The baryo-quarkonium picture for hidden-charm and bottom pentaquarks and LHCb $P_{\rm c}(4380)$ and $P_{\rm c}(4450)$ states

We study baryo-charmonium [$\eta_{\rm c}$- and $J/\psi$-$N^*$, $\eta_{\rm c}(2S)$-, $\psi(2S)$- and $\chi_{\rm c}(1P)$-$N$] and baryo-bottomonium [$\eta_{\rm b}(2S)$-, $\Upsilon(2S)$- and $\chi_{\rm b}(1P)$-$N$] bound states, where $N$ is the nucleon and $N^*$ a nucleon resonance. In the baryo-quarkonium model, the five $qqq Q \bar Q$ quarks are arranged in terms of a heavy quarkonium core, $Q \bar Q$, embedded in light baryonic matter, $qqq$, with $q = u$ or $d$. The interaction between the $Q \bar Q$ core and the light baryon can be written in terms of the QCD multipole expansion. The spectrum of baryo-charmonium states is calculated and the results compared with the existing experimental data. In particular, we can interpret the recently discovered $P_{\rm c}(4380)$ and $P_{\rm c}(4450)$ pentaquarks as $\psi(2S)$-$N$ and $\chi_{\rm c2}(1P)$-$N$ bound states, respectively. We observe that in the baryo-bottomonium sector the binding energies are, on average, slightly larger than those of baryo-charmonia. Because of this, the hidden-bottom pentaquarks are more likely to form than their hidden-charm counterparts. We thus suggest the experimentalists to look for five-quark states in the hidden-bottom sector in the $10.4-10.9$ GeV energy region.

The hypothesis of charmonium-nuclei bound states dates back to the early nineties. At that time, it was shown that QCD van der Waals-type interactions, due to multiple gluon-exchange, may provide a strong enough binding to produce charmonium-nuclei bound states if A 4 [29][30][31][32], where A is the atomic mass number. On studying the charmonium-nucleon systems, the interaction, though attractive [O(10) MeV], is too weak to produce a cc-N bound state. Notwithstanding, it is still unclear if a similar interaction may give rise to cc-qqq bound states if the nucleon is replaced by its radial or orbital excitations, or the charmonium ground-state by its radial excitations. These possibilities are worth to be investigated in the baryo-charmonium picture.
By analogy with four-quark hadro-charmonia [33][34][35][36][37][38][39], namely cc-qq states, the baryo-charmonium is a pentaquark configuration, where a compact cc state, ψ, is embedded in light baryonic matter, B [17][18][19]. The interaction between the two components, ψ and B, takes place via a QCD analog of the van der Waals force of molecular physics. It can be written in terms of the multipole expansion in QCD [40], with the leading term being the E1 interaction with chromo-electric field E a .
In the present manuscript, we use the baryocharmonium model to discuss the possible emergence of η c -and J/ψ-N * , η c (2S)-, ψ(2S)-and χ c (1P )-N bound states, where N is the nucleon and N * a nucleon resonance. The energies of baryo-charmonia are computed by solving the Schrödinger equation for the baryocharmonium potential [33,39]. This is approximated as a finite well whose width and size can be expressed as a function of the N (N * ) radius and the charmonium chromo-electric polarizability, α ψψ . The baryocharmonium masses and quantum numbers are compared with the existing experimental data and some tentative assignments are discussed; in particular, we can interpret the recently discovered P c (4380) and P c (4450) pentaquarks as ψ(2S)-N and χ c2 (1P )-N bound states, respectively.
Furthermore, we extend the previous calculations to the bottom sector and calculate the spectrum of bottomonium-N bound states. Our results are compatible with the emergence of 2S and 1P bottomoniumnucleon bound states, with binding energies of the order of a few hundreds of MeV. We observe that in the baryobottomonium sector the binding energies are, on average, slightly larger than those of baryo-charmonia. Because of this, the hidden-bottom pentaquarks are more likely to form than their hidden-charm counterparts. We thus suggest the experimentalists to look for five-quark states in the hidden-bottom sector in the 10.4−10.9 GeV energy region.

II. BARYO-QUARKONIUM HAMILTONIAN
The baryo-quarkonium is a particular pentaquark configuration, where five quarks are arranged in terms of a compact QQ state embedded in light baryonic matter.
The interaction between the quarkonium core, Q, and the gluonic field inside the light-baryon, B, can be written in terms of the QCD multipole expansion [40,41]. In particular, one considers as leading term the E1 interaction with chromo-electric field E [31,33], α ij being the quarkonium chromo-electric polarizability. In order to calculate the baryo-quarkonium masses, one has to compute the expectation value of Eq. (1) on |QB states. The chromo-electric field matrix elements can be calculated in terms of the QCD energy-momentum tensor, θ µ µ ≈ 9 16π 2 E 2 [42]. Its expectation value on a nonrelativistic normalized |B at rest gives the mass of this state [33], The baryo-quarkonium effective potential, V bq , describing the coupling between Q and B, can be approximated by a finite well [33,39] RB 0 where R B is the radius of B [2,43] and α QQ the quarkonium diagonal chromo-electric polarizability. Thus, we have: The kinetic energy term is where k is the relative momentum (with conjugate coordinate r) between Q and B, and µ the reduced mass of the QB system. Finally, the total baryo-quarkonium Hamiltonian is: The baryo-quarkonium quantum numbers are obtained by combining those of the quarkonium core, Q, and light baryon, B, as where the baryo-quarkonium parity is P = (−1) ℓ bq P Q P B , and ℓ bq is the relative angular momentum between Q and B. From now on, unless we indicate an explicit value, we will assume ℓ bq = 0.

