Exotic Hadrons with Heavy Flavors -X, Y, Z and Related States-

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 those states with emphasis on 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 yet undiscovered exotic hadrons.

Z b (10610) + and Z b (10650) + , the KEK B-factory is the only facility which has the ϒ(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 ( * )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.

The Belle experiment
The Belle detector is a 4π 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 C P 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 µm in J/ψ → μ + μ − 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 π 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 ϒ(4S) resonance and thus high statistics B meson data. The accumulated integrated luminosity in that condition is 711 fb −1 , corresponding to 772 million BB pairs. The accelerator has also operated at ϒ(1S), ϒ(2S), and ϒ(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 ϒ(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 (cc) and bottomonium (bb), can be regarded as the reference for discussing exotic hadrons. Among charmonia, J/ψ and ψ(2S) (or ψ in Fig. 1) decays to an e + e − or μ + μ − pair. Summing up these two decay modes, the branching fraction of interest for J/ψ and ψ(2S) amount to 12% and 1.6%, respectively. This enables us to reconstruct them with a low background. ψ(2S) can also be reconstructed by the J/ψπ + π − mode. The J/ψγ mode is useful in reconstructing χ c1 and χ c2 states. Similarly, we can have a clear signature through μ + μ − modes for ϒ(1S), ϒ(2S), and ϒ(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 ±+ and 2 ±+ states. (4) Only C = +1 states can be produced in double charmonia production in e + e − annihilation since J/ψ or ψ(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.
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) → J/ψγ 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.
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/ψπ + π − mode by CDF [31] and Belle [32] using three decay angular variables, as well as the 3π invariant mass spectrum in the J/ψπ + π − π 0 mode by BaBar [33], give a constraint on J PC to be either 1 ++ or 2 −+ , but do not reach a definitive determination.   full five-dimensional amplitude analysis was performed by the LHCb collaboration [34] for the X (3872) → J/ψπ + π − mode and J PC = 1 ++ was unambiguously assigned.
In a tetraquark hypothesis for X (3872), mass splitting due to the mixing between ccuū and ccdd was theoretically predicted [35], M(X h ) − M(X l ) = (7 ± 2)/ cos(2θ) MeV, where M(X 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 ± → X K ± and B 0 → X K 0 . The Belle result for this difference in the J/ψπ + π − mode is found to be (−0.71 ± 0.96 (stat) ± 0.19 (syst)) MeV/c 2 [32], which strongly disfavors the tetraquark interpretation. The BaBar measurement for this quantity is reported to be (2.7 ± 1.6 (stat) ± 0.4 (syst)) MeV/c 2 [36]. Belle also finds no signature for the charged partner state in the J/ψπ ± π 0 mode.
On the other hand, a D 0D * 0 molecule is proposed as very plausible interpretation for X (3872) because of the very narrow width ( < 1.2 MeV) and the mass M = 3871.69 ± 0.17 MeV/c 2 [22] being very close to the D 0D * 0 threshold. Inspired by this argument, B → D 0D * 0 K decays are reconstructed to examine the D 0D * 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/ψπ + π − mode. The resultant B X (3872) → D 0 D * 0 is found to be 10 times as large as B X (3872) → J/ψπ + π − .

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Downloaded from https://academic.oup.com/ptep/article-abstract/2016/6/062C01/2240707 by guest on 29 July 2018 PTEP 2016, 062C01 A. Hosaka et al. The difference between the observed mass and the threshold is regarded as the binding energy and it is small, 0.11 ± 0.21 MeV [22]. This value would give an estimation of the X (3872)'s size, i.e. the distance between D 0 andD * 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/ψ, whose diameter is estimated to be 0.4 fm, and thus the D 0D * 0 mode should have a much larger branching fraction with respect to the J/ψπ + π − 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 pp 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). χ 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 0D * 0 molecule and also contains a sizable χ c1 (2P) component. The pure χ 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/ψ. The χ 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 χ 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.

B(X (3872)→ψ(2S)γ )
B(X (3872)→J/ψγ ) < 2.1 at 90% C.L. [29]. Recently, the LHCb experiment also reported the radiative decays of X (3872) → J/ψγ and ψ(2S)γ . Reconstruction of the ψ, which denotes a J/ψ or ψ(2S) meson, is done by the μ + μ − 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 ψγ K ± and ψγ , where the former peaks at the B ± mass and the latter forms a peak at the X (3872) mass for the signal. It confirms the X (3872) radiative decay in J/ψγ mode; at the same time, it finds evidence of X (3872) → ψ(2S)γ with a significance of 4.4 σ [40]. The ratio of X (3872) → ψ(2S)γ to X (3872) → J/ψγ is obtained to be B(X (3872)→ψ(2S)γ ) B(X (3872)→J/ψγ ) = 2.46 ± 0.64 (stat) ± 0.29 (syst). Neither BaBar nor Belle results contradict this LHCb result within the uncertainties; the resultant branching fraction disfavors an interpretation as a pure DD * 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 8 [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/ψπ + π − invariant mass distribution with the full data sample [43]. After these discoveries, the hadron which is created by ISR and contains a cc constituent but cannot be interpreted to be a charmonium has come to be called "Y ." In the analysis for the ψ(2S)π + π − 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/ψπ + π − or ψ(2S)π + π − final states despite the heavier mass than the D D threshold, contrary to the fact that ψ(3770) decays predominantly to a D 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 cc. Rather, the large π + π − decay with its scalar and isoscalar correlations would be possible from flavor blind gluonic excitations, rendering its interpretation as a hybrid state [17][18][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/ψπ + π − production cross section and confirmed it exhibits a peak around √ 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 √ s in the same center-mass-energy region [48]. Thus, the fraction of J/ψπ + π − production in the total hadronic cross section is estimated to be very high and the resultant partial width Y (4260) → J/ψπ + π − 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

