Abstract

In the past decade, exotic hadrons with charm and bottom flavors have been extensively studied both in experiments and in theories. In this review, we provide topical discussions by selecting $$X,Y,Z$$ particles, to which Belle has made important contributions. These are $$X(3872)$$, $$Y(4260)$$, $$Z_c(4430)^+$$, $$Z_c(3900)^+$$, $$Z_{b}(10610)^+$$, and $$Z_{b}(10650)^+$$. Based on the current experimental observations, we discuss these states with emphasis on the hadronic molecule whose dynamics is governed by chiral symmetry and heavy-quark symmetry of QCD. We also mention briefly various interpretations and some theoretical predictions for the as yet undiscovered exotic hadrons.

Introduction

The nature of the strong interaction does not allow quarks to be “bare” and “alone,” and thus they are confined into hadrons. Many hadron states are categorized into mesons and baryons containing constituent quark–antiquark ($$q \bar q$$) and three quarks ($$qqq$$), respectively [1, 2]. On the other hand, there is no proof in the fundamental theory of the strong interaction, quantum chromodynamics (QCD), to exclude hadrons having other structure than the ordinary mesons and baryons. In fact, various candidates for exotic hadrons have been proposed since the 1970s or earlier, such as tetraquarks having two quarks and two antiquarks as constituents, hybrids containing gluonic excitations as additional degrees of freedom, mesonic molecules which are bound or resonant states of two mesons, and so on [3–8]. However, due to the lack of clear experimental evidence, the existence of exotic hadrons has lacked a “smoking gun” for a long time.

The situation has changed radically due to the high statistics $$e^+e^-$$ collision data accumulated by $$B$$-factory experiments [9]. Originally their data were collected to perform a comprehensive study of the $$CP$$ violation in $$B$$ meson decays. At the same time, the high statistics data with abundant charm and bottom productions brought an ideal playground for the study of heavy hadron spectroscopy. In particular, the states called “$$X$$, $$Y$$, $$Z$$” are thought to be candidates for exotic hadrons, and have been attracting a lot of attention to reveal unvisited areas of QCD.

Such activities were initiated around 2003, which is the year when the first observation of $$X(3872)$$ was reported [10]. Interestingly, an observation of another candidate for an exotic state, the pentaquark $$\Theta ^+$$ baryon, was also reported in the same period, which together with $$X(3872)$$ triggered diverse activities in both theoretical and experimental studies [11, 12]. In this review, we describe the “$$X$$, $$Y$$, $$Z$$” states as best established states discovered by Belle as well as other relevant experiments and discuss their theoretical interpretations.

Historically, exotic states, in fact multiquark states, had already been pointed out by Gell-Mann in his original paper of the quark model in 1964 [1]. Before the discovery of the charm (heavy) quark, the situation for the light quark sector of $$u, d, s$$ quarks was somewhat complicated, due to their light masses which are comparable to the QCD scale of several hundred MeV. On the contrary, the masses of $$c$$ and $$b$$ quarks are large, approximately 1.5 $${\rm GeV}/c^2$$ and 5 $${\rm GeV}/c^2$$, respectively. Because of their large masses well-separated from the QCD scale, their description based on heavy constituent quark wave functions is well established. The success of the quark model with a static potential for heavy quarkonium systems may be explained by non-relativistic QCD (NRQCD) [13] or potential non-relativistic QCD (pNRQCD) [14]. In pNRQCD, the hierarchy of $$m \gg mv \gg mv^2$$ for small velocity $$v$$ justifies the use of a potential for non-relativistically slowly moving heavy quarks.

Fig. 1.

Charmonium spectrum (black solid bars) with some $$X$$, $$Y$$, $$Z$$ states (red solid bars) in comparison with the results of a conventional quark model for $$c \bar c$$ states (blue dashed bars) [15].

Fig. 1.

Charmonium spectrum (black solid bars) with some $$X$$, $$Y$$, $$Z$$ states (red solid bars) in comparison with the results of a conventional quark model for $$c \bar c$$ states (blue dashed bars) [15].

In Fig. 1, a charmonium spectrum is shown with some $$X$$, $$Y$$, $$Z$$ particles. The experimentally observed spectrum is shown by solid bars, and is compared with the predictions of the conventional $$\bar {c}c$$ quark model [15]; naive assignment is also shown. Below the open charm threshold of $$D\bar {D}$$, $$D \bar {D}^*$$, and $$D^* \bar {D}^*$$ the agreement between experiment and theory is remarkable, as anticipated by pNRQCD. On the contrary, near and above the threshold the situation changes: the clear hierarchy of small parameters may no longer hold well. Moreover, the coupling to the open charm (decaying) channels strongly affects the bare $$\bar cc$$ states. Due to confinement, an open charm necessarily requires the creation of a light $$q \bar q$$ pair to form charmed mesons, $$D$$ or $$D^*$$. Thus the presence of extra quark degrees of freedom associated with the energy deposit above the threshold makes the charmonium spectrum much richer [8], where realistic theoretical description requires coupled channel treatment of the quarkonium-like configuration to the open charm mesons.

Among various candidates for exotic hadrons, perhaps $$X(3872)$$ is the most established one experimentally, and is also well studied by theories [16]. The most accepted current picture for $$X(3872)$$ is a molecular state of the $$D$$ and $$\bar {D}^*$$ mesons.1 Because the observed $$X(3872)$$ is very close to the $$D \bar {D}^*$$ threshold, the difference between the masses of charged $$D^{\pm } D^{* \mp }$$ and the neutral $$D^{0} \bar {D}^{* 0}$$ channels has a large influence on the breaking of isospin symmetry. The fact that the neutral channel is much closer to the threshold than the charged one leads to the dominance of the neutral one. This in turn implies a large isospin mixing of $$I =0$$ and $$I=1$$ channels, reasonably explaining the decay branching ratio of $$X(3872) \to J/\psi \ \pi ^+ \pi ^-$$ and $$J/\psi \ \pi ^+ \pi ^- \pi ^0$$. We will discuss $$X(3872)$$ further in Sects. 2.3 and 3.4.1.

Theoretically there are other candidates for exotic hadrons such as tetraquarks, hybrids, or even pure glue-like states. The tetraquark is a multiquark configuration where colored diquarks are bound by a strong color force between them, thus forming a rather tightly bound compact object. So far, we still do not have good evidence for such tetraquarks. The hybrid states contain gluonic excitations, which is a consequence of the colored dynamics of QCD, and therefore their excitation energies are of the order of at least several hundred MeV. Moreover, because the gluons are flavor blind, such excitations couple to flavor singlet channels; in terms of hadrons, for instance, two pions. There are discussions that $$Y(4260)$$ and similar states could be candidates for such hybrid states [17–19], though we need more systematic studies not only in the charm but also in the bottom region.

In general, the above different configurations may mix for a given hadronic state as long as its quantum numbers allow. This is typically what we expect in quantum many-body systems with strong interaction near the threshold. As anticipated, this requires a treatment for the coupled channel problem. In hadron physics, these occur with non-trivial dynamics of QCD, such as color confinement and spontaneous breaking of chiral symmetry. Perhaps one of the aims of hadron physics is to clarify how and where different configurations show up, from which we hope to explain the observed data with a clear physical mechanism. We then expect to approach more fundamental questions of QCD to fill the missing link between quarks and hadrons.

Thus, in hadron physics, experimental information is extremely important, indicating the necessity of data to explore the strong interaction dynamics. In addition to $$X(3872)$$ and $$Y(4260)$$, more states have been well-established; two charged charmonium-like states of $$Z_c(4430)^+$$ and $$Z_c(3900)^+$$,2 as well as the twin charged bottomonia, $$Z_b(10610)^+$$ and $$Z_b(10650)^+$$ [20]. Due to their non-zero charge, the $$Z_{c, b}^+$$ states are truly four quark exotic hadrons. Experimentally, the confirmation of $$Z_c(4430)^+$$ by LHCb is also important: they performed an Argan plot analysis, verifying its resonance nature [21]. This is the first evidence that shows the particle nature among exotic hadrons. Theoretical interpretation, however, does not seem easy and its nature is still open. Concerning $$Z_b(10610)^+$$ and $$Z_b(10650)^+$$, the KEK B-factory is the only facility which has the $$\Upsilon (5S)$$3 data to make the measurements possible. The current observations, such as the peaks in several different final states [20], finding the neutral partner [23], and analysis of decay branching ratios [24] are all consistent with what is expected in the $$B^{(*)} \bar B^*$$ molecule picture.

This paper is organized as follows. In Sect. 2 we review the experimental status, focusing on the results brought by Belle from the early stage activities to the most recent achievements. Anomalous behavior in the $$e^+ e^-$$ reaction rate accompanied by two pion emission, so-called $$Y$$ states, is emphasized both in the charm and bottom sectors, which became the driving force of the new findings. The status of the most important states $$X(3872)$$, $$Y(4260)$$, $$Z_c(3900)^+$$, $$Z_c(4430)^+$$, $$Z_b(10610)^+$$, and $$Z_b(10650)^+$$ are briefly summarized. Section 3 presents the theory discussions. As basic issues of the theoretical ideas, we briefly describe heavy quark symmetry and chiral symmetry with its spontaneous breakdown, as well as important theoretical methods of QCD sum rule and lattice simulations. For interpretations of the exotic states, our discussions do not cover diverse approaches, but mostly concentrate on a rather well accepted interpretation, the molecular picture, where the basic effective degrees of freedom are supplied by the scattering hadrons instead of quarks. This is explained for the examples of $$X(3872)$$ and $$Z_b$$ resonances. We also give a brief review of other possible interpretations for $$Y(4260)$$, $$Z_c(4430)^+$$, and $$Z_c(3900)^+$$, and on recent discussion based on the dynamical treatment of the scattering hadrons. Some future prospects are also discussed. We will then summarize our discussions in Sect. 4.

Experiment

The Belle experiment

The Belle detector is a 4$$\pi $$ general purpose spectrometer with 1.5 T solenoidal magnetic field [25] (see Ref. [26] for a summary of its achievements). It is designed to measure time-dependent $$CP$$ violation in the $$B$$ meson system using (1) a silicon vertex detector (SVD), consisting of double-sided silicon strip detectors (DSSDs) to get $$B$$ meson decay vertices with superb resolution of 61 $$\mu $$m in $$J/\psi \to \mu ^+\mu ^-$$ mode; (2) a central drift chamber (CDC) filled with a half-and-half mixture of He and C$$_2$$H$$_6$$ gas at atmospheric pressure to give a high momentum resolution of 0.36% for 1 GeV/$$c$$ transverse momentum tracks; (3) a charged particle identification system comprising plastic scintillator array based time-of-flight (TOF) counters and aerogel Cerenkov counters (ACC) to identify charged kaons with typically 90% efficiency, with 10% pion misidentification; (4) a high resolution electromagnetic calorimeter based on CsI(Tl) scintillating crystals to realize 5 MeV/$$c^2$$$$\pi ^0$$ mass resolution; and (5) iron flux return instrumented by resistive planer chambers to identify muons and $$K^0_L$$ (KLM). Electron (muon) identification is done with 90% (also 90%) efficiency with a small fake rate of 0.3% (2%).

The KEKB collider [27, 28] stores 8 GeV $$e^-$$ and 3.5 GeV $$e^+$$ beams in the proper rings to produce $$\Upsilon (4S)$$ resonance and thus high statistics $$B$$ meson data. The accumulated integrated luminosity in that condition is 711 fb$$^{-1}$$, corresponding to 772 million $$B\bar {B}$$ pairs. The accelerator has also operated at $$\Upsilon (1S)$$, $$\Upsilon (2S)$$, and $$\Upsilon (5S)$$, up to the integrated luminosities of 6 fb$$^{-1}$$, 25 fb$$^{-1}$$, and 121 fb$$^{-1}$$, respectively. Runs at off-resonance energy as well as the energy scan to go above $$\Upsilon (5S)$$ were also performed. Thanks to the KEKB accelerator world record luminosity performance, each subset corresponds to two (or one) order of magnitude higher statistics than the older experiments. These high statistics data result in an extremely rich and interesting outcome for heavy flavored hadron spectroscopy.

Selections of exotic events

Heavy quarkonium (or simply quarkonium), which is the general term to describe charmonium ($$c\bar {c}$$) and bottomonium ($$b\bar {b}$$), can be regarded as the reference for discussing exotic hadrons. Among charmonia, $$J/\psi $$ and $$\psi (2S)$$ (or $$\psi ^\prime $$ in Fig. 1) decays to an $$e^+e^-$$ or $$\mu ^+\mu ^-$$ pair. Summing up these two decay modes, the branching fraction of interest for $$J/\psi $$ and $$\psi (2S)$$ amount to 12% and 1.6%, respectively. This enables us to reconstruct them with a low background. $$\psi (2S)$$ can also be reconstructed by the $$J/\psi \pi ^+\pi ^-$$ mode. The $$J/\psi \gamma $$ mode is useful in reconstructing $$\chi _{c1}$$ and $$\chi _{c2}$$ states. Similarly, we can have a clear signature through $$\mu ^+\mu ^-$$ modes for $$\Upsilon (1S)$$, $$\Upsilon (2S)$$, and $$\Upsilon (3S)$$ in bottomonia cases. The exotic hadron candidate states mentioned in this report are mostly discovered in the data analyses involving these quarkonia and decays.

For charmonia above the $$D$$ meson pair (open charm) threshold (3.74 GeV/$$c^2$$) and bottomonia above the $$B$$ meson pair (open bottom) threshold (10.56 GeV/$$c^2$$), the strong decays to those heavy meson pairs become dominant and consequently the decay width would be broad unless a special suppression mechanism works. Therefore we can identify an exotic hadron candidate as the quarkonium-like state above the corresponding meson pair threshold satisfying one or more criteria out of the following conditions: (1) large branching fraction to the mode other than the heavy meson pair, (2) extraordinary narrow decay width, (3) special decay mode which cannot be explained by two constituent quarks, (4) the observed mass does not match with any of the predicted ones, and (5) having an electric charge.

Various different reaction processes which can take place at the $$e^+e^-$$ collider would also help a lot to find and identify new quarkonium(-like) states. Depending on the production processes, the allowed or favored quantum numbers such as spin ($$J$$), parity ($$P$$), and charge conjugation ($$C$$) are different:

  1. The charmonium(-like) states of quantum numbers $$J^{PC}=0^{-+}$$, $$1^{--}$$, and $$1^{++}$$ are favored to be produced in two-body $$B$$ meson decays, because the factorization hypothesis works well, i.e. with small final state interaction due to the relatively large $$B$$ mass.

  2. Initial state radiation produces $$J^{PC}=1^{--}$$ states via a virtual photon exchange with varying effective center-mass energy.

  3. Two photon collisions produce $$J^{PC}=0^{\pm +}$$ and $$2^{\pm +}$$ states.

  4. Only $$C=+1$$ states can be produced in double charmonia production in $$e^+e^-$$ annihilation since $$J/\psi $$ or $$\psi (2S)$$ is used to tag a candidate event of this reaction (see Fig. 2).

Such quantum number selection rules are beneficial in discussing the possible interpretation of the observed states.

Fig. 2.

Four production processes in $$e^{+}e^{-}$$ colliders. See the text for an explanation.

Fig. 2.

Four production processes in $$e^{+}e^{-}$$ colliders. See the text for an explanation.

$$X(3872)$$

In 2003, Belle found a very narrow peak in the $$J/\psi \pi ^+\pi ^-$$ mass spectrum at 3872 MeV/$$c^2$$ in $$B^{\pm } \to J/\psi \pi ^+\pi ^- K^{\pm }$$ decays, as shown in Fig. 3, using the data corresponding to an integrated luminosity of 140 fb$$^{-1}$$ which contains 152 million $$B\bar {B}$$ pairs [10]. The discovered state is named $$X(3872)$$. The letter “$$X$$” was chosen because of its extraordinary properties; in spite of its mass being well above the $$D\bar {D}$$ threshold, its decay width is surprisingly narrow, the observed decay mode is $$J/\psi \pi ^+\pi ^-$$, and there was no obvious assignment to a conventional charmonium.

Fig. 3.

The distribution of mass difference between $$J/\psi \pi ^+\pi ^-$$ and $$J/\psi $$ in $$B^{\pm } \to J/\psi \pi ^+\pi ^- K^{\pm }$$ decays. The peak at 0.59 GeV/$$c^2$$ is due to the conventional charmonium, $$\psi (2S)$$. The peak corresponding to the $$X(3872)$$ is indicated by a vertical arrow [10].

Fig. 3.

The distribution of mass difference between $$J/\psi \pi ^+\pi ^-$$ and $$J/\psi $$ in $$B^{\pm } \to J/\psi \pi ^+\pi ^- K^{\pm }$$ decays. The peak at 0.59 GeV/$$c^2$$ is due to the conventional charmonium, $$\psi (2S)$$. The peak corresponding to the $$X(3872)$$ is indicated by a vertical arrow [10].

Immediately after this narrow state was reported, a lot of discussions arose which attempted to give a proper interpretation. What experimentalists should do to reveal $$X(3872)$$'s nature would be to determine its quantum number $$J^{PC}$$. The $$X(3872) \to J/\psi \gamma $$ mode is established by both Belle [29] and BaBar [30] measurements. The Belle result is shown in Fig. 4, and thus it is confirmed that the charge conjugation of $$X(3872)$$ is $$C=+1$$.

Fig. 4.

The $$J/\psi \gamma $$ invariant mass distributions for (left) $$B^{\pm } \to J/\psi \gamma K^{\pm }$$ and (right) $$B^0 \to J/\psi \gamma K^0_S$$ decays [29].

Fig. 4.

The $$J/\psi \gamma $$ invariant mass distributions for (left) $$B^{\pm } \to J/\psi \gamma K^{\pm }$$ and (right) $$B^0 \to J/\psi \gamma K^0_S$$ decays [29].

It is also possible to determine the spin and parity by the angular distribution of decay products of $$X(3872)$$. The studies for the $$J/\psi \pi ^+\pi ^-$$ mode by CDF [31] and Belle [32] using three decay angular variables, as well as the 3$$\pi $$ invariant mass spectrum in the $$J/\psi \pi ^+\pi ^- \pi ^0$$ mode by BaBar [33], give a constraint on $$J^{PC}$$ to be either $$1^{++}$$ or $$2^{-+}$$, but do not reach a definitive determination. A full five-dimensional amplitude analysis was performed by the LHCb collaboration [34] for the $$X(3872) \to J/\psi \pi ^+\pi ^-$$ mode and $$J^{PC}=1^{++}$$ was unambiguously assigned.

In a tetraquark hypothesis for $$X(3872)$$, mass splitting due to the mixing between $$c\bar {c}u\bar {u}$$ and $$c\bar {c}d\bar {d}$$ was theoretically predicted [35], $$M(X_{{\rm h}}) - M(X_{{\rm l}}) = (7 \pm 2)/\cos (2\theta )$$ MeV, where $$M(X_{{\rm h,l}})$$ are the masses of the higher and lower states after the mixing of the two states. The difference is expected to appear as the difference in the $$X$$ masses separately measured in $$B^{\pm } \to X K^{\pm }$$ and $$B^0 \to X K^0$$. The Belle result for this difference in the $$J/\psi \pi ^+\pi ^-$$ mode is found to be ($$-0.71 \pm 0.96\,\mbox {(stat)} \pm 0.19\,\mbox {(syst)}$$) MeV/$$c^2$$ [32], which strongly disfavors the tetraquark interpretation. The BaBar measurement for this quantity is reported to be ($$2.7 \pm 1.6\,\mbox {(stat)} \pm 0.4\,\mbox {(syst)}$$) MeV/$$c^2$$ [36]. Belle also finds no signature for the charged partner state in the $$J/\psi \pi ^{\pm } \pi ^0$$ mode.

On the other hand, a $$D^0 \bar {D}{}^{*0}$$ molecule is proposed as very plausible interpretation for $$X(3872)$$ because of the very narrow width ($$\Gamma < 1.2$$ MeV) and the mass $$\left (M = 3871.69 \pm 0.17\,{\rm MeV}/c^2\right )$$ [22] being very close to the $$D^0 \bar {D}{}^{*0}$$ threshold. Inspired by this argument, $$B \to D^0\bar {D}{}^{*0} K$$ decays are reconstructed to examine the $$D^0 \bar {D}{}^{*0}$$ invariant mass spectrum and a clear excess has been observed [37, 38]; the Belle result is shown in Fig. 5. The observed mass peak is consistent with that determined by the $$J/\psi \pi ^+\pi ^-$$ mode. The resultant $$\mathcal {B}\big (X(3872) \to D^0 \overline {D}{}^{*0}\big )$$ is found to be 10 times as large as $$\mathcal {B}\big (X(3872) \to J/\psi \pi ^+ \pi ^-\big )$$.

Fig. 5.

The $$D^0 \bar {D}{}^{*0}$$ invariant mass distributions for (left) $$\bar {D}{}^{*0} \to \bar {D}{}^0 \gamma $$ and (right) $$\bar {D}{}^{*0} \to \bar {D}{}^0 \pi ^0$$ modes in $$B \to D^0\bar {D}{}^{*0} K$$ decays [38].

Fig. 5.

The $$D^0 \bar {D}{}^{*0}$$ invariant mass distributions for (left) $$\bar {D}{}^{*0} \to \bar {D}{}^0 \gamma $$ and (right) $$\bar {D}{}^{*0} \to \bar {D}{}^0 \pi ^0$$ modes in $$B \to D^0\bar {D}{}^{*0} K$$ decays [38].

The difference between the observed mass and the threshold is regarded as the binding energy and it is small, $$0.11 \pm 0.21$$ MeV [22]. This value would give an estimation of the $$X(3872)$$'s size, i.e. the distance between $$D^0$$ and $$\bar {D}{}^{*0}$$ mesons, of 10 fm (see Sect. 3.4.1 for details). In this case, the entire volume of $$X(3872)$$ would become about 1000 times larger than that for the $$J/\psi $$, whose diameter is estimated to be 0.4 fm, and thus the $$D^0 \bar {D}{}^{*0}$$ mode should have a much larger branching fraction with respect to the $$J/\psi \pi ^+\pi ^-$$ mode. Small binding energy raises another question in the molecular state hypothesis, because the $$X(3872)$$ must be so fragile that it would be hard to explain the observed production rate in the high energy $$p\bar {p}$$ collisions at the Tevatron [39].

It is therefore plausible that $$X(3872)$$ could be an admixture of a molecular state and charmonium having the same quantum numbers (Sect. 3.4.1). $$\chi _{c1}(2P)$$ is an unconfirmed charmonium with $$J^{PC}=1^{++}$$, and its original mass is expected to be about 3920 MeV/$$c^2$$. In the admixture scenario, the $$X(3872)$$ is comprised mainly of the $$D^0 \bar {D}{}^{*0}$$ molecule and also contains a sizable $$\chi _{c1}(2P)$$ component. The pure $$\chi _{c1}(2P)$$ state itself must be a compact object, and thus it is possible to explain the production in high energy collisions at either Tevatron or LHC. It can also explain the significant branching fractions to the final states containing a $$J/\psi $$. The $$\chi _{c1}(2P)$$ has not yet been observed—the admixture hypothesis also gives an explanation: we can think of the $$X(3872)$$ partner state, which is perpendicular to $$X(3872)$$, by containing $$\chi _{c1}(2P)$$ as the main component and it would become several hundreds of MeV/$$c^2$$ heavier than the lower state. This larger mass allows a strong decay by emitting pion(s), and consequently it may become so broad a state that it is difficult to recognize as a resonance in the currently available data statistics. Hence, the admixture hypothesis has not been contradicted in the compilation of the experimental results.

The radiative decay of $$X(3872) \to \psi (2S) \gamma $$ would be suppressed in a pure meson–meson molecule hypothesis because of a small $$Q$$-value, while it can be significant if $$X(3872)$$ is the unconfirmed charmonium of $$\chi _{c1}(2P)$$. BaBar reported that $$\mathcal {B}(X(3872) \to \psi (2S) \gamma )$$ is three times as large as $$J/\psi \gamma $$, $$\frac {\mathcal {B}(X(3872) \to \psi (2S) \gamma )} {\mathcal {B}(X(3872) \to J/\psi \gamma )}=3.4\pm 1.4$$ [30]. But Belle found no evidence and gave an upper limit of $$\frac {\mathcal {B}(X(3872) \to \psi (2S) \gamma )} {\mathcal {B}(X(3872) \to J/\psi \gamma )} < 2.1$$ at 90% C.L. [29]. Recently, the LHCb experiment also reported the radiative decays of $$X(3872) \to J/\psi \gamma $$ and $$\psi (2S) \gamma $$. Reconstruction of the $$\psi $$, which denotes a $$J/\psi $$ or $$\psi (2S)$$ meson, is done by the $$\mu ^+ \mu ^-$$ mode only, because of the difficulty in achieving enough reconstruction efficiencies for other decay modes. Signal extraction is done by an extended maximum likelihood fit in the two-dimensional space of the invariant masses of $$\psi \gamma K^{\pm }$$ and $$\psi \gamma $$, where the former peaks at the $$B^{\pm }$$ mass and the latter forms a peak at the $$X(3872)$$ mass for the signal. It confirms the $$X(3872)$$ radiative decay in $$J/\psi \gamma $$ mode; at the same time, it finds evidence of $$X(3872) \to \psi (2S) \gamma $$ with a significance of 4.4 $$\sigma $$ [40]. The ratio of $$X(3872) \to \psi (2S) \gamma $$ to $$X(3872) \to J/\psi \gamma $$ is obtained to be $$\frac {\mathcal {B}(X(3872) \to \psi (2S) \gamma )} {\mathcal {B}(X(3872) \to J/\psi \gamma )}=2.46\pm 0.64\,\mbox {(stat)}\pm 0.29\,\mbox {(syst)}$$. Neither BaBar nor Belle results contradict this LHCb result within the uncertainties; the resultant branching fraction disfavors an interpretation as a pure $$D\bar {D}^*$$ molecule and supports the charmonium and molecular admixture state hypothesis.

The functional hypotheses described above are the main progress in both theory and experiment in the recent years to give an interpretation for the $$X(3872)$$ state, which will be discussed in some detail in Sect. 3.4.1.

$$Y(4260)$$ and some similar states in initial state radiation

In $$e^+ e^-$$ colliders, one of the beam particles may emit a photon and consequently $$e^+ e^-$$ annihilation takes place at a lower effective center mass energy. This reaction is called the initial state radiation (ISR) or radiative return, and produces $$J^{PC}=1^{--}$$ states via a virtual photon at any reachable invariant masses. In 2005, BaBar reported a resonance at 4260 MeV/$$c^2$$ in the $$J/\psi \pi ^+ \pi ^-$$ final state [41], and this state is named $$Y(4260)$$. Belle confirmed the same resonance, and another cluster of events at 4008 MeV/$$c^2$$ was seen [42]. Figure 6 shows the $$J/\psi \pi ^+ \pi ^-$$ invariant mass distribution with the full data sample [43]. After these discoveries, the hadron which is created by ISR and contains a $$c\bar {c}$$ constituent but cannot be interpreted to be a charmonium has come to be called “$$Y$$.”

Fig. 6.

Invariant mass distributions of the $$J/\psi \pi ^+\pi ^-$$ in ISR events. Points with error bars are data and the shaded histograms are the normalized $$J/\psi $$ mass sidebands for background estimation. The solid line shows the best fit with two coherent resonances and background while the dashed and dot-dashed lines are solutions I and II. The distributions on a logarithmic vertical scale are shown in the inset and the peak at 3686 MeV/$$c^2$$ is the $$\psi (2S) \to J/\psi \pi ^+\pi ^-$$ contribution [43].

Fig. 6.

Invariant mass distributions of the $$J/\psi \pi ^+\pi ^-$$ in ISR events. Points with error bars are data and the shaded histograms are the normalized $$J/\psi $$ mass sidebands for background estimation. The solid line shows the best fit with two coherent resonances and background while the dashed and dot-dashed lines are solutions I and II. The distributions on a logarithmic vertical scale are shown in the inset and the peak at 3686 MeV/$$c^2$$ is the $$\psi (2S) \to J/\psi \pi ^+\pi ^-$$ contribution [43].

In the analysis for the $$\psi (2S) \pi ^+\pi ^-$$ channel, BaBar reported a relatively broad resonance with mass 4324 MeV/$$c^2$$ and width 172 MeV/$$c^2$$ [44], while Belle found two peaks at 4361 MeV/$$c^2$$ and 4664 MeV/$$c^2$$ [45]. The last result from BaBar [46] confirmed those two peaks, and thus their first result is interpreted as the hypothesis that two resonances could not be separately recognized because of less statistics. It is unusual that the $$Y$$ state decays to $$J/\psi \pi ^+\pi ^-$$ or $$\psi (2S) \pi ^+\pi ^-$$ final states despite the heavier mass than the $$D\overline {D}$$ threshold, contrary to the fact that $$\psi (3770)$$ decays predominantly to a $$D\overline {D}$$ pair. In addition, only one state is predicted among unconfirmed $$J^{PC}=1^{--}$$ charmonia above 4 GeV/$$c^2$$. Therefore these “$$Y$$” states are thought to have a different structure from the conventional charmonium, as an orbital excitation of $$c \bar c$$. Rather, the large $$\pi ^+ \pi ^-$$ decay with its scalar and isoscalar correlations would be possible from flavor blind gluonic excitations, rendering its interpretation as a hybrid state [17–19]. A short review of $$Y(4260)$$ with various interpretations is discussed in Sect. 3.4.3.1.

The CLEO experiment also performed measurement of the $$J/\psi \pi ^+\pi ^-$$ production cross section and confirmed it exhibits a peak around $$\sqrt {s}=4.26$$ GeV/$$c^2$$ [47], while BES measurement shows that there is a dip in the total hadronic cross section as a function of $$\sqrt {s}$$ in the same center-mass-energy region [48]. Thus, the fraction of $$J/\psi \pi ^+\pi ^-$$ production in the total hadronic cross section is estimated to be very high and the resultant partial width $$\Gamma \big (Y(4260) \to J/\psi \pi ^+\pi ^-\big )$$ should be more than 1.0 MeV (at 90% C.L.), which is too large to interpret it as an ordinary charmonium. This $$Y(4260)$$ and related states are thought to be similar to the bottomonium-like exotics described in Sect. 2.7. Such a point of view lead us to search for charged charmonium-like exotics as an intermediate state of $$Y(4260)$$ decays, as discussed later.

$$Z_c(4430)^+$$ and some similar states

In either molecular state or tetraquark hypothesis, the number of constituent quarks is four. If a quarkonium-like state has an electric charge, this is a firm signature that it has four constituent quarks. The most straightforward way to search for the charged state is looking for a peak in the invariant mass spectrum for a charmonium and a charged pion system. In 2007, Belle reported a peak at 4430 MeV/$$c^2$$ in the $$\psi (2S) \pi ^+$$ invariant mass spectrum for $$B \to \psi (2S) \pi ^+ K$$ three-body decays [49], as shown in Fig. 7.

Fig. 7.

The invariant mass spectrum for $$\psi (2S) \pi ^+$$ in $$B \to \psi (2S) \pi ^+ K$$ decays. A significant peak is observed at 4430 MeV/$$c^2$$; it is called $$Z_c(4430)^+$$. Superimposed lines show the Breit–Wigner lineshape for the $$Z_c(4430)^+$$ signal and the phase-space-like non-resonant $$\psi (2S) \pi ^+$$ component. The shaded histogram shows the estimated background distribution by the $$B$$ decay signal sideband events [49].

Fig. 7.

