I. INTRODUCTION

One of the greatest advances of particle physics in the 21 st century has been the discovery and confirmation of scores of exotic multiquark hadrons [1]. Most of these states appear in the hidden-charm sector (both nonstrange and strange, both tetraquarks and pentaquarks), but others have been identified in the bottomoniumlike system and even in the open di-charm system (T þ cc , [2,3]). All of these states share the property of containing heavy (c, b) quarks, a feature that has been instrumental in identifying them as exotic. For example, a state like Z c ð3900Þ þ with mass that lies among the conventional charmonium states (and which decays to conventional charmonium, in this case J=ψ) but that also carries nonzero electric charge is clearly exotic, since its minimal valencequark content is ccu d. Likewise, quark potential models have quite successfully predicted the entire cc spectrum up to the open-flavor D D threshold, and so the numerous J PC ¼ 1 --states appearing in addition to the predicted conventional charmonium ψ states [such as Yð4230Þ] are interpreted as exotic.

Part of this success in the heavy-quark sector is by experimental design (the B-factory experiments BABAR and Belle produced large, clean samples of B → cc decays. BESIII is a tau-charm factory. LHCb is constructed to produce enormous data sets of hadronic decays). But of course, huge samples of lighter (u, d, s) hadrons have been collected for decades, and yet precious few clear signals of exotic hadrons have emerged. 1 The quark-flavor universality of QCD demands that any dynamical phenomena observed for heavy quarks should appear for light quarks as well. But the appearance of heavy quarks Q provides several advantages over their lighter counterparts in the production of experimentally identifiable exotic candidates [4]. They satisfy Λ QCD =m Q ≪ 1, which leads to relatively larger mass splittings between multiplets and hence more cleanly separated states (whereas light-quark hadrons form a dense forest of states up to ≈2.5 GeV). Their Fermi kinetic energy is suppressed by Λ QCD =m Q , making it easier for the heavy quarks to nucleate clusters with the light quarks, as well as often suppressing decay widths, and as noted above in the case of Z c ð3900Þ þ , the simple presence of both heavy and light quarks allows more valence-quark combinations that can be unambiguously identified as exotic.

The utility of heavy quarks in forming identifiable multiquark exotics motivates the development of the dynamical diquark model [5,6]. Here, the heavy quarks nucleate diquarks δ ≡ ðQqÞ¯3 and antidiquarks δ ≡ ð Q qÞ 3 via the attractive 3 × 3 → 3 short-distance color interaction (and its conjugate). As long as the δ and δ quasiparticles form with characteristic sizes smaller than the δδ separation-which can be realized, for example, in high-energy collider environments-then the full resulting tetraquark state can assume a δδ configuration. Likewise, pentaquarks can be formed in this model through the formation of a diquark and a triquark, θ ≡ ½ Qðq 1 q 2 Þ¯3 3 [7].

This particular implementation of diquark substructure within multiquark hadronic states differs greatly from its original application [8], which addressed peculiarities in the spectrum of light scalar mesons such as f 0 ð980Þ and a 0 ð980Þ. For example, the distinction between "good" (J P ¼ 0 þ ) and "bad" (J P ¼ 1 þ ) light diquarks [9] central to studies of light tetraquarks is suppressed by effects of Oð1=m Q Þ for diquarks containing a heavy quark Q.

The original dynamical diquark model has successfully been applied to multiple flavor and spin sectors: the ccq q0 multiplet average masses [10], the fine structure of the ccq q0 ground-state [11] and orbitally excited [12] multiplets, b bq q0 and ccss [13], cccc [14] and other all-heavy tetraquarks [15], ccs q [16], T þþ cc [17], and ccq 1 q 2 q 3 pentaquarks [18]. In each case where data are available, the model is supported by experimental findings; furthermore, it provides multiple predictions for yet-unseen states.

