Hadronic molecules

Hadronic molecules

Feng-Kun Guo fkguo@itp.ac.cn CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China    Christoph Hanhart c.hanhart@fz-juelich.de Institute for Advanced Simulation, Institut für Kernphysik and Jülich Center for Hadron Physics, Forschungszentrum Jülich, D-52425 Jülich, Germany    Ulf-G. Meißner meissner@hiskp.uni-bonn.de Helmholtz-Institut für Strahlen- und Kernphysik and Bethe Center for Theoretical Physics, Universität Bonn, D-53115 Bonn, Germany Institute for Advanced Simulation, Institut für Kernphysik and Jülich Center for Hadron Physics, Forschungszentrum Jülich, D-52425 Jülich, Germany    Qian Wang wangqian@hiskp.uni-bonn.de Helmholtz-Institut für Strahlen- und Kernphysik and Bethe Center for Theoretical Physics, Universität Bonn, D-53115 Bonn, Germany    Qiang Zhao zhaoq@ihep.ac.cn Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China Theoretical Physics Center for Science Facilities, Chinese Academy of Sciences, Beijing 100049, China    Bing-Song Zou zoubs@itp.ac.cn CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
July 12, 2019

A large number of experimental discoveries especially in the heavy quarkonium sector that did not at all fit to the expectations of the until then very successful quark model led to a renaissance of hadron spectroscopy. Among various explanations of the internal structure of these excitations, hadronic molecules, being analogues of light nuclei, play a unique role since for those predictions can be made with controlled uncertainty. We review experimental evidences of various candidates of hadronic molecules, and methods of identifying such structures. Nonrelativistic effective field theories are the suitable framework for studying hadronic molecules, and are discussed in both the continuum and finite volumes. Also pertinent lattice QCD results are presented. Further, we discuss the production mechanisms and decays of hadronic molecules, and comment on the reliability of certain assertions often made in the literature.


I Introduction

With the discovery of the deuterium in 1931 and the neutron in 1932, the first bound state of two hadrons, i.e., the deuteron composed of one proton and one neutron, became known. The deuteron is very shallowly bound, by a mere MeV per nucleon, i.e. it is located just below the neutron-proton continuum threshold. Furthermore, it has a sizeable spatial extension. These two features can be used for defining a hadronic molecule. A more precise definition will be given in the course of this review.

Then the first meson, the pion, as the carrier particle of the nuclear force proposed in 1935 by Yukawa was discovered in 1947, followed by the discovery of a second meson, the kaon, in the same year. Since then, many different hadrons have been observed. Naturally hadronic molecules other than the deuteron have been expected. The first identified meson-baryon molecule, i.e., the resonance composed of one kaon and one nucleon, was predicted by Dalitz and Tuan in 1959 Dalitz and Tuan (1959) and observed in the hydrogen bubble chamber at Berkeley in 1961 Alston et al. (1961) several years before the quark model was proposed. With the quark model developed in the early 1960s, it became clear that hadrons are not elementary particles, but composed of quarks and antiquarks. In the classical quark model, a baryon is composed of three quarks and a meson is composed of one quark and one antiquark. In this picture, the resonance would be an excited state of a three-quark () system with one quark in an orbital -wave excitation. Ten years later, the theory of the strong interactions, Quantum Chromodynamics (QCD), was proposed to describe the interactions between quarks as well as gluons. The gluons are the force carriers of the theory that also exhibit self-interactions due to the non-abelian nature of the underlying gauge group, SU(3), where denotes the color degree of freedom. In QCD the basic constituents of the hadrons are both quarks and gluons. Therefore, the structure of hadrons is more complicated than the classical quark model allows. There may be glueballs (which contain only valence gluons), hybrids (which contain valence quarks as well as gluons) and multiquark states (such as tetraquarks or pentaquarks). Note, however, that in principle the quark model also allows for certain types of multiquark states Gell-Mann (1964).

While the classical quark model is very successful in explaining properties of the spatial ground states of the flavor SU(3) vector meson nonet, baryon octet and decuplet, it fails badly even for the lowest spatial excited states in both meson and baryon sectors.

