Radiative Transitions of Charmoniumlike Exotics in the Dynamical Diquark Model

Using the dynamical diquark model, we calculate the electric-dipole radiative decay widths to X(3872) of the lightest negative-parity exotic candidates, including the four I = 0, J C = 1−− (“Y ”) states. The O(100–1000 keV) values obtained test the hypothesis of a common substructure shared by all of these states. We also calculate the magnetic-dipole radiative decay width for Zc(4020) → γX(3872), and find it to be rather smaller (< 10 keV) than its predicted value in molecular models.


I. INTRODUCTION
The number of new heavy-quark exotic-hadron candidates, presumptive tetraquark and pentaquark states, increases every year. In the past 18 years, over 40 candidates have been observed at multiple facilities and their hosted experiments. However, no single theoretical picture to describe the structure of these states has emerged as an undisputed favorite. Both the broad scope of experimental results and competing theoretical interpretations have been reviewed by many in recent years [1][2][3][4][5][6][7][8][9][10][11].
Among these competing physical approaches, the dynamical diquark picture [12] was developed to provide a mechanism through which diquark (δ)-antidiquark (δ) states could persist long enough to be identified as such experimentally. Diquarks are formed through the attractive channels 3 ⊗ 3 →3 [δ ≡ (Qq)3] and3 ⊗3 → 3 [δ ≡ (Qq ) 3 ] between color-triplet quarks. In this physical picture, the heavy quark Q must first be created in closer spatial proximity to a light quark q than to a light antiquarkq (and vice versa forQ). This initial configuration provides an opportunity for the formation of fairly compact δ andδ quasiparticles, in distinction to an initial state in which the strongly attractive 3 ⊗3 → 1 coupling immediately leads to (Qq )(Qq) meson pairs. Second, the large energy release of the production process (from a heavy-hadron decay or in a collider event) drives apart the δ-δ pair before immediate recombination into a meson pair can occur, creating an observable resonance. A similar mechanism extends the picture to pentaquark formation [13], by means of using color-triplet "antitriquarks"θ ≡ [Q3(q 1 q 2 )3] 3 .
This physical picture was subsequently developed into the dynamical diquark model [14]: The separated δ-δ pair is connected by a color flux tube, whose quantized states are best described in terms of the potentials computed using the Born-Oppenheimer (BO) approximation. These are the same potentials as appear in QCD lattice gauge-theory simulations that predict the spectrum of heavyquarkonium hybrid mesons [15][16][17][18][19]. The BO potentials are introduced into coupled Schrödinger equations that are solved numerically in order to produce predictions for the δ-δ spectrum, as shown in Ref. [20]. As one of the primary results of that work, all the observed exotic candidates are shown to be accommodated within the ground-state BO potential Σ + g , with the specific multiplets in order of increasing average mass being 1S, 1P , 2S, 1D, and 2P . A full summary of the BO potential notation is presented in Ref. [14].
The mass spectrum and preferred decay modes (organized by eigenstates of heavy-quark spin) of the 6 isosinglets and 6 isotriplets comprising the ccqq positiveparity Σ + g (1S) multiplet (where q, q ∈ {u, d}) were studied in Ref. [21]. This was the first work to differentiate I = 0 and I = 1 states in a diquark model. The specific model of Ref. [21] naturally produces scenarios in which X(3872) is the lightest Σ + g (1S) state, and moreover predicts that the lighter of the two I = 1, J P C = 1 +− states in Σ + g (1S) [Z c (3900)] naturally decays almost exclusively to J/ψ and the heavier one [Z c (4020)] to h c , as is observed. The model of Ref. [21] uses a 3-parameter Hamiltonian consisting of a common multiplet mass, an internal diquark-spin coupling, and a long-distance isospinand spin-dependent coupling (analogous to π exchange) between the light quark q in δ and light antiquarkq in δ. Similar conclusions using QCD sum rules have been obtained in Ref. [22]. The dynamical diquark model was developed further through the corresponding analysis [23] of the negativeparity ccqq Σ + g (1P ) multiplet and its 28 constituent isomultiplets (14 isosinglets and 14 isotriplets), which includes precisely four Y (I = 0, J P C = 1 −− ) states. In this case, the simplest model has 5 parameters: the 3 listed above, plus spin-orbit and tensor terms. An earlier diquark analysis using a similar Hamiltonian, but not including isospin dependence, appears in Ref. [24].
