Disentangling different structures in heavy-light four-quark states

Models proposed to explain recently discovered heavy-light four-quark states already assume certain internal structures, i.e. the (anti)quark constituents are grouped into diquark/antidiquark clusters, heavy-meson/light-meson clusters (hadrocharmonium) or heavy-light meson molecules. We propose and use an approach to four-quark states based on Dyson-Schwinger and Bethe-Salpeter equations that has the potential to discriminate between these models. We study heavy-light $cq\bar{q}\bar{c}$ and $cc\bar{q}\bar{q}$ four-quark states with $q=u,d,s$ and quantum numbers $I(J^{PC})=0(1^{++}),1(1^{+-}),0(0^{++})$ and $1(0^+),0(1^+),1(1^+)$. We identify the dominant components of the ground states with these quantum numbers and suggest candidates for corresponding experimental states. Most notably, we find strong heavy-light meson-meson and negligible diquark-antidiquark components in all $cq\bar{q}\bar{c}$ states, whereas for $cc\bar{q}\bar{q}$ states diquarks are present.

In the past two decades a number of highly interesting states have been identified in the charmonium and bottomonium energy regions that cannot be accommodated for in the conventional quark model for mesons made of a quark and an antiquark. Since the quark model is otherwise extremely successful in predicting spectra of heavy QQ states (with Q = c, b), these exceptional states are considered to be exotic hadrons. Some of them carry electromagnetic charge and thus may be naturally explained as four-quark states QQqq (q = u, d, s) with a light charged quark-antiquark pair in addition to the overall neutral QQ component. Thus, four-quark states are generally considered as promising candidates to explain the structure and properties of these exotic hadrons, see e.g. [1][2][3][4][5][6][7] for recent review articles.
There is, however, no agreement on the internal structure of these four-quark states. Model approaches usually assume some kind of internal clustering from the start. One possibility, the hadroquarkonium picture [8], suggests a heavy quark and antiquark grouped together in a tight core surrounded by the light qq pair. This is motivated by the experimental observation of final states with a specific charmonium state and light hadrons. The second possibility is the clustering of constituents in diquarkantidiquark (dq-dq) components which interact via colored forces, see e.g. [1] for a review. A third possibility, especially relevant for states close to open-charm thresholds, is the meson-molecule picture of arrangements into pairs of D ( * ) D ( * ) mesons that interact with each other by short-and/or long-range forces [5].
It is important to note that these possibilities are not mutually exclusive: In general, every experimental state may be a superposition of components with a different structure and the 'leading' component may be different on a case-by-case basis. It is therefore of utmost importance to develop theoretical approaches to QCD that can * e-mail: paul.wallbott@physik.uni-giessen.de † e-mail: gernot.eichmann@tecnico.ulisboa.pt ‡ e-mail: christian.fischer@physik.uni-giessen.de deal with and distinguish between all these possibilities. Lattice QCD is one such approach and has made interesting progress so far, see [9][10][11][12][13][14][15][16][17][18] and references therein. Nevertheless most simulations are still performed at an exploratory level using light quarks with unphysical large masses. Functional methods, on the other hand, have been restricted to four-quark states with equal masses [19,20] or specific quantum numbers [21]. In this work we present a generalization of the functional approach to four-quark states that has the potential to systematically address and compare heavy-light states in different flavor combinations and with different J P C quantum numbers. Based on a well-studied and understood truncation of the underlying quark-gluon interaction, we work with an approximated version of the four-body Faddeev-Yakubovsky equations that takes into account the two-body correlations that lead to the internal clustering described above. We apply the resulting formalism to the experimentally interesting cqqc hiddencharm states with quantum numbers J P C = 0(1 ++ ), 1(1 +− ) and 0(0 ++ ), which are carried by the X(3872) [22,23], the neutral Z(3900) [24] and (likely) the X(3915) [25], respectively. Furthermore, we discuss four-quark states with open charm ccqq in the channels 1(0 + ), 0(1 + ) and 1(1 + ). Currently there are no experimental candidates for these states but searches are underway. Corresponding states in the heavier bottom-quark region received a lot of attention in recent years since they are promising candidates for deeply bound and narrow states, see e.g. [15,16,26,27]. It is certainly interesting to see whether this is still the case for the experimentally more easily accessible open-charm states.
