Spectrum of the Hidden-Bottom and the Hidden-Charm/Strange Exotics in the Dynamical Diquark Model

The lightest hidden-bottom tetraquarks in the dynamical diquark model fill an $S$-wave multiplet consisting of 12 isomultiplets. We predict their masses and dominant bottomonium decay channels using a simple 3-parameter Hamiltonian that captures the core fine-structure features of the model, including isospin dependence. The only experimental inputs needed are the corresponding observables for $Z_b(10610)$ and $Z_b(10650)$. The mass of $X_b$, the bottom analogue to $X(3872)$, is highly constrained in this scheme. In addition, using lattice-calculated potentials we predict the location of the center of mass of the $P$-wave multiplet and find that $Y(10860)$ fits well but the newly discovered $Y(10750)$ does not, more plausibly being a $D$-wave bottomonium state. Using similar methods, we also examine the lowest $S$-wave multiplet of 6 $c\bar c s\bar s$ states, assuming as in earlier work that $X(3915)$ and $Y(4140)$ are members, and predict the masses and dominant charmonium decay modes of the other states. We again use lattice potentials to compute the centers of mass of higher multiplets, and find them to be compatible with the masses of $Y(4626)$ ($1P$) and $X(4700)$ ($2S$), respectively.


I. INTRODUCTION
The modern study of hadrons that manifest exotic valence-quark content has produced numerous surprises in both experiment and theory, as reviewed in Refs. [1][2][3][4][5][6][7][8][9][10][11]. As of this writing, more than 40 candidates have been observed in the heavy-quark sector. However, the fundamental organizing principle underlying their spectroscopy has proved elusive, unlike the clear structure derived from quark-potential models in the conventional cc and bb sectors [12].
For instance, one may attempt to model multiquark exotics using the original molecular picture of two conventional hadrons bound via light-meson (e.g., π) exchange [13,14]. This approach can provide some guidance regarding which thresholds might be expected to support a molecule [15,16]. However, hadronic molecules lack a regularly spaced spectrum because the pattern of mass splittings among the light and heavy-light hadrons acting as their constituents is itself nontrivial, being obscured by the specifics of strong-interaction dynamics. In addition, a composite state of a given width cannot form if either constituent hadron has a larger width, and it remains unclear whether molecular formation is limited to the case in which the constituents are in a relative S wave. Indeed, calculating the detailed properties of hadronic molecules appears to require the careful consideration of a variety of near-threshold effects such as cusps and rescattering diagrams [5].
The J P C = 1 ++ X(3872), the first heavy-quark exotic state discovered [17], is touted as the ne plus ultra of hadronic molecules, and no reasonable researcher can deny that the absurdly small splitting m X(3872) −m D 0 − m D * 0 = +0.01±0. 18 MeV indicates the controlling influence of the state D 0D * 0 (charge conjugates understood) over the nature of the resonance. And yet, the very proximity of X(3872) to threshold indicates that it is almost certainly not a "traditional" molecule of the type described above, but rather its precise mass eigenvalue relies in an intrinsic way upon threshold effects. Several observed features of X(3872) point to a complicated structure; for example, its substantial collider prompt production rate suggests that X(3872) possesses a tightly bound component, but the suppression of this rate with increasing charged-particle multiplicity in pp collisions as compared to that of ψ(2S) [18] suggests that X(3872) may more easily dissociate in a dense particle environment, as one expects for a molecule. A typical resolution of this conundrum is to suppose that the 1 ++ conventional charmonium state χ c1 (2P ), predicted by potential models to lie around 3925 MeV [19] but conspicuously absent from the data, mixes to a significant degree with a D 0D * 0 state to form the physical X(3872).
The χ c1 (2P ) is not the only example of a tightly bound state that can mix with X(3872). Diquark models also produce a single isoscalar 1 ++ tetraquark state as one of their lowest hidden-charm excitations, appearing in the color-attractive arrangement (cq)3(cq) 3 [20]. Typical diquark δ ≡ (cq)3 masses of ∼ 1.9 GeV naturally produce such a 1 ++ δ-δ state in the vicinity of 3.9 GeV [21]. X(3872) might actually, in the end, prove to be a perfect storm of a D 0D * 0 molecular state enhanced by threshold effects, mixing with the otherwise isolated conventional charmonium χ c1 (2P ) state and the lowest-lying isoscalar 1 ++ δ-δ state.
