Light tetraquark states with the exotic quantum number $J^{PC} = 3^{-+}$

We apply the method of QCD sum rules to study the $s q \bar s \bar q$ tetraquark states with the exotic quantum number $J^{PC} = 3^{-+}$, and extract mass of the lowest-lying state to be $2.33^{+0.19}_{-0.16}$ GeV. To construct the relevant tetraquark currents we need to explicitly add the covariant derivative operator. Our systematical analysis on their relevant interpolating currents indicates that: a) this state well decays into the $P$-wave $\rho\phi/\omega\phi$ channel but not into the $\rho f_2(1525)/\omega f_2(1525)/\phi f_2(1270)$ channels, and b) it well decays into the $K^*(892) \bar K_2^*(1430)$ channel but not into the $P$-wave $K^*(892) \bar K^*(892)$ channel.


I. INTRODUCTION
There have been many candidates of exotic hadrons observed in particle experiments, which can not be well explained in the traditional quark model. However, many of them do not have "exotic" quantum numbers, making them not so easy to be clearly identified as exotic hadrons [1][2][3][4][5][6][7][8][9][10]. Accordingly, the hadrons with "exotic" quantum numbers are of particular interests, such as the mesons with the spin-parity quantum numbers J P C = 0 −− , 0 +− , 1 −+ , 2 +− , and 3 −+ , etc. These mesons can not be explained as traditionalqq states any more, while their possible interpretations are hybrid states and tetraquark states, etc.
In this paper we shall investigate the exotic quantum number J P C = 3 −+ , and we shall investigate the qsqs (q = u/d) tetraquark states with such a quantum number. They may exist in the energy region around 2.0 GeV. With a large amount of J/ψ sample, the BESIII Collaboration are carefully examining the physics happening in this energy region [11][12][13][14][15][16][17]. So do the Belle-II [18] and GlueX [19] experiments. Hence, these states are potential exotic hadrons to be observed in future experiments.
However, there has not been any theoretical study directly on this subject. In Ref. [20] the authors used the one-boson-exchange model to study the D * D * 2 molecular state of J P C = 3 −+ . They found that the isoscalar (I = 0) state has the most attractive potential, suggesting that this D * D * 2 molecular state of J P C = 3 −+ may exist, and the K * (892)K * 2 (1430) molecular state of J P C = 3 −+ might also exist. Besides, there was a Lattice QCD study on the 3 −+ glueball, but this was done forty years ago [21].
In this paper we shall investigate the qsqs (q = u/d) tetraquark state with the exotic quantum number J P C = 3 −+ . We shall use the method of QCD sum rules, and the same approach has been applied in Refs. [22][23][24] to study the ssss tetraquark states of J P C = 1 ±− . Different from those studies, in the present study we need to explicitly add the covariant derivative operator in order to construct the qsqs (q = u/d) tetraquark currents of J P C = 3 −+ . This will be detailedly discussed in the next section.
This paper is organized as follows. In Sec. II, we systematically construct the qsqs (q = u/d) tetraquark currents with the exotic quantum number J P C = 3 −+ . Then we use them to perform QCD sum rule analyses in Sec. III, and perform numerical analyses in Sec. IV. The results are summarized and discussed in Sec. V, where we discuss their special decay behavior.

