Multiquark Hadrons - A New Facet of QCD
I review some selected aspects of the phenomenology of multiquark states discovered in high energy experiments. They have four valence quarks (called tetraquarks) and two of them are found to have five valence quarks (called pentaquarks), extending the conventional hadron spectrum which consists of quark-antiquark mesons and baryons. Multiquark states represent a new facet of QCD and their dynamics is both challenging and currently poorly understood. I discuss various approaches put forward to accommodate them, with emphasis on the diquark model.
Deutsches Elektronen-Synchrotron DESY,
D-22607 Hamburg, Germany
Keywords: Exotic Hadrons, Tetraquarks, Pentaquarks, Hadron Molecules
Ever since the discovery of the state by Belle in 2003 , a large number of multiquark states has been discovered in particle physics experiments (see recent reviews ). Most of them are quarkonium-like states, in that they have a or a component in their Fock space. A good fraction of them is electrically neutral but some are singly-charged. Examples are , , , , in the hidden charm sector, and , and , in the hidden bottom sector. The numbers in the parentheses are their masses in MeV. Of these, is a pentaquark state, as its discovery mode requires a minimal valence quark content . The others are tetraquark states, with characteristic decays, such as , , , , and . No doubly-charged multiquark hadron has been seen so far, though some are expected, such as , in the tetraquark scenario discussed below.
Deciphering the underlying dynamics of the multiquark states is a formidable challenge and several models have been proposed to accommodate them. They include, among others, cusps , which assume that the final state rescatterings are enough to describe data, and as such there is no need for poles in the scattering matrix. This is the minimalist approach, in particular, invoked to explain the origin of the charged states and . If proven correct, one would have to admit that all this excitement about new frontiers of QCD is “much ado about nothing”.
A good majority of the interested hadron physics community obviously does not share this agnostic point of view, and dynamical mechanisms have been devised to accommodate the new spectroscopy. One such model put forward to accommodate the exotic hadrons is hadroquarkonium, in which a forms the hard core surrounded by light matter (light states). For example, the hadrocharmonium core may consist of states, and the light degrees of freedom can be combined to accommodate the observed hadrons . This is motivated by analogy with the good old hydrogen atom which explained a lot of atomic physics. A variation on this theme is that the hard core quarkonium could be in a color-adjoint representation, in which case the light degrees of freedom are also a color-octet to form an overall singlet.
Next are hybrid models, the basic idea of which dates back to circa 1994 based on the QCD-inspired flux-tubes, which predict exotic states of both the light and heavy quarks. Hybrids are hadrons formed from the valence quarks and gluons, for example, consisting of . In the context of the hadrons, hybrids have been advanced as a model for the state , which has a small annihilation cross section, But, hybrids have been offered as templates for other exotic hadrons as well .
Another popular approach assumes that the multiquark states are meson-meson and meson-baryon bound states, with an attractive residual van der Waals force generated by mesonic exchanges . This hypothesis is in part supported by the closeness of the observed exotic hadron masses to the respective meson-meson (meson-baryon) thresholds. In many cases, this leads to very small binding energy, which imparts them a very large hadronic radius. This is best illustrated by , which has an S-wave coupling to (and its conjugate) and has a binding energy MeV. Such a hadron molecule will have a large mean square separation of the constituents fm, where the quoted radius corresponds to a binding energy MeV. This would lead to small production cross-sections in hadronic collisions , contrary to what has been observed in a number of experiments at the Tevatron and the LHC. In some theoretical constructs, this problem is mitigated by making the hadron molecules complicated by invoking a hard (point-like) core. In that sense, such models resemble hadroquarkonium models, discussed above. In yet others, rescattering effects are invoked to substantially increase the cross-sections . Theoretical interest in hadron molecules has remained unabated, and there exists a vast and growing literature on this topic with ever increasing sophistication, a sampling of which is referenced here .