In the meson sector, the lowest spatial excited SU(3) nonet is supposed to be the lowest scalar nonet which includes the , the , the and the . In the classical constituent quark model, these scalars should be states, where denotes the orbital angular momentum, with the as an state, the as an state and the as mainly an state. This picture, however, fails to explain why the mass of the is degenerate with the instead of being close to the, as it is the case of the and the in the vector nonet. Instead, this kind of mass pattern can be easily understood in the tetraquark picture Jaffe (1977a) or in a scenario where these states are dynamically generated from the meson-meson interaction Weinstein and Isgur (1982); Janssen et al. (1995); Oller et al. (2000), with the and the coupling strongly to the channel with isospin 0 and 1, respectively.

In the baryon sector, a similar phenomenon seems also to be happening Zou (2008). In the classical quark model, the lowest spatial excited baryon is expected to be a () state with one quark in an orbital angular momentum state to have spin-parity . However, experimentally, the lowest negative parity resonance is found to be the , which is heavier than two other spatial excited baryons: the and the . This is the long-standing mass reversal problem for the lowest spatial excited baryons. Furthermore, it is also difficult to understand the strange decay properties of the , which seems to couple strongly to the final states with strangeness Liu and Zou (2006), as well as the strange decay pattern of another member of the -nonet, the , which has a coupling to much larger than to and according to its branching ratios listed in the tables in the Review of Particle Physics by the Particle Data Group (PDG) Patrignani et al. (2016). All these difficulties can be easily understood by assuming large five-quark components in them Zou (2008); Liu and Zou (2006); Helminen and Riska (2002) or considering them to be dynamically generated meson-baryon states Oller et al. (2000); Kaiser et al. (1995); Oset and Ramos (1998); Oller and Meißner (2001); Inoue et al. (2002); Garcia-Recio et al. (2004); Hyodo et al. (2003); Magas et al. (2005); Huang et al. (2007); Bruns et al. (2011).

No matter which configurations are realized in multiquark states, such as colored diquark correlations or colorless hadronic clusters, the mass and decay patterns for the lowest meson and baryon nonets strongly suggest that one must go beyond the classical, so-called quenched, quark model. The unquenched picture has been further supported by more examples of higher excited states in the light quark sector, such as the as a molecule Törnqvist (1994), and by many newly observed states with heavy quarks in the first decade of the new century, such as the as a molecule or tetraquark state, as molecule or tetraquark state Chen et al. (2016a). In fact, the possible existence of hadronic molecules composed of two charmed mesons was already proposed 40 years ago by Voloshin and Okun Voloshin and Okun (1976) and supported by Törnqvist later within a one-pion exchange model Törnqvist (1994).

However, although many hadron resonances were proposed to be dynamically generated states from various hadron-hadron interactions or multiquark states, most of them cannot be clearly distinguished from classical quark model states due to tunable ingredients and possible large mixing of various configurations in these models. A nice example is the already mentioned . Until 2010, i.e., 40 years after it was predicted and observed as the molecule, the PDG Nakamura et al. (2010) still claimed that “the clean spectrum has in fact been taken to settle the decades-long discussion about the nature of the — true 3-quark state or mere threshold effect? — unambiguously in favor of the first interpretation.” Only after many delicate analyses of various relevant processes, the PDG Patrignani et al. (2016) now acknowledges the two-pole structure of the Oller and Meißner (2001) and thus a dynamical generation is most probable.

One way to unambiguously identify a multiquark state (including hadronic molecular configurations) is the observation of resonances decaying into a heavy quarkonium plus a meson with nonzero isospin made of light quarks or plus a baryon made of light quarks. Since 2008, several such states have been claimed, six states, two states and two states — details on the experimental situation are given in the next section. Among these newly claimed states, the two states are quite close to the predicted hadronic molecular states Wu et al. (2010); Wang et al. (2011b); Yang et al. (2012b); Xiao et al. (2013a). However, many of those states are challenged by some proposed kinematic explanations, such as threshold cusp effects Bugg (2011); Swanson (2015), triangle singularity effects Chen et al. (2013); Wang et al. (2013a); Guo et al. (2015c), etc. Some of these claims were challenged in Ref. Guo et al. (2015b) where strong support is presented that at least some of the signals indeed refer to -matrix poles.

