s) = k 0 + k ′0 represents the sum of the incoming and outgoing energies of the meson. In the center-of mass frame (CM) where √ s is the total CM energy, M the mass of the target, and m the mass of the pseudo-Nambu-Goldstone meson. V WT is the tree level on-shell interaction. The normalization we use is such that the corresponding T -matrix for elastic scattering is related to the scattering amplitude by where k is the CM momentum. 2 For three flavors and baryons in the 1 2 + (nucleon) octet, the Hamiltonian density of the WT interaction takes the form (we assume exact SU(3) symmetry for simplicity) [48] where φ and B are the meson and baryon matrices in the adjoint representation of SU(3). On account of the trace cyclic property, and neglecting the meson momentum, the WT interaction can be recast as From the symmetry group point of view, the interaction in Eq. (2.5) is, schematically, where adjoint and singlet refers to the adjoint and singlet representations of the flavor group. This corresponds with the structure in Eq. (2.1). M , M † , B and B † refer to the matrices of annihilation and creation operators of meson and baryons (see below).
It is worth noticing that the eigenvalues of the relevant operator in Eq. (2.1), 2J i P J i T , can be written using the quadratic Casimir operator of SU(N F ) (see e.g. [96]) where µ, µ P and µ T are, respectively, the SU(N F ) irreducible representations (irreps) of the system, the Nambu-Goldstone boson (i.e., the adjoint representation) and the target. We use the normalization C 2 (adjoint; N ) = N for SU(N ). In our convention a positive eigenvalue indicates repulsion and a negative one attraction.
B. Spin-flavor extended Weinberg-Tomozawa interaction Next we turn to spin-flavor (SF) symmetry, SU(2N F ), [97][98][99]. This symmetry has been phenomenologically successful in the classification of lowest-lying hadrons as well as in uncovering regularities present in the masses and other hadron properties [100,101]. This is particularly true for baryons, a fact that can be understood from the large N C (number of colors) limit of QCD. In that limit SF becomes exact for the baryon sector [102]. As for mesons, the lowest-lying states can also be classified quite naturally according to SF multiplets, but the symmetry works worse for the meson spectrum. A prime example of this is provided by the pion and rho mesons. They belong to the same multiplet of SU (6) and this would require these two mesons to be approximately degenerated in mass. Also, the pion is a collective state identified as the pseudo-Nambu-Goldstone boson of the spontaneously broken approximate chiral symmetry, whereas the rho meson mass fits well with a pair of constituent quark-antiquark. Vector dominance also suggests that the rho meson should belong to the same chiral representation of the vector current (8, 1) + (1,8), which is different from the chiral representation of the pion, (3, 3 * ) + (3 * , 3). The apparent conflict was solved by Caldi and Pagels in two insightful papers [103,104], where a number of related puzzles are clarified. These authors noted that chiral and SF symmetries are compatible, as they can be regarded as subgroups of a larger symmetry group, SU(2N F ) L × SU(2N F ) R , actually a realization of the Feynman-Gell-Mann-Zweig algebra [105]. In their solution, the SF extended chiral symmetry is spontaneously broken, the rho meson being a dormant Goldstone boson of this breaking. The collective nature of the rho meson has been confirmed in lattice QCD [106]. As it is well known, exact SF invariance is not compatible with exact relativistic invariance [107]. In the Caldi-Pagels scenario, vector mesons acquire mass through SF-breaking relativistic corrections which restore Poincaré invariance.
While not an immediate consequence of the QCD Lagrangian, SF symmetry emerges in some limits, such as large N C for baryons, as already noted, and partially also in heavy quark limits. We will turn to this point below. The lack of exact relativistic invariance is not unusual in other treatments involving hadron or quark interactions to form new hadrons, either bound states or resonances. A good example would be the successful "relativized quark model" of Isgur and coworkers [108,109]. Similarly, in our approach the breaking of relativistic invariance is very mild: the spin is treated as just an internal label (as another kind of flavor) and our fields are effectively spinless as regards to the kinematics. So we have fully relativistic kinematics with SF as a purely internal symmetry. This entails the following. Because angular momentum can be transferred between orbital and spin components, and spin is not conserved under Lorentz transformations, a strict relativistic treatment requires fields with different spin to behave differently. In turn this yields differences in the off-shell propagators for each spin and this breaks strict SF symmetry. To the extent that we consider on-shell particles the various fields behave in the same way, and only the off-shell baryon-meson propagator (or loop function) would depend on the spin of the particles involved. We disregard this effect. 3 On the other hand, because we consider pure S-wave interactions, the spin is separately conserved, and also, near threshold, the case of interest to us, the specifically relativistic properties of the spin become irrelevant. It should be noted that even approaches with formally relativistic Lagrangians are in practice subject to simplifying approximations in vertices and propagators which break exact relativistic invariance without relevant phenomenological implications. Also we remark that SF invariance is just our starting point for modeling the interaction. Modifications will be introduced below to account for other established properties of QCD, and more importantly, we use physical values for hadron masses and meson decay constants in our kinematics.
The compatibility between SF and chiral symmetries implies that the WT interaction can be extended to enjoy SF invariance, and this can be done in a unique way [84]. For the on-shell vertex the extension is simply where J i M and J i B are the SU(2N F ) generators on mesons and baryons. Mesons consists of 0 − (P ) and 1 − (V ) lowest-lying states, while baryons contain 1 2 + (B) and 3 2 + (B * ) lowest-lying states. When this interaction is restricted to the sector P B → P B it reproduces the standard WT off B targets. Its SF extension automatically yields the standard WT for P B * → P B * (hence the compatibility between the two symmetries). Additionally, the extended interaction provides contact S-wave vertices for V B → V B, V B * → V B * , P B ↔ V B, P B ↔ V B * , P B * ↔ V B, P B * ↔ V B * , and V B ↔ V B * . As we have tried to argue above, these new vertices are well defined predictions of an approximate emergent symmetry of hadrons. So we adopt them as our starting point to describe interactions involving vector mesons.
