Spectrum of Hidden-Charm, Open-Strange Exotics in the Dynamical Diquark Model

The observation by BESIII and LHCb of states with hidden charm and open strangeness ($c\bar c q\bar s$) presents new opportunities for the development of a global model of heavy-quark exotics. Here we extend the dynamical diquark model to encompass such states, using the same values of Hamiltonian parameters previously obtained from the nonstrange and hidden-strange sectors. The large mass splitting between $Z_{cs}(4000)$ and $Z_{cs}(4220)$ suggests substantial SU(3)$_{\rm flavor}$ mixing between all $J^P \! = \! 1^+$ states, while their average mass compared to that of other sectors offers a direct probe of flavor octet-singlet mixing among exotics. We also explore the inclusion of $\eta$-like exchanges within the states, and find their effects to be quite limited. In addition, using the same diquark-mass parameters, we find $P_c(4312)$ and $P_{cs}(4459)$ to fit well as corresponding nonstrange and open-strange pentaquarks.


I. INTRODUCTION
The spectrum of known heavy-quark exotic hadrons continues to expand frequently, with about 50 candidates observed to date. Almost all have a valence light-flavor content consisting of only q ≡ u and d quarks, but very recently some states with open strangeness have been detected in both the open-charm [1,2] (cdsu) and hiddencharm sectors. In the latter, both a pentaquark [3] (ccuds) state and tetraquark [4,5] (ccus) states have been observed. 1 Multiple reviews summarizing both experimental and theoretical advances in this field have appeared in recent years [6][7][8][9][10][11][12][13][14][15][16]. Several competing theoretical frameworks (di-hadron molecular states, bound states of diquarks, threshold enhancements, etc.) have been developed for years, but no single scheme has yet emerged as a dominant paradigm to explain all the new states, analogous to the way that quark-potential models successfully elucidate the conventional cc and bb sectors [17].
Since a number of the observed exotics decay to final states like J/ψ φ or D ( * ) sJD ( * ) sJ , they possess a presumptive ccss valence content. The advent ofccsq states thus introduces an intermediate case between ccqq and ccss cases, and therefore not only provides an opportunity to examine whether a particular theoretical picture can successfully incorporate data from all of these flavor sectors, but also examines the manifestation of lightquark SU(3) flavor for the first time outside of conventional mesons and baryons.
Quite an extensive body of theoretical work on the hidden-charm, open-strange hadrons exists. For example, in the meson sector whose study forms the bulk of this work, several papers [19][20][21][22][23][24] predate the experimental observations, while multiple studies followed the announcement of the BESIII result but preceded the appearance of the LHCb paper , and yet others appeared subsequent to the LHCb results [18,[48][49][50][51][52][53][54][55]. As one may imagine, this body of work encompasses multiple approaches, including molecular and diquark models, chiral-quark models, and QCD sum rules, among others.
The present work uses the dynamical diquark model, initially introduced in Ref. [56] as a theoretical picture to explain how relatively compact color-triplet diquark quasiparticle pairs can form spatially extended tetraquark states, and extended in Ref. [57] to describe pentaquarks as color-triplet diquark-triquark quasiparticle bound states. The picture is developed in Ref. [58] into a predictive model by noting that the static interaction potential between the heavy color-triplet quasiparticles is the same one as appearing in lattice simulations of heavy quarkonium and its hybrids. The multiplet band structure for ccqq and ccqqq states is studied numerically in Ref. [59]; the fine structure of the groundstate (S-wave) ccqq multiplet is examined numerically in Ref. [60] and that of the P -wave multiplet appears in Ref. [61]. An analogous study of the bbqq and ccss systems is presented in Ref. [62], and the cccc states are investigated in Ref. [63]. Radiative transitions between exotic states are computed in Ref. [64].