Further experimental as well as theoretical studies are necessary to settle the question which of the claimed states indeed exist. Nevertheless the observation of at least some of these new states opens a new window for the study of multiquark dynamics. Together with many other newly observed states in the heavy quarkonium sector, they led to a renaissance of hadron spectroscopy. Among various explanations of the internal structure of these excitations, hadronic molecules, being analogues of the deuteron, play a unique role since for those states predictions can be made with controlled uncertainty, especially for the states with one of or both hadrons containing heavy quark(s). In fact most of these observed exotic candidates are indeed closely related to open flavor -wave thresholds. To study these hadronic molecules, both nonrelativistic effective field theories and pertinent lattice QCD calculation are the suitable frameworks. Especially, Weinberg’s famous compositeness criterion Weinberg (1963a, b) (and extensions thereof), which pinned down the nature of the deuteron as a proton-neutron bound state, is applicable here. The pole location in the corresponding hadron-hadron scattering -matrix could also shed light on the nature of the resonances as extended hadronic molecules or compact states.

The revival of hadron spectroscopy is also reflected in a number of review articles. A few years ago, Klempt and his collaborators have given two broad reviews on exotic mesons Klempt and Zaitsev (2007) and baryons Klempt and Richard (2010). Other more recent pertinent reviews include Brambilla et al. (2011); Olsen (2015); Oset et al. (2016); Chen et al. (2016a, 2017c); Esposito et al. (2016b); Lebed et al. (2017); Hosaka et al. (2016); Dong et al. (2017); Olsen et al. (2017). Among various theoretical models for these new hadrons, we mainly cite those focusing on hadronic molecules and refer the interesting readers to the above mentioned comprehensive reviews for more references on other models.

This paper is organized as follows: In Sec. II, we discuss the experimental evidences for states that could possibly be hadronic molecules. In Sec. III, after a short review of the basic -matrix properties, we give a general definition of hadronic molecules and discuss related aspects. Then, in Sec. IV, nonrelativistic effective field theories tailored to investigate hadronic molecules are formulated, followed by a brief discussion of hadronic molecules in lattice QCD in Sec. V. Sec. VI is devoted to the discussion of phenomenological manifestations of hadronic molecules, with a particular emphasis on clarifying certain statements from the literature that have been used to dismiss certain states as possible hadronic molecules. We end with a short summary and outlook in Sec. VII. We mention that this field is very active, and thus only references that appeared before April 2017 are included.

Ii Candidates of hadronic molecules — experimental evidences

State -wave threshold(s) [MeV] Decay mode(s) [branching ratio(s)]
 Peláez (2016) 111The mass and width are derived from the pole position quoted in Ref. Peláez (2016) via . [dominant]
[possibly seen]
) [seen]
Table 1: Mesons that contain at most one heavy quark that cannot be easily accommodated in the quark model. Their quantum numbers , masses, widths, the nearby -wave thresholds, , where we add in brackets , and the observed decay modes are listed in order. The data without references are taken from the 2016 edition of the Review of Particle Physics Patrignani et al. (2016).
State -wave threshold(s) [] Observed mode(s) (branching ratios)
Aaij et al. (2017a, b)
Aaij et al. (2017a, b)
Aaij et al. (2017a, b)
Aaij et al. (2017a, b)
Table 2: Same as Table 1 but in the charmonium and bottomonium sectors. A blank in the fifth column means that there is no relevant nearby -wave threshold.

In this section we briefly review what is known experimentally about some of the most promising candidates for exotic states. Already the fact that those are all located close to some two-hadron continuum channels indicates that the two-hadron continuum is of relevance for their existence. We will show that many of those states are located near -wave thresholds in both light and heavy hadron spectroscopy, which is not only a natural property of hadronic molecules, which are QCD bound states of two hadrons (a more proper definition will be given in Sec. III.2), but also a prerequisite for their identification as will be discussed in Sec. III. In the course of this review, we will present other arguments why many of these states should be considered as hadronic molecules and what additional experimental inputs are needed to further confirm this assignment.