The Hamiltonian corresponding to the vertex in Eq. (2.8) can be written for any number of flavors and colors N C [85]. For the physical case N C = 3 The indices A, B, . . ., denote spin and flavor, and so they take 2N F values. Φ A B (x) is the meson field, a 2N F × 2N F Hermitian matrix which contains the fields of 0 − (pseudoscalar) and 1 − (vector) mesons. This matrix is not traceless; for later convenience it includes the SU(2N F ) singlet meson (the mathematical η 1 ). The contribution of η 1 to Φ is proportional to the identity matrix and so it does not couple in H sf WT . The normalization of Φ(x) is such that a mass term (with a common mass m for all mesons) would read 1 2 m 2 tr(Φ 2 ). B(x) is the baryon field. B ABC is a completely symmetric tensor, that is, in the irrep [3] of SU(2N F ). It has 56 components for N F = 3, and 120 components for N F = 4, and contains the lowest-lying baryons with J P = 1 2 + and The Hamiltonian H sf WT has precisely the same structure displayed in Eq. (2.6), this time for the SF group SU(2N F ). 4 The predictions of the SF extended WT model for N F = 3 have been worked out in [87] for baryonic resonances, and in [110] for the mesonic version. Applications involving charm have been given in [81][82][83].
Before closing this subsection, we note that the eigenvalues of the relevant operator 4J i M J i B in Eq. (2.8) can also be written using the quadratic Casimir operator of SU(2N F ) sectors involving charm. At the same time, new specific symmetries of non relativistic type arise when heavy quarks are involved [78][79][80]. In the heavy quark limit the number of charm quarks and the number of charm antiquarks are separately conserved. This implies a symmetry U c (1) × Uc(1). Likewise, the terms in the QCD Hamiltonian which depend on the heavy quark or antiquark spin are suppressed, being of order 1/m h , where m h is the mass of the heavy quark. Therefore, in the heavy quark limit, arbitrary rotations of the spin carried by the c quarks and, independently, of the spin carried by thec antiquarks, would leave unchanged the energy of the hadronic state. 5 This implies a symmetry SU c (2) × SUc(2) in the heavy quark limit. These invariances are aspects of heavy-quark spin symmetry (HQSS). In what follows we refer to SU c (2) × SUc(2) × U c (1) × Uc(1) as the HQSS group. The approximate HQSS reflects on the hadronic spectrum and for charm it has a level of accuracy similar to that of flavor SU(3). Taking HQSS into account implies that the model described in the previous subsection, the SF extended WT or just SU(8)-WT model (for four flavors), has to be slightly modified. In order to keep the model simple, we will impose exact HQSS on it. The alternative would be to introduce instead 1/m h suppressions in some amplitudes, but such improvement is beyond the scope of the present work.
First, it should be noted that SF by itself already guarantees HQSS in many sectors. Consider for instance, the couplings involving the channels N D and N D * . These channels are related through HQSS since there should be invariance under rotations of the c quark spin (leaving the light quarks unrotated), and this mixes D and D * . But the same invariance is already implied by SF, which requires symmetry under independent rotations of spin for each flavor separately. The only cases where SF does not by itself guarantee HQSS is when there are simultaneously c quarks andc antiquarks: SF implies invariance under equal rotations for c andc, but HQSS requires also invariance when the two spin rotations are different.
To be more specific, let us consider baryon-meson channels, and let N c be the number of c quarks and Nc the number ofc antiquarks. N c ranges from 0 to 4, and Nc from 0 to 1. SF guarantees HQSS in the sectors (N c , Nc) = (0, 0), (0, 1), (1, 0), (2, 0), (3, 0), (4, 0), but not in the sectors (1, 1), (2, 1), (3, 1), (4,1). As compared to the former sectors, the latter ones contain extra cc pairs. For the present discussion, we refer collectively to these sectors as sectors with "hidden charm", regardless of whether they have net charm or not. The hidden charm sectors are the main subject of the present work. We note that, for S-wave interactions (the ones of interest here), even SU(6) SF, rather than SU (8), is sufficient to guarantee HQSS in the sectors without hidden charm: a rotation of the single heavy quark (or antiquark) can be produced by a light sector rotation followed by a global rotation, without changing the energy. In order words, in those sectors and for S-wave, any SF invariant interaction enjoys HQSS automatically.
It is perfectly possible to write down non trivial models enjoying simultaneously SU(8) and HQSS invariances (namely, by requiring SU q (8) × SUq(8)) but they would not reduce to WT in the light sector. Concretely SU(8)-WT conserves C = N c − Nc but not N c and Nc separately. Of course, one could impose this by hand, but it is automatically taken care of by our modified interaction below (Eq. (2.22)). Also, the restrictions of SU(8)-WT to the sectors (N c , Nc) = (1, 1), (2, 1), (3, 1), (4, 1) turn out to violate HQSS.