Our analysis of ccqs exotics here directly interpolates between the analysis of ccqq in Ref. [60] and ccss in Ref. [62], and uses the same numerical inputs. However, we find that a careful treatment of the SU(3) flavor structure introduces one new parameter, related to octetsinglet mixing. In addition, we allow for the possibility of an η-like exchange between the diquarks analogous to the π-like exchange already present in the original model [60], but show that its effects are quite limited by constraints from the phenomenology of the ccqq sector. We find that the known phenomenology of the open-strange sector does indeed follow from that of the other sectors, despite superficially appearing quite different. We also carry out an analogous exercise for nonstrange and openstrange hidden-charm pentaquark states [P cs (4459) currently being the only known example of the latter], and obtain remarkably satisfactory results. This paper is organized as follows. In Sec. II we define the multiplets of states in terms of eigenstates of both good diquark spin and good heavy-quark/lightquark spins. Section III presents the Hamiltonian for the ccqq and ccss sectors, now including a possible term from η-like exchanges, and computes all relevant matrix elements. Section IV performs the same analysis for the ccqs sector, and discusses possible mixing between multiplets whose nonstrange members carry opposite C parity. In Sec. V we discuss the effects of octet-singlet mixing on the analysis and present numerical results, and in Sec. VI we summarize and conclude.

II. STATES OF THE MODEL
A cataloguing of the QQqq or QQq q 1 q 2 states in the dynamical diquark model, where q, q , q i ∈ {u, d}, first appears in Ref. [58]. The same notation, with small modifications, is applied to ccss in Ref. [62] and to cccc in Ref. [63]. All confirmed exotic candidates to date have successfully been accommodated within the lowest (Σ + g ) Born-Oppenheimer potential of the gluon field connecting the heavy diquark [δ ≡ (Qq)]-antidiquark [δ ≡ (Qq )] or diquark-triquark [θ ≡ (Q(q 1 q 2 ))] quasiparticles. In all cases, δ,δ,θ are assumed to transform as color triplets (or antitriplets) and each quasiparticle contains no internal orbital angular momentum.
In the case of QQqq , the classification scheme then begins with 6 possible core states in which the quasiparticle pair lie in a relative S wave. Indicating the total spin s of a diquark δ,δ by s δ , sδ and using a subscript on the full state to indicate its total spin, one obtains the spectrum Since 4 quark angular momenta are being combined, one may transform these states into other convenient bases by means of 9j angular momentum recoupling coefficients. In particular, in the basis of good total heavy-quark (QQ) and light-quark (qq ) spin, the transformation reads with [s] ≡ 2s + 1 signifying the multiplicity of a spin-s state. Using Eqs. (4) and (5), one then obtains In this work it is especially convenient to employ a basis of states carrying a unique value of s QQ and of s qq .
These states are X 1 and X 2 [as seen in Eqs. (6)], and Including (u, d) light-quark flavor produces 12 states: 6 each with I = 0 and I = 1, and spin structures in the form of Eqs. (4), (6), or (7). The basis of Eqs. (7) in particular is ideal for discussing SU(3) flavor multiplets: A state component like 1 us is a pure flavor octet that transforms under spin and flavor analogously to K * + (although in a diquark model it comprises a mixture of color-singlet and color-octet components). The full SU(3) flavor structure of the multiplet Σ + g (1S) thus consists of 6 octets and 6 singlets. A study of the possible mixing of states with the same J P between different SU(3) flavor octets, or of octetsinglet mixing, form two principal theory innovations of this work.
The QQqq states in the multiplet Σ + g (1S) are sufficient to accommodate all particles considered in this work. However, we note that Ref. [58] also provides a classification of orbitally excited states (the multiplets Σ + g (nP ) appearing in Ref. [61]), as well as states in excited-glue Born-Oppenheimer potentials such as Π + u (which are exotic analogues to hybrid mesons), and pentaquark states QQq q 1 q 2 .

