Mass and Decay Constant of $I=1/2$ Scalar Meson In QCD Sum Rule

We calculate the mass and decay constant of $I=1/2$ scalar mesons composed of quark-antiquark pairs based on QCD sum rule. The quauk-antiquark pairs can be $s\bar{q}$ or $q\bar{s}$ ($q=u,d$) in quark model, the quantum numbers of spin and orbital angular momentum are S=1, L=1. We obtain the mass of the ground sate in this channel is $1.410\pm 0.049$GeV. This result favors that $K_{0}^{\ast}(1430)$ is the lowest scalar state of $s\bar{q}$ or $q\bar{s}$. We also predict the first excited scalar resonance of $s\bar{q}$ is larger than 2.0 GeV.


Introduction
Glueball and scalar mesons should exist according to QCD and quark model. Some scalar mesons below 2 GeV have been observed, such as, i) for isospin I = 0, 1 states: f 0 (600) or σ, a 0 (980), f 0 (980), f 0 (1370), f 0 (1500), f 0 (1710); ii) for I = 1 2 states: κ(900) and K * 0 (1430) [1,2,3,4]. The number of these scalar mesons exceeds the particle states which can be accommodated in one nonet in the quark model. It is believed that there are two nonets below and above 1 GeV [5,6]. The components of the meson states in each nonet have not been completely determined yet. For the scalar mesons below 1 GeV there are several interpretations. They are interpreted as meson-meson molecular states [7] or multi-quark states qqqq [8], etc.. However, from the theoretical point of view there must be quark-antiquark SU(3) scalar nonet. Therefore it is important to determine the masses of the ground states of qq with quantum number J P = 0 + based on QCD. For isospin I = 0, 1 states different quark flavor may mix, and scalar qq states may also mix with scalar glueball if they have the same quantum number of J P C and similar masses [9]- [14]. Some authors have tried to determine the mixing angles of the glueball with qq scalar mesons by using decay patterns of some scalar mesons [14]- [17]. These works imply that glueball possibly mix with qq scalar mesons. For I = 1/2 states, they cannot mix with glueball because they have strange quantum number. The physical state is directly the sq and qs bound state. Therefore the mass of the ground state of sq or qs can be determined without necessity for considering mixing effect.
In this paper, we calculate the mass and decay constant of I = 1/2 scalar meson with QCD sum rule. We find that it is impossible to obtain sq scalar meson mass below 1 GeV from QCD sum rule. The most favorable result for the mass of sq scalar meson is 1.410±0.049GeV. Therefore, if κ(900) is sq scalar bound state, this would be a big problem for QCD. This problem can be solved by assuming that κ(900) is irrelevant to sq scalar channel, < 0|sq|κ(900) >∼ 0, and K * 0 (1430) is the scalar ground state of sq or qs. With this assumption, calculation based on QCD will be consistent with experiment. Therefore, our result favors that K * 0 (1430) is the lowest scalar bound state of sq. If this is correct, then from the approximate SU(3) flavor symmetry, the masses of the other J P = 0 + mesons in the scalar nonet should be slightly above or below 1.4 GeV. This result would imply that scalars with masses below 1 GeV are not dominated by quark-antiquark pairs. This is consistent with the calculation of lattice QCD which implies that a nonet of quark-qntiquark scalars is in the range 1.2-1.6 GeV [22].
The remaining part of this paper is organized as follows. In Section 2, we briefly introduce the process to calculate the scalar meson with QCD sum rule and get the Wilson coefficients for the corresponding two-point scalar current correlation function. Section 3 is devoted to numerical analysis and conclusion.