Four-body equation. The homogeneous Bethe-Salpeter equation (BSE) shown in Fig. 1 has the form where Γ is the BS amplitude, K is the four-quark interaction kernel that contains all possible two-, threeand four-body interactions, and G 0 is the product of four dressed (anti)quark propagators; see [19][20][21]  all loop momenta. Eq. (1) holds at a given pole position of the offshell qqqq scattering matrix T , which satisfies the scattering equation T = K + KG 0 T . Poles on the real axis of the total squared momentum P 2 correspond to bound states, whereas resonances appear as poles in the complex plane on higher Riemann sheets.
In this work we focus entirely on the two-body correlations in K since these generate the internal two-body clusters discussed above. The three-and four-body interactions are neglected for simplicity and need to be studied separately in future work. This leads to where a, a stand for qq,qq or qq pairs and aa is either (12)(34), (13)(24) or (14) (23). The subtraction is necessary to avoid overcounting [19,28,29]. For the two-body kernels we employ the same rainbowladder interaction that is used in the Dyson-Schwinger equation (DSE) for the quark propagator. This truncation has recently been reviewed in [30], where the DSE for the quark propagator is discussed around Eq. (3.18) and the effective interaction in Eqs. (3.95-3.96). We use Λ = 0.72 GeV for the scale parameter, adjusted to reproduce the pion decay constant f π , and η = 1.8 ± 0.2. Together with the current-quark masses, these are the only input parameters in all equations. The construction satisfies chiral constraints such as the Gell-Mann-Oakes-Renner relation, ensures the (pseudo-) Goldstone-boson nature of the pion and has been extensively applied to meson and baryon phenomenology. As discussed in [30], the truncation is well-known to reliably reproduce many properties of pseudoscalar and vector mesons (and, correspondingly, scalar and axialvector diquarks), whereas it falls short in other channels. Since we focus on twobody clusters inside tetraquarks in these channels only, we may expect at least qualitatively reasonable results.
The quantitative reliability of the approach may be judged from the results for meson masses in Table I. We work in the isospin symmetric limit where m D + = m D − = m D 0 . The u/d current-quark mass is fixed by m π , the strange quark mass is chosen such that the sum m Ds +m D * s equals the sum of the experimental values [31] and analogously for the charm quark mass in m D + m D * . The deviations between the theoretical and experimental meson masses are then below 7% in all cases.  mq is the input current-quark mass at a renormalization point µ = 19 GeV in a momentum-subtraction scheme. The column mP S contains the masses of π, D, Ds and ηc, the column mV those of ρ/ω, D * , D * s and J/ψ, and the columns mS and mA list the corresponding diquark masses. The quoted errors are obtained by varying the parameter η = 1.8 ± 0.2.
Four-quark amplitude. The main challenge in solving Eq. (1) for given J P C is the structure of the BS amplitude, which depends on four independent momenta. Its general decomposition can be written as as where the Lorentz-invariant dressing functions f i (. . . ) depend on the ten Lorentz invariant momentum variables that can be constructed from four independent momenta. The tensors τ i are the direct products of Dirac, color and flavor parts. A J = 0 state has 256 linearly independent Dirac tensors and a J = 1 state 768, which are collected in Ref. [20] and the Appendix of [21]. The color part of the amplitude consists of two independent color-singlet tensors and the flavor wave functions depend on the particular system, cf. Appendix A for details.
To extract physical content from the BS amplitude Γ (µ) (p 1 . . . p n ), we observe that the amplitude develops internal two-body clusters, which for heavy-light systems occur in the three different channels corresponding to hadroquarkonia, heavy-light meson-meson components and dq-dq clusters. These clusters may go on-shell provided that the sum of their masses is smaller than the mass of the four-body state. If this occurs in color-singlet channels, the four-quark state becomes a resonance in the two-body hadronic system of the corresponding clusters. But even if the masses of the two-body clusters are large enough such that the probed momenta only come close to the corresponding singularities, this will influence the four-body system. Thus the guiding idea is to represent Γ (µ) (p 1 . . . p n ) in terms of these two-body clusters.