A. ccqq Sector
For hidden heavy-flavor exotics containing only u and/or d light valence quarks, we write the following Hamiltonian: Here, M 0 is the common Σ + g (1S) multiplet mass, which depends only upon the chosen diquark (δ,δ) masses and a central potential V (r) computed numerically on the lattice from pure glue configurations that connect 3 and 3 sources, as employed in Ref. [59]. The second term represents the spin-spin interaction within diquarks, assumed to couple only q ↔ Q andq ↔Q, and κ qQ indicates the strength of this interaction. The prime (flavor) index on the light antiquark has been suppressed throughout Eq. (8), since for the moment we consider q, q to be either a light-quark or strange-quark pair, so that the same value of κ qQ appears for both spin-spin terms. An isospin-spin-dependent interaction of strength V 0 between the light-quark spins, which is modeled on the pion-nucleon coupling and was first introduced in Ref. [60], comprises the third term. These 3 terms form the full set included in the analysis of Ref. [60]. The final term is new to this work; it is modeled on an η-nucleon coupling and evaluates in the relevant flavor sectors to: To compute the mass expressions arising from Eq. (8), let us first abbreviate Then the Hamiltonian matrix elements for the mixed QQqq states of the Σ + g (1S) multiplet, their components arranged in the order s QQ = 0, 1, read Diagonalizing the expressions of Eq. (11) in order of increasing mass and appending the corresponding (already diagonal) expressions for the remaining QQqq states of the Σ + g (1S) multiplet, one obtains using the abbreviations

B. ccss Sector
The Hamiltonian relevant to the ccss sector is identical to the one in Eq. (8), omitting the isospin-dependent V 0 term and performing some q → s relabeling, Equivalently, this expression generalizes the Hamiltonian used in the analysis of Ref. [62] by the inclusion of the whereṼ

IV. HIDDEN-CHARM, OPEN-STRANGE SECTOR
In this sector, the Hamiltonian analogous to Eq. (8) becomes Without loss of generality, we have taken q → s, with the opposite choice q → s simply leading to the antiparticles of those studied here. Then the spin couplings κ sQ and κ qQ are numerically quite distinct, and we compute the mass contributions and with the final expression computed in the same manner as is performed to obtain Eq. (9). A notable feature of the open-strange exotics sector becomes apparent when considering the full SU(3) flavor multiplet structure. QQqq states with I 3 = 0 carry good J P C quantum numbers, and states with different J P C values of course cannot mix with them. Inasmuch as isospin is a nearly exact symmetry, one can extend C parity to a full isospin multiplet by defining the conserved G-parity quantum number (whose eigenvalues for all hadrons are tabulated by the Particle Data Group (PDG) [17]). Specifically, where the C-parity eigenvalue here is that of the I 3 = 0 member of the isomultiplet. One could generalize the concept of G parity to a full SU(3) flavor multiplet, but since the corresponding flavor symmetry is broken, mixing between the open-strange members of multiplets whose I 3 = 0, Y = 0 members have opposite C parities can occur. Indeed, this phenomenon is known among the conventional mesons: For example, the strange partners to the lightest 1 ++ and 1 +− mesons are named K 1A and K 1B respectively, and the observed 1 + strange-meson mass eigenstates K 1 (1270) and K 1 (1400) are believed to be nearly equal admixtures of K 1A and K 1B [17].