An analysis within this model of the 12 isomultiplets comprising the bbqq Σ + g (1S) multiplet and the 6 states of the ccss Σ + g (1S) multiplet appears in Ref. [25]. By using only experimental inputs for the states Z b (10610) and Z b (10650), which includes their masses and relative probability of decay into h b versus Υ states, the entire bbqq mass spectrum is predicted. In particular, the mass of the bottom analogue to X(3872) is highly constrained (≈ 10600 MeV), and the lightest bbqq state (I = 0, J P C = 0 ++ ) lies only a few MeV above the BB threshold. Furthermore, starting with the assumption that X(3915) is the lowest lying ccss state [26] and Y (4140) is the sole J P C = 1 ++ ccss state in Σ + g (1S), the remaining 4 masses in the multiplet are predicted. Emerging naturally in the spectrum is X(4350), a J/ψ-φ resonance seen by Belle [27], while Y (4626) and X(4700) are found to fit well within the Σ + g (1P ) and Σ + g (2S) ccss multiplets, respectively. The dynamical diquark model has also recently been extended to the case in which the light quarks q are replaced with heavy quarks Q to produce fully heavy tetraquark states Sparked by the recent LHCb report of at least one di-J/ψ resonance near 6900 MeV [28], Ref. [29] determined the spectrum of cccc states in the dynamical diquark model. In this system, the minimal model predicts each Swave multiplet to consist of 3 degenerate states (J P C = 0 ++ , 1 +− , 2 ++ ) and 7 P -wave states. X(6900) was found to fit most naturally as a Σ + g (2S) state, with other structures in the measured di-J/ψ spectrum appearing to match C = + members of the Σ + g (1P ) multiplet. In this paper we use the dynamical diquark model to predict radiative transitions between exotic states. So far, very few theoretical papers have investigated exoticto-exotic transitions (and of these papers, only diquark models have been considered [30,31]). One of the distinctive features of the P -wave study in Ref. [23] is the direct calculation of decay probabilities to eigenstates of heavyquark spin. Indeed, Ref. [23] uses the heavy quark-spin content of states as the main criterion for associating observed resonances with particular states in the Σ + g (1P ) multiplet, and identifies using likelihood fits two particularly plausible assignments for the states. Using the same decay probabilities, we calculate here the transition amplitudes for Σ + g (1P ) → γΣ + g (1S). We directly adapt the well-known expression for electric dipole (E1) radiative transitions used to great effect for conventional quarkonium. Since the E1 transition formula depends sensitively upon the initial and final wave functions, a comparison between our predictions and data provides an important test of the hypothesis that the purported Σ + g (1P ) and Σ + g (1S) states, such as in Y (4220) → γX(3872), truly share a common structure. The corresponding magnetic dipole (M1) expression within this model is also presented, in anticipation of the observation of relevant transitions such as Σ + g (2S) → γΣ + g (1S), or even between two Σ + g (1S) states such as Z c (4020) 0 → γX(3872). This paper is organized as follows: In Sec. II we review the current experimental data on transitions between ccqq states. Section III reprises the relevant phenomenological aspects of Ref. [23]. In Sec. IV we calculate the decay widths and decay probabilities for exoticto-exotic radiative transitions associated with two of the more probable P -wave state assignments in Ref. [23]. We conclude in Sec. V.

II. EXPERIMENTAL REVIEW OF EXOTIC-TO-EXOTIC TRANSITIONS
Although the number of exotic-candidate discoveries continues to increase at a remarkable pace, only a handful of exotic-to-exotic decays have been observed to date, through radiative [32,33] and pionic [34][35][36] transitions.
