Analysis of the scalar nonet mesons with the QCD sum rules

In this article, we assume that the nonet scalar mesons below $1\,\rm{ GeV}$ 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 $\mathcal{O}(\alpha_s)$ 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 $50\%$, then obtain the masses and pole residues of the nonet scalar mesons.


Introduction
There are many scalar mesons below 2 GeV, which cannot be accommodated in oneqq nonet, some are supposed to be glueballs, molecular states and tetraquark states [1,2,3,4,5]. In the scenario of molecular states, the scalar states below 1 GeV are taken as loosely bound mesonic molecular states [6], or dynamical generated resonances [7]. On the other hand, in the scenario of tetraquark states, if we suppose the dynamics dominates the scalar mesons below and above 1 GeV are different, there maybe exist two scalar nonets below 1.7 GeV [2,3,4]. The strong attractions between the scalar diquarks and anti-diquarks in relative S-wave maybe result in a nonet tetraquark states manifest below 1 GeV, while the conventional 3 P 0 quark-antiquark nonet mesons have masses about (1.2−1.6) GeV. The well established 3 P 1 and 3 P 2 quark-antiquark nonets lie in the same region. In 2013, Weinberg explored the tetraquark states in the large-N c limit and observed that the existence of light tetraquark states is consistent with large-N c QCD [8]. We usually take the lowest scalar nonet mesons {f 0 /σ(500), a 0 (980), κ 0 (800), f 0 (980)} to be the tetraquark states, and assign the higher scalar nonet mesons {f 0 (1370), a 0 (1450), K * 0 (1430), f 0 (1500)} to be the conventional 3 P 0 quark-antiquark states [2,3,4,9].
We can use QCD sum rules to study the two-quark and tetraquark states. QCD sum rules provides a powerful theoretical tool in studying the hadronic properties, and has been applied extensively to study the masses, decay constants, hadronic form-factors, coupling constants, etc [12,13]. There have been several works on the light tetraquark states using the QCD sum rules [14,15,16,17,18,19,20,21,22]. In Refs. [14,15], 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. [18], 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 Ref. [19], Chen, Hosaka and Zhu study the light scalar tetraquark states with the QCD sum rules in a systematic way. In Ref. [20], 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-quarktetraquark 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.
If we take the diquarks and antidiquarks as the basic constituents, the two isoscalar statesūdud andssū u+dd √ 2 mix ideally, thessū u+dd √ 2 degenerates with the isovector statesssdu,ssū u−dd √ 2 and ssūd 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 Π 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 two-quark 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 represent the scalar diquarks in color anti-triplet, 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 [3,23].
In the following, we perform Fierz re-arrangement to the currents J 4 f0(980) and J 4 a 0 0 (980) both in the color and Dirac-spinor spaces to obtain the result, some components couple potentially to the meson pairs ππ, KK, ηπ, the strong decays f 0 (980) → ππ, KK and a 0 0 (980) → ηπ, KK are Okubo-Zweig-Iizuka super-allowed, which can also be used to study the radiative decays φ(1020) → f 0 (980)γ and φ(1020) → a 0 0 (980)γ through the virtual KK loops. So it is reasonable to assume that the nonet scalar mesons below 1 GeV have some tetraquark constituents.
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 2 , then The contracted parts appear as the normalization factors − qq 3 √ 2 , − ss 6 , − qq 6 and − qq 3 √ 2 in the currents J 2 f0(980) (x), J 2 a0(980) (x), J 2 κ0(800) (x) and J 2 f0(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 Π S (p 2 ) to obtain the hadronic representation [12,13]. 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 Π S (p 2 ) = cos 2 θ Π 44 S (p 2 ) + sin θ cos θ Π 42 S (p 2 ) + sin θ cos θ Π 24 where m, n = 2, 4. We can prove that Π mn S (p 2 ) = Π 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 Π mn S (p 2 ) in perturbative QCD. Firstly, we contract the u, d and s quark fields in the correlation 2 For example, where the α, β, λ and τ are Dirac spinor indexes.
functions Π mn S (p 2 ) with Wick theorem, and obtain the results: where where q = u, d [13]. We take the assumption of vacuum saturation for the higher dimension vacuum condensates and factorize the higher dimension vacuum condensates into lower dimension vacuum condensates [12], for example, qqqq ∼ qq qq , qqqg s σGq ∼ qq qg s σGq , where q = u, d, s. Factorization works well in large N c limit, in reality, N c = 3, some (not much) ambiguities maybe originate from the vacuum saturation assumption. In Fig.1, we show the Feynman diagrams containing theqq 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 σGq ′ 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 ij (x), D ij (x) and S ij (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 the q iqj and s isj 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 σGq and sg s σGs . Some terms involving the mixed condensates qg s σGq and sg s σGs 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 Π 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 the 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.   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 , αsGG π 2 , αsGG π qg s σGq have the dimensions 6, 8, 9 respectively, but they are the vacuum expectations of the operators of the or- 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 [22]. As there are normalization factors qq 2 in the correlation functions Π 22 S (p) , we count those perturbative terms as of the order qq 2 , and 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, ρ 42 We differentiate Eq.(21) with respect to − 1 M 2 , then eliminate the pole residues λ S , and obtain the QCD sum rules for the masses, 3 Numerical results and discussions 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 f0(980) = 1.9 GeV 2 , s 0 a0(980) = 1.8 GeV 2 , s 0 κ0(800) = 1.7 GeV 2 and s 0 f0(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 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 σGq ′ 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.3-4, we can draw the conclusion tentatively that the condensates qq q ′ g s σGq ′ of dimension 8 play an important role. The conclusion is compatible with the observation of Ref. [18], that there exists no evidence of the couplings of the tetraquark states to the pure light scalar nonet mesons [18]. 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 exist 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 σGq ′ with q, q ′ = u, d, s, see the Feynman diagrams in Figs.1-2. The contributions of the vacuum condensates qq q ′ g s σGq ′ of dimension 8 can be 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.
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 [17]. In this article, we exclude the contaminations of the continuum states by the truncation s 0 S , see Eq.(34), although the truncation s 0 S cannot lead to the pole contribution larger than (or about) 50% in all the Borel windows. Such a situation is in contrary to the hiddencharm and hidden-bottom tetraquark states and hidden-charm pentaquark states, where the two heavy quarks Q andQ stabilize the four-quark systems qq ′ QQ and five-quark systems qq ′ q ′′ QQ, and result in QCD sum rules satisfying the pole dominance condition [25,26,27].
In Fig.7 % for the 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.8-9, 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.
There exists a compromise between the minimal masses and the maximal pole contributions, and in the following two paragraphs we will show that the mixing angles θ 0 S are optimal values.  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.

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.