However, as already noted, the presence of heavy quarks is central to the argument for the physical plausibility of the dynamical diquark model. The key question to be addressed in this paper is the extent to which the model can provide a meaningful description of multiquark exotic states (specifically tetraquarks) in which the heaviest quarks are strange. Of course, since m s =Λ QCD ¼ Oð1Þ, the usual arguments are no longer valid, and thus, this work provides tests of the most marginal cases to which the original δδ picture might apply. Nevertheless, lattice-QCD studies such as [19,20] have demonstrated the existence of hybrid mesons with strange-quark content that feature explicit gluonic excitations coupled to the ss pair. While such simulations do not explicitly invoke a color flux-tube picture, they clearly show that nontrivial gluonic degrees of freedom persist in the strange sector and contribute to hadron structure. These results support the physical plausibility of extended gluonic fields mediating interactions beyond simple quark models, which is a key assumption underlying Born-Oppenheimer-inspired frameworks such as the dynamical diquark model. Although hybrids and tetraquarks differ in their quark content and color structure, the demonstrated persistence of gluonic-excitation effects lends credence to the existence of exotic tetraquark states with strange quarks that utilize this structure.

Such states are expected to have masses in the range of 2.0-2.5 GeV and are particularly relevant to the experimental programs of BESIII, JLab (particularly GlueX), and eventually the EIC. They likely coexist amongst excited conventional ss mesons, ssg hybrids, and glueballs, making a detailed study to disentangle these possibilities a project of both current theoretical and experimental interest. Since our numerical inputs (e.g., diquark masses) are taken from calculations using other approaches that respect QCD dynamics (lattice, QCD sum rules, quark models), our primary message is this: If the flavor universality of QCD allows the same dynamical mechanisms to persist in the strange sector, then this work gives their spectrum and the mass range in which to look for them.

This paper is structured as follows. In Sec. II, we enumerate and discuss experimentally observed mesons in the region of 2 GeV that potentially could contain a significant ssss or ssq q component, such as ϕð2170Þ, f 2 ð2340Þ, and Xð2370Þ. Section III describes the dynamical diquark model, the formalism in which the ssss and ssq q systems are analyzed, and introduces the minimal set of fine-structure effects responsible for the level structure of these systems. The numerical inputs used to produce our phenomenological analysis, as well as the detailed results of that analysis, appear in Sec. IV. We summarize our findings and describe future prospects for the study of these states in Sec. V.

II. POSSIBLE ssss AND ssq q CANDIDATES

The (unflavored) hadron spectrum above 2 GeV presented in the Particle Data Group (PDG) summary table [1] contains several meson resonances that challenge the traditional quark-antiquark (q q) picture. Isoscalar meson resonances such as ϕð2170Þ (hidden ss, J PC ¼ 1 --), f 0 (0 þþ ), and f 2 (2 þþ ) may incorporate exotic structures such as glueballs, hybrids, or tetraquarks.

The state ϕð2170Þ [or Yð2175Þ] was first observed by the BABAR Collaboration [21] in the process e þ e -→ ϕf 0 ð980Þ, and was subsequently confirmed by BES [22] and Belle [23]. This resonance has been confirmed by the BESIII Collaboration in several independent channels, notably e þ e -→ ϕf 0 ð980Þ [24], ϕη [25], and ϕη 0 [26]. These channels exhibit clear resonant structures around 2.17 GeV in the invariant mass spectra, in agreement with the earlier observations by BABAR, BES, and Belle. The consistency of mass and width measurements across these processes, summarized by the PDG as 2146 AE 6 MeV and 106 þ24 -18 MeV, respectively [1], strongly supports the existence of this resonance. The BESIII results suggest a nontrivial internal structure for ϕð2170Þ, making it a strong candidate for exotic interpretations such as a tetraquark state or a hybrid meson, rather than as a simple radial excitation of the vector meson ϕð1020Þ. Its internal structure has been shown to conflict with a hybrid-meson hypothesis in the analysis of Ref. [26]. In addition, the mass of the lowest 1 -- glueball candidate in lattice QCD (LQCD) is estimated to be much higher, around 3.8 GeV [27][28][29].

The 2 þþ f 2 ð1910Þ and f 2 ð1950Þ resonances have long been considered potential non-q q candidates. f 2 ð1950Þ, in particular, is a well-established tensor resonance with an unusually large width of 464 AE 24 MeV.

The observations that the resonance f 0 ð2020Þ decays to η 0 η 0 [30] and that f 2 ð2010Þ decays to KK [31] and ϕϕ [32], suggest that both states likely possess significant ss content. The proximity of the ϕϕ threshold at 2.039 GeV indicates the possibility that one or both of the resonances could be either a ϕϕ molecular state or a hybrid meson resulting from the gluonic excitation of an ss pair.