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 ψ(2S)π + invariant mass spectrum for B → ψ(2S)π + K three-body decays [49], as shown in Fig. 7.
BaBar analyzed the same B decay process and found the effect corresponding only to 1.9 σ statistical significance, thus an upper limit was presented [50], stating that two B-factories' results are not statistically contradicting each other. In B → ψ(2S)π + K decays, major contributions come from ordinary quasi-two-body B decays, B → ψ(2S)K * where K * → K π + . In Belle's first discovery paper [49], B → ψ(2S)K * (892) and B → ψ(2S)K * (1430) dominant regions were removed according to the K π + invariant mass. However, it is important to see if the potential interference between those B → ψ(2S)K * decays and other featureless processes such as nonresonant three-body B decays to ψ(2S)π + K may exhibit a peak-like projection in ψ(2S)π + . 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 σ . In addition, the most recent Belle study using a four-dimensional full amplitude analysis ofB 0 → ψ(2S)π + K − decays constrains the spin and parity of the Z c (4430) + . Here, the ψ(2S) is reconstructed via its e + e − or μ + μ − 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 π , M 2 ψ(2S)π , θ ψ(2S) , and φ, where M 2 K π M 2 ψ(2S)π is the squared mass of the K π (ψ(2S)π ) system, θ ψ(2S) is the ψ(2S) helicity angle defined as the angle 10 between the daughter lepton's momentum in the ψ(2S) rest frame and the ψ(2S) momentum in the mother B rest frame, and φ is the angle between the ψ(2S) decay plane and the plane containing the K and π 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 π 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 σ 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 σ 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 ψ(2S)π bins has been performed and plotted in an Argand diagram. The phase is found to increase as a function of M 2 ψ(2S)π ; 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 = 181 ± 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.
InB 0 → χ c1 π + K − decay, Belle reported two resonance-like structures in the χ c1 π + mass spectrum [53]. From a Dalitz plot analysis, a fit gives the two resonance-like objects' masses and widths: 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 theB 0 → J/ψπ + K − decay. In order to resolve possible intermediate states, an amplitude analysis which is the same approach as Z c (4300) + → ψ(2S)π + is performed, taking 10 possible K * resonances into account. The Z c (4430) + → J/ψπ + is found to be evident and, in addition, one more charged charmoniumlike state of Z c (4200) + → J/ψπ + is observed with a significance of 6.2 σ . Its mass and width are obtained to be M = 4196 +31 −29 +17 −13 MeV/c 2 and = 370 ± 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 ϒ(nS)π + π − (n = 1, 2, 3) production at ϒ(5S) led to the discovery of Z b (10610) + and Z b (10650) + in ϒ(nS)π + and h b (m P)π + (m = 1, 2) final states, as will be discussed in Sect. 2.8.2. Since Y (4260) has an anomalous high rate of J/ψπ + π − production as described in Sect. 2.4, it can be interpreted as an analog in the charmonium(-like) sector. In this point of 11  The invariant mass spectrum for the larger one among two possible combinations of the J/ψ and one charged pion (M max (π J/ψ)) in the Y (4260) → J/ψπ + π − 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/ψ background by the normalized sideband events [43].
view, the charged bottomonium-like particles' appearance at ϒ(5S) inspired us to visit the J/ψπ In a Y (4260) → J/ψπ + π − event, there are two combinations of the J/ψ and one charged pion. Among them, the one having the larger invariant mass, M max (π J/ψ), is taken. The M max (π J/ψ) distribution is shown in Fig. 8. We see a significant enhancement at 3900 MeV/c 2 with 5.2 σ significance. Its mass and width are obtained to be M = 3895.5 ± 6.6 ± 4.5 MeV/c 2 and = 63 ± 24 ± 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.
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 max (π J/ψ) around 3.9 GeV/c 2 and obtained the mass M = 3899.0 ± 3.6 ± 4.9 MeV/c 2 and width = 46 ± 10 ± 20 MeV [55]. Hence, this charged charmonium-like state Z c (3900) + is confirmed.
Because of the similarity between Y (4260) → J/ψπ + π − and ϒ(5S) → ϒ(nS)π + π − , as well as the discovery of Z b (10610) + and Z b (10650) + states, one possible hypothesis is regarding Z c (3900) + as a (DD * ) + molecular state. Using BES III data collected at √ s = 4.26 GeV, the e + e − → (DD * ) ± π ∓ 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 π D recoil mass spectrum. In both D 0 D * − and D +D * 0 cases, the clear enhancement in the DD * mass spectrum just above the threshold is significant. The extracted resonance parameters are pole mass M pole = 3883.9 ± 1.5 ± 4.2 MeV/c 2 and width pole = 24.8 ± 3.3 ± 11.0 MeV. Here, Z c (3885) + denotes this structure in the (DD * ) + final state. Assuming that the Z c (3885) + is same as the Z c (3900) + in the J/ψπ + mode, the ratio of partial decay widths is obtained to be Z c (3885) + → (DD * ) + / Z c (3900) + → J/ψπ + = 6.2 ± 1.1 ± 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]. 12 More searches for analogous reactions have been performed. Exploiting a partial reconstruction technique, the e + e − → D * D * ± π ∓ process is reconstructed and an enhancement just above the D * D * ± 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 ± 2.6 ± 3.7 MeV/c 2 and = 24.8 ± 5.6 ± 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 − → π + π − 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 → γ η c followed by the η c decays in 16 exclusive hadronic final states. The spectrum of the invariant mass of the π + h c system, M π + h c , shows a distinct structure and it is referred to as Z c (4020) + with a mass of 4022.9 ± 0.8 ± 2.7 MeV/c 2 and width of 7.9 ± 2.7 ± 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.