In order to implement HQSS in the model let us analyze its content. We extract the trivial kinematic part and work directly in the space with only spin and flavor as degrees of freedom. Let This operator can be written in terms of meson and baryon operators [81,85], and it contains two distinct mechanisms which stem from expanding the meson commutator in Eq. (2.9), Here M A B and B ABC are the annihilation operators of mesons and baryons, respectively, with ABC is a completely symmetric tensor. They are normalized as Note that in the SU(8)-WT model, the η 1 (SU(8) singlet meson) does not couple and could be ignored, however, this meson has to be present in the corrected interaction since it mixes with the other mesons under HQSS. Schematically, representing the quark and antiquark operators by So, an upper index in M or B represents the SF of a quark to be annihilated, whereas in M † it represents that of an antiquark to be created. Likewise, a lower index in M † or B † represents the SF of a quark to be created while in M it represents that of an antiquark to be annihilated. From this identification it is immediate to interpret the two mechanisms H ex and H ac in terms of quark and antiquark propagation. The two mechanisms involved, H ex and H ac are displayed in Fig. 1. In H ex (exchange) the quark with spin-flavor A is transferred from the meson to the baryon, as is the quark with label B from the baryon to the meson. On the other hand, in H ac (annihilation-creation) an antiquark with spin-flavor B in the meson annihilates with a similar quark in the baryon, with subsequent creation of a quark and an antiquark with spin-flavor A. In both mechanisms the quarks or antiquarks C, D and E are spectators from the point of their spin-flavor (the ubiquitous gluons are not explicited). Also in both mechanisms effectively a meson is exchanged. In passing we note that the OZI rule is automatically fulfilled as regards to the exchanged meson. OZI rule violating mechanisms would be of the type depicted in Fig. 2 and are not present in WT. It appears that H ac can violate HQSS when the annihilation or creation of qq pairs involves heavy quarks, whereas H ex would not be in conflict with HQSS. This is indeed correct. To expose this fact more clearly, let us consider the symmetries of these two interaction mechanisms. Let N F = 4 and let U be a matrix of SF SU (8). Upper indices transform with U † and lowers indices with U Therefore (with obvious matrix/tensor notation) The indices are so contracted that H ex and H ac are both invariant under these SU (8) transformations. However, HQSS requires also invariance when the charm quark and the charm antiquark receive different rotations. To examine this, let us consider the transformation under U ∈ SU q (8) andŪ ∈ SUq(8), i.e., different transformations for quarks and antiquarks (previously U =Ū ). In this case and therefore Clearly, the mechanism H ex , which depends on the combination M M † , is still invariant under this larger group SU q (8) × SUq(8). 6 It certainly preserves SF and HQSS. On the other hand, H ac depends on the combination M † M which transforms withŪ , while BB † transforms with U . H ac is SF invariant (U =Ū ) but not HQSS invariant. A simple solution to enforce HQSS with minimal modifications is to remove just the offending contributions in H ac , which come from creation or annihilation of charm quark-antiquark pairs. This implies to remove the interaction when the labels A or B are of heavy type in H ac . 7 That is, we adopt the following modified version of the H ac mechanism The indices with hat are restricted to SU (6). By construction N c and Nc are exactly conserved in H ′ ac . Also SU c (2)×SUc(2) is conserved: when U andŪ act only on the heavy sector M †Â C M CB and BB DE B †Â DE are unchanged. So HQSS is preserved. On the other hand, when U =Ū and this matrix acts on the light sector, H ′ ac is unchanged, so it enjoys SU(6) symmetry. Exact SF SU(8) and flavor SU(4) is no longer present. Presumably this breaking of SU(4) is comparable to the breaking through the kinematics due to the substantially heavier mass of the charmed quark.
To summarize, our model (in all sectors) is given by This model fulfills some desirable requirements: i) It has symmetry SU(6) × HQSS, i.e., SF symmetry in the light sector and HQSS in the heavy sector, the two invariances being compatible. ii) It reduces to SU(6)-WT in the light sector, so it is consistent with chiral symmetry in that sector. And, iii) is a minimal modification that preserves simplicity and does not introduce new adjustable parameters.

D. The model in the various sectors
We can analyze the model in the different (N c , Nc) sectors, which as already noted, do not mix due to HQSS. In all the sectors without hidden charm, namely, (N c , Nc) = (0, 1), (0, 0), (1, 0), (2, 0), (3, 0), (4, 0), H ′ ac produces the same amplitudes as H ac when the latter is restricted to the corresponding sector. Indeed these interactions vanish unless the state contains a quark-antiquark pair with quark and antiquark of the same type. In the absence of hidden charm, the pair must be light and in this case the two operators produce the same result. This is consistent with our previous observation that, when there are only heavy quarks or heavy antiquarks but not both, SF already implies HQSS. So in all these sectors, our model produces the same amplitudes as SU(8)-WT after removing channels involving hidden charm. This observation has been applied in [81][82][83]88].
It is noteworthy that, in the sectors (N c , Nc) = (0, 1) and (4, 0), corresponding to C = −1 and C = 4, H ′ ac = H ac = 0 as they do not contain light quark-antiquark pairs. Also these two sectors cannot couple to any other (N c , Nc) sector in the baryon-meson case. Therefore for them, our model coincides directly with SU(8)-WT.
