D-wave excited tetraquark states with and

We study the mass spectra of D-wave excited tetraquark states with and in both symmetric and antisymmetric color configurations using the QCD sum rule method. We construct the D-wave diquark-antidiquark type of tetraquark interpolating currents in various excitation structures with . Our results support the interpretation of the recently observed resonance as a D-wave tetraquark state with in the or excitation mode, although some other possible excitation structures cannot be excluded exhaustively within theoretical errors. Moreover, our results provide the mass relations and for the positive and negative -parity D-wave tetraquarks, respectively. We suggest searching for these possible D-wave tetraquarks in both the hidden-charm channels and , as well as open-charm channels such as and .

Abstract: We study the mass spectra of D-wave excited tetraquark states with and in both symmetric and antisymmetric color configurations using the QCD sum rule method. We construct the D-wave diquark-antidiquark type of tetraquark interpolating currents in various excitation structures with . Our results support the interpretation of the recently observed resonance as a D-wave tetraquark state with in the or excitation mode, although some other possible excitation structures cannot be excluded exhaustively within theoretical errors. Moreover, our results provide the mass relations and for the positive and negative -parity D-wave tetraquarks, respectively. We suggest searching for these possible D-wave tetraquarks in both the hidden-charm channels and , as well as open-charm channels such as and .
Keywords: tetraquark states, exotic states, QCD sum rules DOI: 10.1088/1674-1137/acbf2c The history of multiquarks can be traced to 1964, when Gell-Mann and Zweig proposed such configurations in building the quark model [1,2]. Although the existence of tetraquarks and pentaquarks has long been speculated, it has rarely, if ever, been proven. The scenario has changed since 2003, owing to the observation of numerous charmoniumlike/bottomoniumlike XYZ states [3], hidden-charm states [4][5][6][7], doubly-charm states [8,9], and fully-charm tetraquark [10] states, which cannot be well explained within the traditional quark model. They are very good candidates for tetraquark and pentaquark states. Details regarding the experimental as well as theoretical progress can be found in review papers [11][12][13][14][15][16][17]. In 2017, the LHCb Collaboration observed four structures, i.e., , , and , in the decay process [18,19], among which and were confirmed to be consistent with previous measurements performed by the CDF Collaboration [20,21], CMS Collaboration [22], D0 Collaboration [23], and BABAR Collaboration [24], while and were new resonances. In-J/ψϕ X(4140) X(4274) cscs X(4500) X(4700) cscs spired by these structures observed in the invariant mass spectrum, and were considered to be the tetraquark ground states, whereas and were interpreted as the tetraquark excited states, in various theoretical methods [25][26][27][28][29][30][31][32][33]. can be assigned as the S-wave tetraquark ground states with , the state may be interpreted as the D-wave excited tetraquark state. In Ref. [28], the authors calculated the masses of the excited hidden-charm tetraquarks without internal diquark excitation (λ-mode excitation) by using the relativistic quark model. The mass of the D-wave tetraquark with was calculated to be approximately 4.8 GeV. The same λ-mode excited D-wave tetraquarks were also studied in the color flux-tube model with masses of approximately 4.9 and 5.2 GeV for the and color structures, respectively [35]. In Ref. [29], the D-wave tetraquarks were investigated in different excitation modes by considering internal excited diquarks (ρ-mode excitation) in the relativistic quark model. The masses of the ρ-mode D-wave tetraquarks with were predicted as 4.6-4.7 GeV, which are far lower than those of the λ-mode tetraquarks. Additionally, the authors of Ref. [36] calculated the mass of the ground state of the S-wave tetraquark to be approximately 4.6 GeV according to the QCD sum rules, which is far higher than those obtained in Ref. [25]. In Ref. [37], was also considered as the axialvector radial excited tetraquark state. According to the above analyses and theoretical investigations, the newly observed state may be explained as a ρ-mode excited -wave tetraquark with . In this work, we systematically study the mass spectra of the D-wave with and in both color symmetric and antisymmetric configurations within the framework of QCD sum rules [38,39]. We investigate the D-wave tetraquarks in different excitation structures, including the ρmode and λ-mode.
The remainder of this paper is organized as follows. In Sec. II, we construct the nonlocal D-wave interpolating currents for tetraquark states with and in various excitation structures and color configurations. In Sec. III, we introduce the formalism of tetraquark QCD sum rules and calculate the two-point correlation functions and spectral densities for all currents. We perform numerical analyses to extract the full mass spectra of these D-wave tetraquark states in Sec. IV. The last section presents a summary.
In this section, we construct the D-wave tetraquark interpolating currents with and . The tetraquark is composed of diquark and antidiquark fields. By analogy with the heavy baryon system, the orbital angular momentum of the tetraquark can be decomposed into , where ( ) represents the internal orbital angular momentum for the ( ) field, and represents the orbital angular momentum between the diquark and antidiquark fields. It is convenient to denote the orbital excitation of the tetraquark system as , as shown in Fig. 1. The D-wave excited tetraquarks are the excitations with . There exist several different excitation structures for the D-wave tetraquarks: , , , , , . We study all these D-wave tetraquarks by constructing the interpolating currents with the same structures and quantum numbers.
[cs] [cs] SU (3) The color structure of a diquark-antidiquark tetraquark operator can be expressed via symmetry: (3 ⊗ 3) [cs] ⊗ (3 ⊗3) [cs] =(6 ⊕3) [cs] ⊗ (3 ⊕6) [cs] 6 cs ⊗6cs3 cs ⊗ 3cs in which the color singlet structures come from the and terms, which are denoted as the color symmetric and antisymmetric configurations, respectively. In this work, we consider both these color configurations. We use only the S-wave good diquark field with to compose the D-wave tetraquark currents by inserting covariant derivative operators. For example, one can obtain a ρ-mode P-wave diquark field with where is the covariant derivative, the subscripts are color indices, denotes the charge conjugate operator, and T represents the transpose of the quark fields. The corresponding charge conjugate antidiquark fields are To compose the λ-mode excited tetraquark operator, one should insert the covariant derivative operator between the diquark and antidiquark fields.
cscs J PC = 1 ++ Considering both the symmetric and antisymmetric color configurations, we construct the D-wave interpolating tetraquark currents with as , , , , , , , , , cscs J PC = 1 +− and the D-wave interpolating tetraquark currents with as , , , , ) , , , , , , , where . The interpolating cur-rents with the superscripts "S" and "A" denote the sym- metric and antisymmetric color structures, which are abbreviated as and , respectively, hereinafter. The excitation structures , color configurations, and quantum numbers for these interpolating currents are presented in Table 1. The abbreviation ( ) indicates that the corresponding current contains two λ-orbital (ρ-orbital) momentums with an antisymmetric/symmetric color structure, while indicates that the current contains one λ-orbital momentum and one ρ-orbital momentum with an antisymmetric/symmetric color structure. In the following, we investigate the mass spectra for the D-wave tetraquarks by using these interpolating currents. Among the currents belonging to the and struc- tures, we only study the ones, because the currents would yield the same results in our calculations.