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 ϒ(5S) region [59,60]. The corresponding particle (denoted as Y b hereafter) has J PC = 1 −− , and a large decay rate into the ϒ(nS)π + π − (n = 1, 2, 3) final state is expected. Therefore, Belle studied the decays ϒ(5S) → ϒ(nS)π + π − using 21.7 fb −1 data accumulated at the ϒ(5S) peak.
Assuming that ϒ(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 ϒ(1S)π + π − and ϒ(2S)π + π − channels. The measured partial widths are about 100 times larger than those of ϒ(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/ψπ π was measured at around Y (4260). This led to the idea that there could be a Y b state different from ϒ(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 √ 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 − → ϒ(nS)π + π − (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 13   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 ϒ(nS)π + π − decays were measured to be 9 ± 4 MeV/c 2 in mass and −15 +11 −12 MeV/c 2 in width, which correspond to 2.0 σ including the systematic error.
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 ± 3.9 MeV/c 2 in mass and 5.2 ± 8.1 MeV/c 2 in width, which corresponds to about the 2 σ 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 ϒ(nS)π + π − not being proved to be different particles. However, the anomalously large rates to ϒ(nS)π + π − remain a mystery. e + e − → h c π + π − also increase around Y (4260) in a similar way as e + e − → J/ψπ + π − , 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 π + h c system of the e + e − → h c π + π − process [58]. Indication of the existence of the Y b state also leads to an idea of large rates of e + e − → h b (m P)π + π − (m = 1, 2) at ϒ(5S). The h b (m P) had not been observed yet and was hoped to be observed in this process. Belle searched for h b (m P) states in missing mass distribution against π + π − using the 121.4 fb −1 data sample accumulated at ϒ(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 ϒ(1S)π + π − and ϒ(2S)π + π − final states. In order to suppress e + e − → qq (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 − → π + π − ϒ(nS)[→ μ + μ − ] samples, where the ϒ(nS) mass is calculated as the missing mass against π + π − . The result is shown in Fig. 12. In addition to clear peaks of ϒ(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 (m P)π + π − are of a similar order as those of ϒ(nS)π + π − in spite of the expectation that h b (m P)π + π − is suppressed because a spin flip of the bottom quark is required.

Observation of Z b (10610) + and Z b (10650) +
In order to understand the abnormal features in the e + e − → ϒ(nS)π + π − and e + e − → h b (m P)π + π − processes at ϒ(5S), more detailed studies of the final particles were performed. The invariant mass distributions of intermediate states of the three-body ϒ(nS)π + π − 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 ϒ(nS)π + π − now show clear structures of two resonances in ϒ(nS)π mass distribution, as shown in Fig. 13 [20]. 15    The invariant mass distributions of h b (m P)π + (m = 1, 2) are obtained by the yields of ϒ(5S) → h b (m P)π + π − signals in bins of the h b (m P)π + mass, where the h b (m P)π + mass is calculated as the missing mass against π − (M miss (π − )). For each bin, the signal yield is obtained by fitting the 16 Table 1. Summary of masses and widths of Z b s in ϒ(nS)π + (n = 1, 2, 3) decays [67] and h b (m P)π + (m = 1, 2) decays [20].
Parameter Parameter distribution of missing mass against π + π − , 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 ϒ(5S) → ϒ(nS)π + π − (n = 1, 2, 3). The spin and parity of two Z b resonances are determined by the full amplitude analysis with angular distributions in six dimensions for processes ϒ(5S) → ϒ(nS)π + π − with ϒ(nS) → μ + μ − [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 ± are forbidden because Z b decays to both ϒ(nS)π and h b (m P)π ). The fit favors both Z b s with 1 + over all other combinations of J P values with significances more than 6 σ . 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.
The masses, widths, and other quantities for the two Z b states obtained for five transitions are summarized in Table 1  10607 The masses of Z b (10610) + and Z b (10650) + are very close to the sums of the masses of BB * and B * B * , respectively. It should be noted that the relative phase between the two Z b amplitudes is consistent with zero for ϒ(5S) → ϒ(nS)π + π − and with π for ϒ(5S) → h b (m P)π + π − , which would be expected from the B ( * )B * molecular model [68].
The proximity of the Z b (10610) + and Z b (10650) + masses to the sum of the B ( * )B * masses indicates that these states would be B ( * )B * molecular states and decay to B ( * )B * , as will be discussed in Sect. 2.8.2. Rather large branching fractions in ϒ(5S) → BB * π decay were found with a data sample of 23.6 fb −1 [69]. The intermediate structure of ϒ(5S) → B ( * )B( * ) π three-body decays were studied using a 121.4 fb −1 data sample [24]. We reconstructed one B meson with B + → J/ψ K + , B + → D 0 π + , B 0 → J/ψ K * 0 , B 0 → D − π + , and B 0 → D * − π + . The reconstructed B + or B 0 candidates were combined with π − candidates and the missing mass against the Bπ combination was calculated. The missing mass distribution is shown in Fig. 16. The two peaks correspond to BB * π and B * B * π signals with signal yields of 184 ± 19 and 82 ± 11 events, respectively. The B ( * )B * invariant mass was calculated as a missing mass against π for events in the BB * π and B * B * π signal regions. The distributions are shown in Fig. 17. The fits to the distribution yielded a significant signal of The relative branching fractions of Z + b decaying to observed channels (ϒ(nS)π + , h b (m P)π + , and B ( * )B * ) were calculated from the transition rates to three-body final states and the fraction of Table 2 gives the branching fractions assuming that the sum of these modes is 100%. The dominant decay modes are BB * for Z b (10610) + and B * B * for Z b (10650) + , respectively.

