$\eta_{\rm c}$- and $J/\psi$-isoscalar meson bound states in the hadro-charmonium picture

We study $\eta_{\rm c}$- and $J/\psi$-isoscalar meson bound states in the hadro-charmonium picture. In the hadro-charmonium, the four $q\bar q c \bar c$ quarks are arranged in terms of a compact charm-anticharm pair, $c \bar c$, embedded in light hadronic matter, $q \bar q$, with $q = u$, $d$ or $s$. The interaction between the charmonium core and the light matter can be written in terms of the multipole expansion in QCD, with the leading term being the $E1$ interaction with chromo-electric field ${\bf E}^a$. The spectrum of $\eta_{\rm c}$- and $J/\psi$-isoscalar meson bound states is calculated and the results compared with the existing experimental data.


I. INTRODUCTION
Recent discoveries by Belle and BESIII Collaborations of charged and neutral exotic quarkonium-like resonances, which do not fit into a traditional quarkantiquark interpretation, have driven new interest in theoretical and experimental searches for exotics. Charged states, like Z c (3900) [1,2], Z c (4025) [3], Z b (10610) and Z b (10650) [4], have similar features and must be made up of four valence quarks because of their exotic quantum numbers. There are also several examples of neutral exotic quarkonium-like resonances, the so-called X states, whose unusual properties do not fit into a quarkantiquark classification [5].
The hadro-charmonium is a tetraquark configuration, where a compact cc state (ψ) is embedded in light hadronic matter (X ) [34]. The interaction between the two components, ψ and X , 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 [51][52][53], with the leading term being the E1 interaction with chromo-electric field E a .
In the present manuscript, we calculate the spectrum of η c -and J/ψ-isoscalar meson bound states under the hadro-charmonium hypothesis. The qqcc masses are computed by solving the Schrödinger equation for the hadro-charmonium potential [34]. This is approximated as a finite well whose width and size can be expressed as a function of the chromo-electric polarizability, α ψψ , and light meson radius. The chromo-electric polarizability is estimated in the framework of the 1/N c expansion [53,59]. Finally, the hadro-charmonium masses and quantum numbers are compared with the existing experimental data. Some tentative assignments are also discussed.

II. A MASS FORMULA FOR THE HADRO-CHARMONIUM
The hadro-charmonium is a tetraquark configuration, where a compact cc state (ψ) is embedded in light hadronic matter (X ) [34]. The interaction between the charmonium core, ψ, and the gluonic field inside the light-meson, X , can be written in terms of the QCD multipole expansion [51][52][53], considering as leading term the E1 interaction with chromo-electric field E a [34,62].
The effective Hamiltonian we consider is the same describing a ψ 2 → ψ 1 transition in the chromo-electric field. It can can be written as [63] where is the chromo-electric polarizability. It is expressed in terms of the Green function G of the heavy-quark pair in a color octet state (having the same color quantum numbers as a gluon), the relative coordinate between the quark and the antiquark, r, and the difference between the color generators acting on them, ξ a = t a 1 − t a 2 . A schematic representation of a hidden-flavor ψ 1 → ψ 2 + h transition in the QCD multipole expansion approach is given in Fig. 1 In order to calculate the hadrocharmonium masses, we have to compute the expectation value of Eq. (2) on the charmonium state |ψ , i.e. the diagonal chromo-electric polarizability α ψψ , and also the diagonal matrix elements A. Diagonal chromo-electric polarizability In the following, we discuss three possible prescriptions for the diagonal chromo-electric polarizabilities, α ψψ .
2. Alternately, one can calculate the chromo-electric polarizability by considering quarkonia as pure Coulombic systems. While this is a very good approximation in the case of bb states, one may object that it is questionable in the case of charmonia.
The perturbative result in the framework of the 1/N c expansion is [53,59] α ψψ (nS) = 16πn 2 c n a 3 0 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, with α s being the QCD running coupling constant; finally, is the Bohr radius of nonrelativistic charmonium [41], with C F = As discussed in the following, the value of α ψψ (1S) gives rise to hadro-charmonium states with binding energies O(10 − 100) MeV. On the contrary, the largeness of α ψψ (2S) gives rise to unphysical states, characterized by negative masses. A possible explanation is the following: 2S are larger than 1S cc states; thus, the QCD multipole expansion, where one assumes the quarkonium size to be much smaller than the soft-gluon wave-length, is not applicable anymore.
Here, the condition singlet| ξ a ξ b |singlet = 2 3 δ ab is used, because the operator ξ a turns a singlet state into an octet one, and vice-versa (only the octet states contribute), and is the color-octet Green's function. Here, E ψ and E ν kℓ are charmonium and string-vibrational state [73,74] energies. After introducing the propagator of Eq. (8) in (7), the chromo-electric polarizability calculation essentially reduces to evaluating dipole matrix elements between quarkonium and stringvibrational states.
The product E a i E a i in Eq. (1) can be re-written using the anomaly in the trace of the energy-momentum tensor θ µν in QCD [71], where B a i is the chromo-magnetic field. If we neglect the contribution due to the chromo-magnetic fields, which is expected to be smaller than the chromo-electric one [63], Eq. (9) can be re-written as: The expectation value of the operator θ µ µ on a generic state X is given by [34] where a non-relativistic normalization for X , X |X = 1, is assumed.
C. An Hamiltonian for the hadro-charmonium The effective potential V hc , describing the coupling between ψ and X , can be approximated as a finite well [34] where is the radius of the light meson X [9]. Thus, we have: By analogy with calculations of the interaction between heavy quarkonia and the nuclear medium [62][63][64], we get a potential that is a constant square well inside the light meson X and null outside. We can estimate the order of magnitude of the strength of V hc by introducing into Eq. (14) typical values for R X and M X . If we take R X = 0.5 fm, M X = 1 GeV and α ψψ from Eq. (3), we get a potential well with a depth of the order of 250 MeV. The Hamiltonian of the hadro-charmonium system also contains a kinetic energy term, where k is the relative momentum (with conjugate coordinate r) between ψ and X , and µ the reduced mass of the ψX system. The total hadro-charmonium Hamiltonian is thus:

III. RESULTS AND DISCUSSION
Below, we calculate the spectrum of η c -and J/ψisoscalar meson bound states in the hadro-charmonium picture by solving the eigenvalue problem of Eq. The calculated hadro-charmonium spectrum is shown in Table I; here, we also try some tentative assignments to experimental X states. See [49, Table I].
The hadro-charmonium quantum numbers are shown in the third column of Table I. They are obtained by combining those of the charmonium core, ψ, and light meson, X , as  where the hadro-charmonium P -and C-parity are given by: P = (−1) L hc and C = (−1) L hc +S hc .
Finally, it is worth noticing that: I) The quantum number assignments in Table I for several states are not uni-vocal. A possible way to distinguish between them is to calculate the hadro-charmonium main decay amplitudes and compare the theoretical results with the data; II) The results strongly depend on the chromo-electric polarizability, α ψψ . Up to now, the value of α ψψ cannot be fitted to the experimental data; it has to be estimated phenomenologically. Because of this, it represents one of the main sources of theoretical uncertainty on the results; III) In the calculation of X | θ µ µ (q = 0) |X matrix elements on light mesons, X , the contributions due to the chromo-magnetic field, B a , are neglected. This may represent another source of theoretical uncertainties; IV) By combining ψ and X quantum numbers, several J P C hc configurations are obtained. Thus, once the value of the J/ψ and η c chromo-electric polarizability is measured (and thus the main source of theoretical uncertainties removed), it would be interesting to introduce spin-orbit and spin-spin corrections in order to split the degenerate configurations.