The tensor (J PC ¼ 2 þþ ) resonances f 2 ð2300Þ and f 2 ð2340Þ are of distinct interest in the study of exotic hadrons, especially glueballs. They have both been observed in radiative J=ψ decays, particularly in channels such as J=ψ → γϕϕ [32] and γηη [33], where substantial gluonic couplings are expected. In particular, LQCD simulations [27][28][29] predict the lightest tensor glueball to lie near 2.4 GeV, and the masses of these f 2 states are consistent with such expectations. On the other hand, the observed decay channels of f 2 ð2300Þ, namely, ϕϕ, Λ Λ, and K K, suggest a significant ss content. The resonance f 2 ð2340Þ exhibits a notably broad width (331 MeV), suggesting an exotic nature beyond the conventional q q picture. Nevertheless, the measured production rate of f 2 ð2340Þ radiative decays such as via J=ψ → γη 0 η 0 [30] is significantly lower than the production rate expected from LQCD for a pure tensor glueball [34]. Recent phenomenological analysis [35] based upon a chiral effective model [36] suggests that another broad 2 þþ state, f 2 ð1950Þ, could be a glueball-rich resonance. Despite such insights, a definitive interpretation of these states remains elusive due to an abundance of overlapping resonances combined with limited experimental resolution.

We also comment on the X states with masses ≤ 2.5 GeV observed by the BESIII Collaboration. The state Xð2370Þ is given in the PDG particle listing, but not in the summary table [1]. Initial evidence for this state emerged in 2011 through analysis of the decay channel J=ψ → γπ þ π -η 0 [37]. Subsequent studies by BESIII have confirmed the existence of Xð2370Þ in additional decay modes, including J=ψ → γK þ K -η 0 and J=ψ → γK 0 S K0 S η 0 [38]. These observations allowed for detailed measurements of the resonance's properties such as its mass and width, and determined its J PC to be 0 -þ [39]. Its mass makes it a plausible candidate for the 0 -þ glueball predicted by LQCD, which occurs near the somewhat higher value of 2.6 GeV [27][28][29]. The BESIII Collaboration also reported two other resonances with masses closer to the LQCD prediction (with undetermined J PC , but 0 -þ being at least somewhat favored): Xð2500Þ [32] and Xð2600Þ [40]. One further pseudoscalar state in the PDG light-unflavored meson list is Xð1835Þ. It was first reported by BESII in the radiative decay J=ψ → γp p [41], later confirmed in J=ψ → γπ þ π -η 0 [42], and its quantum numbers 0 -þ were determined in Ref. [43].

Continued investigation into hidden-strange tetraquark candidates over a broader range of energies and decay modes-beyond just the ϕπ channel [44] in the case of isovector (Z s ) candidates-is crucial for disentangling their internal structure and for distinguishing them from conventional or hybrid ss resonances.

III. THE DYNAMICAL DIQUARK MODEL

The dynamical diquark model [5,6] provides a theoretical framework for describing exotic hadrons-such as tetraquarks and pentaquarks-as bound systems of compact diquarks and antidiquarks (or triquarks), dynamically confined by the gluonic field. Unlike static constituent models, this approach treats the relative motion of the color sources explicitly, allowing a more realistic description of spatial separation and interaction dynamics.

A central ingredient of the model is the application of the Born-Oppenheimer (BO) approximation, in which the quickly changing gluonic field is treated separately from the slower motion of the heavy constituents. For a given gluonic excitation labeled by quantum numbers Λ ϵ η (e.g., Σ þ g , Π - u ) defined later in this section, the diquark-antidiquark pair at separation r moves in an effective potential V Λ ϵ η ðrÞ determined by LQCD simulations.

To calculate the mass spectrum of such exotic states, one solves the radial Schrödinger equation for the relative motion of the diquark (δ) and antidiquark ( δ) in the corresponding BO potential,

where μ ¼ m δ m¯δ m δ þm¯δ is the reduced mass of the δδ system, L is their relative orbital angular momentum, and V Λ ϵ η ðrÞ is the BO potential extracted from LQCD for the given gluonic excitation. The choice of BO potential depends upon the quantum numbers of the gluonic field, which are classified using the symmetries of the cylindrical group D ∞h , analogous to those used in diatomic molecular physics.

The total mass of the tetraquark state is then given by

where m δ and m¯δ are the constituent diquark and antidiquark masses, and E nL is the eigenvalue obtained from solving Eq. ( 1).

