^{1}

^{1}

^{1}

^{3}.

In recent years, the discovery in quarkonium spectrum of several states not predicted by the naive quark model has awakened a lot of interest. A possible description of such states requires the enlargement of the quark model by introducing quark-antiquark pair creation or continuum coupling effects. The unquenching of the quark models is a way to take these new components into account. In the spirit of the Cornell Model, this is usually done by coupling perturbatively a quark-antiquark state with definite quantum numbers to the meson-meson channel with the closest threshold. In this work we present a method to coupled quark-antiquark states with meson-meson channels, taking into account effectively the nonperturbative coupling to all quark-antiquark states with the same quantum numbers. The method will be applied to the study of the X(3872) resonance and a comparison with the perturbative calculation will be performed.

Constituent quark models (CQM) have been extremely successful in describing the properties of hadrons such as the spectrum and the magnetic moments. However, since the earliest days of the hadron spectroscopy, it was realized [

The components beyond the naive constituent quark model became more relevant since 2003, when the

The

A different point of view from the references mentioned above is presented in Ref. [

Some models, as our previous work in Ref. [

In an attempt to improve these caveats, we have developed a new scheme in which the contributions of the complete tower of radial excitations corresponding to a given

The paper is organized as follows. In Section

The constituent quark model used in this work has been extensively described elsewhere [

The Goldstone boson field matrix

In the heavy quark sector chiral symmetry is explicitly broken and this type of interaction does not act. However it constrains the model parameters through the light meson phenomenology and provides a natural way to incorporate the pion exchange interaction in the open charm dynamics.

Below the chiral symmetry breaking scale quarks still interact through gluon exchanges described by the Lagrangian

Lattice calculations have shown that, as far as the quarks get separated, virtual quark anti-quark pairs tend to modify the confinement potential, giving rise at some scale to a breakup of the color flux-tube [

Although over the time the procedure to incorporate new Fock components to the

In practice what is done is to assume that the first Fock component, namely, the

As stated in the introduction, this method has two important shortcomings. First of all, it focuses the study on the modification of the

For these reasons, we have developed a new scheme in which the contributions of all states are initially taken into account, being the dynamics the responsible of selecting the contribution of each bound state.

The Hamiltonian we consider

Instead of expanding the wave function of the

The meson wave functions to be used all along this work will be expressed using the Gaussian Expansion Method [

To introduce higher Fock components in the

We must notice that, although the Gaussian Expansion Method of the

In order to couple both sectors, we use the QCD-inspired

Finally, in the context of the

Now, we use Eq. (

Then, the coefficients

The previous coupled channel equations can be solved in a more elegant way through the

Using the

In the previous equation we identify

In order to find molecular states above and below thresholds in the same formalism we have to analytically continue all the potentials for complex momenta. Therefore, resonances are solutions of Eq. (

The molecular wave function is related with the

The partial decay widths can be defined through the complete S-matrix of the mix channel, as detailed in [

In order to compare the results of the proposed scheme with the perturbative one we perform a similar calculation as in Ref. [

Isospin symmetry is explicitly broken taking the experimental threshold difference into account in our equations and solving for the charged and the neutral components.

One can see in Table

| Mass [MeV] |
---|---|

| 3503.9 |

| 3947.4 |

Mass and channel probabilities for the three states in the present approach using the two values of the

| | | | | I=0 | I=1 |
---|---|---|---|---|---|---|

| | | | | | |

0.260 | | | | | | |

| | | | | | |

| ||||||

3944.58 | | | | | | |

0.218 | 3871.76 | | | | | |

3478.55 | | | | | |

Decomposition of the

| | | | | | |
---|---|---|---|---|---|---|

| | | | | | |

0.260 | | | | | | |

| | | | | | |

| ||||||

3944.58 | | | | | | |

0.218 | 3871.76 | | | | | |

3478.55 | | | | | |

One can analyze the isospin content of the

As in Ref. [

The scenario drawn by these results consists of two bare states (a molecule and a

Although not addressed in Ref. [

Whereas

Our

The obtained value for the first ratio is

In this work we present a method to coupled quark-antiquark states with meson-meson channels, taking into account effectively the nonperturbative coupling to all quark-antiquark states with the same quantum numbers. Instead of expanding the wave function of the

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This work has been funded by Ministerio de Economía, Industria y Competitividad under Contract No. FPA2016-77177-C2-2-P. Pablo G. Ortega acknowledges the financial support from Spanish MINECO’s Juan de la Cierva-Incorporación program, Grant Agreement No. IJCI-2016-28525.

^{+}

^{−}

_{b}(3

^{ PC }=1

^{−}hidden charm resonances

^{3}P

_{0}model applied to the decay of mesons into two mesons

_{c}(2

_{b}(3