On the existence of exotic and non-exotic multiquark meson states

To obtain an exact solution of a four-body system containing two quarks and two antiquarks interacting through two-body terms is a cumbersome task that has been tackled with more or less success during the last decades. We present an exact method for the study of four-quark systems based on the hyperspherical harmonics formalism that allows us to solve it without resorting to further approximations, like for instance the existence of diquark components. We apply it to systems containing two heavy and two light quarks using different quark-quark potentials. While QQn̄n̄ states may be stable in nature, the stability of QQ̄nn̄ states would imply the existence of quark correlations not taken into account by simple quark dynamical models. The discoveries on several fronts [1], of unusual charmonium states like X(3872) and Y (4260) and open-charm mesons with unexpected masses like D∗ sJ(2317) and D ∗ 0(2308), have re-invigorated the study of hadronic resonances. Any debate on the possible multiquark structure of meson resonances should be based on our capability to find an exact solution of the four-body problem [2]. Theoretical predictions often differ because of the approximation method used. A powerful tool to solve a few-particle system is an expansion of the trial wave function in terms of hyperspherical harmonics (HH) basis functions. In Ref. [3] a generalization of the HH formalism to study four-quark systems in an exact way was presented. Due to their actual interest and having in mind that systems with unequal masses are more promising [2], we will center our attention on QQn̄n̄ and QQ̄nn̄ states (n stands for a light quark and Q for a heavy one). We will analyze the possible existence of compact four-quark bound states using two standard quark-quark interactions, a Bhaduri-like potential (BCN) [4] and a constituent quark model considering boson exchanges (CQC) [5]. Both interactions fulfill the requirement of giving a reasonable description of meson and baryon spectroscopy. Assuming non-relativistic quantum mechanics we solve the E-mail address: [email protected] 2 On the existence of exotic and non-exotic multiquark meson states four-body Schrödinger equation. The grand angular momentum K is the main quantum number in our expansion and the calculation is truncated at some K value. Further details of the numerical method can be found in Ref. [3]. In spite of the shortcomings of the methods used to study four-quark systems, in the past, many four-quark bound states have been suggested. To analyze their stability against dissociation, parity and total angular momentum must be preserved. Additionally, C−parity is a good quantum number for cc̄nn̄ and the Pauli principle must be fulfilled in the ccn̄n̄ case. The corresponding thresholds can be computed by adding the meson masses of the dissociation channel. Four-quark states will be stable under strong interaction, and therefore very narrow, if their total energy lies below all allowed two-meson thresholds. Sometimes, results of four-quark calculations have been directly compared to experimental thresholds. In this case one could misidentify scattering wave functions as bound states. When they are referred to the thresholds within the same model, theoretical predictions do not imply an abundance of multiquark states in the data. Table 1. Energy (MeV) and probability of the different color components for the cc̄nn̄ J = 1 both for CQC and BCN models. The last rows indicate the lowest theoretical two-meson thresholds. P11 (P88) stands for the probability of singlet-singlet (octet-octet) color components.