STATES
In this section, we introduce the method of QCD sum rules for the hidden-charm tetraquark states. The twopoint correlation functions for the tensor currents can be written as where is the polarization function related to the spin-1 intermediate state, and represents other tensor structures relating to different hadron states. The tensor current can couple to the spin-1 physical state X through are coupling constants, is the antisymmetical tensor, and is the polarization tensor. At the hadron level, the two-point correlation function can be written as where we use the form of the dispersion relation, and denotes the physical threshold. The imaginary part of the correlation function is defined as the spectral function, which is usually evaluated at the hadron level by inserting intermediate hadron states where we have adopted the usual parametrization of onepole dominance for the ground state X and a continuum contribution. Researchers have investigated the excited mesons [40][41][42], baryons [43], and tetraquarks [44][45][46] in QCD sum rules by using the non-local interpolating currents under the "pole+continuum" approximation. The spectral density can also be evaluated at the quarkgluon level via the operator product expansion (OPE). To pick out the contribution of the lowest lying resonance in (12), the QCD sum rules are established as  (6) and (7).
which is the function of two parameters and . We discuss the details of obtaining suitable parameter working regions in QCD sum rule analyses in next section. Using the operator production expansion method, the twopoint function can also be evaluated at the quark-gluonic level as a function of various QCD parameters, such as QCD condensates, quark masses, and the strong coupling constant . To evaluate the Wilson coefficients, we adopt the heavy quark propagator in the momentum space and the strange quark propagator in the coordinate space: where and . In this work, we evaluate the Wilson coefficients of the correlation function up to dimension ten condensates at the leading order of . We find that the calculations are highly complex owing to the existence of the covariant derivative operators. The results of spectral functions are too lengthy to present here; thus, they are provided in the Appendix.