In addition to this leading-order (muliplet-average) mass spectrum obtained [10] by solving the radial Schrödinger equation Eq. ( 1), fine-structure effects arising from spindependent interactions between δ and δ must be included in order to reproduce the observed multiplet splittings [11,12]. These effects, including spin-spin, spin-orbit, and tensor couplings, are treated perturbatively and induce the following terms within the Hamiltonian:

with multiplet-average Hamiltonian H 0 , diquark-spin operators s δ ≡ s q þ s s , s¯δ ≡ s q þ s s, total quark spin operator S ≡ s δ þ s¯δ, and orbital angular momentum operator L. The tensor operator S δ δ 12 is defined by

For simplicity, isospin effects are neglected in this work (and indeed, are irrelevant for ssss states). Such operators are well known, and their matrix elements have been computed in many places. In the context of the dynamical diquark model, they are collected in Refs. [12,14]. For completeness, we restate them here,

where J is the quantum number corresponding to

where ½j ≡ 2j þ 1, parentheses containing six elements denote a 3j symbol, and braces denote a 6j or 9j symbol, as indicated by the number of elements. The reduced matrix elements of the canonically normalized angular-momentum generators j are given by

The operators σ in Eq. ( 7) are twice the relevant canonically normalized angular-momentum generators, thus generalizing the familiar relation for spin-1 2 Pauli matrices. In the dynamical diquark model, each state is characterized by radial quantum number n, orbital angular momentum L, total quark spin S, and total angular momentum J quantum numbers, where J ¼ L þ S. These quantum numbers are combined with the Born-Oppenheimer (BO) quantum numbers Λ ϵ η , where Λ denotes the projection of the gluonic angular momentum onto the δδ axis r (Λ ¼ 0; 1; 2; …, corresponding to Σ; Π; Δ; …), η ¼ gðuÞ labels even(odd) behavior under CP inversion through the midpoint of r (the gluon-field flux tube), and ϵ ¼ AE for Λ ≥ 1 is the eigenvalue of reflection through any plane containing r.

The full state is expressed in the following form:

where the spectroscopic notation 2Sþ1 L J identifies the spin and orbital structure, and Λ ϵ η specifies the BO potential.foot_0 Phenomenologically, all known heavy exotics to date can be accommodated just within the lowest orbitals of the groundstate BO multiplet Σ þ g , specifically Σ þ g ð1S; 1P; 2S; 2PÞ [10]. Thus, the excited BO potentials (Π þ g , etc.) are relevant to a full analysis of all possible states [6] (and represent the hybrid analogues for tetraquark states), but for this work we restrict to only the lowest BO potential, Σ þ g . Then, each such state can be labeled by J PC , where the parity (P) and charge-conjugation (C) quantum numbers (for neutral tetraquarks) are given by

In the S-wave case (L ¼ 0), all 6 ssq q states (including two states apiece for J PC ¼ 0 þþ and 1 þ-) have P ¼ þ1. They are denoted in the basis js δ ; s¯δ > J as follows [45]:

The s q q þ s ss eigenvalues required by Eq. ( 10) to compute the C eigenvalues given in Eqs. (11) are obtained by recoupling the spins of the quarks within s δ , s¯δ to the basis s q q; s ss through the application of 9j symbols [6],

hðs q s qÞs q q; ðs s s sÞs ss ; Sjðs q s s Þs δ ; ðs qs sÞsδ; Si ¼ ð½s q q½s ss ½s δ ½s¯δÞ 1=2

> < > :

s q s q s q q s s s s s ss s δ s¯δ S

> = > ;

:

The color-3 structure of each diquark, as assumed by the dynamical diquark model, contributes to antisymmetry under particle exchange. Thus, whenever a diquark contains two identical quarks, such as for the fully strange tetraquark δ δ ¼ ðssÞðs sÞ, the remaining quantum numbers for each of δ and δ must be completely symmetric under quark exchange in order to satisfy the Pauli exclusion principle. Assuming that δ and δ both lie in the spatial ground-state (hence symmetric) configuration, then both are restricted to be spin-symmetric axial-vector diquarks (s δ ¼ s¯δ ¼ 1). In this case, the only surviving three S-wave states in Eqs. (11) for the fully strange tetraquarks are X 0 0 ; Z 0 ; X 2 . Note, in particular, the absence of an S-wave 1 þþ state.

