Exotic resonances due to $η$ exchange
Exotic resonances due to exchange
School of Physics and Astronomy
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, Tel Aviv 69978, Israel
Enrico Fermi Institute and Department of Physics
University of Chicago, 5620 S. Ellis Avenue, Chicago, IL 60637, USA
The meson and several related states appear to be, at least in part, hadronic molecules in which a heavy flavored meson (such as ) is bound to another heavy meson (such as ). Although not the only effect contributing to the binding, pion exchange seems to play a crucial role in generating the longest-range force between constituents. Mesons without and light quarks (such as ) cannot exchange pions, but under suitable circumstances can bind as a result of exchange. Channels in which this mechanism is possible are identified, and suggestions are made for searches for the corresponding molecular states, including a manifestly exotic baryonic resonance decaying into .
PACS codes: 12.39.Hg, 12.39.Jh, 14.20.Pt, 14.40.Rt
The discovery more than a dozen years ago of an extremely narrow resonance, , right at the threshold, inaugurated a flurry of observations of charmonium-like and bottomonium-like resonances similarly correlated with thresholds. A number of these could be identified as possessing a significant “molecular” component, in which a heavy charmed or bottom hadron was bound to an anticharmed or anti-bottom hadron [2, 3]. When these hadrons possess light quarks, the longest-range force between them is single-pion exchange, in analogy with the deuteron which binds via exchange of pions and other light mesons [4, 5, 6, 7, 8, 9]. The question then arises as to whether a related mechanism can play a role in binding heavy hadrons which contain no quarks. In this note we identify potential channels in which exchange is the longest-range force, and can thus form bound states with quark content such as . We predict masses based on the proximity to thresholds of charmed-antistrange and anticharmed-strange pairs. Such a proximity is a widespread feature of S-wave structures .
There have been observations [11, 12, 13, 14, 15] or failures to observe [16, 17, 18] a resonance at 4140 MeV, which does not correspond to any known threshold. Both and exchange were considered in a work identifying the 4140 MeV state as a molecule , with predicted and masses highly dependent on an arbitrary cutoff parameter. Such a molecule was also considered in Ref. , where the binding was due to , , and exchange. The large binding energy in these two works is somewhat suspicious in view of the short range of these potentials. A recent work explains the 4140 MeV state as a mixture of 10% , 10% , and 80% . If the existence of the resonance at 4140 MeV is confirmed, it is likely to be due to an additional mechanism, beyond the exchange discussed here.
forbidden by symmetry.
Proximity of these two channels may lead to binding. See text.
Cannot be produced in because of kinematic mass limit.
The pseudoscalar cannot couple to a pair of scalar or pseudoscalar mesons. Thus some channels will receive a contribution to their binding from exchange, while others will not. In Table 1 we summarize possible resonances involving two mesons, with special attention to those which can be produced in decays of the form , i.e., states below about 4786 MeV. We take the masses MeV, MeV, MeV, MeV, MeV, MeV, MeV, MeV, and MeV from Ref. . Thresholds involving two mesons are compared with the and thresholds in Fig. 1.
We now discuss the sign of the forces due to exchange in some of the lowest-mass channels in which binding is possible.
(i) : This channel is analogous to if one replaces a or quark with an or quark. Hence the binding due to exchange for the combination should be of the same sign as it is for the , which is generally acknowledged as having a significant component of the combination . The range, of course, will be smaller by a factor of than it is for pion exchange. As the threshold is 36 MeV below , and just below , the most one can expect is an enhancement in the and spectra near threshold.
(ii) : The related channel was analyzed in Ref. , where it was concluded that the most attractive channel was the one with . This was a consequence of the expectation values
where the most attractive channel for a interaction is the one with the largest value of . In the present case, in which the isospin factor is absent, the most attractive channel will be that with . Thus, exchange between and should give rise to a resonance near 4224 MeV decaying to .
(iii) : The forces due to exchange will be equal and opposite for eigenstates of the matrix
in the channels (cf. the discussion of in Ref. ). The eigenstates have positive and negative , and thus . The attractive channel, with , can decay to . One would then see a resonance near 4286 MeV with decaying to . Indeed, the CDF Collaboration has evidence for a state at MeV decaying to , identified as a molecule in Refs.  and .
(iv) and : The proximity of these two channels means that mixing between them due to exchange may be possible, with an interaction of the form (3). One should then expect a resonance near 4429 MeV decaying to . The mixing will produce two eigenstates of opposite , with attractive in the channel.
(v) We have included in the discussion even though it is not as narrow as the other states, having a width of MeV. Any resonance involving it will be at least as broad, such as the predicted state around 4540 MeV with . The potential is again of the form (3), with the lower-lying eigenstate having .
(vi) Arguments similar to those in (iii) may be applied to states near 4572, 4647, 4684, and 4777 MeV. In each case exchange gives an attractive force in one or more channels with , giving resonances which can decay to .
If it turns out that exchange can indeed lead to resonances, then analogous meson-baryon resonances should also exist, by the same reasoning as in . A prerequisite is that both the meson and the baryon must be heavy, and at least one of them should not couple to pions. The simplest example is a resonance, with quark content . The relevant threshold is at 4398.6 MeV.
If such a resonance does exist, its best chance of being formed is in decay. The decay is Cabibbo favored. The mass of is 5619.5 MeV, so approximately 1221 MeV needs to be carried off, e.g., by an extra pair or, as recently suggested , by an . The resonance can decay through quark rearrangement to , with -value of approximately 186 MeV. The most promising discovery channel is then
where one looks for a resonance around 4400 MeV.
When quarks are absent, exchange indeed seems to be the longest-range single-particle-exchange force available to form hadronic molecules of two systems containing heavy quarks. It will be interesting to see if the dynamics of this formation is sufficiently sensitive to exchange that the predicted states are observed.
We thank Tomasz Skwarnicki for many helpful comments on the manuscript. The work of J.L.R. was supported in part by the U.S. Department of Energy, Division of High Energy Physics, Grant No. DE-FG02-13ER41958.
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