# Light Tetraquark State Candidates

Authors: Zhi-Gang Wang
Journal: Advances in High Energy Physics
 (2020)

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## Abstract

In this article, we study the axialvector-diquark-axialvector-antidiquark type scalar, axialvector, tensor, and vectorsss¯s¯ tetraquark states with the QCD sum rules. The predicted mass mX=2.08±0.12GeV for the axialvector tetraquark state is in excellent agreement with the experimental value 2062.8±13.1±4.2MeV from the BESIII collaboration and supports assigning the new X state to be a sss¯s¯ tetraquark state with JPC=1+−. The predicted mass mX=3.08±0.11GeV disfavors assigning ϕ2170 or Y2175 to be the vector partner of the new X state. As a byproduct, we obtain the masses of the corresponding qqq¯q¯ tetraquark states. The light tetraquark states lie in the region about 2GeV rather than 1GeV.


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## Body

## 1. Introduction

Recently, the BESIII collaboration studied the processJ/ψ⟶ϕηη′ and observed a structure X in the ϕη′ mass spectrum [1]. The fitted mass and width are mX=2002.1±27.5±15.0MeV and ΓX=129±17±7MeV, respectively, with assumption of the spin-parity JP=1−, the corresponding significance is 5.3σ, while the fitted mass and width are mX=2062.8±13.1±4.2MeV and ΓX=177±36±20MeV, respectively, with assumption of the spin-parity JP=1+, the corresponding significance is 4.9σ. The X state was observed in the ϕη′ decay model rather than in the ϕη decay model; they may contain a large sss¯s¯ component; in other words, it may have a large tetraquark component. In Ref. [2], Wang et al. assign the X state to be the second radial excitation of h11380. In Ref. [3], Cui et al. assign X to be the partner of the tetraquark state Y2175 with the JPC=1+−.We usually assign the lowest scalar nonet mesonsf0500,a0980,κ0800,f0980 to be tetraquark states and assign the higher scalar nonet mesons f01370,a01450,K0∗1430,f01500 to be the conventional 3P0 quark-antiquark states [4–6]. In Ref. [7], we take the nonet scalar mesons below 1GeV as the two-quark-tetraquark mixed states and study their masses and pole residues with the QCD sum rules in detail and observe that the dominant Fock components of the nonet scalar mesons below 1GeV are conventional two-quark states. The light tetraquark states may lie in the region about 2GeV rather than lie in the region about 1GeV.In this article, we take the axialvector diquark operators as the basic constituents to construct the tetraquark current operators to study the scalar (S), axialvector (A), tensor (T), and vector (V) tetraquark states with the QCD sum rules and explore the possible assignments of the new X state. We take the axialvector diquark operators as the basic constituents because the favored configurations from the QCD sum rules are the scalar and axialvector diquark states [8–10]; the current operators or quark structures chosen in the present work differ from that in Ref. [3] completely.The article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of thesss¯s¯ tetraquark states in Section 2; in Section 3, we present the numerical results and discussions; Section 4 is reserved for our conclusion.


