$\chi^{\vphantom\dagger}_{c0}(3915)$ As the Lightest $c\bar c s \bar s$ State

The state $\chi^{\vphantom\dagger}_{c0}(3915)$ has recently been demoted by the Particle Data Group from its previous status as the conventional $c\bar c$ $2 {}^3P_0$ state, largely due to the absence of expected $D\bar D$ decays. We propose that $\chi^{\vphantom\dagger}_{c0}(3915)$ is actually the lightest $c\bar c s \bar s$ state, and calculate the spectrum of such states using the diquark model, identifying many of the observed charmoniumlike states that lack open-charm decay modes as $c\bar c s \bar s$. Among other results, we argue that $Y(4140)$ is a $J^{PC} = 1^{++}$ $c\bar c s \bar s$ state that has been not been seen in two-photon fusion largely as a consequence of the Landau-Yang theorem.


I. INTRODUCTION
The past 13 years have been a time of remarkable growth in experimental reports of hadronic states, particularly in the charmonium and bottomonium sectors. Starting with Belle's observation of the X(3872) in 2003 [1], almost 30 new states with masses lying in these regions have been reported. Until last year's observation of the baryonic P + c states by LHCb [2], all of the observed states were mesonic. Since this counting does not include new conventional quarkonium states discovered in the interim, such as the cc χ c2 (2P ) [3,4], all of these states are considered exotic. These additional exotic states have been suggested in numerous papers to be gluon hybrids, kinematical threshold effects, di-meson molecules, compact charmonium embedded in a light-quark cloud (hadrocharmonium), and diquarkantidiquark states (Ref. [5] gives an exhaustive recent review of work in these areas).
Evidence has steadily mounted that at least some of the mesonic exotics are tetraquarks, and the baryonic exotics are pentaquarks. For example, the Z + (4430) state first observed in 2008 [6] is charmoniumlike but also charged, so that its minimum valence quark content is ccud. But the confirmation experiment by LHCb [7] also measured a rapid phase variation of the Z + (4430) production amplitude near the peak mass, which is characteristic of true resonant scattering behavior. Similar observations were carried out for the P + c states [2]. 1 The definitive separation of exotic from conventional states is not always trivial, however. The X(3872) has the same J P C = 1 ++ as the yet-unseen cc state χ c1 (2P ), but its mass is several tens of MeV lower than expected. Moreover, Γ X(3872) < 1.2 MeV, while the χ c1 (1P ), its ostensible radial ground state, has a width Γ = 0.84 MeV, almost as large. The χ c1 (1P ) has a mass 360 MeV lower, so one expects the χ c1 (2P ) to have all of the 1P state's decay modes (as well as many additional ones), but with much more phase space, and hence a substantially larger width.
The J P C = 0 ++ state χ c0 (3915) is an even trickier example. Its mass lies very close to quark-potential model predictions for that of the yet-unseen cc state χ c0 (2P ). As described in detail below, it is produced in γγ fusion, as one would expect for the χ c0 (2P ), and χ c0 (3915) was briefly hailed by the Particle Data Group (PDG) [12] as the missing cc state χ c0 (2P ). However, the current absence of the expected dominant D ( * )D( * ) decay modes speaks against a cc interpretation, and indeed, also against a ccqq interpretation (q = u, d).
In this work, we therefore propose that χ c0 (3915) is the lightest hidden-charm, hidden-strangeness (ccss) tetraquark state. Our analysis is performed assuming the diquark-antidiquark model first proposed in Ref. [13] and applied to ccss states in Ref. [14] (where the lightest ccss state was indeed found to have J P C = 0 ++ ). Since the advent of those two papers, many new exotic states have been observed, and the model was improved recently to reflect the new data in Ref. [15]. Our analysis, therefore, develops this improved version of the diquark model for ccss states, under the assumption that χ c0 (3915) is their ground state.