Let us turn now to the sectors with hidden charm. These are (N c , Nc) = (1, 1), (2, 1), (3, 1), (4,1). For all these sectors H ′ ac vanishes. The reason is that in these sectors the relevant quark-antiquark pair (quark and antiquark with equal labels) is necessarily of heavy type, and such amplitude has been removed from H ′ ac . (Note that H ac does not vanish in these sectors.) So for the hidden charm sectors H ′ WT reduces to the exchange mechanism H ex . The interaction is effectively H ex for the four hidden charm sectors and also for C = −1 and C = 4. This has the immediate consequence that the interaction H ′ WT has only two eigenvalues, namely −2 (attractive) and 6 (repulsive), in those sectors. 8 This follows from the fact that H ex has a large (accidental) symmetry group, SU q (8) × SUq (8), which produces only two large multiplets of degenerated states in the baryon-meson coupled-channels space. Under SU q (8) × SUq(8) the baryons fall in the irrep (120, 1) (120 being the symmetric representation of three quarks with 8 possible spin-flavor states). 9 Likewise, the mesons belong to (8, 8 * ) (being qq states). Therefore, the baryon-meson states form two SU q (8) × SUq(8) multiplets: These two eigenvalues are also present in the original SU(8)-WT interaction (namely, λ 2520 = 6 and λ 4752 = −2 from Eq. (2.11)) since the two interactions coincide for C = −1 or C = 4. It can also be noted that under SF SU (8), these multiplets reduce as follows Of course, these are the same SU(8) irreps obtained from 120⊗ 63 (baryon-meson except η 1 ) and 120⊗ 1 (baryon-η 1 ). Therefore, the interaction can be written using SU(8) Clebsch-Gordan coefficients by means of 11 H ex = 6(P ′ 120 + P 2520 ) − 2(P 120 + P 168 + P 4752 ). (2.26) We emphasize that the large multiplets (330, 8 * ) and (630, 8 * ) are not realized in our model. First of all, they contain sectors without hidden charm, for which the interaction does not reduce to the exchange mechanism H ex . And second, the eigenvalues 6 and −2 refer only to the driving operator 4J i M J i B . The vertex V in Eq. (2.22) depends also on hadron masses and meson decay constants, for which we use physical values in V and in the propagators. The values that we use for these magnitudes are collected in Table II of [83].

E. Analysis of the hidden charm sectors
Here we consider hidden charm sectors with C = 0, 1, 2, 3, i.e. Nc = 1 and N c = 1, 2, 3, 4, respectively. We want to classify the possible states under the symmetry group SU(6) × HQSS, with HQSS = SU c (2) × SUc(2) × U c (1) × Uc(1). Since in the hidden charm sectors there is exactly one heavy antiquark, it is not necessary to specify the irrep of the factor SUc(2) × Uc(1) and we can use the notation R 2Jc+1,C for the irreps of SU(6) × HQSS, R being the SU(6) irrep of the light sector, C the charm quantum number and J c the total spin carried by the one or more c quarks (not including the spin of thec antiquarks). The corresponding dimension is dim R × (2J c + 1) × 2 (the last factor coming from the two possible spin states of thec).
Subsequently, we study the breaking of light SF down to SU(3) × SU(2) keeping HQSS, and enumerate the number of attractive channels in each (C, r, J) sector, where r is the SU(3) irrep and J the total spin.
In practice we will assume exact isospin and spin SU(2) I × SU(2) J , as well as conservation of each flavor, but not exact SU(3) and HQSS, for the baryons and mesons composing the coupled-channels space. Therefore the sectors are labeled by (C, S, I, J), S being the strangeness quantum number and I the isospin. This implies a further breaking of each (C, r, J) sector into (C, S, I, J) subsectors.

C = 2
For the hidden charm sector with C = 2 there are two baryon-meson quark structures, (ℓcc)(cc) and (ccc)(ℓc). The space has dimension 120, with the following reduction and eigenvalues: The SU(3) × HQSS multiplets are as follows: In this case the SU(6) × HQSS multiplets are not reduced further under SU(3) × HQSS. Calling λ 1 the eigenvalue of the repulsive 6 4,2 , λ 2 that of the attractive 6 4,2 , and λ 3 the one of 6 2,2 , yields the following spectra for the three (C, r, J) sectors: This produces the following expected maximum number of states: three in 3 2 , three in 3 4 and one in 3 6 It can be noted in the present case, C = 2 with hidden charm, the assumption of SU(6) invariance does not add anything (does not reduce the number of parameters) on top of that of SU(3). The reason is that here HQSS automatically implies SF: there is just one light quark and rotations of it can be produced by combining global rotations with heavy quark rotations.

C = 3
For C = 3 there is just one quark structure, (ccc)cc. The dimension is 16. The SU(6) × HQSS reduction is with eigenvalues The SU(3) × SU(2) J reduction is: where λ 1 denotes the eigenvalue corresponding to 1 5,3 , and λ 2 that of 1 3,3 . So no states will be produced in 1 6 , and up to one state is expected in 1 2 and 1 4 . Some of the results of this subsection are summarized in Table I.

F. Lagrangian form of the interaction
In the strict heavy quark limit, the total spin J, and the separate spins of the light, heavy quarks and heavy antiquarks subsystems are conserved, when only S-wave interactions are considered. Moreover, the matrix elements of QCD Hamiltonian depend only on the spin-flavor quantum numbers of the light degrees of freedom. Each n-fold degenerated SU(6) or SU(3) multiplet implies the specification of n(n + 1)/2 parameters which are coupling constants of the corresponding operators present in the interaction. From the previous analysis it follows that the number of independent operators in the hidden charm sectors with C = 0, 1, 2, 3 is 4, 5, 4, 2, respectively, for a generic interaction if SU(6) × HQSS invariance is assumed and 12, 10, 4, 2 if the symmetry is reduced to SU(3) × HQSS.
In what follows we will focus on the sector with hidden charm and C = 0. For this sector we will write down the most general (modulo kinematical factors) S-wave Lagrangian consistent with SU(3)×HQSS for the baryon-meson coupledchannels space. This Lagrangian contains 12 operators and our model gives well defined values for the corresponding coupling constants. We should note that we actually compute the matrix elements of our interaction using directly the previous expressions, either in terms of projectors using Clebsch-Gordan coefficients, or of hadron creation and annihilation operators in spin-flavor space, by taking Wick contractions. Nevertheless, writing the interaction in field-theoretical Lagrangian form is highly interesting in order to make contact with alternative approaches in the literature.