In the exotics sector, the nonstrange X 1 (1 ++ ) states cannot mix withZ,Z (1 +− ) due to G-parity conservation. However, their open-strange 1 + partners can mix, leading to richer phenomenological possibilities. To wit: The mass expressions obtained from Eq. (17), prior to diagonalization, read The elements of the matrices for 0 + are again arranged in order of increasing heavy-quark spin. However, those for 1 + are arranged in the order corresponding to increasing mass eigenvalues for their nonstrange partners in the hidden-charm sector: X 1 ,Z ,Z. The mass eigenvalues for the 0 + sector read wherẽ V QQqs The exact expressions for the 1 + eigenvalues are of course complicated roots of a cubic equation, but anticipating that V 8 κ qQ κ sQ , one may perform a perturbative expansion in V 8 to compute approximate values: V. ANALYSIS

A. Flavor SU(3) Multiplets and Mixing
The original analysis of ccss states in Ref. [65], as well as its updated form in Ref. [62], takes the ccss states to be completely unmixed with those in the ccqq sector, where q, q ∈ {u, d}. If, on the other hand, SU(3) flavor is exact, then the states should fill octets and singlets of the flavor symmetry. Specifically, the flavor structure of ccqq states, now allowing q, q ∈ {u, d, s}, can be discussed using the same framework that applies to conventional qq mesons. The I = 0, I 3 = 0, Y = 0 unmixed octet and singlet combinations are, as usual, respectively. In the lightest (J P C = 0 −+ ) meson multiplet, these states correspond to η and η , respectively, which remain largely unmixed because the octet states are pseudo-Nambu-Goldstone bosons whose masses vanish in the chiral limit, while the singlet has a nonzero mass in this limit due to the anomalous breaking of the axial U(1) symmetry of massless QCD. Heavier meson multiplets, however, support much larger SU(3) flavor mixing between the octet and singlet combinations. For example, the next-lightest (1 −− ) multiplet features the ω and φ as its I = 0 states, which appear to be nearly ideally mixed into the flavor combinations 1 √ 2 (uū+dd) and ss, respectively. The appearance of only the J/ψ φ decay mode for most of the purported ccss candidates inspired the implicit adoption of an idealmixing ansatz in Refs. [62,65].
Moreover, the approach of treating I = 0 states in the ccqq sector as containing no ss component, which is implicit in Refs. [59,60,62], also introduces a hidden assumption of octet-singlet mixing into the analysis. The fact that ρ 0 (I = 1, pure octet) and ω (I = 0, ideally mixed) are nearly degenerate in mass, and likewise for Z c (3900) 0 (I = 1, pure octet) and X(3872) (I = 0), suggests that substantial octet-singlet flavor mixing is needed to understand the spectrum of both 1 −− conventional mesons and Σ + g (1S) hidden-charm exotic mesons. Isospin symmetry then links the remaining Y = 0 states (ρ ± , Z c (3900) ± ). In the case of exotics, the values of We illustrate this mixing effect using a toy example well known from elementary quantum mechanics: Ignore fine-structure effects and let the unmixed mass parameters M 8 (pure octet) and M 1 (pure singlet) be degenerate, M = M 8 = M 1 , in a 2-level system with an octetsinglet mass-mixing parameter ∆. Then the resulting mass eigenvalues are M ∓ ∆, and the mixing angle of the system is maximal, 45 • . More generally, the lower mass eigenvalue is always smaller than the smaller diagonal element, whether or not the unmixed octet and singlet mass parameters are equal. In our case, the value of M ccqq 0 from the previous analyses of Refs. [59,60,62,65] is assumed to refer to ideally mixed states; and since one expects the exotics observed thus far (which are used to extract M 0 values) to be the lightest ones that exist, the derived M ccqq values should therefore be slightly lower than one determined entirely from the pure-octet ccqs sector. This expectation, in fact, is precisely what occurs, as we see below.

B. ccqq Sector and V8
The analysis of the ccqq sector here closely follows that of Ref. [60], and especially Ref. [62]. The 3 primary inputs are the PDG averages [17] m X(3872) = 3871.69 ± 0.17 MeV , m Zc(3900) = 3888.4 ± 2.5 MeV , m Zc(4020) = 4024.1 ± 1.9 MeV , (27) with only the value for Z c (3900) changing slightly since the previous analyses. Since the Hamiltonian of Eq. (8) now has 4 parameters, the system is underdetermined. However, one further constraint arises from noting the strong charmonium decay preference [17] of Z c (3900) to J/ψ, and Z c (4020) to h c , suggesting that these Z c states are nearly pure s cc = 1 and s cc = 0 eigenstates, respectively. Defining P as the s QQ = 1 probability content of the lower-mass 1 +− , I = 1 eigenstate of Eqs. (12) [i.e., the square of the off-diagonal component of the unitary matrix diagonalizingM I=1 1 +− in Eqs. (11)], one obtains which means that V 8 can be expressed as a function of P (and the parameters V 0 and κ qQ ). Using this constraint with the mass expressions in Eqs. (12), the most convenient combinations of the 3 masses in Eqs. (27) are .