Since we are interested in specific quantum numbers with experimental candidates for four-quark states, we draw on existing information on the decay channels of these states and construct our representation along the content displayed in Table II. For example, in the heavylight meson sector we took into account combinations of the I(J P ) = 1/2(0 − ) multiplets (D, D * ) and their anti-particles (D,D * ), omitting the heavy combination D * D * . Note that since we work in the isospin-symmetric limit the charged and neutral states are mass-degenerate and both taken into account. The construction of the τ (µ) i (p 1 . . . p 4 ) for the configurations qqqq, cqqc and ccqq is detailed in App. A. We construct the Dirac parts according to the dominant two-body clusters and combine them with appropriate color and flavor wave functions such that the charge-conjugation and Pauli exchange symmetries are respected. As a result, one populates a physically motivated subset of all possible basis elements. Whereas in this work we restrict ourselves to the combinations displayed in Table II, in principle one could systematically proceed and construct a complete basis for the four-body amplitude with entangled Dirac, color and flavor tensors including all possible meson and diquark channels (i.e., also those with higher total angular momentum).
Hidden-charm states. We first discuss our results for the hidden-charm cqqc four-quark states in the I(J P C ) = 0(0 ++ ) and 1(1 +− ) channels. The results for the 0(1 ++ ) state can be found in [21], where we also described our procedure to estimate (part of) the error of the calculation. In that case we found a dominant heavylight meson component, whereas the hadrocharmonium component is rather weak and the diquark component has almost no effect at all and can be neglected.
A similar pattern arises in the scalar 0(0 ++ ) case displayed in the top panel of Fig. 2. We show the mass evolution of the four-quark state when the mass of the cc pair is fixed and the mass of the other qq pair is varied from the charm mass (rightmost vertical dashed line) to the strange and light quark masses (other two vertical lines). We compare calculations with hadrocharmonium content only (squares), heavy-light meson components only (circles) and a combination (triangles). The full calculation is marked with crosses. For large masses, the dq-dq component has almost no effect on the results, whereas the hadrocharmonium component gives only mild corrections to the leading heavy-light meson components. The diquark corrections become somewhat more prominent for small masses, however without changing the general picture. Due to the sizeable error bar we cannot discriminate between a bound state and a resonance. The masses quoted in Table III are obtained from a linear fit to the mass evolution at larger quark masses; only for 0(0 ++ ) the direct calculation yields a mass below any threshold. I(J P C ) cnnc cssc I(J P ) ccnn ccss 0(0 ++ ) 3.20 (11) 3.36(10) 0(0 + ) -3.95 (10) 3.50(42) 3.59 (30) 1(0 + ) 3.80(10) -0(1 ++ ) 3.92 (7) 4.07 (6)  The general picture changes somewhat for the 1(1 +− ) state shown in the bottom of Fig. 2. Once again we find negligible diquark components and a strong heavy-light meson component, but also non-negligible contributions from the hadrocharmonium component. It is interesting to compare this behavior to the 1 ++ channel. From Table II the only difference between the two states are the hadrocharmonium components. The J/ψ π component in the 1 +− channel is significantly lighter than the J/ψ ω component in the 1 ++ channel, which lifts the degeneracy between the two states and leads to a lighter mass of the 1 +− (which is, however, opposite to the current experimental situation). We have also tested further components, which can contribute to the axialvector 1(1 +− ) state. We found the D * D * components to be negligible; however, the η c ρ component is sizable and enhances the mass splitting once included. Further studies in this direction are necessary.
The resulting masses are collected in Table III. It is interesting to compare our results with expectations from the literature. In Ref. [32] heavy-quark symmetry has been used to predict patterns for molecular states. This led to the identification of the X(3872) with a molecular state in the 1 ++ channel and the neutral Z c (3900) with a molecular state in the 1 +− channel. Our results agree with this identification: for the X(3872) we expect an almost pure heavy-light meson state which is then natural to expect to sit very close to the DD * threshold. For the Z c (3900), however, we find non-negligible corrections from other components, which may shift the physical state away from the threshold. For the scalar channel no predictions have been made in [32]. The lightest scalar molecule, if it exists, would be expected at the DD threshold. This is indeed the case for our scalar state, which sits in the region of the threshold of our D mesons (cf. Table I). Thus the mass pattern emerging in Table III is in line with our observation of heavy-light meson dominance in all cqqc states studied so far.