The method
To calculate the mass of scalar sq or qs meson, the two-point correlation function should be taken as On one hand, the correlation function can be expressed based on the dispersion relation in terms of hadron states whereÎ m Π(s) is the imaginary part of the two-point correlation function, which can be obtained by inserting a complete set of quantum states |n n| into For the scalar states S, its decay constant f S can be defined through where m S is the mass of the scalar state. Based on Eq.(2)-Eq.(4), and explicitly separating out the lowest scalar state, the correlation function can be expressed as where ρ h (s) expresses the contribution of higher resonances and continuum state, s 0 is the threshold of higher resonances and continuum state. On the other hand, the correlation function can be expanded in terms of operator-product expansion at large negative value of q 2 .
where C i , i = 0, 3, 4, 5, 6, · · · are Wilson coefficients, I is the unit operator, ΨΨ is the local Fermion field operator of light quarks, G a αβ is gluon strength tensor, Γ and Γ ′ are the matrices appearing in the procedure of calculating the Wilson coefficients.
Matching Π h (q 2 ) with Π QCD (q 2 ) we can get the equation which relates mass of scalar meson with QCD parameters and a few condensate parameters. In order to suppress the contribution of higher resonances and that of condensate terms, we make Borel transformation over q 2 in both sides of the equation, the Borel transformation is defined aŝ After assuming quark-hadron duality, i.e., by assuming that the contribution of higher resonance and continuum states can be approximately cancelled by the perturbative integration over the threshold s 0 [23], the resulted sum rules for the mass and decay constant of the scalar meson are where

Numerical analysis and conclusion
The numerical parameters used in this paper are taken as [18,21] qq = −(0.24 ± 0.01GeV) 3 , For the choice of Borel parameter M 2 , as in [18,24], we define f thcorr (M 2 ) as m(M 2 ) in Eq. (8)  . To get reliable prediction of the mass in QCD sum rule, f cont should be limited to above 90% to suppress the contribution of higher resonance and continuum, and f nopower (M 2 ) be limited to less than 10% deviation from 1, which can ensure condensate contribution much less than perturbative contribution.
There are two low mass scalar meson states with isospin I = 1/2 and strange number |S| = 1 found in experiment. They are κ(900) with mass m κ about 800 ∼ 900MeV [2,3,4], and K * 0 (1430) with mass m(K * 0 (1430)) = 1.412 ± 0.006GeV [1]. In theory, taking appropriate value for the threshold parameter s 0 , one can separate out the contribution of the lowest resonance in QCD sum rule. We vary the value of the threshold parameter s 0 , and find that it is impossible to obtain the mass of κ(900) with the sum rule in Eq. (8). There is no stable 'window' for the Borel parameter in this mass region. Therefore, if κ(900) is the lowest scalar state in the sq channel, it would be a big problem for QCD sum rule. However, if we increase the value of s 0 , i.e., for s 0 = 4.0 ∼ 4.8 GeV 2 , we does find the stable 'window' for Borel parameter, which is shown in Fig.2. The resulted stable window is in the range 1.0 < M 2 < 1.2 GeV 2 . Fig.2(a) shows that between the arrows A and B, both the contributions of condensate and higher resonance are less than 10%. So in this region, the operator product expansion is effective, and the assumption of quarkhadron duality does not seriously affect the numerical result, which means that QCD sum rule can give reliable prediction in this parameter space. For the uncertainty of higher α s correction for the perturbative diagram and the condensate parameters. The variation of Borel parameter yields ±1.8% uncertainty for the mass, s 0 yields ±2.0%, α s correction gives ±2.2%, the uncertainty caused by the condensate parameters is less than 0.6%. All the uncertainties are added quadratically. The energy scale for the α s (µ) correction is taken to be µ = M. In the stable window, the range of Borel parameter is 1.0 < M 2 < 1.2GeV 2 , therefore α s (M) ∼ 0.5. We checked that the contribution of the α s correction at first order is about 2.2%, which is not large. This can be understood because most contribution of the α s correction is cancelled between the numerator and denominator of Eq.