## 2. QCD Sum Rules for thesss¯s¯ Tetraquark States

We write down the two-point correlation functionsΠμναβp and Πp firstly
(1)Πμναβp=i∫d4xeip·x0∣TJμνxJαβ†0∣0,Πp=i∫d4xeip·x0∣TJ0xJ0†0∣0,where Jμνx=J2,μνx, J1,μνx,
(2)J2,μνx=εijkεimn2sTjxCγμskxs¯mxγνCs¯Tnx+sTjxCγνskxs¯mxγμCs¯Tnx,J1,μνx=εijkεimn2sTjxCγμskxs¯mxγνCs¯Tnx−sTjxCγνskxs¯mxγμCs¯Tnx,J0x=εijkεimnsTjxCγμskxs¯mxγμCs¯Tnx,where i, j, k, m, and n are color indexes and C is the charge conjugation matrix. Under charge conjugation transform C^, the currents Jμνx and J0x have the properties
(3)C^J2,μνxC^−1=+J2,μνx,C^J1,μνxC^−1=−J1,μνx,C^J0xC^−1=+J0x.The doubly strange diquark operators(4)sTjCΓsk=12sTjCΓsk−sTkCΓsj=12εijksTjCΓskwith Γ=γμ, σμν in color antitriplet 3¯c and
(5)sTjCΓsk=12sTjCΓsk+sTkCΓsjwith Γ=1, γ5, γμγ5 in color sextet 6c satisfy Fermi-Dirac statistics. On the other hand, the scattering amplitude for one-gluon exchange is proportional to
(6)λa2ijλa2kl=−13δijδkl−δilδkj+16δijδkl+δilδkj,where
(7)εmikεmjl=δijδkl−δilδkj.λa is the Gell-Mann matrix. The negative sign in front of the antisymmetric antitriplet 3¯c indicates that the interaction is attractive, which favors formation of the diquarks in color antitriplet. The positive sign in front of the symmetric sextet 6c indicates that the interaction is repulsive, which disfavors formation of the diquarks in color sextet. The diquark states which couple potentially to the sTjCsk, sTjCγ5sk, and sTjCγμγ5sk operators in color sextet 6c are expected to have larger masses than the diquark states which couple potentially to the sTjCγμsk and sTjCσμνsk operators in color antitriplet 3¯c. We prefer the diquark operators in color antitriplet 3¯c to the diquark operators in color sextet 6c in constructing the tetraquark current operators. Up to now, the scalar and axialvector diquark states in color antitriplet 3¯c have been studied with the QCD sum rules [8–10]. In our previous studies, we observed that the pseudoscalar and vector diquark states in color antitriplet 3¯c are not favored configurations and cannot lead to stable QCD sum rules, which are not included in Ref. [8]. The tensor diquark states, which have both JP=1+ and 1− components, have not been studied with the QCD sum rules yet. We can draw the conclusion tentatively that the most favored quark configuration is the axialvector diquark operator εijksTjCγμsk. In Ref. [3], Cui et al. choose the pseudoscalar diquark operator in color sextet 6c and vector antidiquark operator in color antisextet 6¯c and axialvector diquark operator in color antitriplet 3¯c and tensor antidiquark operator in color triplet 3c to construct the axialvector currents to study the axialvector tetraquark states. In Ref. [11], we choose the color octet-octet type vector four-quark current to study Y2175; Fierz rearrangement of this current cannot lead to a diquark-antidiquark type tensor component. In the present work, we choose the axialvector diquark (antidiquark) operators in color antitriplet 3¯c (triplet 3c) to construct the tensor current, which is expected to couple potentially to the lowest tetraquark states, to study both the axialvector and vector tetraquark states. The quark configuration in the present work differs completely from that in Ref. [3] and Ref. [11]; it is interesting to study the new quark configuration. Furthermore, the conclusion of the present work differs completely from that of Ref. [3].At the hadronic side, we can insert a complete set of intermediate hadronic states with the same quantum numbers as the current operatorsJμνx and J0x into the correlation functions Πμναβp and Πp to obtain the hadronic representation [11–14]. After isolating the ground state contributions of the scalar, axialvector, vector, and tensor tetraquark states, we get the results
(8)Π2,μναβp=λXT2mXT2−p2g~μαg~νβ+g~μβg~να2−g~μνg~αβ3+⋯=Π2+pg~μαg~νβ+g~μβg~να2−g~μνg~αβ3+⋯,Π1,μναβp=λ~XA2mXA2−p2p2gμαgνβ−p2gμβgνα−gμαpνpβ−gνβpμpα+gμβpνpα+gναpμpβ+λ~XV2mXV2−p2−gμαpνpβ−gνβpμpα+gμβpνpα+gναpμpβ+⋯=Π1+p2p2gμαgνβ−p2gμβgνα−gμαpνpβ−gνβpμpα+gμβpνpα+gναpμpβ+Π1−p2−gμαpνpβ−gνβpμpα+gμβpνpα+gναpμpβ,Πp=Π0+p2=λXS2mXS2−p2+⋯,where g~μν=gμν−pμpν/p2; the subscripts 2+, 1+, 1−, and 0+ denote the spin-parity JP of the corresponding tetraquark states. The pole residues λX and λ~X are defined by
(9)0∣J2,μν0∣XTp=λXTεμν,0∣J1,μν0∣XAp=λ~XAεμναβεαpβ,0∣J1,μν0∣XVp=λ~XVεμpν−ενpμ,0∣J00∣XSp=λXS,where εμν and εμ are the polarization vectors of the tetraquark states.Now we contracts quarks in the correlation functions with the Wick theorem; there are four s-quark propagators; if two s-quark lines emit a gluon by itself and the other two s-quark lines contribute a quark pair by itself, we obtain operator GGs¯ss¯s, which is of order Oαsk with k=1 and of dimension 10. In this article, we take into account that the vacuum condensates up to dimension 10 and k≤1 in a consistent way. For the technical details, one can consult Refs. [7, 15]. Once the analytical expressions of the QCD spectral densities are obtained, we take the quark-hadron duality below the continuum thresholds s0 and perform Borel transform with respect to the variable P2=−p2 to obtain the QCD sum rules:
(10)λX2exp−mX2T2=∫0s0dsρsexp−sT2,where ρs=ρSs, ρAs, ρVs, and ρTs.
(11)ρSs=s43840π6−13smss¯gsσGs384π4+2ss¯s23π2−17s¯ss¯gsσGs48π2+s2192π4αsGGπ+19mss¯s96π2αsGGπ−16mss¯s33δs+s¯gsσGs2192π2δs−s¯s224αsGGπδs,(12)ρAs=s411520π6−s2mss¯s12π4+smss¯gsσGs9π4+4ss¯s29π2−5s¯ss¯gsσGs18π2−s22304π4αsGGπ+3mss¯s64π2αsGGπ−32mss¯s39δs−2s¯s227αsGGπδs,(13)ρVs=s411520π6+s2mss¯s12π4−7smss¯gsσGs72π4−2ss¯s29π2+5s¯ss¯gsσGs18π2+s2768π4αsGGπ−79mss¯s1728π2αsGGπ+16mss¯s39δs−2s¯s281αsGGπδs−s¯gsσGs218π2δs,(14)ρTs=s45376π6−3s2mss¯s20π4+29smss¯gsσGs96π4+8ss¯s29π2−37s¯ss¯gsσGs48π2−11s21920π4αsGGπ+43mss¯s864π2αsGGπ−64mss¯s39δs−4s¯s227αsGGπδs,and λXA/V=mXA/Vλ~XA/V.We derive equation (10) with respect to τ=1/T2, then obtain the QCD sum rules for the masses of the tetraquark states through a fraction
(15)mX2=−∫0s0dsd/dτρsexp−τs∫0s0dsρsexp−τs.