In Table III we also list the masses of the charm-strange cssc states extracted from the mass evolution. For I = 0 these correspond to observable states, whereas for I = 1 they are unphysical. In the 1 ++ channel there is an ex- perimental candidate, namely the X(4140) with a mass only slightly above the upper range of our error bar. Provided this identification holds, we predict a strong heavylight meson component of the X(4140), even though it is not overly close to the D sD * s threshold. In addition, we find a corresponding state in the 0 ++ channel, although the large error bars in this case make a prediction of its mass rather imprecise. moves only slightly above threshold for decreasing quark masses. Whereas the AS diquark contribution in the I = 0 case is negligible compared to the DD * contribution, the AA diquark component for I = 1 has a significant impact in pushing the mass evolution up above threshold. In the graph for 0(1 + ) we also included the D * D * component explicitly, which is negligible as in all other channels. The resulting mass hierarchy between the isosinglet and isotriplet states is as expected from heavy quark symmetry [26]. Our extrapolated values for the masses are in the ballpark expected from other approaches, see e.g. [26,27,33] and Refs. therein. In the open-charm case, the ccss states (with I = 0) must be read off from the ccqq curves with I = 1 since those have the same wave-function components (cf. Table VI). As a consequence, several slots in Table III are empty because they do not support physical states. Moreover, there is a large gap between the light and strange state in the 0(1 + ) channel but it comes again with a sizeable mass uncertainty.
Conclusions. In this work we have studied and compared heavy-light four-quark states in the charm energy region. We developed a dynamical framework that takes into account all possible combinations of internal twobody clusters and is therefore able to decide dynamically whether pictures from effective field theory and models (meson molecule, hadrocharmonium, diquarkantidiquark) are realized. For hidden charm, in all cases considered we do not find a sizeable diquark-antidiquark component. Instead, the heavy-light meson component is favoured, with channel-dependent negligible (for 0(1 ++ )) or small but significant (0(0 ++ ) and 1(1 +− )) contributions from the hadrocharmonium component. The situation for open charm is similar; the dominant contribution is the heavy-light meson component although significant corrections from the diquark-antidiquark component arise. Although the masses of the charm quarks are far from static, mass patterns expected from heavyquark symmetry are visible: In the open-charm sector we observe the expected mass hierarchy and difference between the I = 0 and I = 1 axialvector channels [26].
The observed mass pattern in the hidden charm sector resembles the one expected of the lowest-lying multiplet of states in the hadronic molecular approach [32]. In order to make further contact with heavy-quark symmetry and lattice QCD, it would be interesting to further increase the masses of the heavy quarks and explore the bottomonium sector. This is technically challenging and therefore left for future studies. A potential caveat of the present formalism is that it does not take into account potentially important effects from mixing with ordinary cc-states in the I = 0 channels [10,14]. Again, this is left for future work. In this appendix we describe the construction of the tetraquark amplitude Γ (µ) αβγδ (p 1 , p 2 , p 3 , p 4 ) in detail. It is the direct product of Dirac, color and flavor parts; here, {α, β, γ, δ} either represent the Dirac indices or absorb all Dirac, color and flavor indices to avoid clutter. If we consider the color parts alone, we work with latin indices {A, B, C, D}. We also suppress the Lorentz index µ that distinguishes between J = 0 and J = 1 states.
We consider the three different physical systems shown in Fig. 4: a state with four identical quarks qqqq, a hidden-charm system cqqc, and an open-charm state ccqq. Depending on how we connect the indices, each system can come in a diquark-antidiquark (dq-dq) configuration (12)(34) and two meson-meson configurations (13) (24) and (14)(23). In terms of diquark and meson clusters, the hidden-charm system cqqc contains two heavy-light diquarks in (12) Depending on which system we study, the amplitude is subject to certain symmetry constraints, namely Pauli antisymmetry in (12) or (34) and charge-conjugation symmetry in (13) (24) or (14) (23). Pauli antisymmetry in (12) or (34) means where a permutation of all (Dirac, color and flavor) indices is understood. Likewise, charge-conjugation symmetry in (13)(24) amounts to and in (14) (23) to where C = γ 4 γ 2 is the charge-conjugation matrix and the signs ± determine the C parity of the state. The amplitude for a qqqq system satisfies all four relations, whereas a hidden-charm amplitude cqqc is only subject to Eq. (A4) and an open-charm configuration ccqq only satisfies Eqs. (A1-A2).