For the P-wave states, P ¼ -1. The possible total angular momenta are J ¼ jL -Sj; …; L þ S, where the total quark spin S here is just J from the prior S-wave case. For S ¼ 0 (states based upon X 0 0 ), J can only be 1, and hence, one obtains a state [using Eqs. (10)] with J PC ¼ 1 --. For S ¼ 1 (states based upon Z 0 ), the allowed values are J ¼ 0, 1, 2, giving states with J PC ¼ ð0; 1; 2Þ -þ . For S ¼ 2 (states based upon X 2 ), the allowed values are J ¼ 1, 2, 3, giving the states J PC ¼ ð1; 2; 3Þ --. We note that two distinct 1 --states arise in this multiplet, one from S ¼ 0 and one from S ¼ 2. One thus obtains the following seven distinct P-wave states:

This enumeration of states is identical to that for the cccc system obtained in Ref. [14]. In the case of isoscalar ssq q tetraquarks, the number of possible P-wave states increases to 14 [6]. This set consists of states with multiple copies, 2 × ð0; 1; 2Þ -þ , ð2 × 2; 4 × 1Þ --, in addition to ð0; 3Þ --, all of which arise from combining distinct total quark-spin eigenvalues S ¼ 0, 1, 2 with orbital angular momentum L ¼ 1. Notably, the exotic quantum numbers 0 --, which are forbidden for both conventional q q and ssss mesons, become accessible for ssq q.

For the D-wave states, P ¼ þ1. For S ¼ 0 (states based upon X 0 0 ), J can only be 2, and hence, one obtains a state [using Eqs. (10)] with J PC ¼ 2 þþ . For S ¼ 1 (states based upon Z 0 ), the allowed values are J ¼ 1, 2, 3, giving the states J PC ¼ ð1; 2; 3Þ þ-. For S ¼ 2 (states based upon X 2 ), the allowed values are J ¼ 0, 1, 2, 3, 4, giving the states J PC ¼ ð0; 1; 2; 3; 4Þ þþ . We note that two distinct 2 þþ states arise in this multiplet, one from S ¼ 0 and one from S ¼ 2. One thus obtains the following nine distinct D-wave states:

This enumeration of states is identical to that for the cccc system obtained in Ref. [14]. For the isoscalar ssq q system, seven additional states are anticipated [6],

which duplicate J PC values already present in Eq. ( 14).

Further refinements, such as identifying hierarchies in the sizes of fine-structure interactions, serve to improve the predictions, at least in terms of correctly ordering the masses of these states within multiplets of the dynamical diquark model.

IV. RESULTS

To produce tetraquarks within the dynamical diquark model, we adopt a variant of the Cornell potential that incorporates a Coulombic term at small δδ separation r, a linear confining term at large r, and an offset V 0 ,

where the parameters extracted from LQCD calculations [46] (similar results can be obtained from Ref. [47]) and from the relativistic quark model (RQM) [48] are presented in Table I.

In accordance with the RQM framework [48], the value of α depends upon the number of active quark flavors in the β function and the specific values of quark masses (m q ¼ 0.33 GeV, m s ¼ 0.50 GeV); from Ref. [48], we use α ¼ 0.948 for the ðsqÞðs qÞ diquark potential and α ¼ 0.975 for the ðssÞðs sÞ diquark potential. The ðsqÞ diquark mass is approximated using the spin-averaged form,

as obtained from the corresponding diquark masses [48] m S¼0 δ¼ðsqÞ ¼ 0.948 GeV and m S¼1 δ¼ðsqÞ ¼ 1.069 GeV. Similar also arise from QCD sum rules [49]. We use the findings of Ref. [50] to estimate diquark masses within the context of LQCD. In summary, (1) The mass difference between a 0 þ diquark and pair of quarks q in the chiral limit (with inverse lattice spacing a -1 ¼ 2.12 GeV) is given by

Approximating the (sq) constituent quark mass as m q ¼ 0.4 GeV (roughly averaging m q and m s ), Eq. ( 18) yields an estimate for the 0 þ ðsqÞ diquark mass,

(2) The mass difference between the 1 þ ("bad") and 0 þ ("good") (sq) diquarks is given by m S¼1 δ¼ðsqÞ -m S¼0 δ¼ðsqÞ ¼ 0.162 GeV: ð20Þ

TABLE I. Numerical values of the parameters in Eq. (16). Potential V 0 (GeV) α σ (GeV 2 ) LQCD [46] -0.383 0.297 0.216 RQM [48] -0.300 0.948, 0.975 0.180