In Tables 1 and  2 we present the current status for exotic candidates in the meson sector. Exotic candidates in the baryon sector are listed later in Tab. 4. Besides the standard properties we also quote for each state the nearest relevant -wave threshold as well as its distance to that threshold. Note that only thresholds of narrow states are quoted since these are the only ones of relevance here Guo and Meißner (2011). Otherwise, the bound system would also be broad Filin et al. (2010). In addition, as a result of the centrifugal barrier one expects that if hadronic molecules exist, they should first of all appear in the -wave which is why in this review we do not consider - or higher partial waves although there is no principle reason for the non-existence of molecular states in the -wave.

ii.1 Light mesons

ii.1.1 Scalars below 1 GeV

The lowest -wave two-particle thresholds in the hadron sector are those for two pseudoscalar mesons, , , , and . Those channels carry scalar quantum numbers. The pion pair is either in an isoscalar or an isotensor state, and the isovector state is necessarily in a -wave. It turns out that there is neither a resonant structure in the isotensor nor in the isospin -wave, however, there are resonances observed experimentally in all other channels. According to the conventional quark model, a scalar meson of with carries one unit of orbital angular momentum. Thus, the mass range of the lowest scalars is expected to be higher than the lowest pseudoscalars or vectors of which the orbital angular momentum is zero. However, the lightest scalars have masses below those of the lightest vectors. Moreover, the mass ordering of the lightest scalars apparently violates the pattern of other nonets: Instead of having the isovectors to be the lowest states, the isovector states are almost degenerate with one of the isoscalar states, , and those are the heaviest states in the nonet. The other isoscalar, , also known as , has the lightest mass of the multiplet and an extremely large width. The strange scalar , also known as , has a large width as well. All these indicate some nontrivial substructure beyond a simple description.

The mass ordering of these lightest scalars is seen as a strong evidence for the tetraquark scenario proposed by Jaffe in the 1970s Jaffe (1977a, b). Meanwhile, they can also be described as dynamically generated states through meson-meson scatterings Pennington and Protopopescu (1973); Au et al. (1987); Morgan and Pennington (1993); Peláez (2016). For a theoretical understanding of the pole it is crucial to recognize that as a consequence of the chiral symmetry of QCD the scalar isoscalar interaction is proportional to at the leading order (LO) in the chiral expansion. Here, denotes the pion mass (decay constant). As a result, the LO scattering amplitude has already hit the unitarity bound for moderate energies necessitating some type of unitarization, which at the same time generates a resonance-like structure Meißner (1991). This observation, deeply nested in the symmetries of QCD, has indicated the significance of the interaction for the light scalar mesons. The history and the modern developments regarding the was recently very nicely reviewed in Peláez (2016). Similar to the isoscalar scalar generated from the scattering, the whole light scalar nonet appears naturally from properly unitarized chiral amplitudes for pseudoscalar-pseudoscalar scatterings Oller et al. (1998, 1999); Gomez Nicola and Peláez (2002). Similar conclusions also follow from more phenomenological studies Weinstein and Isgur (1990); Janssen et al. (1995). One of the most interesting observations about and is that their masses are almost exactly located at the threshold. The closeness of the threshold to and and their strong -wave couplings makes both states good candidates for molecular states Weinstein and Isgur (1990); Baru et al. (2004).

ii.1.2 Axial vectors , and implications of the triangle singularity

The -wave pseudoscalar meson pair scatterings can be extended to -wave pseudoscalar-vector scatterings and vector-vector scatterings where again dynamically generated states can be investigated. The -wave pseudoscalar-vector scatterings can access the quantum numbers , while the vector-vector scatterings give , and . This suggests that some of the states with those quantum numbers can be affected by the -wave open thresholds if their masses are close enough to the thresholds. Or, it might be possible that such scatterings can dynamically generate states as discussed in the literature Lutz and Kolomeitsev (2004b); Roca et al. (2005); Geng and Oset (2009). Note that not all states found in these studies survive once a more sophisticated and realistic treatment as outlined in Ref. Gülmez et al. (2017) is utilized.

In addition, the quark model also predicts regular states in the same mass range such that it appears difficult to identify the most prominent structure of the states.

