# Comment on “Possibility of Deeply Bound Hadronic Molecules from Single Pion Exchange”

###### pacs:

12.39.-x, 13.75.Lb, 14.40.Gx, 21.30.FeIn Ref. close1 deeply bound systems of a pair of two open charm mesons, for simplicity here called and , were predicted. It was assumed that one of the two constituents, here , had to have a large width, dominated by the -wave decay . Since this large width implies a large coupling to pions in the transition potential, the authors concluded, backed by a variational calculation close1 and a solution of a Schrödinger equation close2 , that in some channels the attraction in the system must be sufficiently strong to produce deeply bound states. In this comment we present arguments that those states should not exist. In short our reasoning goes as follows: if has a significant decay width, this not only means that the interaction via pion exchange should be strong, but also that in the dynamical equations for scattering, the width has to be included as well as the three–body cuts due to intermediate states. That the former effect alone will already strongly distort the resonance shape was discussed recently in Ref. finitewidth . For the case at hand here we found from an explicit calculation, which reproduces the results of Ref. close2 once the approximations of that paper are imposed, that taking into account both aforementioned effects completely removes any signal of bound states. Thus, we find that as soon as the coupling is sufficiently strong to produce a bound state it is at the same time necessarily sufficiently strong to provide the state with such a large width that it becomes unobservable. This connection is unavoidable for the interplay of the various components is a consequence of unitarity AAY .

In order to make the arguments given more quantitative we now focus on the example of a possible bound system in the isoscalar–vector channel. To visualize our findings we present the predicted invariant mass distribution assuming all three particles to emerge from a point source. To do the calculation we convolute the resulting transition matrix element with the proper three–body phase space as well as the spectral function. For the latter quantity in the phase space integration for all calculations we use the correct expression, regardless what approximations are used in the scattering equation. The parameters underlying the calculation are 2427 MeV and 453 MeV for the mass and the width of the , respectively, where the latter is consistent with the strength of the potential given in Ref. close2 . No form factors were used. In Fig. 1 we show as the dashed line the result of our calculation once all approximations of Refs. close1 ; close2 are imposed. The spectral distribution clearly shows the lowest two states as very sharp peaks. The binding energies are 227 MeV and 12 MeV, respectively, in agreement with the claims of Ref. close2 . Then we add in the imaginary part of the potential as derived, but dropped in Ref. close1 , as well as a constant width for the . This leads to the dot–dashed result in Fig. 1. As one can see, both resonance signals are completely gone.

In addition, this simplified calculation is already very close to the full result, as given by solid line, which is obtained by solving the Lippmann-Schwinger type equation for system including the full dynamics (in particular, the cut) with relativistic pions as well as the full energy dependence of the potential and of the width. Moreover, going to the full calculation does not introduce any new parameter, since all individual contributions are linked through two– and three–body unitarity. Thus, the deeply bound hadronic molecules, advocated in Refs. close1 ; close2 do not exist.

###### Acknowledgements.

Supported by the Helmholtz Association, DFG, EU HadronPhysics2, RFFI, grants NSh-4961.2008.2 & NSh-4568.2008.2, “Rosatom”, “Dynasty”, ICFPM, and FCT.## References

- (1) F. Close and C. Downum, Phys. Rev. Lett. 102, 242003 (2009).
- (2) F. Close et al., arXiv:1001.2553 [hep-ph].
- (3) C. Hanhart et al. arXiv:1002.4097 [hep-ph].
- (4) R. Aaron at al., Phys. Rev. 174, 2022 (1968).