Along the way we predict the full spectrum of ccss states, noting several whose properties match those of observed exotics remarkably well. For example, the Y (4140) observed in B decays appears as an enhancement in the J/ψ φ spectrum, exactly as expected for a ccss state, but it has not yet appeared in γγ fusion experiments. Our model neatly accommodates a J P C = 1 ++ state at 4140 MeV, which is forbidden by the Landau-Yang theorem [16,17] from coupling to a two-photon state.
This paper is organized as follows. In Sec. II we review the measured properties of the χ c0 (3915) to motivate the proposal that it and several other exotics may be ccss states. Section III introduces the diquark-antidiquark model used and develops its spectrum of ccss states. We analyze our results in Sec. IV by comparing to the known exotics spectrum, pointing out both successes and shortcomings of the results. In Sec. V we present a brief discussion and conclude.
II. χ c0 (3915) AND OTHER POTENTIAL ccss STATES An understanding of the exotic charmoniumlike spectrum remains elusive, to say the least, from both experimental and theoretical viewpoints (See Ref. [5] for a thorough review and Ref. [18] for perspectives on future prospects.). With respect to the current work, the most interesting state is χ c0 (3915), which was discovered by Belle in 2005 [19] as a J/ψ ω enhancement in the process B → J/ψ ωK [and was originally labeled Y (3940)], and confirmed by BaBar [20,21]. However, Belle found no evidence for D * 0D0 decays of the state [22]. In 2010, Belle discovered [23] the state X(3915) in γγ → J/ψ ω, and BaBar subsequently confirmed the result [24], establishing furthermore that the state has J P C = 0 ++ , so that its name under the conventional scheme should be χ c0 . However, again, no evidence for a peak near 3915 MeV in D ( * )D( * ) was found in B → D ( * )D( * ) K decays at Belle [25] or BaBar [26]. The shared J/ψ ω decay mode and proximity in mass and width for these two states has led them to be identified as the same state, currently called χ c0 (3915). Its mass and width are currently given as [12]: In fact, the establishment of J P C = 0 ++ for χ c0 (3915) immediately suggested that the state is actually the first radial excitation χ c0 (2P ) of the known conventional charmonium state χ c0 (1P ), the 2P state mass being predicted in quark potential models to lie in the range 3842-3916 MeV [27][28][29]. The 2P identification was also briefly espoused by the Particle Data Group (PDG) [12] (in its online form). However, this identification was questioned by Refs. [29][30][31]; their objections amount to: i) The mass splitting between the established χ c2 (2P ) (3927 MeV) and χ c0 (3915) is rather smaller than expected from quark potential models; ii) The true cc χ c0 (2P ) should decay copiously to D ( * )D( * ) (the D 0D * 0 threshold lies at 3872 MeV, and the D 0D0 threshold lies at 3730 MeV); iii) As a charmonium-to-charmonium process, the decay χ c0 (2P ) → J/ψ ω is Okubo-Zweig-Iizuka (OZI) suppressed and would be expected to occur less frequently than is observed. In fact, Ref. [31] showed that the tension between ii) and iii) if χ c0 (3915) is assumed to be χ c0 (2P ) leads to incompatible bounds on the branching fraction B(χ c0 (2P ) → J/ψ ω). As a result of these objections, the PDG currently refers to the state as χ c0 (3915).
Some comments regarding the J P C assignment in γγ fusion are in order. If the photons are both transversely polarized, then the Landau-Yang theorem [16,17] forbids the resonance from having spin one. Of course, the photons at Belle and BaBar are produced from e + e − collisions, and longitudinally-polarized off-shell photons can evade this constraint. However, the photon virtuality in this case scales with m e , which is much smaller than the other mass scales in the process. The difference between the longitudinal and timelike photon polarizations (the latter of which gives an exactly vanishing contribution to physical amplitudes due to the Ward identity) then vanishes with m e , meaning that longitudinal photon contributions also vanish in this limit. Noting both P and C conservation in QED and using Bose symmetry, the allowed quantum numbers for resonances formed in e + e − → γγ → X are therefore indeed either 0 ++ or 2 ++ .