The variant diquark model used in this work is the so-called "dynamical" diquark model, which was developed [22] to address the issue of how δ-δ states persist long enough to be observed, rather than their quarks instantly recombining through the more strongly attractive arXiv:2005.07100v2 [hep-ph] 18 May 2020 dict masses of the remaining 10 states and their preferred heavy-quark decay channels with surprising accuracy. The analysis of the ccss states builds upon that of Ref. [33] [which assumes that X(3915) and Y (4140) are ccss states] to reflect the current state of data and to develop a better understanding of the underpinnings of the model. The negative-parity Σ + g (1P ) multiplet consists of 28 states for bbqq and 14 states for ccss, but only a very small number of candidates have been observed for each type; nevertheless, we use the approach of Ref. [21] to predict the centers of mass of the Σ + g (1P ) and Σ + g (2S) multiplets, and find that most of the candidates lie in the anticipated mass regions [the exception being Y (10750), which we argue to be a conventional bottomonium state].
This paper is organized as follows: In Sec. II we review the current data on bbqq and ccss candidates. Section III reprises the analysis of Ref. [30], as applied to these sectors. The naming scheme for levels comprising the Σ + g (1S) multiplets is defined in Sec. IV, and we present explicit expressions for their masses in terms of the model parameters. Numerical analysis of states in the bbqq and the ccss sectors appears in Section V, where both mass eigenvalues and mixing parameters relevant to heavy-quark decay modes are predicted. We conclude in Sec. VI.

A. The bbqq Sector
Of all hidden-bottom states thus far observed, only a handful are exotic candidates, which are summarized in Table I. The most familiar examples are the I = 1, J P C = 1 +− states Z b (10610) and Z b (10650). Their proximity to the thresholds for BB * (10604.2 ± 0.3 MeV) and B * B * (10649.4 ± 0.4 MeV), respectively, suggests a natural identification as molecular states. These states also possess hidden-charm analogues Z c (3900) and Z c (4020) that carry the same quantum numbers, which indeed lie near the DD * and D * D * thresholds, respectively. Nevertheless, Z c (3900) and Z c (4020) were found in Ref. [30] to serve naturally as the I = 1, J P C = 1 +− members of the ground-state Σ + g (1S) multiplet of the dynamical diquark model, and so in this work we interpret the two Z b states analogously. Furthermore, both Z b (10610) and Z b (10650), like the Z c states, have been observed to decay to both closed [35] [Υ(nS), h b (nP )] and open [36] [B ( * )B * ] heavy-flavor states. However, the Z b and Z c states differ in one important regard: The observed charmonium decays of Z c (3900) to date all have total charmquark spin s cc = 1 (i.e., that of J/ψ), while those of Z c (4020) have s cc = 0 (i.e., that of h c ), and obtaining this idealized mixing in a natural way is one of the central results of Ref. [30]. However, a glance at Table I shows that the Z b system is rather different: Both Z b states decay to states with s bb = 0 and 1 (Υ and h b ) with fairly comparable branching ratios.  [12]. Also included is the recently observed Y (10750) [34]. Both the particle name most commonly used in the literature and its label as given in the PDG are listed. Only bottomonium decays are listed, and branching ratios are given where available.
Production and Decay The current experimental situation for the hiddenbottom sector also differs from the hidden-charm sector in one obvious respect: In the latter, the most obvious and best-studied state is the neutral 1 ++ X(3872). However, the hidden-bottom analogue X b (I G = 0 + , J P C = 1 ++ ) has not yet been observed, despite a number of searches [37][38][39]. Partly, this absence reflects the relative difficulty of probing the hidden-bottom sector with limited energy (e.g., at the original Belle Experiment operating at a center-of-momentum energy equal to the Υ(4S) mass [37]) or limited only to the decay channel Υ(1S)π + π − (at the LHC [38,39]), which has opposite G-parity to that expected for X b . It would be truly surprising in both molecular models and diquark models (as well as in coupled-channel and QCD sum-rule approaches) were the X b state to fail to exist; as a re-sult, a great deal of theoretical effort has been invested in studying the conjectured X b [14,16,[37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55]. Bounding the possible range for the X b mass and determining whether any hidden-bottom exotics can be even lighter constitute a major goal of this work. Table I presents two further observed exotic candidates, both with J P C = 1 −− . The Y (10750) was recently observed at Belle [34], and has already been studied as a diquark state [56] and within QCD sum rules [57]. Additionally, Y (10750) and the remaining exotic candidate Y (10860) have been argued to be conventional bottomonium states [58,59]. One should note, however, that Y (10860) (like the Z b states) has both s bb = 0 and s bb = 1 decay modes (thus violating heavy-quark spin symmetry in its decays if it is conventional bottomonium). In contrast, Y (10750) has thus far been observed to decay only to Υ(nS). In addition, Y (10750) lies only 100 MeV above the Z b states, which would indicate a much smaller 1P -1S splitting (assuming they share a related structure) than between corresponding bottomonium states (e.g., m χ bJ (1P )−m Υ(1S) > 400 MeV). We argue in Sec. V that Y (10860) is well suited to being a Σ + g (1P ) excitation of Σ + g (1S) states like Z b (10610) and Z b (10610), but Y (10750) is not.

