Analysis of the scalar nonet mesons with QCD sum rules

In this article, we assume that the nonet scalar mesons below 1GeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\,\mathrm { GeV}$$\end{document} are the two-quark–tetraquark mixed states and study their masses and pole residues using the QCD sum rules. In the calculation, we take into account the vacuum condensates up to dimension 10 and the O(αs)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(\alpha _s)$$\end{document} corrections to the perturbative terms in the operator product expansion. We determine the mixing angles, which indicate the two-quark components are much larger than 50%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$50~\%$$\end{document}, then we obtain the masses and pole residues of the nonet scalar mesons.

We can use QCD sum rules to study the two-quark and tetraquark states. QCD sum rules provide a powerful theoretical tool in studying the hadronic properties, and they have been applied extensively to study the masses, decay constants, hadronic form factors, coupling constants, etc. [19][20][21]. There have been several works on the light tetraquark states using the QCD sum rules [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40]. In Refs. [22][23][24], the scalar nonet mesons below 1 GeV are taken to be the tetraquark states consist of scalar diquark pairs and studied with the QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of dimension 6. In Ref. [29], Lee carries out the operator product expansion by including the vacuum condensates up to dimension 8, and observes no evidence of the couplings of the tetraquark currents to the light scalar nonet mesons. In Refs. [30][31][32], Chen, Hosaka and Zhu study the light scalar tetraquark states with the QCD sum rules in a systematic way. In Ref. [33], Sugiyama et al. study the non-singlet scalar mesons a 0 (980) and κ 0 (800) as the two-quark-tetraquark mixed states with the QCD sum rules, and observe that the tetraquark currents predict lower masses than the two-quark currents, and the tetraquark states occupy about (70-90) % of the lowest mass states.
In this article, we assume that the scalar nonet mesons below 1 GeV are the two-quark-tetraquark mixed states and study their properties with the QCD sum rules in a systematic way by taking into account the vacuum condensates up to dimension 10 and the O(α s ) corrections to the dimension zero terms in the QCD spectral densities in the operator product expansion.
The article is arranged as follows: we derive the QCD sum rules for the scalar nonet mesons in Sect. 2; in Sect. 3, we present the numerical results and discussions; and Sect. 4 is reserved for our conclusions.

The scalar nonet mesons with the QCD sum rules
In the scenario of conventional two-quark states, the structures of the scalar nonet mesons in the ideal mixing limit can be symbolically written as In the scenario of tetraquark states, the structures of the scalar nonet mesons in the ideal mixing limit can be symbolically written as [2][3][4] If we take the diquarks and antidiquarks as the basic constituents, the two isoscalar statesūdud andssū u+dd √ 2 mix ideally,ssū u+dd √ 2 degenerates with the isovector statesssdu, ssū u−dd √ 2 andssūd naturally. The mass spectrum is inverted compare to the traditionalqq mesons. The lightest state is the non-strange isosinglet, the heaviest states are the degenerate isosinglet and isovector states with hiddenss pairs, the four strange states lie in between.
In this article, we take the scalar nonet mesons to be the two-quark-tetraquark mixed states, and write down the two-point correlation functions S ( p), where S = f 0 (980), a 0 0 (980), κ + 0 (800), f 0 (500), and , , the currents J 4 S (x) and J 2 S (x) are tetraquark and twoquark operators, respectively, and couple potentially to the tetraquark and two-quark components of the scalar nonet mesons, respectively, the θ S are the mixing angles. In the currents J 4 S (x), the i, j, k, ... are color indices and C is the charge conjugation matrix, the i jk u T j (x)Cγ 5 d k (x), i jk u T j (x)Cγ 5 s k (x), and i jk d T j (x)Cγ 5 s k (x) represent the scalar diquarks in the color antitriplet, the corresponding antidiquarks can be obtained by charge conjugation. The one-gluon exchange force and the instanton induced force can result in significant attractions between the quarks in the scalar diquark channels [3,41].