From Eqs. (29), one extracts The case V 8 = 0, which (in effect) is imposed in Ref. [62], becomes The only input used in Ref. [62] beyond those of Eqs. (27) is the discrete choice of the larger P value in Eq. (31) to recognize the Z c charmonium decay preferences noted above.
In fact, Eq. (30) places a rather strong constraint upon V 8 . While any P ∈ [0, 1] is in principle allowed, values of P smaller (larger) than the smaller (larger) root in Eq. (31) lead to negative values of V 8 -and, for sufficiently small values of P , values of V 8 that are also larger in magnitude than V 0 . Inasmuch as the accompanying operators in Eq. (8) represent π-like and η-like exchanges, respectively, one expects the analogy to the dynamics of true π and η exchanges between nucleons (from, e.g., chiral perturbation theory) to hold. Under this assumption, the η-like exchange should be attractive like the π-like exchange; hence V 8 , like V 0 , should be positive. However, genuine η exchange is also weaker than π exchange, both due to the η's larger mass and larger decay constant. We therefore take V 8 > 0 to be a natural constraint of the model, which requires P to lie between the roots given in Eq. (31). According to Eq. (30), within this range V 8 reaches a maximum at P = 1 2 1 + 1 √ 2 = 0.854, at which We therefore expect the allowed range of V 8 , as determined by the known phenomenology of the ccqq sector, to have a modest effect compared to that provided by the other parameters in Eq. (8). Using Eqs. (29), one obtains for the other Hamiltonian parameters: The values of M 0 , κ qc , and V 0 obtained for both V 8 = 0 and for an optimized V 8 value obtained below from the ccss spectrum [in Eqs. (45)] are presented in Table I. The full spectrum of masses for the ccqq Σ + g (1S) multiplet appears in Table II.
Using the value of M ccqq 0 from Table I A value spanning this spread is presented in Table I. Again, these results for V 8 = 0 differ from those in Ref. [62] only through a small shift in the tabulated PDG value of m Zc(3900) in Eq. (27).

C. ccss Sector: κsc and V8
The signature process used in Ref. [62] to identify ccss exotics is the decay mode J/ψ φ, although some candidates are identifiable through D s -type meson-pair decays. Furthermore, the analysis of Ref. [62] argues that the J P C = 1 ++ X(4274) is an excellent candidate for It has definitively been seen to couple only to γγ and J/ψ ω; with respect to the latter mode, note that X(3915) lies below the J/ψ φ threshold, so that φ → ω mixing is proposed in Ref. [65] to be responsible for the J/ψ ω decay mode. Furthermore, a recent lattice calculation [69] predicts the existence of a 0 ++ state in this mass region that has a strong coupling to D sDs but a weak coupling to DD. The mass used in this work is the PDG value [17]: In the previous analysis [62], the ccss spectrum obtained for the multiplet is very simple. Referring to Eqs. (15), the assumption that κ sc V 8 > 0 leads to the spectrum (in increasing order of mass): which reduces to 3 degenerate sets in the case V 8 = 0, as listed in Ref. [62]. In particular, the lighter 0 ++ state clearly lies far below the others, with the 1 ++ state (and the lighter 1 +− ) being intermediate in mass, and the 2 ++ and heavier 0 ++ (and 1 +− ) states lying close together at a larger mass value. The overall effect of Eq. (36) is to split the Σ + g (1S) multiplet into 3 roughly equally spaced (by 2κ sc ) clusters of ccss states.