The invariant mass spectrum for $$\psi (2S) \pi ^+$$ in $$B \to \psi (2S) \pi ^+ K$$ decays. A significant peak is observed at 4430 MeV/$$c^2$$; it is called $$Z_c(4430)^+$$. Superimposed lines show the Breit–Wigner lineshape for the $$Z_c(4430)^+$$ signal and the phase-space-like non-resonant $$\psi (2S) \pi ^+$$ component. The shaded histogram shows the estimated background distribution by the $$B$$ decay signal sideband events [49].

BaBar analyzed the same $$B$$ decay process and found the effect corresponding only to 1.9 $$\sigma $$ statistical significance, thus an upper limit was presented [50], stating that two $$B$$-factories' results are not statistically contradicting each other. In $$B \to \psi (2S) \pi ^+ K$$ decays, major contributions come from ordinary quasi-two-body $$B$$ decays, $$B \to \psi (2S) K^*$$ where $$K^* \to K \pi ^+$$. In Belle's first discovery paper [49], $$B \to \psi (2S) K^*(892)$$ and $$B \to \psi (2S) K^*(1430)$$ dominant regions were removed according to the $$K \pi ^+$$ invariant mass. However, it is important to see if the potential interference between those $$B \to \psi (2S) K^*$$ decays and other featureless processes such as non-resonant three-body $$B$$ decays to $$\psi (2S) \pi ^+ K$$ may exhibit a peak-like projection in $$\psi (2S) \pi ^+$$. Hence, Belle reanalyzed the same data with a Dalitz analysis approach, which is a fit featuring all possible three-body $$B$$ decay amplitudes written as a function on the Dalitz plane [51]. The result obtained by this advanced analysis technique turned out to have a significance of 6.4 $$\sigma $$. In addition, the most recent Belle study using a four-dimensional full amplitude analysis of $$\bar {B}^0 \to \psi (2S) \pi ^+ K^-$$ decays constrains the spin and parity of the $$Z_c(4430)^+$$. Here, the $$\psi (2S)$$ is reconstructed via its $$e^+e^-$$ or $$\mu ^+\mu ^-$$ decay channels and the amplitude of this $$B$$ decay is expressed by the helicity formalism in a four-dimensional parameter space of $$M^2_{K\pi }$$, $$M^2_{\psi (2S)\pi }$$, $$\theta _{\psi (2S)}$$, and $$\phi $$, where $$M^2_{K\pi } \left (M^2_{\psi (2S)\pi }\right )$$ is the squared mass of the $$K\pi $$ ($$\psi (2S) \pi $$) system, $$\theta _{\psi (2S)}$$ is the $$\psi (2S)$$ helicity angle defined as the angle between the daughter lepton's momentum in the $$\psi (2S)$$ rest frame and the $$\psi (2S)$$ momentum in the mother $$B$$ rest frame, and $$\phi $$ is the angle between the $$\psi (2S)$$ decay plane and the plane containing the $$K$$ and $$\pi $$ momentum vectors. Each of the different intermediate two-body states is represented by the Breit–Wigner, and the sum of those contributions forms the total decay amplitude. In total, six possible $$K^*$$ resonances, a non-resonant $$K\pi $$ component, and a charged $$Z_c$$ state are taken into account. The fit result favors $$J^P = 1^+$$ over the $$0^-$$, $$1^-$$, and $$2^+$$ hypothesis at 3.4 $$\sigma $$ or more [52]. Finally, in 2014, LHCb reported confirmation of the resonant structure of $$Z_c(4430)^+$$ [21] by performing the same amplitude analysis as Belle. As a result, they ruled out the $$J^P=0^-$$, $$1^-$$, $$2^+$$, and $$2^-$$ hypotheses by more than 9 $$\sigma $$ relative to $$1^+$$, thus now the quantum number is unambiguously determined. Exploiting the large $$B$$ decay signal statistics brought by a $$pp$$ collision environment, a fit to obtain the $$Z_c(4430)^+$$ amplitude in six $$M^2_{\psi (2S) \pi }$$ bins has been performed and plotted in an Argand diagram. The phase is found to increase as a function of $$M^2_{\psi (2S) \pi }$$; this is consistent with the expectation from a Breit–Wigner formula, and thus the $$Z_c(4430)^+$$ is confirmed to be a real resonance. Its mass and width are determined to be $$M = 4478^{+15}_{-18}$$ MeV/$$c^2$$ and $$\Gamma = 181\pm 31$$ MeV, respectively, in the latest world average [22].

The discovery of $$Z_c(4430)^+$$ is striking, and leads us to the knowledge that exotic hadrons can really be formed. However, it is not yet easy to reach a convincing interpretation for the $$Z_c(4430)^+$$ for the following reasons: (1) its mass is so far away from any charmed meson pair threshold that a molecular state hypothesis can hardly be recognized as a plausible scenario, (2) neither another decay mode nor a partner state has been reported. Further searches for other decay modes and partner states from the experimental side are still necessary to settle the argument about this charged state's structure. Various theoretical proposals are briefly discussed in Sect. 3.4.3.2.

In $$\bar {B}^0 \to \chi _{c1} \pi ^+ K^-$$ decay, Belle reported two resonance-like structures in the $$\chi _{c1} \pi ^+$$ mass spectrum [53]. From a Dalitz plot analysis, a fit gives the two resonance-like objects' masses and widths: $$M_1 = 4051$$ MeV/$$c^2$$, $$\Gamma _1 = 82$$ MeV, $$M_2 = 4248$$ MeV/$$c^2$$, $$\Gamma _2 = 177$$ MeV. Meanwhile, BaBar reported that the distribution obtained in the same $$B$$ decay mode can be explained without these charged charmonium-like objects [54]. Thus, attempts to make a confirmation with higher statistics are necessary for these states. Belle also visited the $$\bar {B}^0 \to J/\psi \pi ^+ K^-$$ decay. In order to resolve possible intermediate states, an amplitude analysis which is the same approach as $$Z_c(4300)^+ \to \psi (2S) \pi ^+$$ is performed, taking 10 possible $$K^*$$ resonances into account. The $$Z_c(4430)^+ \to J/\psi \pi ^+$$ is found to be evident and, in addition, one more charged charmonium-like state of $$Z_c(4200)^+ \to J/\psi \pi ^+$$ is observed with a significance of 6.2 $$\sigma $$. Its mass and width are obtained to be $$M = 4196^{+31}_{-29} {}^{+17}_{-13}$$ MeV/$$c^2$$ and $$\Gamma = 370 \pm 70 ^{+70}_{-132}$$ MeV, where the first and second errors are statistical and systematic, respectively, and $$J^P = 1^+$$ is the most favored quantum number.

These results tell us that multi-body $$B$$ decays are functional as the source of the exotic charmonium-like states, and extraction of information often requires Dalitz or more sophisticated amplitude analysis techniques.

$$Z_c(3900)^+$$ and some similar states

The large rate for $$\Upsilon (nS) \pi ^+\pi ^-$$ ($$n=1,2,3)$$ production at $$\Upsilon (5S)$$ led to the discovery of $$Z_b(10610)^+$$ and $$Z_b(10650)^+$$ in $$\Upsilon (nS) \pi ^+$$ and $$h_b(mP) \pi ^+$$ ($$m=1,2$$) final states, as will be discussed in Sect. 2.8.2. Since $$Y(4260)$$ has an anomalous high rate of $$J/\psi \pi ^+\pi ^-$$ production as described in Sect. 2.4, it can be interpreted as an analog in the charmonium(-like) sector. In this point of view, the charged bottomonium-like particles' appearance at $$\Upsilon (5S)$$ inspired us to visit the $$J/\psi \pi ^+$$ intermediate state in $$Y(4260) \to J/\psi \pi ^+\pi ^-$$.

In a $$Y(4260) \to J/\psi \pi ^+ \pi ^-$$ event, there are two combinations of the $$J/\psi $$ and one charged pion. Among them, the one having the larger invariant mass, $$M_{\rm max}(\pi J/\psi )$$, is taken. The $$M_{\rm max}(\pi J/\psi )$$ distribution is shown in Fig. 8. We see a significant enhancement at 3900 MeV/$$c^2$$ with 5.2 $$\sigma $$ significance. Its mass and width are obtained to be $$M= 3895.5\pm 6.6\pm 4.5$$ MeV/$$c^2$$ and $$\Gamma = 63\pm 24\pm 26$$ MeV, where the first and second errors are statistical and systematic, respectively. This structure can be interpreted as a new charged charmonium-like state.

Fig. 8.

The invariant mass spectrum for the larger one among two possible combinations of the $$J/\psi $$ and one charged pion ($$M_{\rm max}(\pi J/\psi )$$) in the $$Y(4260) \to J/\psi \pi ^+ \pi ^-$$ candidates. A significant peak is seen around 3900 MeV/$$c^2$$ and it is called $$Z_c(3900)^+$$. Superimposed lines show the Breit–Wigner lineshape for the $$Z_c(3900)^+$$ signal and the phase-space-like background. The shaded histogram shows the estimated non-$$J/\psi $$ background by the normalized sideband events [43].

Fig. 8.

The invariant mass spectrum for the larger one among two possible combinations of the $$J/\psi $$ and one charged pion ($$M_{\rm max}(\pi J/\psi )$$) in the $$Y(4260) \to J/\psi \pi ^+ \pi ^-$$ candidates. A significant peak is seen around 3900 MeV/$$c^2$$ and it is called $$Z_c(3900)^+$$. Superimposed lines show the Breit–Wigner lineshape for the $$Z_c(3900)^+$$ signal and the phase-space-like background. The shaded histogram shows the estimated non-$$J/\psi $$ background by the normalized sideband events [43].

The BES III experiment at the Beijing Electron Positron Collider performed a dedicated run setting the center mass energy at 4260 MeV to collect 525 pb$$^{-1}$$. From this data sample, they also found the structure in $$M_{\rm max}(\pi J/\psi )$$ around 3.9 GeV/$$c^2$$ and obtained the mass $$M=3899.0\pm 3.6\pm 4.9$$ MeV/$$c^2$$ and width $$\Gamma = 46\pm 10\pm 20$$ MeV [55]. Hence, this charged charmonium-like state $$Z_c(3900)^+$$ is confirmed.

Because of the similarity between $$Y(4260) \to J/\psi \pi ^+\pi ^-$$ and $$\Upsilon (5S) \to \Upsilon (nS) \pi ^+\pi ^-$$, as well as the discovery of $$Z_b(10610)^+$$ and $$Z_b(10650)^+$$ states, one possible hypothesis is regarding $$Z_c(3900)^+$$ as a $$(D \bar {D}{}^*)^+$$ molecular state. Using BES III data collected at $$\sqrt {s} = 4.26$$ GeV, the $$e^+e^- \to (D \bar {D}{}^*)^{\pm } \pi ^{\mp }$$ process is visited by a partial reconstruction technique requiring a charged pion and a $$D$$ meson to tag the candidate events. The $$D^*$$ signal is identified as the corresponding peak in the $$\pi D$$ recoil mass spectrum. In both $$D^0 D^{*-}$$ and $$D^+ \bar {D}{}^{*0}$$ cases, the clear enhancement in the $$D \bar {D}{}^*$$ mass spectrum just above the threshold is significant. The extracted resonance parameters are pole mass $$M_{\rm pole} = 3883.9 \pm 1.5 \pm 4.2$$ MeV/$$c^2$$ and width $$\Gamma _{\rm pole} = 24.8 \pm 3.3 \pm 11.0$$ MeV. Here, $$Z_c(3885)^+$$ denotes this structure in the $$(D \bar {D}{}^*)^+$$ final state. Assuming that the $$Z_c(3885)^+$$ is same as the $$Z_c(3900)^+$$ in the $$J/\psi \pi ^+$$ mode, the ratio of partial decay widths is obtained to be $$\Gamma \left (Z_c(3885)^+ \to (D\bar {D}{}^*)^+\right )/\Gamma \left ( Z_c(3900)^+ \to J/\psi \pi ^+\right ) = 6.2 \pm 1.1 \pm 2.7$$, which is considerably smaller than that for conventional charmonium states and thus implies a very different mechanism in the $$Y(4260)$$-$$Z_c^+$$ system [56].

More searches for analogous reactions have been performed. Exploiting a partial reconstruction technique, the $$e^+e^- \to \big (D^* \bar {D}{}^*\big )^{\pm } \pi ^{\mp }$$ process is reconstructed and an enhancement just above the $$\big (D^*\bar {D}{}^*\big )^{\pm }$$ threshold has been observed. This enhancement is called $$Z_c(4025)^+$$ and the fit assuming that it is a resonance described by an $$S$$-wave Breit–Wigner lineshape gives the mass and width to be $$M = 4026.3 \pm 2.6 \pm 3.7$$ MeV/$$c^2$$ and $$\Gamma = 24.8 \pm 5.6 \pm 7.7$$ MeV, respectively. This can be regarded as a candidate for the charm sector counterpart of the $$Z_b(10650)^+$$ state [57]. The $$e^+e^- \to \pi ^+\pi ^- h_c$$ process is also studied at 13 center mass energies from 3.900 GeV to 4.420 GeV by a full reconstruction approach with the radiative decay mode of $$h_c \to \gamma \eta _c$$ followed by the $$\eta _c$$ decays in 16 exclusive hadronic final states. The spectrum of the invariant mass of the $$\pi ^+ h_c$$ system, $$M_{\pi ^+ h_c}$$, shows a distinct structure and it is referred to as $$Z_c(4020)^+$$ with a mass of $$4022.9\pm 0.8\pm 2.7$$ MeV/$$c^2$$ and width of $$7.9\pm 2.7\pm 2.6$$ MeV [58].

These results suggest a similar special mechanism in the $$J^{PC}=1^{--}$$ state production followed by the decay to a quarkonium and a charged pion pair to produce a charged object as an intermediate state in both charmonium and bottomonium cases. However, as described in Sect. 2.8, all the $$B\overline {B}{}^*$$, $$B^* \overline {B}{}^*$$, $$\Upsilon (nS) \pi ^+$$ ($$n=1,2,3$$), and $$h_b(mP)\pi ^+$$ ($$m=1,2$$) modes are commonly possessed by the two charged states of $$Z_b(10610)^+$$ and $$Z_b(10650)^+$$ at $$\Upsilon (5S)$$, while several different charged charmonium-like structures are found depending on the final state in the $$Y(4260)$$ region. Proper theoretical approaches to explain such a difference between charmonium-like and bottomonium-like cases are still awaited. Current theoretical studies for $$Z_c(3900)^+$$ and related discussions are briefly given in Sect. 3.4.3.3.

$$Y_b$$ state

As described Sect. 2.4, the discovery of $$Y(4260)$$ in the initial state radiation process [41] and the following similar states [42, 44, 45] are thought to be candidates for exotic states. The corresponding exotic hadrons where a charm and an anti-charm quark are replaced by a bottom and an anti-bottom quark are expected to have masses around the $$\Upsilon (5S)$$ region [59, 60]. The corresponding particle (denoted as $$Y_b$$ hereafter) has $$J^{PC} = 1^{--}$$, and a large decay rate into the $$\Upsilon (nS) \pi ^+\pi ^-$$ ($$n = 1, 2, 3$$) final state is expected. Therefore, Belle studied the decays $$\Upsilon (5S) \to \Upsilon (nS) \pi ^+\pi ^-$$ using 21.7 fb$$^{-1}$$ data accumulated at the $$\Upsilon (5S)$$ peak.

Assuming that $$\Upsilon (5S)$$ is an ordinary bottomonium above the open bottom threshold, only a few events were expected with this amount of data. However, Belle observed a clear and anomalously large signal in the $$\Upsilon (1S) \pi ^+\pi ^-$$ and $$\Upsilon (2S) \pi ^+\pi ^-$$ channels. The measured partial widths are about 100 times larger than those of $$\Upsilon (n'S)$$ ($$n' = 2, 3, 4$$) resonances. The result was presented at the Hadron 2007 conference [61].

Although the inclusive energy dependence does not show a peak-like structure, it showed a characteristic structure when the exclusive process going to $$J/\psi \pi \pi $$ was measured at around $$Y(4260)$$. This led to the idea that there could be a $$Y_b$$ state different from $$\Upsilon (5S)$$ with a different mass around 10.9 GeV/$$c^2$$. In order to check this hypothesis, an energy scan was proposed and performed in December 2007. The data were taken at six energy points from $$\sqrt {s}$$ = 10.83 GeV to 11.02 GeV with a total integrated luminosity of 8.1 fb$$^{-1}$$. In addition, there were nine scan points with an integrated luminosity of about 30 pb$$^{-1}$$ each, in order to measure the hadronic line shape with more data points. The energy dependence of the cross section $$e^+e^- \to \Upsilon (nS)\pi ^+\pi ^-$$ ($$n = 1, 2, 3$$) is shown in Fig. 9 [62]. A peak structure is observed for all three final states. The line shapes of the cross sections are fitted by a common Breit–Wigner function, and the mass and width values are compared with those obtained from the fit to the hadronic cross section (Fig. 10). The differences between the Breit–Wigner parameters determined from hadronic decay and those from $$\Upsilon (nS)\pi ^+\pi ^-$$ decays were measured to be $$9 \pm 4$$ MeV/$$c^2$$ in mass and $$-15 ^{+11}_{-12}$$ MeV/$$c^2$$ in width, which correspond to 2.0 $$\sigma $$ including the systematic error.

Fig. 9.

The energy-dependent cross sections for $$e^+e^- \to \Upsilon (nS)\pi ^+\pi ^-$$ ($$n=1, 2, 3$$) processes normalized to the leading-order $$e^+e^- \to \mu ^+\mu ^-$$ cross sections. The results of the fits are shown as smooth curves. The vertical dashed line indicates the energy at which the hadronic cross section is maximal [62].

Fig. 9.

The energy-dependent cross sections for $$e^+e^- \to \Upsilon (nS)\pi ^+\pi ^-$$ ($$n=1, 2, 3$$) processes normalized to the leading-order $$e^+e^- \to \mu ^+\mu ^-$$ cross sections. The results of the fits are shown as smooth curves. The vertical dashed line indicates the energy at which the hadronic cross section is maximal [62].

Fig. 10.

The energy-dependent cross sections for $$e^+e^- \to b\bar b$$ processes normalized to the leading-order $$e^+e^- \to \mu ^+\mu ^-$$ cross sections. The horizontal dotted line is the non-interfering non-resonance contribution [62].

Fig. 10.

The energy-dependent cross sections for $$e^+e^- \to b\bar b$$ processes normalized to the leading-order $$e^+e^- \to \mu ^+\mu ^-$$ cross sections. The horizontal dotted line is the non-interfering non-resonance contribution [62].

In 2010, from May to June, which is a last run period of Belle experiment, a new energy scan was performed. Data were taken at additional 16 energy points with about 1 fb$$^{-1}$$ each from 10.63 to 11.02 GeV. Also, a fine energy scan was made with 5 MeV intervals with 50 pb$$^{-1}$$ at each of 61 points from 10.75 to 11.05 GeV. The new data together with the previously taken data were analyzed. The results on the differences in the two Breit–Wigner parameters were $$9.2 \pm 3.9$$ MeV/$$c^2$$ in mass and $$5.2 \pm 8.1$$ MeV/$$c^2$$ in width, which corresponds to about the 2 $$\sigma $$ level [63]. Here, note that we could use only a relatively simple model comprised of the sum of two Breit–Wigner functions and a flat non-resonant component. Potentially its imperfect ability to describe hadronic cross sections may limit the reliability of the parameters obtained. We cannot totally exclude such an issue as a reason for the resonance particle decaying to hadrons and that decaying to $$\Upsilon (nS)\pi ^+\pi ^-$$ not being proved to be different particles. However, the anomalously large rates to $$\Upsilon (nS)\pi ^+\pi ^-$$ remain a mystery.

$$Z_b(10610)^+$$ and $$Z_b(10650)^+$$

Observation of $$h_b(1P)$$ and $$h_b(2P)$$

In addition to anomalously large rates to $$\Upsilon (nS)\pi ^+\pi ^-$$ ($$n = 1, 2, 3$$) at $$\Upsilon (5S)$$, another exotic feature was found, again motivated by the analogy with $$Y(4260)$$. CLEO observed that the rates of $$e^+e^- \to h_c \pi ^+\pi ^-$$ also increase around $$Y(4260)$$ in a similar way as $$e^+e^- \to J/\psi \pi ^+\pi ^-$$, with comparable magnitudes [64]. As mentioned in Sect. 2.6, BES III found a charged charmonium-like state at 4200 MeV/$$c^2$$ in the $$\pi ^+ h_c$$ system of the $$e^+e^- \to h_c \pi ^+\pi ^-$$ process [58]. Indication of the existence of the $$Y_b$$ state also leads to an idea of large rates of $$e^+e^- \to h_b(mP) \pi ^+\pi ^-$$ ($$m = 1,2$$) at $$\Upsilon (5S)$$. The $$h_b(mP)$$ had not been observed yet and was hoped to be observed in this process. Belle searched for $$h_b(mP)$$ states in missing mass distribution against $$\pi ^+\pi ^-$$ using the 121.4 fb$$^{-1}$$ data sample accumulated at $$\Upsilon (5S)$$ that includes about 100 fb$$^{-1}$$ taken in the period from October 2008 to the end of 2009 in addition to the one used for discovery of anomalously large transition rates to $$\Upsilon (1S) \pi ^+\pi ^-$$ and $$\Upsilon (2S) \pi ^+\pi ^-$$ final states. In order to suppress $$e^+e^- \to q\bar q$$ ($$q = u,d,s,c$$) continuum background, the ratio of the second to zeroth Fox–Wolfrum moments, $$R_2$$, is required to be less than $$0.3$$ [65]. The missing mass distribution shown in Fig. 11 is fitted with a polynomial background function plus bottomonium resonance states with their masses being free parameters. The widths of the resonance peaks are calibrated by exclusively reconstructed $$e^+e^- \to \pi ^+\pi ^- \Upsilon (nS) [\to \mu ^+\mu ^-]$$ samples, where the $$\Upsilon (nS)$$ mass is calculated as the missing mass against $$\pi ^+\pi ^-$$. The result is shown in Fig. 12. In addition to clear peaks of $$\Upsilon (1,2,3S)$$ at the expected masses, clear peaks of $$h (1P)$$ and $$h_b(2P)$$ were discovered [66]. As seen in Fig. 12, the rates of $$h_b(mP) \pi ^+\pi ^-$$ are of a similar order as those of $$\Upsilon (nS) \pi ^+\pi ^-$$ in spite of the expectation that $$h_b(mP) \pi ^+\pi ^-$$ is suppressed because a spin flip of the bottom quark is required.

Fig. 11.

The missing mass distribution for the selected $$\pi ^+\pi ^-$$ pairs (solid histogram). Vertical lines indicate the locations of the $$\Upsilon (1S)$$, $$h_b(1P)$$, $$\Upsilon (2S)$$, $$h_b(2P)$$, and $$\Upsilon (3S)$$ signals [66].

Fig. 11.

The missing mass distribution for the selected $$\pi ^+\pi ^-$$ pairs (solid histogram). Vertical lines indicate the locations of the $$\Upsilon (1S)$$, $$h_b(1P)$$, $$\Upsilon (2S)$$, $$h_b(2P)$$, and $$\Upsilon (3S)$$ signals [66].

Fig. 12.

The missing mass spectrum with the background subtracted (points with errors) with the signal component of the fit function overlaid (smooth curve). The vertical lines indicate the boundaries of the fit regions [66].

Fig. 12.

The missing mass spectrum with the background subtracted (points with errors) with the signal component of the fit function overlaid (smooth curve). The vertical lines indicate the boundaries of the fit regions [66].

Observation of $$Z_b(10610)^+$$ and $$Z_b(10650)^+$$

In order to understand the abnormal features in the $$e^+e^- \to \Upsilon (nS) \pi ^+\pi ^-$$ and $$e^+e^- \to h_b(mP) \pi ^+\pi ^-$$ processes at $$\Upsilon (5S)$$, more detailed studies of the final particles were performed. The invariant mass distributions of intermediate states of the three-body $$\Upsilon (nS) \pi ^+\pi ^-$$ system were checked when these processes were first observed with 21.7 fb$$^{-1}$$ data, but no clear structure was seen due to low statistics. With a five times larger data sample of 121.4 fb$$^{-1}$$, Dalitz plots of $$\Upsilon (nS) \pi ^+\pi ^-$$ now show clear structures of two resonances in $$\Upsilon (nS) \pi $$ mass distribution, as shown in Fig. 13 [20]. Amplitude analysis of these Dalitz plot distributions revealed two resonance states at masses of 10610 MeV/$$c^2$$ and 10650 MeV/$$c^2$$ in all three transitions $$\Upsilon (5S) \to \Upsilon (nS)\pi ^+\pi ^-$$ ($$n = 1, 2, 3$$) with significances of well above 10 $$\sigma $$, as shown in Fig. 14.

Fig. 13.

Dalitz plots for $$\Upsilon (nS)\pi ^+\pi ^-$$ events of (left) $$\Upsilon (1S)$$; (middle) $$\Upsilon (2S)$$; (right) $$\Upsilon (3S)$$ [20].

Fig. 13.

Dalitz plots for $$\Upsilon (nS)\pi ^+\pi ^-$$ events of (left) $$\Upsilon (1S)$$; (middle) $$\Upsilon (2S)$$; (right) $$\Upsilon (3S)$$ [20].

Fig. 14.

$$\Upsilon (nS)\pi $$ invariant mass distributions of $$\Upsilon (nS)\pi ^+\pi ^-$$ events for (left) $$\Upsilon (1S)$$; (middle) $$\Upsilon (2S)$$; (right) $$\Upsilon (3S)$$. Points with error bars are data and hatched histograms show background. Solid histograms are fits with $$J^P = 1^+$$ assigned to both $$Z_b$$ states and dashed histograms with $$J^P = 2^+$$ (see text) [20].

Fig. 14.

$$\Upsilon (nS)\pi $$ invariant mass distributions of $$\Upsilon (nS)\pi ^+\pi ^-$$ events for (left) $$\Upsilon (1S)$$; (middle) $$\Upsilon (2S)$$; (right) $$\Upsilon (3S)$$. Points with error bars are data and hatched histograms show background. Solid histograms are fits with $$J^P = 1^+$$ assigned to both $$Z_b$$ states and dashed histograms with $$J^P = 2^+$$ (see text) [20].

The invariant mass distributions of $$h_b(mP)\pi ^+$$ ($$m =1,2$$) are obtained by the yields of $$\Upsilon (5S) \to h_b(mP)\pi ^+\pi ^-$$ signals in bins of the $$h_b(mP)\pi ^+$$ mass, where the $$h_b(mP)\pi ^+$$ mass is calculated as the missing mass against $$\pi ^-$$ ($$M_{\rm miss}(\pi ^-)$$). For each bin, the signal yield is obtained by fitting the distribution of missing mass against $$\pi ^+\pi ^-$$, as described previously. The resulting mass distributions of signal yields are shown in Fig. 15 [20]. The two peaks of resonances are clearly seen and their masses and widths are consistent with those obtained from $$\Upsilon (5S) \to \Upsilon (nS)\pi ^+\pi ^-$$ ($$n = 1, 2, 3$$).

Fig. 15.

(a) $$h_b(1P)$$ and (b) $$h_b(2P)$$ yields as a function of $$M_{\rm miss}(\pi )$$ (points with error bars) and results of the fit (histogram) [20].

Fig. 15.

(a) $$h_b(1P)$$ and (b) $$h_b(2P)$$ yields as a function of $$M_{\rm miss}(\pi )$$ (points with error bars) and results of the fit (histogram) [20].

The spin and parity of two $$Z_b$$ resonances are determined by the full amplitude analysis with angular distributions in six dimensions for processes $$\Upsilon (5S) \to \Upsilon (nS)\pi ^+\pi ^-$$ with $$\Upsilon (nS) \to \mu ^+\mu ^-$$ [67]. The maximum likelihood fits with six-dimensional amplitude are performed for all possible hypotheses of $$J^P$$ for $$Z_b(10610)^+$$ and $$Z_b(10610)^+$$, each with $$J^P = 1^+$$, $$1^-$$, $$2^+$$, and $$2^-$$ ($$0^\pm $$ are forbidden because $$Z_b$$ decays to both $$\Upsilon (nS)\pi $$ and $$h_b(mP)\pi $$). The fit favors both $$Z_b$$s with $$1^+$$ over all other combinations of $$J^P$$ values with significances more than $$6\,\sigma $$. The discrimination among different $$J^P$$ values mainly arises from an interference between $$Z_b$$ and other amplitudes in the Dalitz plot and angular distributions as shown in Fig. 14.

Table 1.

Summary of masses and widths of $$Z_b$$s in $$\Upsilon (nS)\pi ^+$$$$(n = 1, 2, 3)$$ decays [67] and $$h_b(mP)\pi ^+$$$$(m = 1,2)$$ decays [20].

