Fully-strange tetraquark states with the exotic quantum numbers J PC = 0 + − and 2 + −

We study the fully-strange tetraquark states with the exotic quantum numbers J PC = 0 + − and 2 + − . We construct their corresponding diquark-antidiquark interpolating currents, and apply the QCD sum rule method to calculate both their diagonal and off-diagonal correlation functions. The obtained results are used to construct some mixing currents that are nearly non-correlated, from which we extract the masses of the lowest-lying states to be M 0 + − = 2 . 47 +0 . 33 − 0 . 44 GeV and M 2 + − = 3 . 07 +0 . 25 − 0 . 33 GeV. We apply the Fierz rearrangement to transform the diquark-antidiquark currents to be the combinations of meson-meson currents, and the obtained Fierz identities indicate that these two states may be searched for in the P -wave ϕ (1020) f 0 (1710) /ϕ (1020) f ′ 2 (1525)( → ϕK ¯ K/ϕππ ) channels.

In this paper we shall investigate the fully-strange tetraquark states with the exotic quantum numbers J P C = 0 +− and 2 +− through the method of QCD sum rules.Recently, we have applied this method to study the fully-strange tetraquark states of J P C = 0 −+ /1 ±± /4 +− in Refs.[84][85][86][87][88][89][90].In the present study we shall explicitly add the covariant derivative operator in order to construct the fully-strange tetraquark currents of J P C = 0 +− and 2 +− .We shall construct some diquarkantidiquark interpolating currents, and apply the QCD sum rule method to calculate both their diagonal and off-diagonal correlation functions.The obtained results will be used to construct some mixing currents that are nearly non-correlated, from which we shall extract the masses of the lowest-lying states to be M 0 +− = 2.47 +0.33  −0.44 GeV , M 2 +− = 3.07 +0.25 −0.33 GeV .
With a large amount of J/ψ sample, BESIII are carefully studying the physics happening around this energy region [91][92][93][94][95][96][97], and so do the Belle-II [98] and GlueX [99].Therefore, the above fully-strange tetraquark states of J P C = 0 +− and 2 +− are potential exotic hadrons to be observed in the future experiments.The present study would provide not only complementary information for possible counter-candidates in the charm sector cccc [100][101][102], but also systematic understanding of exotics in a wider flavor region.This paper is organized as follows.In Sec.II we construct the fully-strange tetraquark currents with the exotic quantum numbers J P C = 0 +− and 2 +− .These currents are used in Sec.III to perform QCD sum rule analyses, where we calculate both their diagonal and offdiagonal correlation functions.Based on the obtained results, we use the diquark-antidiquark currents to perform numerical analyses in Sec.III A, and use their mixing currents to perform numerical analyses in Sec.III B. Sec.IV is a summary.

II. TETRAQUARK CURRENTS
In this section we construct the fully-strange tetraquark currents with the exotic quantum numbers J P C = 0 +− and 2 +− .Note that these two quantum numbers can not be simply reached by using one quark arXiv:2307.07819v2[hep-ph] 18 Oct 2023 field and one antiquark field, and they can not be reached by only using two quark fields and two antiquark fields neither.Actually, we need two quark fields and two antiquark fields together with one or more derivatives to reach them.
We have systematically constructed three independent diquark-antidiquark currents in Ref. [90] using two quark fields and two antiquark fields together with two derivatives: Here and the symbol S denotes symmetrization and subtracting trace terms in the set Following a similar procedure for η α1α2α3α4 4 +− ;1/2/3 , for J P C = 0 +− and 2 +− currents, we can use the spin-0 and spin-2 projection operators rather than the symmetrization operator S: For J P C = 0 +− , we can construct three independent diquark-antidiquark currents: For J P C = 2 +− , we can also construct three independent diquark-antidiquark currents: The internal orbital angular momenta contained in the above diquark-antidiquark currents are all where L is the total orbital angular momentum, l ρ and l ρ ′ are the momenta inside the diquark and antidiquark respectively, and l λ is the momentum between the diquark and antidiquark, as depicted in Fig. 1.Among the above diquark-antidiquark currents, η  .We use lρ and l ρ ′ to denote the momenta inside the diquark and antidiquark respectively, and l λ to denote the momentum between them.We use l ′ ρ and l ′ ρ ′ to denote the momenta inside the two mesons, and l ′ λ to denote the momentum between them.The Fierz identities given in Eqs.(23) indicate that the internal orbital angular momenta contained in the diquark-antidiquark currents η Moreover, the currents η ••• 0/2/4 +− ;1 are constructed by using the S-wave diquark field s T a Cγ µ s b of J P = 1 + , and the currents η ••• 0/2/4 +− ;3 are constructed by using the diquark field s T a Cσ µν s b of J P = 1 ± that contains both the S-and P -wave components, so they may lead to better QCD sum rule results.Oppositely, the currents η ••• 0/2/4 +− ;2 are constructed by using the P -wave diquark field s T a Cγ µ γ 5 s b of J P = 1 − , so their predicted masses are probably larger.The results of Ref. [90] have partly verified these analyses, as summarized in Table I.
Besides the diquark-antidiquark configuration, we can also investigate the meson-meson configuration.We have constructed three independent meson-meson currents of J P C = 4 +− in Ref. [90]: We can similarly construct three independent mesonmeson currents of J P C = 0 +− : and three independent meson-meson currents of J P C = 2 +− : × sa γ α1 γ 5 As depicted in Fig. 1, the internal orbital angular momenta contained in the above meson-meson currents are all where l ′ ρ and l ′ ρ ′ are the momenta inside the two mesons, and l ′ λ is the momentum between them.We can apply the Fierz rearrangement to derive the relations between η We shall use these Fierz identities to study the decay behaviors at the end of this paper.