II. INTERPOLATING CURRENTS
In this section we construct the qsqs (q = u/d) tetraquark currents with the exotic quantum number J P C = 3 −+ . This quantum number is exotic, and can not be simply composed by using one quark and one antiquark. Moreover, we can not use only two quarks and two antiquarks without derivatives, and two quarks and two antiquarks together with at least one derivative are necessary to reach such a quantum number.
Besides, the ssss tetraquark currents of J P C = 3 −+ can not be constructed using two quarks and two antiquarks with just one derivative; in the present study we shall not investigate the qqqq tetraquark currents, since the widths of the qsqs tetraquark states (if exist) are probably narrower, making them easier of being observed.
First let us consider the diquark-antidiquark [qs][qs] construction. In principle, the derivative can be either inside the diquark/antidiquark field or between the diquark and antidiquark fields, i.e., where a · · · d are color indices, and the sum over repeated indices is taken; Γ 1,2,3,4 are Dirac matrices. However, we find that only the former construction can reach the quantum number J P C = 3 −+ .
Altogether we find six non-vanishing diquarkantidiquark currents of J P C = 3 −+ : where S denotes symmetrization and subtracting the trace terms in the set {α 1 α 2 α 3 }. Three of them η 1,3,5 α1α2α3 have the antisymmetric color structure (qs)3 C (qs) 3C , and the other three η 2,4,6 α1α2α3 have the symmetric color structure (qs) 6C (qs)6 C . Considering that the diquark fields s T a Cγ µ s b /s T a Cγ µ γ 5 s b /s T a Cσ µν s b have the quantum numbers J P = 1 + /1 − /1 ± respectively, the first current η 1 α1α2α3 has the most stable internal structure and may lead to the best sum rule result.
Besides the above diquark-antidiquark currents, we can construct six color-singlet-color-singlet mesonicmesonic currents of J P C = 3 −+ : The former three ξ 1,2,3 α1α2α3 have the quark combination [qq] [ss], and the derivatives are between the two quarkantiquark pairs; the latter three ξ 4,5,6 α1α2α3 have the quark combination [qs] [sq], and the derivatives are inside the quark-antiquark pairs. This difference is useful when in-vestigating their decay properties, which will be discussed in Sec. V.
We can further construct six color-octet-color-octet mesonic-mesonic currents, which can be related to the above color-singlet-color-singlet mesonic-mesonic currents through the Fierz transformation. Moreover, we can apply the Fierz transformation to derive the relations between diquark-antidiquark and mesonic-mesonic currents: Therefore, these two constructions are equivalent, and in the following we shall only use η 1···6 α1α2α3 to perform QCD sum rule analyses.

III. QCD SUM RULE ANALYSIS
In this section we use the currents η 1···6 α1α2α3 to perform QCD sum rule analyses. We assume that they couple to some exotic state X through where f X is the decay constant and ǫ α1α2α3 is the traceless and symmetric polarization tensor, satisfying: In this expressiong µν = g µν − q µ q ν /q 2 , and S ′ denotes symmetrization and subtracting the trace terms in the sets {α 1 α 2 α 3 } and {β 1 β 2 β 3 }. Based on Eq. (15), we study the two-point correlation function at both hadron and quark-gluon levels.
At the hadron level we use the dispersion relation to express Eq. (17) as: where ρ(s) is the spectral density. Then we parameterize it using one pole dominance for the ground state X and a continuum contribution: At the quark-gluon level we insert η 1···6 α1α2α3 into Eq. (17) and calculate it using the method of operator product expansion (OPE). After performing the Borel transformation to Eq. (17) at both hadron and quark-gluon levels, we can approximate the continuum using the spectral density above a threshold value s 0 , and obtain the sum rule equation We can use it to further evaluate M X , the mass of X, through, In the present study we have calculated OPEs up to the tenth dimension, including the perturbative term, the strange quark mass, the gluon condensate, the quark condensate, the quark-gluon mixed condensate, and their combinations: Based on these expressions, we shall perform numerical analyses in the next section.
Taking the current η 1 α1α2α3 as an example, whose sum rules are given in Eq. (22). First we investigate the convergence of OPE, which is the cornerstone of a reliable QCD sum rule analysis. We require the D = 10 term to be less than 5%: As shown in Fig. 2 using the solid curve, the lower bound of the Borel mass is determined to be M 2 B > 1.32 GeV 2 , when setting s 0 = 7.2 GeV 2 .
Then we investigate the one-pole-dominance assumption by requiring the pole contribution (PC) to be larger than 45%: As shown in Fig. 2 using the dashed curve, the upper bound of the Borel mass is determined to be M 2 B < 1.45 GeV 2 , when setting s 0 = 7.2 GeV 2 .
Altogether we obtain the Borel window to be 1.32 GeV 2 < M 2 B < 1.45 GeV 2 when setting s 0 = 7.2 GeV 2 . Redoing the same procedures by changing s 0 , we find that there are non-vanishing Borel windows as long as s 0 > 6.7 GeV 2 .
Finally, we study the mass dependence on M B and s 0 . We show the mass M X in Fig. 3 with respective to these two parameters. It is stable around s 0 ∼ 7.2 GeV 2 , and its dependence on M B is weak in the Borel window 1.32 GeV 2 < M 2 B < 1.45 GeV 2 . Accordingly, we choose the working regions to be 6.2 GeV 2 < s 0 < 8.2 GeV 2 and 1.32 GeV 2 < M 2 B < 1.45 GeV 2 , where the mass M X is evaluated to be M η1 = 2.33 +0. 19 −0. 16 GeV .
Here the central value corresponds to s 0 = 7.2 GeV 2 and M 2 B = 1.38 GeV 2 , and the uncertainty comes from M B , s 0 , and various quark and gluon parameters listed in Eqs. (29).
Similarly, we use η 2 α1α2α3 to perform numerical analyses. We show the mass extracted in Fig. 4 as a function of the threshold value s 0 (left) and the Borel mass M B (right). After extracting the working regions to be 6.6 GeV 2 < s 0 < 8.6 GeV 2 and 1.33 GeV 2 < M 2 B < 1.48 GeV 2 , we obtain where the central value corresponds to s 0 = 7.6 GeV 2 and M 2 B = 1.40 GeV 2 . The same procedures are applied to analyses the currents η 5 α1α2α3 and η 6 α1α2α3 , but the masses extracted from them are significantly larger than those from η 1 α1α2α3 and η 2 α1α2α3 . We summarize all the results in Table I.