Parameter
 
$$\Upsilon (1S)\pi ^+$$ $$\Upsilon (2S)\pi ^+$$ $$\Upsilon (3S)\pi ^+$$ 
$$Z_b(10610)^+$$ $$M$$ (MeV/c$$^2$$$$10608.5 \pm 3.4 ^{+3.7}_{-1.4}$$ $$10608.1 \pm 1.2 ^{+1.5}_{-0.2}$$ $$10607.4 \pm 1.5 ^{+0.8}_{-0.2}$$ 
 $$\Gamma $$ (MeV/c$$^2$$$$18.5 \pm 5.3 ^{+6.1}_{-2.3}$$ $$20.8 \pm 2.5 ^{+0.3}_{-2.1}$$ $$18.7 \pm 3.4 ^{+2.5}_{-1.3}$$ 
$$Z_b(10650)^+$$ $$M$$ (MeV/c$$^2$$$$10656.7 \pm 5.0^{+1.1}_{-3.1}$$ $$10650.7 \pm 1.5 ^{+0.5}_{-0.2}$$ $$10651.2 \pm 1.0 ^{+0.4}_{-0.3}$$ 
 $$\Gamma $$ (MeV/c$$^2$$$$12.1 ^{+11.3 +2.7}_{-4.8 -0.6}$$ $$14.2 \pm 3.7 ^{+0.9}_{-0.4}$$ $$9.3 \pm 2.2 ^{+0.3}_{-0.5}$$ 
 Relative phase (deg) $$67 \pm 36 ^{+24}_{-52}$$ $$-10 \pm 13 ^{+34}_{-12}$$ $$-5 \pm 22 ^{+15}_{-33}$$ 
 Parameter $$h_b(1P)\pi ^+$$ $$h_b(2P)\pi ^+$$  
$$Z_b(10610)^+$$ $$M$$ (MeV/c$$^2$$$$10605 \pm 2 ^{+3}_{-1}$$ $$10599 ^{+6 +5}_{-3 -4}$$  
 $$\Gamma $$ (MeV/c$$^2$$$$11.4 \pm ^{+4.4 +2.1}_{-3.9 -1.2}$$ $$13 ^{+10 +9}_{-3 -4}$$  
$$Z_b(10650)^+$$ $$M$$ (MeV/c$$^2$$$$10654 \pm 3 ^{+1}_{-2}$$ $$10651 ^{+2 +3}_{-3 -2}$$  
 $$\Gamma $$ (MeV/c$$^2$$$$20.9 \pm ^{+5.4 +2.1}_{-4.7 -5.7}$$ $$19 \pm 7 ^{+11}_{-7}$$  
 Relative phase (deg) $$187 ^{+44 +3}_{-57 -12}$$ $$181 ^{+65 +74}_{-105-109}$$  
Parameter
 
$$\Upsilon (1S)\pi ^+$$ $$\Upsilon (2S)\pi ^+$$ $$\Upsilon (3S)\pi ^+$$ 
$$Z_b(10610)^+$$ $$M$$ (MeV/c$$^2$$$$10608.5 \pm 3.4 ^{+3.7}_{-1.4}$$ $$10608.1 \pm 1.2 ^{+1.5}_{-0.2}$$ $$10607.4 \pm 1.5 ^{+0.8}_{-0.2}$$ 
 $$\Gamma $$ (MeV/c$$^2$$$$18.5 \pm 5.3 ^{+6.1}_{-2.3}$$ $$20.8 \pm 2.5 ^{+0.3}_{-2.1}$$ $$18.7 \pm 3.4 ^{+2.5}_{-1.3}$$ 
$$Z_b(10650)^+$$ $$M$$ (MeV/c$$^2$$$$10656.7 \pm 5.0^{+1.1}_{-3.1}$$ $$10650.7 \pm 1.5 ^{+0.5}_{-0.2}$$ $$10651.2 \pm 1.0 ^{+0.4}_{-0.3}$$ 
 $$\Gamma $$ (MeV/c$$^2$$$$12.1 ^{+11.3 +2.7}_{-4.8 -0.6}$$ $$14.2 \pm 3.7 ^{+0.9}_{-0.4}$$ $$9.3 \pm 2.2 ^{+0.3}_{-0.5}$$ 
 Relative phase (deg) $$67 \pm 36 ^{+24}_{-52}$$ $$-10 \pm 13 ^{+34}_{-12}$$ $$-5 \pm 22 ^{+15}_{-33}$$ 
 Parameter $$h_b(1P)\pi ^+$$ $$h_b(2P)\pi ^+$$  
$$Z_b(10610)^+$$ $$M$$ (MeV/c$$^2$$$$10605 \pm 2 ^{+3}_{-1}$$ $$10599 ^{+6 +5}_{-3 -4}$$  
 $$\Gamma $$ (MeV/c$$^2$$$$11.4 \pm ^{+4.4 +2.1}_{-3.9 -1.2}$$ $$13 ^{+10 +9}_{-3 -4}$$  
$$Z_b(10650)^+$$ $$M$$ (MeV/c$$^2$$$$10654 \pm 3 ^{+1}_{-2}$$ $$10651 ^{+2 +3}_{-3 -2}$$  
 $$\Gamma $$ (MeV/c$$^2$$$$20.9 \pm ^{+5.4 +2.1}_{-4.7 -5.7}$$ $$19 \pm 7 ^{+11}_{-7}$$  
 Relative phase (deg) $$187 ^{+44 +3}_{-57 -12}$$ $$181 ^{+65 +74}_{-105-109}$$  

The masses, widths, and other quantities for the two $$Z_b$$ states obtained for five transitions are summarized in Table 1. The average masses and widths are $$M_{Z_b(10610)^+} = 10607.2 \pm 1.1$$ MeV/$$c^2$$, $$M_{Z_b(10650)^+} = 10651.3 \pm 0.8$$ MeV/$$c^2$$, $$\Gamma _{Z_b(10610)^+} = 18.0 \pm 2.2$$ MeV/$$c^2$$, and $$\Gamma _{Z_b(10650)^+} = 11.3 \pm 1.8$$ MeV/$$c^2$$. The masses of $$Z_b(10610)^+$$ and $$Z_b(10650)^+$$ are very close to the sums of the masses of $$B\bar B^*$$ and $$B^* \bar B^*$$, respectively. It should be noted that the relative phase between the two $$Z_b$$ amplitudes is consistent with zero for $$\Upsilon (5S) \to \Upsilon (nS)\pi ^+\pi ^-$$ and with $$\pi $$ for $$\Upsilon (5S) \to h_b(mP)\pi ^+\pi ^-$$, which would be expected from the $$B^{(*)}\bar B^*$$ molecular model [68].

$$\Upsilon (5S) \to \big [B^{(*)}\bar B^*\big ]^\pm \pi ^\mp $$

The proximity of the $$Z_b(10610)^+$$ and $$Z_b(10650)^+$$ masses to the sum of the $$B^{(*)}\bar B^*$$ masses indicates that these states would be $$B^{(*)}\bar B^*$$ molecular states and decay to $$B^{(*)}\bar B^*$$, as will be discussed in Sect. 2.8.2. Rather large branching fractions in $$\Upsilon (5S) \to B\bar B^*\pi $$ decay were found with a data sample of 23.6 fb$$^{-1}$$ [69]. The intermediate structure of $$\Upsilon (5S) \to B^{(*)}\bar B^{(*)}\pi $$ three-body decays were studied using a 121.4 fb$$^{-1}$$ data sample [24]. We reconstructed one $$B$$ meson with $$B^+ \to J/\psi K^+$$, $$B^+ \to \bar {D}^0 \pi ^+$$, $$B^0 \to J/\psi K^{*0}$$, $$B^0 \to D^- \pi ^+$$, and $$B^0 \to D^{*-} \pi ^+$$. The reconstructed $$B^+$$ or $$B^0$$ candidates were combined with $$\pi ^-$$ candidates and the missing mass against the $$B\pi $$ combination was calculated. The missing mass distribution is shown in Fig. 16. The two peaks correspond to $$B\bar {B}^*\pi $$ and $$B^*\bar {B}^*\pi $$ signals with signal yields of $$184 \pm 19$$ and $$82 \pm 11$$ events, respectively. The $$B^{(*)}\bar {B}^*$$ invariant mass was calculated as a missing mass against $$\pi $$ for events in the $$B\bar {B}^*\pi $$ and $$B^*\bar {B}^*\pi $$ signal regions. The distributions are shown in Fig. 17. The fits to the distribution yielded a significant signal of $$Z_b(10610)^+$$ ($$8\,\sigma $$) in $$B\bar {B}^*\pi $$ and $$Z_b(10650)^+$$ ($$6.8\,\sigma $$) in $$B^*\bar {B}^*\pi $$ transitions, while a marginal signal of $$Z_b(10650)^+$$ is seen in the $$B\bar {B}^*\pi $$ transition.

Fig. 16.

The distribution of missing mass against $$B\pi $$ ($$rM(B\pi )$$) for selected candidates (red histogram) and background (hatched histogram) [24].

Fig. 16.

The distribution of missing mass against $$B\pi $$ ($$rM(B\pi )$$) for selected candidates (red histogram) and background (hatched histogram) [24].

Fig. 17.

The distribution of missing mass against $$\pi $$ ($$rM(\pi )$$) for (a) $$\Upsilon (5S) \to B\bar {B}^*\pi $$ and (b) $$\Upsilon (5S) \to B^*\bar {B}^*\pi $$ candidate events. Points with error bars are data, the solid line is the result of the fit with $$Z_b$$s only, the dashed line is the fit to the pure non-resonant amplitude, the dotted line is the fit to a single $$Z_b$$ state plus a non-resonant amplitude, and the dash-dotted line is the fit to two $$Z_b$$ states and a non-resonant amplitude. The hatched histogram represents the background component normalized to the estimated number of background events [24].

Fig. 17.

The distribution of missing mass against $$\pi $$ ($$rM(\pi )$$) for (a) $$\Upsilon (5S) \to B\bar {B}^*\pi $$ and (b) $$\Upsilon (5S) \to B^*\bar {B}^*\pi $$ candidate events. Points with error bars are data, the solid line is the result of the fit with $$Z_b$$s only, the dashed line is the fit to the pure non-resonant amplitude, the dotted line is the fit to a single $$Z_b$$ state plus a non-resonant amplitude, and the dash-dotted line is the fit to two $$Z_b$$ states and a non-resonant amplitude. The hatched histogram represents the background component normalized to the estimated number of background events [24].

The relative branching fractions of $$Z_b^+$$ decaying to observed channels ($$\Upsilon (nS)\pi ^+$$, $$h_b(mP)\pi ^+$$, and $$B^{(*)}\bar {B}^*$$) were calculated from the transition rates to three-body final states and the fraction of intermediate $$Z_b^+$$ states. Table 2 gives the branching fractions assuming that the sum of these modes is 100%. The dominant decay modes are $$B\bar {B}^*$$ for $$Z_b(10610)^+$$ and $$B^*\bar {B}^*$$ for $$Z_b(10650)^+$$, respectively.

Table 2.

Branching fractions (%) of $$Z_b^+$$s in different channels [24].

Channel $$Z_b(10610)^+$$ $$Z_b(10650)^+$$ 
$$\Upsilon (1S)\pi ^+$$ $$0.61 \pm 0.28$$ $$0.19 \pm 0.09$$ 
$$\Upsilon (2S)\pi ^+$$ $$4.19 \pm 1.51$$ $$1.54 \pm 0.69$$ 
$$\Upsilon (3S)\pi ^+$$ $$2.49 \pm 0.96$$ $$1.81 \pm 0.75$$ 
$$h_b(1P)\pi ^+$$ $$4.40 \pm 2.17$$ $$10.3 \pm 5.5$$ 
$$h_b(2P)\pi ^+$$ $$6.26 \pm 3.76$$ $$19.0 \pm 9.3$$ 
$$B^+\bar B^{*0} + \bar B^0B^{*+}$$ $$82.0 \pm 3.5$$ — 
$$B^{*+}\bar B^{*0} $$ — $$67.2 \pm 7.1$$ 
Channel $$Z_b(10610)^+$$ $$Z_b(10650)^+$$ 
$$\Upsilon (1S)\pi ^+$$ $$0.61 \pm 0.28$$ $$0.19 \pm 0.09$$ 
$$\Upsilon (2S)\pi ^+$$ $$4.19 \pm 1.51$$ $$1.54 \pm 0.69$$ 
$$\Upsilon (3S)\pi ^+$$ $$2.49 \pm 0.96$$ $$1.81 \pm 0.75$$ 
$$h_b(1P)\pi ^+$$ $$4.40 \pm 2.17$$ $$10.3 \pm 5.5$$ 
$$h_b(2P)\pi ^+$$ $$6.26 \pm 3.76$$ $$19.0 \pm 9.3$$ 
$$B^+\bar B^{*0} + \bar B^0B^{*+}$$ $$82.0 \pm 3.5$$ — 
$$B^{*+}\bar B^{*0} $$ — $$67.2 \pm 7.1$$ 

Neutral partner of $$Z_b^+$$ states

Since $$Z_b^+$$ states decay to $$\Upsilon (nS)\pi ^+$$, $$h_b(mP)\pi ^+$$, and $$B^{0(*)}\bar B^{*+}$$, they are expected to be isovector states ($$I = 1$$). Their neutral partners were searched for in $$\Upsilon (5S) \to \Upsilon (nS)\pi ^0\pi ^0$$ transitions. The analysis was performed in a similar way as $$\Upsilon (5S) \to \Upsilon (nS)\pi ^+\pi ^-$$, replacing $$\pi ^\pm $$ by $$\pi ^0$$ [23]. The results provide the first observation of $$\Upsilon (5S) \to \Upsilon (nS)\pi ^0\pi ^0$$ transitions with their rates being consistent with half of $$\Upsilon (5S) \to \Upsilon (nS)\pi ^+\pi ^-$$ rates, as expected from the isospin relation. Amplitude analysis of the Dalitz plot distributions has revealed $$Z_b(10610)^0$$ in $$\Upsilon (2S)\pi ^0\pi ^0$$ and $$\Upsilon (3S)\pi ^0\pi ^0$$ transitions with a significance of $$6.5\,\sigma $$, as shown in Fig. 18. No significant signal is observed for $$Z_b(10650)^0$$ due to insufficient statistics. This result confirms that the isospin of the $$Z_b$$ state is one.

Fig. 18.

$$\Upsilon (nS)\pi ^0$$ invariant mass distributions of $$\Upsilon (nS)\pi ^0\pi ^0$$ events for (left) $$\Upsilon (1S)$$; (middle) $$\Upsilon (2S)$$; (right) $$\Upsilon (3S)$$. Points with error bars are data and hatched histograms show background. Solid red and blue histograms show the fits with and without $$Z_b^0$$s, respectively [23].

Fig. 18.

$$\Upsilon (nS)\pi ^0$$ invariant mass distributions of $$\Upsilon (nS)\pi ^0\pi ^0$$ events for (left) $$\Upsilon (1S)$$; (middle) $$\Upsilon (2S)$$; (right) $$\Upsilon (3S)$$. Points with error bars are data and hatched histograms show background. Solid red and blue histograms show the fits with and without $$Z_b^0$$s, respectively [23].

Table 3.

New unconventional state candidates in the $$c\bar {c}$$ and $$b\bar {b}$$ regions, ordered by mass.$$^*$$

State $$m$$ (MeV) $$\Gamma $$ (MeV) $$J^{PC}$$ Process (mode) References  
$$X(3872)$$ 3871.69$$\pm $$0.17 $$<$$1.2 $$1^{++}$$ $$B\to K (\pi ^+\pi ^-J/\psi )$$ Belle [10, 32], BaBar [36], 
     LHCb [34, 73
    $$p\bar p\to (\pi ^+\pi ^- J/\psi )+ \ldots $$ CDF [31, 74, 75], D0 [76
    $$e^+ e^- \to \gamma (\pi ^+ \pi ^- J/\psi )$$ BES III [77
    $$B\to K (\omega J/\psi )$$ Belle [78], BaBar [33
    $$B\to K (D^{*0} \bar {D}^0)$$ Belle [38, 79], BaBar [37
    $$B\to K (\gamma J/\psi )$$ and Belle [29], BaBar [30], 
    $$B\to K (\gamma \psi (2S))$$ LHCb [40
$$Z_c(3900)^+$$ $$3888.7\pm 3.4$$ 35$$\pm $$$$1^{+}$$ $$e^+e^- \to (J/\psi ~\pi ^+) \pi ^-$$ Belle [43], BES III [55
    $$e^+e^-\to (D \bar {D}^*)^+ \pi ^-$$ BES III [56
$$X(3915)$$ $$3915.6\pm 3.1$$ 28$$\pm $$10 $$0/2^{?+}$$ $$B\to K (\omega J/\psi )$$ Belle [80], BaBar [33
    $$e^+e^-\to e^+e^- (\omega J/\psi )$$ Belle [81], BaBar [82
$$X(3940)$$ $$3942^{+9}_{-8}$$ $$37^{+27}_{-17}$$ $$?^{?+}$$ $$e^+e^-\to J/\psi (D D^*)$$ Belle [83
    $$e^+e^-\to J/\psi \ (\ldots )$$ Belle [84
$$Y(4008)$$ $$3891 \pm 42$$ 255$$\pm $$42 $$1^{--}$$ $$e^+e^-\to \gamma (\pi ^+\pi ^-J/\psi )$$ Belle [42, 43
$$Z_c(4050)^+$$ $$4051^{+24}_{-43}$$ $$82^{+51}_{-55}$$ $$ B\to K (\pi ^+\chi _{c1}(1P))$$ Belle [53], BaBar [54
$$X(4050)^+$$ $$4054 \pm 3$$ $$45 $$ $$ e^+ e^- \to (\pi ^+ \psi (2S)) \pi ^-$$ Belle [85
$$Y(4140)$$ $$4143.4\pm 3.0 $$ $$15^{+11}_{-\ 7}$$ $$?^{?+}$$ $$B\to K (\phi J/\psi )$$ CDF [72],D0 [86
$$X(4160)$$ $$4156^{+29}_{-25} $$ $$139^{+113}_{-65}$$ $$?^{?+}$$ $$e^+e^- \to J/\psi (D \bar {D}^*)$$ Belle [83
$$Z_c(4200)^+$$ $$4196^{+35}_{-32}$$ 370$$^{+99}_{-149}$$ $$ B\to K (\pi ^+ J/\psi )$$ Belle [87
$$Z_c(4250)^+$$ $$4248^{+185}_{-\ 45}$$ 177$$^{+321}_{-\ 72}$$ $$ B\to K (\pi ^+\chi _{c1}(1P))$$ Belle [53], BaBar [54
$$Y(4260)$$ $$4263\pm 5$$ 108$$\pm $$14 $$1^{--}$$ $$e^+e^-\to \gamma (\pi ^+\pi ^- J/\psi )$$ BaBar [41, 88], CLEO [89], 
     Belle [42, 43
    $$e^+e^-\to (\pi ^+\pi ^- J/\psi )$$ CLEO [47], BES III [56
    $$e^+e^-\to (\pi ^0\pi ^0 J/\psi )$$ CLEO [47
$$X(4350)$$ $$4350.6^{+4.6}_{-5.1}$$ $$13.3^{+18.4}_{-10.0}$$ ?$$^{?+}$$ $$e^+e^-\to e^+e^- (\phi J/\psi )$$ Belle [90
$$Y(4360)$$ $$4361 \pm 13$$ 74$$\pm $$18 $$1^{--}$$ $$e^+e^-\to \gamma (\pi ^+\pi ^- \psi (2S))$$ BaBar [44], Belle [45, 85
$$Z_c(4430)^+$$ $$4485^{+36}_{-25}$$ $$200^{+49}_{-58}$$ $$1^+$$ $$B\to K (\pi ^+\psi (2S))$$ Belle [49, 51, 52], 
     BaBar [50], LHCb [21
    $$B\to K (\pi ^+ J/\psi )$$ Belle [87], BaBar [50
$$X(4630)$$ $$4634^{+\ 9}_{-11}$$ $$92^{+41}_{-32}$$ $$1^{ --}$$ $$e^+e^- \to \gamma (\Lambda _c^+ \Lambda _c^-)$$ Belle [91
$$Y(4660)$$ 4664$$\pm $$12 48$$\pm $$15 $$1^{--}$$ $$e^+e^-\to \gamma (\pi ^+\pi ^- \psi (2S))$$ Belle [45
$$Z_b(10610)^+$$ 10607.2$$\pm $$2.0 18.4$$\pm $$2.4 $$1^+$$ $$e^+e^- \to (b\bar {b}~\pi ^+) \pi ^-$$ Belle [20
$$Z_b(10610)^0$$ 10609$$\pm 4\pm $$NA $$1^{+-}$$ $$e^+e^- \to (\Upsilon (2,3S) \pi ^0) \pi ^0$$ Belle [23
$$Z_b(10650)^+$$ 10652.2$$\pm $$1.5 11.5$$\pm $$2.2 $$1^+$$ $$e^+e^- \to (b\bar {b}~\pi ^+) \pi ^-$$ Belle [20
$$Y_b(10888)$$ 10888.4$$\pm $$3.0 30.7$$^{+8.9}_{-7.7}$$ $$1^{--}$$ $$e^+e^-\to (\pi ^+\pi ^- \Upsilon (nS))$$ Belle [60, 62
State $$m$$ (MeV) $$\Gamma $$ (MeV) $$J^{PC}$$ Process (mode) References  
$$X(3872)$$ 3871.69$$\pm $$0.17 $$<$$1.2 $$1^{++}$$ $$B\to K (\pi ^+\pi ^-J/\psi )$$ Belle [10, 32], BaBar [36], 
     LHCb [34, 73
    $$p\bar p\to (\pi ^+\pi ^- J/\psi )+ \ldots $$ CDF [31, 74, 75], D0 [76
    $$e^+ e^- \to \gamma (\pi ^+ \pi ^- J/\psi )$$ BES III [77
    $$B\to K (\omega J/\psi )$$ Belle [78], BaBar [33
    $$B\to K (D^{*0} \bar {D}^0)$$ Belle [38, 79], BaBar [37
    $$B\to K (\gamma J/\psi )$$ and Belle [29], BaBar [30], 
    $$B\to K (\gamma \psi (2S))$$ LHCb [40
$$Z_c(3900)^+$$ $$3888.7\pm 3.4$$ 35$$\pm $$$$1^{+}$$ $$e^+e^- \to (J/\psi ~\pi ^+) \pi ^-$$ Belle [43], BES III [55
    $$e^+e^-\to (D \bar {D}^*)^+ \pi ^-$$ BES III [56
$$X(3915)$$ $$3915.6\pm 3.1$$ 28$$\pm $$10 $$0/2^{?+}$$ $$B\to K (\omega J/\psi )$$ Belle [80], BaBar [33
    $$e^+e^-\to e^+e^- (\omega J/\psi )$$ Belle [81], BaBar [82
$$X(3940)$$ $$3942^{+9}_{-8}$$ $$37^{+27}_{-17}$$ $$?^{?+}$$ $$e^+e^-\to J/\psi (D D^*)$$ Belle [83
    $$e^+e^-\to J/\psi \ (\ldots )$$ Belle [84
$$Y(4008)$$ $$3891 \pm 42$$ 255$$\pm $$42 $$1^{--}$$ $$e^+e^-\to \gamma (\pi ^+\pi ^-J/\psi )$$ Belle [42, 43
$$Z_c(4050)^+$$ $$4051^{+24}_{-43}$$ $$82^{+51}_{-55}$$ $$ B\to K (\pi ^+\chi _{c1}(1P))$$ Belle [53], BaBar [54
$$X(4050)^+$$ $$4054 \pm 3$$ $$45 $$ $$ e^+ e^- \to (\pi ^+ \psi (2S)) \pi ^-$$ Belle [85
$$Y(4140)$$ $$4143.4\pm 3.0 $$ $$15^{+11}_{-\ 7}$$ $$?^{?+}$$ $$B\to K (\phi J/\psi )$$ CDF [72],D0 [86
$$X(4160)$$ $$4156^{+29}_{-25} $$ $$139^{+113}_{-65}$$ $$?^{?+}$$ $$e^+e^- \to J/\psi (D \bar {D}^*)$$ Belle [83
$$Z_c(4200)^+$$ $$4196^{+35}_{-32}$$ 370$$^{+99}_{-149}$$ $$ B\to K (\pi ^+ J/\psi )$$ Belle [87
$$Z_c(4250)^+$$ $$4248^{+185}_{-\ 45}$$ 177$$^{+321}_{-\ 72}$$ $$ B\to K (\pi ^+\chi _{c1}(1P))$$ Belle [53], BaBar [54
$$Y(4260)$$ $$4263\pm 5$$ 108$$\pm $$14 $$1^{--}$$ $$e^+e^-\to \gamma (\pi ^+\pi ^- J/\psi )$$ BaBar [41, 88], CLEO [89], 
     Belle [42, 43
    $$e^+e^-\to (\pi ^+\pi ^- J/\psi )$$ CLEO [47], BES III [56
    $$e^+e^-\to (\pi ^0\pi ^0 J/\psi )$$ CLEO [47
$$X(4350)$$ $$4350.6^{+4.6}_{-5.1}$$ $$13.3^{+18.4}_{-10.0}$$ ?$$^{?+}$$ $$e^+e^-\to e^+e^- (\phi J/\psi )$$ Belle [90
$$Y(4360)$$ $$4361 \pm 13$$ 74$$\pm $$18 $$1^{--}$$ $$e^+e^-\to \gamma (\pi ^+\pi ^- \psi (2S))$$ BaBar [44], Belle [45, 85
$$Z_c(4430)^+$$ $$4485^{+36}_{-25}$$ $$200^{+49}_{-58}$$ $$1^+$$ $$B\to K (\pi ^+\psi (2S))$$ Belle [49, 51, 52], 
     BaBar [50], LHCb [21
    $$B\to K (\pi ^+ J/\psi )$$ Belle [87], BaBar [50
$$X(4630)$$ $$4634^{+\ 9}_{-11}$$ $$92^{+41}_{-32}$$ $$1^{ --}$$ $$e^+e^- \to \gamma (\Lambda _c^+ \Lambda _c^-)$$ Belle [91
$$Y(4660)$$ 4664$$\pm $$12 48$$\pm $$15 $$1^{--}$$ $$e^+e^-\to \gamma (\pi ^+\pi ^- \psi (2S))$$ Belle [45
$$Z_b(10610)^+$$ 10607.2$$\pm $$2.0 18.4$$\pm $$2.4 $$1^+$$ $$e^+e^- \to (b\bar {b}~\pi ^+) \pi ^-$$ Belle [20
$$Z_b(10610)^0$$ 10609$$\pm 4\pm $$NA $$1^{+-}$$ $$e^+e^- \to (\Upsilon (2,3S) \pi ^0) \pi ^0$$ Belle [23
$$Z_b(10650)^+$$ 10652.2$$\pm $$1.5 11.5$$\pm $$2.2 $$1^+$$ $$e^+e^- \to (b\bar {b}~\pi ^+) \pi ^-$$ Belle [20
$$Y_b(10888)$$ 10888.4$$\pm $$3.0 30.7$$^{+8.9}_{-7.7}$$ $$1^{--}$$ $$e^+e^-\to (\pi ^+\pi ^- \Upsilon (nS))$$ Belle [60, 62

$$^*$$The mass and width values are as quoted in PDG2014 [22]. $$X(3945)$$ and $$Y(3940)$$ have been subsumed under $$X(3915)$$ due to compatible properties and it is interpreted as $$\chi _{c0}(2P)$$ in PDG2014 [22]. The states known as $$X(3823)$$ [70] and $$Z(3930)$$ [71] are omitted because interpretations for these states were established as $$\psi _2(1D)$$ and $$\chi _{c2}(2P)$$, respectively, shortly after their discoveries. The original $$Y(4008)$$ measurement [42] is superseded by the latest one [43] and its mass and width values are taken from the latter. Since there are no public average values for $$Y(4140)$$ mass and width, the CDF [72] values are written. The $$b\bar {b}$$ in $$Z_b(10610)^+$$ and $$Z_b(10650)^+$$ decay modes represent $$\Upsilon (nS)$$$$(n=1,2,3)$$ or $$h_b(mP)$$$$(m=1,2)$$. $$Z_b(10610)^0$$ width and $$J^P$$ are assumed to be the same as $$Z_b(10610)^+$$.

Experiment summary

Thanks to the world highest luminosity provided by the KEKB asymmetric-energy $$e^+e^-$$ collider, the Belle experiment has recorded high statistics data including 772 million $$B\bar {B}$$ pairs at $$\Upsilon (4S)$$ as well as the $$\Upsilon (5S)$$ data corresponding to an integrated luminosity of 121.4 fb$$^{-1}$$. Accumulation of these excellent quality data is essential for all the discoveries of the $$X$$, $$Y$$, $$Z$$ states. The molecular states turned out to be playing an important role near the thresholds. When the state is neutral and isosinglet, possible mixing with the heavy quarkonia possessing same quantum number also affects markedly the properties of the observed states. As already discussed, interpreting $$X(3872)$$ as an admixture of $$D\bar {D^*}$$ and $$\chi _{c1}(2P)$$ would show no conflict with all the available experimental results. The $$B^* \bar {B}^{(*)}$$ molecule picture works well to describe the charged bottomonium-like states, $$Z_b(10610)^+$$ and $$Z_b(10650)^+$$; the co-existence of $$\Upsilon (nS) \pi ^+\pi ^-$$ ($$n=1, 2, 3$$) and $$h_b(mP) \pi ^+ \pi ^-$$ ($$m=1,2$$) usually does not take place because of the suppression to require the heavy quark's spin flip. The large branching fraction to the $$B^* \bar {B}^{(*)}$$ final state also supports the molecular hypothesis.

Reaching such very plausible interpretations would be remarkable progress made in recent years. In total, seven charged quarkonium-like states have been reported; they cannot be interpreted as usual mesons but should have at least four constituents of $$Q\bar {Q}u\bar {d}$$, where $$Q\bar {Q}$$ detonates $$c\bar {c}$$ or $$b\bar {b}$$. Compilation of the relevant activities leads us to predictions of the partner states and their possible decay modes. Here, we should also not forget the many results brought by other experiments: BaBar, CLEO, BES III, CDF, D0, and LHCb. All these have also contributed greatly to making clear the existence of many states or pointing out the necessity of further confirmation. All the available information as of today is summarized in Table 3.

Determination of the still unsettled quantum numbers and various partner state searches require more statistics. Continuing to exploit the possibilities to access various production mechanisms as well as a variety of decay modes with higher statistics data is the aim of the upgrade of the $$B$$-factory project, Belle II experiment's mission in hadron spectroscopy, and friendly competition with other running experiments (BES III, LHCb, and some other projects) is expected to provide many interesting discoveries.

Theory

Overview

In this chapter, we discuss theoretical methods from some fundamentals to applications to the $$X,Y,Z$$ exotic hadrons. We emphasize the roles of heavy quarks, which are essential to understand the properties of these states.

Let us start with quarkonium spectroscopy, which has been successfully described phenomenologically by quark models. The method has been working well particularly in the mass region below the thresholds of open heavy mesons, such as $$D\bar {D}$$, $$D\bar {D}^{\ast }$$, and $$D^{\ast }\bar {D}^{\ast }$$ (see Fig. 1). There are two essential ingredients in the phenomenological quark model: (i) heavy quarks as effective degrees of freedom for constituents, and (ii) a quark potential which has two important components of the color Coulomb force at short distances and of the confinement (linear) force at long distances [92]. Phenomenologically, the potential is supplemented by the spin-dependent and velocity-dependent forces. The important static part of potential has been nicely reproduced by lattice QCD calculations [93], and in this respect the model has a correspondence to QCD.

Recent experimental observations of $$X,Y,Z$$ exotic hadrons (Sect. 2) have shown, however, that those successes are limited to below the thresholds of open heavy hadrons. Near and above the thresholds, many exotic states have been found, whose properties cannot be well understood by the conventional quark model with the incorporation of (i) and (ii). Thus, various ideas have been proposed with using degrees of freedom other than heavy quarks, such as light constituent quarks, and constituent gluons. In the early stages, for instance, theoretical studies were performed in Refs. [94–96] where the light degrees of freedom were incorporated in hadronic dynamics through $$D^{(\ast )}\bar {D}^{(\ast )}$$ mesons, and exotic hadrons were predicted in the form of their hadronic molecules. Due to the lack of experimental information at that time, however, not much progress was made.

We can understand at least in a qualitative manner how the introduction of light degrees of freedom is essential to understand the $$X,Y,Z$$ exotic hadrons. As for the conventional quarkonia below the thresholds, orbital motions of the constituent quarks carry (excitation) energy whose scale is much smaller than the heavy quark mass. Hence we may regard them as light degrees of freedom (d.o.f.). For example, the mass difference between $$S$$-wave quarkonia ($$\eta _{c}$$ and $$J/\psi $$) and $$P$$-wave ones ($$h_{c}$$ and $$\chi _{cJ}$$) is about 300 MeV, which is explained by the excitation energies of orbital motions. With this energy the light quark–antiquark pair creation is still suppressed or, slightly more precisely, may occur only virtually. Near and above the thresholds with more excitation energies, however, light quarks can appear as explicit degrees of freedom. Practically they can be constituents of hadrons to form so-called multiquark systems $$\big (\bar {Q}Q\bar {q}q\big )$$ with compact structure or an extended structure of hadronic molecules through the rearrangement of the multiquarks into several hadrons. The mechanism of how various kinds of hadrons are formed is dictated by QCD, in particular by those of quark confinement, mass generation, and chiral symmetry breaking. In other words, by properly describing the properties of exotic hadrons we hope to know better the non-perturbative dynamics of QCD.

Let us turn to some features associated with heavy particle nature. These are the essences of the potential non-relativistic QCD (pNRQCD) [14] and heavy quark spin symmetry [97].

  • Constituent nature: For heavy quarks, the particle number of constituent quarks is well conserved, where quark–antiquark pair creation is suppressed at the QCD energy scale. Thus the concept of constituents works and, therefore, a non-relativistic treatment is justified not only kinematically but also dynamically with the heavy particle number fixed.

  • Potential: Heavy quarks move slowly. Therefore, the potential approximation for the interaction by an instantaneously propagating force works well. Moreover, kinetic energy is suppressed for heavy particles, and even a small attraction can allow bound states or resonances. Their properties are sensitive to the details of the interaction. This feature may provide us with precise information about the interaction between the constituent particles. The inter-quark potential can be supplied directly from QCD in the pNRQCD formalism [14].

