Abstract
We discuss the stability of hidden and open heavy-flavor hadronic states made of either two or three mesons. References are made in passing to studies regarding two and three-body systems containing baryons. We perform a comparative study analyzing the results in terms of quark and hadron degrees of freedom. Compact and molecular states are found to exist in very specific situations. We estimate the decay width for the different scenarios: weak decays for bound states by the strong interaction, and strong decays for hadronic resonances above a decay threshold. The experimental observation of narrow hadrons lying well above their lowest decay threshold is theoretically justified.
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Notes
For instance, going from \(H_0=p^2+x^2\) to \(H_0 + \lambda x\), lowers the ground-state energy from \(E_0=1\) to \(E_0-\lambda ^2/4\), and more generally, breaking parity in \(H=H_\mathrm{even} + H_\mathrm{odd}\) gives \(E < E_\mathrm{even}\).
It is worth to note that Ref. [13] derived the same conclusion for a three-body system of particles with masses \(MM\mu \), with \(M > \mu \). For non-interacting heavy-particles and an slightly attractive mass independent interaction between the light and heavy particles, the binding energy of the three-body system increases rapidly when \(M/\mu \) augments, see Fig. 3 of Ref. [13].
Results obtained with the AL1 model [20].
Note that the vicinity of thresholds is a necessary though not sufficient condition for the existence of a resonance. See Ref. [25] for a thorough and critical analysis.
See, for example, Ref. [46] for a further demonstration of the instability of all-heavy tetraquarks \(QQ\bar{Q} \bar{Q}\) with a rigorous treatment of the few-body problem.
The binding energy reported for the \(T_{bb}\) tetraquark ranges between 90 and 214 MeV.
Note that the three-body channels with \(J=0\) or 1 would couple to two B-meson subsystems where no attraction has been reported [22, 23, 28, 54,55,56,57,58,59,60,61], whereas the \(J=3\) would not contain a two-body subsystem with \(j=1\), the quantum numbers of the \(T_{bb}\) tetraquark. The same reasoning excludes the \(I=3/2\) channels.
Although the Breit–Wigner formula is not very accurate close to threshold; however, we have explicitly checked by analytic continuation of the S-matrix on the second Riemann sheet that at low energy the width follows the expected \(\varGamma \sim E^{1/2}\) behavior, the one shown by Fig. 9.
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Acknowledgements
The authors are deeply indebted to their long-term collaborators T. F. Caramés, E. Hernández, J. -M. Richard and J. Vijande that have participated in some of the issues reviewed in this work. This work has been partially funded by COFAA-IPN (México) and by Ministerio de Ciencia e Innovación and EU FEDER under Contracts No. FPA2016-77177-C2-2-P, PID2019-105439GB-C22 and RED2018-102572-T.
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Garcilazo, H., Valcarce, A. Hidden and Open Heavy-Flavor Hadronic States. Few-Body Syst 61, 24 (2020). https://doi.org/10.1007/s00601-020-01557-1
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DOI: https://doi.org/10.1007/s00601-020-01557-1