The state X(4140) is taken to be an unmistakable ccss candidate, the sole 1 ++ member of the multiplet Σ + g (1S). Therefore, the ccss spectrum should start with X(3915) being the distinct lightest member, a 1 +− state is predicted to appear with a mass near m X(4140) , and a trio of states (0 ++ , 1 +− , 2 ++ ) is predicted to appear at approximately m X(3915) +2(m X(4140) −m X(3915) ). A complication arises, however, with the latest LHCb measurement [5] of X(4140): which should be compared to the PDG average [17], the mass differing by about 1.3 σ (and the width differing radically). LHCb observes X(4140) with a 13 σ total significance. On the other hand, the previous LHCb observation of X(4140) [70] (at a significance of 8.4 σ) forms part of the PDG averages of Eqs. (38), and the mass value obtained in Ref. [70] is much more in line with the average mass value given in Eqs. (38): In fact, the data used in Ref. [70] forms a small subset of the LHCb data reported in Ref. [5]. So how can a measurement using much more data lead to a result with much larger uncertainties? In large part, it arises from a new modeling of the X(4140) lineshape, in which a naive Breit-Wigner profile is replaced with a Flatté form [71]. To incorporate this new development, we reanalyze the PDG mass average of Eqs. (38) by replacing the old LHCb mass measurement of Eqs. (39) with the new one of Eqs. (37), producing the value to be used in our analysis: The state X(4350), although not yet confirmed at the same level of confidence (3.2σ), is seen in γγ → J/ψ φ and thus is an excellent ccss 0 ++ or 2 ++ candidate. Noting that [17] m X(4350) = 4351 ± 5 MeV , and using Eqs. (35) and (40) One then sees (as in Refs. [62,65]) that X(4350) nearly satisifies the equal-spacing rule discussed above, which confirms our previous result that V 8 is numerically small. In fact, at linear order in V 8 , Eqs. (36) give Using m X(4350) from Eq. (41) assuming that X(4350) is 0 ++ , and assuming that X(4350) is 2 ++ . Obtaining these results requires the resolution of a discrete ambiguity to impose the physical expectation κ sc > 0, as discussed in Ref. [62]. The latter solution produces a slightly larger value of V 8 than allowed by Eq. (32), but only by 1.0 σ, and therefore still viable. Nevertheless, for purposes of illustration, we choose Eqs. (45) as the best-fit parameters (also included in Table I), and use them to compute the full spectrum of masses for the Σ + g (1S) ccss multiplet in Table II The M ccss 0 and κ sc values obtained in Eqs. (45) and (47) differ rather little from those in Ref. [62], in part because the previous work effectively takes V 8 = 0, and also because the inputs of Eqs. (35) and (40) have changed little in the interim.
Feeding the value of nonzero V 8 back into the ccqq expressions given by Eqs. (30) and (33), one obtains the V 8 > 0 values of M ccqq 0 , κ qc , V 0 , and P given in Table I The averaged values for Eqs. (47) and (48) appear in Table I.

D. ccqs Sector
The results obtained from the ccqq and ccss Σ + g (1S) multiplets in the previous two subsections, with parameters collected in Table I, are almost completely sufficient to predict the entire ccqs Σ + g (1S) spectrum. However, as noted in Sec. V A, the fact that the open-strange states are pure SU(3) flavor octet, while ccqq and ccss are assumed to be ideally mixed octet-singlet combinations, means that the value of M ccqs 0 extracted using only inputs from the other sectors is likely to be slightly too low to match observed Z cs masses in Eqs. (2). On the other hand, the fine structure obtained in this sector using values of κ sc , κ sq , and V 8 from Table I should be predicted correctly.
Explicitly, using the m δ (cq) from Eqs. (48), m δ (cs) from Eqs. (47), and the same lattice-calculated potentials V (r) as used previously, we compute a remarkably stable result across simulations. Using this value along with the other parameters in Table I in the 1 + expressions of Eqs. (22), we compute which are lower than the measured values given in Eqs. (2). If, however, we add an offset and so m Z (1) cs and m Z (3) cs beautifully match the observed values in Eqs. (2).