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 BB pairs at ϒ(4S) as well as the ϒ(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 DD * and χ c1 (2P) would show no conflict with all the available experimental results. The B * B( * ) molecule picture works well to describe the charged bottomonium-like states, Z b (10610) + and Z b (10650) + ; the co-existence of ϒ(nS)π + π − (n = 1, 2, 3) and h b (m P)π + π − (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 * 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 QQud, where QQ detonates cc or bb. 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.

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 DD, DD * , and D * D * (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 21/64 Downloaded from https://academic.oup.com/ptep/article-abstract/2016/6/062C01/2240707 by guest on 29 July 2018 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][95][96] where the light degrees of freedom were incorporated in hadronic dynamics through D ( * )D( * ) 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 (η c and J/ψ) and P-wave ones (h c and χ c J ) 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 Q Qqq 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

The basic idea 3.2.1. The QCD Lagrangian
The QCD Lagrangian is well known and is given by Here, the sum f goes over flavors is the gluon field tensor with the color index a and Lorentz indexes μ, ν, and D μ = ∂ μ − g s A a μ T a is the covariant derivative with the gluon field A a μ (a = 1, . . . , N 2 c − 1; μ = 0, . . . , 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 α s μ 2 runs as a function of a momentum scale μ as dictated by the following renormalization group equation with the beta function β(α s ): where the coefficients b n can be calculated at each order of n loops. For instance, at the one-loop level, b 1 = 33 − 2N f / (12π ) with the flavor number N f . At this level, one can solve Eq. (2) analytically to find the solution Hence, α s μ 2 diverges at the QCD scale μ 2 = 2 QCD , where 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. In fact, these inequalities may not be well satisfied for the strange quark with m s ∼ 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) QCD 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 QCD , it is possible to introduce two different small parameters: (light mass)/ QCD and 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 → 0) for up, down, and strange quarks ( f = u, d, s). In this case, chiral fermions of right-and left-handed chiralities 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: This is invariant under separate flavor transformations for the right and left components, with ψ R/L = u R/L , d R/L , s R/L t and with T a a = 1, . . . , N 2 F − 1 being generators of the flavor SU(N F ) 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, 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 (π , K , η mesons) become (approximately) massless, while their chiral symmetry partners, scalar mesons ( f 0 , κ, 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 24 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 ± 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][131][132][133][134][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 Q , the parameter QCD /m 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 Q → ∞, disturbance from the QCD interaction of order 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 μ by the four-velocity v μ , 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 where k μ is a residual momentum whose scale is much smaller than m Q , characterizing the offshell-ness of the heavy quark. In many practical applications, the four-velocity may be chosen as v μ = 1, 0 , 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 μ is defined as for the original heavy quark field Q(x). The factor e im Q v·x indicates that the momentum scale m Q v μ 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 μ . The operator (1 + ¡ 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 in the same frame, by applying the projection operator (1 − ¡ v)/2. We note also that Q v (x) corresponds to the excitation with mass 2m Q , as shown soon. 25 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, where the explicit form of L heavy is and the explicit form of L 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 The 2m Q in the second term indicates that Q v corresponds to the excitation of the mass 2m Q , hence the component Q v should be suppressed for large m Q , so long as we are interested in the heavy quark limit. In fact, using the equation of motion for Q v , and eliminating Q v , we obtain . This is the result at the tree level, because we have used the equation of motion for Q v (x). In order to reach final results consistent with the original QCD at a given order of 1/m 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 Q expansion for heavy quarks as with σ μν = i γ μ , γ ν /2. In the third term in the square brackets in Eq. (16) Let us consider the leading order (LO) as the dominant term in the 1/m Q expansion. The term for the heavy quark at LO, has two important symmetries, spin symmetry and heavy flavor symmetry. The spin symmetry is understood from the invariance under the spin transformation of (2) spin is a spin transformation operator [97]). This is clear because the term v · 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 · 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 SU(2N h ) symmetry including both the spin symmetry and the flavor symmetry, where N h is the number of heavy flavors. N h = 2 for two flavors (e.g. charm and bottom), and N 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 → c eν 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 Q → ∞). In realistic situations, however, the heavy quark masses are still finite, and the heavy quark symmetry is violated at O(1/m Q ). To see the violation, we consider the next-to-leading order (NLO) in the 1/m Q expansion in L HQET : Because m Q differs for each heavy flavor, the first term is not invariant under rotations in the multiheavy-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 → S Q v ). The latter violation is understood, because σ μν 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 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 Q → ∞), where only the LO term in L 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 where S is a heavy quark spin (excluding the angular momentum), and j is the total spin of the light components (u, d, s quarks, their antiquarks, and gluons). Because J and S are conserved quantities, j must also be conserved. Hereafter we call 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 Qq meson composed of a  Fig. 19. A schematic picture of the spin degeneracy of states with total spin J ± = j ± 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. heavy quark Q and a light antiquarkq, provided that the light component contains not only aq but also a multiple number of light quark-antiquark pairs (qq) and gluons (g). Hence, the Qq should be regarded as multi-particle objects whose structure may be denoted by Qq + Qqqq + Qqqqg + · · · schematically. 6 We emphasize that, regardless of the complexity of the light components, j is a well-conserved quantity, because S is conserved in the heavy quark limit and J is also conserved. Thus, j provides us with a good quantum number for the classification of the non-perturbative object composed of the light components [97,136].
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 ≥ 1/2 and J ± = j ± 1/2, are degenerate in mass for j = 0, as shown in Fig. 19. This is because the spin-dependent interaction is suppressed by 1/m Q due to the HQS. We will call those two paired states the HQS doublet. We note that j takes a halfinteger 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.  7 The small values suggest that c , * c baryons and b , * b baryons, respectively, are approximately degenerate, and hence they are regarded as HQS doublets with j = 1. We note that the ground state baryons, c for charm and 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 Q expansion in the HQET Lagrangian (18). The contributions at NLO are roughly estimated as follows. Employing the typical energy scale 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 μ = Mv μ + k μ , the sum of the on-shell part (Mv μ with v 2 = 1) and the off-shell part (k μ ). The latter is a quantity of order O( 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 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 μ = v μ + q μ /M (w-frame), where q μ is a small momentum which is much smaller than M and satisfies O(v · q) = q 2 /M (Fig. 21). Here, because In the w-frame, the momentum p μ is given by p μ = Mw μ + l μ with l μ being defined by l μ = k μ − q μ . 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 μ = v μ + q μ /m 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)  Lastly, we comment on the parametrization of the masses of heavy hadrons. In the 1/m Q expansion, the mass of a heavy hadron is given by where we have defined the low-energy parameters, in the rest frame with v r = 1, 0 , PTEP 2016, 062C01 A. Hosaka et al. Fig. 21. The Lorentz boost from the v-frame to the w-frame. In each frame, the heavy particle has four-velocity v μ and w μ , respectively. The different lengths of the arrows indicate the different spatial components for v μ and w μ , respectively.
denoting the hadron state by H v r [139][140][141][142]. 8 S Q = σ /2 and j are the spin operators for the heavy quark and the brown muck respectively. We have also introduced the Gell-Mann-Low function β(α s ) = μdα s (μ)/dμ, the chromoelectric gluon field E i = −G a 0i T a , and the chromomagnetic gluon field B i = ε i jk G a jk T a (i, j, k = 1, 2, 3; a = 1, . . . , 8). In Eq. (23), the Wilson coefficient c(μ) is determined by matching to QCD at energy scale μ m 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¯ , λ 1 , and λ 2 (μ) depend on each hadronic state.