To proceed, we write the most general tensor basis decomposition of the amplitude as where the Lorentz-invariant dressing functions f i (. . . ) depend on the ten Lorentz invariant momentum variables that can be constructed from four independent momenta. The tensors τ i are the direct products of Dirac, color and flavor parts.
Let us first work out the color tensors. From 3 ⊗ 3 ⊗ 3 ⊗3 = (3 ⊕ 6) ⊗ (3 ⊕6) = 1 ⊕ 1 ⊕ ..., the color part of the amplitude consists of two independent color singlet tensors, which can be taken from the dq-dq (3 ⊗ 3, 6 ⊗6) or either of the meson-meson configurations (1⊗1, 8⊗8), for example: The two tensors in the dq-dq decomposition are linear combinations of these, as well as the remaining octet-octet tensors: The tensors {C 11 , C 88 }, {C 11 , C 88 } and {C3 3 , C 66 } are mutually orthogonal. The color tensors are invariant under both charge-conjugation operations (A3-A4), i.e., they carry a positive sign, whereas either of the two Pauli symmetries (A1) or (A2) transforms them into each other: C 11 ↔ C 11 and therefore C3 3 ↔ −C3 3 . Next, we write down the Dirac-color tensors. A scalar tetraquark has 256 linearly independent Dirac tensors (see Appendix A in Ref. [21] for the full basis construction), which together with the two independent color structures amounts to 512 tensors in total: Here the Greek subscripts are Dirac indices only and we suppressed the color indices as well as those for the full Dirac-color amplitudes φ i ; they are always understood to be in the standard order αβγδ or ABCD. The tensors in (A9) correspond to the dominant 'physical' two-body clusters in terms of mesons and diquarks: φ 1 and φ 2 describe configurations with two pseudoscalar mesons in (13)(24) and (14)(23), φ 3 and φ 4 represent vector-vector meson configurations, φ 5 a (12)(34) diquark-antidiquark system with two scalar diquarks (SS) and φ 6 one with two axialvector diquarks (AA). They are part of a Fierz-complete subset of 16 Dirac tensors that do not depend on any relative momentum, i.e., the 's waves' which do not carry orbital angular momentum. In principle one could continue the list until the basis is complete.
In Table IV we collect the combinations of the Diraccolor tensors φ i and ψ i that carry definite Pauli symmetry. For example, a combination of type Φ ss is fully symmetric under both (12) and (34), Φ sa is symmetric under (12) but antisymmetric under (34), etc. Table V lists the transformation properties of the φ i and ψ i under charge conjugation.
The symmetries of the Dirac-color tensors dictate the symmetries of the remaining flavor tensors. As an example, consider the leading Dirac-color-flavor tensors (γ 5 C) C3 3 F 0 for a scalar diquark and (γ µ C) C3 3 F 1 for an axialvector diquark. Here, F 0 = [ud] is the I = 0 flavor wave function and F 1 denotes the isospin triplet consisting of uu, {ud} and dd, where [. . . ] means antisymmetrization and {. . . } symmetrization. The color tensor C3 3 is Pauli-antisymmetric and the Dirac and flavor parts are either both antisymmetric or both symmetric, which ensures the antisymmetry of the total wave function. As a consequence, scalar diquarks carry isospin I = 0 and axialvector diquarks I = 1. One can also con- Ψss Ψaa Ψsa Ψas  struct diquarks with reversed isospin by including Pauliantisymmetric momentum prefactors p 2 1 − p 2 2 (in analogy to quantum-number exotic mesons), but those prefactors would induce a strong suppression of the amplitudes. Following this argument, we restrict ourselves to dressing functions f i (. . . ) in Eq. (A5) that are even under all symmetries, so that the symmetry properties are carried by the basis tensors τ i alone.