Of course, this model predicts also a third open-strange 1 + state Z (2) cs , which is not, as yet, reported by LHCb. In this regard, we note that the reported mass uncertainty and width of Z cs (4220) in Eqs. (2) are quite large, meaning that subsequent analysis might resolve the peak as two states Z (2) cs and Z (3) cs , as was found for the pentaquark candidates P c (4440) and P c (4457) in Ref. [72]. Small hints of additional structure may be already visible in the LHCb results above 4100 MeV (Fig. 3 of [72], right inset). The relative closeness of Z (2) cs and Z (3) cs in mass follows directly in this model, as can be seen from Eqs. (25), since (Table I) κ qc κ sc . The large mass splitting m Z (3) cs − m Z (1) cs ≈ 2κ sc > 200 MeV is also explained naturally by the model; in this regard, note that such a large mixing would not have occurred without the mixing of strange states between 1 ++ and 1 +− multiplets, as discussed in Sec. IV.
One other criterion may be useful for disentangling the trio of 1 + states Z (i) cs : The eigenvectors of Eqs. (22) couple differently to the heavy-and light-quark-spin eigenstates X 1 [Eqs. (6)],Z , andZ [defined in Eqs. (7)]. Explicitly, in the limit V 8 = 0, the corresponding eigenvectors with respect to this basis are (53) Since onlyZ has s QQ = 0, the third component of each eigenvector indicates the relative strength of the coupling to h c (vs. J/ψ) in decays that conserve heavy-quark spin. We note that Z (2) cs has the largest such coupling: 50% of its decays should be to s QQ = 0 states. Therefore, a prediction of this model is not only that Z cs (4220) resolves into two peaks, but also that the lower state couples particularly strongly to h c .
Using the values of M ccqs 0 , κ sc , κ qc , and V 8 from Table I Table II.
Using P c (4312) to fix M 0 for the Σ + (1P ) multiplet, where [72] m Pc(4312) = 4311.9 ± 6.8 MeV , (56) and using m δ =(cq) obtained from the ccqq states, Ref. [59] computes a value of mθ 1.93 GeV. Repeating the analysis here using the new value of m δ =(cq) from Eq. (48), we find mθ =c(q1q2) = 1884.6 ± 7.5 MeV (JKM) , = 1866.5 ± 7.5 MeV (CPRRW) . (57) As noted in Ref. [59], m δ =(cq) and mθ =[c(q1q2)] are quite close in mass; indeed, mθ is actually slightly smaller than m δ in the new calculation. This peculiarity arises from assigning the lowest observed ccqq states to the groundstate Σ + g (1S) multiplet but assigning the lowest observed pentaquark P c (4312) to the excited Σ + (1P ) multiplet. Should the opposite-parity P c (4380) disappear from future data, then P c (4312) and the other states would become suitable to belong to Σ + (1S), and mθ (absorbing what was orbital excitation energy) would become numerically larger. Even in the current circumstance, however, no obvious physical requirement demands that mθ > m δ . Indeed, one may argue thatθ contains two significant sources of binding energy: within the diquark (q 1 q 2 ), and between this diquark andc, while δ possesses only the first type of binding, thereby allowing mθ m δ . Now suppose that the state P cs (4459) of Eqs. (3) is the open-strange Σ + (1P ) analogue to P c (4312), i.e., a δθ state with δ = (cs), where m δ =(cs) is given in Eqs. (47). This is a stunning result, being less than 2 σ lower than the value in Eqs. (3). Indeed, no reason apart from convenience leads one to take the P c (4312) [as opposed to, say, P c (4457)] and P cs (4459) masses equal to the M 0 values for their respective Σ + (1P ) multiplets, except that they are the lightest ones known. A complete analysis would incorporate fine structure, as is done for the tetraquark sectors, but this exercise has not yet been carried out in the pentaquark sectors of this model, due to a lack of experimental clarity on J P quantum numbers for at least some of the observed states. Nevertheless, the result of Eqs. (58) shows that a single model can, in fact, accommodate exotics in all observed flavor sectors.