III. QCD SUM RULE ANALYSES
The QCD sum rule method has been successfully applied to study various conventional and exotic hadrons in the past fifty years [103][104][105][106][107][108][109][110].In this section we apply this non-perturbative method to study the fullystrange tetraquark currents η 0 +− ;1/2/3 of J P C = 0 +− and η β1β2 2 +− ;1/2/3 of J P C = 2 +− .We use the three currents η 0 +− ;1/2/3 of J P C = 0 +− as examples, and assume that they couple to the states where X n is the state that the current η 0 +− ;i can couple to, N is the number of such states, and f in is the 3 × N matrix for the coupling of the current η 0 +− ;i to the state X n .Then we can investigate the diagonal and offdiagonal correlation functions at both the hadron and quark-gluon levels.
At the hadron level we use the dispersion relation to express Π ij (q 2 ) as where s < = 16m 2 s is the physical threshold.The spectral density ρ phen ij (s) can be generally parameterized for the states X n and a continuum as where M n is the mass of the state X n .At the quark-gluon level we apply the method of operator product expansion (OPE) to calculate Π ij (q 2 ), from which we can extract the OPE spectral density In the present study we have calculated the Feynman diagrams depicted in Fig. 2, where we use the strangeness quark propagator as We have considered the perturbative term, the quark condensate ⟨ss⟩, the gluon condensate ⟨g 2 s GG⟩, the quarkgluon mixed condensate ⟨g s sσGs⟩, and their combinations.We have calculated all the diagrams proportional to g N =0 s and g N =1 s , but we have only partly calculated the diagrams proportional to g N ≥2 s .

A. Single-channel analysis
In this subsection we perform the single-channel analysis by setting ρ ij (s)| i̸ =j = 0.This assumption neglects the off-diagonal correlation functions to make the three currents η 0 +− ;1/2/3 "non-correlated", i.e., any two of them can not mainly couple to the same state X, oth-erwise, Accordingly, we further assume that there are three states X 1,2,3 separately corresponding to the three currents η 0 +− ;1/2/3 through We parameterize the spectral density ρ ii (s) as one pole dominance for the single state X i between the physical threshold s < and the threshold value s 0 as well as a continuum contribution above s 0 .This simplifies Eq. ( 42) to be and the mass M i can be calculated through We use the spectral density ρ 11 (s) extracted from the current η 0 +− ;1 as an example to calculate the mass M 1 of the state X 1 .As given in Eq. ( 46), the mass M 1 depends on two free parameters: the Borel mass M B and FIG.2: Feynman diagrams calculated in the present study.The covariant derivative Dα = ∂α +igsAα contains two terms, and we use the green vertex to describe the latter term.the threshold value s 0 .We consider three aspects to find their proper working regions: a) the OPE convergence, b) the one-pole-dominance assumption, and c) the dependence of the mass M 1 on these two parameters.
Firstly, we consider the OPE convergence and require the D = 12/10/8 terms to be less than 5%/10%/20%, respectively: These conditions demand the Borel mass to be larger than M 2 B ≥ 2.31 GeV 2 , as depicted in Fig. 3. Secondly, we consider the one-pole-dominance assumption and require the pole contribution to be larger than 40%: This condition demands the Borel mass to be smaller than M 2 B ≤ 2.57 GeV 2 when setting s 0 = 14.0 GeV 2 , as depicted in Fig. 3.
Altogether we determine the Borel window to be 2.31 GeV 2 ≤ M 2 B ≤ 2.57 GeV 2 for s 0 = 14.0 GeV 2 .We redo the same procedures and find that the Borel windows exist as long as s 0 ≥ s min 0 = 12.5 GeV 2 .Accordingly, we demand the threshold value s 0 to be slightly larger and choose its working region to be 11.0 GeV 2 ≤ s 0 ≤ 17.0 GeV 2 .
Thirdly, we consider the dependence of the mass M 1 on M B and s 0 .As shown in Fig. 4, the mass M 1 is stable against M B inside the Borel window 2.31 GeV 2 ≤ M 2 B ≤ 2.57 GeV 2 , and its dependence on s 0 is acceptable inside the working region 11.0 GeV 2 ≤ s 0 ≤ 17.0 GeV 2 , where the mass M 1 is calculated to be M 0 +− ;1 = 3.21 +0. 23 −0.28 GeV .
Its uncertainty comes from M B and s 0 as well as various QCD parameters given in Eqs.(42).Note that the mass M 1 has a stability point at around s 0 ∼ 6.0 GeV 2 , as shown in Fig. 4(a).However, there does not exist the Borel window at this energy point.We apply the same method to study the other two J P C = 0 +− currents η 0 +− ;2/3 and the three . The obtained results are summarized in Table I.