Let us focus on the lowest mesons. Despite that these states could be dynamically generated from the resummed chiral interactions Lutz and Kolomeitsev (2004b); Roca et al. (2005), there are various experimental findings consistent with a usual nature of the members of the lightest axial nonet, , , , and  Patrignani et al. (2016). However, two recent experimental observations expose novel features in their decay mechanisms which illustrate the relevance of their couplings to the two-meson continua. The BESIII Collaboration observed an anomalously large isospin symmetry breaking in  Ablikim et al. (2012), which could be accounted for by the so-called triangle singularity (TS) mechanism as studied in Ref. Wu et al. (2012); Aceti et al. (2012). This special threshold phenomenon arises in triangle (three-point loop) diagrams with special kinematics which will be detailed in Sect. IV.1. Physically, it emerges when all the involved vertices in the triangle diagram can be interpreted as classical processes. For it to happen, one necessary condition is that all intermediate states in the triangle diagram, for the example at hand, should be able to reach their on-shell condition simultaneously. As a consequence, the , which is close to the threshold and couples to in an -wave as well, should also have large isospin violations in . This contribution has not been included in the BESIII analysis Ablikim et al. (2012). However, a detailed partial wave analysis suggests the presence of the contribution via the TS mechanism Wu et al. (2013). Moreover, the TS mechanism predicts structures in different -parity and isospin (or parity) channels via the triangle diagrams. The was speculated long time ago to be a molecule from a dynamical study of the three-body system Longacre (1990).

Apart from the , state , one would expect that the TS will cause enhancements in channels with . It provides a natural explanation for the newly observed by the COMPASS Collaboration Adolph et al. (2015) in and  Liu et al. (2016a); Mikhasenko et al. (2015). It should be noted that in Refs. Aceti et al. (2016); Cheng et al. (2016); Debastiani et al. (2017a) the enhancement is proposed to be caused by the together with the TS mechanism and similarly is produced by . However, as shown by the convincing experimental data from MARK-III, BESII, BESIII, and the detailed partial wave analysis of Ref. Wu et al. (2013), the matches the behavior of a genuine state in the channel that is distorted in other channels by an interference with the TS. This appears to be a more consistent picture to explain the existing data and underlying mechanisms Zhao (2017). These issues are discussed further in Sec. VI.

ii.2 Open heavy-flavor mesons

Since 2003, quite a few open heavy-flavor hadrons have been observed experimentally. Some of them are consistent with the excited states predicted in the potential quark model, while the others are not (for a recent review, see Chen et al. (2017c)). Particular interest has been paid to the positive-parity charm-strange mesons and observed in 2003 by the BaBar Aubert et al. (2003) and CLEO Besson et al. (2003) Collaborations. The masses of and are below the and thresholds, respectively, by about the same amount, only 45 MeV (see Table 1 and references therein), which makes them natural candidates for hadronic molecules Barnes et al. (2003); van Beveren and Rupp (2003); Szczepaniak (2003); Kolomeitsev and Lutz (2004); Hofmann and Lutz (2004); Guo et al. (2006, 2007); Gamermann et al. (2007); Faessler et al. (2007); Flynn and Nieves (2007); Cleven et al. (2011b); Wu and Zhao (2012); Cleven et al. (2014a); Albaladejo et al. (2016c), while also other explanations such as -wave states and tetraquarks exist in the literature. We will come back to the properties of these states occasionally in this review. Here, we collect the features supporting the molecular hypothesis:

  • Their masses are about and , respectively, below the predicted and charm-strange mesons by the Godfrey–Isgur quark model Godfrey and Isgur (1985); Di Pierro and Eichten (2001), making them not easy to be accommodated by the conventional states.

  • The mass difference between these two states is equal to the energy difference between the corresponding thresholds. This appears to be a natural consequence in the hadronic molecular scenario, since the involved interactions is approximately heavy quark spin symmetric Guo et al. (2009a).

  • The small width of both and can only be understood if the are isoscalar states 222Negative result was reported in a search for the isospin partner of the  Choi et al. (2015), for then, since both of them are below the thresholds, the only possible hadronic decay modes are the isovector channels and , respectively. The molecular nature together with the proximity to the thresholds leads to a prediction for the width of the states above 100 keV while other approaches give a width about 10 keV Colangelo and De Fazio (2003); Godfrey (2003). These issues are discussed in detail in Sections V.4 and VI.1.3.

  • Their radiative decays, i.e., and , and production in decays proceed via short-range interactions Lutz and Soyeur (2008); Chen et al. (2015a); Cleven et al. (2014a). They are therefore insensitive to the molecular component of the states.

  • As will be discussed in Secs. III.2 and V.4, the scattering length extracted from LQCD calculations Liu et al. (2013a) is compatible with the result extracted in the molecular scenario for based on Weinberg’s compositeness theorem.