The χ c0 (3915) therefore appears to be a supernumerary 0 ++ charmoniumlike state, and very likely a 4-quark state (the lowest 0 ++ hybrid computed by lattice QCD being expected to lie many hundreds of MeV higher [32]). It is most natural to suppose that χ c0 (3915) has the flavor structure of an isosinglet: cc(uū − dd)/ √ 2. Indeed, searches for signals of charged partner states ccud or ccdū [33,34] in the same energy range [actually designed to look for X(3872) isospin partners] have produced no clear signal. 2 Furthermore, such a 4-quark state would seem to have no obvious barrier for decaying into DD, 3 and only have a relatively small p-wave barrier for decay into DD * . The absence of observed open-charm decays of χ c0 (3915) poses a real problem for the 4-quark interpretation.
We propose, therefore, a rather radical solution: The χ c0 (3915) is a ccss state, hence naturally an isosinglet that eschews open-charm decays. It lies just below the D + sD − s threshold (3937 MeV) as well as the J/ψ φ threshold (4116 MeV), and therefore the only OZI-allowed decay (in that no new flavors in a quark-antiquark pair are created or destroyed) open to it is η c η (threshold 3531 MeV). 4 We present a calculation of this width in Sec. IV and argue that it naturally accommodates the value in Eq. (1). The observed decay mode J/ψ ω actually appears to be quite suppressed, being either due 2 We thank S. Olsen for pointing out this very important fact. 3 The only known exception to this statement is if the state is a molecule of two mesons held together primarily through 0 − exchanges, such as by π and η. In that case, Lorentz symmetry plus P conservation of strong interactions forbids decay into two 0 − mesons. 4 Note that no exotic to ηc decays have yet been observed [35].
to ω-φ mixing that is less than ideal (so that ω contains a small amount of valence ss, and φ contains a small amount of valence qq), or double OZI-suppression (ss → g → qq). Furthermore, we assert that χ c0 (3915) is the lightest ccss state; the only lighter charmoniumlike exotic is X(3872), and it decays freely into open-charm states.
A number of higher exotic states have properties amenable to a ccss description, by virtue of having neither obvious isospin partners nor observed opencharm decays. Including the χ c0 (3915), 9 states share these properties: , and Y (4660). This list includes 4 of the 5 states, Y (4008), Y (4260), Y (4360), and Y (4660), observed using initial-state radiation (ISR) production in e + e − annihilation, and therefore necessarily carrying c . Y (4008) and Y (4260) have been seen only in decays containing a J/ψ, while Y (4360) and Y (4660) have been seen only in decays containing a ψ(2S). ISR states curiously also do not appear as obvious peaks in the R(e + e − → hadrons) ratio, unlike the conventional 1 −− charmonium states J/ψ, ψ(2S), ψ(3770), ψ(4040), ψ(4160), and ψ(4415) [12] (Indeed, a local minimum of R appears around 4260 MeV). If this effect reflects the relative difficulty of making extra particles in e + e − annihilation at energies where α s is small [i.e., with α s (m c ) ≃ 0.3, producing not just cc, but ccg or ccqq], then the production of ccss would presumably be even further suppressed due to a mass effect.
The Y (4140), Y (4274), and X(4350) are even better ccss candidates, since they are observed as J/ψ φ enhancements. The Y (4140) was first reported by CDF in the process B → J/ψ φK in 2009 [36], and presented with higher statistics by them in 2011 [37], with other observations in this channel provided by DØ [38] and CMS [39], while LHCb has not yet seen the state, but the disagreement is only at the level of 2σ [40]. Along the way, Refs. [37,39] observed in the same channel the enhancement called Y (4274). Belle, however, using the production mode γγ → J/ψ φ, saw neither Y (4140) nor Y (4274), but instead discovered a new state, X(4350) [41]. A possible explanation for the absence of Y (4140) and Y (4274) in γγ production is of course the Landau-Yang theorem, granted that neither state is J P C = 0 ++ nor 2 ++ . A study of Y (4140), Y (4274), and X(4350) as ccss states using QCD sum rules (but leading to rather different J P C assignments) appears in Ref. [42], while Ref. [43] is a quark-model study predicting Y (4140) to be 1 ++ and notes the importance of the η c η mode.