## 3. Numerical Results and Discussions

We take the standard values of the vacuum condensatesq¯q=−0.24±0.01GeV3, q¯gsσGq=m02q¯q, m02=0.8±0.1GeV2, s¯s=0.8±0.1q¯q, s¯gsσGs=m02s¯s, and αsGG/π=0.012±0.004GeV4 at the energy scale μ=1GeV [12–14, 16] and choose the MS¯ mass msμ=2GeV=0.095±0.005GeV from the Particle Data Group [17] and evolve the s-quark mass to the energy scale μ=1GeV with the renormalization group equation; furthermore, we neglect the small u and d quark masses.We choose suitable Borel parameters and continuum threshold parameters to warrant the pole contributions (PC) are larger than40%, i.e.,
(16)PC=∫0s0dsρsexp−s/T2∫0∞dsρsexp−s/T2≥40%,and convergence of the operator product expansion. The contributions of the vacuum condensates Dn in the operator product expansion are defined by
(17)Dn=∫0s0dsρnsexp−s/T2∫0s0dsρsexp−s/T2,where the subscript n in the QCD spectral density ρns denotes the dimension of the vacuum condensates. We choose the values ∣D10∣~1% to warrant the convergence of the operator product expansion. In Table 1, we present the ideal Borel parameters, continuum threshold parameters, pole contributions, and contributions of the vacuum condensates of dimension 10. In Figure 1, we plot the absolute contributions of the vacuum condensates of dimension n for the central values of the input parameters in the operator product expansion. Although in some cases the contributions of the perturbative terms D0 are not the dominant contributions, the contributions of the vacuum condensates of dimensions 6 and 8 are very large; the hierarchy ∣D6∣≫∣D8∣ warrants the good convergent behavior of the operator product expansion; furthermore, the contributions D7, D9, and D10 are very small. From Table 1 and Figure 1, we can see that the pole dominance is well satisfied and the operator product expansion is well converged; we expect to make reliable predictions.Table 1
The Borel parameters, continuum threshold parameters, pole contributions, contributions of the vacuum condensates of dimension 10, masses, and pole residues of the tetraquark states, where the subscriptsS, A, T, and V denote the scalar, axialvector, tensor, and vector tetraquark states, respectively.
T2GeV2
s0GeV
Pole
∣D10∣
mXGeV
λX10−2GeV5
sss¯s¯S
1.4−1.8
2.65±0.10
40−73%
≪1%
2.08±0.13
2.73±0.56
sss¯s¯A
1.5−1.9
2.65±0.10
41−72%
<1%
2.08±0.12
1.87±0.34
sss¯s¯T
1.5−1.9
2.75±0.10
41−72%
<1%
2.22±0.11
3.02±0.53
sss¯s¯V
2.1−2.7
3.60±0.10
42−73%
≤1%
3.08±0.11
6.47±1.07
qqq¯q¯S
1.2−1.6
2.40±0.10
40−76%
≪1%
1.86±0.11
1.95±0.38
qqq¯q¯A
1.3−1.7
2.40±0.10
40−73%
≤1%
1.87±0.10
1.30±0.22
qqq¯q¯T
1.4−1.8
2.65±0.10
42−74%
≤1%
2.13±0.10
2.58±0.42
qqq¯q¯V
1.9−2.5
3.40±0.10
41−74%
≤2%
2.86±0.11
4.94±0.93Figure 1
The absolute contributions of the vacuum condensates of dimensionn for the central values of the input parameters in the operator product expansion, where the S, A, T, and V denote the scalar, axialvector, tensor, and vector tetraquark states, respectively, (a) and (b) denote the sss¯s¯ and qqq¯q¯ quark constituents, respectively.
(a)
(b)We take into account all uncertainties of the input parameters and obtain the values of the masses and pole residues of thesss¯s¯ tetraquark states, which are shown explicitly in Figure 2 and Table 1. In this article, we have assumed that the energy gaps between the ground state and the first radial state are about 0.6GeV [18–20]. In Figure 2, we plot the masses of the scalar, axialvector, tensor, and vector sss¯s¯ tetraquark states with variations of the Borel parameters at larger regions than the Borel windows shown in Table 1. From the figure, we can see that there appear platforms in the Borel windows.Figure 2
The masses with variations of the Borel parametersT2, where (a), (b), (c), and (d) denote the scalar, axialvector, tensor, and vector tetraquark states, respectively.
(a)
(b)
(c)
