Half a century after the quark model was devised, a number of hadrons appear to challenge its axioms. But are they truly exotic?
Fifty years have passed since Dick Dalitz presented his explicit constituent-quark model at the 1966 International Conference on High Energy Physics in Berkeley, US. Murray Gell Mann and George Zweig independently introduced the quark concept in 1964, and the idea had also been anticipated by André Petermann in a little-known paper received by Nuclear Physics in 1963. But it was Dalitz who developed the model and considered excitations of quarks by analogy with the behaviour of nucleons in atomic nuclei. His primary focus was on the spectroscopy of baryons, which were interpreted as bound states of three quarks. Dalitz realised that the restrictions enforced by the Pauli exclusion principle led to a distinct pattern of supermultiplets. Today, this simple model remains in excellent agreement with experiments, in particular for mesons that comprise a quark–antiquark pair.
Despite its success in matching empirical data, the theoretical underpinning of this non-relativistic model for light hadrons has always been unclear. One of the remarkable features of hadron spectroscopy is that, half a century after the invention of the constituent-quark model, the particle data tables are filled with states that fit with a non-relativistic spectrum almost to the exclusion of anything else. Quarks are but a few MeV in mass, and are therefore surely relativistic when confined within the 1 fm radius of a proton, yet the constituent-quark model treats them as if relativity plays no role.
In the case of mesons, which fit the quark model arguably even better than baryons, this incongruity is especially significant. When Dalitz spoke in 1966, it made sense to emphasise baryons because they outnumbered the known mesons at that time. Following the discovery of charm and heavy flavours in the late 1970s, however, the spectroscopy of mesons flourished and the correlations among a meson’s spin (J), parity (P) and charge conjugation (C) were also found to be in accord with those of a non-relativistic system.
Following Dalitz’s description of the baryon spectrum, Greenberg, Nambu, Lipkin and others noted that the model’s ad-hoc correlation of baryon spins with the constraints of the Pauli principle required some novel degree of freedom, which we call “colour”. The advent of quantum chromodynamics (QCD) in the 1970s provided the rationale for this concept, explaining the existence of quark–antiquark or three-quark combinations in terms of colour-singlet clusters. But QCD did not explain the non-relativistic pattern of states. Feynman, who in his final years devoted his attention to this issue, asserted: “The [non-relativistic] quark model is correct as it explains so much data. It is for theorists to explain why.” Today, physicists still await this explanation. Yet the empirical guide of the quark model is so well established that hadrons outside of this straitjacket are deemed “exotic”.
Although the restriction to colour singlets within QCD explains the existence of qq and qqq hadrons, it raised the question of why the spectroscopy of QCD is so meagre. Colour singlets also allow combinations of pairs of quarks and antiquarks (“tetraquark” mesons), four quarks and an antiquark (“pentaquark” baryons), in addition to states comprised solely of gluons (“glueballs”). Furthermore, combinations called “hybrids” in which the gluonic fields entrapping the quark and antiquark are themselves excited are also theoretically possible within QCD (figure 1). Glueballs, tetraquarks and hybrid mesons, predicted in the late 1970s, can form correlations among a meson’s J, P and C quantum numbers that are forbidden by the non-relativistic model. Indeed, it is the lack of any empirical evidence for such exotic states in the meson spectrum that helped to establish the constituent-quark model in the first place. It is therefore ironic that searches for such states at modern experiments are now being used to establish the dynamic role of gluonic excitations in hadron spectroscopy.
Although QCD is well tested to high precision in the perturbative regime, where it is now an essential tool in the planning and interpretation of experiments, its implications for the strong-interaction limit are far less understood. Forty years after its discovery, and notwithstanding the advent of lattice QCD, hadron physics is still led by empirical data, from which clues to novel properties in the strong interactions may emerge. The search for exotic hadrons is an essential part of this strategy, and in recent years several new hadrons have been discovered that do not fit well within the traditional quark model.
With hindsight, one of the first clues to the existence of quarks came in the 1950s from measurements of cosmic-ray interactions in the atmosphere, which revealed hadrons with unusual production and decay properties. These “strange” hadrons, we now know, contain one or more strange quarks or strange antiquarks, yet history has left us with a perverse convention whereby strange quarks are deemed to carry negative strangeness, and strange antiquarks are positive. Thus mesons can have one unit of strangeness, in either positive or negative amounts, while baryons can have strangeness –1, –2 or –3 (antibaryons, in turn, can have positive strangeness).