The tetraquark operator J 4 S (x) contains a hiddenqq component with q = u, d or s. If we contract the corresponding quark pair in the currents J 4 S (x) and substitute it by the quark condensate, 1 then The contracted parts appear as the normalization factors − qq and J 2 f 0 (500) (x), respectively. We insert a complete set of intermediate states with the same quantum numbers as the current operators J S (x) satisfying the unitarity principle into the correlation functions S ( p 2 ) to obtain the hadronic representation [19][20][21]. After isolating the ground state contributions from the pole terms of the scalar nonet mesons, we get the result where we have used the definitions 0|J S (0)|S = λ S for the pole residues. The correlation functions can be re-written as where m, n = 2, 4. We can prove that mn S ( p 2 ) = nm S ( p 2 ) with the replacements x → −x and p → −p for m = n.
In the following, we briefly outline the operator product expansion for the correlation functions mn S ( p 2 ) in perturbative QCD. First of all, we contract the u, d, and s quark fields in the correlation functions mn S ( p 2 ) with the Wick theorem, and we obtain the results where α, β, λ, and τ are Dirac spinor indices. 42 where where q = u, d [21]. We make the assumption of vacuum saturation for the higher dimension vacuum condensates and factorize the higher dimension vacuum condensates into lower dimension vacuum condensates [19,20], for example, qqqq ∼ qq qq , qqqg s σ Gq ∼ qq qg s σ Gq , where q = u, d, s. Factorization works well in the large N c limit, but in reality, N c = 3, some (not many) ambiguities maybe originate from the vacuum saturation assumption. In Fig. 1, we show the Feynman diagrams containing thē qq annihilations accounting for the mixing of different Fock states. The quark-pair annihilations are substituted by the condensates qq q q and qq q g s σ Gq as there are normalization factors qq in the interpolating currents J 2 S (x). The perturbative part of the quark-pair annihilations must disappear as only the terms qq and q j σ μν q i in the full quark propagators U i j (x), D i j (x), and S i j (x) survive in the limit x → 0, where q = u, d, s. In Eq. (18), we retain the terms q j σ μν q i and s j σ μν s i come from the Fierz re-arrangement of q iq j and s is j to absorb the gluons emitted from other quark lines to form q j g s G a αβ t a mn σ μν q i and s j g s G a αβ t a mn σ μν s i to extract the mixed condensates qg s σ Gq and sg s σ Gs . Some terms involving the mixed condensates qg s σ Gq and sg s σ Gs appear and play an important role in the QCD sum rules; see the second Feynman diagram shown in Fig. 1 and the first two Feynman diagrams shown in Fig. 2.
Then we compute the integrals in the coordinate space to obtain the correlation functions S ( p 2 ), therefore the QCD spectral densities ρ S (s) at the quark level through the dispersion relation, In this article, we approximate the continuum contributions by which contain both perturbative and non-perturbative contributions, we use s 0 S to denote the continuum threshold parameters. For the conventional two-quark scalar mesons, only perturbative contributions survive in such integrals; see Eqs. (26)- (27), (30) and (33).
In this article, we carry out the operator product expansion by including the vacuum condensates up to dimension 10. The condensates g 3 s GGG , α s GG π 2 , α s GG π qg s σ Gq have the dimensions 6, 8, 9, respectively, but they are the vacuum expectations of the operators of the order O(α s ), respectively, their values are very small and discarded. We take the truncations n ≤ 10 and k ≤ 1, the operators of the orders O(α k s ) with k > 1 are discarded. Furthermore, we take into account the O(α s ) corrections to the perturbative terms, which were calculated recently [40]. As there are normalization factors qq 2 in the correlation functions 22 S ( p) , we count those perturbative terms as of the order qq 2 , and we truncate the operator product expansion to the order qq 2 q q , where q, q = u, d, s.