B. The ccss Sector
The most likely hidden-charm/strange (ccss) exotic candidates are listed in Table II. Almost all have been seen exclusively in the decay channel φJ/ψ, which indicates that each either has a valence ccss quark content or is a pure cc state decaying through an Okubo-Zweig-Iizuka (OZI)-suppressed channel. Similar statements apply to the newly observed Y (4626) [60,61], which has been observed to decay thus far only to channels of open charm and strangeness.
X(3915) has been included in Table II as the lightest ccss candidate despite having no observed decays to states of hidden or open strangeness. 2 Upon its discovery, X(3915) was immediately assigned by the Particle Data Group (PDG) as the first radial excitation χ c0 (2P ) of the conventional charmonium state χ c0 (1P ). However, this identification was found to be problematic for several reasons [63][64][65][66]: First, the mass splitting between χ c2 (2P ) and X(3915) (only about 10 MeV [12]) is smaller than the χ c2 (2P )-χ c0 (2P ) splitting expected from quark potential models; furthermore, one would expect χ c0 (2P ) (or a ccqq exotic) to decay prominently into DD, but the dominant observed X(3915) decay channel is actually the OZI-suppressed mode ωJ/ψ. These features led Ref. [33] to suppose that X(3915) is actually a ccss state, its ωJ/ψ decay possibly proceeding by means of a small ss component in ω. Indeed, the subsequent Belle discovery of χ c0 (3860) [67] as a candidate with the expected properties of the missing χ c0 (2P ) sharpens the case for arguing that X(3915) is exotic [68].
S-wave hidden-charm/strange exotics have been discussed by multiple authors , using methods as varied as ordinary (tetra)quark models, diquark models, molecular/rescattering models, and QCD sum rules (as well as combinations of these). Following on the observation of the negative-parity Y (4626), P -wave ccss states have also recently been considered [91,92].
The precise nature of the two 1 ++ states Y (4140) and Y (4274) in Table II is particularly interesting. On one hand, they both appear in the mass range predicted for the conventional 1 ++ charmonium state χ c1 (3P ). One might naively think that since the (missing) χ c1 (2P ) state is expected to be quite wide (> 100 MeV), its radial excitation χ c1 (3P ) should be even wider. However, it has been known for some time that the more complicated wave-function nodal structure of χ c1 (3P ) actually suppresses its width [19] to the same order of magnitude as that of both Y (4140) and Y (4274). So then which one, if either, is the χ c1 (3P )? Studies in which the Y (4140)-Y (4274) sector is described in terms of conventional charmonium appear in Ref. [93][94][95][96][97][98]. Moreover, as seen in these papers and in Refs. [64,76], no true consensus has emerged on the assignment of either one. Additionally, in the simplest diquark models such as the one used in this work, the ground-state Σ + g (1S) multiplet contains only one 1 ++ ccss state [see Eqs. (3)]. In this paper, we show that the most natural assignment identifies Y (4140) as the unique J P C = 1 ++ ccss state and Y (4274) as χ c1 (3P ).

III. MASS HAMILTONIAN
In the most minimal model variant associated with the dynamical diquark picture, bbqq exotics (q, q ∈ {u, d}) connected by a color flux tube in its ground state (the 1S multiplet of the Σ + g BO potential) can be described using a very simple 3-parameter Hamiltonian: Here, M 0 is the common Σ + g (1S) multiplet mass, which depends only upon the chosen diquark [δ ≡ (bq)3 or δ ≡ (bq) 3 ] mass and a central potential V (r) computed numerically on the lattice from pure glue configurations that connect 3 and3 sources, as done in Ref. [21]. M 0 is the lowest eigenvalue of the Schrödinger equation using the Σ + g BO potential V Σ + g (r); higher eigenvalues have also been computed for this potential [e.g., for Σ + g (1P ), Σ + g (2S), etc.], as well as eigenvalues for lattice-computed excited-glue configurations (e.g., for BO potentials Π + u , The second term in Eq. (1) represents the spin-spin interaction within diquarks, assumed to couple only q ↔ b andq ↔b, and κ qb indicates the strength of this interaction. These couplings are singled out as having greater physical effect upon the nature of the state by assuming that δ,δ are at least somewhat separated quasiparticles within the full exotic state, so that their internal spin couplings are expected to be stronger than the ones between δ andδ. This ansatz originates with Ref. [99], and is incorporated into the motivation behind the dynamical diquark picture, as described in the Introduction and discussed in further detail in Refs. [22,30].
The final term in Eq. (1) is an isospin-spin-dependent interaction between the light-quark spins, where V 0 is the strength of the coupling. The exotic candidates, appearing in distinct I = 0 and I = 1 multiplets, undisputedly  [12]. Also included is Y (4626) [60,61]. Both the particle name most commonly used in the literature and its label as given in the PDG are listed.