Lastly, the Y (4230) is an enhancement seen in the process e + e − → χ c0 ω [44]. Should it turn out to be a ccss state, its χ c0 ω decay must proceed through the same ωφ mixing or double-OZI suppression mechanism as suggested for χ c0 (3915).

III. DIQUARK MODELS
Interest in diquark-antidiquark models for light scalar mesons has a long and interesting history (see, e.g., Ref. [45] for a review). The decay patterns for such states obtained from the OZI rule are discussed in Ref. [46], and those from instanton-induced decays are discussed in Ref. [47]. Here, however, we focus on an approach obtained from simple Hamiltonian considerations. 5 The "Type-I" diquark model of Ref. [13] is defined in terms of a Hamiltonian with local spin-spin couplings combined with spin-orbit and purely orbital terms. The orbital angular momentum operator L refers to the excitation between the diquark-antidiquark pair, while orbital excitations within each diquark are ignored. Specializing (for notational simplicity) to 4-quark systems with hidden charm [cq 1 ][cq 2 ], the Hamiltonian reads where m [cq1] and m [cq2] are the diquark masses, H qq SS refers to spin-spin couplings between two quarks (or antiquarks) and therefore refers to spin-spin couplings within either the diquark or antidiquark: H qq SS couples quarks to antiquarks, and therefore induces interactions between the diquark and the antidiquark: and H SL and H L are the spin-orbit and purely orbital terms, respectively: where S is the total quark spin operator. The "Type-II" diquark model [15] is defined by neglecting all spinspin couplings between a quark of the diquark and an antiquark of the antidiquark, i.e., effectively by setting H qq SS = 0. The dynamics binding tetraquark states can be very different from that binding conventional hadrons, so one should not expect a "universal" Hamiltonian to hold for all hadrons.
The most natural basis in which to describe the diquark-antidiquark states is one in which the good quantum numbers are the four quark spins s c , ss, s q1 , sq 2 , diquark spins s [cq1] , s [cq2] , total quark spin S, orbital angular momentum L, and total angular momentum J. One can also recouple the quark spins into s cc , s q1q2 using the Wigner 9j symbol [49] This basis is particularly convenient for identifying the charge conjugation (C) quantum number of the states: The ccss tetraquark states have received a dedicated study only in the Type-I model [14], some years ago when many known exotic charmoniumlike states had not yet been observed. Should ccss tetraquark states be pro- s . The presence of possible ccss states also ameliorates one of the more awkward problems of tetraquark models: If hidden-charm tetraquarks contain light quarks, then one expects either near-degenerate quartets {ccuū, ccdd, ccud, ccdū} or an isosinglet-isotriplet combination of these states, all carrying the same J P C . The original X(3872) exotic discovered at Belle [1] is a J P C = 1 ++ state widely believed to be ccqq, but dedicated searches for such partner states [33,34] have produced no signal. 6 Of course, any states believed to be ccss do not present this problem. Implicit in these diquark models is the assumption of the validity of a Hamiltonian approach, which in turn implies a single relevant time coordinate (as the conjugate variable to the Hamiltonian), and hence a common rest frame for the component quarks. In reality, the quarks can move relativistically, especially since the exotic states are generally created in b-quark decays or colliders, in processes accompanied by the release of large amounts of energy. In particular, the spin of a particle is measured in its rest frame, and therefore the meaning of a spin-spin operator becomes obscured in highly relativistic systems. If needed, the mathematical way forward is to employ a helicity formalism, as was most famously expounded in Ref. [51].