Considering first the radiative decays that form the topic of this work, thus far only E1 transitions (as indicated by changing parity ∆P = −) have been observed in two states at BESIII, the J P C = 1 −− Y (4260) [32] and Y (4220) [33], both seen to decay to a photon and the J P C = 1 ++ X(3872). Indeed, an increasing amount of evidence from BESIII (e.g., in Ref. [37]) suggests that the well-known Y (4260) is actually a collection of resonances, of which Y (4220) is just one component. Observed exotic-to-conventional radiative transitions are also rather few in number, due to the large decay widths of exotics that follows from the dominance of their strong decay modes. To date, only X(3872) → γJ/ψ and γψ(2S), also both E1 transitions, have definitely been seen (e.g., in Ref. [38]). BESIII has also recently announced an interesting negative result [39], an upper limit for Z c (4020) 0 [1 +− ] → γX(3872). Indeed, to date no M1 radiative decay (∆P = +) of any exotic candidate has yet been seen at any experiment.
As for pionic transitions, both BESIII [34] and Belle [35] have observed (indeed, discovered) Z c (3900) ± through Y (4260) → π + π − J/ψ, and BESIII recently observed Z c (3900) 0 via Y (4220) → π 0 π 0 J/ψ [36]. Assuming only a similarity of hadronic structure between various exotic candidates, one may expect several more exoticto-exotic pionic (or other light-meson) transitions to be observed in the future. An essential criterion for how such transitions may best be studied relies on the size of the pion momentum p π in such processes; for example, in the decays listed above, p π ≈ 300 MeV. Processes with smaller p π values may be reliably studied using conventional chiral perturbation theory, while processes with larger p π values would require modifications to the perturbative calculation to improve its convergence. Since the methods associated with radiative transitions (particularly E1 transitions) present fewer computational ambiguities, we defer a study of exotic-to-exotic pionic transitions for future work.
In the dynamical diquark model, all states in the mul- [14]. The current observed properties of the J P C = 1 −− (Y ) states identified with the multiplet Σ + g (1P ), whose spectroscopy is analyzed extensively in Ref. [23], are summarized in Table I.  TABLE I. J P C = 1 −− charmoniumlike exotic-meson candidates catalogued by the Particle Data Group (PDG) [40], which appear in specific spectroscopic identifications within the Σ + g (1P ) multiplet of the dynamical diquark model that are given by the cases presented in Ref. [23]. Both the particle name most commonly used in the literature and its label as given in the PDG are listed.

Particle PDG label I G J P C Mass [MeV] Width [MeV]
Production and decay

III. THEORETICAL REVIEW OF P -WAVE EXOTIC STATES
The full spectroscopy of diquark-antidiquark (δ-δ) tetraquarks and diquark-antitriquark (δ-θ) pentaquarks connected by a gluonic field of arbitrary excitation quantum numbers, and including arbitrary orbital excitations between the δ-δ or δ-θ pair, is presented in Ref. [14]. As discussed in that work, the gluonic-field excitations combined with the quasiparticle sources δ,δ,θ produce states analogous to ordinary quarkonium hybrids; therefore, these states may likewise be classified according to the quantum numbers provided by BO-approximation static gluonic-field potentials. The numerical studies of Ref. [20] show that the exotic analogues to hybrid quarkonium states lie above exotic states within the corresponding BO ground-state potential Σ + g by at least 1 GeV (just as for conventional quarkonium). Since the entire range of observed hidden-charm exotic candidates [not counting cccc candidates such as X(6900)] spans only about 800 MeV [1], it is very likely that all hidden-charm exotic states occupy energy levels within the Σ + g BO potential. All known ccqq candidates can be accommodated by the lowest Σ + g levels: 1S, 1P , 2S, 1D, and 2P , in order of increasing mass [20].
A detailed enumeration of the possible QQqq states, in which the light quarks q,q do not necessarily carry the same flavor, is straightforward for the S wave. Assuming zero relative orbital angular momenta between the quarks, any two naming conventions for the states differ only by the order in which the 4 quark spins are coupled.