Production and decay exhibit nontrivial isospin dependence, thus requiring a term such as this to be included in the Hamiltonian. Its precise form as given in Eq. (1) is of course motivated by that of chiral pion exchanges in hadronic physics, and one plausible interpretation of this operator [30] is to represent the effect of exchanging a Goldstone-boson-like mode across the flux tube connecting the light quarks q andq in δ andδ, respectively. Nevertheless, one could argue for alternate forms that still carry isospin dependence. For example, Refs. [30,31] consider the possibility that the final operator in Eq. (1) couples not to light-quark spins s q,q , but to the full diquark spins s δ ,δ , which would be an appropriate scheme were the diquarks truly pointlike. However, as seen in Refs. [30,31] for the hidden-charm sector, this alternate formulation leads to results inconsistent with experiment, such as degeneracy between X(3872) and its (unobserved) I = 1 partners. The form of Eq. (1) has been presented for use in the bbqq Σ + g (1S) sector, but as indicated above, it was originally used for ccqq [30]. It can be generalized to B c -like exotics bcqq by using the reduced mass obtained from unequal m δ and mδ in the Schrödinger equation and introducing unequal κ qb , κ qc coefficients into the relevant Hamiltonian terms. Equation (1) has also been generalized to the Σ + g (1P ) sector [31] by the addition of spinorbit and (isospin-dependent) tensor couplings.
The ground-state [Σ + g (1S)] hidden-charm/strange (ccss) exotics can be described using an even simpler Hamiltonian, since the states lack isospin dependence: where M 0 and κ sc are defined analogously to the parameters above. This Hamiltonian actually first appeared in Ref. [33], and also included orbital and spin-orbit terms to allow comparison between S-and P -wave states; in the current model, the S-P splitting (as well as the 2S-1S splitting) can be computed directly using the techniques of Ref. [21], as seen in Sec. V. Moreover, subsequent experimental findings that confirm the existence and J P C quantum numbers of relevant states, as well as the discovery of X(4500) and X(4700) ( Table II) and their assignment to the 2S multiplet in a diquark model [77], make a fresh analysis of the ccss sector quite relevant.

IV. MASS FORMULA
The fully general notation for all states in the dynamical diquark model appears in Ref. [24]. Since the current work focuses solely on states in the lowest BO potential Σ + g , and most often those in its lowest multiplet 1S, we can reduce to a much more compact notation. For diquark-antidiquark (δ-δ) states of good total J P C in the S-wave band (i.e., zero orbital angular momentum), the defining notation is: where outer subscripts indicate total quark spin S = J in the absence of orbital angular momentum. The same states may be expressed in any other spin-coupling basis by using angular momentum recoupling coefficients, specifically 9j symbols. For both the simplest evaluation of the final operator in Eq. (1) and for convenient physical interpretation, the most useful alternate basis is that of definite heavy-quark (and light-quark) spin eigenvalues, (QQ)+(qq): where [s] ≡ 2s + 1 denotes the multiplicity of a spin-s state. Using Eqs. (3) and (4), one then obtains A similar recoupling can be used to express these states in terms of equivalent heavy-light meson spins, (qQ)+(qQ). The pairs of states X 0 , X 0 , and Z, Z carry the same values of J P C and can therefore mix. One may define the equivalent heavy-quark spin eigenstates, which are X 1 , X 2 , and Assuming q, q ∈ {u, d}, the Σ + g (1S) multiplet for either ccqq or bbqq then consists of precisely 12 isomultiplets: an isosinglet and an isotriplet corresponding to each of the 6 states in Eqs. (3) or (5) [or as reorganized in Eqs. (6)]. The current PDG nomenclature [12] adopted for the bbqq states is X The corresponding multiplet for B c -like exotics would also contain 12 isomultiplets, but which are no longer C-parity eigenstates. If the light quarks are replaced by an ss pair, then only 6 distinct states remain; in PDG notation, the ccss states are labeled X ( )

A. Bottomoniumlike Exotics
Using the Hamiltonian of Eq. (1) and working (for definiteness) in the heavy-quark spin basis of Eqs. (6), one obtains mass matrices for all 12 isomultiplets of the bbqq Σ + g (1S) multiplet. The cases with nonvanishing offdiagonal elements, for which the entries are arranged in the order s bb = 0, 1, read Diagonalizing these expressions, and appending the expressions for the other states (whose mass matrices are already diagonal), one obtains the mass eigenvalues for all 12 isomultiplets of the bbqq Σ + g (1S) multiplet: where we abbreviate The pairs of states in Eqs. (8) degenerate in J P C are arranged in order of increasing mass.