The flavor wave functions depend on the explicit quark content (qqqq, ccqq or cqqc) and determine the isospin of the state. They also transform under the respective symmetries. Then, for a system which has Pauli symmetry, the Dirac-color amplitudes in Table IV must be combined with appropriate flavor wave functions F to form a totally antisymmetric tensor: If C parity does not apply (such as for ccqq), these are the final Dirac-color-flavor wave functions; otherwise (qqqq) one must find appropriate combinations which also have definite C parity. If the system has no Pauli symmetry and is only subject to charge-conjugation invariance in one channel (cqqc), one can combine the φ i and ψ i directly with the flavor tensors to obtain total wave functions with definite C parity. The resulting Dirac-color-flavor tensors that can be constructed in this way are listed in Table VI, together with the isospin and C parities they carry. In the following we discuss the construction for different quark content in detail.   The C parities of the full wave functions Φ ss F aa and Φ aa F ss from Eq. (A11) can be read off from Table V. Since all φ i as well as F aa and F ss have positive C parity, the same is true for their combinations and thus they describe I(0 ++ ) states, where the isospin depends on the flavor wave functions. Similarly, ψ + 1,2 and ψ 7 have nega-tive C parity so that the combinations Ψ ss F aa , Ψ aa F ss describe I(1 +− ) states. The remaining wave functions Ψ sa F as and Ψ as F sa carry I = 1 due to their flavor parts; their combinations with definite C parity are given by where Ψ ± corresponds to I(J P C ) = 1(1 +± ).
The final Dirac-color-flavor wave functions satisfy all four constraints in Eqs. (A1-A4). Each has an interpretation in terms of a meson-meson or dq-dq configuration. The corresponding physical two-body systems are listed in the right column of Table VI. cccc: For an all-charm state there is only one flavor wave function F ss = cccc which is symmetric under any quark exchange, invariant under both charge-conjugation operations and carries I = 0. As a consequence, only the combinations Φ aa F ss and Ψ aa F ss in Eq. (A11) survive and produce states with quantum numbers 0(0 ++ ) and 0(1 +− ), respectively. Note that due to the symmetry of F ss there is no overlap with the 'σ−like' components Φ ss in the nnnn system. (A14) In this case charge conjugation symmetry does not apply, so that the final wave functions are Φ aa F ss , Ψ aa F ss , Ψ as F sa with quantum numbers 1(0 + ), 1(1 + ) and 0(1 + ), respectively. Note that this does not produce a 0(0 + ) state since there is no entry Φ as in Table IV. ccss: Here the only flavor wave function is F ss , however with I = 0, so that the combinations Φ aa F ss and Ψ aa F ss produce the quantum numbers 0(0 + ) and 0(1 + ). Compared to the previous case, going from light to strange quarks for the same quantum numbers therefore leads to quite different wave-function components.
cnnc: In the hidden-charm system Bose symmetry does not apply. Instead, the flavor wave functions Up to this point we have made two approximations in the construction of the Bethe-Salpeter amplitude. Instead of the complete tensor basis, we have restricted ourselves to the dominant tensors in Eqs. (A9-A10), and we only considered dressing functions f i that are even under the relevant symmetry operations. Those dressing functions, however, still depend on ten independent variables: the total momentum P 2 = (p 1 + p 2 + p 3 + p 4 ) 2 = −M 2 , where M is the mass of the tetraquark, plus nine variables that are dynamical. This is what makes a full numerical solution practically impossible.
In the final step we therefore assume that the momentum dependence of the f i is mainly carried by the symmetric variable and that the remaining angular dependencies can be well described by two-body intermediate poles corresponding to the physical components listed in Table VI. Such a behavior has been found directly in the dynamical solution of the four-body equation for light scalar tetraquarks [20] and it is also the approximation employed in Ref. [21].
We therefore assume where the pole factors depend on the respective tensor component: where we compute all meson and diquark masses from their two-body Bethe-Salpeter equations. Note that these pole factors are invariant under the symmetry operations (A1-A4) that are relevant for the particular system (in the cnnc example only charge conjugation in (14)(23) applies), so we can also attach them directly to the tensors (A9-A10). As a result, only the dependence on the symmetric variable S 0 remains dynamical.
The f i (S 0 ) are therefore the amplitudes that are determined in the solution of the four-body equation shown in Fig. 1, together with the tetraquark mass M . By switching off individual tensor components we can determine their strengths and isolate the dominant contributions. This completes our construction of the tetraquark Bethe-Salpeter amplitude.