Using these values in Eq. ( 17), we obtain

For the ðssÞ case, we use m s ¼ 0.50 GeV and estimate the 1 þ ðssÞ diquark mass using the given a -1 value and Eqs. ( 18) and (20) [the latter assumed to be applicable also for ðssÞ, since LQCD can treat the two s quarks as distinguishable],

The corresponding RQM diquark mass [48] is

We compute the tetraquark spectrum within the framework of the dynamical diquark model, using the values in Eqs. ( 17), ( 21)-( 23) of m δ ¼ m¯δ and the binding energies E nL as inputs to Eq. ( 2). In turn, the E nL eigenvalues are obtained by solving the Schrödinger equation Eq. ( 1) with the potential VðrÞ defined in Eq. ( 16), using the numerical values for its parameters listed in Table I. The resulting spectra for the ðsqÞð q sÞ and ðssÞðs sÞ systems (subsequently labeled as ssq q and ssss, respectively) are presented in Tables II and III (and graphically represented in Figs. 1 and 2), respectively. These results do not include the spin-dependent fine-structure interactions described by Eq. ( 3), as the corresponding couplings sq , V LS , and V T are currently unknown. However, to estimate the impact of such effects, we report the parametric mass splittings of P-and D-wave ssss tetraquarks in Table IV [The S-wave ssss states are degenerate, with common mass M 0 ð1SÞ þ κ ss ], and discuss their pattern later in this section. TABLE II. Masses M 0 and corresponding J PC eigenvalues of ssq q dynamical-diquark states, computed using the given diquark masses m δ and the nL eigenstates of the potential Σ þ g [Eq. ( 16)] obtained from the indicated sources. Also computed are matrix elements indicating the spatial extent of the states. Fine-structure effects have not yet been included.

LQCD [46] 0.710 1.842 0.404 0.535 2 × 0 þþ ; ð1; 2Þ þþ ; 2 × 1 þ- RQM [48] 1.039 1.954 0.272 0.380 Σ þ g ð2SÞ LQCD [46] 0.710 2.592 0.608 0.977 RQM [48] 1.039 2.721 0.504 0.826 Σ þ g ð1PÞ LQCD [46] 0.710 2.311 0.696 0.802 4 × 1 --; 2 × 2 --; 2 × ð0; 1; 2Þ -þ ; ð0; 3Þ -- RQM [48] 1.039 2.536 0.576 0.673 Σ þ g ð2PÞ LQCD [46] 0.710 2.942 0.878 1.182 RQM [48] 1.039 3.082 0.764 1.043 Σ þ g ð1DÞ LQCD [46] 0.710 2.689 0.933 1.026 4 × 2 þþ ; 2 × ð1; 3Þ þþ ; ð0; 4Þ þþ ; 2 × ð1; 2; 3Þ þ- RQM [48] 1.039 2.895 0.815 0.901 TABLE III. Masses M 0 and corresponding J PC eigenvalues of ssss dynamical-diquark states, computed using the given diquark masses m δ and the nL eigenstates of the potential Σ þ g [Eq. ( 16)] obtained from the indicated sources. Also computed are matrix elements indicating the spatial extent of the states. Fine-structure effects have not yet been included.

LQCD [46] 0.950 2.219 0.359 0.478 0 þþ ; 1 þ-; 2 þþ RQM [48] 1.203 2.183 0.245 0.345 Σ þ g ð2SÞ LQCD [46] 0.950 2.909 0.547 0.881 RQM [48] 1.203 2.956 0.471 0.771 Σ þ g ð1PÞ LQCD [46] 0.950 2.656 0.627 0.723 ð0; 1; 2Þ -þ ; ð2 × 1; 2; 3Þ -- RQM [48] 1.203 2.786 0.536 0.628 Σ þ g ð2PÞ LQCD [46] 0.950 3.233 0.793 1.068 RQM [48] 1.203 3.316 0.718 0.983 Σ þ g ð1DÞ LQCD [46] 0.950 3.004 0.843 0.928 ð0; 1; 2 × 2; 3; 4Þ þþ ; ð1; 2; 3Þ þ- RQM [48] 1.203 3.141 0.767 0.849

We also present results for expectation values relevant to the size of the state in Tables II and III. While hri provides an estimate of the overall spatial extent of the state, h1=ri -1 probes the short-distance structure of the wave function and is particularly informative in the presence of a Cornell-type potential. In this framework, the Coulombic term -α=r dominates the short-distance interaction, while the linear term σr governs long-range confinement. Thus, h1=ri -1 effectively measures the average strength of the short-range attraction, while hri captures the extent of the confining color flux tube. Such distinctions are useful for differentiating compact δ δ states from hadronic molecules, which generally exhibit both a larger hri and a larger h1=ri -1 (typically ≳1 fm).