Last, but by no means least, on this list are QCD-based interpretations in which tetraquarks and pentaquarks are genuinely new hadron species . In the large limit of QCD, tetraquarks are shown to exist as poles in the S-matrix, and they may have narrow widths in this approximation, and hence they are reasonable candidates for multiquark states. First attempts using Lattice QCD have been undertaken in which correlations involving four-quark operators are studied numerically. Evidence of tetraquark states in the sense of S-matrix poles using these methods is still lacking. Establishing the signal of a resonance requires good control of the background. In the lattice QCD simulations of multiquark states, this is currently not the case. This may be traced back to the presence of a number of nearby hadronic thresholds and to lattice-specific issues, such as an unrealistic pion mass. More powerful analytic and computational techniques are needed to draw firm conclusions. In the absence of reliable first principle calculations, approximate phenomenological methods are the only way forward. In that spirit, an effective Hamiltonian approach has been often used , in which tetraquarks are assumed to be diquark-antidiquark objects, bound by gluonic exchanges (pentaquarks are diquark-diquark-antiquark objects). This allows one to work out the spectroscopy and some aspects of tetraquark decays. Heavy quark symmetry is a help in that it can be used for the heavy-light diquarks relating the charmonia-like states to the bottomonium-like counterparts. I will be mainly discussing interpretations of the current data based on the phenomenological diquark picture to test how far such models go in describing the observed exotic hadrons and other properties measured in current experiments.
2 The Diquark Model
The basic assumption of this model is that diquarks are tightly bound colored objects and they are the building blocks for forming tetraquark mesons and pentaquark baryons. The diquarks, for which we use the notation , and interchangeably , have two possible SU(3)-color representations. Since quarks transform as a triplet of color SU(3), the diquarks resulting from the direct product , are thus either a color anti-triplet or a color sextet . The leading diagram based on one-gluon exchange is shown below.
The product of the SU(3)-matrices in Fig. 1 can be decomposed as
The coefficient of the antisymmetric representation is , reflecting that the two diquarks bind with a strength half as strong as between a quark and an antiquark, in which case the corresponding coefficient is . The symmetric on the other hand has a positive coefficient, +1/3, reflecting a repulsion. This perturbative argument is in agreement with lattice QCD simulations . Thus, in working out the phenomenology, a diquark is assumed to be an -antitriplet, with the antidiquark a color-triplet. With this, we have two color-triplet fields, quark and anti-diquark or , and two color-antitriplet fields, antiquark and diquark or , from which the spectroscopy of the conventional and exotic hadrons is built.
Since quarks are spin-1/2 objects, a diquark has two possible spin-configurations, spin-0, with the two quarks in a diquark having their spin-vectors anti-parallel, and spin-1, in which case the two quark spins are aligned, as shown in Fig. 2.
They were given the names “good diquarks” and “bad diquarks”, respectively, by Jaffe , implying that in the former case, the two quarks bind, and in the latter, the binding is not as strong. There is some support of this pattern from lattice simulations for light diquarks . However, as the spin-degree of freedom decouples in the heavy quark systems, as can be shown explicitly in heavy quark effective theory context for heavy mesons and baryons, we expect that this decoupling will also hold for heavy-light diquarks with . So, for the heavy-light diquarks, both the spin-1 and spin-0 configurations are present. Also, what concerns the diquarks in heavy baryons (such as and ), consisting of a heavy quark and a light diquark, both and quantum numbers of the diquark are needed to accommodate the observed baryon spectrum.