IV. MASS SUM RULE ANALYSES cscs
In this section, we perform the QCD sum rule analyses for the tetraquark systems. We use the following values of the quark masses and various QCD condensates [3,[47][48][49][50][51][52][53][54][55]:  To ensure the unified renormalization scale in our analyses, we use the renormalization scheme and scale independent mass ratio from PDG [3] to obtain the strange quark mass .
To establish a stable mass sum rule, one should initially find the appropriate parameter working regions, i.e, for the continuum threshold and the Borel mass . The threshold can be determined via the minimized variation of the hadronic mass with respect to the Borel mass . The lower bound on the Borel mass can be fixed by requiring a reasonable OPE convergence, while its upper bound is determined through a sufficient pole contribution. The pole contribution is defined as L 0 where is defined in Eq. (13).
As an example, we use the color antisymmetric current with in the excitation mode to show the details of the numerical analysis. For this current, the dominant non-perturbative contribution to the correlation function comes from the quark condensate, which is proportional to the charm quark mass . Figure 2 shows the contributions of the perturbative term and various condensate terms to the correlation function with respect to when tends to infinity. It is clear that the Borel mass should be large enough to ensure the convergence of the OPE series. In this work, we require that the perturbative term be two times larger than the quark condensate term, providing the lower bound of the Borel mass . The other QCD condensates are far smaller than the quark condensate in this region of . Studying the pole contribution defined in Eq. (17)  bound on the Borel mass, we require the pole contribution to be larger than . As a result, the reasonable Borel window for the current is obtained as . As mentioned previously, the variation of the extracted hadron mass with respect to should be minimized to obtain the optimal value of the continuum threshold . We show the variation of with in the left panel of Fig. 3, from which the optimized value of the continuum threshold can be chosen as . In the right panel of Fig. 3, the mass sum rules are established to be very stable in the above parameter regions of and . The hadron mass for this D-wave tetraquark with can be obtained as   the following hadron mass and quantity to study the stability of the mass sum rules: where represents N definite values for the Borel parameter in the Borel window. According to the above definition, the optimal choice for the continuum threshold in the QCD sum rule analysis can be obtained by minimizing the quantity , which is a function of only . This relation is shown in the right panel of Fig. 4, in which there is a minimum point at approximately . We can thus determine the working range for the continuum threshold to be , as shown in the left panel of  In these analyses, we find that the OPE series for the and belonging to the structure differ significantly from those of other interpolating currents. As shown in the Appendix, the quark condensate does not contribute to the correlation function for any of the currents.
By performing similar analyses, we obtain the numerical results for all the other interpolating currents in Eqs. (6) and (7), and they are presented in Table 2. The extracted hadron masses from and with agree well with the mass of the newly observed resonance , implying that can be interpreted as a D-wave tetraquark state with in the excitation mode of or .
Considering the same physical picture for the and excitation structures, the interpolating currents and exhibit similar mass sum rules. The currents and give almost degenerate hadron masses, as shown in Table 2. To study their mixing effects, we also perform analyses for the mixed currents . Our calculations show that the off-diagonal correlator is nonzero, implying that the currents and may couple to the same hadron state. The same situation arises for the interpolating currents and , which couple to the same tetraquark state. We investigated the mass spectra for the D-wave tetraquark states with and in the framework of QCD sum rules. We constructed the D-wave non-local interpolating tetraquark currents with covariant derivative operators in the excitation structures. The two-point correlation functions were calculated up to dimension ten condensates in the leading order of . We established reliable mass sum rules for all these currents and obtained the mass spectra of Dwave tetraquarks, as shown in Table 2. Our results support the interpretation of the recently observed cscs J PC = 1 ++ 1 +− Table 3. Possible decay channels of the D-wave tetraquark states with and .   structure as a D-wave tetraquark state with in the or excitation mode. However, some other possibilities of the excitation modes cannot be excluded by our results within errors.

V. CONCLUSION AND DISCUSSION
The mass spectra of tetraquark states in different color configurations were studied in Ref. [35], and the results indicated that the masses of color symmetric tetraquarks are lower than those of color antisymmetric tetraquarks in the ground state ( ). Similar results were obtained for the fully heavy tetraquark states [56][57][58]. However, the situation is different for the excited tetraquarks: the masses of color antisymmetric tetraquarks are lower than those of color symmetric tetraquarks. Such behavior is consistent with our results in Table 2 for the D-wave tetraquarks, except for those in the structures with two ρ-mode excitations. In Table 2, the masses for the positive -parity tetraquarks follow the relation , and those for the negative -parity tetraquarks exhibit the relation , which is consistent with the conclusion for P-wave systems [57].
We present the mass spectra of these tetraquarks in comparison with the corresponding two-meson opencharm mass thresholds in Fig. 5. Clearly, these D-wave tetraquarks with and lie above the mass thresholds of , , and . Accordingly, we present their possible decay channels in both the S- wave and P-wave in Table 3. We suggest searching for these D-wave tetraquarks in both the hidden-charm channels and , as well as open-charm channels such as and .