For this purpose it is convenient to organize the hadrons forming multiplets of HQSS into building blocks with well defined HQSS transformation properties [80,[112][113][114]. Specifically, consider a HQSS doublet composed of pseudoscalar meson and vector meson with one heavy quark (e.g. D and D * ). Let M (c) and M (c) µ be the corresponding fields, then (2.43) As usual (±) represent the positive and negative frequency part of the fields, corresponding to purely annihilation for (+) and purely creation for (− where v µ is the heavy hadron velocity (v 2 = 1). We use Bjorken and Drell [115] conventions for the Dirac gammas.
For the hadron fields we use the conventions of [111] and this fixes the relative sign between pseudoscalar and vector (see Appendix A). Likewise, for a HQSS meson doublet with one heavy antiquark For a HQSS meson doublet with one heavy quark and one heavy antiquark (e.g., η c and J/ψ): For a HQSS baryon doublet with exactly one heavy quark (e.g., Σ c and Σ * c ), Under HQSS rotations these fields transform as follows Here S c and Sc are the matrices in Dirac space representing the c orc spin rotation and satisfy S † c,c = γ 0 S −1 c,c γ 0 . All other hadrons without heavy quarks nor antiquarks are HQSS singlets. They have complete fields (positive and negative frequency parts) and are denoted without boldface type.
The hadrons are also organized into SU(3) multiplets. We use fields with labels a, b, c, . . . = 1, 2, 3 (or up, down, strange) in the fundamental or anti-fundamental representations of SU(3), in such a way that For C = 0 with hidden charm, the following SU(3) multiplets are needed HereD 0 represents the HQSS doublet formed byD 0 andD * 0 , etc.
In this case the members of the SU(3) multiplet are HQSS singlets. Further we define ψ as the SU(3) singlet and HQSS doublet containing (η c , J/ψ). In addition, the following light baryon multiplets appear: and with J P = 3/2 + , ∆ abc µ , a Rarita-Schwinger field and a symmetric tensor normalized as The relative phases of all fields here are standard with respect to the conventions adopted in [111] for the rotation, flavor and spin-flavor groups. So for instance, (Ξ + c , Ξ 0 c ) is a standard isospin doublet and (Σ + , Σ 0 , Σ − ) is a standard isovector. For SU(3) (and SU(4)) the convention in [116] is used instead of that in [117]. 14 Regarding parity, we note that the hadrons with spin-parity 1 − or 1/2 + have normal parity, whereas those with 0 − or 3/2 + have abnormal parity. So Σ b a , ψ, Ξ ca , andD a are true tensors while ∆ abc µ , and Σ µ c ab are pseudotensors. Also, Σ b a , Ξ ca , ∆ abc i , and Σ i c ab have large upper components while ψ andD a have large off-diagonal blocks in Dirac space. The γ 5 matrices introduced to preserve parity fit in this scheme.
Next, we write down the 12 most general operators allowed by SU(3)× HQSS in the baryon-meson coupled-channels space, in S-wave and preserving parity. The operator acts on the mesons only and it is introduced in order to produce the correct kinematical dependence in the amplitudes.

A. Unitarization and renormalization scheme
As previously discussed, the baryon-meson interaction is mediated by the extended WT interaction of Eq. (2.22) that fulfills SU(6) × HQSS and it is consistent with chiral symmetry in the light sector. The final expression for the potential to be used throughout this work is where k 0 i and k ′ j 0 are the CM energies of the incoming and outgoing mesons, respectively, and f i and f j are the decay constants of the meson in the i-channel and j-channel. 15 We use the hadron masses and meson decay constants compiled in Table II of Ref. [83]. In particular, f J/Ψ is taken from the width of the J/Ψ → e − e + decay, that is, f J/Ψ = 290 MeV and we set f ηc = f J/Ψ , as predicted by HQSS and corroborated in the lattice evaluation of Ref. [118]. The D ij are the matrix elements of H ′ WT , Eq. (2.22), for the various hidden charm CSIJ sectors previously discussed. Those for the strangeless hidden charm C = 0 case, for which H ′ WT = H ex , and that we will discuss in what follows are given in Appendix B. 16 In order to calculate the scattering amplitudes, T ij , we solve the on-shell Bethe-Salpeter equation (BSE), using the matrix V CSIJ as kernel: where G CSIJ is a diagonal matrix containing the baryon-meson propagator for each channel. Explicitly with µ CSI = m 2 th + M 2 th , where m th and M th , are, respectively, the masses of the meson and baryon producing the lowest threshold (minimal value of m th + M th ) for each CSI sector, independent of the angular momentum J. This renormalization scheme was first proposed in Refs. [65,66] and it was successfully used in Refs. [56,81,83]. A recent discussion on the regularization method can be found in Ref. [60].
The dynamically-generated baryon resonances appear as poles of the scattering amplitudes on the complex energy √ s plane. One has to check both first and second Riemann sheets. The poles of the scattering amplitude on the first Riemann sheet that appear on the real axis below threshold are interpreted as bound states. The poles that are found on the second Riemann sheet below the real axis and above threshold are identified with resonances 17 . The mass and 15 As compared to our previous work of Ref. [83], we have • and, also for this latter reason, moved the (E + M )/(2M ) factors included in the potential used in [83] to the definition of the loop function in Eq. (3.3).
the width of the state can be found from the position of the pole on the complex energy plane. Close to the pole, the scattering amplitude behaves as The mass M R and width Γ R of the resonance result from √ s R = M R − i Γ R /2, while g j e iφj (modulus and phase) is the coupling of the resonance to the j-channel.