III. CHROMO-ELECTRIC POLARIZABILITY
In this section, we depict two different procedures for the quarkonium diagonal chromo-electric polarizabilities.
A. Chromo-electric polarizabilities of charmonia as pure Coulombic systems There are several possible approaches for the quarkonium diagonal chromo-electric polarizability. One possibility is to calculate it by considering quarkonia as pure Coulombic systems. The perturbative result in the framework of the 1/N c expansion is [44] Here, n is the radial quantum number; c 1 = 7 4 and c 2 = 251 8 ; N c = 3 is the number of colors; g c = √ 4πα s ≃ 2.5, and α s is the QCD running coupling constant at the charm quark mass-scale; finally, is the Bohr radius of nonrelativistic charmonium [37], with color factor C F = N 2 c −1 2Nc , and m c = 1.5 GeV is the charm quark mass. A nonperturbative calculation of the chromo-electric polarizability was carried out in Refs. [45,46]. The result is where ℓ is the orbital angular momentum, β = 4 3 α s and ǫ nℓ 's are numerical coefficients, with ǫ 10 = 1.468. The use of Eqs. (8,9) or (10) provides the same result. One obtains [39] α Coul and B. Chromo-electric polarizabilities from charmonium-nucleon scattering lengths The second approach to extract charmonium diagonal chromo-electric polarizabilities is to fit them to results of charmonium-nucleon scattering lengths. The latter can be written as [47,Eq. (104)] where M N is the nucleon mass and µ the charmoniumnucleon reduced mass. The results are shown in Table I. It is interesting to observe that the calculated values of α ψψ (1S) span a wide interval, α ψψ ∈ [0.25 − 3.8] GeV −3 .
In particular, a global fit to both differential and total cross sections from available data on J/ψp scattering provides a value a pJ/ψ = −0.046 ± 0.005 fm [48], which is consistent with the value a N ηc = −0.05 fm from Ref. [31]. The corresponding value of the binding energy of J/ψ in nuclear matter is ≈ 3 MeV, which is close to the deuteron binding energy. On the other hand, the binding energy of J/ψ in nuclear matter was found to be 21 MeV for α ψψ (1S) = 2 GeV −3 [32,49], corresponding to a scattering length of a N J/ψ = −0.37 fm. An even larger value for the charmonium-nucleon scattering length was obtained by means of quenched lattice QCD calculations, a N ψ ≈ −0.7 fm [50].
Finally, we extract the values of the chromo-electric polarizabilities of 2S and 1P charmonia. That of 2S states can be estimated as four times the ratio between c 2 = 251 8 and c 1 = 7 4 . One gets The chromo-electric polarizability of 1P charmonia can be estimated by means of Eq. (10) and Ref.
[45, Table  1]. This means where ǫ 20 = 1.585 and ǫ 21 = 0.998. It is still not possible to fit α ψψ (nℓ) to the experimental data. So, as previously discussed, α ψψ (nℓ)'s have to be estimated phenomenologically. This could be one of the main sources of theoretical uncertainty on our results.

C. Chromo-electric polarizabilities of bottomonia
Finally, we calculate bottomonium diagonal chromoelectric polarizabilities by considering them as pure Coulombic systems. See Eqs. (8) and (9), where we substitute the charm-quark mass, m c , with the bottom one, m b = 5.0 GeV, and evaluate α s at the m b mass-scale. We get and Similar values are obtained by using the nonperturbative results of Refs. [45,46]. On the contrary, if we define the bottomonium Bohr radius as [17,18] a 0 = 16π we get and The chromo-electric polarizabilities of 1P bottomonia can be estimated using Eq. (15); we get and