The observed by the BaBar Collaboration Aubert et al. (2006a) presents another example of an interesting charm-strange meson. It decays into both and with similar branching fractions Patrignani et al. (2016). One notices that the difference between the mass and the threshold is similar to that between the and . Assuming the to be a hadronic molecule, an -wave bound state with quantum numbers was predicted to have a mass  MeV, consistent with that of the , in Guo and Meißner (2011), where the ratio of its partial widths into the and also gets naturally explained. As a result of heavy quark spin symmetry, a hadronic molecule with and a mass of around 2.91 GeV was predicted in Guo and Meißner (2011). A later analysis by the LHCb Collaboration suggests that this structure corresponds to two states: with and with  Aaij et al. (2014b). Regular interpretations for these two states have been nicely summarized in Chen et al. (2017c).

The most recently reported observation of an exotic singly-heavy meson candidate is a narrow structure in the invariant mass distribution, dubbed as , by the D0 Collaboration Abazov et al. (2016). Were it a hadronic state, it would be an isovector meson containing four different flavors of valence quarks . However, the peak is located at only about 50 MeV above the threshold. The existence of a tetraquark, whether or not being a hadronic molecule, at such a low mass is questioned from the quark model point of view in Burns and Swanson (2016), and, more generally, from chiral symmetry and heavy quark flavor symmetry in Guo et al. (2016d). Both the LHCb Aaij et al. (2016c) and CMS CMS Collaboration (2016) Collaborations quickly reported negative results on the existence of in their data sets. An alternative explanation for the observation is necessary. One possibility is provided in Ref. Yang et al. (2017b). Because of these controversial issues with the , we will not discuss this structure any further.

ii.3 Heavy quarkonium-like states:

The possibility of hadronic molecules in the charmonium mass region was suggested in Voloshin and Okun (1976); De Rujula et al. (1977) only a couple of years after the “November Revolution” due to the discovery of the . Such an idea became popular after the discovery of the famous by Belle in 2003 Choi et al. (2003).

Since then, numerous other exotic candidates have been found in the heavy quarkonium sector as listed in Table 2. In fact, it is mainly due to the observation of these structures that the study of hadron spectroscopy experienced a renaissance. The naming scheme currently used in the literature for these states assigns isoscalar states as and the isoscalar (isovector) states with other quantum numbers are named as . Note that the charged heavy quarkonium-like states , , , and are already established as being exotic, since they should contain at least two quarks and two anti-quarks with the hidden pair of or providing the dominant parts of their masses.

Figure 1: -wave open charm thresholds and candidates for exotic states in charmonium sector. Red solid (blue dashed) horizontal lines indicate the thresholds for nonstrange (strange) meson pairs. Two additional thresholds involving a charmonium and are also shown in the figure as green dotted lines. The data are taken from Ref. Patrignani et al. (2016). The exotic candidates are listed as black dots and green triangles with the latter marking the states to be discussed here. Here , , and mean , , and , respectively.

In the heavy quarkonium mass region, there are quite a few -wave thresholds opened by narrow heavy-meson pairs. In the charmonium mass region, the lowest-lying thresholds are , and . They are particularly interesting for understanding the and states which can couple to them in an -wave. The relevant quantum numbers are thus and (for more details, see Section IV.2). The -wave thresholds for the exotic candidates are also shown in Table 2. In addition, the exotic candidates in the charmonium sector and the -wave open-charm thresholds are shown in Fig. 1. Here, the thresholds involving particles with a large width,  MeV, have been neglected.

Since only -wave hadronic molecules with small binding energies are well-defined (Sec. III.2), in the following, we will focus on those candidates, i.e., , , , in the charmonium sector and , in the bottomonium sector. All of them have extremely close-by -wave thresholds except for the , as will be discussed below. For the experimental status and phenomenological models of other exotic candidates, we refer to several recent reviews Swanson (2006); Eichten et al. (2008); Brambilla et al. (2014); Esposito et al. (2015b); Lebed et al. (2017); Richard (2016); Chen et al. (2016a); Esposito et al. (2016b) and references therein.


In 2003, the Belle Collaboration reported a narrow structure in the invariant mass distribution in  Choi et al. (2003) process. It was confirmed shortly after by BaBar Aubert et al. (2005c, 2008) in collisions, and by CDF Acosta et al. (2004); Abulencia et al. (2006, 2007); Aaltonen et al. (2009) and D0 Abazov et al. (2004) in collisions. Very recently LHCb also confirmed its production in collisions Aaij et al. (2012, 2013, 2014a, 2015c) and pinned down its quantum numbers to , which are consistent with the observations of its radiative decays Abe et al. (2005); Aubert et al. (2006b); Bhardwaj et al. (2011) and multipion transitions Abulencia et al. (2006); Abe et al. (2005); del Amo Sanchez et al. (2010). The negative result of searching for its charged partner in decays Aubert et al. (2005b) indicates that the is an isosinglet state.