Theoretical methods
In this section, we discuss several theoretical methods for analyzing the exotic hadrons: the quark model (Sect

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 andqq, 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 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 8 In Ref. [142], H v r is introduced as the hadron state whose normalization factor is consistent with those in Refs. [140,141].

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Downloaded from https://academic.oup.com/ptep/article-abstract/2016/6/062C01/2240707 by guest on 29 July 2018 (m u,d 400 MeV and m s 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 semiquantitative 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 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 × 2 = 6 remaining degrees of freedom by flavor (u, d, s) and spin (↑ and ↓), whose components are denoted by (u ↑, u ↓, d ↑, d ↓, s ↑, s ↓). 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 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 where 6 * is the complex conjugate of 6, corresponding to an antiquark. We obtain the flavor singlet and octet states both for spin 0 and 1. The baryons are given by 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 The short-range part V (S) i j for quarks i and j is given by where α s is the strong coupling constant, m i are quark masses, s i = σ i /2 spin operators ( σ i are the Pauli matrices), F i = λ i /2 color SU(3) operators ( λ i are the Gell-Mann matrices), S i j = 3 s i ·r s j ·r − s i · s j the tensor operators withr = r /| r |, and l i the angular momentum. The long-range part V (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, plays an important role in the mass splittings in S-wave hadrons. We note that F i · F j takes −4/3 for theqq pair (color singlet) and −2/3 for the qq pair (color antitriplet). Similarly, s i · s j takes −3/4 for spin 0 and 1/4 for spin 1 inqq (qq) pairs. We define F i = λ i /2 and s i = σ i /2 for the Gell-Mann matrices λ i and the Pauli matrices σ i . We summarize the values of λ i · λ j as and those of σ i · σ j as  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 l · s and tensor interactions as shown in Eq. (27). 32 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 α s used in the quark model takes a value close to one. Hence, the short-range part V (S) i j 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 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 (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 nonrelativistic quark model [150][151][152][153][154][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 (1405), which is the lowest state among negative parity baryons, but is overestimated by about 100 MeV in the quark model. Thus, the (1405) state has been investigated extensively as a candidate for theK N 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 (qq). As proposed by Jaffe, however, the tetraquark picture (qqqq) 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), κ(800), f 0 (980), and a 0 (980). The mass ordering observed in experiments is M( f 0 (600)) < M(κ(800)) < M( f 0 (980)) M(a 0 (980)).
When two quarks (qq; quark and antiquark) form the light scalar mesons, the mass ordering is naively given as by using isospin symmetry and the larger mass of the strange quark than up and down quarks. However, this clearly contradicts (32 (34) and can naturally explain the mass ordering in (32). Now, in the quark model, excitation energies are brought either by orbital motions or by an additional qq pair, where the quark "q" is the constituent quark (Fig. 22). In general, hadrons are PTEP 2016, 062C01 A. Hosaka et al. superpositions of the minimal quark configuration and other multi-quark configurations, such as qq + qqqq. 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 cc and bb , an interesting feature is that For instance, we remember that there is evidence that the light scalar mesons are likely to be qqqq, 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 QQ 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. 12 It is interesting to note that Gell-Mann has already suggested that qqqqq components can exist in addition to the normal qqq in baryons, and qqqq components could exist in addition to the normal qq 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 Q expansion in the given coordinate frame.

