Abstract
Heavy hadron molecules were first theorized from a crude analogy with the deuteron and the nuclear forces binding it, a conjecture which was proven to be on the right track after the discovery of the X(3872). However, this analogy with nuclear physics has not been seriously exploited beyond a few calculations in the two- and three-body sectors, leaving a great number of possible theoretical consequences unexplored. Here we show that nuclear and heavy hadron effective field theories are formally identical: using a suitable notation, there is no formal difference between these two effective field theories. For this, instead of using the standard heavy superfield notation, we have written the heavy hadron interactions directly in terms of the light quark degrees of freedom. We give a few examples of how to exploit this analogy, e.g. the calculation of the two-pion exchange diagrams. Yet the most relevant application of the present idea is the conjecture of exotic nuclear landscapes, i.e. the possibility of few heavy hadron bound states with characteristics similar to those of the standard nuclei.
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Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: Being a theoretical manuscript, there are no specific data associated to the ideas I discuss.].
Notes
In particular this potential explains why the \(Z_c\) and \(Z_c'\) resonances appear in pairs [24], why the same happens for the \(Z_b\) and \(Z_b'\) resonances [21, 55, 86], and why we have a hidden-charm and hidden-bottom version of them [24]. Besides, this potential also leads to the prediction of a \(2^{++}\)\(D^* {\bar{D}}^*\) molecular partner of the X(3872) [68, 69], though this partner has not been experimentally observed. Recently a similar HQSS contact-range potential has been derived for the \(P \varSigma _Q\), \(P^* \varSigma _Q\), \(P \varSigma _Q^*\) and \(P^* \varSigma _Q^*\) molecules [32], which in turn has been proven to be surprisingly useful to explain the recently observed LHCb pentaquark trio [20] as hadronic molecules belonging to the same HQSS multiplet [33].
We notice in passing that as for now the only doubly heavy baryon that has been experimentally observed is the \(\varXi _{cc}^{++}\) by the LHCb [87].
For the charmed-bottom baryons, besides the \(\varXi _{bc}\) and \(\varXi _{bc}^*\) configurations in which the spin of the heavy diquark is \(S_H=1\), there is also a \(\varXi _{bc}'\) configuration in which the heavy diquark spin is \(S_H=0\). For the \(\varXi _{bc}'\) the rule is \(\langle \varXi _{bc}' | \varvec{\sigma }_L | \varXi _{bc}' \rangle = \varvec{\sigma }\).
Actually, this can be easily taken into account by writing the contact-range Lagrangian with projectors, i.e. \({\mathcal {L}} = \sum _\alpha C_\alpha (N^T P_{\alpha } N)^{\dagger } (N^T P_{\alpha } N)\), with \(\alpha \) the spin-isospin channel we are considering and \(P_{\alpha }\) a suitable projector (plus a similar expression for the light-quark subfield).
For the two-nucleon case, which are fermions, we end up with two S-wave coupling, while for the two heavy-hadron case, which can be either fermions (\(\varXi _{QQ}\), \(\varXi _{QQ}^*\)) or bosons (P, \(P^*\)), we end up with four S-wave couplings.
The reduced mass of a system of two \(\varXi _{cc}\) baryons is about twice the one of a system of two charmed mesons. The square of the axial coupling is nine times smaller. Compounded together, the conclusion is that OPE is 4–5 times weaker in the \(\varXi _{cc} \varXi _{cc}\) system than in the \(D^* D^*\) one.
Here it is interesting to notice that the \(\varXi _{cc}^{++}\) is about half the size of the proton: while the electromagnetic radius of the latter is about \(\sqrt{{\langle r^2_{e.m.} \rangle }_p} \simeq 0.83\,\mathrm{fm}\) [109], for the former we have \(0.4\,\mathrm{fm}\) according to the lattice calculation of Ref. [110].
This implies that the charming triton is closer to the universal limit than the standard triton, as its size is comparatively larger in relation to its constituents.
Notice that we are not making the explicit distinction between the three- and four-body Efimov effect that is sometimes done in the literature.
Preliminary numerical explorations are being conducted by Kirscher, König and Yang, though no definite conclusion has been reached yet.
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Acknowledgements
I would like to thank Johannes Kirscher, Sebastian König, Ubirajara van Kolck and Jerry Yang for discussions concerning the semi-unitary limit. This work is partly supported by the National Natural Science Foundation of China under Grant No. 11735003, the fundamental Research Funds for the Central Universities, and the Thousand Talents Plan for Young Professionals.
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Valderrama, M.P. Heavy hadron molecules in effective field theory: the emergence of exotic nuclear landscapes. Eur. Phys. J. A 56, 109 (2020). https://doi.org/10.1140/epja/s10050-020-00099-8
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DOI: https://doi.org/10.1140/epja/s10050-020-00099-8