  • Spin symmetry: Heavy quark spins are conserved in the heavy quark limit. It is the exact symmetry of QCD for heavy quarks where they interact through vector (gauge) gluons. Thus, just as in non-relativistic theories, the spin of the heavy quark is conserved in addition to the total hadron spin.

Because of the above features, more predictions about heavy hadrons have been made theoretically than so far observed. For example, tetraquark mesons $$T_{cc}$$ with charm number two ($$|C|=2$$) are considered to be stable states against decays by the strong interaction.4 If they exist, they are the ground states of hadrons with charm number two and baryon number zero.

In this section, we present theory discussions with some examples for exotic hadrons. In Sect. 3.2, we describe the basic theoretical ideas for the study of heavy hadrons, starting from the QCD Lagrangian. First we briefly discuss chiral symmetry, which is important for light quark dynamics. Then we turn to discussions of heavy quark dynamics. In particular, we present how the heavy quark (spin) symmetry shows up in normal hadrons as well as in exotic ones, with an emphasis that the heavy quark symmetry holds not only at the level of quark dynamics, but also at the level of hadron dynamics. In Sect. 3.3, we discuss several theoretical methods including the quark model (Sect. 3.3.1), heavy-hadron chiral effective theories (Sect. 3.3.2), models for hadronic molecules (Sect. 3.3.3), QCD sum rules (Sect. 3.3.4), and lattice QCD studies (Sect. 3.3.5). In Sect. 3.4, we apply the idea of the hadronic molecule to the description of $$X(3872)$$ (Sect. 3.4.1), and $$Z_{b}(10610)$$ and $$Z_{b}(10650)$$ (Sect. 3.4.2). We briefly summarize the current theoretical status for other states of $$Y(4260)$$, $$Z_c^+(4430)$$, and $$Z_c^+(3900)$$, and mention the recent studies based on dynamical treatments (Sect. 3.4.3). As future prospects, in Sect. 3.5, we discuss possible new hadronic and nuclear systems, which have been predicted theoretically but not yet observed in experiments. We consider the stability of $$D^{(\ast )}D^{(\ast )}$$ hadronic molecules (Sect. 3.5.1), and the $$T_{cc}$$ state (Sect. 3.5.2).

The basic idea

The QCD Lagrangian

The QCD Lagrangian is well known and is given by  

(1)
\begin{equation} \mathcal{L}_{\rm{QCD}} = -\frac{1}{4}F_{\mu\nu}^{a} F^{a\,\mu\nu} + \sum_{f} \bar{\psi}_{f} \left(i \diagup \hskip -10pt{D} - m_{f}\right) \psi_{f}. \end{equation}
Here, the sum $$\sum _{f}$$ goes over flavors $$f$$, $$F^{a}_{\mu \nu }=\partial _{\mu } A^{a}_{\nu } - \partial _{\nu } A^{a}_{\mu } - g_{\rm s} f^{abc} A^{b}_{\mu } A^{c}_{\nu }$$ is the gluon field tensor with the color index $$a$$ and Lorentz indexes $$\mu , \nu $$, and $$D_{\mu } = \partial _{\mu } - g_{\rm s} A_{\mu }^{a}T^{a}$$ is the covariant derivative with the gluon field $$A^{a}_{\mu }$$ ($$a=1,\dots ,N_{\rm c}^{2}-1$$; $$\mu =0,\ldots ,3$$; $$N_c$$ is the number of colors).

The Lagrangian (1) looks similar to that of QED, but is dynamically very much different. In QCD, it is known that the strong coupling constant $$\alpha _{{\rm s}}\left (\mu ^2\right )$$ runs as a function of a momentum scale $$\mu $$ as dictated by the following renormalization group equation with the beta function $$\beta (\alpha _{{\rm s}})$$:  

(2)
\begin{equation} \mu^2 \frac {d\alpha_{{\rm s}}}{d\mu^2} = \beta\left(\alpha_{{\rm s}}\right) \equiv -\left(b_1 \alpha_{{\rm s}}^2 + b_2 \alpha_{{\rm s}}^3 + \cdots \right), \end{equation}
where the coefficients $$b_n$$ can be calculated at each order of $$n$$ loops. For instance, at the one-loop level, $$b_1 = \left (33 - 2 N_f\right )/\left (12 \pi \right )$$ with the flavor number $$N_f$$. At this level, one can solve Eq. (2) analytically to find the solution  
(3)
\begin{equation} \alpha_{{\rm s}}\left(\mu^2\right) = \frac{4 \pi}{\left( 11 - \frac{2}{3} N_f\right) \ln\left(\frac{\mu^2}{\Lambda_{{\rm QCD}}^2}\right) }. \end{equation}
Hence, $$\alpha _{{\rm s}}\left (\mu ^2\right )$$ diverges at the QCD scale $$\mu ^2 = \Lambda _{\rm QCD}^2$$, where $$\Lambda _{\rm QCD}$$ is several hundred MeV. The emergence of the dimensional scale brings the rich structure in the QCD vacuum.5 Physically, due to the self coupling of the gluons, the forces between the colored quarks and gluons increase as their distances increase. This causes non-trivial phenomena such as color confinement, mass generation of hadrons, and spontaneous breaking of chiral symmetry. Thus, at the hadronic scale, we need essentially a non-perturbative method.

Now, the six kinds of quark have very different bare (current) masses: $$u$$, $$d$$, and $$s$$ quarks are light while $$c$$, $$b$$, and $$t$$ quarks are heavy in comparison with $$\Lambda _{\rm QCD}$$; $$m_u, m_d, m_s \ll \Lambda _{\rm QCD} \ll m_c, m_b, m_t$$. In fact, these inequalities may not be well satisfied for the strange quark with $$m_s \sim $$ 100 MeV, but for the discussion up to the bottom region (without top which is too far from the current discussions and is the scale where the weak interaction is no longer weak) $$\Lambda _{\rm QCD} \simeq 200$$–300 MeV, and so at least the inequality holds. The different mass scales for the light and heavy quarks provide us with different structures in hadron dynamics. In fact, because of the hierarchy of the quark masses and $$\Lambda _{{\rm QCD}}$$, it is possible to introduce two different small parameters: (light mass)/$$\Lambda _{{\rm QCD}}$$ and $$\Lambda _{{\rm QCD}}$$/(heavy mass). These two parameters may be used as a small parameter for perturbation series in QCD. For light hadrons, chiral perturbation is the relevant one associated with chiral symmetry with its spontaneous breaking, and for heavy hadrons spin symmetry is the one due to heavy quark symmetry. We will see that, for the discussion of exotic hadrons, both symmetries are useful and important.

Chiral symmetry

Let us consider the massless limit ($$m_{f} \rightarrow 0$$) for up, down, and strange quarks ($$f=u,d,s$$). In this case, chiral fermions of right- and left-handed chiralities  

(4)
\begin{equation} \psi_{{\rm R}f} = \frac{1+\gamma_{5}}{2} \psi_{f}, \quad \psi_{{\rm L}f} = \frac{1-\gamma_{5}}{2} \psi_{f} \end{equation}
may decouple. This is shown explicitly by the QCD Lagrangian, which, by ignoring the mass term, can be written as the sum of the right- and left-handed terms:  
(5)
\begin{equation} \bar{\psi}_{f} \left(i \diagup \hskip -10pt{D} - m_{f}\right) \psi_{f} \rightarrow \bar{\psi}_{{\rm R}f} i \diagup \hskip -10pt{D} \psi_{{\rm R}f} + \bar{\psi}_{{\rm L}f} i \diagup \hskip -10pt{D} \psi_{{\rm L}f}. \end{equation}
This is invariant under separate flavor transformations for the right and left components, $${\rm SU}(N_{\rm F})_{\rm R} \times {\rm SU}(N_{\rm F})_{\rm L}$$:  
(6)
\begin{equation} \psi_{\rm R} \rightarrow e^{i \theta_{\rm R}^{a}T^{a}} \psi_{\rm R}, \quad \psi_{\rm L} \rightarrow e^{i \theta_{\rm L}^{a}T^{a}} \psi_{\rm L}, \end{equation}
with $$\psi _{\rm R/L} = \big (u_{\rm R/L}, d_{\rm R/L}, s_{\rm R/L}\big )^{\rm t}$$ and with $$T^{a} \big (a=1,\ldots ,N_{\rm F}^{2}-1\big )$$ being generators of the flavor $${\rm SU}(N_{\rm F})_{\rm R/L}$$ symmetry for the right- and left-handed components, respectively.

Chiral symmetry manifests for the QCD Lagrangian of the light flavor sector, while it may not for the vacuum. It is known that non-trivial configurations of the gluons field, instantons, may provide interactions which mix the left- and right-handed quarks. When the interaction is sufficiently strong, the left and right quark–antiquark pairs condense,  

(7)
\begin{equation} \left\langle \bar{\psi}_{f} \psi_{f} \right\rangle = \left\langle \bar \psi_{\rm L} \psi_{\rm R} \right\rangle + \left\langle \bar \psi_{\rm R} \psi_{\rm L} \right\rangle \neq 0, \end{equation}
causing the breaking of the chiral (left- and right-handed) symmetry of the vacuum. This resolves the degeneracy of hadrons of opposite parities, leading to a rich/complicated spectrum of hadrons. For instance, the Nambu–Goldstone bosons of negative parity ($$\pi $$, $$K$$, $$\eta $$ mesons) become (approximately) massless, while their chiral symmetry partners, scalar mesons ($$f_0$$, $$\kappa $$, and $$a_{0}$$ mesons), remain massive. The ground state nucleon is of positive parity, while its negative parity partner appears at about 600 MeV above the nucleon [128].

An interesting feature of hadron dynamics is that mesons can mediate interactions between colorless hadrons. Such an interaction is regarded as a residual interaction after saturating the color charges, and is considered to be relatively weak as compared to the force between the colored quarks and gluons. Among various meson exchanges, of particular importance is the pion exchange interaction. There are two features in the pion exchange interaction:

  • It is a long distance force, because of the small mass of the Nambu–Goldstone boson.

  • It generates the strong tensor force that mixes different angular momenta, $$L$$ and $$L\pm 2$$.

Those two features lead to rich structures in many-body hadron systems. A typical example is the atomic nucleus, where a variety of modes are realized, e.g., single particle motions, collective motions such as surface vibrations and rotations, alpha-clustering, spin- and isospin-correlations, and so on. In the early developments of nuclear physics, these modes have been separately studied in a phenomenological manner [129]. Recently, however, there have been attempts at a systematic description in a unified manner, where the pion exchange force plays crucial roles [130–135]. What we are interested in here are the hadrons that are many-body systems of quarks and gluons, in particular, exotic states containing both heavy and light quarks. Once light quarks exist, they can couple to the pion, and cause the strong tensor force. The possible pion exchange force as well as the presence of heavy quarks may give us as rich a structure as in atomic nuclei, which is one of the main issues in the present discussions.

Heavy quark symmetry

Let us ask what will happen when a quark becomes very heavy as compared to the QCD scale of several hundred MeV. For sufficiently large mass $$m_{\rm Q}$$, the parameter $$\Lambda _{{\rm QCD}}/m_{\mathrm Q}$$ is regarded as a small parameter for a systematic expansion of the strong interaction dynamics in powers of it. In this subsection, we briefly discuss the usefulness of such expansions for the charm and bottom sectors, and apply them to the study of heavy hadron phenomenology (see Refs. [97, 136] for more precise information).

Let us consider a system which contains one heavy quark. In the limit $$m_{\rm Q} \to \infty $$, disturbance from the QCD interaction of order $$\Lambda _{{\rm QCD}}$$ may be neglected, and so the heavy quark can remain almost on-shell, though it is confined. Thus we introduce the on-shell momentum $$m_Q v^\mu $$ by the four-velocity $$v^\mu , v^2 = 1$$. In reality, a finitely heavy quark is disturbed by the QCD interaction and the momentum is changed from its on-shell value to  

(8)
\begin{equation} p^{\mu} = m_{\rm Q} v^{\mu} + k^{\mu},\end{equation}
where $$k^{\mu }$$ is a residual momentum whose scale is much smaller than $$m_{\rm Q}$$, characterizing the off-shell-ness of the heavy quark. In many practical applications, the four-velocity may be chosen as $$v^{\mu }=\big (1,\vec {0}\big )$$, the one in the rest frame of the heavy quark.

According to the decomposition of the momentum, the effective field for the heavy quark with four-velocity $$v^{\mu }$$ is defined as  

(9)
\begin{equation} Q_{v}(x) = e^{im_{\rm Q}v\cdot x} \frac{1+ \diagup \hskip -10pt{v} }{2} Q(x)\end{equation}
for the original heavy quark field $$Q(x)$$. The factor $$e^{im_{\rm Q}v\cdot x}$$ indicates that the momentum scale $$m_{\rm Q} v^{\mu }$$ is extracted from $$Q(x)$$, so that the energy of the effective heavy quark $$Q_{v}$$ is defined by subtracting the mass. As a result, the effective field has only a residual momentum $$k^{\mu }$$. The operator $$(1+ \diagup \hskip -10pt{v} )/2$$ is the projection operator to select the positive energy component of $$Q(x)$$.

Similarly, we can define the effective field for the heavy antiquark as  

(10)
\begin{equation} \mathcal{Q}_{v}(x) = e^{im_{\rm Q}v\cdot x} \frac{1- \diagup \hskip -10pt{v} }{2} Q(x), \end{equation}
in the same frame, by applying the projection operator $$(1- \diagup \hskip -10pt{v} )/2$$. We note also that $$\mathcal { Q}_{v}(x)$$ corresponds to the excitation with mass $$2m_{\rm Q}$$, as shown soon.

With the above setup, we separate the heavy quark part from the light quark and gluon part in the QCD Lagrangian, as a sum of the heavy quark part and the light quark and gluon part,  

(11)
\begin{equation} \mathcal{L}_{\rm{QCD}} = \mathcal{L}_{{\rm heavy}} + \mathcal{L}_{{\rm light}}, \end{equation}
where the explicit form of $$\mathcal {L}_{{\rm heavy}} $$ is  
(12)
\begin{equation} \mathcal{L}_{{\rm heavy}} = \bar{Q} \big(i \diagup \hskip -10pt{D} -m_{\rm Q}\big) Q, \end{equation}
and the explicit form of $$\mathcal {L}_{\rm {light}}$$ is omitted because the contributions from light quarks and gluons are irrelevant to the heavy quark spin symmetry. As for the heavy part, in the first term, a summation over heavy quark flavors can be taken, if multiple heavy flavors (e.g. charm and bottom) are considered. In the present cases, we consider only a single heavy flavor. By using the positive energy field operator $$Q_{v}$$ in Eq. (10), let us rewrite the heavy quark part as  
(13)
\begin{equation} \mathcal{L}_{{\rm heavy}} = \bar{Q}_{v} v \cdot i D Q_{v} -\bar{\mathcal{Q}}_{v} \left(v \cdot i D + 2m_{\rm Q}\right) \mathcal{Q}_{v} +\bar{Q}_{v} i \diagup \hskip -10pt{D} _{\perp}\mathcal{Q}_{v} +\bar{\mathcal{Q}}_{v} i \diagup \hskip -10pt{D} _{\perp} Q_{v}. \end{equation}
The $$2m_Q$$ in the second term indicates that $$\mathcal {Q}_{v}$$ corresponds to the excitation of the mass $$2m_{\rm Q}$$, hence the component $$\mathcal {Q}_{v}$$ should be suppressed for large $$m_{\rm Q}$$, so long as we are interested in the heavy quark limit. In fact, using the equation of motion for $$\mathcal {Q}_{v}$$,  
(14)
\begin{equation} (v \cdot i D + 2m_{\rm Q}) \mathcal{Q}_{v} = i \diagup \hskip -10pt{D} _{\perp}{Q}_{v}, \end{equation}
and eliminating $$\mathcal {Q}_{v}$$, we obtain  
(15)
\begin{align} \mathcal{L}_{{\rm heavy}} &=\bar{Q}_{v} \left( v \cdot i D + i \diagup \hskip -10pt{D} _{\perp}\frac{1}{2m_{\rm Q} + v \cdot i D} i \diagup \hskip -10pt{D} _{\perp}\right) Q_{v} \nonumber \\ &=\bar{Q}_{v} v \cdot i D Q_{v} + \bar{Q}_{v} \frac{(i D_{\perp})^2}{2m_{\rm Q}} Q_{v} - g_{\rm s} \bar{Q}_{v} \frac{\sigma_{\mu \nu}G^{\mu \nu}}{4m_{\rm Q}} Q_{v}+ \mathcal{O}\left(1/m_{\rm Q}^{2}\right)\!, \end{align}
with $$D_{\perp }^{\mu }=D^{\mu } -v^{\mu } \, v \cdot D$$ [97]. This is the result at the tree level, because we have used the equation of motion for $$\mathcal {Q}_{v}(x)$$. In order to reach final results consistent with the original QCD at a given order of $$1/m_{\rm Q}$$, we need to include the quantum corrections given by the Wilson coefficients. As a result, we obtain the effective Lagrangian in the $$1/m_{\rm Q}$$ expansion for heavy quarks as  
(16)
\begin{align} \mathcal{L}_{\rm{HQET}} = \sum_{\rm Q} \left[ \bar{Q}_{v} v \cdot i D Q_{v} + \bar{Q}_{v} \frac{(i D_{\perp})^2}{2m_{\rm Q}} Q_{v}- c(\mu) g_{{\rm s}} \bar{Q}_{v}\frac{\sigma_{\mu \nu}G^{\mu \nu}}{4m_{\rm Q}} Q_{v}+ \mathcal{O}\big(1/m_{\rm Q}^{2}\big) \right] + \mathcal{L}_{{\rm light}}, \end{align}
with $$\sigma ^{\mu \nu }=i\big [\gamma ^{\mu },\gamma ^{\nu }\big ]/2$$. In the third term in the square brackets in Eq. (16), $$c(\mu )$$ is the Wilson coefficient which is determined by matching to QCD at an energy scale $$\mu $$ due to quantum corrections. On the other hand, the first and second terms are not affected by the quantum corrections, because the invariance under the velocity rearrangement (i.e. the Lorentz boost valid up to $$\mathcal { O}(1/m_{\rm Q})$$) is maintained. In other words, the quantum corrections are protected from the Lorentz symmetry up to $$\mathcal { O}(1/m_{\rm Q})$$, as explained later.

The effective theory given by Eq. (16) is called the heavy quark effective theory (HQET). The important feature is that the Lagrangian for the heavy quark is given as a power expansion series of $$1/m_{\rm Q}$$. This feature enables us to discuss systematically at each $$\mathcal {O}\big (1/m_{\rm Q}^{n}\big )$$ ($$n=0,1,\dots $$) for sufficiently large $$m_{\rm Q}$$.

Let us consider the leading order (LO) as the dominant term in the $$1/m_{\rm Q}$$ expansion. The term for the heavy quark at LO,  

(17)
\begin{equation} \sum_{\rm Q} \bar{Q}_{v} v \cdot i D Q_{v}, \end{equation}
has two important symmetries, spin symmetry and heavy flavor symmetry. The spin symmetry is understood from the invariance under the spin transformation of $$Q_{v}$$, $$Q_{v} \mapsto S Q_{v}$$, where $$S \in \rm {SU}(2)_{\rm spin}$$ is a spin transformation operator [97]). This is clear because the term $$v \cdot i D$$ does not contain the Dirac (and hence spin) matrices. The heavy flavor symmetry, a unitary transformation for mixing the different heavy flavors, is also understood from the fact that the term $$v \cdot i D$$ does not depend on the species of the heavy quark $$Q$$ and hence it is regarded as a unit matrix in the heavy flavor space. Consequently, the Lagrangian (17) has $${\rm SU}(2N_{\rm h})$$ symmetry including both the spin symmetry and the flavor symmetry, where $$N_{\rm h}$$ is the number of heavy flavors. $$N_{\rm h}=2$$ for two flavors (e.g. charm and bottom), and $$N_{\rm h}=1$$ for a single flavor (e.g. charm or bottom). The heavy flavor symmetry is used for studies of the weak interaction, for instance for the process $$b \rightarrow c\,e\,\bar {\nu }_{e}$$ [97], but this is not covered in the present article.

As discussed above, the heavy quark symmetry is exact in the heavy quark limit ($$m_{\rm Q} \to \infty $$). In realistic situations, however, the heavy quark masses are still finite, and the heavy quark symmetry is violated at $$\mathcal {O}(1/m_{\rm Q})$$. To see the violation, we consider the next-to-leading order (NLO) in the $$1/m_{\rm Q}$$ expansion in $$\mathcal {L}_{\rm {HQET}}$$:  

(18)
\begin{equation} \sum_{\rm Q} \left[\bar{Q}_{v} \frac{(i D_{\perp})^2}{2m_{\rm Q}} Q_{v}- c(\mu) g_{{\rm s}} \bar{Q}_{v} \frac{\sigma_{\mu \nu}G^{\mu \nu}}{4m_{\rm Q}} Q_{v}\right]\!. \end{equation}
Because $$m_{\rm Q}$$ differs for each heavy flavor, the first term is not invariant under rotations in the multi-heavy-flavor space—the heavy flavor symmetry is violated. Moreover, the second term is not invariant under both the flavor symmetry and the spin transformation ($$Q_{v} \mapsto SQ_{v}$$). The latter violation is understood, because $$\sigma ^{\mu \nu }$$ contains Dirac matrices proportional to the Pauli matrices in the rest frame. The violation of the spin symmetry is known in the spin interaction by the magnetic moment of an electron in quantum electrodynamics (QED). The present case is a version of the non-Abelian gauge theory. It is worthwhile noting that the spin symmetry violation occurs at the NLO of the heavy mass expansion for heavy fermions whenever they couple to gauge fields (either Abelian or non-Abelian).

In the following discussion, we will consider only a single flavor ($$N_{\rm h}=1$$), and concentrate on the strong interaction, considering only the spin symmetry. This is the heavy quark spin symmetry, or the heavy quark symmetry (HQS) in short.

Now, let us come back again to the heavy quark limit ($$m_{{\rm Q}} \rightarrow \infty $$), where only the LO term in $$\mathcal {L}_{\rm HQET}$$ plays a role. In this limit, the spin of the heavy quark is a well-conserved quantity. Let us define the total spin (including simultaneously the angular momenta and spins) for hadrons as  

(19)
\begin{equation} \vec{J} = \vec{S} + \vec{j}, \end{equation}
where $$\vec {S}$$ is a heavy quark spin (excluding the angular momentum), and $$\vec {j}$$ is the total spin of the light components ($$u$$, $$d$$, $$s$$ quarks, their antiquarks, and gluons). Because $$\vec {J}$$ and $$\vec {S}$$ are conserved quantities, $$\vec {j}$$ must also be conserved. Hereafter we call $$\vec {j}$$ the “brown muck” spin. Originally, the word “muck,” implying garbage or scattered useless things, was used because the structure of the light component is not important, but only the conserved total light spin $$j$$ is essential for the discussion of heavy quark symmetry [97, 136]. In hadron dynamics at low energy, however, the properties of the brown muck light components turn out to be important. For example, let us consider a $$Q\bar {q}$$ meson composed of a heavy quark $$Q$$ and a light antiquark $$\bar {q}$$, provided that the light component contains not only a $$\bar {q}$$ but also a multiple number of light quark–antiquark pairs ($$\bar {q}q$$) and gluons ($$g$$). Hence, the $$Q\bar {q}$$ should be regarded as multi-particle objects whose structure may be denoted by $$Q\bar {q}+Q\bar {q}\bar {q}q + Q\bar {q}\bar {q}qg+\cdots $$ schematically.6 We emphasize that, regardless of the complexity of the light components, $$\vec {j}$$ is a well-conserved quantity, because $$\vec {S}$$ is conserved in the heavy quark limit and $$\vec {J}$$ is also conserved. Thus, $$\vec {j}$$ provides us with a good quantum number for the classification of the non-perturbative object composed of the light components [97, 136].

Fig. 19.

A schematic picture of the spin degeneracy of states with total spin $$J_{\pm } = j \pm 1/2$$ in the heavy quark mass limit (left) and the small mass splitting for finite heavy quark mass (right). The short and long arrows indicate the spins of the heavy quarks and the light components, respectively.

Fig. 19.

A schematic picture of the spin degeneracy of states with total spin $$J_{\pm } = j \pm 1/2$$ in the heavy quark mass limit (left) and the small mass splitting for finite heavy quark mass (right). The short and long arrows indicate the spins of the heavy quarks and the light components, respectively.

Fig. 20.

A schematic picture of the mass splitting between the $$J^{P}=0^{-}$$ and $$1^{-}$$ mesons, $$\pi $$$$\rho $$, $$K$$$$K^{\ast }$$, $$D$$$$D^{\ast }$$, and $$B$$$$B^{\ast }$$.

Fig. 20.

A schematic picture of the mass splitting between the $$J^{P}=0^{-}$$ and $$1^{-}$$ mesons, $$\pi $$$$\rho $$, $$K$$$$K^{\ast }$$, $$D$$$$D^{\ast }$$, and $$B$$$$B^{\ast }$$.

As an application of heavy quark spin conservation to hadron spectroscopy, we find that the two states of heavy hadrons with brown muck spins, $$j \ge 1/2$$ and $$J_{\pm }=j\pm 1/2$$, are degenerate in mass for $$j \neq 0$$, as shown in Fig. 19. This is because the spin-dependent interaction is suppressed by $$1/m_{{\rm Q}}$$ due to the HQS. We will call those two paired states the HQS doublet. We note that $$j$$ takes a half-integer value for mesons, and an integer value for baryons. For $$j=0$$ for a baryon, there is only a single state with $$J_{+}= 1/2$$. We call this state the HQS singlet.

Such classifications according to spin symmetry are found in several heavy hadrons. For examples, we consider $$D$$ and $$D^{*}$$ mesons for charm and $$B$$ and $$B^{*}$$ mesons for bottom, whose mass splittings are small: about 140 MeV for the former and about 45 MeV for the latter, as shown in Fig. 20. It may be worthwhile to compare those values with the large mass splittings in light hadrons: about 630 MeV for $$\pi $$ and $$\rho $$ mesons, and about 390 MeV for $$K$$ and $$K^{\ast }$$ mesons. The small mass splittings suggest that $$(D^*,D)$$ mesons and $$(B^*,B)$$ mesons, respectively, are approximately degenerate and are regarded as an HQS doublet with $$j=1/2$$. Examples for the baryon sector are $$\Sigma _{c}$$ and $$\Sigma _{c}^{\ast }$$ baryons for charm and $$\Sigma _{b}$$ and $$\Sigma _{b}^{\ast }$$ baryons for bottom. Their mass splittings are about 65 MeV and 20 MeV, respectively.7 The small values suggest that $$\big (\Sigma _{c}, \Sigma _{c}^{\ast }\big )$$ baryons and $$\big (\Sigma _{b}, \Sigma _{b}^{\ast }\big )$$ baryons, respectively, are approximately degenerate, and hence they are regarded as HQS doublets with $$j=1$$. We note that the ground state baryons, $$\Lambda _{c}$$ for charm and $$\Lambda _{b}$$ for bottom, are regarded as HQS singlets, because the light component would have $$j=0$$ in the corresponding state in the heavy limit.

Next, let us consider the NLO terms in the $$1/m_{\rm Q}$$ expansion in the HQET Lagrangian (18). The contributions at NLO are roughly estimated as follows. Employing the typical energy scale $$\Lambda _{\rm QCD} \simeq 200$$–300 MeV, we find $$\Lambda _{\rm QCD}/m_{\rm c}=0.15$$$$0.23$$ ($$m_{\rm c}=1.3$$ GeV) for charm and $$\Lambda _{\rm QCD}/m_{\rm b}=0.04$$$$0.06$$ ($$m_{\rm b}=4.7$$ GeV) for bottom. To discuss a general formulation for $$1/M$$ corrections, let us remember that the coordinate frame with four-velocity $$v(v$$-frame) has been defined for the heavy particle with momentum $$p^{\mu } = M v^{\mu } + k^{\mu }$$, the sum of the on-shell part ($$Mv^{\mu }$$ with $$v^{2}=1$$) and the off-shell part ($$k^{\mu }$$). The latter is a quantity of order $$\mathcal {O}(\Lambda _{\rm QCD})$$ (cf. Eq. (8)). An important observation, however, is that those two parts are separated uniquely only in the heavy quark limit. When the contribution from $$\mathcal { O}(1/M)$$ is taken into account, the choice of the frame is not unique. Applying a small Lorentz boost, instead of the $$v$$-frame, we can consider another frame with four-velocity $$w^{\mu }=v^{\mu }+q^{\mu }/M$$ ($$w$$-frame), where $$q^{\mu }$$ is a small momentum which is much smaller than $$M$$ and satisfies $$\mathcal {O}(v\cdot q) = q^{2}/M$$ (Fig. 21). Here, because $$w^{2}=(v+q/M)^2=v^{2}+\mathcal {O}(1/M^2)$$, $$w^{\mu }$$ is properly normalized as $$w^{2}=1$$ up to $$\mathcal {O}(1/M)$$. In the $$w$$-frame, the momentum $$p^{\mu }$$ is given by $$p^{\mu } = M w^{\mu }+l^{\mu }$$ with $$l^{\mu }$$ being defined by $$l^{\mu } = k^{\mu }-q^{\mu }$$. Such a transformation is called “velocity rearrangement” (VR) [137, 138].

The velocity rearrangement can be applied to any heavy particles with an arbitrary spin. For example, the heavy quark field $$Q_{v}(x)$$ defined in the $$v$$-frame is related by the Lorentz transformation to $$Q_{w}(x)$$ in the $$w$$-frame with $$w^{\mu }=v^{\mu }+q^{\mu }/m_{{\rm Q}}$$. Any Lagrangian of the heavy particles has to be constrained by the invariance under velocity rearrangement, which may be called “velocity rearrangement invariance” (VRI). This gives useful constraints in constructing the effective Lagrangians at NLO, as presented explicitly in Sect. 3.3.2. In fact, the Lagrangian of the HQET (16) is invariant under velocity rearrangement up to $$\mathcal {O}(1/m_Q)$$.

Fig. 21.

The Lorentz boost from the $$v$$-frame to the $$w$$-frame. In each frame, the heavy particle has four-velocity $$v^{\mu }$$ and $$w^{\mu }$$, respectively. The different lengths of the arrows indicate the different spatial components for $$v^{\mu }$$ and $$w^{\mu }$$, respectively.

Fig. 21.

The Lorentz boost from the $$v$$-frame to the $$w$$-frame. In each frame, the heavy particle has four-velocity $$v^{\mu }$$ and $$w^{\mu }$$, respectively. The different lengths of the arrows indicate the different spatial components for $$v^{\mu }$$ and $$w^{\mu }$$, respectively.