From a dynamical point of view, one can imagine the heavy-quark diquark and antidiquark to be fairly compact objects (tenths of a fm) 7 that achieve a substantial separation (1 fm or more) due to the large energy release, before being forced to hadronize due to confinement. In this "dynamical diquark picture" [53], the implicit rest-frame approximations of Refs. [13,15] are not wholly satisfactory, but they should nevertheless provide a lowest-order set of expectations for the spectrum of fully dynamical tetraquark states produced via the diquark-antidiquark mechanism. Moreover, the dynamical diquark picture explains why exotics have only become clearly visible in the heavy-quark sector: In the light-quark sector, the diquark-antidiquark pair never achieve sufficient separation for clear identification. In the intermediate ss case, one may discern some hints of diquark structure [54,55].
Diquark structure, via the attractive channel of two color-3 quarks into a color-3 diquark, has also successfully been used to explain the P + c pentaquark states, both in the original formulation [56] and the dynamical picture [57].
With the formalism established, it is a simple matter to enumerate the [cs][cs] diquark-antidiquark states and compute their masses using Eqs. (2)-(5). One finds the 6 s-wave and 14 p-wave states listed in Table I. The results in Table I of Ref. [14] are analogous, but once again, use a different model (as well as different numerical inputs). The mass formula obtained in the Type-II model is concise. Since q 1 = q 2 = s, the diquark masses are equal, and only one distinct spin-spin coupling κ [cs] appears: Abbreviating   (7)] to form eigenstates of C. State names used in Ref. [15] are also listed, and masses are obtained from Eq. (10).
from which the mass expressions given in the last column of Table I follow. The additional Type-I terms of Eq.
(3) could also be computed, for example, by first diagonalizing the states in a more convenient basis, using recoupling formulas like Eq. (6); however, as seen in Ref. [14], the typical contributions from these terms appear to be no more than about 20 MeV, which we can treat as a systematic uncertainty in our mass predictions. This uncertainty is indicated henceforth by the use of the symbol "≃". Using the results of Table I, one can quickly establish the mass hierarchy of states. Assuming only that k > 0 [expected from Eq. (3) to hold, inasmuch as vector diquarks are heavier than scalar diquarks], the lightest s-wave state is X 0 = |0 ++ 1 , which we naturally identify with the 0 ++ state χ c0 (3915), and hence M 0 = 3918.4 MeV [12] ≃ 3920 MeV. One also expects B ≥ 0, or else orbitally excited states would actually be lower in mass than s-wave states. Lastly, the spin-orbit coefficient α was argued in Ref. [15] to be positive, so that masses increase with L and S [as seen in Eq. (8)]; an interesting feature of this choice, as noted in Ref. [58], is that with this inverted spin-orbit coupling, states of higher J but other quantum numbers the same are lighter [compare, e.g.,

IV. ANALYSIS
The strategy for the fit is now quite straightforward. The ccss spectrum depends upon 4 parameters: the multiplet base mass M 0 , the orbital excitation coefficientB, the spin-orbit coefficient α, and the diquark spin-spin coefficient k. We have noted that 9 candidate exotics may be used to fix these parameters, and that the s-and pwave bands consist of 20 states. Therefore, the goal is to achieve a fit that predicts as many of the 9 exotics as possible, while not predicting any of the 20 − 9 = 11 states with unseen J P C values to occur in mass regions where they likely would already have been observed.
Of particular note is that only 4 1 −− states occur in the s and p waves; Ref. [15] notes that one more 1 −− state, labeled Y 5 , occurs in the f wave (s [cs] = s [cs] = 1, S = 2, L = 3), but it is most likely much heavier than the others considered here due to its high orbital excitation. That being said, at least 4 1 −− candidate states have already been observed in ISR processes: Y (4008), Y (4260), Y (4360), and Y (4660) (although Y (4008) has only been seen by Belle [59,60]). In addition, the Y (4230) seen by BESIII in e + e − → χ c0 ω [44] is necessarily a 1 −− state if formed in the s wave. On the other hand, lattice calculations, while still not in full agreement, concur that no more than one 1 −− charmonium hybrid should occur below 4.5 GeV (see, e.g., Ref. [32], which predicts it to lie at a mass of 4216 ± 7 MeV).