V. SUMMARY AND DISCUSSIONS
In this paper we use the method of QCD sum rules to study light tetraquark states with the exotic quantum number J P C = 3 −+ . We find that two quarks and two antiquarks together with at least one derivative are necessary to reach such a quantum number; besides, the quark content can be qsqs (q = u/d), but can not be ssss.
Altogether we have constructed six diquarkantidiquark interpolating currents, where the derivative can only be inside the diquark/antidiquark field, i.e., η = q ↔ Ds qs + qs q ↔ Ds . (33) We use them to perform QCD sum rule analyses, and the results are summarized in Table I. The lowest mass, is extracted from the current η 1 α1α2α3 , which is defined in Eq. (4). From its definition, we clearly see that it contains one "good" diquark of s qs = 1 and one "good" antidiquark of sqs = 1 [32], with one of them orbitally excited: |J P C = 3 −+ ; s qs = sqs = 1; l qs = 1 or lqs = 1 . (34) Since the derivative can not be between the diquark and antidiquark fields, this combination is the most stable one, phenomenologically.
In the present study we have also constructed six meson-meson interpolating currents, as defined in Eqs. (9)(10)(11)(12)(13). Three of them have the quark combination [qq] [ss], and the derivative is between the two quarkantiquark pairs, Hence, a special decay behavior of the sqsq tetraquark states with J P C = 3 −+ is that: a) they well decay into the the P -wave (qq) S−wave (ss) S−wave final states but not into the (qq) S−wave (ss) P −wave or (qq) P −wave (ss) S−wave final states, and b) they well decay into the (qs) S−wave (sq) P −wave final states but not into the P -wave (qs) S−wave (sq) S−wave final states. Especially, we use the Fierz transformation given in Eq. (14) to investigate the light sqsq tetraquark state defined in Eq. (34). It is well coupled by the current η 1 α1α2α3 , and its mass has been calculated to be 2.33 +0. 19 −0.16 GeV. Its isospin can be either I = 0 or I = 1, which can not be differentiated in the present study. It has a special decay behavior that: a) it well decays into the P -wave ρφ/ωφ channel but not into the ρf 2 (1525)/ωf 2 (1525)/φf 2 (1270) channels, and b) it well decays into the K * (892)K * 2 (1430) channel but not into the P -wave K * (892)K * (892) channel. Note that some of these features can also be derived by analysing quantum numbers of the initial and final states.
This state lies very close to the K * (892)K * 2 (1430) threshold. Theoretically, it is not so easy to differentiate them, since we do not well understand the K * 2 (1430) meson yet. However, experimentally, one may be able to do this, since the K * (892) and K * 2 (1430) mesons are both not very narrow, i.e., Γ K * (892) = 50.3 ± 0.8 MeV and Γ K * 2 (1430) = 98.5 ± 2.7 MeV [1]. We propose to investigate the P -wave ρφ/ωφ channel in future BE-SIII, Belle-II, and GlueX experiments. If there existed a narrower resonance of J P C = 3 −+ , it would be more likely to be a compact sqsq tetraquark state other than a K * (892)K * 2 (1430) molecular state. For completeness, in the present study we have also studied its partner state with the quark content qqqq, whose mass is extracted to be 2.27 +0. 28 −0.17 GeV. To end this paper, we note that the BESIII Collaboration are possibly able to analyses some of the above decay channels simultaneously. For example, in Ref. [17] they performed a partial-wave analysis for the process e + e − → K + K − π 0 π 0 . They analysed the four subprocesses K + (1460)K − , K + 1 (1400)K − , K + 1 (1270)K − , and K * + (892)K * − (892), where they clearly observed the φ(2170)/Y (2175) in the former two processes but not in the latter two processes.