The most salient feature of the is that its mass coincides exactly with the threshold Patrignani et al. (2016)333Here we use the updated “OUR AVERAGE” values in PDG2016 for the masses:  MeV,  MeV, and  MeV from the and modes Patrignani et al. (2016).


which indicates the important role of the in the dynamics. That this should be the case can be seen most clearly from the large branching fraction Gokhroo et al. (2006); Aushev et al. (2010) (see Table 2)


although the mass is so close to the and thresholds. These experimental facts lead naturally to the interpretation of the as a hadronic molecule Törnqvist (2003),444 See also, e.g., Törnqvist (2004); Swanson (2004b); Close and Page (2004); Pakvasa and Suzuki (2004); Wong (2004); Voloshin (2004b); Swanson (2004a); AlFiky et al. (2006); Braaten and Lu (2007); Fleming et al. (2007); Liu et al. (2009a); Dong et al. (2009); Ding et al. (2009); Zhang and Huang (2009); Wang et al. (2010); Lee et al. (2009); Gamermann et al. (2010); Mehen and Springer (2011); Nieves and Valderrama (2011); Lee et al. (2011); Nieves and Valderrama (2012); Li et al. (2013d); Li and Zhu (2012); Sun et al. (2012b, a); Guo et al. (2013b); Hidalgo-Duque et al. (2013a); Wang and Wang (2013); Yamaguchi et al. (2013); Guo et al. (2014a); He (2014); Zhao et al. (2014); Karliner and Rosner (2015a); Baru et al. (2015a); Jansen et al. (2015); Baru et al. (2015b); Molnar et al. (2016); Yang et al. (2017a). which had been predicted by Törnqvist with the correct mass a decade earlier Törnqvist (1994). As will be discussed in Section VI, precise measurements of the partial widths of the processes and are particularly important in understanding the long-distance structure of the . In the hadronic molecular scenario, one gets a tremendously large scattering length of  fm, c.f. Eq. (18). However, a precision measurement of its mass is necessary to really distinguish a molecular from, e.g., a tetraquark scenario Maiani et al. (2005); Esposito et al. (2015b). This will be discussed further in Sec. III.2.3 and in Sec. III.3.

Other observables are also measured which could be sensitive to the internal structure of the . The ratio of branching fractions

was measured to be by Belle Abe et al. (2005) and by BaBar del Amo Sanchez et al. (2010). The value about unity means a significant isospin breaking because the three and two pions are from the isoscalar  Abe et al. (2005); del Amo Sanchez et al. (2010) and from the isovector  Abulencia et al. (2006), respectively. Notice that there is a strong phase space suppression on the isospin conserved three-pion transition through the channel. The fact that the molecular scenario of provides a natural explanation for the value of will be discussed in Sec. VI.1.3.

The experimental information available about the radiative decays of the is Aaij et al. (2014a)


A value larger than 1 for this ratio was argued to favor the interpretation Swanson (2004a) over the hadronic molecular picture. This, however, is not the case Mehen and Springer (2011); Guo et al. (2015a) as will be demonstrated in Sec. VI.

The production rates of in and decays was measured by BaBar Aubert et al. (2006c), i.e.,


We show in Sec. VI that this value is also consistent with a molecular nature of the .

One expects mirror images of charmonium-like states to be present in the bottomonium sector. The and states to be discussed in the next subsection suggest that such phenomena do exist. The analogue of the in the bottom sector, , has not yet been identified. A search for the was carried out by the CMS Collaboration, but no signal was observed in the channel Chatrchyan et al. (2013b). However, as pointed out in Ref. Guo et al. (2013b) before the experimental results and stressed again in Refs. Guo et al. (2014e); Karliner and Rosner (2015b) afterwards, the decay requires an isospin breaking which should be strongly suppressed due to the extremely small mass differences between the charged and neutral bottomed mesons and the large difference between the threshold and the , thresholds. In contrast, other channels such as ,  Guo et al. (2013b, 2014e); Karliner and Rosner (2015b) and  Li and Wang (2014) should be a lot more promising for an search.