B. Symmetry breaking
As it was already pointed out in the subsection II E, we classify states under the symmetry group SU(6) × HQSS, and consider the breaking of the light SF SU(6) to SU(3) × SU J l (2). Subsequently we break the SU(3) light flavor group to SU(2) isospin symmetry group, preserving the HQSS, and finally we break the HQSS. Thus, we assume exact isospin, total spin and flavor conservation. The symmetry breaking is performed by adiabatic change of hadron masses and meson weak decay constants, as it was previously done in Ref. [83]. At each symmetric point, the hadron masses and meson decay constants are averaged over the corresponding group multiplets. Further, we introduce three parameters, x, x ′ and x ′′ that are changed from 0 to 1, to gradually break the symmetry from SU(6) × HQSS down to SU(3) × HQSS, then to SU(2) × HQSS, and finally down to SU(2) isospin, respectively: In this way we can assign SU(3) and SU(6) representation labels to each found resonance, and also identify the HQSS multiplets. We will show below a diagram (Fig. 3) with the evolution of the hidden charm N and ∆ pole positions as the various symmetries are gradually broken.

IV. CHARMLESS AND STRANGELESS HIDDEN CHARM SECTOR: THE N AND ∆ STATES
In this work we will only discuss results on hidden charm baryon resonances with total charm C = 0 and strangeness S = 0. Other sectors with charm different from zero will be studied elsewhere.
In this sector, we find several I = 1/2 and I = 3/2 states, which correspond to N -like and ∆-like states, respectively (here we use the same notation as in Refs. [74,75]). All these states have odd parity and different values (J = 1/2, 3/2 and 5/2) of total angular momentum. The list of coupled-channels and the corresponding coefficients D IJ ij can be found in the first six tables of Appendix B.
In this hidden charm sector and in the SU(6)× HQSS limit, we saw (Eqs. (2.27) and (2.28)) that the group structure of the HQSS-constrained extension of the WT interaction developed in this work consists of two 56 2,0 and one 70 2,0 representations. One of the 56 2,0 multiplets and the 70 2,0 one are attractive. Thus, from the decomposition in Eq. (2.29) (see also Table I), we could expect up to a total of ten N -like and seven ∆-like resonances. 18 Because of the breaking of the SU(6) × HQSS symmetry, due to the use of physical hadron masses and meson decay constants, we only find seven heavy N and five heavy ∆ states in the physical Riemann sheets. They have masses around 4 GeV and most of them turn out to be bound. The remaining missing states show up in unphysical Riemann sheets. The evolution of all states as we gradually break the symmetry from SU(6) × HQSS down to SU(3) × HQSS, then to SU(2) × HQSS, and finally down to SU(2) isospin, is depicted in Fig. 3. Thanks to this latter study, we could assign SU(6) × HQSS and SU(3) × HQSS labels to each of the predicted resonances, which are all of them collected in Tables II and IV, and could also identify two HQSS multiplets in each isospin sector.
A. N states (C = 0, S = 0, I = 1/2 ) As mentioned above, the model predicts the existence of seven heavy nucleon resonances: three states with the spin-parity J P = 1 2 − , also three states with 3 2 − sectors, and one state with J P = 5 2 − . Their masses, widths and couplings to the different channels are compiled in Table II. • J = 1/2: In this sector, there are seven coupled channels, with the following threshold energies (in MeV): • J = 3/2: In this sector, there are five coupled channels, with the following threshold energies: From the group decomposition of the SU(6) × HQSS representations, we could expect up to a maximum of five states with spin J = 1/2 (see Table I): one state from each of the two J = 1/2 octets encoded in the attractive 56 2,0 representation, and three states corresponding to the each of the 8 2 octets that appear in the reduction of the 70 2,0 representation [Eq. (2.29)]. However, the two poles related to the 56 2,0 representation appear in an unphysical Riemann sheet, at the physical point (i.e., at the point of the evolution when the hadron masses and meson decay constants attain their physical values). As it can be seen from Fig. 3, these poles dissappear from the physical sheet when we pass from the SU(3) × HQSS limit to the SU(2) × HQSS one. Indeed, we could observe how the (8 2 ) 2,0 ⊂ (56 2 ) 2,0 pole almost coincides with the threshold value of the degenerated N η c and N J/ψ channels in the first steps of this evolution until it finally disappears. On the other hand, the (8 2 ) 2,0 ⊂ (56 2 ) 2,0 pole also gives rise to an octet of J = 3/2 states (see Eq. (2.29)), which is also lost at the physical point. Thus, for J = 3/2 we are also left only with the three baryon resonances stemming from the 70 2,0 representation, one from (8 2 ) 2,0 , and two from (8 4 ) 2,0 . The J = 5/2 state is originated also from this latter multiplet.
From the above discussion, it is clear that the N -like resonances found in this work, and collected in Table II, form two HQSS multiplets. In the first one the light degrees of freedom have quantum numbers (8 2 ) 2,0 ⊂ (70 2 ) 2,0 . This multiplet is formed by the three first resonances of the table (two with spin 1/2 and third one with spin 3/2) that correspond to the blue states in Fig. 3. They only differ in how the light sector spin is coupled to the spin of the cc pair. The second HQSS multiplet corresponds to (8 4 ) 2,0 ⊂ (70 2 ) 2,0 quantum numbers for the light sector, and it consists of the four remaining states in Table II (displayed in green in Fig. 3) one with spin 1/2, two with spin 3/2, and another one with spin 5/2.
The members of each HQSS multiplet are nearly degenerate, but not totally because we also break the HQSS by the use of physical hadron masses.
A word of caution is needed here. The mass of the J = 5/2 resonance is around 4027.2 MeV. In this sector there is only one channel (Σ * cD * , with threshold equal to 4526.3 MeV), thus this state is around five hundred MeV bound. We expect our model to work well close to threshold, and therefore, in this case, interaction mechanisms neglected here and involving higher partial waves could be relevant for determining the actual properties of this resonance.