.2.3. This is schematically expressed as
where the sum is taken over the spin combinations with appropriate Clebsh-Goldan coefficients, denoted by C αβ;γ . For example, we consider the mesons composed of a heavy quark Q and a light antiquarkq with conserved spin and parity j P = 1/2 − , baryon number −1/3, and color antitriplet, denoted by P v (x) with spin 0 and P * μ v (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 * μ v (x) are degenerate in mass, and they are treated as heavy quark spin multiplets where (1 + ¡ v)/2 is the projection operator as introduced in Eq. (9) for the positive energy state of the heavy quark Q in the Qq meson. More explicitly, the meson fields P * μ v and P v have the matrix structure where Q v is the heavy quark field in the v-frame as defined by Eq. (9) andq 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 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 where the trace is taken over the Dirac matrices (see Eq. (36)), D μ is the chiral covariant derivative, , and V μ (x) and A μ (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 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 35 with q/M = w − v. This has not only the phase e −iq·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] H which is given by the Lorentz boost from the v-frame to the (v + q/M)-frame up to O(1/M ), where q μ is supposed to be the residual momentum in H v (x). This has the interesting property namely a change only by the phase factor e −iq·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 O(1/M ) [137,138,142]: where λ, g 1 , and g 2 are new coupling constants at 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 γ μ γ 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 Q expansion for heavy quarks, when we retain the expansions up to NLO [159]. This is because the relation M = m Q + ( being the energy from the light d.o.f., of the order of QCD ) is inverted to 1/M = 1/m Q + O 1/m 2 Q . From the HQET Lagrangian (15), we remember that there are two terms at O 1/m Q : the heavy quark spin conserving and non-conserving terms. 15 It was shown in the HQET that, at O 1/m Q , the former is related to PTEP 2016, 062C01 A. Hosaka et al. color-electric gluon fields, and the latter with color-magnetic gluons [139][140][141] (cf. Sect. 3.2.3). Thus, the terms at 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 andD 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 (σ ) and vector (ρ, ω) mesons [160,161].
Let us consider the P ( * )P ( * ) systems, where P and P * (P andP * ) are (Qq) spin 0 and (Qq) spin 1 mesons ((Qq) spin 0 and (Qq) spin 1 ) with S-wave. The interaction Lagrangian for a P ( * ) 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 ( * ) mesons. From Eq. (39), we obtain the π P P * and π P * P * vertices The πPP * and πP * P * vertices are obtained by changing the sign of the π P P * and π P * P * vertices in Eqs. (44) and (45). It is important to note that the strength of the interaction vertex is the same for π P P * and π P * P * . This is a consequence of the heavy quark symmetry. 37 The one-pion-exchange potential (OPEP) for PP * → P * P and P * P * → P * P * is given from the vertices (44) and (45) as and the OPEP for PP → P * P * and PP * → P * P * is Here, three polarizations are possible for P * as defined by ε (±) = ∓1/ √ 2, ±i/ √ 2, 0 and ε (0) = (0, 0, 1), and the spin-one operator T is defined by k . As a convention, we assign ε (λ) for an incoming vector particle and ε (λ) * for an outgoing vector particle. The tensor operators are defined by The functions C π (r ) and T π (r ) in the central and tensor potentials, respectively, have asymptotic behaviors at long distance: These are modified by the form factor at short distance [160][161][162]. τ 1 and τ 2 are isospin operators for P 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 PP( 1 S 0 ) for the initial state, P * P * ( 5 D 0 ) for the intermediate state, and PP( 1 S 0 ) for the final state. 16 We note that the spins of the heavy (anti)quarks Q andQ 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)quarksq 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 PP 1 S 0 ↔ P * P * 5 D 0 . The tensor force in the OPEP provides a strong attraction. We note that, in order to switch on the tensor force, P and P * (orP andP * ) must PTEP 2016, 062C01 A. Hosaka et al. be mixed in the process. In the present case, the mass difference between P and P * (orP andP * ) is zero in the heavy quark limit, and hence they can easily be mixed. Thus, the P-P * (P-P * ) mixing accompanying the OPEP is important in this system. We emphasize that the P-P * (P-P * ) 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 * (P-P * ) 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.
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 DD * 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 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 ω meson exchange contributes as an attraction for P ( * )P ( * ) . 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. 39 The idea of the QCD sum rule is to extract physical quantities by comparing the correlation function computed in two different methods [167][168][169]. On the QCD side, one calculates a correlation function of currents, 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 ρ meson operator can be chosen as A, B =qγ μ τ q, where τ is an isospin matrix. The computation can be done by using the Wilson's operator product expansion (OPE) at short distances [170], where 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 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 → y, are isolated by the Wilson coefficients C i (x − y), and the operators 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 , For instance, for A =ψγ μ ψ, 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 → y or p 2 → ∞, by powers of 2 QCD / 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 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 ρ(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 ρ(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 ρ(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,
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 σ ( f 0 ) meson, there are five independent currents [171], where C is the charge conjugation matrix in Dirac space, C = iγ 0 γ 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 T a Cγ 5 d b in the first equation of (60)) can be written equivalently in terms ofqq 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) color3 c diquarks and antidiquark, (2) a color 6 c diquarks and antidiquark, (3) a DD * molecular type, and (4) a (cc)(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 α s corrections. They found a reasonable Borel window to reproduce the experimental data for the mass. They also studied the coupling X (3872) → J/ψω 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 DD * molecular type currents and hybrid currents ofcGc, 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 0D * [175] and J/ψ 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]. J P = 1 + by LHCb, QCD sum rule analyses were performed by assuming J P = 0 − and by employing D 1D * 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 π/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.   [183,184]. Right: Heavy flavor (cb) mesons, data taken from [22].
in Euclidean space-time, 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.
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 π , K , and φ mesons are used as inputs while in the BMW2008 the masses of π and K mesons and 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. 43 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 ofcc, DD * , and diquarkantidiquark 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 thecc term. These observations seem consistent with the admixture picture of molecular andcc 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 thecc 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 3 P 1 state ofcc, its properties are not explained by a simplecc configuration.
As compared to the other states of the same L S multiplets 3 P 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 andD * , Most notably, the decay rate of [78] Br(X (3872) → J/ψ + 3π(ω)) Br(X (3872) → J/ψ + 2π(ρ)) 1 (63) 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 0D * 0 threshold. These observations have naturally led to the idea of a hadronic molecule picture of D 0D * 0 -D ± D * ∓ . 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