In the diquark basis, defined by coupling in the order (qQ)+(qQ), the 6 possible states are denoted by [30]: where outer subscripts indicate total quark spin S. The same states may be expressed in any other basis by using angular momentum recoupling coefficients in the form of the relevant 9j symbol. For the purposes of this work, the most useful alternate basis is that of definite heavy-quark (and light-quark) spin, (QQ)+(qq): where [s] ≡ 2s + 1 denotes the multiplicity of a spin-s state. Using Eqs. (1) and (2), one then obtains Reference [23] extends this analysis by examining the P -wave multiplet, whose mass spectrum is dictated by the most minimal 5-parameter Hamiltonian: where M 0 is the common mass of the multiplet, κ qQ represents the strength of the spin-spin coupling within each diquark, V LS is the spin-orbit coupling strength, V 0 is the isospin-dependent coupling, 2 V T represents the tensor coupling, and S (qq) 12 is the tensor operator defined as The well-known tabulated expressions for matrix elements of S 12 (e.g., in Ref. [41]) hold neither in the basis of s qq , s QQ spins nor s δ , sδ spins, but rather in the basis of total light-quark angular momentum J qq : Assuming that δ andδ have no internal orbital excitation so that L qq = L, the matrix elements of S (qq) 12 are most easily computed in the J qq basis, with results that are then related back to the s qq , s QQ basis by means of recoupling using 6j symbols: Using this expression, S (qq) 12 matrix elements for all relevant states are tabulated in Ref. [23].
The experimental status of the P -wave J P C = 1 −− exotic candidates remains in flux, with BESIII providing the majority of the most recent data. With reference to the information presented in Table I, we have already noted that the analysis of the BESIII Collaboration [37] favors the interpretation of Y (4260) as a superposition of states, the lowest component of which is Y (4220). They identify the higher component with Y (4360), although the previous mass measurements of this state given in Table I are rather higher, and one of several scenarios considered in Ref. [23] proposes that Y (4360) and Y (4390) are the same state, while the higher-mass component in Ref. [37] can be interpreted as a distinct "Y (4320)". Alternately, if the only lower states are Y (4220), Y (4360), and Y (4390), then Y (4660) becomes the fourth I = 0, 1 −− candidate state in Σ + g (1P ). With the mass spectrum of these charmoniumlike states not yet entirely settled, Ref. [23] also employs information on their preferred charmonium decay modes as classified by heavy-quark spin: ψ(s QQ = 1) or h c (s QQ = 0). Assuming heavy-quark spin symmetry as expressed by the conservation of s QQ in the decays, the heavy-quark spin content P s QQ of each state becomes an invaluable diagnostic in disentangling the J P C = 1 −− spectrum. For example, from Table I one sees that Y (4220) decays to both ψ states and h c , while if Y (4360) and Y (4390) are in fact one state, the same can be said for them as well. Reference [23] also introduces a parameter designed to enforce the goodness-of-fit to a particular value f of P s QQ , which in the case of s QQ = 0 reads In terms of the parameters P s QQ , f , and , the 5 cases discussed in Ref. [23] designed to represent a variety of interpretations of the current data are: 1. Y (4220), Y (4260), Y (4360), Y (4390) masses are as given in the PDG (Table I) We previously suggested the importance of heavyquark spin-symmetry (s QQ ) conservation in the decays of exotics, particularly for Z c (3900) and Z c (4020), but also for several other exotic candidates that to date have only been observed to decay to charmonium states carrying one specific value of s QQ (e.g., to ψ or to h c ). We assume that a state like Y (4220) is able to decay to channels with either value of s QQ due to the initial state being a mixture of s QQ eigenstates, rather than to the value of s QQ changing in the decay process through a heavy-quark spin-symmetry violation. In addition, in this analysis we take the well-known radiative transition selection rules to apply to the light degrees of freedom, which carry the total angular momentum J qq defined in Eq. (6). As usual, the operators defining E1 and M1 transitions transform as J P = 1 − and J P = 1 + , respectively.
Explicit expressions for radiative transitions between quarkonium states (themselves transcribed from textbook atomic-physics formulae) appear in the literature (e.g., Ref. [45]), and may readily be adapted to the present case. In particular, the quarkonium orbital angular momentum L is replaced with J qq , and the heavy quark mass m Q is replaced with the diquark mass m δ . For E1 partial widths, one has where The labels i and f refer to initial and final states, respectively. The initial exotic state QQqq , of mass M (QQqq ) i , decays in its rest frame into a final exotic state with the same flavor content and energy E (QQqq ) f , and a photon of energy E γ . α is the fine-structure constant. ψ denotes radial wave functions of the exotic hadrons, and r is the spatial separation between the δ-δ pair centers. Q δ is the total electric charge (in units of proton charge) to which the photon couples; in Ref. [31], the diquarks are treated as pointlike, in which case one simply takes Q δ = Q Q +Q q .