To interpret our results, we compare the predicted tetraquark masses with available experimental candidates and examine their compatibility with observed resonances. Additionally, we contrast our findings with those obtained from alternative theoretical approaches, such as LQCD, QCD sum rules, and constituent-quark models, in order to assess the robustness and distinctive features of the dynamical-diquark framework.

(a) 1S ssq q Tetraquarks. The mass spectrum and quantum numbers presented in Table II suggest that the observed 2 þþ mesons f 2 ð1910Þ and f 2 ð1950Þ are promising candidates for Σ þ g ð1SÞ ssq q tetraquark states. This assignment gains further plausibility when considering the difficulty of placing these states within conventional q q meson nonets (Sec. 63 of Ref. [1]). In particular, the large f 2 ð1950Þ decay width (464 MeV) and its observation in the K K decay channel [53] suggest that this meson possesses exotic quark content, with a significant ss component.

It is worth mentioning that the Σ þ g ð1SÞ mass prediction using the LQCD potential of Ref. [46] aligns with that of Xð1835Þ (which is also close to the p p threshold). Nevertheless, its quantum numbers 0 -þ [54] (in particular, P ¼ -1) disfavor its interpretation as an S-wave ssq q tetraquark within the dynamical diquark model.

(b) 1S ssss Tetraquarks. The predicted mass eigenvalue and allowed J PC values in Table III suggest that one of the observed 2 þþ mesons f 2 ð2300Þ and f 2 ð2340Þ is a ssss 1S tetraquark. It is worth noting that the latter state is also predicted to be an ssss tetraquark in Ref. [55].

(c) 1P ssq q Tetraquarks. The mass eigenvalue for Σ þ g ð1PÞ listed in Table II is close to that of Xð2370Þ as well as to Xð2500Þ and Xð2600Þ (depending upon the particular potential used), although the quantum numbers J PC require experimental confirmation. A recent potentialmodel analysis [56] shows good agreement with our results, predicting the lightest P-wave states 0 -þ and 1 --to lie at 2.3 GeV and 2.6 GeV, respectively. According to a recent QCD sum-rule analysis [57], Xð2370Þ is also interpreted as a 0 -þ P-wave ssq q tetraquark. However, Ref. [55] interprets Xð2500Þ as a 0 -þ ssss FIG. 2. Spectrum of ssss tetraquarks within the dynamical diquark model (red lines), not yet including fine-structure effects, compared to the results of calculations in a potential model [51] (triangles) and using QCD sum rules [52] FIG. 1. Spectrum of ssq q tetraquarks within the dynamical diquark model, not yet including fine structure. The specific mass eigenvalues presented in this figure are computed using the potential of Ref. [46].

parameters, V LS ≈ 29 AE 5 MeV and V T ≈ -1 AE 2 MeV, can be obtained from QCD sum-rule results for the tetraquark mass spectrum given in Ref. [52]. The corresponding values obtained in a charm-sector fit [12] are 43-45 MeV, 43-49 MeV and 3-6 MeV, respectively.

In the D-wave sector, Table IV and the assumption V LS ≫ V T > 0 indicate that the two lightest states are 0 þþ and 1 þþ , the two heaviest are 4 þþ and 3 þ-, and two degenerate pairs occur: 1 þ-and 2 þþ [the latter based upon X 2 in Eq. ( 11)] and 2 þþ (based upon X 0 0 ) and 3 þþ .

V. CONCLUSIONS

The dynamical diquark model treats multiquark hadrons as being composed of color-triplet diquark (and triquark) quasiparticle subcomponents that can briefly achieve a static configuration describable in terms of the Born-Oppenheimer approximation. Key in the plausibility of the model is that the diquarks are substantially heavier than the mass equivalent of the light degrees of freedom in the color flux tube connecting them.

Here, we have pushed the model to the limit of its feasibility by treating diquarks containing strange quarks as heavy, in order to explore what vestiges of the model persist in the 1.9-3.2 GeV region. In this mass range, several meson resonances with unusual properties have been observed, such as ϕð2170Þ, f 2 ð2340Þ, and Xð2370Þ, and are often interpreted as candidates for tetraquarks, hybrids, or glueballs.