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To illustrate the molecular picture for X (3872) with a large isospin violation, let us start with a two-channel model of D 0D * 0 and D ±D * ∓ . The model Hamiltonian is given by where B 1,2 are the unperturbed binding energies of the neutral and charged states, D 0 D * 0 and D ± D * ∓ , respectively, and V their interaction matrix element, Good isospin symmetry implies that B 1 B 2 , and so In this case the eigenstates of the Hamiltonian (64) are a superposition of |B 1 and |B 2 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 B 1 B 2 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 1/4 1 (see Eq. (73)), and so Thus the eigenstates are (almost) purely |B 1 or |B 2 , corresponding to the isospin breaking states. So far, the discussions have been made under the assumption that a suitable interaction for DD * 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 * Dπ 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 DD * systems, vector meson exchange (ρ and ω) 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. 45 The possible existence of a DD * molecular state does not necessarily explain the (seeming) absence of χ c1 (2P). Because the χ c1 (2P) state can definitely be accessible by acc pair, it would be natural to consider a model with thecc pair and a DD * molecule.
With those observations, Takizawa and Takeuchi proposed to describe X (3872) as an admixture state of charmonium and a D D * molecule [196], The basic assumption of their model is the coupling between the charmoniumcc and the molecule DD * , which is parametrized as where g and are the coupling constant and the form factor, respectively. This interaction is commonly used for both the neutral and charged DD * molecules. The Hamiltonian to be diagonalized is then where K is the non-relativistic kinetic energy of the corresponding DD * state. In their model, the interactions between DD * are not considered in order to minimize the ambiguity, but with the emphasis on their coupling to thecc state. Such a coupling effectively plays the role of attractive interaction if thecc mass is larger than the DD * 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 = 0.5 GeV. The coefficients c i are given as This in turn gives the probabilities indicating non-negligible isospin violation as implied in (63). The amount of thecc 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 0D * 0 origin only slightly below the threshold. The effect of another pole of charged D ±D * ∓ origin is not seen, while the evidence of thecc core origin with mass 3.95 GeV remains as a wide (almost flat) background. This explains the absence of a clear peak ofcc-like state at the position expected in the quark model. Finally, to show another unique feature of X (3872) let us look at its wave function. Suppose that the bound states |B 1 and |B 2 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 PTEP 2016, 062C01 A. Hosaka et al.  [196].
written for the outside of the interaction range as where μ is the reduced mass of the two particles. We find the following elementary relation: from which one can estimate the size of the bound state using the reduced mass of the DD * system (∼1 GeV) and the binding energy. Thus, from (62) 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 ∼ 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
• 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. These problems can be resolved by assuming the molecular structure of Z b (10610) as BB * and Z b (10650) as B * B * [68]. In particular, the last point of the violation of heavy quark spin symmetry can be explained by the following decomposition: 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 andB * are where the notation of spin coupling [s 1 , s 2 ] s is introduced. Equations (77)  There have been many theoretical studies of Z b s assuming the B ( * ) molecule structure [68,[198][199][200][201][202][203][204][205][206][207][208][209][210][211][212][213][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 ( * ) andB * 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 π , ρ, and ω 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 ≤ 2 were investigated. For a given set of quantum numbers J PC 2S+1 L J , several coupled channels are possible, as pointed out in Sect. 3.3.3. In particular, the coupling of different angular momenta L , L ± 2 is important due to the tensor force of the one pion exchange. We summarize coupled channels for various molecular states in Table 4.
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 BB * 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.
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 BB * and B * B * states have 48 Table 4. Possible coupled channels for low-lying B ( * )B( * ) states with J PC (J ≤ 2) [213]. *

J PC
Components W b2 , W b2 * 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.  [213]. Some states are labeled as Z and W , as shown in Table 5. been confirmed, as well as other branching ratios into various channels. These are also useful in testing the molecular structure of Z b s.
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 BB * or B * B * , (2) pion emission by B * , and then (3) BB * or BB merging into the final state bottomonium ϒ. Various coupling constants needed to calculate these Feynman diagrams are evaluated by other mesonic processes, and then the decay rates of ϒ(nS)π have been studied. By introducing a phenomenological form factor of the initial vertex for the Z b BB * and Z b B * B * , it was shown that the experimental decay ratios were explained qualitatively. In particular, the small branching ratio decaying into ϒ(1S)π , which seems opposite to what is naively expected from the 49 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 ( * )B( * ) 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.  [204,213].
again the phase space and form factor effects) where the W b0 , W b1 , W b1 , W b2 , and W b2 states are summarized in Table 5. For S-wave pion decays into ϒ(1S), the selection rules allow only the transition from the 1 +− h b state while others are forbidden (dotted arrow in Fig. 29). Finally, let us look at the productions of the above five W b states in the one pion emission decay of the ϒ(5S) state as denoted by the double arrows in Fig. 29. These are inversion processes of their decays into ϒ(1S) with one pion emission. Here the one pion is combined with the W b s, while in their decays it is combined with ϒ(1S), and so the resulting heavy quark decompositions differ. The result of the production rates P of W b s is  [216], a molecule of a charmonium and a light flavor meson [217,218], and baryonium of c¯ c [219]. A molecular picture of D 1 (2420)D was also tested with the decay dynamics to Z c (3900) and to J ψππ [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 ππ, 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.