To obtain the mixing angles, one must first derive the corresponding normalized eigenvectors for the 4 mixed pairs of states with J P C = 0 ++ , 1 +− . Further denoting the normalized eigenvectors collected into columns of uni-tary matrices R read The probability P of the lighter mass eigenstate in each mixed case to be measured to have heavy-quark spin eigenvalue s bb = 1, which is simply obtained by squaring the 1,2 element in each matrix of Eqs. (11), is given by Assuming that heavy-quark symmetry is unbroken in the decays of these states, the P values give the relative branching ratios for the lighter mass eigenstate in each case to decay into a bottomonium state with s bb = 1 (Υ,

B. Hidden-Charm/Strange Exotics
Using the Hamiltonian of Eq. (2) and working (for definiteness) in the heavy-quark spin basis of Eqs. (6), one obtains mass matrices for all 6 states of the ccss Σ + g (1S) multiplet. The cases with nonvanishing off-diagonal elements, for which the entries are arranged in the order s cc = 0, 1, read Diagonalizing these expressions, and appending the expressions for the other states (whose mass matrices are already diagonal), one obtains the mass eigenvalues for all 6 states of the ccss Σ + g (1S) multiplet: The pairs of states in Eqs. (14) degenerate in J P C are arranged in order of increasing mass. Note at this point we have not constrained the spin-spin coupling κ sc to assume a positive value.
To obtain the mixing angles, one must first derive the corresponding normalized eigenvectors for the 2 mixed pairs of states with J P C = 0 ++ , 1 +− . Collected into columns of unitary matrices R, the eigenvectors read The probability P of the lighter mass eigenstate in each mixed case to be measured to have heavy-quark spin eigenvalue s cc = 1, which is simply obtained by squaring the 1,2 element in each matrix of Eqs. (15), is given by Assuming that heavy-quark symmetry is unbroken in the decays of these states, the P values give the relative branching ratios for the lighter mass eigenstate in each case to decay into a charmonium state with s cc = 1 (ψ, χ c ) vs. s cc = 0 (η c , h c ).

A. ccqq Exotics Redux
The masses of the 12 isomultiplets in the bbqq Σ + g (1S) multiplet depend upon only 3 Hamiltonian parameters: M 0 , κ qb , and V 0 , as seen in Eqs. (8)-(9). A similar, but not identical, analysis of the 12 ccqq Σ + g (1S) isomultiplets appears in Ref. [30] (with, of course, κ qb → κ qc , and different M 0 and V 0 numerical values for the ccqq and bbqq systems). There, the masses of the 3 states X(3872), Z c (3900), and Z c (4020) [12] are used as inputs, and the mixing angles of 0 ++ and 1 +− states are allowed to vary under the reasoning that any additional operators omitted from the minimal 3-parameter form have small numerical coefficients and would leave the mass spectrum stable, but could nevertheless substantially change the precise values of the mixing angles. Using the additional phenomenological observation that X I=0 1 [corresponding to X(3872)] appears to be the lightest ccqq state, Ref. [30] obtained M 0 = 3988.75 MeV, κ qc = 17.76 MeV, V 0 = 33.10 MeV.
(17) From these values, Ref. [30] found that Z c (3900) decays almost exclusively to J/ψ and Z c (4020) to h c , in full accord with current observations. However, one may just as easily adopt the strict 3parameter form of Eq.
and the phenomenologically unacceptably small value P scc=1 [Z c (3900)] = 0.631. One learns from this exercise that the value of P , even if not precisely measured, serves as a decisive input to the model. But one also finds, using the fit values of Eqs. (18) in the minimal 3-parameter model, that X I=0 1 is no longer the lightest ccqq state; X I=0 0 (the 0 ++ isosinglet) assumes that status, with This prediction is remarkable, in that it overlaps with the observed mass 3862 +50 −35 MeV of the conventional charmonium χ c0 (2P ) candidate [67], which shares the same quantum numbers. The large observed width 201 +180 −110 MeV indicates unimpeded S-wave decays into DD pairs (threshold ≈ 3740 MeV) for either χ c0 (2P ) or X I=0 0 , and indeed, the observed χ c0 (3860) could be a mixture of the two. Table I shows that only 2 out of 12 bbqq candidates in the positive-parity Σ + g (1S) multiplet have been observed to date, both with (I G ) J P C = (1 + ) 1 +− : Z b (10610) and Z b (10650). Two known masses for a model with 3 Hamiltonian parameters hardly seems sufficient input to draw many conclusions, but the results of the previous subsection indicate that using the s bb = 1 content P I=1 1 +− , s bb =1 of Z b (10610) can be helpful. Indeed, we define

B. Bottomoniumlike Exotics
where the definitions of V 4 and P I=1 1 +− , s bb =1 are the same as in Eqs. (9) and (12), respectively, and for definiteness our numerical analysis uses the mass of the charged Z b (10650). Using these definitions, one may express the original parameters in Eq. (1) as Given a particular numerical value for P , the only remaining ambiguity in predicting the entire Σ + g (1S) mass spectrum is the sign of κ qb . With reference to Eq. (1), κ qb > 0 indicates a scenario in which the spin-singlet diquark δ ≡ (qb) is lighter than the spin-triplet, which is the expectation of virtually every model. Thus making the mild assumption that κ qb > 0, the formulas of Eqs. (8)-(9) for the mass eigenstates (indicated henceforth by overlines, with primes for the heavier of states that are degenerate in J P C ) then read  (1S) bbqq multiplet, using the Hamiltonian of Eq. (1) as evaluated using Eqs. (21), (23), and (24). Boldface indicates the measured Z b mass inputs. where we abbreviate The expressions in Eqs. (12) for the heavy-quark spin content of the remaining mixed states then assume the forms and all observables for the entire Σ + g (1S) multiplet are now expressed as functions of the single parameter P ≡ P I=0 Z,s bb=1 , which varies between 0 and 1; the only numerical inputs are the Z b (10610) and Z b (10650) masses.