Lastly, we comment on the parametrization of the masses of heavy hadrons. In the $$1/m_{\rm Q}$$ expansion, the mass of a heavy hadron is given by  

(20)
\begin{equation} M_{H} = m_{Q} + \bar{\Lambda} - \frac{\lambda_{1}}{2m_{Q}} + 4\vec{S}_{Q} \cdot \vec{j} \, \frac{\lambda_{2}(\mu)}{2m_{Q}} + \mathcal{O}\left(1/m_{Q}^2\right), \end{equation}
where we have defined the low-energy parameters, in the rest frame with $$v_{{\rm r}}=\big (1,\vec {0}\big )$$,  
(21)
\begin{align} \bar{\Lambda} &= \frac{1}{2M_{{\rm H}}} \left\langle \tilde{{\rm H}}_{v_{{\rm r}}} \right| \frac{\beta(\alpha_{{\rm s}})}{4\alpha_{{\rm s}}} G^{2} \left| \tilde{{\rm H}}_{v_{{\rm r}}} \right\rangle, \end{align}
 
(22)
\begin{align} \frac{\lambda_{1}}{m_{Q}} &= -\left\langle {\rm H}_{v_{{\rm r}}} \right| \bar{Q}_{v_{{\rm r}}} g_{{\rm s}} \vec{x} \cdot \vec{E} Q_{v_{{\rm r}}} \left| {\rm H}_{v_{{\rm r}}} \right\rangle, \end{align}
 
(23)
\begin{align} 8 \vec{S}_{Q} \cdot \vec{j} \lambda_{2}(\mu) &= \tfrac{1}{2} c(\mu) \left\langle {\rm H}_{v_{{\rm r}}} \right| \bar{Q}_{v_{{\rm r}}} g_{{\rm s}} \vec{\sigma} \cdot \vec{B} Q_{v_{{\rm r}}} \left| {\rm H}_{v_{{\rm r}}} \right\rangle, \end{align}
denoting the hadron state by $$\left | {\rm H}_{v_{{\rm r}}} \right \rangle $$ [139–142].8$$\vec {S}_{Q}=\vec {\sigma }/2$$ and $$\vec {j}$$ are the spin operators for the heavy quark and the brown muck respectively. We have also introduced the Gell-Mann–Low function $$\beta (\alpha _{{\rm s}})=\mu {d}\alpha _{{\rm s}}(\mu )/{d}\mu $$, the chromoelectric gluon field $$E^{i}=-G^{a\,0i}T^{a}$$, and the chromomagnetic gluon field $$B^{i}=\varepsilon ^{ijk}G^{a\,jk}T^{a}$$ ($$i,j,k=1,2,3$$; $$a=1,\dots ,8$$). In Eq. (23), the Wilson coefficient $$c(\mu )$$ is determined by matching to QCD at energy scale $$\mu \simeq m_{{\rm Q}}$$. The first equation (21) originates from the scale anomaly in the trace of the energy–momentum tensor in QCD [140, 141]. The parametrization (20) holds generally in any hadronic states with a heavy quark, provided that the values $$\bar {\Lambda }$$, $$\lambda _{1}$$, and $$\lambda _{2}(\mu )$$ depend on each hadronic state.

Theoretical methods

In this section, we discuss several theoretical methods for analyzing the exotic hadrons: the quark model (Sect. 3.3.1), the heavy hadron chiral effective theory (Sect. 3.3.2), the hadronic molecule model (Sect. 3.3.3), the QCD sum rules (Sect. 3.3.4), and lattice QCD simulations (Sect. 3.3.5).

Quark model

The quark model was introduced by Gell-Mann and Zweig independently to explain the new particles found around the 1960s in addition to the nucleons and pions. In his famous paper entitled “A schematic model of baryons and mesons” in 1964 [1], Gell-Mann showed that hadrons are classified systematically by a few constituent particles, named quarks “$$q$$.” Baryons and mesons are then composed of $$qqq$$ and $$\bar {q}q$$, respectively, and the light hadrons are explained by multiplets of SU(3) flavor symmetry for $$u$$, $$d$$, and $$s$$ quarks. In QCD, the SU(3) flavor symmetry is considered to be a consequence of spontaneous breaking of chiral symmetry, where light current quarks acquire approximately similar masses of order $$\Lambda _{\rm QCD}$$. In reality, the $$s$$ quark has a heavier mass than the $$u$$ and $$d$$ quarks. But the small breaking of flavor symmetry was successfully applied to explain mass splittings among an SU(3) multiplet, which is known as the Gell-Mann–Okubo mass formula.

Here we emphasize that the “quark” in the quark model is not the same as the quark in the QCD Lagrangian (1). The “quarks” in the former are the constituent particles with dressed masses ($$m_{u,d} \simeq 400$$ MeV and $$m_{s} \simeq 500$$ MeV) which are dynamically generated as a result of the spontaneous breaking of chiral symmetry in the QCD vacuum.9 In this sense, the constituent quarks are composite particles with some structure.

Although the relationship between the quark model and QCD is not clear, it works well in a semi-quantitative manner, not only for the ground state, but also for the classification of low-lying excited states. Some of the excited states are described by orbital motions of the constituent quarks and/or by spin–spin interaction, and so on. Thus, the quark model provides us with a simple picture for the internal structures of hadrons.

The quark model carries important aspects of low energy QCD dynamics, mostly based on flavor and spin symmetries [145, 146]. Both of them are, however, not exact symmetries of QCD, but empirically they work well by approximate degeneracy of baryon octet states of spin 1/2 and decuplet states of spin 3/2. These symmetries may be explained due to spontaneous breaking of chiral symmetry generating constituent masses for $$u$$, $$d$$, and $$s$$ quarks with an almost equal amount of order $$\Lambda _{{\rm QCD}}$$. The quark model is often also used for studies of new hadron states. The deviations from quark model predictions often trigger new ideas for dynamics which are not incorporated by the quark model.

In the quark model the wave functions are given as products of color, flavor, spin, and orbital parts:10 

(24)
\begin{equation} (\mbox{color}) \times (\mbox{flavor}) \times (\mbox{spin}) \times (\mbox{orbital motion}). \end{equation}
Let us consider the ground states where all quarks and antiquarks are in the lowest $$S$$-wave states. Then, because hadrons are color singlets, there are $$3\times 2=6$$ remaining degrees of freedom by flavor ($$u$$, $$d$$, $$s$$) and spin ($$\uparrow $$ and $$\downarrow $$), whose components are denoted by $$(u\uparrow ,u\downarrow ,d\uparrow ,d\downarrow ,s\uparrow ,s\downarrow )$$. An approximate flavor and spin symmetries imply that these six states are regarded as components of the extended flavor–spin symmetry SU(6).

In this symmetry, quarks belong to the fundamental representation $$\textbf {6}=(3,1/2)$$ (the latter notation means 3 for the flavor triplet and 1/2 for the spin doublet) in the SU(6) symmetry. Then hadron states are given as higher-dimensional representations (multiplets) of the SU(6) symmetry. The mesons are  

(25)
\begin{equation} \textbf{6} \times \textbf{6}^{\ast} = \textbf{1} + \textbf{35} = (1,0) + (1,1) + (8,0) + (8,1), \end{equation}
where $$\textbf {6}^{\ast }$$ is the complex conjugate of $$\textbf { 6}$$, corresponding to an antiquark. We obtain the flavor singlet and octet states both for spin 0 and 1. The baryons are given by  
(26)
\begin{equation} \textbf{6} \times \textbf{6} \times \textbf{6} = \textbf{20}_{\rm A} + \textbf{70}_{\rm MA} + \textbf{70}_{\rm MS} + \textbf{56}_{\rm S}. \end{equation}
However, because the color d.o.f. are antisymmetric in color singlet states, the flavor and spin d.o.f. must be symmetric and only $$\textbf {56}_{\rm S}=(8,1/2)+(10,3/2)$$ is allowed for the ground state baryons. These are the baryons of the flavor octet with spin 1/2 and flavor decuplet with spin 3/2, as anticipated. As another static property, the magnetic moments of hadrons can be explained. The excited hadrons with $$P$$-wave, $$D$$-wave, $$F$$-wave, and so on are classified according to the $$\mbox {SU(6)} \times \mbox {O(3)}$$ symmetry with orbital excitations incorporated by $$\mbox {O(3)}$$.

The above classification of the hadrons may be regarded as the “taxonomy” of multiplets based on the SU(6) symmetry. To understand the quantitative properties of hadrons, we need to further consider the quark dynamics. The study by Rújula, Georgi, and Glashow in their paper “Hadron masses in gauge theory” in 1975 played a seminal role in connecting the quark model with QCD [148]. Considering the one-gluon exchange between two quarks, they proposed a potential of the form  

(27)
\begin{equation} V = \alpha_{\rm s} \sum_{i>j} V^{\rm(S)}_{ij} + V^{\rm(L)}.\end{equation}
The short-range part $$V^{\rm (S)}_{ij}$$ for quarks $$i$$ and $$j$$ is given by  
(28)
\begin{align} V^{\rm(S)}_{ij} &= \vec{F}_{i} \cdot \vec{F}_{j} \left[ \frac{1}{r} - \frac{1}{2m_{i}m_{j}} \left( \frac{\vec{p}_{i} \cdot \vec{p}_{j}}{r} + \frac{\big(\vec{r} \cdot \vec{p}_{i}\big) \big(\vec{r} \cdot \vec{p}_{j}\big)}{r^{3}} \right)- \frac{\pi}{2} \delta^{(3)}(r) \left( \frac{1}{m_{i}^{2}} + \frac{1}{m_{j}^{2}} \right) \right. \nonumber \\ &\quad - \frac{1}{m_{i}m_{j}} \left(\frac{8\pi}{3} \vec{s}_{i} \cdot \vec{s}_{j} \delta^{(3)}(r) + \frac{1}{r^{3}} S_{ij} \right) \nonumber \\ &\quad - \left.\frac{1}{2r^{2}} \left(\frac{1}{m_{i}^{2}} \vec{l}_{i} \cdot \vec{s}_{i} - \frac{1}{m_{j}^{2}} \vec{l}_{j} \cdot \vec{s}_{j} + \frac{2}{m_{i}m_{j}} \left( \vec{l}_{i} \cdot \vec{s}_{i} - \vec{l}_{j} \cdot \vec{s}_{j} \right) \right)\right], \end{align}
where $$\alpha _{\rm s}$$ is the strong coupling constant, $$m_{i}$$ are quark masses, $$\vec {s}_{i}=\vec {\sigma }_{i}/2$$ spin operators ($$\vec {\sigma }_{i}$$ are the Pauli matrices), $$\vec {F}_{i}=\vec {\lambda }_{i}/2$$ color SU(3) operators ($$\vec {\lambda }_{i}$$ are the Gell-Mann matrices), $$S_{ij} = 3 \big (\vec {s}_{i} \cdot \hat {r}\big ) \big (\vec {s}_{j} \cdot \hat {r}\big ) - \vec {s}_{i} \cdot \vec {s}_{j}$$ the tensor operators with $$\hat {r} = \vec {r}/|\vec {r}\,|$$, and $$\vec {l}_{i}$$ the angular momentum. The long-range part $$V^{\rm (L)}$$ is responsible for confinement and is given by a linear potential, such as the Cornel potential [145].

Among several terms, the color-spin dependent potential in the short-range part,  

(29)
\begin{equation} V_{{\rm color-spin}} = - \frac{8\pi}{3} \sum_{i,j} \frac{\alpha_{\rm s}}{m_{i}m_{j}} \vec{F}_{i} \cdot \vec{F}_{j} \, \vec{s}_{i} \cdot \vec{s}_{j} \, \delta^{(3)}(r),\end{equation}
plays an important role in the mass splittings in $$S$$-wave hadrons. We note that $$\vec {F}_{i} \cdot \vec {F}_{j}$$ takes $$-4/3$$ for the $$\bar {q}q$$ pair (color singlet) and $$-2/3$$ for the $$qq$$ pair (color antitriplet). Similarly, $$\vec {s}_{i} \cdot \vec {s}_{j}$$ takes $$-3/4$$ for spin 0 and $$1/4$$ for spin 1 in $$\bar {qq}$$ ($$qq$$) pairs. We define $$\vec {F}_{i} = \vec {\lambda }_{i}/2$$ and $$\vec {s}_{i}=\vec {\sigma }_{i}/2$$ for the Gell-Mann matrices $$\vec {\lambda }_{i}$$ and the Pauli matrices $$\vec {\sigma }_{i}$$. We summarize the values of $$\vec {\lambda }_{i} \cdot \vec {\lambda }_{j}$$ as  
(30)
\begin{align} \vec{\lambda}_{i} \cdot \vec{\lambda}_{j} &= \begin{cases} -\tfrac{8}{3} & {\rm for} \quad \bar{\textbf{3}}_{c} \\ \tfrac{4}{3} & {\rm for} \quad \textbf{6}_{c} \end{cases} \quad {\rm in} \ {qq,} \quad {\rm and} \quad \begin{cases} -\tfrac{16}{3} & {\rm for} \quad \textbf{1}_{c} \\ \tfrac{2}{3} & {\rm for} \quad \textbf{8}_{c} \end{cases} \quad {\rm in} \ q\bar{q}, \end{align}
and those of $$\vec {\sigma }_{i} \cdot \vec {\sigma }_{j}$$ as  
(31)
\begin{equation} \vec{\sigma}_{i} \cdot \vec{\sigma}_{j}= \begin{cases} -3 & {\rm for} \quad S=0 \\ 1 & {\rm for} \quad S=1 \end{cases}.\end{equation}
These spin-dependent interactions order the masses as $$M(S=1)>M(S=0)$$ for mesons and $$M(S=3/2)>M(S=1/2)$$ for baryons. Moreover, due to the color-dependent part, the mass differences in mesons and baryons are related by $$M(1)-M(0)>M(3/2)-M(1/2)$$, which is consistent with the experimental data. We note that in this inequality the color factor is important; without it, the inequality would become reversed to be inconsistent with experimental data.

Next, we consider excited hadrons with various orbital motions. In this case, we need to consider interactions depending on orbital states, in particular $$\vec {l} \cdot \vec {s}$$ and tensor interactions as shown in Eq. (27). Then, the quark model has been shown to reproduce well properties of many excited hadrons with only a few input parameters.

Several comments are in order. First, the coupling constant $$\alpha _{\rm s}$$ used in the quark model takes a value close to one. Hence, the short-range part $$V^{\rm (S)}_{ij}$$ may not be regarded simply as a result of perturbation of QCD. Second, the quark mass used in this model is a dynamical mass (constituent mass) which is of order $$\Lambda _{{\rm QCD}}$$ (a few hundred MeV). These masses are heavier than those of current quarks in the QCD Lagrangian (a few MeV). Although it is generally believed that the large dynamical mass is due to spontaneous breaking of chiral symmetry, the relevance of the “quarks” used in the quark model to the “quarks” in the QCD Lagrangian is not yet clear. Third, the long-range part $$V^{({\rm L})}$$ indicates confinement through linearly increasing potential. Such behavior has been confirmed by lattice QCD simulations [93]. The origin of the confinement potential and chiral symmetry breaking has been studied in lattice QCD by analyzing the quark (Dirac) spectral modes [149]. They pointed out that the low-energy modes contribute to chiral symmetry breaking while for the confinement higher modes seem to play a role.

In spite of the phenomenological model setting, whose relationship to QCD is not necessarily clear, the quark model has been successfully applied to hadron spectroscopy, especially in the non-relativistic quark model [150–155]. In the present context, we refer to [15] for the charmonium mass spectrum which is used to draw Fig. 1.

However, the quark model has still left unresolved questions in hadron spectroscopy. For example, a longstanding problem is the mass of $$\Lambda (1405)$$, which is the lowest state among negative parity baryons, but is overestimated by about 100 MeV in the quark model. Thus, the $$\Lambda (1405)$$ state has been investigated extensively as a candidate for the $$\bar KN$$ molecule (for a recent review, see Ref. [156]).11 Another example is the mass ordering of scalar mesons. In the quark model, the scalar mesons are naively considered to be $$P$$-wave states composed of quark and antiquark ($$q\bar {q}$$). As proposed by Jaffe, however, the tetraquark picture ($$qq\bar {q}\bar {q}$$) can naturally explain the mass ordering, where the $$qq=ud$$, $$ds$$, $$su$$ diquarks are the building blocks instead of the constituent quarks [3, 4]. For example, let us consider the light scalar mesons: $$f_{0}(600)$$, $$\kappa (800)$$, $$f_{0}(980)$$, and $$a_{0}(980)$$. The mass ordering observed in experiments is  

(32)
\begin{equation} M(f_{0}(600)) < M(\kappa(800)) < M(f_{0}(980)) \simeq M(a_{0}(980)).\end{equation}
When two quarks ($$q\bar {q}$$; quark and antiquark) form the light scalar mesons, the mass ordering is naively given as  
(33)
\begin{equation} M\big(u\bar{u}+d\bar{d}\big) \simeq M\big(u\bar{d}\big) < M\big(u\bar{s}\big) < M\big(s\bar{s}\big) \end{equation}
by using isospin symmetry and the larger mass of the strange quark than up and down quarks. However, this clearly contradicts (32). On the other hand, when four quarks with a diquark pair ($$[qq]\big [\bar {q}\bar {q}\big ]$$; diquark $$[qq]=[ud], [ds], [su]$$ and antidiquark $$\big [\bar {q}\bar {q}\big ]=\big [\bar {u}\bar {d}\big ], \big [\bar {d}\bar {s}\big ], \big [\bar {s}\bar {u}\big ]$$ with spin 0 and antitriplet in SU(3) flavor symmetry) are considered as fundamental blocks, the mass ordering becomes  
(34)
\begin{equation} M\left([ud]\left[\bar{u}\bar{d}\right]\right) < M\left([ud]\left[\bar{d}\bar{s}\right]\right) < M(\left[su]\left[\bar{s}\bar{u}\right]+[ds]\left[\bar{d}\bar{s}\right]\right) \simeq M\left([su]\left[\bar{s}\bar{u}\right]-[ds]\left[\bar{d}\bar{s}\right]\right), \end{equation}
and can naturally explain the mass ordering in (32).

Fig. 22.

A schematic picture of the quark model.

Fig. 22.

A schematic picture of the quark model.

Now, in the quark model, excitation energies are brought either by orbital motions or by an additional $$q \bar q$$ pair, where the quark “$$q$$” is the constituent quark (Fig. 22). In general, hadrons are superpositions of the minimal quark configuration and other multi-quark configurations, such as $$q\bar {q}+qq\bar {q}\bar {q}$$.12 Multi-quark components suggest a variety of hadron structures including hadronic molecules which are dynamically generated through hadron interactions. Whether excited states are described as orbital excitations or as multi-quarks is one of the relevant questions for hadron structure, especially for exotic hadrons, which should eventually be solved by QCD.

Turning to quarkonium states of heavy flavors $$\big (c\bar {c}$$ and $$b\bar {b}\big )$$, an interesting feature is that many low-lying excited states have rather small decay widths below the open flavor threshold $$\big (\bar {D}^{(\ast )}D^{(\ast )}\ {\rm and}\ \bar {B}^{(\ast )}B^{(\ast )}\big )$$. The decay modes are indeed restricted due to the isosinglet nature of charmonia and bottomonia. For example, a single pion emission can occur only by small isospin breaking. Therefore, the excited charmonia and bottomonia are well described by a $$Q\bar {Q}$$ component with either node excitations or orbital excitations without additional light (anti)quarks and gluons like $$q\bar {q}$$, $$q\bar {q}g$$ below the open flavor threshold. This situation is very much different from that of light hadrons. For instance, we remember that there is evidence that the light scalar mesons are likely to be $$qq\bar {q}\bar {q}$$, while in the standard quark model, they are assigned as $$^{3}P_{0}$$ states of $$P$$-wave excitation. This is one of the reasons why charmonia and bottomonia bring us to precise mass spectroscopy. Recently, they have been investigated by QCD effective theories, such as potential non-relativistic QCD (pNRQCD), where the interaction between heavy quarks is introduced by the velocity expansion [157].13 This formalism enables us to systematically relate the phenomenological quark model for quarkonia to QCD.

Recent experimental discoveries of $$X,Y,Z$$ exotic hadrons, however, have opened the question about the applicability of the quark model around and above the thresholds. In addition to the conventional $$Q\bar {Q}$$ picture, a multi-quark and/or the hadronic molecule configurations should be taken into account. These are the issues which will be discussed in the following.

Heavy hadron chiral effective theory

In Sect. 3.2.3, we have discussed that the spin of a heavy quark is conserved in the heavy quark limit, and the total spin of the brown muck, namely the light component in the heavy hadron, is also conserved. In this subsection, we discuss the heavy hadron effective theory in terms of the heavy quark symmetry for the heavy quark and chiral symmetry for the brown muck, the light degrees of freedom. In order to present the concrete form of the equations, we concentrate on the heavy meson effective theory (HMET) [97, 158]. The basic idea will be straightforwardly applicable to the heavy baryon effective theory.

We define the effective field $$H_{\gamma }$$ for the heavy meson with spin $$\gamma $$, which is composed of the heavy quark ($$Q_{\alpha }$$) with spin 1/2 ($$\alpha = \pm 1/2$$) and the brown muck ($$\bar {q}_{\beta }$$) with a spin component $$\beta $$ as introduced in Sect. 3.2.3. This is schematically expressed as  

(35)
\begin{equation} H_{\gamma} = \sum_{\alpha,\beta} C_{\alpha \beta; \gamma} Q_{\alpha} \bar{q}_{\beta},\end{equation}
where the sum is taken over the spin combinations with appropriate Clebsh–Goldan coefficients, denoted by $$C_{\alpha \beta ;\gamma }$$. For example, we consider the mesons composed of a heavy quark $$Q$$ and a light antiquark $$\bar {q}$$ with conserved spin and parity $$j^\mathcal { P}=1/2^{-}$$, baryon number $$-1/3$$, and color antitriplet, denoted by $$P_{v}(x)$$ with spin 0 and $$P_{v}^{\ast \, \mu }(x)$$ with spin 1. Here, $$v$$ is the conserved four-velocity of the heavy meson, which is regarded as that of the heavy quark in the heavy quark limit. Moreover, in this limit $$P_{v}(x)$$ and $$P_{v}^{\ast \, \mu }(x)$$ are degenerate in mass, and they are treated as heavy quark spin multiplets  
(36)
\begin{equation} H_{v}(x) = \frac{1+ \diagup \hskip -10pt{v} }{2} \left(\gamma_{\mu}P_{v}^{\ast\mu}(x) + i \gamma_{5} P_{v}(x) \right)\!,\end{equation}
where $$(1+ \diagup \hskip -10pt{v} )/2$$ is the projection operator as introduced in Eq. (9) for the positive energy state of the heavy quark $$Q$$ in the $$Q \bar q$$ meson. More explicitly, the meson fields $$P_{v}^{\ast \,\mu }$$ and $$P_{v}$$ have the matrix structure  
(37)
\begin{equation} P_{v}^{\ast\,\mu} \sim Q_{v}\gamma^{\mu}\bar{q}, \quad P_{v} \sim Q_{v} \gamma_{5} \bar{q}, \end{equation}
where $$Q_{v}$$ is the heavy quark field in the $$v$$-frame as defined by Eq. (9) and $$\bar {q}$$ is the brown muck field. The transformation properties are important in the construction of the effective Lagrangian. The heavy quark spin for the heavy quark and chiral transformation for the brown muck leads to the change of the field as  
(38)
\begin{equation} H_{v}(x) \rightarrow S H_{v}(x) U_{V}^{\dagger},\end{equation}
where $$S$$ is the heavy quark spin operator and $$U_{V}$$ is the chiral transformation operator. The effective Lagrangian for heavy hadrons is constructed systematically according to both heavy quark mass expansion and chiral order expansion.

First, we consider the leading order in both of them. For the effective field $$H_{v}(x)$$ defined in Eq. (36), the effective Lagrangian at leading order (LO) is given as  

(39)
\begin{equation} \mathcal{L}_{\rm{HMET}}^{{\rm LO}} = - {\rm Tr}\, \bar{H}_{v}(x) i v \cdot D H_{v}(x) + g \, {\rm Tr} \, \bar{H}_{v}(x) H_{v}(x) \gamma_{\mu} \gamma_{5} A^{\mu}(x),\end{equation}
where the trace is taken over the Dirac matrices (see Eq. (36)), $$D^{\mu }$$ is the chiral covariant derivative, $$D^{\mu } = \partial ^{\mu } - V^{\mu }(x)$$, and $$V^{\mu }(x)$$ and $$A^{\mu }(x)$$ are the vector and axial-vector currents of pions, respectively [97, 136, 158]. The coupling constant $$g$$ is a free parameter which should be determined from experimental data or by non-perturbative methods for QCD. We can verify that Eq. (39) is invariant under the transformation (38).

Second, we consider the effective Lagrangian at the next-to-leading order (NLO). To include the $$1/M$$ corrections with $$M$$ being the mass of the heavy particle, we recall the velocity rearrangement invariance in Sect. 3.2.3. In this scheme, we define the effective fields $$H_{v}(x)$$ and $$H_{w}(x)$$ in the two different frames with four-velocities $$v$$ and $$w$$. Because the Lorentz boost between the $$v$$-frame and the $$w$$-frame is taken into account up to $$\mathcal {O}(1/M)$$, the Lorentz transformation between $$H_{v}(x)$$ and $$H_{w}(x)$$ is also taken into account up to this order. Then, we construct the effective Lagrangian invariant under the velocity rearrangement. However, it is not easy to find out practically the velocity rearrangement invariant terms, because the form of the Lorentz transformation between $$H_{v}(x)$$ and $$H_{w}(x)$$ is not so simple in general. For $$Q\bar {q}$$ mesons, for example, the heavy fields $$H_{v}(x)$$ and $$H_{w}(x)$$ are related as [137, 138]  

(40)
\begin{equation} H_{v}(x) = \left( H_{w}(x) - \frac{1}{2M} \left[ \diagup \hskip -10pt{q} ,H_{w}(x) \right] \right) e^{-iq\cdot x} + \mathcal{O}\left(1/M^2\right)\!, \end{equation}
with $$q/M=w-v$$. This has not only the phase $$e^{-iq\cdot x}$$, but also the additional term in the parentheses, in which the latter causes the difficulty in finding the invariant effective Lagrangian. To avoid this difficulty, instead of $$H_{v}(x)$$, we introduce the modified field as [137, 138, 142]  
(41)
\begin{equation} {\mathcal H}_{v} = H_{v} + \frac{1}{2M} \left(i\overrightarrow{ \diagup \hskip -10pt{D} }/ H_{v} - H_{v} i\overleftarrow{ \diagup \hskip -10pt{D} }- 2v \cdot iD H_{v} \right) + \mathcal{O}\left(1/M^2\right)\!, \end{equation}
which is given by the Lorentz boost from the $$v$$-frame to the $$(v+q/M)$$-frame up to $$\mathcal {O}(1/M)$$, where $$q^{\mu }$$ is supposed to be the residual momentum in $$H_{v}(x)$$. This has the interesting property  
(42)
\begin{equation} {\mathcal H}_{v} = e^{-i q\cdot x} {\mathcal H}_{w} + \mathcal{O}\left(1/M^2\right)\!, \end{equation}
namely a change only by the phase factor $$e^{-i q\cdot x}$$. This enables us to easily construct the effective Lagrangian with velocity rearrangement invariance [137, 138]. Another $$1/M$$ correction is provided by the heavy quark spin violation, independent of the velocity rearrangement.

Including all the $$1/M$$ contributions, we obtain the form of the effective Lagrangian up to $$\mathcal { O}(1/M)$$ [137, 138, 142]:  

(43)
\begin{align} {\mathcal L}_{{\rm HMET}}^{{\rm LO}+{\rm NLO}} &= - {{\rm Tr}}\, \overline{H}_{v} v \cdot iD H_{v} - {{\rm Tr}}\, \overline{H}_{v} \frac{(iD)^2}{2M} H_{v} + \frac{\lambda}{M} {\rm Tr}\, \overline{H}_{v} \sigma^{\mu\nu} H_{v} \sigma_{\mu\nu} \nonumber\\ &\quad+ \left( g+\frac{g_1}{M} \right) {{\rm Tr}}\, \overline{H}_{v} H_{v} \gamma_{\mu} \gamma_{5} \mathcal{A}^{\mu} \nonumber \\ &\quad+ \frac{g}{2M} \left( {{\rm Tr}}\, v \cdot iD \overline{H}_{v} H_{v} \gamma_{\mu} \gamma_{5} \mathcal{A}^{\mu} - {{\rm Tr}}\, \overline{H}_{v} v \cdot iD H_{v} \gamma_{\mu} \gamma_{5} \mathcal{A}^{\mu} \right) \nonumber \\ &\quad+ \frac{g}{4M} \varepsilon_{\mu\nu\rho\sigma} \left( {{\rm Tr}}\, iD^{\nu} \overline{H}_{v}H_{v} \sigma^{\rho\sigma} \mathcal{A}^{\mu} - {{\rm Tr}}\, \overline{H}_{v} iD^{\nu} H_{v} \sigma^{\rho\sigma} \mathcal{A}^{\mu} \right) \nonumber \\ &\quad+ \frac{g_2}{M} {{\rm Tr}}\, \overline{H}_{v} \gamma_{\mu} \gamma_{5} H_{v} \mathcal{A}^{\mu} + {\mathcal O}\left(1/M^2\right)\!, \end{align}
where $$\lambda $$, $$g_{1}$$, and $$g_{2}$$ are new coupling constants at $$\mathcal {O}(1/M)$$.14 We note that fifth and sixth terms on the right-hand side are proportional to $$g$$. This indicates that they are connected with the terms at LO due to the velocity rearrangement invariance. We note, however, that this is not the case in general. For instance, the terms proportional to $$g_{1}$$ and $$g_{2}$$ are not constrained by the velocity rearrangement invariance. More importantly, the term proportional to $$g_{2}$$ breaks the heavy quark spin symmetry, due to the factor $$\gamma _{\mu } \gamma _{5}$$ which plays the role of the spin for the heavy $$Q$$. Therefore, we have the heavy quark spin conserving part ($$g$$ and $$g_{1}$$) and the heavy quark spin non-conserving part ($$g_{2}$$).

Lastly we comment on the relation between the heavy hadron effective theory, such as the HMET, and the heavy quark effective theory (HQET). It is interesting to note that the $$1/M$$ expansion for heavy hadrons has a correspondence to the $$1/m_{\rm Q}$$ expansion for heavy quarks, when we retain the expansions up to NLO [159]. This is because the relation $$M=m_{\rm Q}+\Lambda $$ ($$\Lambda $$ being the energy from the light d.o.f., of the order of $$\Lambda _{\rm QCD}$$) is inverted to $$ 1/M = 1/m_{\rm Q} + \mathcal {O}\big (1/m_{\rm Q}^{2}\big )$$. From the HQET Lagrangian (15), we remember that there are two terms at $$\mathcal {O}\big (1/m_{\rm Q}\big )$$: the heavy quark spin conserving and non-conserving terms.15 It was shown in the HQET that, at $$\mathcal { O}\big (1/m_{\rm Q}\big )$$, the former is related to color-electric gluon fields, and the latter with color-magnetic gluons [139–141] (cf. Sect. 3.2.3). Thus, the terms at $$\mathcal { O}(1/M)$$ in heavy hadrons are also related to the color-electric and -magnetic gluons. This property is useful in studying the role of gluon dynamics in the vacuum and in a finite temperature and density medium [142].

Hadronic molecule model

The quarkonium is a simple system composed of a heavy quark and a heavy antiquark. Let us consider how the inter-quark interaction changes at different distance scales (Fig. 23). We have seen that gluon exchange is a dominant force at short distances, as shown in Eq. (28). When the distance between heavy quarks become larger, light (constituent) quark–antiquark pairs can be created virtually from the vacuum, and the light quark components can appear as effective degrees of freedom around the thresholds. One can treat light components by hadron models, where the open heavy hadrons are regarded as the effective degrees of freedom. They are, for instance, $$D$$ and $$\bar {D}$$ mesons, and their interactions are provided by meson exchanges. Here we discuss in detail the one-pion-exchange force as the most dominant interaction at long distances. It is straightforward to include further massive mesons such as scalar ($$\sigma $$) and vector ($$\rho $$, $$\omega $$) mesons [160, 161].

Fig. 23.