3.4.3.2.
Z c (4430) + . The observation of this state is the first evidence of a genuinely exotic state as its charge requires a multiquark component,ccud. Of course, in principle, it is possible to reach this state withoutcc, which is, however, unlikely to the same extent that J/ψ is not regarded as a light quark and antiquark pair, e.g.,ūu. 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 ψ(2S)π + 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][229][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.

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 ( * ) D ( * ) 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][261][262], and in lattice simulations [262][263][264][265][266][267][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). 53 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 qqQQ. When two clusters of qQ 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:P ( * )P ( * ) with P ∼ (qQ) spin 0 andP * ∼ (qQ) spin 1 .
In this case,P ( * ) andP ( * ) 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 ofP ( * )P ( * ) with I (J P ) = 0(1 + ) has four components, with 2S+1 L J denoting a state of spin S, angular momentum L, and total spin J . Thus, we have the following 4 × 4 matrix for the interaction, with the functions C π (r ) and T π (r) in the central and tensor potentials defined in Eqs. (54) and (55) (r is the distance between the two mesons) from the π exchange potential [94,124]. The Hamiltonian is given by where K = K P P * 0 , K P * P * 1 , K P P * 0 , K P * P * 1 is the term for the kinetic energy and M = (0, 0, m P * P , m P * P ) with theP-P * mass splitting m P * P = m P * − m P . The energy is measured from thePP * 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 I P = 0 1 + , as shown in Fig. 30. In our calculation, we obtain a bound state with the mass 3813 MeV (binding energy 60 MeV from theDD * threshold) for 0 1 + . 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).

T cc
When twoP ( * ) mesons approach closer, diquarks qq andQQ may be developed more to form an exotic tetraquark qqQQ. We call this state T Q Q to distinguish it from theP ( * )P ( * ) hadronic molecules.
T Q Q 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 (q) and a heavy (anti)quark 54  Q (Q) is suppressed by O(1/m Q ) with a heavy quark mass m Q . The interaction between two heavy (anti)quarks is further suppressed by O 1/m 2 Q . Therefore, the mass of T Q Q could be lower than the mass of the possible strong decay channel of two qQ mesons, if the qq interaction is attractive: In this case, the only possible decay is the weak decay of heavy quarks. We should note that the situation differs for qq QQ states. If qq behaves as the Nambu-Goldstone bosons (π , K , and η mesons) with small mass, M(qq) ∼ 0, we expect the following mass relations: and hence observe that the qq QQ state is not likely to be the ground state of the system. To estimate the mass of T Q Q , we assume that the i j quark pair interact though the color-magnetic interaction (cf. Eq. (29)), with dynamical (constituent) quark mass m i and spin operator 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 1500 MeV and m b 4700 MeV (=m Q ) as compared to light quarks of mass m u m d 300 MeV and m s 500 MeV. For quark-quark interaction, from the mass splittings among light hadrons with different spins such as N -, -, cc , and bb , we find C B /m 2 u = 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) (color3 c ). Thus the lowest mass of T Q Q is given as M qqQQ = 2m Q + 2m q + V qq (S = 0), which gives a lighter mass than M qQ + M qQ 2m Q + 2m q . The quantum numbers of the lowest T Q Q should be I J P = 0 1 + . The total spin J = 1 is due to the S-waveQQ pair of color 3 c , satisfying the Pauli principle. More realistically, we need to consider the contributions of order 1/m q m Q for qQ and of order 1/m 2 Q forQQ. Finally, we obtain the binding energy of T cc to be about 80 MeV when measured from the D D * threshold, and about 124 MeV for T bb from B B * [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 D D * scattering in the I J P = 0 1 + 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 3 c configuration forQQ (this T cc is denoted by T Q Q [3 c ]), we may consider the exotic color "antisextet" configuration forQQ: T Q Q 6 c [270]. The qq QQ pair of 6 c 6 c receives attraction in the S = 1 (isospin-singlet) channel with the strength reduced to 1/6 of that of the3 c qq pair (see Eqs. (29), (30), and (31)). Being of the same total quantum numbers, those two T Q Q s may mix, but the mixing should be suppressed due to the small spin-flip betweenQQ (S = 0) in T Q Q 3 c andQQ(S = 1) in T Q Q 6 c . The estimate of the mass of T Q Q 6 c , however, indicates that T Q Q 6 c may not be bound. For example, the mass of T cc 6 c (Q = c) is 54 MeV above the D D * threshold, and it can decay to D and D * in S-wave. Thus, some special mechanisms may be required for the T cc 6 c 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 Q Q , as well as the P ( * ) P ( * ) states discussed in the previous section, in experimental facilities is an important issue. One possible method to investigate the double charm production (cccc) is collider experiments. In fact, there are several reports on observations of double charmonia, where cccc is combined to two cc pairs in the final states. It will then be naturally expected that the cccc is also combined to produce double charm hadrons containing cc and others with twocq. 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 Q Q 3 c and T Q Q 6 c 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 Q Q 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 56/64 Downloaded from https://academic.oup.com/ptep/article-abstract/2016/6/062C01/2240707 by guest on 29 July 2018 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 quark-quark (Q Q) interaction with color non-singlet (3 c ) channels. Furthermore, spectroscopy of the triply heavy baryon Q Q Q will be important for understanding the color-confinement force in three quarks Q Q Q. 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][277][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 Q → ∞. 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 Q corrections beyond the leading order. The symmetry-breaking effects will be important, particularly when the mass is close 57/64 Downloaded from https://academic.oup.com/ptep/article-abstract/2016/6/062C01/2240707 by guest on 29 July 2018 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 DD * molecular states, and to regard Z b 10610 + and Z b 10650 + as BB * and B * 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 ϒ(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 4430 + and Z c 3900 + 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.