In fact, sufficient data exists to go even further: An examination of the exclusive Υ-and h b -channel branching ratios in Table I reveals some interesting effects. First, the branching ratios to Υ(1S) are the smallest among bottomonium decays for both Z b (10610) and Z b (10650), and the branching ratios to h b (2S) are the largest. Noting from simple quark-potential models that Υ(1S) has by far the most spatially compact bottomonium wave function while h b (2P ) has the largest of those kinematically allowed in the Z b decays, one is led to the qualitative conclusion that the Z b states are not spatially compact. Moreover, h b (2P ) has a complicated wave function with not only angular but radial nodes, suggesting initial Z b wave functions that are similarly nonuniform in their spatial density. For our immediate purposes, however, the most interesting feature arises in a direct comparison of the branching ratios for individual Υ and h b channels, noting that the phase space factors for exclusive Z b (10610) and Z b (10650) decay modes are almost identical. With the possible exception of the Υ(3S), the Υ branching ratios of the Z b (10610) appear to be a factor of about 3 times larger than those of the Z b (10650), and the h b branching ratios of the Z b (10610) appear to be a factor of about 3 times smaller than those of the Z b (10650). One is therefore led to the natural estimate P ≈ 3/4.
In addition, the last of Eqs. (22) shows that the sign of P− 1 2 directly gives the sign of V 0 . Since as mentioned below its definition in Eq. (1), the V 0 term is motivated [30] by its similarity in form to the attractive pion interaction in hadronic physics, the value of P ≈ 3/4 obtained above gives V 0 > 0 and suggests a similar interaction in the Σ + g (1S) multiplet for both charmoniumlike and bottomoniumlike states. In fact, independently of V 0 , the values of M 0 and κ qb obtained solely from the Z b masses are numerically very stable over the whole allowed range of P : Of special note, the allowed values of κ qb in this range of P are numerically very close to those obtained in Eq. (18) for κ qc , indicating a common physical origin for both. In contrast, the allowed values of V 0 are several times smaller in the hidden-bottom sector, reflecting the expectation that V 0 contains a coefficient scaling inversely with a power of the heavy-quark mass and therefore being smaller for bottomoniumlike than for charmoniumlike states.
Inserting the values of parameters determined in Eqs. (28) from these three choices of P , we obtain predictions for masses of all 12 isomultiplets of the bbqq Σ + g (1S) multiplet in Table III. The most notable feature of these results is the remarkably small numerical variation of individual state mass predictions over the whole range 1/4 ≤ P ≤ 3/4.
Another way to visualize these results is to impose our expectation that P ≥ 1/2 and consider the entire range 1/2 ≤ P ≤ 1. We then plot the results from combining Eqs. (21), (23), and (24) for all 12 Σ + g (1S) isomultiplet masses in Fig. 1. The ordering of the states in this range of P is remarkably stable. Of particular note: Over most of the allowed range for P , the isosinglet J P C = 0 ++ statē X I=0 0 is lightest, and its isotriplet partnerX I=1 0 is second lightest. Both lie above the threshold (≈ 10560 MeV) of their expected dominant BB decay channel but not excessively so, suggesting that reasonably narrow 0 ++ bbqq states will be discovered in future experiments. Meanwhile,X b ≡ X I=0 1 , the bbqq analogue to the X(3872), only becomes the second-lightest bbqq state for values of P very close to 1 (which is what occurs in the ccqq system). More interestingly,X I=0 1 lies at most only a few MeV below the BB * threshold (≈ 10605 MeV) over almost the whole range 1/2 ≤ P ≤ 1, and thus analogously to X(3872) in its relation to DD * ,X I=0 1 will need to be analyzed by considering the impact of BB * threshold effects. Explicitly, we predict The heavy-quark spin structure of the mixed eigenstates can also be computed solely as functions of P , according to Eqs. (24)- (25). The results are presented in Fig. 2. We find in the range 1/2 ≤ P ≤ 1 thatX I=0 0 decays preferentially to Υ or χ b ,Z I=0 to h b or η b , and the proportion forX I=1 0 depends sensitively upon the precise value of P .