Alternately, one may argue that the diquarks δ are of sufficient spatial extent that the photon couplings to the distinct quarks in δ should add through incoherent diagrams, in which case one takes Q 2 δ = Q 2 Q + Q 2 q . In our calculation we use the first option, but note in addition that a Y state, being an isosinglet, contains an equal superposition of u and d quarks. We thus take (11) Other schemes give rise to coefficients that differ from this value only at O (1).
The corresponding expression for M1 partial widths, involving no change in parity but a flip of the heavy-quark spin s QQ (hence breaking heavy-quark spin symmetry), reads This expression is presented here for completeness, in light of the current lack of experimental evidence for such transitions. However, in Sec. IV we use it to calculate the expected radiative width for the yet-unobserved [39] transition Z c (4020) 0 → γX(3872).

IV. ANALYSIS AND RESULTS
Possible assignments of observed Y states to members of the Σ + g (1P ) multiplet in this model are described by the 5 cases discussed extensively in Ref. [23] and summarized in Sec. III. Of these cases, all have excellent goodness-of-fit values χ 2 min /d.o.f. except Case 3; however, we argue this case and Case 5 to be the most phenomenologically relevant ones, since they enforce the important physical constraint that both Y (4220) and Y (4390) are observed (see Table I) to have substantial couplings to h c (s QQ = 0). Since Σ + g (1S) contains only one I = 0, J P C = 1 −− state with s QQ = 0, the requirement of providing a substantial component of this state to both of the well-separated Y (4220) and Y (4390) mass eigenstates is one of the primary obstacles to achieving a good fit.
Case 5 relieves the tension of Case 3 by identifying, as discussed in Sec. II, a new state "Y (4320)" from the data of Ref. [37]. In addition, Cases 1, 2, 3, and 5 all predict the sole I = 1, J P C = 0 −− state in Σ + g (1P ) to lie in the range 4220-4235 MeV, which agrees well with the unconfirmed state Z c (4240) carrying these quantum numbers that is observed in the LHCb paper [46] confirming the existence of Z c (4430).
Case 4 also satisfies the Y (4220)/Y (4390) s QQ = 0 criterion, but additionally assigns the rather high-mass Y (4660) to the Σ + g (1P ) multiplet; the cost is a much higher prediction (≈ 4440 MeV) for the mass of the Σ + g (1P ) I = 1, J P C = 0 −− state, in conflict with the value of m Zc(4240) .
We therefore single out the fits of Cases 3 and 5 for the decomposition of Y states with respect to the total light-quark angular momentum J qq in Tables II and III, respectively. For completeness, we also provide the corresponding information for Cases 1, 2, and 4 in Table IV.
Using the mass eigenvalues for the Y states in Table I, the state decompositions according to J qq in Tables II, III, and IV, the coefficient factors in Eq. (10), and the effective squared-charge Q 2 δ from Eq. (11), one may calculate the E1 radiative partial decay widths for Σ + g (1P ) → γΣ + g (1S) transitions from Eq. (9). The only nontrivial new input to the calculation is that of the transition matrix element ψ f |r|ψ i . Using the numerical methods for solving Schrödinger equations developed in Ref. [20], and particularly the fits performed in Ref. [25] to obtain the fine structure of the Σ + g (1S) multiplet, the optimal diquark mass is found to be as one varies over the static gluonic-field potentials Σ + g obtained in the lattice calculations of Refs. [15][16][17][18][19]. We then compute the relevant matrix element to be TABLE II. Decomposition of Y (I = 0, J P C = 1 −− ) charmoniumlike exotic candidates into a basis of good light-quark spin sqq, heavy-quark spin s QQ , and total light-quark angular momentum Jqq, performed for the 4 experimentally observed candidate states as described in Case 3 above and in Ref. [23]. A minus sign on the probability (−|P |) means that the corresponding amplitude is −|P | 1/2 , the same convention as is used for Clebsch-Gordan coefficients by the PDG [40].  (13) and (14) only refer to variation over different lattice simulations, and do not take into account other much more significant potential sources of uncertainty, such as effects due to finite diquark size. Nevertheless, such effects were shown [25] to change expectation values like r no more than 10%, a value that we adopt as a benchmark uncertainty for all Γ values computed here.