(d)From Table1, we can see that the uncertainties of the masses δMX are small, while the uncertainties of the pole residues δλX are large, for example, δMX/MX=6% and δλX/λX=21% for the scalar sss¯s¯ tetraquark state. We obtain the tetraquark masses from a fraction, see equation (14); the uncertainties originating from the input parameters in the numerator and denominator are almost canceled out with each other, so the net uncertainties of the tetraquark masses are very small. In this article, we have neglected the perturbative Oαs corrections. For the traditional two-quark light mesons, the perturbative Oαs corrections amount to multiplying the perturbative terms with a factor 1+11/3αs/π for the JPC=0+−, 0++ mesons, 1+αs/π for the JPC=1−−, 1++, 1+− mesons, and 1−αs/π for the JPC=2++ mesons [14]. Now we estimate the possible uncertainties due to neglecting the perturbative Oαs corrections by multiplying the perturbative terms with a factor 1+−1~4αs/π. The additional uncertainties δMX and δλX are shown in Table 2. From the table, we can see again that the uncertainties of the mass δMX are small, while the uncertainties of the pole residues δλX are large, for example, δMX/MX=−1%+2% and δλX/λX=−7%+23% for the scalar sss¯s¯ tetraquark state. In the QCD sum rules for the X, Y, Z states, which are excellent candidates for the compact tetraquark states or loosely bound molecular states, the uncertainties of the masses are less than or about 6% [21]. Ref. [21] is the most recent review.Table 2
The possible uncertainties induced by the perturbativeOαs corrections, where the subscripts S, A, T, and V denote the scalar, axialvector, tensor, and vector tetraquark states, respectively.
δmXGeV
δλX10−2GeV5
sss¯s¯S
−0.02+0.04
−0.18+0.64
sss¯s¯A
−0.02+0.03
−0.09+0.33
sss¯s¯T
−0.01+0.03
−0.18+0.63
sss¯s¯V
−0.06+0.03
−0.45+1.62
qqq¯q¯S
−0.01+0.04
−0.10+0.35
qqq¯q¯A
−0.01+0.03
−0.05+0.18
qqq¯q¯T
−0.01+0.03
−0.14+0.51
qqq¯q¯V
−0.02+0.02
−0.37+1.27The predicted massmX=2.08±0.12GeV for the axialvector tetraquark state is in excellent agreement with the experimental value 2062.8±13.1±4.2MeV from the BESIII collaboration [1], which supports assigning the new X state to be an axialvector-diquark-axialvector-antidiquark type sss¯s¯ tetraquark state. The predicted mass mX=3.08±0.11GeV for the vector tetraquark state lies above the experimental value of the mass of ϕ2170 or Y2175, mϕ=2188±10MeV, from the Particle Data Group, and disfavors assigning ϕ2170 or Y2175 to be the vector partner of the new X state. If ϕ2170 have a tetraquark component, it may have color octet-octet component [11]. As a byproduct, we obtain the masses and pole residues of the corresponding qqq¯q¯ tetraquark states, which are shown in Table 1. The present predictions can be confronted to the experimental data in the future.Now we perform Fierz rearrangement to the currents both in the color and Dirac-spinor spaces(18)J0=2s¯ss¯s+2s¯iγ5ss¯iγ5s+s¯γαss¯γαs−s¯γαγ5ss¯γαγ5s,J1,μν=2is¯ss¯σμνs−s¯σμνγ5ss¯iγ5s+iεμναβs¯γαγ5ss¯γβs,J2,μν=122s¯γμγ5ss¯γνγ5s−2s¯γμss¯γνs+2gαβs¯σμαss¯σνβs+gμνs¯ss¯s+s¯iγ5ss¯iγ5s+s¯γαss¯γαs−s¯γαγ5ss¯γαγ5s−12s¯σαβss¯σαβs.The diquark-antidiquark type currents can be rearranged into currents as special superpositions of color singlet-singlet type currents, which couple potentially to the meson-meson pairs or molecular states; the diquark-antidiquark type tetraquark states can be taken as special superpositions of meson-meson pairs and embody the net effects. The decays to their components are Okubo-Zweig-Iizuka supper-allowed; we can search for those tetraquark states in the decays(19)XS⟶η′η′,f0980f0980,ϕ1020ϕ1020,XA/V⟶f0980h11380,ϕ1020η′,ϕ1020ϕ1020,XT⟶η′η′,f0980f0980,ϕ1020ϕ1020.


## 4. Conclusion

In this article, we construct the axialvector-diquark-axialvector-antidiquark type currents to interpolate the scalar, axialvector, tensor, and vectorsss¯s¯ tetraquark states, then calculate the contributions of the vacuum condensates up to dimension 10 in the operator product expansion and obtain the QCD sum rules for the masses and pole residues of those tetraquark states. The predicted mass mX=2.08±0.12GeV for the axialvector tetraquark state is in excellent agreement with the experimental value, mX=2062.8±13.1±4.2MeV, from the BESIII collaboration and supports assigning the new X state to be an axialvector-diquark-axialvector-antidiquark type sss¯s¯ tetraquark state. The predicted mass mX=3.08±0.11GeV for the vector tetraquark state lies above the experimental value of the mass of ϕ2170, mϕ=2188±10MeV, from the Particle Data Group and disfavors assigning ϕ2170 to be the vector partner of the new X state. As a byproduct, we also obtain the masses and pole residues of the corresponding qqq¯q¯ tetraquark states. The present predictions can be confronted to the experimental data in the future.



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