In this lecture, we will be mostly discussing heavy-light diquarks, and following the discussion above, we construct the interpolating diquark operators for the two spin-states of such diquarks (here ) :
In the non-relativistic (NR) limit, these states are parametrized by Pauli matrices: and We will characterize a tetraquark state with total angular momentum by the state vector showing the diquark spin and the antidiquark spin . Thus, the tetraquarks with the following diquark-spin and angular momentum have the Pauli forms:
2.1 NR Hamiltonian for Tetraquarks with hidden charm
For the heavy quarkonium-like exotic hadrons, we work in the non-relativistic limit and use the following effective Hamiltonian to calculate the tetraquark mass spectrum
where is the diquark mass, the second term above is the spin-spin interaction involving the quarks (or antiquarks) in a diquark (or anti-diquark), the third term depicts spin-spin interactions involving a quark and an antiquark in two different shells, with the fourth and fifth terms being the spin-orbit and the orbit-orbit interactions, involving the quantum numbers of the tetraquark, respectively. For the -states, these last two terms are absent. For illustration, we consider the case and display the individual terms in :
The usual angular momentum algebra then yields the following form:
2.2 Low-lying and -wave tetraquark states in the and sectors
The states in the diquark-antidiquark basis and in the and basis are related by Fierz transformation. The positive parity -wave tetraquarks are given in terms of the six states listed in Table 1 (charge conjugation is defined for neutral states). These states are characterized by the quantum number , hence their masses depend on just two parameters and , leading to several predictions to be tested against experiments. The -wave states are listed in Table 2. The first four of them have , and the fifth has , and hence is expected to be significantly heavier.
The parameters appearing on the r.h. columns of Tables 1 and 2 can be determined using the masses of some of the observed states, and their numerical values are given in Table 3. Some parameters in the and sectors can also be related using the heavy quark mass scaling .
Typical errors on the masses due to parametric uncertainties are estimated to be about 30 MeV. As we see from table 4, there are lot more hadrons observed in experiments in the charmonium-like sector than in the bottomonium-like sector, with essentially three entries , and in the latter case. There are several predictions in the charmonium-like sector, which, with the values of the parameters given in the tables above, are in the right ball-park aaaI thank Satoshi Mishima for providing these estimates.. It should be remarked that these input values, in particular for the quark-quark couplings in a diquark, , are larger than in the earlier determinations of the same by Maiani et al. . Better agreement is reached with experiments assuming that diquarks are more tightly bound than suggested from the analysis of the baryons in the diquark-quark picture, and the spectrum shown here is in agreement with the one in the modified scheme. Alternative calculations of the tetraquark spectrum based on diquark-antidiquark model have been carried out .
The exotic bottomonium-like states are currently rather sparse. The reason for this is that quite a few exotic charmonium-like states were observed in the decays of -hadrons. This mode is obviously not available for the hidden states. They can only be produced in hadro- and electroweak high energy processes. Tetraquark states with a single quark can, in principle, also be produced in the decays of the mesons, as pointed out recently . As the and cross-section at the LHC are very large, we anticipate that the exotic spectroscopy involving the open and hidden heavy quarks is an area where significant new results will be reported by all the LHC experiments. Measurements of the production and decays of exotica, such as transverse-momentum distributions and polarization information, will go a long way in understanding the underlying dynamics.
As a side remark, we mention that recently there has been a lot of excitement due to the D0 observation of a narrow structure , consisting of four different quark flavors , found through the decay mode. However, this has not been confirmed by the LHCb collaboration , despite the fact that LHCb has 20 times higher sample than that of D0. This would have been the first discovery of an open -quark tetraquark state. They are anticipated in the compact tetraquark picture , and also in the hadron molecule framework . We wait for more data from the LHC experiments.
We now discuss the three observed exotic states in the bottomonium sector in detail. The hidden state with was discovered by Belle in 2007 in the process just above the . The branching ratios measured are about two orders of magnitude larger than anticipated from similar dipionic transitions in the lower states and (for a review and references to earlier work, see Brambilla et al .). Also the dipion invariant mass distributions in the decays of are marked by the presence of the resonances and . This state was interpreted as a P-wave tetraquark . Subsequent to this, a Van Royen-Weiskopf formalism was used in which direct electromagnetic couplings with the diquark-antidiquark pair of the was assumed. Due to the -wave nature of the , with a commensurate small overlap function, the observed small production cross-section in was explained. In the tetraquark picture, is the analogue of the state , also a -wave, which is likewise found to have a very small production cross-section, but decays readily into . Hence, the two have very similar production and decay characteristics, and, in all likelihood, they have similar compositions.