There exist previous works on hidden-charm odd-parity nucleon states, also named crypto-exotic hadronic states. These studies can be divided in two types. Namely, those based on a constituent quark description of the resonances, and those where they are described as baryon-meson bound molecules or resonating states. Some of the predictions of these other models are compiled in Table III. The baryon-meson coupled-channel calculations by Hofmann and Lutz for J P = 1/2 − in Ref. [65] and for J P = 3/2 − in Ref. [66] rely on a model of zero-range t-channel exchange of light vector mesons, based on chiral and large N C considerations, and supplemented with SU(4) input in some vertices. This model is used as the driving interaction of pseudo-scalar mesons with the J P = 1/2 + , 3/2 + baryon ground states. After solving the BSE using a renormalization scheme similar to that proposed here, some 1/2 − , 3/2 − -resonances were dynamically generated. Vector mesons in the coupled-channel space were omitted in those early studies, thus channels like Σ cD * or Λ cD * were not considered. More recently baryon-meson calculations using a hidden-gauge model have been carried out in Refs. [74][75][76] is also taken into account in [74][75][76], which leads to additional and degenerate J P = 1/2 − and 3/2 − hidden-charm nucleons. However, the J = 3/2 + baryons are not included at all, and thus some channels like Σ * cD * are excluded. The main difference among our scheme and the hidden-gauge models is the definition of the coupled-channels space. We consider simultaneously pseudoscalar meson-baryon (P B) and vector meson-baryon (V B) channels, with J P = 1/2 + and 3/2 + baryons. However in the approaches of Refs. [74][75][76] all interaction terms of the type P B → V B are neglected. Furthermore, channels with J P = 3/2 + baryons are not considered either. The potential used in [74][75][76] for the P B → P B transitions, with J P = 1/2 + baryons, is similar to that derived here. However, there exist important differences in all transitions involving vector mesons. When restricting our model to the P B → P B sector, we still do not obtain the same results as in Refs. [74][75][76]. This is mainly due to i) the use of a different renormalization scheme and, ii) the presence in these latter works of a suppression factor in those transitions that involve a t-channel exchange of a heavy charm vector meson.
However, when we use our full space, the inclusion of a similar suppression factor in our HQSS kernel is not quantitatively relevant for the dynamical generation of the resonances. Note that HQSS does not require the presence of such suppression factor. In summary, when we compare our approach with the other molecular-type ones, we observe in our model a rich structure of resonances due to the many channels cooperating to create them. Our states are much more lighter that those predicted in Refs. [74][75][76], though significantly less bound that the crypto-exotic baryons reported in Refs. [65,66].
Finally, we will pay attention to the recent work of Ref. [119]. There a constituent quark model is used to describe isospin I = 1/2 baryons with uudcc quark content. The mass spectra is evaluated with three types of hyperfine interactions: color-magnetic interaction (CM) based on one-gluon exchange, chiral interaction (FS) based on meson exchange, and instanton-induced interaction (INST) based on the non-perturbative QCD vacuum structure. The FS (CM) model predicts the lowest (highest) mass for each state. Results for the FS and CM models are displayed in Table III. In all cases, the mass predicted by the INST model (not displayed in the table) lies between the values predicted by the other two models. Our results are closer to those predicted by the FS model, specially for the lowest lying states.  Table IV  • J = 1/2: In this sector, there are four coupled channels, with the following threshold energies (in MeV):  Comparison of the whole spectrum of hidden-charm nucleons (or crypto-exotic nucleons) with odd-parity and angular momentum L = 0 predicted by our model with some results from previous models. In all cases, masses and couplings (g) to the dominant channels (when available) are shown in sequential rows. In the hidden gauge model of Ref. [76], the numerical values of the couplings are not given. In this case, we indicate with a symbol X the elements of the coupled channel space used to generate each resonance. On the other hand, in the case of the predictions of this work, in the column "Model" we give the HQSS multiplet. Besides, we have also omitted the small couplings to the N ηc and N J/ψ channels that can be seen in Table II.
We obtain three ∆(J = 1/2) states as expected from the group decomposition of the SU(6) × HQSS representations (see Table I): one state from each of the two J = 1/2 decuplets encoded in the attractive 70 2,0 representation, and a further state corresponding to the J = 1/2 decuplet that appears in the reduction of the 56 2,0 representation [Eq. (2.29)]. The evolution of the corresponding poles is shown in Fig. 3.
The pole that corresponds to (10 4 ) 2,0 ⊂ 56 2,0 (light magenta circles in Fig. 3) has a mass quite close to the ∆η c and ∆J/ψ degenerated thresholds, between the SU(6) × HQSS and the SU(2) × HQSS symmetric points. Later, while moving to the SU(2) isospin symmetric point, the spin 1/2 ∆ resonance keeps having a mass close to the ∆J/ψ threshold, and ends up with a final mass of 4306.2 MeV (the ∆J/ψ threshold is at 4306.9 MeV). However, the spin 5/2 and the two spin 3/2 states, that are also originated from this (10 4 ) 2,0 ⊂ 56 2,0 pole, essentially disappear. One of the J = 3/2 states still shows up as a cusp very close to the ∆J/ψ threshold, and it has been included in the table. The second state with spin 3/2 (light magenta triangles in Fig. 3) and the spin 5/2 one appear as small unnoticeable peaks right at the ∆η c and ∆J/Ψ thresholds, respectively.
From the discussion above, the (10 4 ) 2,0 ⊂ 56 2,0 HQSS multiplet could be incomplete. However the three ∆ states (dark magenta circles for the two J P = 1/2 − states and dark magenta triangles for the J P = 3/2 − resonance in Fig. 3) that stem from the (10 2 ) 2,0 ⊂ 70 2,0 configuration of the light degrees of freedom turn out to be quite bound. Indeed, we find binding energies of at least 250 (150) MeV in the spin 1/2 (3/2) sector. These three states, nearly degenerate, form a clear HQSS multiplet.