IV. BARYO-CHARMONIA AND THE Pc(4380) AND Pc(4450) PENTAQUARKS
In this section, we give results for the binding energies of charmonium-N and N * bound states. The previous observables are computed by using the values of the charmonium chromo-electric polarizabilities from Sec. III.
The spectrum of η c -and J/ψ-N * , η c (2S)-, ψ(2S)and χ c (1P )-N bound states is calculated in the baryocharmonium picture by solving the eigenvalue problem of Eq. (6). See Table II and Fig. 1    The results strongly depend on the values of charmonium diagonal chromo-electric polarizabilities, α ψψ (nℓ). These values are not defined unambiguously, but span a wide interval. Up to now, α ψψ (nℓ)'s cannot be fitted to the experimental data; they have to be estimated phenomenologically. Because of this, they represent one of the main sources of theoretical uncertainty on our results. We have thus decided to present two sets of results for the baryo-charmonium spectrum.
In the second case, we use α Coul ψψ (1S) = 4.1 GeV −3 and α Coul ψψ (2S) = 296 GeV −3 . These values are extracted by considering charmonia as pure Coulombic systems [44][45][46]. Similar values of α Coul ψψ (1S) are obtained by fitting the 1S chromo-electric polarizability to the quenched lattice QCD result for the charmonium-nucleon scattering length of Ref. [50]. Thus, the baryo-charmonium bound states will give rise due to the interaction between N * and 1S charmonia, α Coul ψψ (2S) being too large [39]. It is worth noticing that we are able to make at least a clear assignment in the α scatt ψψ (2S) case. In particular, the P c (4380) pentaquark is interpreted as a ψ(2S)⊗N bound state with J P = 3 2 − quantum numbers. Besides, we can also speculate on assigning the P c (4450) to a χ c2 (1P )⊗N baryo-charmonium state although, in this second case, the theoretical prediction for the mass falls outside the experimental mass interval.
However, our predictions do not agree with the baryocharmonium results of Refs. [17,18], where the P c (4450) pentaquark is interpreted as a ψ(2S) ⊗ N bound state. This difference could be related to the different choices on α ψψ (2S): our values are calculated with 18 GeV −3 and theirs with 12 GeV −3 .

V. BARYO-BOTTOMONIUM PENTAQUARK STATES
Below, we calculate the spectrum of η b (2S)-, Υ(2S)and χ b (1P )-N bound states in the baryo-bottomonium picture by solving the eigenvalue problem of Eq. (6). The results are enlisted in Table III. The baryo-bottomonium quantum numbers, shown in the third column of Table  III, are obtained by means of the prescription of Eq. (7). In order to show that the binding energies of the bb − qqq system are strongly dependent on the values of the chromo-electric polarizabilities, we present two sets of results for the spectrum of this system. The results are listed on the fourth column of Table III. In the first case, we use α ΥΥ (2S) and α ΥΥ (1P ) from Eqs. (20) and (22), respectively; in the second case, the values of α ΥΥ (2S) and α ΥΥ (1P ) are given by Eqs. (17) and (21), respectively. The two sets of α ΥΥ 's values are calculated by using two different definitions of the Bohr radius of bottomonium; see Eqs. (9) and (18).
Our results for the baryo-bottomonium pentaquarks span a wide energy interval, 10.4 − 10.9 GeV. The presence of a heavier (nonrelativistic) QQ pair is expected to make the system more stable: this is why the hiddenbottom pentaquarks are more tightly bounded than their hidden-charm counterparts. See Tables II and III. Our conjecture agrees with an unitary coupled-channel model [55], and with a molecular model [11] approach. For this reason, after the recent observation of hidden-charm P c (4380) and P c (4450) pentaquarks, we suggest to experimentalists that looking for pentaquark states in the hidden-bottom sector may be essential on the analysis of new bound states. Moreover, in some cases, the baryobottomonium potential well is deep enough to give rise to a ground state plus its excited state, as can be observed in the case of η b (2S) ⊗ N and Υ(2S) ⊗ N from the lower half of Table III, though the binding energy of the excited state is just a few tens of MeV. The emergence of these excitations is a consequence of the value of the bottomonium chromo-electric polarizability. If the value of α ΥΥ (2S) is decreased from 33 to 23 GeV −3 , these excitations disappear.
Once the values of the 2S and 1P bottomonium chromo-electric polarizabilities are fitted to available experimental data, it will be interesting to discuss the possible emergence of deeply bound baryo-bottomonium pentaquarks by using more realistic values of α ΥΥ (nℓ).

VI. CONCLUSION
We adopted the baryo-charmonium model to discuss the possible emergence of η c -and J/ψ-N * , η c (2S)-, ψ(2S)-and χ c (1P )-N bound states, where N is the nucleon and N * a nucleon resonance. The energies of baryocharmonia were computed by solving the Schrödinger equation for the baryo-charmonium potential [33,39], which was approximated as a finite well whose width and size could be expressed as a function of the N (N * ) radius and the charmonium chromo-electric polarizability, α ψψ . The baryo-charmonium masses and quantum numbers were compared with the existing experimental data, so that we could interpret the recently discovered P c (4380) and P c (4450) pentaquarks as ψ(2S) ⊗ N and χ c2 (1P ) ⊗ N baryo-charmonia, respectively.
We also provided results for bottomonium-nucleon bound states, where the beauty partners of the LHCb pentaquarks, P b , were found to be more deeply bound. In some cases, the potential well describing the interaction between the bottomonium core and the baryonic matter was found to be deep enough to give rise to a ground-plus excited state.
For this reason, we believe that it is more probable to detect hidden-bottom pentaquarks than their hiddencharm counterparts. We suggest to experimentalists to look for five-quark states in the hidden-bottom sector in the 10.4 − 10.9 GeV energy region.
In conclusion, a possible way to disentangle the in-