Also of note is that the neutral states so far lacking open-charm decays appear to fall into at least two distinct classes based upon their widths: Only Y (4008) and Y (4260) have widths > 100 MeV. One may suppose that one or both of these states are ccqq (hence possessing many more open channels and thus a larger width) or ccg hybrids (so that OZI suppression of ss annihilation is absent). The Y (4260) has been observed in the 6-quark modes J/ψ ππ and J/ψ K + K − , which speaks against a hybrid structure, and the J/ψ ππ channel speaks against a ccss structure [but see further discussion on Y (4008) later in this section]. In addition, a recent study [61] has calculated that the rate for the radiative transition Y (4260) → γX(3872) not only suggests that Y (4260) is a ccqq state like X(3872), but also that both states are compatible with having the same diquark-antidiquark wave function, except that Y (4260) carries an additional unit of orbital angular momentum. II: Predictions for s-wave (the first 6 entries) and p-wave (the remaining 14 entries) ccss diquark-antidiquark state masses (in MeV), following the notation from Table I. Inputs are labeled by "*". A "?" indicates when more than one good assignment is possible. In terms of Eq. (10), the parameter fit values in MeV are M0 = 3920, k = 220,B = α = 90. State Pred. Mass Observed

B. s-Wave States
Of little ambiguity is the necessity of assigning χ c0 (3915) the role of the ccss band ground state X 0 = |0 ++ 1 , which according to Table I immediately fixes the parameter M 0 ≃ 3920 MeV. The full set of mass predictions is presented in Table II.
Beyond this start, however, hints from the exotic state decay modes become essential. Perhaps the other states most essential to describe as ccss are those observed to decay into J/ψ φ, namely, Y (4140), Y (4274), and X(4350). Assuming that Y (4140) is the X 1 = |1 ++ , then using Table I one chooses k ≃ 220 MeV, which not only resolves the absence of this state from γγ production via the Landau-Yang theorem, but also allows immediate prediction of all the other masses in the s-wave band. In particular, one finds a degenerate state Z = |1 +− 1 at 4140 MeV and another Z ′ = |1 +− 2 at 4360 MeV; note that the known neutral isotriplet J P C = 1 +− states Z 0 c (4025), Z 0 c (3900) lie rather lower in mass. Additionally, one finds two more degenerate states at 4360 MeV, X ′ 0 = |0 ++ 2 and X 2 = |2 ++ . Either of these is an excellent candidate for the X(4350) found in γγ production.
Returning to the Y (4140), one may use Eq. (6) to find that the state X 1 has solely s cc = s ss = 1 content. At the quark level, one expects γγ fusion to produce one of the quark-antiquark pairs first (and necessarily with J P C = 0 ++ or 2 ++ ), and the other pair to be produced as the result of bremsstrahlung from one of the initial quarks. Thus, even at the quark level, one sees the production of such a state to be problematic. According to Table I or II, the s-wave states are highly degenerate and obey a simple equal-spacing rule (in k). Note that no s-wave state therefore carries a mass close to that of Y (4274), reported by CDF [37] as 4274.4 +8.4 −6.7 ± 1.9 MeV, and by CMS [39] as 4313.8 ± 5.3 ± 7.3 MeV. Fitting to the p-wave states requires input from the ISR state masses, as discussed below. Then, the sole potential candidate for the first mass is |2 −+ 2 at 4270 MeV, while the second mass can be accommodated by either |0 −− or |0 −+ 1 at 4320 MeV. In the first case, a lighter |2 −+ 1 state occurs at 4050 MeV, which lies below the 4116 MeV J/ψ φ threshold and therefore could easily have escaped detection to now. In all cases, however, the fact that none of these states have J P C = 0 ++ or 2 ++ means that they cannot be created in γγ fusion, in agreement with observation.