The dynamical diquark model, by limiting inter-diquark color forces to occur in the QCD color triplet, predicts a very specific ordering of radial and orbital (nL) excitations when one chooses any definite potential-energy function connecting them (in this work, taken from lattice-QCD simulations and from quark models). We have also extracted numerical values for the masses of ðsqÞ and ðssÞ diquarks from such previous works and used these potentials and masses to solve the resulting Schrödinger equations. We find, for example, that f 2 ð1950Þ is a good 1S ssq q candidate, either f 2 ð2300Þ or f 2 ð2340Þ fits well as a 1S ssss candidate, states such as Xð2500Þ are plausible 1P ssq q candidates, and ϕð2170Þ is more likely a 1P ssq q than a 1P ssss state.

The limitation to (antisymmetric) color-triplet diquarks also greatly restricts the specific spectra of J PC quantum numbers in each nL multiplet, and this specificity provides strong constraints, particularly in the ssss sector due to the Pauli exclusion principle. Here, the model predicts the existence of precisely three approximately degenerate states in nS multiplets (including the ground-state 1S multiplet), with J PC ¼ 0 þþ ; 1 þ-, and 2 þþ . Under mild assumptions on the nature of the spin-dependent couplings, we also predict the mass ordering of states within nP and nD ssss multiplets.

This work presents the results of a deliberately simplistic model. In particular, we have ignored effects due to the coupling of ssq q and ssss resonances to di-meson thresholds, including di-meson pairs (e.g., K ðÃÞ KðÃÞ or ϕη 0 ) to which the tetraquark states may decay.

Such modifications may be incorporated rigorously, generalizing the adiabatic nature of the Born-Oppenheimer approximation to produce the so-called diabatic dynamical diquark model [68]. This generalization is ideally suited to studying decay widths [69] and resonant line shapes [70] of such states and constitutes a major thrust for future work.

These effects can be quite dramatic. As shown in the cc sector [68], the exotic state Xð3872Þ has a dominant D DÃ component, comprising approximately 91% of its structure, but still has a non-negligible δ δ contribution even though the di-meson threshold is only about 0.04 MeV higher than the Xð3872Þ mass. Including the di-meson coupled channels can actually reduce the mass eigenvalue by tens of MeV. The same analyses show that, when the S-wave δ δ component dominates, the state exhibits a preferential coupling to S-wave di-meson channels over D-wave channels, even when the D-wave threshold lies closer to the mass eigenvalue. A similar situation is expected in the S-wave ss tetraquark sector.

In the cc sector, a diabatic coupled-channel approach is also used to compute mass shifts and decay widths resulting from open di-meson thresholds [69]. Notably, the mass corrections for J PC ¼ 2 þþ states remain below 10 MeV, and the widths are typically sub-MeV, indicating minimal impact from continuum coupling. In contrast, 1 --states exhibit more substantial continuum dressing: Their widths can reach tens of MeV, as seen for hidden-charm vectors (Y) decaying into open-charm meson pairs. These quantitative findings illustrate that while tensor states are relatively robust against continuum effects, vector states [like ϕð2170Þ] may be significantly modified. For instance, Ref. [71] employs a multichannel T-matrix formalism to demonstrate that the ϕð2170Þ resonance can be dynamically generated through strong coupling between a confined ss seed state and open di-meson channels such as K Ã K1 ð1270Þ. This interaction produces a resonance near the observed mass despite the substantial K 1 decay width, and the diabatic model likely produces a similar result.

We also focused exclusively upon tetraquarks in this paper. However, as noted above, the dynamical diquark model can also be used to produce pentaquark states as diquark-triquark composites. An application of this approach to the hidden-strangeness baryon sector can address phenomena such as the cusplike structure seen near 1900 MeV in γp → K þ Σ 0 by the BGOOD Experiment [72].

Meanwhile, a detailed calculation of the fine-structure corrections required to resolve the multiplets studied here requires either robust predictions from lattice QCD, quark models, or QCD sum rules, or additional hints from experiment. With regard to the latter, all measurements of such states, whether they eventually turn out to be hidden-strangeness tetraquarks, hybrids, or glueballs (or even peculiar highly excited conventional mesons) are of great interest to ongoing experimental efforts at BESIII and JLab and in future experiments at the EIC.