Having completed the analysis of the Σ + g (1S) multiplet, we now use the techniques of Ref. [21] to compute the center of mass M 0 for any other multiplet. The results of Eqs. (28) indicate that M 0 (1S) = 10630 MeV, with an uncertainty of no more than 5 MeV. Using this mass eigenvalue in a Schrödinger equation with the lattice-computed potentials of Refs. [25][26][27][28][29], one finds the diquark δ ≡ (bq)3 and its antiparticleδ ≡ (bq) 3 to have mass where the range indicates the effect of varying over potentials taken from the different lattice simulations. In turn, these m δ ,δ values serve as inputs used to compute other multiplet mass eigenvalues, and we predict One immediately notes that of the two remaining bbqq candidates in Table I (both with negative parity), Y (10860) lies about 70 MeV below M 0 (1P ) and thus uncontroversially fits into the 1P multiplet. 3 On the other hand, Y (10750) does not fit well into this scheme; indeed, it is only about 100 MeV heavier than the 1S state Z b (10650). The nP -nS average mass splitting for conventional bottomonium is about 450 MeV for n = 1 and 250 MeV for n = 2, suggesting that Y (10750) is not sufficiently heavy to be a Σ + g (1P ) bbqq state. However, it was noted even in the discovery paper [34] that Y (10750) is a natural candidate for a higher conventional Υ state, likely identifying with a missing Υ(nD) state, and possibly mixing with Υ(nS) states, although the exact composition remains a matter of debate [58,59]. In support of this view, note from Table I that only Y (10750) → Υ (but not h b ) decay modes have been observed to date, thus promoting the hypothesis of a pure s bb = 1 state, as expected for conventional bottomonium.

C. Hidden-Charm/Strange Exotics
As noted in the Introduction, the ccss sector was first considered using a model with separated (cs) and (cs) diquarks in Ref. [33]. The possibility that the lightest ccss state is X(3915) was introduced in that work, a reprise of the arguments in favor of this assignment appearing in Sec. II B. We also noted that two strong candidates for the sole 1 ++ state in the ccss Σ + g (1S) multiplet, Y (4140) and Y (4274), have been experimentally confirmed (Table II), but also that the conventional charmonium state χ c1 (3P ) is predicted to have a mass and a width comparable to those observed for the two candidates. Indeed, the early calculation of Ref. [19] predicts The Hamiltonian introduced in Ref. [33] restricted to the Σ + g (1S) multiplet is actually identical to the one given in Eq. (2). In Ref. [33] it was introduced as a purely phenomenological construct, but in this work it is seen to be the direct expression of the dynamical diquark model, and mass splittings between different BO multiplets can be computed using lattice-calculated potentials, as in Ref. [21].
A nagging difficulty with the X(3915) has been an ambiguity in its measured J P C quantum numbers. As suggested in Table II and discussed by the PDG [12], the original 0 ++ assignment relies on the assumption of dominance by a particular γγ helicity component in X(3915) production, and if this assumption is relaxed then the assignment 2 ++ is also possible.
Using the measured masses in Table II, we therefore obtain fits to the Hamiltonian of Eq. (2) under two alternate assumptions: that the X(3915) is the lighter of the two 0 ++ states in Σ + g (1S), or that it is the sole 2 ++ state. For the moment we also assign Y (4140) to be the sole 1 ++ state, supposing by default that Y (4274) is χ c1 (3P ). The results of fits with both X(3915) assignments are presented in Table IV. In either case, the spectrum is quite simple, consisting of only 3 distinct (and equally spaced) mass eigenvalues for the 6 states.
A stunning feature of Table IV is that both assignments predict a 0 ++ state at the mass of the X(3915), which suggests one remarkable scenario in which the observed X(3915) is actually a mixture of 0 ++ and 2 ++ states. Furthermore, the third distinct mass in either case, 4375.2 MeV, lies quite close to that of the X(4350), another ccss candidate in Table IV. Confirmation of this state and a precise measurement of its mass and J P quantum numbers (C = + is known) at Belle II will be quite incisive.