Before leaving the s-wave band, let us note interesting properties of the χ c0 (3915) under this assignment. We have seen in the previous section that its mass lies just below the D + sD − s threshold 3937 MeV. However, it is extremely problematic to identify χ c0 (3915) as a D sDs molecule (which was proposed in Ref. [62]) held together by meson exchanges, again using a fact noted in the previous section: D + s andD − s are J P = 0 − states, and coupling to a 0 − meson (presumably η) is forbidden by Lorentz symmetry plus P invariance. Should χ c0 (3915) prove to be a ccss state, it is almost certainly not a hadronic molecule. The closeness of the χ c0 (3915) mass to the D + sD − s threshold need not be considered an unnatural coincidence, as the so-called "cusps" due to such thresholds have been shown to be effective in attracting nearby states, in particular for heavy-quark states [63,64].
Second, we have noted that the only OZI-allowed and phase-space allowed decay mode for a ccss state of this mass is η c η. We propose that this is the dominant χ c0 (3915) decay mode. The recombination of quark spins for the X 0 state according to Eq. (6) gives meaning that the J/ψ modes, if kinematically allowed, are more probable by a factor 3. Likewise, the η wave function is only fractionally ss: The decay χ c0 (3915) → η c η is otherwise a simple 2-body decay of a scalar to two (pseudo)scalars, and therefore its width is of the form where M is the χ c0 (3915) mass and p = 665.0 MeV is the magnitude of the spatial momentum for the 2body decay. The invariant amplitude M is seen to have dimensions of mass; with Γ = 20 MeV, one finds |M| = 3.4 GeV. When the suppression factors suggested by Eqs. (11)- (12) are removed, the "natural" amplitude for the process is about 8.3 GeV, a substantial number that suggests the sole decay already observed, χ c0 (3915) → J/ψ ω, can occur at a reasonable rate if the ω contains a phenomenologically acceptable ss component. For example, if the non-ideal mixing ǫ of ω is parametrized as then using Eq. (11) and the same value of |M|, one finds Γ(χ c0 (3915) → J/ψ ω) = 29.9 sin 2 ǫ MeV, which for, e.g., ǫ = 10 −3 gives Γ = 29.9 eV. As mentioned above, the size of the J/ψ ω branching fraction for χ c0 (3915), given in Ref. [23] in the form is considered too large to be compatible with the expected size of OZI-suppressed decays of conventional charmonium. If χ c0 (3915) is a ccss state, then OZI violation is evaded if the decay mode is accomplished through the presence of a small valence ss component in the ω, which means non-ideal ω-φ mixing. This effect has been considered in heavy-quark decays such as D + s → ωe + ν e [65]. It might, however, be more complicated in the 4-quark environment in the sense that ω-φ mixing influenced by final-state interactions can have a significantly different strength than in exclusive processes in which ω is the only hadron present.

C. p-Wave States
Let us now turn to the p waves. We have already fixed 2 of the 4 model parameters, M 0 and k, from the s waves. When including the p waves, we find that the fits best representing the known spectrum and introducing fewer light unknown states leave out Y (4260) and keep Y (4008). We have remarked above that these are the two widest neutral charmoniumlike states, and are therefore the best candidates for ccqq, and also that the mode Y (4260) in particular is very unlikely to be purely ccss. Therefore, in the fit we present in Table II, the Y (4260) is excluded.
The closeness of the Y (4008) and χ c0 (3915) masses has an additional peculiar effect. If one identifies Y (4008) as the lightest J P C = 1 −− ccss state Y 1 = |1 −− 2 , then the fit in Table II givesB = α, or using Eq. (9), B c = 0 in the original notation of Eq. (5), which means that the only orbital coupling appears through the spin-orbit term.
In fact, the actual fit in Table II does not choose Y (4008) as an input, but rather chooses Y (4230) = Y 2 = |1 −− 1 and Y (4360) = Y 3 = |1 −− 3 to fixB = α = 90 MeV. Then, the prediction of Y (4008) as Y 1 and Y (4660) as Y 4 = |1 −− 4 is noteworthy. An additional feature commending this choice is that Eq. (6) can again be used to show that Y 2 contains only terms in which s cc = s ss = 1, very much in agreement with the Y (4230) so far being seen only in the χ c0 ω channel: The preferred decay mode would be χ c0 φ, but its threshold is 4434 MeV, so again we suggest that Y (4230) is a ccss state that can decay via ω-φ mixing.