The current status of is unclear. Subsequent to the discovery of , Belle undertook high-statistics scans for the ratio , and also measured more precisely the ratios . No results are available on from BaBar, so we discuss the analysis reported by Belle. The two masses, measured through , and , measured through , now differ by slightly more than 2, yielding MeV. From the mass difference alone, these two could very well be just one and the same state, namely the canonical - an interpretation adopted by the Belle collaboration . On the other hand, it is now the book keeping of the branching ratios measured at or near the , which is puzzling. This is reflected in the paradox that direct production of the as well as of states have essentially no place in the Belle counting , as the branching ratios of the are already saturated by the exotic states and their isospin partners). In our opinion, an interpretation of the Belle data based on two resonances and is more natural, with having the decays expected for the bottomonium -state above the threshold, and the decays of , a tetraquark, being the source of the exotic states seen. As data taking starts in a couple of years in the form of a new and expanded collaboration, Belle-II, cleaning up the current analysis in the and region should be one of their top priorities. In the meanwhile, the 2007 discovery of stands, not having been retracted by Belle, at least as far as I know.
Thus, there is a good case that and , while having the same quantum numbers and almost the same mass, are different states. As already mesntioned, this is hinted by the drastically different decay characteristics of the dipionic transitions involving the lower quarkonia -states, such as , on one hand, and similar decays of the , on the other. These anomalies are seen both in the decay rates and in the dipion invariant mas spectra in the modes. The large branching ratios of , as well as of , are due to the Zweig-allowed nature of these transitions, as the initial and final states have the same valence quarks. The final state in decays requires the excitation of a pair from the vacuum. Since, the light scalars , are themselves tetraquark candidates , they are expected to show up in the invariant mass distributions, as opposed to the corresponding spectrum in the transition (see Fig. 4). Subsequent discoveries of the charged states and , found in the decays , leading to the final states , , , and , give credence to the tetraquark interpretation, as discussed below.
2.3 Heavy-Quark-Spin Flip in
We summarize the relative rates and strong phases measured by Belle in the process , with and in Table 5. For ease of writing we shall use the notation and for the two charged states. Here no assumption is made about the nature of , it can be either or . Of these, the decay involves both a resonant (i.e., via ) and a direct component, but the other four are dominated by the resonant contribution. One notices that the relative normalizations are very similar and the phases of the differ by about compared to the ones in . At the first sight this seems to violate the heavy-quark-spin conservation, as in the initial state , which remains unchanged for the in the final state, i.e., it involves an transition, but as for the , this involves an transition, which should have been suppressed, but is not supported by data.
It has been shown that this contradiction is only apparent . Expressing the states and in the basis of definite and light quark spins, it becomes evident that both the and have and components,
Defining ( is the effective couplings at the vertices and )
we note that within errors, Belle data is consistent with the heavy quark spin conservation, which requires . The two-component nature of the and is also the feature which was pointed out earlier for in the context of the direct transition . To determine the coefficients and , one has to resort to : transitions
The analogous effective couplings are
Dalitz analysis indicates that proceed mainly through the resonances and , though has a significant direct component, expected in tetraquark interpretation of . A comprehensive analysis of the Belle data including the direct and resonant components is required to test the underlying dynamics, which yet to be carried out. However, parametrizing the amplitudes in terms of two Breit-Wigners, one can determine the ratio from . For the , we get for the averaged quantities:
For the , we get
Within errors, the tetraquark assignment with is supported, i.e.,
It is interesting that similar conclusion was drawn in the ‘molecular’ interpretation of the and .
The Fierz rearrangement used in obtaining second of the above relations would put together the and fields, yielding
Here, the labels and could be viewed as indicating and mesons, respectively, leading to the prediction and , which is not in agreement with the Belle data . However, this argument rests on the conservation of the light quark spin, for which there is no theoretical foundation. Hence, this last relation is not reliable. Since and are rather close in mass, and there is an issue with the unaccounted direct production of the and states in the Belle data collected in their vicinity, we conclude that the experimental situation is still in a state of flux and look forward to its resolution with the consolidated Belle-II data.
2.4 Drell-Yan mechanism for vector exotica production at the LHC and Tevatron
The exotic hadrons having can be produced at the Tevatron and LHC via the Drell-Yan process