A baryon with positive strangeness (or an antibaryon with negative strangeness) is therefore classed as exotic. The minimal configuration for such a baryon would involve four quarks together with the strange antiquark, giving a total of five and the technically incorrect name of “pentaquark”. A claim to have found such a state – the θ(1540) – made headlines nearly two decades ago but is now widely disregarded. The scepticism was not that a pentaquark exists, since QCD can accommodate such a state, but that it appeared to be anomalously stable. More recently, the LHCb experiment at CERN’s Large Hadron Collider (LHC) reported decays of the Λb pentaquark-like baryon that revealed similar structures with a mass of around 4.4 GeV (CERN Courier September 2015 p5). These have normal strong-interaction lifetimes and have been interpreted as clusters of three quarks plus a charm–anticharm pair. Whether these are genuinely compact pentaquarks, or instead bound states of a charmed baryon and a meson or some other dynamic artefact, they do appear to qualify as “exotic” in that they do not fit easily into a traditional three-constituent picture.
There have also been interesting meson sightings at lepton colliders in recent decades. Electron–positron annihilation above energies of 4 GeV in numerous experiments reveals a series of peaks in the total cross-section that are consistent with radial excitations of the fundamental cc J/ψ meson: the ψ(2S), ψ(4040), ψ(4160) and ψ(4415), which are non-exotic and fit within the non-relativistic spectrum. Evidence for exotic mesons has come from data on specific final states, notably those containing a J/ψ with one or more pions, which have revealed several novel states. Historically, the first clue for an exotic charmonium meson of this type above a mass of 4 GeV came around a decade ago from the BaBar experiment at SLAC in the US. Analysing the process e+e– → J/ψππ, researchers there found a clear resonant-like structure dubbed Y(4260), which has no place in the qq spectrum because its mass lies between the ψ(4160) and ψ(4415) cc states. More remarkably, this state decays into charmonium and pions with a standard strong-interaction width of the order of 100 MeV rather than 100 keV, which is more typical for such a channel.
The clue to the nature of this meson appears to be that the mass of the Y meson (4260 MeV) is near the threshold for the production of DD1 – the combination of pseudoscalar (D) and axial (D1) charmed mesons (figure 2). This is the first channel in e+e– annihilation where charmed meson pairs can be produced with no orbital angular momentum (i.e. via S-wave processes). Thus at threshold there is no angular-momentum barrier against a DD1 pair being created effectively at rest, and rearranging their constituents into the form of J/ψ and light flavours (the latter then seeding pions). Thus the structure could simply be a threshold effect rather than a true resonance, or an exotic “molecule” made of D and D1 charmed mesons.
The decay of the Y(4260) into J/ψππ reveals a manifestly exotic structure. The J/ψπ± channel is electrically charged with a pronounced peak called Z(3900), as reported by both the BESIII experiment in China and Belle in Japan in 2013. Another sharp peak observed by BESIII – the Z(4020) – appears in the flavour-exotic channel containing a pion and a charmonium meson. Since it can carry electric charge, this state must contain ud (or du) in addition to its cc content, and therefore cannot be explained as a bound state of a single quark and antiquark. In principle, these states should be accessible in decays of B mesons, but there is no sign of them so far.
Nonetheless, B decays are a source of further exotic structures. For example, the invariant-mass spectrum of B → K π±ψ(2S) contains a structure called the Z(4430) observed by Belle and LHCb in the ψ(2S)π invariant-mass spectrum, which contains both hidden charm and isospin and hence must contain (at least) two quarks and two antiquarks. These features first need to be established as genuine and not artefacts associated with some specific production process. Their appearance and decay in other channels would help in this regard, while the observation of analogous signals for other combinations of flavour may also signpost the underlying dynamics. If real, these states are the product of charmonium cc– and light-quark basis states (a summary of charmonium candidates can be seen in figure 3).
Proceed with caution
It is clear that peaks are being found that cannot be interpreted as qqq or qq clusters. But one should not leap to the conclusion that we have discovered some fundamentally novel state built from, say, diquarks and antidiquarks or, for baryons, a pentaquark. A qq qq “tetraquark”, for example, looks less exotic when trivially rewritten as qqqq, which is suggestive of two bound conventional mesons. Indeed, these could be the two mesons in the invariant mass of which the peak was seen. Unless the peak is seen in different channels, and ideally in different production mechanisms, one should be cautious.
For example, when three or more hadrons are produced in a single decay it is common to discover peaks in invariant-mass spectra just above the two-body thresholds. These are not resonances, although papers on the arXiv preprint server are full of models built on the assumption that they are. Instead, the peaks likely arise due to competition between two effects. First, phase space opens up for the production of the two-body channel, but as the invariant mass increases, the chance of this exclusive two-body mode dies off because the probability for the wavefunctions of the two hadrons to overlap decreases. Any peak seen within a few hundred MeV of such a threshold is most likely to be the accidental result of this phenomenon. Such “cusps” have been proposed as explanations of several recent exotic candidates, such as the Z(3900) and Z(10610) spotted at BESIII and Belle, among others. Whether the tetraquark candidates X(4274), X(4500) and X(4700) recently observed at LHCb, in addition to the X(4140) found by the CDF experiment at Fermilab in 2009, herald the birth of a new QCD spectroscopy or are examples of more mundane dynamics such as cusps, is also the subject of considerable debate. In short, if a peak occurs above a two-body threshold in a single channel: beware.