These two fits, however, have a major difference that selects one as more relevant to the spirit of the dynamical diquark model. If X(3915) is 0 ++ , then one obtains while taking X(3915) to be 2 ++ gives We have already noted in Sec. V B that the diquark spinspin coupling κ qQ is positive in virtually every model, so the scenario of Eq. (34) leading to a large, negative value of κ sc and the X(3915) being a degenerate 0 ++ -2 ++ combination seems phenomenologically less appealing. The large value of κ sc obtained in Eq. (33) as compared to κ cq in Eq. (18) or κ qb in Eq. (28) (a factor of 5-6) suggests that the lighter constituent of the diquark δ has a significantly greater influence on the size of the spin-spin coupling within δ than does the flavor of the heavy quark. One may argue that the s quark, being much heavier than u or d, allows δ to be substantially more compact, thus enhancing the effects of spin couplings within δ.
Turning now to the identity of the sole 1 ++ state, we consider the alternate possibility that Y (4274) is a ccss state and Y (4140) is χ c1 (3P ). Then the third distinct mass eigenvalue in the fits of Table IV becomes 4629.6 MeV, a much higher value than in the previous fit, and completely unsuitable for the X(4350).
Using the methods of Ref. [21] and the inputs of Eq. (33) [taking X(3915) as the unique lightest state and Y (4140) as the sole 1 ++ state in the ccss Σ + g (1S) multiplet], we obtain m δ = mδ = 2063.7-2085. 5 MeV , In comparison with the remaining states of Table II, the Σ + g (1P ) multiplet center of mass lies extraordinarily close to that of Y (4626), while X(4500) is somewhat light to serve as a Σ + g (2S) state [plausibly, it could even be the heavier Σ + g (1S) 0 ++ state], but X(4700) works well as the lighter 0 ++ state in the Σ + g (2S) multiplet. Had Y (4274) instead been used for these fits, the results would have been hundreds of MeV higher, reinforcing the conclusion that Y (4140) works much better as a ccss state and Y (4274) as χ c1 (3P ).

VI. CONCLUSIONS
This paper expands upon the work of Refs. [21,30,31] to incorporate the hidden-bottom (bbqq ) and hiddencharm/strange (ccss) sectors into the dynamical diquark model, primarily (but not exclusively) for the states that lie in their respective ground-state [Σ + g (1S)] multiplets. Starting from a Hamiltonian with only 3 parameters (for bbqq ) or 2 parameters (for ccss) that describes the fine structure within each multiplet of the model, we obtain explicit, closed-form expressions for all 12 bbqq isomultiplet masses and all 6 ccss masses.
In the bbqq sector, the masses of the Z b (10610) and Z b (10650) combined with their relative preferences to decay to Υ or h b states are sufficient to highly constrain all other masses and heavy-quark-spin decay-mode preferences in the Σ + g (1S) multiplet. In particular, the lightest states carry J P C = 0 ++ and lie only a few 10's of MeV above the BB threshold, and thus may have observably small widths. The 1 ++ analogue of the X(3872) is predicted to lie in an especially constrained range (10598-10607 MeV), near the BB * threshold.
In a redux of the ccqq sector (following on Ref. [30]), we find that the 3-parameter Hamiltonian also predicts an isoscalar 0 ++ state that is lighter than X(3872), but with exactly the right mass to merge with the conventional charmonium χ c0 (2P ) candidate at 3860 MeV. Moreover, the fit values of the diquark internal spin cou-pling κ qb in the bbqq sector and κ qc in the ccqq sector are numerically equal, but both are much smaller than κ sc in the ccss. The isospin-dependent couplings V 0 in the bbqq and ccqq sectors are both positive, having the same sign as the corresponding pion-exchange operator in hadronic physics.
Once the center of mass for the Σ + g (1S) multiplet is determined from this analysis, we use potentials calculated in lattice simulations to compute the corresponding centers for higher multiplets, such as Σ + g (1P ) and Σ + g (2S). We find that Y (10860) works well as a bbqq 1P state but Y (10750) is too light, very likely being primarily a D-wave conventional bottomonium state.
In the ccss sector, we find it possible to identify X(3915) as a 2 ++ state, but only if the diquark spin coupling κ sc has opposite sign to the positive one nearly universally accepted. Thus the assignment J P C = 0 ++ is much more natural in the dynamical diquark model. Additionally, X(4350) emerges directly as a ccss state. We also find that Y (4140) is much more likely the sole 1 ++ Σ + g (1S) ccss state and Y (4274) is the conventional charmonium state χ c1 (3P ). Computing higher center-ofmultiplet masses, we find that Y (4626) fits the Σ + g (1P ) multiplet well and X(4700) [but not X(4500)] fits the Σ + g (2S) multiplet well.
To summarize, the dynamical diquark model produces a large number remarkable results, both in the fine structure of individual multiplets by employing an extremely simple model, and in the calculated splittings between multiplets, by using potentials calculated from first principles on the lattice. It further produces interesting physical insights in multiple sectors of exotic states, thus far including ccqq , ccss, and bbqq . One could similarly analyze hidden-charm/open-strange states, B c -like exotics, pentaquarks, and other possibilities.