Once the analytical QCD spectral densities are obtained, then we can take the quark-hadron duality below the continuum thresholds s 0 S and perform the Borel transformation with respect to the variable P 2 = −p 2 , finally we obtain the QCD sum rules, We differentiate Eq. (21) with respect to − 1 M 2 , then we eliminate the pole residues λ S and obtain the QCD sum rules for the masses,

Numerical results and discussions
In For s 0 S , it is reasonable to take any values satisfying the relation, m gr + gr 2 ≤ s 0 S ≤ m 1st − 1st 2 , where the gr and 1st denote the ground state and the first excited state (or the higher resonant state), respectively. The s 0 S lies between the two Breit-Wigner resonances, if we parameterize the scalar mesons with the Breit-Wigner masses and widths. More explicitly, 2 , In Table 1, we show the Breit-Wigner masses and widths of the scalar mesons from the Particle Data Group explicitly [1].  Based on the values in Table 1, we can choose the largest continuum threshold parameters s 0 f 0 (980) = 1.9 GeV 2 , s 0 a 0 (980) = 1.8 GeV 2 , s 0 κ 0 (800) = 1.7 GeV 2 , and s 0 f 0 (500) = 1.6 GeV 2 tentatively to take into account all the ground state contributions and avoid the possible contaminations from the higher resonances f 0 (1370), a 0 (1450), K * 0 (1430), and f 0 (1500). In Fig. 3, we plot the masses of the scalar mesons as pure tetraquark states with variations of the Borel parameter M 2 , where the central values of other parameters are taken. From the figure, we can see that if we exclude the contributions of the condensates qq q g s σ Gq with q, q = u, d, s, the predicted masses m S increase monotonously and quickly with increase of the Borel parameters M 2 at the value M 2 < 0.9 GeV 2 , then increase slowly and reach the values m f 0 (980) = 1.06 GeV, m a 0 (980) = 1.03 GeV, m κ 0 (800) = 0.99 GeV, m f 0 (500) = 0.96 GeV at the value M 2 = 3.3 GeV 2 . It is possible to reproduce the experimental data with fine tuning the continuum threshold parameters. However, if we include the contributions of the condensates qq q g s σ Gq , the predicted masses m S are amplified greatly. The m S decrease monotonously and quickly with increase of the Borel parameters M 2 below some spe-cial values, for example, M 2 < 1.2 GeV 2 for the f 0 (980) and a 0 (980), then decrease slowly and reach the values m S ≥ 1.4 GeV at the value M 2 = 3.3 GeV 2 . It is impossible to reproduce the experimental data by fine tuning the continuum threshold parameters. In Fig. 4, we plot the contributions of different terms in the operator product expansion with variations of the Borel parameters M 2 for the scalar nonet mesons as the pure tetraquark states. From the figure, we can see that the convergent behavior of the operator product expansion is very bad, for example, the condensates qq q g s σ Gq of dimension 8 with q, q = u, d, s have too large negative values at the region M 2 ≥ 1.2 GeV 2 . From Figs. 3 and 4, we can draw the conclusion tentatively that the condensates qq q g s σ Gq of dimension 8 play an important role. The conclusion is compatible with the observation of Ref. [29] that there exists no evidence of the couplings of the tetraquark states to the pure light scalar nonet mesons [29]. Now we set the mixing angles θ S to be 90 • in the QCD spectral densities ρ S (s) in Eq. (22), and take the scalar nonet mesons to be pure two-quark states. In Fig. 5, we plot the masses of the scalar mesons as pure two-quark states with variations of the Borel parameters M 2 , the same parameters as that in Fig. 3 are taken. From the figure, we can see that the predicted masses m S ≈ (0.85-1.14) GeV at the value M 2 = (0.5-3.3) GeV 2 , there also exists some difficulty to reproduce the experimental data approximately by fine tuning the continuum threshold parameters. In Fig. 6, we plot the contributions of different terms in the operator product expansion with variations of the Borel parameters M 2 for the scalar nonet mesons as the pure two-quark states. From the figure, we can see that the convergent behavior of the operator product expansion is very good, the main contributions come from the perturbative terms, which are of dimension 6 according to the normalization factors qq 2 and ss 2 .