Since the D ( * )+ sD ( * )− s thresholds occur at 3937 MeV, 4081 MeV, and 4224 MeV, one would expect these "fallapart" modes to be the dominant ones for many of these states, particularly higher ones such as Y (4660). However, it is worth noting that the best current data for e + e − → D ( * )+ sD ( * )− s [66] is only sensitive to the conventional charmonium ψ states; none of the exotics have yet been seen to decay to charm-strange states. Moreover, should the dynamical diquark picture [53] hold, such that more highly energetic states entail greater separation of the diquarks and therefore suppressed hadronization matrix elements, one then has a natural mechanism for suppressing their decay widths beyond naive expectations.
Lastly, this work presents only one of many possible fits to the known exotic states lacking open-charm decays. Several other possibilities can occur, such as, e.g., identifying the high-mass 1 −− Y (4660) state as the first in the f -wave (L = 3) band (s [cs] = s [cs] = 1, S = 2, called Y 5 in Ref. [15]).

V. CONCLUSIONS
Based on interesting patterns in the phenomenology of the charmoniumlike states observed to date, we propose that the J P C = 0 ++ state χ c0 (3915) is the lightest ccss state. Its lack of observed D ( * )D( * ) decays argue against it being either the conventional cc state χ c0 (2P ) or a light-quark containing ccqq exotic state, and its single known decay mode J/ψ ω can be understood as the ω having a small (non-ideal mixing) ss component.
Furthermore, as a ccss state lying slightly below the D sDs threshold, the χ c0 (3915) is very unlikely to be a loosely bound molecule, and we therefore analyze it as a diquark-antidiquark state. Indeed, a state with J P C = 0 ++ in the mass region ∼ 3900 MeV is precisely where the lightest ccss state was expected in previous studies. Importantly, even if χ c0 (3915) turns out not to be ccss, states with this quark content should certainly appear in the same mass range as some of those already observed. To emphasize: One expects ccss states to occur in the same range as other charmoniumlike states; and even if the particular assignments in this paper are later disfavored, the analysis leading to Table I still holds. Under the current hypothesis, however, some remarkable identifications arise. The Y (4140), a J/ψ φ enhancement seen in B decays, is naturally a 1 ++ ccss state which, by the Landau-Yang theorem, is naturally absent from γγ production experiments (as is the case). The X(4350), Y (4274), and several of the J P C = 1 −− Y states arise naturally at masses predicted for ccss states, and no unwanted extra states that would already likely have been observed appear to occur.
The most flexible part of the identification-both experimentally and theoretically-occurs in the 1 −− sector: If so many of these states are ccss, what has happened to the expected ccqq states? We have argued that Y (4260) is almost certainly ccqq and is quite broad; one can imag-ine that the higher ones are broader still, and thus difficult to discern. Indeed, the very broad Y (4008) might also be ccqq, and either the true lowest 1 −− ccss state is obscured by it, or does not occur until it appears as Y (4230). In any case, subsequent experiments will certainly clarify the true nature of the full spectrum, and ccss states will certainly play a role.
During the finalization of this paper, DØ announced [67] the observation of a new state in the channel B 0 s π ± , while a preliminary analysis by LHCb found no evidence for such a state [68]. Such a novel exotic flavor structure, a tetraquark with only one heavy quark (bsud for π + ), is expected to produce two states close in mass (with J P = 0 + , 1 + ) due to heavy-quark fine structure. In particular, if confirmed, it would be the first tetraquark not simply of the bbqq or ccqq type, which makes studies of new flavor structures like ccss all the more timely. Indeed, Ref. [67] suggests the same type of tetraquark paradigm as discussed here as being the most likely structure.