One observes from Table V that Table I, one sees that were Y (4660) truly a 1P state, then its large phase space for radiative decay to X(3872) [evident from the E 3 γ factor of Eq. (9)] would generate a radiative branching fraction of at least several percent.
The transition matrix element of Eq. (14) has already been noted to apply to all Σ + g (1P ) → γΣ + g (1S) transi-   Table II, except now performed for the 4 experimentally observed candidate states as described in Case 5 above and in Ref. [23]. tions. The only observed hidden-charm tetraquark candidates with P = − apart from the Y states are Z c (4240) and Y (4626); the latter has thus far been observed to decay only to various D s meson pairs [47,48], and therefore is very likely a ccss state [25]. As for Z c (4240), only its charged isobar has yet been observed, but assuming the existence of a degenerate Z c (4240) 0 , one may input its quantum numbers s QQ = 1, J qq = 1, J = 0 [23] into Eq. (9) to obtain Γ Z c (4240) 0 → γX(3872) = 503 keV .

V. CONCLUSIONS
In this paper we have calculated exotic-to-exotic hadronic radiative transitions using the dynamical diquark model. The most phenomenologically relevant final state is X(3872), which is a member of the model's hiddencharm ground-state multiplet Σ + g (1S). We use the results from a recent study [23] of this model for the lowest P -wave multiplet [Σ + g (1P )] of hidden-charm tetraquark states, in which the Σ + g (1P ) states are identified with the observed I = 0, J P C = 1 −− (Y ) states according to a variety of scenarios, based upon both their mass spectra and preferred decay modes to eigenstates of heavy-quark spin (e.g., J/ψ vs. h c ). We calculate E1 and M1 transition amplitudes for Σ + g (1P ) → γΣ + g (1S) and Σ + g (1S) → γΣ + g (1S) processes, respectively, and present corresponding values for the radiative decay widths of a number of particular exclusive channels.
This analysis shows that if Y (4220) and X(3872) have a similar underlying diquark structure, then one expects Γ Y (4220)→γX(3872) ≈ 100 keV. Moreover, similar values (albeit somewhat larger due to increased γ phase space) are expected for the heavier Y states in Σ + g (1P ). The extreme possibility of Y (4660) belonging to the 1P multiplet would lead to a γX(3872) branching fraction of several percent, and so the absence of such a remarkably large signal would appear to relegate Y (4660) instead to the Σ + g (2P ) multiplet. Furthermore, we found that the observed but unconfirmed Z c (4240), a candidate for the sole I = 1, J P C = 0 −− state in Σ + g (1P ), should have a substantial (≈ 500 keV) radiative decay width to X(3872) through its neutral isobar, and therefore this decay is a good candidate for future experimental investigation. Indeed, many of the Σ + g (1P ) states have not yet been observed, offering multiple potential future tests of the model. M1 transitions within a single multiplet, such as Z c (4020) 0 → γX(3872), produce much narrower widths (< 10 keV in this model), and can provide sensitive tests of substructure (e.g., diquarks vs. meson molecules).
One may also study exotic-to-exotic radiative transitions in other heavy-quark sectors (e.g., hidden-bottom or ccss exotics). Indeed, Eqs. (9) and (12) are general for any tetraquark state in the diquark-antidiquark configuration. For example, Ref. [25] calculates the mass of X b [the hidden-bottom analogue to X(3872)] to lie in a rather narrow range m X b ∈ [10598, 10607] MeV, only slightly below the observed Z b (10610) 0 . The M1 transition Z b (10610) 0 → γX b is thus expected from Eq. (12) to produce a tiny [O(eV) or less] radiative width, owing to not only the small phase space, but also the larger (bcontaining) diquark mass. We conclude that even very coarse experimental results in other sectors can be decisive in verifying or falsifying particular models.    Tables II-III, except now performed for the 4 experimentally observed candidate states as described in Cases 1, 2, and 4 above and in Ref. [23].