A schematic picture for the interaction between heavy quarks. (i) At short distances, gluon exchange is the dominant force. (ii) At middle distances, light quark–antiquark pairs appear as effective degrees of freedom around the threshold energy. (iii) At long distances, the constituent becomes heavy hadrons and meson exchange becomes the dominant force.

Fig. 23.

A schematic picture for the interaction between heavy quarks. (i) At short distances, gluon exchange is the dominant force. (ii) At middle distances, light quark–antiquark pairs appear as effective degrees of freedom around the threshold energy. (iii) At long distances, the constituent becomes heavy hadrons and meson exchange becomes the dominant force.

Let us consider the $$P^{(\ast )}\bar {P}^{(\ast )}$$ systems, where $$P$$ and $$P^{\ast }$$ ($$\bar {P}$$ and $$\bar {P}^{\ast }$$) are $$(Q\bar {q})_{\rm spin \, 0}$$ and $$(Q\bar {q})_{\rm spin \,1}$$ mesons ($$(\bar {Q}q)_{\rm spin \, 0}$$ and $$(\bar {Q}q)_{\rm spin \, 1}$$) with $$S$$-wave. The interaction Lagrangian for a $$P^{(\ast )}$$ meson with axial-vector current induced by pions was constructed in Sect. 3.3.2. It naturally gives the interaction vertices for pions and $$P^{(\ast )}$$ mesons. From Eq. (39), we obtain the $$\pi PP^*$$ and $$\pi P^* P^*$$ vertices  

(44)
\begin{align} \mathcal{L}_{\pi PP^*} &= 2 \frac{g}{f_\pi}\left(P^\dagger_a P^\ast_{b\,\mu}+P^{\ast\,\dagger}_{a\,\mu}P_b\right)\partial^\mu\hat{\pi}_{ab}, \end{align}
 
(45)
\begin{align} \mathcal{L}_{\pi P^*P^*} &= 2 i \frac{g}{f_\pi}\epsilon^{\alpha \beta \mu \nu} v_{\alpha} P^{\ast\,\dagger}_{a\,\beta}P^\ast_{b\,\mu}\partial_{\nu} \hat{\pi}_{ab}. \end{align}
The $$\pi \bar {P} \bar {P}^{\ast }$$ and $$\pi \bar {P}^{\ast } \bar {P}^{\ast }$$ vertices are obtained by changing the sign of the $$\pi P P^{\ast }$$ and $$\pi P^{\ast } P^{\ast }$$ vertices in Eqs. (44) and (45). It is important to note that the strength of the interaction vertex is the same for $$\pi P P^{\ast }$$ and $$\pi P^{\ast } P^{\ast }$$. This is a consequence of the heavy quark symmetry.

The one-pion-exchange potential (OPEP) for $$P\bar {P}^{\ast } \rightarrow P^{\ast }\bar {P}$$ and $$P^{\ast }\bar {P}^{\ast } \rightarrow P^{\ast }\bar {P}^{\ast }$$ is given from the vertices (44) and (45) as  

(46)
\begin{align} V^{\pi}_{P_{1}\bar{P}_{2}^{\ast} \rightarrow P_{1}^{\ast}\bar{P}_{2}} & = -\left( \sqrt{2} \frac{g}{f_{\pi}} \right)^{2} \frac{1}{3} \left[ \vec{\varepsilon}_{1}^{\,\ast} \cdot \vec{\varepsilon}_{2} \, C_{\pi}(r) + S_{\varepsilon_{1}^{\ast},\varepsilon_{2}} \, T_{\pi}(r) \right] \vec{\tau}_{1} \cdot \vec{\tau}_{2}, \end{align}
 
(47)
\begin{align} V^{\pi}_{P_{1}^{\ast}\bar{P}_{2}^{\ast} \rightarrow P_{1}^{\ast}\bar{P}_{2}^{\ast}} & = -\left( \sqrt{2} \frac{g}{f_{\pi}} \right)^{2} \frac{1}{3} \left[ \vec{T}_{1} \cdot \vec{T}_{2} \, C_{\pi}(r) + S_{T_{1},T_{2}} \, T_{\pi}(r) \right] \vec{\tau}_{1} \cdot \vec{\tau}_{2}, \end{align}
and the OPEP for $$P\bar {P} \rightarrow P^{\ast }\bar {P}^{\ast }$$ and $$P\bar {P}^{\ast } \rightarrow P^{\ast }\bar {P}^{\ast }$$ is  
(48)
\begin{align} V^{\pi}_{P_{1}\bar{P}_{2} \rightarrow P_{1}^{\ast}\bar{P}_{2}^{\ast}} &= -\left( \sqrt{2} \frac{g}{f_{\pi}} \right)^{2} \frac{1}{3} \left[ \vec{\varepsilon}_{1}^{\,\ast} \cdot \vec{\varepsilon}_{2}^{\, \ast} \, C_{\pi}(r) + S_{\varepsilon_{1}^{\ast},\varepsilon_{2}^{\ast}} \, T_{\pi}(r) \right] \vec{\tau}_{1} \cdot \vec{\tau}_{2}, \end{align}
 
(49)
\begin{align} V^{\pi}_{P_{1}\bar{P}_{2}^{\ast} \rightarrow P_{1}^{\ast}\bar{P}_{2}^{\ast}} &= \left( \sqrt{2} \frac{g}{f_{\pi}} \right)^{2} \frac{1}{3} \left[ \vec{\varepsilon}_{1}^{\,\ast} \cdot \vec{T}_{2} \, C_{\pi}(r) + S_{\varepsilon_{1}^{\ast},T_{2}} \, T_{\pi}(r) \right] \vec{\tau}_{1} \cdot \vec{\tau}_{2}. \end{align}
Here, three polarizations are possible for $$P^{*}$$ as defined by $$\vec {\varepsilon }^{(\pm )} = \left (\mp 1/\sqrt {2}, \pm i/\sqrt {2}, 0 \right )$$ and $$\vec {\varepsilon }^{(0)} = \left (0, 0, 1\right )$$, and the spin-one operator $$\vec {T}$$ is defined by $$T_{\lambda ' \lambda }^{i}=i \varepsilon ^{ijk} \varepsilon _{j}^{(\lambda ')\dagger } \varepsilon _{k}^{(\lambda )}$$. As a convention, we assign $$\vec {\varepsilon }^{\,(\lambda )}$$ for an incoming vector particle and $$\vec {\varepsilon }^{\,(\lambda )\ast }$$ for an outgoing vector particle. The tensor operators are defined by  
(50)
\begin{align} S_{\varepsilon_{1}^{\ast},\varepsilon_{2}} &= 3 \left( \vec{\varepsilon}^{\,(\lambda_{1})\ast} \cdot \hat{r} \right) \left( \vec{\varepsilon}^{\,(\lambda_{2})} \cdot \hat{r} \right) - \vec{\varepsilon}^{\,(\lambda_{1})\ast} \cdot \vec{\varepsilon}^{\,(\lambda_{2})}, \end{align}
 
(51)
\begin{align} S_{T_{1},T_{2}} &= 3 \left( \vec{T}_{1} \cdot \hat{r} \right) \left( \vec{T}_{2} \cdot \hat{r} \right) - \vec{T}_{1} \cdot \vec{T}_{2}, \end{align}
 
(52)
\begin{align} S_{\varepsilon_{1}^{\ast},\varepsilon_{2}^{\ast}} &= 3 \left( \vec{\varepsilon}^{\,(\lambda_{1})\ast} \cdot \hat{r} \right) \left( \vec{\varepsilon}^{\,(\lambda_{2})\ast} \cdot \hat{r} \right) - \vec{\varepsilon}^{\,(\lambda_{1})\ast} \cdot \vec{\varepsilon}^{\,(\lambda_{2})\ast}, \end{align}
 
(53)
\begin{align} S_{\varepsilon_{1}^{\ast},T_{2}} &= 3 \left( \vec{\varepsilon}^{\,(\lambda_{1})\ast} \cdot \hat{r} \right) \left( \vec{T}_{2} \cdot \hat{r} \right) - \vec{\varepsilon}^{\,(\lambda_{1})\ast} \cdot \vec{T}_{2}. \end{align}

The functions $$C_{\pi }(r)$$ and $$T_{\pi }(r)$$ in the central and tensor potentials, respectively, have asymptotic behaviors at long distance:  

(54)
\begin{align} C_{\pi}(r) &\simeq \frac{1}{r} e^{-m_{\pi}r}, \end{align}
 
(55)
\begin{align} T_{\pi}(r) &\simeq \left( 1 + \frac{3}{m_{\pi}r} + \frac{3}{(m_{\pi}r)^{2}} \right) \frac{1}{r} e^{-m_{\pi}r}. \end{align}
These are modified by the form factor at short distance [160–162]. $$\vec {\tau }_{1}$$ and $$\vec {\tau }_{2}$$ are isospin operators for $$P^{(\ast )}_{1}$$ and $$\bar {P}^{(\ast )}_{2}$$.

To clarify the role of the OPEP, as an example, we consider the $$J^{PC}=0^{+-}$$ state shown in Fig. 24. In the figure, we show $$P\bar {P}(^{1}S_{0})$$ for the initial state, $$P^{\ast }\bar {P}^{\ast }(^{5}D_{0})$$ for the intermediate state, and $$P\bar {P}(^{1}S_{0})$$ for the final state.16 We note that the spins of the heavy (anti)quarks $$Q$$ and $$\bar {Q}$$ are not changed in the heavy quark limit, while the spins of light quarks can be changed. We therefore concentrate on the alignment of spins of light (anti)quarks $$\bar {q}$$ and $$q$$ (called the light spin). In the process in Fig. 24, the light spins and orbital angular momentum are changed by the tensor force from the OPEP in such a way that $$P\bar {P}\big (^{1}S_{0}\big ) \leftrightarrow P^{\ast }\bar {P}^{\ast }\big (^{5}D_{0}\big )$$. The tensor force in the OPEP provides a strong attraction. We note that, in order to switch on the tensor force, $$P$$ and $$P^{\ast }$$ (or $$\bar {P}$$ and $$\bar {P}^{\ast }$$) must be mixed in the process. In the present case, the mass difference between $$P$$ and $$P^{\ast }$$ (or $$\bar {P}$$ and $$\bar {P}^{\ast }$$) is zero in the heavy quark limit, and hence they can easily be mixed. Thus, the $$P$$$$P^{\ast }$$ ($$\bar {P}$$$$\bar {P}^{\ast }$$) mixing accompanying the OPEP is important in this system.

We emphasize that the $$P$$$$P^{\ast }$$ ($$\bar {P}$$$$\bar {P}^{\ast }$$) mixing is a characteristic phenomena seen in heavy quark systems. In light quark systems, the mixing between the pseudoscalar meson and the vector meson is not important, because their mass splittings are larger, as presented in Fig. 20. We also note that the $$P$$$$P^{\ast }$$ ($$\bar {P}$$$$\bar {P}^{\ast }$$) mixing is realized by the OPEP with strong tensor force. The existence of the long-range force in the heavy systems may be contrasted with the interaction among the light Nambu–Goldstone bosons being supplied by a short-range force, such as the Tomozawa–Weinberg interaction [163, 164]. Thus, the simultaneous appearance of heavy quark spin symmetry and the OPEP is a unique feature in heavy quark systems.

Fig. 24.

A schematic picture of the interaction by one-pion exchange potential between $$P^{(\ast )}$$ and $$\bar {P}^{(\ast )}$$ mesons in $$J^{PC}=0^{+-}$$ (cf. Table 4 for $$B^{(\ast )}\bar {B}^{(\ast )}$$ mesons). The up and down spins for the light (red) and heavy components (black) are denoted in the circles.

Fig. 24.

A schematic picture of the interaction by one-pion exchange potential between $$P^{(\ast )}$$ and $$\bar {P}^{(\ast )}$$ mesons in $$J^{PC}=0^{+-}$$ (cf. Table 4 for $$B^{(\ast )}\bar {B}^{(\ast )}$$ mesons). The up and down spins for the light (red) and heavy components (black) are denoted in the circles.

In the literature, there are pioneering works based on hadron molecules where the importance of pion exchange was pointed out for meson–meson systems [96, 165]. It was shown that the $$S$$-wave and $$D$$-wave mixing in the tensor force plays an important role in inducing the strong attraction to make the systems bound, as in the case of the deuteron. Such meson–meson systems were called “deusons.” It is important to note that the existence of $$D\bar {D}^{\ast }$$ molecules was predicted in Ref. [96], with a mass very close to X(3872). In the present work, the importance of the OPEP is discussed for $$Z_b(10610)$$ and $$Z_b(10650)$$ in Sect. 3.4.

For more realistic studies, we should also consider short-range forces by heavier meson exchanges like scalar and vector mesons, or by short-distance mechanisms such as quark exchanges. For example, the $$\omega $$ meson exchange contributes as an attraction for $$P^{(\ast )}\bar {P}^{(\ast )}$$. It has been argued that such contributions could be minor in comparison to the strong tensor force by the OPEP [160, 161]. On the other hand, there are opposite discussions which assert that the contribution from the OPEP is rather small [166]. To analyze the interplays by long- and short-range forces will be left for future works.

QCD sum rules

In the following two sections, we would like to briefly discuss two theoretical methods, QCD sum rule and lattice simulations. Both of them are based on QCD to extract hadron properties directly. So far, their applicability, especially to exotic hadrons, is rather limited. However, they provide useful information on their properties.

The idea of the QCD sum rule is to extract physical quantities by comparing the correlation function computed in two different methods [167–169]. On the QCD side, one calculates a correlation function of currents,  

(56)
\begin{equation} \Pi (p) = \int d^4x\ e^{ipx} \left\langle 0\right| T\big(A(x)\bar B(0)\big) \left|0\right\rangle, \end{equation}
in the asymptotic (deep Euclidean) region, where the perturbation method is available. The operators $$A$$ and $$B$$ are the currents which create the physical states that we are interested in. For instance, the $$\rho $$ meson operator can be chosen as $$A, B = \bar q \gamma _\mu \vec \tau q$$, where $$\vec \tau $$ is an isospin matrix. The computation can be done by using the Wilson's operator product expansion (OPE) at short distances [170],  
(57)
\begin{equation} \lim_{x \to y} A(x) \bar B(y) \to \sum_i C_i (x-y) \mathcal{O}_i ((x+y)/2),\end{equation}
where $$\mathcal {O}_i$$ are local operators and $$C_i (x-y) $$ the Wilson's coefficients, which can be calculated perturbatively. The indices summed over $$i$$ are ordered by the dimensions of the operators $$\mathcal {O}_i$$. Examples of such operators for hadrons written in terms of the quark and gluon fields are given in Ref. [169]. In this decomposition, the singularities at short distances, $$x \to y$$, are isolated by the Wilson coefficients $$C_i (x-y)$$, and the operators $$\mathcal {O}_i$$ are regular. After taking the vacuum matrix elements, due to translational invariance, they reduce to constant values; this characterizes the non-perturbative nature of the QCD vacuum at long distances.

Now, it is known that the asymptotic behaviors of the $$C_i$$s are determined up to a logarithmic factor by the canonical dimensions of the operators $$d_A$$, $$d_B$$, and $$d_i$$,  

(58)
\begin{equation} C_i(x-y) \xrightarrow[|x-y| \ll 1/\Lambda]{} |x-y|^{d_i - d_A - d_B} \left(1 + \mathcal{O}(x\Lambda)\right)\!. \end{equation}
For instance, for $$A = \bar \psi \gamma _\mu \psi $$, $$d_A=3$$. Therefore, the terms of higher-dimensional operators (with larger $$d_i$$) are expected to be suppressed in the deep Euclidean region, $$x \to y$$ or $$p^2 \to \infty $$, by powers of $$\Lambda _{\rm QCD}^2/p^2$$. It is known that the OPE may break due to non-perturbative effects at short distances, such as instantons [167]. This occurs at a dimension as high as $$d=12$$. In practice, the computation up to such high dimensions is not very easy.

On the phenomenological side, using the analyticity, the correlation function is expressed by a spectral function which is related to observables of hadrons. In momentum space, it takes the form  

(59)
\begin{equation} \Pi (p) = \int^\infty_{0} ds\ \frac{\rho(s)}{s-p^2 - i \epsilon}.\end{equation}
In general, the integral has contributions from bound states, resonant states, and continuum. A parametrization is usually made with a singly isolated pole for a bound or resonant state, and a continuum term. The latter is defined to be the region $$s > s_0$$ (threshold parameter), where it is assumed that the spectral density $$\rho (s)$$ can be identified with the imaginary part of the correlation function, thus calculable by OPE. By comparing the two expressions (57) and (59), one expects to extract the information of the resonance state by optimizing the parameters in $$\rho (s)$$. The phenomenological expression involves integration (sum) over the physical region and so is named the sum rule.

Comparing the two correlation functions, the parameters in $$\rho (s)$$ are optimized by requiring several conditions for a reliable sum rule. In the widely used Borel analysis, the Borel mass $$M$$ and the threshold $$s_0$$ are such parameters. The Borel mass $$M$$ suppresses the continuum contributions to extract better low-energy information by enhancing the contribution from the resonance as compared to that from the continuum. By varying the parameters $$M$$ and $$s_0$$, the extracted resonance mass should depend only mildly on them. Furthermore, in the spectral sum on the phenomenological side, the resonance (so-called pole) contributions must be sufficiently large as compared to the continuum contributions.

Given the above general remarks, let us turn to the discussion of exotic hadrons with multiquarks. One of the unique features is that currents for multiquark states are of higher dimensions, because they couple to multiple numbers of quarks and antiquarks. This requires an OPE expansion of (57) with more terms, with various vacuum condensates of higher-dimensional operators. In principle, they are independent and we need more inputs for them. One way to reduce the number of input parameters is the vacuum saturation, where the vacuum expectation values of higher-dimensional operators are approximated by the products of those of lower-dimensional operators.

Another point to be considered is the fact that there are several independent currents available for a given set of quantum numbers. For instance, for scalar and isoscalar quantum numbers corresponding to the $$\sigma (f_0)$$ meson, there are five independent currents [171],  

(60)
\begin{align} S^\sigma &= \left(u_a^T C \gamma_5 d_b\right)\left(\bar{u}_a \gamma_5 C \bar{d}_b^T - \bar{u}_b \gamma_5 C \bar{d}_a^T\right)\!,\notag\\ V^\sigma &= \left(u_a^T C \gamma_{\mu} \gamma_5 d_b\right)\left(\bar{u}_a \gamma^{\mu}\gamma_5 C \bar{d}_b^T - \bar{u}_b \gamma^{\mu}\gamma_5 C \bar{d}_a^T\right)\!,\notag\\ T^\sigma &= \left(u_a^T C \sigma_{\mu\nu} d_b\right)\left(\bar{u}_a\sigma^{\mu\nu} C \bar{d}_b^T + \bar{u}_b \sigma^{\mu\nu} C\bar{d}_a^T\right)\!,\notag\\ A^\sigma &= \left(u_a^T C \gamma_{\mu} d_b\right)\left(\bar{u}_a \gamma^{\mu} C \bar{d}_b^T + \bar{u}_b \gamma^{\mu} C \bar{d}_a^T\right)\!,\notag\\ P^\sigma &= \left(u_a^T C d_b\right)\left(\bar{u}_a C \bar{d}_b^T - \bar{u}_b C \bar{d}_a^T\right)\!, \end{align}
where $$C$$ is the charge conjugation matrix in Dirac space, $$C = i \gamma ^0 \gamma ^2$$. Any of the above currents, and hence any linear combinations of them, can couple to scalar mesons. If the QCD sum rule worked in a reasonable manner, the results for the physical states should not depend on the choice of the currents. In practice, this is not the case, because some of them couple more strongly to the physical states than others. Another remark is that the above currents formed by the products of diquarks (for instance $$u_a^T C \gamma _5 d_b$$ in the first equation of (60)) can be written equivalently in terms of $$\bar qq$$ binaries. The two sets of five currents are related to each other by linear transformations [171].

Up to now, there have been many theoretical works for the study of exotic hadrons using the QCD sum rule [172]. Recently, in Ref. [173], $$X(3872)$$ was studied in the sum rule by using several different types of four quark currents. They considered four types of currents made of (1) color $$\bar {\mathbf 3}_{\mathrm c}$$ diquarks and antidiquark, (2) a color $${\mathbf 6}_{\mathrm c}$$ diquarks and antidiquark, (3) a $$D\bar {D}^*$$ molecular type, and (4) a $$(\bar cc) (\bar qq)$$ type. Although their analysis was made up to six dimensions of OPE, they employed a double ratio sum rule to reduce ambiguities coming from higher-order $$\alpha _{{\rm s}}$$ corrections. They found a reasonable Borel window to reproduce the experimental data for the mass. They also studied the coupling $$X(3872) \to J/\psi \omega $$ using the above four currents. The results seem to be very sensitive to the choice of currents, and therefore a definitive conclusion for the nature of $$X(3872)$$ is not easily drawn. In Ref. [174], analysis was made for $$X(3872)$$ by using $$D\bar {D}^*$$ molecular type currents and hybrid currents of $$\bar c G c$$, where $$G$$ is the gluon field, with their mixing. They investigated the mixing of the two terms, and found a reasonable Borel window with a finite amount of mixing.

QCD sum rules have been also applied to other exotic states. $$Y(4260)$$ was analyzed by employing a molecular type current of $$D_0 \bar {D}^*$$ [175] and $$J/\psi f_0$$ [176]. Their results, however, are inconsistent with each other. A consistent result was then obtained when mixed currents of a charmonium and diquark–antidiquark current were employed [177]. For $$Z(4430)$$, prior to the confirmation of $$J^P = 1^+$$ by LHCb, QCD sum rule analyses were performed by assuming $$J^P = 0^-$$ and by employing $$D_1 \bar {D}^*$$ molecular and diquark–antidiquark currents [178]. An attempt was also made by using a diquark–antidiquark current of $$J^P=1^-$$, but the resulting mass was overestimated by about 400 MeV. Recently, analysis was made using a diquark–antidiquark current of $$J^P=1^+$$, where two states were simultaneously studied for both $$Z(4430)$$ and $$Z(3900)$$ [179]. Thus $$Z(4430)$$ was regarded as a radial excitation of $$Z(3900)$$.

At present, it does not seem easy to make conclusive statements for exotic multiquark states in the QCD sum rule. Nevertheless, the QCD sum rule is useful to test at least the consistency of hadronic states with QCD, and further, in some cases, some of their dynamical contents. Keeping these advantages, there have been several recent attempts to improve the sum rule. One was to employ the maximum entropy method for the physical spectral function to be optimized with the OPE correlation function [180]. Another was to extend kinematic variables to complex ones [181]. Both aimed to improve the extraction of physical information from the sum rule equation. Application to exotic states would be an interesting future problem.

Lattice QCD

The original motivation of the lattice QCD is to develop a scheme for the study of a non-Abelian gauge theory for quarks and gluons, where the coupling constant grows indefinitely at a certain energy scale, as in Eq. (3). It is a field theory which is defined on a discretized Euclidean space-time. Due to a finite lattice distance $$a$$, the ultraviolet momentum is cut at $$\pi /a$$, and there is no divergence problem. By making the space-time volume finite, the number of degrees of freedom becomes finite, and the theory renders, in principle, an unambiguous framework without relying on perturbation expansion. In this regard, it is often said that lattice QCD is the first principle method. The path-integration for quantized fields on a lattice is performed as a numerical simulation and has close analogy with the method in statistical physics.

In practice, the lattice simulations require a huge amount of computational resources. Therefore, at each level of computer power, various methods have been invented to obtain the best results within a limited amount of computer resources. For a recent review, we refer, for example, to the textbook of Ref. [182].

The important problems that lattice QCD can attack in hadron physics are:

  • Spectroscopy: Reproduce observed hadron masses such as pions and protons, and describe various form factors probed by electromagnetic, weak, and strong interactions. In the context of the present article, it is important to predict new hadrons, or establish whether, for instance, multiquark hadrons exist or not.

  • Interaction: Explain hadron interactions such as the nuclear force, and describe other various forces which are not easily investigated by actual experiments.

  • Non-perturbative dynamics of vacuum: Confinement and chiral symmetry breaking are the most important issues to be explained.

  • QCD matter: The phase structure of hadronic matter and the nature of phase transitions. Associated to this, hadron properties under extreme conditions of finite temperature and density are also important.

In fact, all of these are important problems for hadron physics. Here, we briefly present the recent status of lattice QCD relevant to our discussions. Perhaps the most successful is in the masses of ground state hadrons and their interactions. Hadron masses are studied by the correlation function in Euclidean space-time,  

(61)
\begin{equation} \langle 0 | T J_h(x) J_h(0) |0\rangle \to \exp(-E_0 T). \end{equation}
Here, $$J_h$$ is a hadronic current which couples to a hadron $$h$$, and the separation $$x$$ can be taken as $$x = (T, 0, 0, 0)$$. By inserting the complete set of the Hamiltonian, the correlation function is expressed by a sum over all eigenstates with the weight factor $$\exp (-E_0 T)$$. For sufficiently large $$T$$, the correlation function is then dominated by the state of the lowest energy $$E_0$$, which is identified with the mass of the ground state $$h$$.

Fig. 25.

Masses of low-lying hadrons. Left: Light flavor ($$uds$$) hadrons from [183, 184]. Right: Heavy flavor ($$cb$$) mesons, data taken from [22].

Fig. 25.

Masses of low-lying hadrons. Left: Light flavor ($$uds$$) hadrons from [183, 184]. Right: Heavy flavor ($$cb$$) mesons, data taken from [22].

The current state-of-the-art achievements are shown in Fig. 25. The left panel shows the masses of light flavor ($$u, d, s$$) hadrons by quenched (CP-PACS2000) [183] and full QCD (BMW2008) [184] calculations as compared with experimental data (see also Ref. [185]). For CP-PACS2000, the masses of the $$\pi $$, $$K$$, and $$\phi $$ mesons are used as inputs while in the BMW2008 the masses of $$\pi $$ and $$K$$ mesons and $$\Xi $$ baryon are. These hadrons are ground states in that they are the lowest-mass states for given flavor quantum numbers, and in the quark model their wave functions are given by the lowest $$S$$-wave states. Remarkably, both the quenched and full QCD results agree reasonably well with the experimental data, while the agreement of the full QCD results is generally better. Now, lattice simulation has become a precision science at the level of a few percent or less for those states classified as orbitally ground states by the quark model; even electromagnetic effects can be calculated very precisely [186].

The heavy hadron spectrum is shown in the right panel of Fig. 25. In this case, several excited states are shown with good agreement with experimental data, but all of them are below their decay channel thresholds of open flavors as shown by the dashed lines.

Resonances may be assigned as orbitally excited states in the conventional quark model. Manipulating the interpolating fields compatible with the SU(6) symmetry on the lattice, excited states have been studied. The signals were observed in a manner consistent with what is expected in the quark model [187]. Furthermore, wave functions are also extracted for the Roper resonance of the nucleon where they clearly observe monopole-type oscillations [188]. Quantitatively, however, as compared to the accuracy of the ground states, excited states cannot be precisely determined, suffering from finite volume effects as the pion mass approaches the physical value. This implies that excited states are extended spatially more than the ground states, especially at the physical point with the light pion. To see the uncertainties that arise from some technical issues we refer to Refs. [187, 189], where resonance masses are computed at different pion masses and extrapolated to the physical point. Typically there are uncertainties of some hundreds of MeV.

For $$X(3872)$$, a lattice calculation was performed by using an interpolator of $$\bar {c}c$$, $$D \bar {D}^*$$, and diquark–antidiquark type [190, 191]. They claim that a signal was observed near the experimental mass region for the $$I=0$$ channel. Also, they found that the signal was seen when they included the $$\bar {c}c$$ term. These observations seem consistent with the admixture picture of molecular and $$\bar {c}c$$ core which we discuss in Sect. 3.4.1. Other applications of the lattice method to other $$X,Y,Z$$ states will be mentioned in Sect. 3.4.3.1.

Examples

In this subsection, we discuss $$X(3872)$$ and $$Z_b(10610)^+, Z_b(10650)^+$$ as typical examples of hadronic molecules to apply the idea explained in Sect. 3.3.3. For $$X(3872)$$, we also consider an admixture of the $$\bar cc$$ component. After concrete and detailed discussions for these states, we briefly summarize various interpretations for other states discussed in this article: $$Y(4260)$$, $$Z_c(4430)^+$$, and $$Z_c(3900)^+$$. Some remarks on possible dynamical treatment are also addressed.

$$X(3872)$$

As discussed in Sect. 2.3, observation of the $$X(3872)$$ triggered the present activities for exotic hadrons. Thus this state has been studied most extensively both experimentally and theoretically. Although the spin and parity of $$X(3872)$$ have been by now confirmed to be $$J^{PC} = 1^{++}$$, which can be accessible by a $$^3P_1$$ state of $$\bar cc$$, its properties are not explained by a simple $$\bar c c$$ configuration.

As compared to the other states of the same $$LS$$ multiplets $$^3P_J(n=2)$$, the mass of $$X(3872)$$ is significantly lighter than what is predicted as the charmonium. Its mass is located very close to the threshold of the masses of $$D$$ and $$\bar {D}^*$$,  

(62)
\begin{align} M(X(3872)) &= 3871.69\,{\rm MeV},\notag\\ M\left(D^0 +D^{*0}\right) &= 1864.84 + 2006.96 = 3871.80\,{\rm MeV}, \notag\\ M\left(D^{\pm} +D^{* \mp}\right) &= 1869.61 + 2010.29 = 3879.87\,{\rm MeV}. \end{align}
Most notably, the decay rate of [78]  
(63)
\begin{equation} \frac{{{\rm Br}}(X(3872) \to J/\psi + 3 \pi (\omega))} {{{\rm Br}}(X(3872) \to J/\psi + 2 \pi (\rho))} \simeq 1 \end{equation}
cannot be explained by the isosinglet charmonium unless there is a mechanism of large isospin violation. If we look at the mass values in Eq. (62), we immediately realize the very special situation where the scale of the isospin violation (mass difference in neutral and charged modes) is significantly larger than the location of $$X(3872)$$ from the $$D^0 \bar {D}^{*0}$$ threshold.

These observations have naturally led to the idea of a hadronic molecule picture of $$D^0\bar {D}^{*0}$$$$D^{\pm }D^{*\mp }$$. In the threshold region where a new channel opens, the new degrees of freedom in the channel will dominate the dynamics of the system. If there is a suitable interaction they may form a bound or resonant state near the threshold which may have a very different character from what is expected for the ordinary states located far from the threshold. The so-called threshold phenomena have been known for a long time—in nuclear physics, for instance, where alpha-clusters may play the role of effective degrees of freedom rather than the nucleons. A well-known example is the Hoyle state of $$^{12}C$$, which can be explained by a three-alpha structure of small binding rather than a single particle excitation of nucleons [192, 193].