We turn on the mixing angles θ S = 0 • , 90 • and take into account all the Feynman diagrams which contribute to the condensate qq q g s σ Gq with q, q = u, d, s; see the Feynman diagrams in Figs. 1 and 2. The contributions of the vacuum condensates qq q g s σ Gq of dimension 8 can be canceled out completely with the ideal mixing angles θ 0 S , which results in much better convergent behavior in the operator product expansion.
In this article, we choose the mixing angles θ S = θ 0 S , then impose the two criteria (i.e. pole dominance and convergence of the operator product expansion) of the QCD sum rules on the two-quark-tetraquark mixed states, and search for the optimal values of the Borel parameters M 2 and continuum threshold parameters s 0 S . The resulting Borel parameters (or Borel windows), continuum threshold parameters and pole contributions of the scalar nonet mesons are shown in Table  2 explicitly. Table 2 The Borel parameters (or Borel windows), continuum threshold parameters and pole contributions of the QCD sum rules for the scalar nonet mesons as the two-quark-tetraquark mixed states  From Table 2, we can see that the upper bound of the pole contributions can reach (51-69) %, the pole dominance condition is satisfied marginally. If we intend to obtain QCD sum rules for the light tetraquark states with the pole contributions larger than 50 %, we should resort to multi-pole plus continuum states to approximate the phenomenological spectral densities, include at least the ground state plus the first excited state, and postpone the continuum threshold parameters s 0 S to much larger values [28]. In this article,  (11)(12)(13)(14)(15)(16), (7)(8)(9)(10), (19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29), and (16)(17)(18)(19)(20)(21)(22)) % for f 0 (980), a 0 (980), κ 0 (800), and f 0 (500), respectively, where the total contributions are normalized to be 1. The operator product expansion is well convergent in the Borel windows shown in Table 2.
Now we can see that it is reasonable to extract the masses from the QCD sum rules by choosing the Borel parameters and continuum threshold parameters shown in Table 2. In Figs. 8 and 9, we plot the masses and pole residues of the scalar nonet mesons as the two-quark-tetraquark mixed states with variations of the Borel parameters in the Borel windows by taking into account the uncertainties of the input parameters. From the figures, we can see that the platforms are very flat, the predictions are reliable. In Table  3, we present the masses and pole residues of the scalar nonet mesons as the two-quark-tetraquark mixed states, where all uncertainties of the input parameters are taken into account. Fig. 8 The masses of the scalar nonet mesons as the two-quark-tetraquark mixed states with variations of the Borel parameters Fig. 9 The pole residues of the scalar nonet mesons as the two-quark-tetraquark mixed states with variations of the Borel parameters Table 3 The masses and pole residues of the scalar nonet mesons as the two-quark-tetraquark mixed states The contributions of the other intermediate meson-loops to the correlation functions S ( p 2 ) can be studied in the same way.

Conclusion
In this article, we assume that the nonet scalar mesons below 1 GeV are the two-quark-tetraquark mixed states and study their masses and pole residues using the QCD sum rules. In calculation, we take into account the vacuum condensates up to dimension 10 and the O(α s ) corrections to the perturbative terms, and neglect the condensates which are vacuum expectations of the operators of the order O(α >1 s ), in the operator product expansion. We choose the ideal mixing angles, which can lead to good convergent behavior in the operator product expansion, the resulting two-quark components are much larger than 50 %. Then we impose the two criteria (i.e. pole dominance and convergence of the operator product expansion) of the QCD sum rules, search for the optimal values of the Borel parameters and continuum threshold parameters, and obtain the masses and pole residues of the nonet scalar mesons. The predicted masses are compatible with the experimental data, while the pole residues can be used to study the hadronic coupling constants and form factors.