The models based on vector meson exchange, naturally predict a suppression factor in the baryon-meson amplitudes involving exchange of charm, from the propagator of the exchanged heavy vector meson. In the heavy quark limit, the suppression factor is of the order of 1/m H . 19 Therefore, in that limit, one expects a quenching of order M V /M D * for the charm exchanging amplitudes of those models. (Of course, the true factor for large but finite physical heavy g∆η c ∼ 0.1, g ∆J/ψ = 0.8, g Σ * cD = 0.2, g ΣcD * = 0.2, 3/2 (cusp) g Σ * cD * = 0.1 hadron masses needs not exactly coincide with this heavy quark limit estimate.) Our model is not directly based on exchange of vector mesons. Nevertheless, as commented above, we have verified that adding such suppression by hand in the charm exchanging amplitudes does not have an impact on our results. Even a factor (M V /M D * ) 2 , proposed in the literature [68] has a very small effect in the position of the resonances we find. Presumably, this is due to the fact that the relevant channels have a small coupling. An exception comes from the two very weakly bound ∆ resonances from the 56 2,0 irrep, which disappear due to the suppression of their dominant channel ∆J/ψ.

V. CONCLUSIONS
In the present work we develop a model for the interaction of lowest-lying 1/2 + and 3/2 + baryons with 0 − and 1 − mesons, including light and heavy flavors. The interaction is of zero range and it is modeled as a suitable extension of the Weinberg-Tomozawa term to include, besides chiral symmetry, spin-flavor symmetry in the light sector and heavy quark spin symmetry. These symmetries are only broken in our model to the extent that they are broken at level of the physical masses and meson decay constants. The OZI rule is also automatically implemented. Our extended WT model, Eq. (2.22), contains no adjustable parameters, although some ambiguity is present through the choice of renormalization prescription, as in all other hadronic molecular approaches. The model has been applied previously to the light sector and to charm or bottom sectors with a single heavy quark. Here we show that it admits a natural realization in sectors with hidden charm in such a way that HQSS is preserved. In particular, the spin of c quarks and the spin ofc antiquarks are separately conserved.
We have carried out a detailed analysis of the hidden charm sectors (i.e., with cc pairs) with C = 0, 1, 2, 3 and their breaking as the symmetry is lifted from SU(6) × HQSS to SU(3) × HQSS (and then to SU(2) × HQSS and SU(2) of isospin). This allows to count the expected number of bound states or resonances, and to classify them into multiplets corresponding to the various symmetries. Taking the eigenvalues λ ′ s as undetermined (free) parameters, the results of Eqs. (2.30), (2.35), (2.38) and (2.42), for the C = 0, C = 1, C = 2 and C = 3 hidden charm sectors respectively, are general. Indeed, these equations fixes the most general structure of eigenvalues that can be deduced from SU(3) × HQSS. The rest of undetermined parameters, not fixed by this latter symmetry, accounts for nondiagonal transitions between multiplets with the same SU(3) × SU J ℓ (2) SF quantum numbers for the light degrees of freedom. Further, we have translated this general discussion of the group structure allowed by SU(3) × HQSS into Lagrangian form, for the charmless hidden-charm sector. This makes the HQSS of the model explicit and it allows to compare with other models in the literature. Finally, we have found the couplings of the HQSS effective Lagrangians of Eqs. (2.57)-(2.66) for the particular case of our extended WT model. This constitutes an additional check of its compatibility with HQSS.
We have analyzed the charmless and strangeless sector, where we have dynamically generated several N and ∆ states. Actually, we predict the existence of seven N -like and five ∆-like states with masses around 4 GeV, most of them as bound states. These states form heavy-quark spin multiplets, which are almost degenerate in mass. The N states form two HQSS multiplets. The lowest one has the light quark flavor-spin content coupled to 8 2 . Since thecc pair can couple to spin S cc = 0, 1, this HQSS multiplet consists of three nucleon states with J = 1/2, 1/2, and 3/2, and masses around 3930 MeV. On the other hand, the highest HQSS nucleon-like multiplet contains four resonances with J = 1/2, 3/2, 3/2 and 5/2, and masses around 4000 MeV. In this case, these states are originated from the 8 4 SF light configuration. These two SU(3) × HQSS multiplets arise from the 70-plet of SU(6) × HQSS. There are no N physical states coming from the 56-plet. With regards to ∆ states, we find two multiples with very different average masses, because in this case they are originated from different SU(6)×HQSS representations. The ∆ multiplet coming from the (10 2 ) 2,0 ⊂ 70 2,0 irrep is formed by 3 states (J = 1/2, 1/2, 3/2) with an average mass of 4035 MeV. Besides, we find only two (J = 1/2, 3/2) ∆ resonances at the physical point out of the four states originated from the (10 4 ) 2,0 ⊂ 56 2,0 in the SU(6) limit. These two states are nearly degenerate, with a mass of 4306 MeV.
When we compare our approach with the other molecular-type ones, we observe in our model a rich structure of resonances due to the many channels cooperating to create them. Our states are much more lighter that those predicted in the hidden-gauge scheme [74][75][76], though significantly less bound that the crypto-exotic baryons reported in the zero range vector meson exchange model of Refs. [65,66]. Moreover, we have presented the first prediction for exotic hidden-charm ∆-like resonances within a molecular baryon-meson scheme.
In comparison with the quark model of Ref. [119], we find that our results are closer to those predicted by the FS hyperfine interaction discussed in Ref. [119], specially for the lowest lying states.
The predicted new resonances definitely cannot be accommodated by quark models with three constituent quarks and they might be looked for in the forthcomingPANDA experiment at the future FAIR facility.