To illustrate the molecular picture for $$X(3872)$$ with a large isospin violation, let us start with a two-channel model of $$D^0 \bar {D}^{*0}$$ and $$D^\pm \bar {D}^{*\mp }$$. The model Hamiltonian is given by  

(64)
\begin{equation} H = \left(\begin{matrix} B_1 & V_{12} \\ V_{21} & B_2 \end{matrix}\right)\!, \end{equation}
where $$B_{1,2}$$ are the unperturbed binding energies of the neutral and charged states, $$D^0D^{*0}$$ and $$D^{\pm }D^{*\mp }$$, respectively, and $$V$$ their interaction matrix element,  
(65)
\begin{equation} V_{12} = V_{21} = \langle B_1 | V |B_2 \rangle . \end{equation}
Good isospin symmetry implies that $$B_1 \simeq B_2$$, and so  
(66)
\begin{equation} |B_1 - B_2| \ll |V_{12}|. \end{equation}
In this case the eigenstates of the Hamiltonian (64) are a superposition of $$|B_1\rangle $$ and $$|B_2\rangle $$ of almost equal weights, corresponding to good isospin states. On the contrary, suppose that the binding energy $$B_1$$ becomes much smaller than the other, $$B_2$$, as $$X(3872)$$ seems to be in such a case: $$ 0 \simeq B_1 \ll B_2 \simeq {\rm a\ few\ MeV} $$. If the binding energy $$B_1$$ is sufficiently small, one expects (as the actual wave functions, Eqs. (62), imply), that the matrix element $$V_{12}$$ is getting smaller proportional to $$B_1^{1/4}$$ (see Eq. (73)), and so  
(67)
\begin{equation} |B_1 - B_2| \gg |V_{12}|. \end{equation}
Thus the eigenstates are (almost) purely $$|B_1\rangle $$ or $$|B_2\rangle $$, corresponding to the isospin breaking states.

So far, the discussions have been made under the assumption that a suitable interaction for $$D \bar {D}^*$$ forms a state near the threshold. Although near-threshold states are sensitive to input conditions, the basic interaction itself is not understood to that accuracy. For instance, the $$D^*D\pi $$ coupling strength obtained by experimentally observed $$D^*$$ decay is about twice as large as is expected from SU(4) symmetry.

The importance of the one pion exchange potential (OPEP) for $$X(3872)$$ was pointed out in Ref. [96] and later in Refs. [162] for other exotic states. The OPEP is a robust consequence of chiral symmetry and its spontaneous breaking for light quarks, and so is the case for the $$D$$ and $$D^*$$ mesons. In fact, many examples have been discussed for low-lying states of heavy hadron systems which are formed by the attractive interaction generated by the OPEP. In particular, the coupled channel effect induced by the tensor structure of the OPEP is particularly important, as is known for the deuteron [96, 194].

On the other hand, in the so-called chiral unitary model only a short-range interaction is employed without the one pion exchange [195]. This is based on the observation that for the channels which allow quark–antiquark annihilation such as $$D \bar {D}^*$$ systems, vector meson exchange ($$\rho $$ and $$\omega $$) provides very strong attraction. In practice, the short-range nature of the interaction necessitates the introduction of a cutoff parameter which controls the strength of quantum corrections. By choosing the parameter suitably, the model can generate bound and/or resonant states in the threshold region. The determination of the interaction between hadrons is therefore crucial to making a realistic description for the new hadrons, and is a good touchstone in particular for $$X(3872)$$. Further studies based on QCD with a non-perturbative method are indeed needed, for example by lattice simulations.

The possible existence of a $$D \bar {D}^*$$ molecular state does not necessarily explain the (seeming) absence of $$\chi _{c1}(2P)$$. Because the $$\chi _{c1}(2P)$$ state can definitely be accessible by a $$\bar c c$$ pair, it would be natural to consider a model with the $$\bar c c$$ pair and a $$D \bar {D}^*$$ molecule.

With those observations, Takizawa and Takeuchi proposed to describe $$X(3872)$$ as an admixture state of charmonium and a $$DD^*$$ molecule [196],  

(68)
\begin{equation} |X\rangle = c_1 |\bar cc\rangle + c_2 \left|D^0 \bar{D}^{*0}\right\rangle + c_3 \left|D^{\pm} \bar{D}^{*\mp}\right\rangle . \end{equation}
The basic assumption of their model is the coupling between the charmonium $$\bar cc$$ and the molecule $$D\bar {D}^*$$, which is parametrized as  
(69)
\begin{equation} \left\langle D\bar{D}^*\right| V \left| \bar c c\right\rangle = \frac{g}{\sqrt{\Lambda}} \frac{\Lambda^2}{q^2 + \Lambda^2}, \end{equation}
where $$g$$ and $$\Lambda $$ are the coupling constant and the form factor, respectively. This interaction is commonly used for both the neutral and charged $$D \bar {D}^*$$ molecules. The Hamiltonian to be diagonalized is then  
(70)
\begin{equation} H = \left(\begin{matrix} m_{\bar cc} & V & V \\ V & m_{D^0\bar{D}^{*0}} + K & 0 \\ V & 0 & m_{D^\pm\bar{D}^{*\mp}} + K \end{matrix}\right)\!, \end{equation}
where $$K$$ is the non-relativistic kinetic energy of the corresponding $$D \bar {D}^*$$ state. In their model, the interactions between $$D \bar {D}^*$$ are not considered in order to minimize the ambiguity, but with the emphasis on their coupling to the $$\bar cc$$ state. Such a coupling effectively plays the role of attractive interaction if the $$\bar cc$$ mass is larger than the $$D \bar {D}^*$$ masses. As shown in Ref. [196], a small coupling $$V$$ indeed generates a state corresponding to $$X(3872)$$ at the right position.

The actual numerical results depend on the choice of parameters. Here we present one of their typical results, where the two parameters in the transition interaction $$V$$ in Ref. [196] are chosen as $$g = 0.03$$ and $$\Lambda = 0.5$$ GeV. The coefficients $$c_i$$ are given as  

(71)
\begin{equation} c_1 = 0.293,\quad c_2 = -0.92,\quad c_3 =- 0.259.\end{equation}
This in turn gives the probabilities  
(72)
\begin{equation} P\big(\bar cc\big) = 8 \%, \quad P\big(\bar I=0\big) = 85 \% ,\quad P\big(\bar I=1\big) = 7 \%, \end{equation}
indicating non-negligible isospin violation as implied in (63). The amount of the $$\bar {c}c$$ component is not very large, but is important in explaining the production rate of $$X(3872)$$ through initial hard processes [197]. The result (72) can be seen well in the spectral function as in Fig. 26, showing a sharp narrow peak at the position of $$X(3872)$$ generated by the pole of the neutral $$D^0 \bar {D}^{*0}$$ origin only slightly below the threshold. The effect of another pole of charged $$D^\pm \bar {D}^{*\mp }$$ origin is not seen, while the evidence of the $$\bar {c} c$$ core origin with mass 3.95 GeV remains as a wide (almost flat) background. This explains the absence of a clear peak of $$\bar {c}c$$-like state at the position expected in the quark model.

Fig. 26.

Spectral function for the $$X(3872)$$ channel [196].

Fig. 26.

Spectral function for the $$X(3872)$$ channel [196].

Finally, to show another unique feature of $$X(3872)$$ let us look at its wave function. Suppose that the bound states $$|B_{1}\rangle $$ and $$|B_{2}\rangle $$ are loosely bound, meaning that their spatial size is sufficiently larger than the interaction range. In such a case, (assuming $$S$$-wave) the normalized wave function can be written for the outside of the interaction range as  

(73)
\begin{equation} \psi_B = \sqrt{\frac{\kappa}{2\pi}} \frac{e^{-\kappa r}}{r},\quad \kappa = \sqrt{2 \mu B}, \end{equation}
where $$\mu $$ is the reduced mass of the two particles. We find the following elementary relation:  
(74)
\begin{equation} \big\langle r^2 \big\rangle^{1/2} = \frac{1}{\sqrt{2} \kappa}, \end{equation}
from which one can estimate the size of the bound state using the reduced mass of the $$D \bar {D}^*$$ system ($${\sim }1$$ GeV) and the binding energy. Thus, from (62), $$B\big (D^\pm \bar {D}^{*\mp }\big ) \sim 8$$ MeV, $$B\big (D^0 \bar {D}^{*0}\big ) \sim 0.1$$ MeV, the sizes (diameters) of the $$D \bar {D}^*$$ bound states are  
(75)
\begin{equation} r_{\rm charged} = 1.1\;{\rm fm},\quad r_{\rm neutral} = 10\;{\rm fm}. \end{equation}
The charged and neutral states are significantly different in size; the neutral one extends further than the charged one due to the smaller binding energy. The diameter of 10 fm corresponds almost to that of a nucleus of mass number around $$A \sim 120$$, which is unexpectedly large as a single hadron. These sizes as well as the composition in the $$X(3872)$$ wave function (71) give an estimation consistent with the large isospin violation.

$$B$$ meson molecules for $$Z_b(10610)^+$$ and $$Z_b(10650)^+$$

As discussed in Sect. 2.8.2, the observation of these two resonances was made in the anomalously large decay rate of the $$\Upsilon (5S)$$ resonance accompanied by two pions. Then it has led to the idea of hadronic molecules again. There are plenty of reasons for it. They are:

  • Their masses are very close to the $$B \bar B^*$$ and $$B^* \bar B^*$$ thresholds, indicating loosely bound or resonant states of these mesons:  

    (76)
    \begin{equation} M(B) = 5280 \; {\rm MeV}, \quad M(B^*) = 5325 \; {\rm MeV}. \end{equation}

  • It does not seem easy to explain the existence of the two states with the same quantum numbers and with a small width in a conventional approach.

  • Decays into both $$\Upsilon \big (^3S_1(n)\big ) \pi $$ ($$n = 1,2,3$$) and $$h_b\big (^1P_1(n)\big ) \pi $$ ($$n = 1,2$$) indicate a large violation of heavy quark spin symmetry, because $$\Upsilon $$ and $$h_b$$ have opposite heavy spin structures of either parallel ($$\Upsilon $$) or antiparallel ($$h_b$$).

These problems can be resolved by assuming the molecular structure of $$Z_b(10610)$$ as $$B \bar B^*$$ and $$Z_b(10650)$$ as $$B^* \bar B^*$$ [68]. In particular, the last point of the violation of heavy quark spin symmetry can be explained by the following decomposition:  

(77)
\begin{align} \left|Z_b(10610)\right\rangle &\simeq \left|B \bar B^*\right\rangle = \frac{1}{\sqrt{2}} \left| 0_H^- \otimes 1_l^-\right\rangle - \frac{1}{\sqrt{2}} \left| 1_H^- \otimes 0_l^-\right\rangle, \nonumber \\ \left|Z_b(10650)\right\rangle &\simeq \left|B^* \bar B^*\right\rangle = \frac{1}{\sqrt{2}} \left| 0_H^- \otimes 1_l^-\right\rangle + \frac{1}{\sqrt{2}} \left| 1_H^- \otimes 0_l^-\right\rangle, \end{align}
where the subscripts $$H$$ and $$l$$ indicate that the spin values (0 or 1) are formed by two heavy and light quarks, respectively. Using these labels, the spin structures of $$B$$ and $$\bar B^*$$ are  
(78)
\begin{equation} \left| B\right\rangle \sim \left| \left[1/2_H, 1/2_l\right]^0 \right\rangle,\quad \left| B^*\right\rangle \sim \left| \left[1/2_H, 1/2_l\right]^1\right\rangle, \end{equation}
where the notation of spin coupling $$[s_1, s_2]^s$$ is introduced. Equations (77) are also regarded as the Fiertz transformations of the four quark spins of $$B\bar {B}^*$$ and $$B^*\bar B^*$$ into the pair of heavy quark spins and light quark spins, namely spin recoupling. An important observation here is that the $$B \bar B^*$$ and $$B^* \bar B^*$$ states have both heavy quark spin 0 and 1 components with equal weights. This explains the decay of $$Z_b$$s into both $$\Upsilon $$ and $$h_b$$ states.

There have been many theoretical studies of $$Z_b$$s assuming the $$B^{(*)}$$ molecule structure [68, 198–214]. Among them, we briefly summarize the work in Ref. [213]. The starting point is to set up a suitable interaction between the heavy $$B^{(*)}$$ and $$\bar B^*$$ mesons. This has been provided by a one boson exchange potential (OBEP) model derived from the Lagrangians of heavy quark and chiral symmetries as described in Sect. 3.3.3. The coupling strengths are determined by several mesonic processes which are derived by these Lagrangians. Thus we have a one-boson-exchange model of $$\pi $$, $$\rho $$, and $$\omega $$ mesons. The form factor can be estimated from a similar OBEP model for the deuteron, where the cutoff parameter is scaled in (inverse) proportion to hadron size (nucleon to $$B$$ meson sizes).

In Ref. [213], low-lying states of angular momentum up to $$L \leq 2$$ were investigated. For a given set of quantum numbers $$J^{PC} \big (^{2S+1}L_J\big )$$, several coupled channels are possible, as pointed out in Sect. 3.3.3. In particular, the coupling of different angular momenta $$L, L\pm 2$$ is important due to the tensor force of the one pion exchange. We summarize coupled channels for various molecular states in Table 4.

Table 4.

Possible coupled channels for low-lying $$B^{(\ast )}\bar {B}^{(\ast )}$$ states with $$J^{PC}$$ ($$J \le 2$$) [213].$$^*$$

$$J^{PC}$$ Components  
$$0^{+-}$$ —  
$$0^{++}$$ $$B\bar {B}(^{1}S_{0})$$, $$B^{\ast }\bar {B}^{\ast }(^{1}S_{0})$$, $$B^{\ast }\bar {B}^{\ast }(^{5}D_{0})$$  
$$0^{--}$$ $$\frac {1}{\sqrt {2}} \left (B\bar {B}^{\ast }+B^{\ast }\bar {B} \right )(^{3}P_{0})$$ $$W_{b0}$$ 
$$0^{-+}$$ $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }-B^{\ast }\bar {B} \right )(^{3}P_{0})$$, $$B^{\ast }\bar {B}^{\ast }(^{3}P_{0})$$  
$$1^{+-}$$ $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }-B^{\ast }\bar {B} \right ) (^{3}S_{1})$$, $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }-B^{\ast }\bar {B} \right ) (^{3}D_{1})$$, $$B^{\ast }\bar {B}^{\ast }(^{3}S_{1})$$, $$B^{\ast }\bar {B}^{\ast }(^{3}D_{1})$$ $$Z_b$$ 
$$1^{++}$$ $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }+B^{\ast }\bar {B} \right ) (^{3}S_{1})$$, $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }+B^{\ast }\bar {B} \right )(^{3}D_{1})$$, $$B^{\ast }\bar {B}^{\ast }(^{5}D_{1})$$  
$$1^{--}$$ $$B\bar {B}(^{1}P_{1})$$, $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }+B^{\ast }\bar {B} \right )(^{3}P_{1})$$, $$B^{\ast }\bar {B}^{\ast }(^{1}P_{1})$$, $$B^{\ast }\bar {B}^{\ast }(^{5}P_{1})$$, $$B^{\ast }\bar {B}^{\ast }(^{5}F_{1})$$ $$W_{b1}$$, $$W_{b1}^\prime $$ 
$$1^{-+}$$ $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }-B^{\ast }\bar {B} \right )(^{3}P_{1})$$, $$B^{\ast }\bar {B}^{\ast }(^{3}P_{1})$$  
$$2^{+-}$$ $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }-B^{\ast }\bar {B} \right )(^{3}D_{2})$$, $$B^{\ast }\bar {B}^{\ast }(^{3}D_{2})$$  
$$2^{++}$$ $$B\bar {B}(^{1}D_{2})$$, $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }+B^{\ast }\bar {B} \right )(^{3}D_{2})$$, $$B^{\ast }\bar {B}^{\ast }(^{1}D_{2})$$ 
 $$B^{\ast }\bar {B}^{\ast }(^{5}S_{2})$$, $$B^{\ast }\bar {B}^{\ast }(^{5}D_{2})$$, $$B^{\ast }\bar {B}^{\ast }(^{5}G_{2})$$  
$$2^{-+}$$ $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }-B^{\ast }\bar {B} \right )(^{3}P_{2})$$, $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }-B^{\ast }\bar {B} \right )(^{3}F_{2})$$, $$B^{\ast }\bar {B}^{\ast }(^{3}P_{2})$$, $$B^{\ast }\bar {B}^{\ast }(^{3}F_{2})$$  
$$2^{--}$$ $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }+B^{\ast }\bar {B} \right )(^{3}P_{2})$$, $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }+B^{\ast }\bar {B} \right )(^{3}F_{2})$$, $$B^{\ast }\bar {B}^{\ast }(^{5}P_{2})$$, $$B^{\ast }\bar {B}^{\ast }(^{5}F_{2})$$ $$W_{b2}$$, $$W_{b2}^\prime $$ 
$$J^{PC}$$ Components  
$$0^{+-}$$ —  
$$0^{++}$$ $$B\bar {B}(^{1}S_{0})$$, $$B^{\ast }\bar {B}^{\ast }(^{1}S_{0})$$, $$B^{\ast }\bar {B}^{\ast }(^{5}D_{0})$$  
$$0^{--}$$ $$\frac {1}{\sqrt {2}} \left (B\bar {B}^{\ast }+B^{\ast }\bar {B} \right )(^{3}P_{0})$$ $$W_{b0}$$ 
$$0^{-+}$$ $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }-B^{\ast }\bar {B} \right )(^{3}P_{0})$$, $$B^{\ast }\bar {B}^{\ast }(^{3}P_{0})$$  
$$1^{+-}$$ $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }-B^{\ast }\bar {B} \right ) (^{3}S_{1})$$, $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }-B^{\ast }\bar {B} \right ) (^{3}D_{1})$$, $$B^{\ast }\bar {B}^{\ast }(^{3}S_{1})$$, $$B^{\ast }\bar {B}^{\ast }(^{3}D_{1})$$ $$Z_b$$ 
$$1^{++}$$ $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }+B^{\ast }\bar {B} \right ) (^{3}S_{1})$$, $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }+B^{\ast }\bar {B} \right )(^{3}D_{1})$$, $$B^{\ast }\bar {B}^{\ast }(^{5}D_{1})$$  
$$1^{--}$$ $$B\bar {B}(^{1}P_{1})$$, $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }+B^{\ast }\bar {B} \right )(^{3}P_{1})$$, $$B^{\ast }\bar {B}^{\ast }(^{1}P_{1})$$, $$B^{\ast }\bar {B}^{\ast }(^{5}P_{1})$$, $$B^{\ast }\bar {B}^{\ast }(^{5}F_{1})$$ $$W_{b1}$$, $$W_{b1}^\prime $$ 
$$1^{-+}$$ $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }-B^{\ast }\bar {B} \right )(^{3}P_{1})$$, $$B^{\ast }\bar {B}^{\ast }(^{3}P_{1})$$  
$$2^{+-}$$ $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }-B^{\ast }\bar {B} \right )(^{3}D_{2})$$, $$B^{\ast }\bar {B}^{\ast }(^{3}D_{2})$$  
$$2^{++}$$ $$B\bar {B}(^{1}D_{2})$$, $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }+B^{\ast }\bar {B} \right )(^{3}D_{2})$$, $$B^{\ast }\bar {B}^{\ast }(^{1}D_{2})$$ 
 $$B^{\ast }\bar {B}^{\ast }(^{5}S_{2})$$, $$B^{\ast }\bar {B}^{\ast }(^{5}D_{2})$$, $$B^{\ast }\bar {B}^{\ast }(^{5}G_{2})$$  
$$2^{-+}$$ $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }-B^{\ast }\bar {B} \right )(^{3}P_{2})$$, $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }-B^{\ast }\bar {B} \right )(^{3}F_{2})$$, $$B^{\ast }\bar {B}^{\ast }(^{3}P_{2})$$, $$B^{\ast }\bar {B}^{\ast }(^{3}F_{2})$$  
$$2^{--}$$ $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }+B^{\ast }\bar {B} \right )(^{3}P_{2})$$, $$\frac {1}{\sqrt {2}} \left ( B\bar {B}^{\ast }+B^{\ast }\bar {B} \right )(^{3}F_{2})$$, $$B^{\ast }\bar {B}^{\ast }(^{5}P_{2})$$, $$B^{\ast }\bar {B}^{\ast }(^{5}F_{2})$$ $$W_{b2}$$, $$W_{b2}^\prime $$ 

$$^*$$The names of some resonances which emerge by solving the coupled channel problem ($$Z$$s and $$W$$s) are shown in the last column; see also Table 5.

The coupled channel problems can be treated non-relativistically by solving the Schrödinger equation, which is a good approximation for the heavy $$B^{(*)}$$ meson molecules with small binding or resonance energies. The latter should be used as a consistency check after obtaining the solutions. The results are summarized in Fig. 27, where theory predictions are shown by horizontal bars together with (the only) two experimental data corresponding to $$Z_b$$s.

It is interesting to see that the OBEP model for the $$B\bar B^*$$ interaction not only reproduces the observed twin $$Z_b$$s (at least qualitatively) but also predicts many weakly bound and resonant states near the threshold regions. A priori, this is not trivial since the energies of those states depend sensitively on the properties of the interaction. It is emphasized that the model parameters here are not fixed to reproduce the twin $$Z_b$$s but from the other conditions. In addition, two more twin states are predicted for $${I^G} J^{PC} = 1^+ 1^{--},\ 1^+ 2^{--}$$. Therefore, the molecular structure of $$Z_b$$ states implies a rich spectrum which can be studied experimentally, a challenge in the future.

Fig. 27.

Low-lying states of $$B \bar {B}^*$$ molecules [213]. Some states are labeled as $$Z$$ and $$W$$, as shown in Table 5.

Fig. 27.

Low-lying states of $$B \bar {B}^*$$ molecules [213]. Some states are labeled as $$Z$$ and $$W$$, as shown in Table 5.

The molecular structure of the $$Z_b$$ resonances also shows its evidence in their decay properties. As discussed in Sect. 2.8.2, large branching ratios of $$Z_b$$ decaying into $$B\bar B^*$$ and $$B^*\bar B^*$$ states have been confirmed, as well as other branching ratios into various channels. These are also useful in testing the molecular structure of $$Z_b$$s.

Fig. 28.

Decay of $$Z_b$$ through a hadron triangle diagram.

Fig. 28.

Decay of $$Z_b$$ through a hadron triangle diagram.

In Ref. [202], a theoretical estimate was made by assuming that the decays occur through triangle mesonic diagrams, as shown in Fig. 28. Physically this corresponds to the process of (1) $$Z_b$$'s dissociation into $$B\bar B^*$$ or $$B^* \bar B^*$$, (2) pion emission by $$B^*$$, and then (3) $$B\bar B^*$$ or $$B\bar B$$ merging into the final state bottomonium $$\Upsilon $$. Various coupling constants needed to calculate these Feynman diagrams are evaluated by other mesonic processes, and then the decay rates of $$\Upsilon (nS) \pi $$ have been studied. By introducing a phenomenological form factor of the initial vertex for the $$Z_b B \bar B^*$$ and $$Z_b B^* \bar B^*$$, it was shown that the experimental decay ratios were explained qualitatively. In particular, the small branching ratio decaying into $$\Upsilon (1S)\pi $$, which seems opposite to what is naively expected from the largest phase space volume for the $$1S$$ state, was explained [202]. We may consider the same mechanism for the decay of $$\Upsilon (5S)$$ to $$\pi Z_{b}'$$s together with the large decay rate associated with two pion emission. This is an important task in the future to clarify the nature of $$\Upsilon (5S)$$.

The decay properties also provide a good opportunity to test the heavy quark spin symmetry, which leads to heavy quark spin selection rules, rigorous relations in the limit of heavy quark mass. The prescription to derive them is to decompose the physical states into heavy quark basis [202, 215]. The model-independent relations are then derived for transitions among the states having the same orbital structure (dynamical property) of light degrees of freedom. For illustration, let us consider radiative decays of $$Z_b$$s into $$\chi _{b J} \gamma $$ ($$J=0,1,2$$), as shown in Fig. 29 by solid lines. In general, for such decays $$M1$$ and $$M2$$ transitions are possible. However, it turns out that only $$M1$$ transitions are available for the present decays. To see this point, let us perform a heavy light decomposition for the final state [204]:  

(79)
\begin{align} \left| \chi_{b0} \; \gamma\big(M1\big) \right\rangle &= \big(1_H^- \otimes 1_l^-\big)_{\chi_{b0}} \otimes \big(0^+_H \otimes 1_l^+\big)_\gamma \nonumber \\ &= \frac{1}{3} \big(1_H^- \otimes 0_l^-\big)_{J=1} - \frac{1}{\sqrt{3}} \big(1_H^- \otimes 1_l^-\big)_{J=1} + \frac{\sqrt{5}}{3} \big(1_H^- \otimes 2_l^-\big)_{J=1}, \end{align}
and similarly,  
(80)
\begin{align} \left| \chi_{b1} \; \gamma\big(M1\big) \right\rangle &= -\frac{1}{\sqrt{3}} \big(1_H^- \otimes 0_l^-\big)_{J=1} + \frac{1}{2} \big(1_H^- \otimes 1_l^-\big)_{J=1} + \frac{\sqrt{15}}{6} \big(1_H^- \otimes 2_l^-\big)_{J=1},\nonumber \\ \left| \chi_{b2} \; \gamma\big(M1\big) \right\rangle &= \frac{\sqrt{5}}{3} \big(1_H^- \otimes 0_l^-\big)_{J=1} - \frac{\sqrt{5}}{\sqrt{6}} \big(1_H^- \otimes 1_l^-\big)_{J=1} + \frac{1}{6} \big(1_H^- \otimes 2_l^-\big)_{J=1}. \end{align}

Fig. 29.

Various transitions where heavy quark selection rules are applied [204, 213]. The black solid bars are normal bottomonia, the red solid bars are $$B^{(\ast )}\bar {B}^{(\ast )}$$ molecules. The solid arrows are for the gamma transitions, dashed lines are for one pion emission in $$P$$-wave, and the dotted line for one pion emission in $$S$$-wave.

Fig. 29.

Various transitions where heavy quark selection rules are applied [204, 213]. The black solid bars are normal bottomonia, the red solid bars are $$B^{(\ast )}\bar {B}^{(\ast )}$$ molecules. The solid arrows are for the gamma transitions, dashed lines are for one pion emission in $$P$$-wave, and the dotted line for one pion emission in $$S$$-wave.

The transition occurs through a matrix element of a total rank zero operator $$\mathcal {O}_{M1}$$, $$\langle \chi _{bL} \; \gamma (M1) | \mathcal {O}_{M1} | Z_b\rangle $$, and thus the coefficients of (77), (79), and (80) imply the ratio  

(81)
\begin{equation} \Gamma(Z_b \to \chi_{b0} \gamma) : \Gamma(Z_b \to \chi_{b1} \gamma) : \Gamma(Z_b \to \chi_{b2} \gamma) = 1:3:5, \end{equation}
when possible difference of phase space volume and form factor dependence are ignored.

Other molecular partners of $$Z_b$$, one $$0^{--}$$ state, two $$1^{--}$$ states, and $$2^{--}$$ states, denoted by $$W_{bJ}$$, decay into $$P$$-wave of $$\Upsilon (1S)\pi $$, rendering another example subject to heavy quark selection rules, because all of these states have the same orbital structure of $$P$$-wave excitations of $$B\bar B^*$$ molecules (indicated by dashed arrows in Fig. 29). The heavy quark decomposition of the above five molecular states and the $$P$$-wave $$\Upsilon (1S) \pi $$ states were performed in Refs. [68, 204]17 to find the ratio (excluding again the phase space and form factor effects)  

(82)
\begin{equation} \Gamma(W_{b0}) : \Gamma\left(W_{b1}^\prime\right) : \Gamma\left(W_{b1}\right) : \Gamma\left(W_{b2}^\prime\right) : \Gamma\left(W_{b2}\right) = 4:1:1:3:1, \end{equation}
where the $$W_{b0}$$, $$W_{b1}^\prime $$, $$W_{b1}$$, $$W_{b2}^\prime $$, and $$W_{b2}$$ states are summarized in Table 5. For $$S$$-wave pion decays into $$\Upsilon (1S)$$, the selection rules allow only the transition from the $$1^{+-}$$$$h_b$$ state while others are forbidden (dotted arrow in Fig. 29).

Table 5.

Predicted bottomonium-like states [204, 213].

$$W_{bJ}$$ $$I^{G}(J^{PC})$$ Main component ($$^{2S+1}L_{J}$$Mass (MeV) 
$$W_{b0}$$ $$1^{+}(0^{--})$$ $$(B\bar {B}^{\ast } + B^{\ast }\bar {B})$$ ($$^{3}P_{0}$$10594 
$$W_{b1}^\prime $$ $$1^{+}(1^{--})$$ $$(B\bar {B}^{\ast } + B^{\ast }\bar {B})$$ ($$^{3}P_{1}$$10617 
$$W_{b1}$$ $$1^{+}(1^{--})$$ $$B\bar {B}$$ ($$^{1}P_{1}$$10566 
$$W_{b2}^\prime $$ $$1^{+}(2^{--})$$ $$B^{\ast }\bar {B}^{\ast }$$ ($$^{5}P_{2}$$10649 
$$W_{b2}$$ $$1^{+}(2^{--})$$ $$(B\bar {B}^{\ast } + B^{\ast }\bar {B})$$ ($$^{3}P_{2}$$10606 
$$W_{bJ}$$ $$I^{G}(J^{PC})$$ Main component ($$^{2S+1}L_{J}$$Mass (MeV) 
$$W_{b0}$$ $$1^{+}(0^{--})$$ $$(B\bar {B}^{\ast } + B^{\ast }\bar {B})$$ ($$^{3}P_{0}$$10594 
$$W_{b1}^\prime $$ $$1^{+}(1^{--})$$ $$(B\bar {B}^{\ast } + B^{\ast }\bar {B})$$ ($$^{3}P_{1}$$10617 
$$W_{b1}$$ $$1^{+}(1^{--})$$ $$B\bar {B}$$ ($$^{1}P_{1}$$10566 
$$W_{b2}^\prime $$ $$1^{+}(2^{--})$$ $$B^{\ast }\bar {B}^{\ast }$$ ($$^{5}P_{2}$$10649 
$$W_{b2}$$ $$1^{+}(2^{--})$$ $$(B\bar {B}^{\ast } + B^{\ast }\bar {B})$$ ($$^{3}P_{2}$$10606 

Finally, let us look at the productions of the above five $$W_b$$ states in the one pion emission decay of the $$\Upsilon (5S)$$ state as denoted by the double arrows in Fig. 29. These are inversion processes of their decays into $$\Upsilon (1S)$$ with one pion emission. Here the one pion is combined with the $$W_b$$s, while in their decays it is combined with $$\Upsilon (1S)$$, and so the resulting heavy quark decompositions differ. The result of the production rates $$P$$ of $$W_b$$s is  

(83)
\begin{equation} P(W_{b0}) : P(W_{b1}^\prime) : P(W_{b1}) : P(W_{b2}^\prime) : P(W_{b2}) = 4: 18: 9: 18: 24. \end{equation}

Quick view of $$Y(4260)$$, $$Z_c(4430)^+$$, $$Z_c(3900)^+$$, and dynamical treatments

In the previous two subsections we have emphasized molecular aspects for $$X(3872)$$, $$Z_b(10610)$$, and $$Z_b(10650)$$. In this subsection, we quickly look at different ideas and studies for $$Y(4240)$$, $$Z_c(4430)^+$$, and $$Z_c(3900)^+$$, though our discussions cannot include a complete list of the previous works. These ideas can also be applied to $$X(3872)$$, $$Z_b(10610)$$, and $$Z_b(10650)$$. As we will see, the fact that there have appeared by now many different ideas implies that the level of our understanding of these exotic particles is not yet satisfactory. The difficulty lies in the fact that they couple to various hadronic channels above the open charm threshold. To perform appropriate theoretical studies, their interactions must be known, which is not currently the case.

$$Y(4260)$$

The mass of this state lies between the expected $$\psi (3S)$$ ($${\sim } 4040$$ MeV) and $$\psi (4S)$$ ($${\sim } 4420$$ MeV) states. Near this energy region, there are no open and hidden charm hadronic channels. The most prominent feature is that it decays into $$J/\psi \pi \pi $$, while the open charm decay rate is suppressed, which is in contrast to the ordinary $$\psi $$ resonances. Thus there have been many proposals such as the orbitally excited tetraquark [216], a molecule of a charmonium and a light flavor meson [217, 218], and baryonium of $$\Lambda _c \bar \Lambda _c$$ [219]. A molecular picture of $$D_{1}(2420)\bar {D}$$ was also tested with the decay dynamics to $$Z_c(3900)$$ and to $$J\psi \pi \pi $$ [220, 221]. A lattice calculation was performed by using an interpolating field of molecular type of open charm, claiming that there is a resonance-like structure [222]. To explain the decay associated with $$\pi \pi $$, the so-called hadro-charmonium was proposed [223, 224]. Yet, conventional charmonium explanations were also discussed [225, 226]. In Ref. [19], Zhu argued that many of these proposals had some difficulties in being consistent with observed data. He concluded that one plausible candidate was a hybrid with a gluon constituent [17, 18]. In view of these discussions, however, while experimental observation is conclusive, theoretical understanding of this state is not so yet.

$$Z_c(4430)^+$$

The observation of this state is the first evidence of a genuinely exotic state as its charge requires a multiquark component, $$\bar c c u \bar d$$. Of course, in principle, it is possible to reach this state without $$\bar c c$$, which is, however, unlikely to the same extent that $$J/ \psi $$ is not regarded as a light quark and antiquark pair, e.g., $$\bar u u$$. Experimentally, as discussed in Sect. 2.5, it is important that the LHCb establishes it as a resonance with phase rotation in their amplitude analysis for the $$\psi (2S)\pi ^+$$ system [21].