Enter the deuson
More interesting for exotic-hadron studies are peaks that lie just below threshold. Such states are well known in the baryon sector, the deuteron being a good example. The nuclear force driven by pion exchange that binds neutrons and protons inside the atomic nucleus should also occur between pairs of mesons, at least for those that are stable on the timescale of the strong interaction. Thus on purely phenomenological and conservative grounds, we should anticipate meson molecules (or, by analogy with the deuteron, “deusons”), which would take us beyond the simple quark-model spectroscopy. The Y(4260) could be an example of such a state, since both DD1 and D*D0 S-wave thresholds lie in this region and pion exchange may play a role in linking the two channels (figure 4). If these states are indeed deusons then there should also be partners with isospin. Establishing whether these structures are singletons or have siblings is therefore another important step in identifying their dynamical origins.
The first sign of deusons may be expected in the axial-vector channel formed from a pseudoscalar and vector charmed (or bottom) meson. This is because pion exchange can occur between a pair of vector mesons or as an exchange force between a pseudoscalar-vector combination, but not within a state of two pseudoscalars as this would violate parity conservation. The enigmatic state X(3872), which was first observed in B decays by Belle in 2003 and occurs at the D0 D*0 + cc threshold, has long been a prime candidate for a deuson. If so, there should be analogous states in the BB* as well as charm-bottom flavour mixtures and perhaps siblings with two units of charm or bottom. Whether these states have charged partners is one of many model-dependent details. That some of these states should occur seems unavoidable, however, and if doubly charmed states exist they should be produced at the LHC.
Whereas for baryons the attractive forces arise in the exchange or “t channel”, for pairs of mesons there can also be contributions due to qq annihilation in the direct s-channel. In QCD this can also mask the search for glueballs: for example, the scalar glueball of lattice QCD predicted at a mass of around 1.5 GeV mixes with the nonet of scalar qq states in this very region. The pattern of these scalars empirically is consistent with such dynamics.
Scalar mesons are interesting not least because the theoretical interest in multiquark or molecular states originated in such particles 40 years ago, after Robert Jaffe noticed that the chromo-magnetic QCD forces are powerfully attractive in the nonet of light-flavoured scalar mesons. Intriguingly, this idea has remained consistent with the observed nonet of scalars below 1 GeV ever since. The main question that remains unresolved is to what extent these states are dominantly formed from coloured diquarks and their antidiquarks, or are better described as molecular states formed from colour-singlet π and K mesons.
LHCb in particular has shown that it is possible to identify light scalars among the decay debris of heavy-flavoured mesons, offering a new opportunity to investigate their nature and dynamics. Indeed, the kinematic reach of the LHC potentially enables a multitude of information to be obtained about heavy-flavoured mesons in both conventional and exotic combinations. We might therefore hope that information about exotic mesons will be extended into different flavour sectors to help identify the source of the binding.
In general, the simple qq picture of mesons appears to remain remarkably robust so long as there are no nearby prominent channels for pair production of hadrons in the S-wave channel. “Exotic” mesons and baryons seem to correlate with some S-wave channel sharing quantum numbers with a nominal qq state and causing the appearance of a state near the corresponding S-wave threshold. In some of these cases, but not all, the familiar forces of conventional nuclear physics play a role, and the multi-particle events at the LHC have the kinematic reach to include all combinations of non-strange, strange, charm and bottom mesons. How many of these can in practice be identified is the challenge, but identifying the dynamics of states “beyond qq” may depend on it.
In conclusion, these exotic states need to be studied in different production mechanisms and in a variety of decay channels. A genuine resonant state should appear in different modes, whereas a structure that appears in a single production mechanism and a unique decay channel is suggestive of some dynamical feature that is not truly resonant. While interesting in its own right, such a state is not “exotic” in the sense of hadron spectroscopy.
As for truly exotic states, there are different levels of exoticity. For flavoured hadrons: the least exotic are meson analogues of nuclei – “deusons” driven by pion exchange between pairs of mesons. Next are “hybrids”: states anticipated in QCD where the gluonic degrees of freedom are excited in the presence of quarks and/or antiquarks. Finally, the most exotic of all would be colour-singlet combinations of compact diquarks, which are allowed in principle by QCD and would lead to a rich spectroscopy. At present their status is like the search for extraterrestrial life: while one feels that in the richness of nature such entities must exist, they seem reluctant to reveal themselves.