The development of the dynamical diquark model to date has focused primarily on spectroscopy, and to a 1 ++ 3872.2 ± 7.8 · · · 3993.8 ± 5.6 · · · 4146.7 ± 3.5 · · · 4225.6 ± 7.5 · · · 2 ++ 3923.4 ± 7.8 · · · 4045.0 ± 5.6 · · · 4366.3 ± 3.5 · · · 2 + 4220.1 ± 7.5 · · · lesser extent on identifying the dominant quarkonium decay channels. Detailed quantitative calculations of strong decay widths, particularly for open-heavy-flavor channels, have not yet been attempted, because a precise description of couplings between diquark and hadronhadron configurations has not yet been developed. Qualitative statements to explain the relative narrowness of exotic states have appeared since the initial description of the picture in Refs. [56,57]; they originate from the significant spatial separation between the diquark or triquark quasiparticles, which hinders the rearrangement of their component quarks into color-singlet hadrons. Such effects may need to be quite potent in the pentaquarks, since P c (4312), P c (4440), P c (4457), P cs (4459) all have surprisingly small widths ( 20 MeV). In addition, the proximity of these states to D ( * ) Σ ( * ) c or D ( * ) Ξ c thresholds has been noted since the original experimental papers, and predicted earlier in molecular models [74][75][76][77]. The next phase of the development of the model will address the effect of mixing between diquark configurations and hadron-hadron thresholds, thus providing critical insight into both the decay properties of exotics and a connection to the successes of hadronic-molecule pictures.

VI. CONCLUSIONS
This work shows that the newly observed hiddencharm, open-strange exotic-hadron candidates Z cs (4000), Z cs (4220), and P cs (4459) fit naturally into the dynamical diquark model. Notably, the same lattice-simulated potential V (r) between two heavy, color-triplet sources in the lowest Born-Oppenheimer configuration Σ + (g) is seen to apply to all cases studied here.
Among tetraquarks, the same Hamiltonian parameters, with numerical values obtained from the ccqq (q, q being u or d) and ccss exotic Σ + g (1S) multiplets, successfully predict masses in the ccqs sector. In particular, the large Z cs (4220)-Z cs (4000) mass splitting emerges naturally as consequences of both the large (cs) diquark internal spin-spin coupling κ sc and the mixing of open-strange members of J P C = 1 ++ and 1 +− multiplets, the latter an effect seen in conventional hadron physics for strange mesons such as K 1A and K 1B . The model also predicts a third 1 + Z cs state lying not far below 4200 MeV.
The overall multiplet-average mass M 0 for ccqs states also receives a shift modification compared to those for ccqq and ccss states, since the former are pure SU(3) flavor octet states, while states in the latter sets are assumed in the numerical analysis to be ideally mixed octet-singlet combinations. The size of this shift ∆M ccqs 0 is found to be numerically not large, at most a few 10's of MeV.
In addition, the model in all sectors has been expanded to allow not only a π-like interaction operator between the diquarks (as in previous studies), but an η-like interaction operator as well. The numerical size of the coefficient V 8 of this operator is found to be much smaller than that (V 0 ) for π-like interactions, and has a fairly minimal effect on the hadron spectra. Mass predictions for all states in the Σ + g (1S) multiplet for each flavor content are presented.
Among pentaquarks, a crude calculation taking the nonstrange P c (4312) as a base state for the positiveparity multiplet Σ + (1P ) constructed of a diquarktriquark pair (cq)[c(ud)], and replacing the (cq) diquark with a (cs) diquark, produces a state very close in mass to that of P cs (4459).
As new exotic hadrons continue to be uncovered-a rather safe expectation, considering the rate of observational advances over the past few years-more opportunities for sharpening our understanding of their mass spectrum and transitions will emerge. Whether or not a diquark-based spectrum provides the eventual global picture for these states, the dynamical diquark model supplies a definite road map for the sort of spectrum to expect.