ii.3.2 , and ,

From an analysis of the processes in 2011 the Belle Collaboration reported the discovery of two charged states decaying into with and with  Bondar et al. (2012). Their line shapes in a few channels are shown in Fig. 2. A later analysis at the same experiment allowed for an amplitude analysis where the quantum numbers were strongly favored Garmash et al. (2015).555The existence of an isovector state with exactly these quantum numbers was speculated long time ago for explaining the puzzling transition Voloshin (1983); Anisovich et al. (1995). The effects in dipion transitions among states were reanalyzed using the dispersion technique recently in Chen et al. (2016f, 2017d). This, together with the fact that the and have masses very close to the and thresholds, respectively, makes both excellent candidates for hadronic molecules Bondar et al. (2011)666See also, e.g., Zhang et al. (2011); Yang et al. (2012a); Danilkin et al. (2012); Sun et al. (2011); Cleven et al. (2011a); Ohkoda et al. (2012b); Li et al. (2013b); Dong et al. (2013a); Wang and Huang (2014); Wang (2014b); Dias et al. (2015); Karliner and Rosner (2015a). This statement finds further support in the observation that both states also decay by far most probably into and , respectively Garmash et al. (2016) (see Tab. 3).777The branching fractions were measured by assuming that these channels saturate the decay modes and using the Breit–Wigner (BW) parameterization for the structures Garmash et al. (2016). However, there could be non-negligible modes such as the , and the branching fractions measured in this way for near-threshold states should not be used to calculate partial widths by simply multiplying with the BW width. This point is discussed in detail in Chen et al. (2016f) for the case. The neutral partner is so far observed only for the lighter state Krokovny et al. (2013). Very recently, the Belle Collaboration reported the invariant mass distributions of and channels at energy region Abdesselam et al. (2016), see Fig. 3, clearly showing a resonant enhancement in the mass region. However, due to the limited statistics it is impossible to judge whether there are two peaks or just one.

Figure 2: Measured line shapes of the two states in the , and channels Garmash et al. (2016) and a fit using the parameterization of Refs. Hanhart et al. (2015); Guo et al. (2016a).
Figure 3: The missing mass spectra for and channels in the region. The solid and dashed histograms are the fits with the signal fixed from the analysis and with only a phase space distribution, respectively. Taken from Ref. Abdesselam et al. (2016).
channel of () of
Table 3: The reported branching fractions of the known decay modes of and  Garmash et al. (2016) with the statistical and systematical uncertainties in order.

Employing sums of BW functions for the resonance signals the experimental analyses gave masses for both states slightly above the corresponding open flavor thresholds together with narrow widths. It seems in conflict with the hadronic molecular picture, and was claimed to be consistent with the tetraquark approach Esposito et al. (2016a). It is therefore important to note that a recent analysis based on a formalism consistent with unitarity and analyticity leads for both states to below-threshold pole positions Hanhart et al. (2015); Guo et al. (2016a).888Notice that this, however, does not exclude the possibility of above-threshold poles. In the used parameterization, the contact terms are taken to be constants. The possibility of getting an above-threshold pole is available once energy-dependence is allowed in the contact terms. Nevertheless, the analyses at least show that the below-threshold-pole scenario is consistent with the current data.

A few years after the discovery of and in the Belle experiment, the BESIII and Belle Collaborations almost simultaneously claimed the observation of a charged state in the charmonium mass range,  Ablikim et al. (2013a); Liu et al. (2013b). It was shortly after confirmed by a reanalysis of CLEO-c data Xiao et al. (2013b), and its neutral partner was also reported in Refs. Xiao et al. (2013b); Ablikim et al. (2015e). Soon after these observations, the BESIII Collaboration reported the discovery of another charged state  Ablikim et al. (2013b), and its neutral partner was reported in Ref. Ablikim et al. (2014c). These charmonium-like states show in many respects similar features as the heavier bottomonium-like states discussed in the previous paragraphs, although there are also some differences. On the one hand, while the is seen in the channel and is seen in , there is no clear signal of in and in , although in the latter case there might be some indications of . This pattern might reflect a strong mass dependence of the production mechanism Wang et al. (2013b). On the other hand, in analogy to and ,