B. Multi-channel analysis
In this subsection we perform the multi-channel analysis by taking into account the off-diagonal correlation functions that are actually non-zero, i.e., ρ ij (s)| i̸ =j ̸ = 0. To see how large they are, we choose s 0 = 9.0 GeV 2 and TABLE I: QCD sum rule results for the fully-strange tetraquark states with the exotic quantum numbers J P C = 0/2/4 +− , extracted from the diquark-antidiquark currents η (52) This indicates that η 0 +− ;1 and η 0 +− ;2 are strongly correlated with each other, as depicted in Fig. 5.
To diagonalize the 3 × 3 matrix ρ ij (s), we construct three mixing currents J 0 +− ;1/2/3 : where T 0 +− is the transition matrix.We use the method of operator product expansion to extract the spectral densities ρ ′ ij (s) from the mixing currents J 0 +− ;1/2/3 .After choosing we obtain at s 0 = 9.0 GeV 2 and M 2 B = 1.50 GeV 2 .Therefore, the off-diagonal terms of ρ ′ ij (s) are negligible and the three mixing currents J 0 +− ;1/2/3 are nearly noncorrelated around here, as depicted in Fig. 5. Besides, Eq. ( 55) indicates that the QCD sum rule result extracted from J 0 +− ;3 is non-physical around here due to its negative correlation function.
Since the off-diagonal terms of ρ ′ ij (s) are negligible around 0 = 9.0 GeV 2 and M 2 B = 1.50 GeV 2 , the procedures used in the previous subsection can be applied to study the three mixing currents J 0 +− ;1/2/3 .We summarize the obtained results in Table I.Especially, the mass extracted from the current J 0 +− is significantly reduced to Similarly, we can investigate the three J P C = 2 +− currents η β1β2 2 +− ;1/2/3 .We construct three mixing currents J β1β2 2 +− ;1/2/3 that are nearly non-correlated at around s 0 = 13.0GeV 2 and M 2 B = 2.0 GeV 2 : where We apply the QCD sum rule method to study the mixing currents J β1β2 2 +− ;1/2/3 , and the obtained results are summarized in Table I.Especially, the mass extracted from the current J β1β2 2 +− ;1 is the lowest: For completeness, we also summarize in Table I the QCD sum rule results obtained in Ref. [90] using the three J P C = 4 +− currents η α1α2α3α4 4 +− ;1/2/3 as well as their mixing currents J α1α2α3α4 4 +− ;1/2/3 that are nearly non-correlated at around s 0 = 11.0GeV 2 and M 2 B = 1.85 GeV 2 : where

IV. SUMMARY AND DISCUSSIONS
In this paper we apply the QCD sum rule method to study the fully-strange tetraquark states with the exotic quantum numbers J P C = 0 +− and 2 +− .We explicitly add the covariant derivative operator to construct some diquark-antidiquark interpolating currents, and apply the method of operator product expansion to calculate both their diagonal and off-diagonal correlation functions.Based on the obtained results, we construct some mixing currents that are nearly non-correlated.
We use both the diquark-antidiquark currents and their mixing currents to perform QCD sum rule analyses.The obtained results are summarized in Table I.Especially, we use the mixing currents J 0 +− ;1 and J β1β2 2 +− ;1 to derive the masses of the lowest-lying J P C = 0 +− and 2 +− states to be M 0 +− = 2.47 +0.33  −0.44 GeV , M 2 +− = 3.07 +0.25 −0.33 GeV .
In this paper we also construct some fully-strange meson-meson currents of J P C = 0 +− and 2 +− , which are related to the diquark-antidiquark currents through the Fierz rearrangement.We can use these meson-meson currents and their mixing currents to perform QCD sum rule analyses.The results extracted from these mixing currents are the same.

FIG. 4 :
FIG.4:The mass M1 of the state X1 with respect to (a) the threshold value s0 and (b) the Borel mass MB.In the subfigure (a) the short-dashed/solid/long-dashed curves are obtained by setting M 2 B = 2.31/2.44/2.57GeV 2 , respectively.In the subfigure (b) the short-dashed/solid/long-dashed curves are obtained by setting s0 = 11.0/14.0/17.0GeV 2 , respectively.These curves are obtained using the spectral density ρ11(s) extracted from the current η 0 +− ;1 .