From the discovery of this state until 2013 when the spin and parity were determined as $$J^P = 1^+$$, there were many speculative discussions on $$Z_c(4430)^+$$ (with incorrect $$J^P$$ assignment). In Ref. [227], possible threshold behaviors were discussed because the mass of 4430 MeV is close to the mass of $$D^*$$ and $$D_1(2420)$$. Naively, $$S$$-wave states of these mesons provide $$J^P = 0^-$$, $$1^-$$, $$2^-$$, because $$J^P$$ of $$D^*$$ and $$D_1(2420)$$ are $$1^-$$ and $$1^+$$, respectively. There were discussions based on molecular assumptions [228–230] and QCD sum rules with different types of interpolating fields [178, 231, 232], and lattice QCD calculations were also performed [233]. Many of these studies gave positive results, indicating the difficulty in the theoretical study.

After the spin and parity were determined to be $$J^P = 1^+$$, together with the discovery of $$Z_c^+(3900)$$ (see the discussions below), several theoretical studies have appeared. Tetraquark calculation was performed with spin–spin interactions included [234]. It was also postulated that $$Z_c^+(4430)$$ could be a radial excitation of $$Z_c(3900)$$ [179]. Recently, dynamical analyses in scattering processes have been performed based on a coupled channel model [235, 236]. As we shall discuss below, perhaps we need more dynamical studies to determine the nature of the observed peak.

$$Z_c(3900)^+$$

As discussed in Sect. 2.6, the peak structure was first observed in BES III [55], and later confirmed by Belle [43], in the $$J\psi \pi ^\pm $$ spectrum. Because of the similarity of the mass, the state, if it exists as a resonance, may be regarded as an isospin partner of $$X(3872)$$. Once again, as another candidate for a charged, and hence genuine, exotic state, many theoretical discussions appeared, including tetraquark [237–239] and $$D\bar {D}^{\ast }$$ molecule [240–244]. For these models to be more established, dynamical and symmetry considerations should be useful. They include large-$$N_{c}$$ aspects in the presence of heavy quarks [245], the validity of the potential treatment [246], four-body dynamics [247], hadro-charmonium [248], and stability of diquarks [249]. QCD sum rule calculations were also performed using different types of interpolating fields [250–254], indicating the existence of the state. Lattice calculations were also performed for masses and scattering properties [190, 255, 256].

Dynamical treatments

In the previous short sections, we have seen many theoretical works, while it would be fair to say that we have not yet reached a satisfactory level of understanding. Empirically, as discussed in Sect. 1, exotic hadrons appear near and above the threshold where many open hadron channels can enter the game. A dynamical approach is needed where these channels are properly included with reliable interactions between them. These include two-body (and three- and more-body) interactions and their coupling to the genuine exotic (and one-body) states of tetraquark, pentaquark, and so on. Eventually, these setups at the phenomenological level should be replaced by QCD. At the same time, however, we must be able to have reasonable descriptions in terms of hadrons, as they are directly observed states.

Once we have various channels coupled in a quantum mechanical manner, we must consider carefully the effect of threshold, resonances, interference, and so on. These also require proper treatment of the scattering amplitude under suitable conditions such as unitarity. Different natures of observed peaks can then be properly analyzed and identified with a resonance peak or a cusp peak. Experimentally, this can be done by analyzing the phase dependence of the amplitude near the energies of the peak position.

In the context of $$X,Y,Z$$, a triangle diagram which couples a quarkonium-like state (hidden heavy quark) with open heavy quark mesons was studied to show explicitly the threshold effects [257]. The same idea was applied to the peaks of $$Z_c^+(3900)$$ [258]. A more general discussion was also made in Ref. [259]. At this moment, however, without enough knowledge of the interactions between relevant hadron channels, we still need more careful analysis.

Future prospects

So far we have discussed the new experimental observations implying hadronic molecules in several cases. Hadrons themselves are indeed useful bases as effective degrees of freedom in the sense that they are not only the possible constituent particles inside hadrons but also the asymptotic states that are directly observed. In this section, we pick up several more exotic states that have been predicted in theory but not yet found in experiments.

First, we discuss the bound/resonant states of $$D^{(\ast )}D^{(\ast )}$$ molecule states, namely the doubly charmed mesons (Sect. 3.5.1). These states have no annihilation channel, and hence should be regarded as genuinely exotic states.

In addition to the hadronic molecules, however, there are other pictures of exotic hadrons, such as compact multiquark states. Here, colored diquarks are essential ingredients because of the strongly attractive force. The diquarks have been discussed for many years, phenomenologically in spectroscopy, productions, and decays [3, 4, 260–262], and in lattice simulations [262–268]. Although clear evidence for their presence and nature has not been established, it is very important to clarify the properties as colorful constituents of hadronic matter for further discussions. The diquarks can provide the basic degrees of freedom as constituents of exotic hadrons. We apply the diquark picture to the doubly charmed mesons, and discuss their properties (Sect. 3.5.2).

The properties of the colorful substructure (diquark and/or brown muck) are also important in matter states under extreme conditions, such as high temperature, densities, and so on. These are the current issues in relation to the physics of relativistic heavy ion collisions and neutron stars. In fact, the important role of diquarks in dense matter has been known as the realization of color superconductivity at high density (see Ref. [269] for a review).

Doubly charmed molecules

Let us consider a system of two light and two heavy antiquarks $$qq\bar {Q}\bar {Q}$$. When two clusters of $$q\bar {Q}$$ are developed and are separated sufficiently far, say 1 fm, it is natural to regard those two clusters as heavy mesons and as the effective degrees of freedom for the molecular structure: $$\bar {P}^{(\ast )}\bar {P}^{(\ast )}$$ with $$\bar {P} \sim (q\bar {Q})_{\rm spin\, 0}$$ and $$\bar {P}^{\ast } \sim (q\bar {Q})_{\rm spin \,1}$$.

In this case, $$\bar {P}^{(\ast )}$$ and $$\bar {P}^{(\ast )}$$ interact through a long-distance force of meson exchanges. The most important one is the pion exchange force, as discussed in previous sections. For molecular states, the wave function of $$\bar {P}^{(\ast )}\bar {P}^{(\ast )}$$ with $$I(J^{P})=0(1^{+})$$ has four components,  

(84)
\begin{equation} \frac{1}{\sqrt{2}} \left( \bar{P} \bar{P}^{\ast} - \bar{P}^{\ast} \bar{P} \right) \left(^{3}{\rm S}_{1}\right), \quad \frac{1}{\sqrt{2}} \left( \bar{P} \bar{P}^{\ast} - \bar{P}^{\ast} \bar{P} \right) \left(^{3}{\rm D}_{1}\right), \quad \bar{P}^{\ast}\bar{P}^{\ast} \left(^{3}{\rm S}_{1}\right), \quad \bar{P}^{\ast}\bar{P}^{\ast} \left(^{3}{\rm D}_{1}\right), \end{equation}
with $$^{2S+1}L_{J}$$ denoting a state of spin $$S$$, angular momentum $$L$$, and total spin $$J$$. Thus, we have the following $$4 \times 4$$ matrix for the interaction,  
(85)
\begin{equation} V_{0(1^{+})}^{\pi}(r) = \left(\begin{matrix} -C_{\pi}(r) & \sqrt{2} \, T_{\pi}(r) & 2 \, C_{\pi}(r) & \sqrt{2} \, T_{\pi}(r) \\ \sqrt{2} \, T_{\pi}(r) & -C_{\pi} - T_{\pi}(r) & \sqrt{2} \, T_{\pi}(r) & 2 \, C_{\pi}(r) - T_{\pi}(r) \\ 2 \, C_{\pi}(r) & \sqrt{2} \, T_{\pi}(r) & -C_{\pi}(r) & \sqrt{2} \, T_{\pi}(r) \\ \sqrt{2} \, T_{\pi}(r) & 2 \, C_{\pi}(r) - T_{\pi}(r) & \sqrt{2} \, T_{\pi}(r) & -C_{\pi}(r) - T_{\pi}(r) \end{matrix}\right)\!,\end{equation}
with the functions $$C_{\pi }(r)$$ and $$T_{\pi }$$(r) in the central and tensor potentials defined in Eqs. (54) and (55) ($$r$$ is the distance between the two mesons) from the $$\pi $$ exchange potential [94, 124]. The Hamiltonian is given by  
(86)
\begin{equation} H_{0(1^{+})}^{{\rm HM}}(r) = K(r) + \Delta M + V_{0(1^{+})}^{\pi}(r),\end{equation}
where $$K = \left (K_{0}^{PP^{\ast }}, K_{1}^{P^{\ast }P^{\ast }}, K_{0}^{PP^{\ast }}, K_{1}^{P^{\ast }P^{\ast }}\right )$$ is the term for the kinetic energy and $$\Delta M = \left (0,0,\Delta m_{P^{\ast }P},\Delta m_{P^{\ast }P}\right )$$ with the $$\bar {P}$$$$\bar {P}^{\ast }$$ mass splitting $$\Delta m_{P^{\ast }P}= m_{P^{\ast }}-m_{P}$$. The energy is measured from the $$\bar {P}\bar {P}^{\ast }$$ threshold. See Ref. [124] for further details, including the numerical settings of the parameters.

By diagonalaizing Eq. (86), we find the masses and decay widths for $$I\big (I^{P}\big )=0\big (1^{+}\big )$$, as shown in Fig. 30. In our calculation, we obtain a bound state with the mass 3813 MeV (binding energy 60 MeV from the $$\bar {D}\bar {D}^{*}$$ threshold) for $$0\big (1^{+}\big )$$. Therefore, a simple molecular model for double charm can accommodate a stable bound state against strong decay. Similarly, we also find other states in several channels (see Ref. [124] for details).

Fig. 30.

The predicted mass spectrum of doubly charmed molecules and $$T_{cc}$$. The red (blue) bars indicate the mass position and decay widths for $$I=0$$ ($$I=1$$) obtained from the hadronic model [124]. The solid (dashed) black bar indicates the mass position of the $$T_{cc}[\textbf {3}_{c}]$$ ($$T_{cc}[\bar {\textbf {6}}_{c}]$$) with $$I(J^{P})=0(1^{+})$$ obtained from the diquark model [116, 117]. The numbers (in parentheses) in the figure indicate the masses (decay widths) in units of MeV.

Fig. 30.

The predicted mass spectrum of doubly charmed molecules and $$T_{cc}$$. The red (blue) bars indicate the mass position and decay widths for $$I=0$$ ($$I=1$$) obtained from the hadronic model [124]. The solid (dashed) black bar indicates the mass position of the $$T_{cc}[\textbf {3}_{c}]$$ ($$T_{cc}[\bar {\textbf {6}}_{c}]$$) with $$I(J^{P})=0(1^{+})$$ obtained from the diquark model [116, 117]. The numbers (in parentheses) in the figure indicate the masses (decay widths) in units of MeV.

$$T_{cc}$$

When two $$\bar {P}^{(\ast )}$$ mesons approach closer, diquarks $$qq$$ and $$\bar Q \bar Q$$ may be developed more to form an exotic tetraquark $$qq\bar {Q}\bar {Q}$$. We call this state $$T_{QQ}$$ to distinguish it from the $$\bar {P}^{(\ast )}\bar {P}^{(\ast )}$$ hadronic molecules.

$$T_{QQ}$$ could become stable against the strong decay in a diquark picture; the argument for the stability goes as follows. Apart from the confinement force, the dominant interaction energy is supplied by the light diquark $$qq$$. The interaction between a light (anti)quark $$q$$ ($$\bar {q}$$) and a heavy (anti)quark $$Q$$ ($$\bar {Q}$$) is suppressed by $$\mathcal {O}(1/m_{Q})$$ with a heavy quark mass $$m_{Q}$$. The interaction between two heavy (anti)quarks is further suppressed by $$\mathcal {O}\big (1/m_{Q}^{2}\big )$$. Therefore, the mass of $$T_{QQ}$$ could be lower than the mass of the possible strong decay channel of two $$q\bar {Q}$$ mesons, if the $$qq$$ interaction is attractive:  

(87)
\begin{equation} M\big(qq\bar{Q}\bar{Q}\big) < M\big(q\bar{Q}\big) + M\big(q\bar{Q}\big). \end{equation}
In this case, the only possible decay is the weak decay of heavy quarks. We should note that the situation differs for $$q\bar {q}Q\bar {Q}$$ states. If $$q\bar {q}$$ behaves as the Nambu–Goldstone bosons ($$\pi $$, $$K$$, and $$\eta $$ mesons) with small mass, $$M(q\bar {q}) \sim 0$$, we expect the following mass relations:  
(88)
\begin{equation} M\big(q\bar{q}\big) + M\big(\bar{Q}Q\big) < M\big(q\bar{q}Q\bar{Q}\big) < M \big(q\bar{Q}\big) + M\big(\bar{q}Q\big), \end{equation}
and hence observe that the $$q\bar {q}Q\bar {Q}$$ state is not likely to be the ground state of the system.

To estimate the mass of $$T_{QQ}$$, we assume that the $$ij$$ quark pair interact though the color-magnetic interaction (cf. Eq. (29)),  

(89)
\begin{equation} V_{ij}(S) = C_{H} \vec{s}_{i} \cdot \vec{s}_{j} \frac{1}{m_{i}m_{j}}, \end{equation}
with dynamical (constituent) quark mass $$m_{i}$$ and spin operator $$\vec {s}_{i}$$ [116, 117]. $$S$$ is the total spin of quarks $$i$$ and $$j$$. Obviously, such an interaction is suppressed for heavy quarks of mass $$m_{c}\simeq 1500$$ MeV and $$m_{b} \simeq 4700$$ MeV ($${=}m_{Q}$$) as compared to light quarks of mass $$m_{u} \simeq m_{d} \simeq 300$$ MeV and $$m_{s} \simeq 500$$ MeV. For quark–quark interaction, from the mass splittings among light hadrons with different spins such as $$N$$$$\Delta $$, $$\Lambda $$$$\Sigma $$, $$\Lambda _{c}$$$$\Sigma _{c}$$, and $$\Lambda _{b}$$$$\Sigma _{b}$$, we find $$C_{B}/m_{u}^{2}=193$$ MeV $$(H=B)$$ reproduces the observed mass splittings.18 The most attractive energy is in the $$ud$$ diquark with spin-singlet ($$S=0$$) and isospin-singlet ($$I=0$$) (color $$\bar {\textbf {3}}_{c}$$). Thus the lowest mass of $$T_{QQ}$$ is given as  
(90)
\begin{equation} M\left(qq\bar{Q}\bar{Q}\right) = 2m_{Q} + 2m_{q} + V_{qq}(S=0), \end{equation}
which gives a lighter mass than $$M\big (q\bar {Q}\big )+M\big (q\bar {Q}\big ) \simeq 2m_{Q} + 2m_{q}$$. The quantum numbers of the lowest $$T_{QQ}$$ should be $$I\big (J^{P}\big )=0\big (1^{+}\big )$$. The total spin $$J=1$$ is due to the $$S$$-wave $$\bar {Q}\bar {Q}$$ pair of color $$\textbf {3}_{c}$$, satisfying the Pauli principle. More realistically, we need to consider the contributions of order $$1/m_{q}m_{Q}$$ for $$q\bar {Q}$$ and of order $$1/m_{Q}^{2}$$ for $$\bar {Q}\bar {Q}$$. Finally, we obtain the binding energy of $$T_{cc}$$ to be about 80 MeV when measured from the $$DD^{\ast }$$ threshold, and about 124 MeV for $$T_{bb}$$ from $$BB^{\ast }$$ [116, 117].

The binding energy estimated in the diquark model is eventually of the same order as that obtained in the hadronic model (Fig. 30). However, the mechanisms are quite different from each other: the pion exchange interaction for the former, and the color–spin interaction in diquarks for the latter. Each model simplifies the real QCD, and the truth must be in between the two of them. Recently, numerical simulation in lattice QCD has reported the attraction for $$DD^{*}$$ scattering in the $$I\big (J^{P}\big )=0\big (1^{+}\big )$$ channel corresponding to that in the $$T_{cc}$$ channel [127]. The final conclusion is, however, not yet available due to the large pion mass in the lattice QCD simulations.

In addition to the color triplet $$\textbf {3}_{c}$$ configuration for $$\bar {Q}\bar {Q}$$ (this $$T_{cc}$$ is denoted by $$T_{QQ}[\textbf {3}_{c}]$$), we may consider the exotic color “antisextet” configuration for $$\bar {Q}\bar {Q}$$: $$T_{QQ}\big [\bar {\textbf {6}}_{c}\big ]$$ [270]. The $$qq$$$$\big (\bar {Q}\bar {Q}\big )$$ pair of $$\textbf {6}_{c}\big (\bar {\textbf {6}}_{c}\big )$$ receives attraction in the $$S=1$$ (isospin-singlet) channel with the strength reduced to $$1/6$$ of that of the $$\bar {\textbf {3}}_{c}$$$$qq$$ pair (see Eqs. (29), (30), and (31)). Being of the same total quantum numbers, those two $$T_{QQ}$$s may mix, but the mixing should be suppressed due to the small spin-flip between $$\bar {Q}\bar {Q}$$ ($$S=0$$) in $$T_{QQ}\big [\textbf {3}_{c}\big ]$$ and $$\bar {Q}\bar {Q}(S=1)$$ in $$T_{QQ}\big [\bar {\textbf {6}}_{c}\big ]$$. The estimate of the mass of $$T_{QQ}\big [\bar {\textbf {6}}_{c}\big ]$$, however, indicates that $$T_{QQ}\big [\bar {\textbf {6}}_{c}\big ]$$ may not be bound. For example, the mass of $$T_{cc}\big [\bar {\textbf {6}}_{c}\big ]$$ ($$Q=c$$) is 54 MeV above the $$DD^{\ast }$$ threshold, and it can decay to $$D$$ and $$D^{\ast }$$ in $$S$$-wave. Thus, some special mechanisms may be required for the $$T_{cc}\big [\bar {\textbf {6}}_{c}\big ]$$ to be a quasi-stable state as a resonance. Nevertheless, the different color structures are reflected in the differences in the production processes, and hence they may give a hint to understand the color structure inside the hadrons, as mentioned below.

The production of $$T_{QQ}$$, as well as the $$P^{(\ast )}P^{(\ast )}$$ states discussed in the previous section, in experimental facilities is an important issue. One possible method to investigate the double charm production ($$cc\bar {c}\bar {c}$$) is collider experiments. In fact, there are several reports on observations of double charmonia, where $$cc\bar {c}\bar {c}$$ is combined to two $$c\bar {c}$$ pairs in the final states. It will then be naturally expected that the $$cc\bar {c}\bar {c}$$ is also combined to produce double charm hadrons containing $$cc$$ and others with two $$\bar {c}q$$. Using this picture, in theoretical studies, the production rate to produce $$T_{cc}$$ in electron–positron collisions was numerically estimated in a framework based on NRQCD [13]. The interesting result is that the properties of the productions of $$T_{QQ}\big [\textbf {3}_{c}\big ]$$ and $$T_{QQ}\big [\bar {\textbf {6}}_{c}\big ]$$ are different in momentum and angle dependence in the differential cross sections [270]. This may give us an experimental way to study the internal “color-exotic” structure in exotic hadrons. Another possible method to observe $$T_{QQ}$$ is to investigate the production of hadrons in relativistic heavy ion collisions. There, an abundance of quarks is produced, and some of them could be combined to exotic hadrons through hadronization processes. The numerical estimates of yields of exotic hadrons, including $$T_{cc}$$ and other exotic states, was given based on the coalescence model and on the statistical model for LHC and RHIC [271, 272]. Here, the interesting conclusion is that the yields of exotic hadrons are sensitive to the formation processes: compact objects such as multi-quarks will be produced in quark coalescence, and extended objects such as hadronic molecules will be produced in hadron coalescence. Experimental observations of exotic hadrons in relativistic heavy ion collisions may unveil the structures of exotic hadrons in the coming future.

Baryon-rich exotics

So far we have discussed the $$X,Y,Z$$ hadrons with baryon number zero. The heavy quark symmetry plays the essential role, not only for the $$X,Y,Z$$ hadrons, but also in heavy flavor baryon and heavy flavor nuclei with finite baryon number. The important thing is the decomposition of the spin structure in the heavy quark limit, as we have discussed in the cases of the $$X,Y,Z$$ hadrons. In the heavy flavor baryons with light quarks, $$Qqq$$, the light degrees of freedom can be regarded as diquarks. Diquarks are composite states of two quarks with color non-singlet combinations. Although the existence of diquarks has been discussed since the time Gell-Mann invented quarks, there has still been no significant establishment of their existence or their properties. According to heavy quark symmetry, we can separate the dynamics of two light quarks, $$qq$$, as a diquark from the heavy quark $$Q$$. Therefore, the study of heavy flavor baryons will provide us with a unique opportunity to investigate diquark dynamics [273, 274]. Along this line, the heavy flavor nuclei, as an extension from one baryon number state to multiple baryon number states, are also interesting topics [215, 275]. In this case, the heavy quark symmetry will affect the spectrum of the heavy flavor nuclei, and the light degrees of freedom will provide us with new dynamical degrees of freedom in describing the several low modes in nuclei. It is important to note that the essential degrees of freedom are given by the heavy quark spin and the brown muck spin in the heavy quark limit, where the latter includes the nucleon–hole pairs around the Fermi surface in nuclei. Thus, the heavy quark symmetry is also related to the breaking of rotation symmetry in deformed nuclei, where nucleon–hole pairs are created collectively in the ground state. The separation of the effective degrees of freedom by the heavy quark spin is a new phenomenon, which cannot be seen in light flavor nuclei. The applications to finite baryon number systems with heavy flavor should be explored both in experimental and theoretical studies in the future.

As subjects towards heavier systems, double charm hadrons ($$|C|=2$$) are important. They would provide us with information about the heavy quarkquark ($$QQ$$) interaction with color non-singlet ($$\textbf {3}_{c}$$) channels. Furthermore, spectroscopy of the triply heavy baryon $$\Omega _{QQQ}$$ will be important for understanding the color-confinement force in three quarks $$QQQ$$. Those topics will open a new horizon for the study of heavy hadron systems.

Theory summary

The question about how hadrons are composed from quarks has been a longstanding problem, since Gell-Mann and Zweig proposed quarks as the most fundamental particles in the strong interaction. It is remarkable that, in the heavy flavor sector, the recent discoveries of $$X,Y,Z$$ exotic hadrons, especially the charged $$Z$$ states, have provided motivation to study the existence of multi-quark states.

In quantum systems, it is generally important to identify the effective degrees of freedom at each different energy scale, and construct the effective Lagrangian for them appropriately. A successful example is the chiral perturbation theory at low energies, where the effective degrees of freedom are given by the Nambu–Goldstone bosons generated by the dynamical breaking of chiral symmetry in the vacuum [276–278]. In heavy quarkonia, on the other hand, the heavy quark and antiquark are well defined as the effective degrees of freedom due to the suppression of the quark–antiquark annihilation processes, and the interaction is provided by the quark potentials like the color–Coulomb and confinement potentials.

In the $$X,Y,Z$$ hadrons, however, the questions about the effective degree of freedom and the effective interaction working for them are still under debate. One of the keys to approaching these questions is the heavy quark symmetry, where each heavy quark spin and brown muck spin are conserved in the heavy quark limit, $$m_{\rm Q} \rightarrow \infty $$. The heavy quark symmetry holds for any hadron systems, including multi-quark states, hadronic molecule states, and nuclear systems, and hence it will give us a systematic way to understand them. We also need to investigate the $$1/m_{\rm Q}$$ corrections beyond the leading order. The symmetry-breaking effects will be important, particularly when the mass is close to thresholds. In particular, if there are open hadronic channels nearby, they couple with each other and influence the scattering amplitudes, showing non-trivial behaviors associated with the bound states, resonant states, cusps, and so on. For the analysis of the $$X,Y,Z$$ hadrons, these dynamical effects are very important, and will be an issue in future developments.

Summary

Discovery of $$X(3872)$$ followed by the series of “$$X,Y,Z$$” particles has opened a new era of hadron spectroscopy. A lot of effort on both experimental and theoretical sides has been invested to give proper understanding of these states. Revealing their structures directly corresponds to the answer to the fundamental question, “What are the most suitable degrees of freedom to form hadrons?”, when hadrons are composite systems of quarks and gluons that are strongly renormalized by QCD interactions. The so-called constituent quarks, gluons, and even their composites such as diquarks may be such degrees of freedom. Although this was a longstanding question before the current activities started, it is amusing to see that the question is revisited by the findings of the $$X,Y,Z$$ states containing heavy quarks.

Experimentally, not only the identification of peak positions but also dynamical properties such as decays and branching ratios have been accumulated. The latter properties are particularly sensitive to internal structures and so are useful to answer the equation raised above. As a consequence of these activities, the role of the molecular states has been recognized to interpret $$X(3872)$$ as the admixture of $$J^{PC}=1^{++}$$ charmonium and $$D\bar {D}^{*}$$ molecular states, and to regard $$Z_b\big (10610\big )^+$$ and $$Z_b\big (10650\big )^+$$ as $$B\bar {B}^*$$ and $$B^* \bar {B}^*$$ molecular states, respectively. Yet there are several states which are well established by experiments, while little is understood theoretically. They are, for instance, $$Y(4260)$$ and $$\Upsilon (5S)$$. Concerning them, the mechanism of a large two-pion emission rate does not seem to be explained easily, while this property has triggered the discovery of many exotic states. The structure of the charged states $$Z_c\big (4430\big )^+$$ and $$Z_c\big (3900\big )^+$$ is not yet clear either. This is partly the reason that we did not discuss these important states in this article; we simply mention that further understanding of them will bring yet further important steps in hadron physics.

Inspired by the progress, strategies to perform various partner searches have been determined for the quarkonium(-like) states. For that purpose, higher statistics data is strongly desired because of the anticipated dominant decay modes and consequent signal yield expectations. Yet further activities are expected for double charmed tetraquarks, double charmed molecules, heavy flavored baryons, and even heavy flavored nuclear matter. Theoretically, further mutual developments of the first-principle methods and effective approaches will become more important. All of them bring challenges, not only for their own sake, but also to answer the fundamental question. After all, their composite nature leads to their properties depending on the environments with which they interact. This is partly the reason that different approaches are indeed required for the study of hadron physics.

Much more information is hoped for from next-generation intensity frontier projects such as Belle II, upgraded LHCb, and J-PARC experiments. At the same time, effective theory as well as lattice QCD approaches are exhibiting remarkable progress. Using the “$$X,Y,Z$$” states as interesting objects to be discussed, the non-perturbative QCD mechanisms continue to be revealed.

Funding

Open Access funding: SCOAP3.

Acknowledgments

The authors would like to acknowledge the support provided by Grant-in-Aid for Scientific Research on Priority Areas “Elucidation of New Hadrons with a Variety of Flavors” (X00: 21105001, A01: 21105002, E01: 21105006) from the Japan Society for the Promotion of Science (JSPS). This work is also supported in part by Grant-in-Aid for Science Research (C) 26400273 (A. H.), 25247036 (S. Y.), and by Grant-in-Aid for Science Research (B) 15K17641 (S. Y.).

1
To be precise, both $$D\bar {D}^*$$ and $$\bar {D}D^*$$ channels can couple. In what follows, whenever one of these configurations is indicated, it is understood that the other one is also included unless otherwise specified.
2
Hereafter, inclusion of the charge conjugate state is implied unless otherwise stated.
3
In this paper we employ the notation $$\Upsilon (5S)$$; it is denoted as $$\Upsilon (10860)$$ in the particle data [22].
4
See Refs. [98–121] for the compact tetraquark picture, Refs. [94, 96, 122–124] for the extended hadronic molecule picture, and Refs. [125–127] for the lattice QCD result.
5
We should remember that there is no scale parameter in the original QCD Lagrangian (1). The existence of $$\Lambda _{\rm QCD}$$ is a consequence of the quantum fluctuations in the QCD vacuum.
6
Here, $$q$$ stands for the current quark with bare mass, which is different from the constituent quark in the quark model in Sect. 3.3.1.
7
We note that the corresponding hadrons in light flavors are nucleon and $$\Delta $$, and $$\Sigma $$ and $$\Sigma ^{\ast }$$, whose mass splittings are about 290 MeV and 195 MeV, respectively.
8
In Ref. [142], $$\big |\tilde {{\rm H}}_{v_{{\rm r}}} \big \rangle $$ is introduced as the hadron state whose normalization factor is consistent with those in Refs. [140, 141].
9
There are studies of dynamical quarks from QCD: see Ref. [143] in the Schwinger–Dyson approach, and Ref. [144] in lattice QCD simulation.
10
More details of the discussion are found in Refs. [145–147].
11
$$\Lambda (1405)$$ as a $$\bar {K}N$$ bound state was suggested in the early days by Dalitz and Tuan [7].
12
It is interesting to note that Gell-Mann has already suggested that $$qqqq\bar {q}$$ components can exist in addition to the normal $$qqq$$ in baryons, and $$qq\bar {q}\bar {q}$$ components could exist in addition to the normal $$q\bar {q}$$ in mesons [1].
13
It is important to distinguish pNRQCD from HQET (Sect. 3.2.3). In the former, the relative motion of the two heavy quarks is given in the velocity expansion, and in the latter, the hadron dynamics is treated by the $$1/m_{{\rm Q}}$$ expansion in the given coordinate frame.
14
We consider only the contributions to the axial-vector coupling. For the vector coupling, see Ref. [138].
15
They are the second and third terms, respectively, in the last line in Eq. (15).
16
To be precise, more channels can couple to the state of this quantum number. See Table 4 for the $$B^{(\ast )}\bar {B}^{(\ast )}$$ mesons case.
17
See also Refs. [204, 214] for gamma decays.
18
For quark–antiquark interaction, we can similarly obtain $$C_{M}/m_{u}^{2}=635$$ MeV ($$H=M$$) to reproduce the meson mass splitting $$\pi $$$$\rho $$, $$K$$$$K^{\ast }$$, $$D$$$$D^{\ast }$$, and $$B$$$$B^{\ast }$$.

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