(8). We use 2.2% to estimate the uncertainty caused by the higher order α s corrections. On one hand, it is impossible to obtain the mass of lower scalar state κ(900) from QCD sum rule for sq channel. If regard κ(900) as sq scalar bound state, it would be a big problem for QCD. On the other hand, QCD sum rule can give most favorable mass which is consistent with the mass of K * 0 (1430). Therefore it is acceptable to assume that κ(900) is irrelevant to sq scalar bound state, and < 0|sq|κ(900) >∼ 0 With this assumption, K * 0 (1430) can be accepted as the lowest scalar bound state of sq. Then there will be no problem between QCD and experiment. One may still be afraid that there are contributions of the lower mass state κ(900) mixed in the result of eq.(13) in fact. If this is indeed the case, the result of the sum rule may be some weighted average of the two resonances of κ(900) and K * 0 (1430). Therefore this situation should be carefully checked. Because the sum rule for the mass of the scalar bound state in eqs. (8), (10) and (11) includes the spectrum integration s 0 (m 1 +m 2 ) 2 ds, in principle one can lower the value of s 0 to separate the lowest bound state. Therefore, we checked what result for the mass can be got by lower the value of s 0 within the stable window Fig.2(a). The result is shown in Fig.3. It shows that for any value of s 0 , the possible mass is large than 960MeV, m(sq) > 960 MeV (15) Therefore the possible effect of κ(900) can be safely ruled out in the sum rule result in eq. (13). Note that the most recent experimental result for the mass of κ(900) from E791 collaboration is m κ = 797 ± 19 ± 42MeV [3]. If K * 0 (1430) is the ground state of sq or qs, from the approximate SU(3) flavor symmetry, the masses of the other J P = 0 + mesons in the scalar nonet should be also around 1.4 GeV. This implies that the scalars with masses less than 1 GeV, i.e., f 0 (600), a 0 (980), f 0 (980) etc., can not be dominated by quark-antiquark bound states. This is consistent with the calculation of lattice QCD which implies that a nonet of quark-antiquark scalars is in the region 1.2-1.6 GeV [22].
Our result can be further checked by experiment. From the threshold parameter s 0 , we can predict that the mass of the first excited resonance in sq scalar channel should be larger than √ s 0 , that is This prediction can be tested by experiment.
Next we discuss the decay constant of the two-quark scalar bound state sq. From the above analysis, we take the threshold parameter s 0 = 4.0 ∼ 4.8 GeV 2 . Consider K * 0 (1430) as the only resonance below 2 GeV in the sq scalar channel, we can obtain the decay constant of K * 0 (1430) as a function of Borel parameter M 2 (see eq. (9)). The numerical result is shown in Fig.4.  Fig.4 shows that the decay constant is very stable. The determined stable 'window' is still in 1.0 < M 2 < 1.2GeV 2 , where the continuum and condensate contribution are restricted to be less than 15% and 4%, respectively. Within this stable window, the decay constant of K * 0 (1430) is The variation of s 0 yields ±30% uncertainty for the decay constant, α s correction gives ±20%, the uncertainties caused by the condensate parameters and the variation of Borel parameter are less than 0.3% and 0.1%, respectively. All the uncertainties are added quadratically to give the error bar in the above result.
Again we should check what will happen if we consider two resonances κ(900) and K * 0 (1430) existing below 2 GeV in our sum rule analysis. Therefore we add one more resonance into eq.(5), then matching Π h (q 2 ) with Π QCD (q 2 ) in eq.(7). By assuming quark-hadron duality to cancel the contribution of higher resonance and continuum above 2 GeV, and making Borel transformation in both sides, we get the Borel improved matching equation where R 2 has been given in eq. (11), and m S1 , m S2 are fixed to be the masses of κ(900) and K * 0 (1430), m S1 = 900MeV, m S2 = 1410MeV. f S1 and f S2 are the decay constants of the relevant scalar mesons.
Differentiate both sides of eq.(18) with the operator d/dM 2 , we can get another equation where R 1 is defined in eq.(10). With eqs. (18) and (19), we can obtain From the above result we can perform the numerical analysis for the decay constants in the two-resonance ansatz. The numerical result is shown in Fig.5. From Fig.5, we can see that both the two decay constants are unstable as a function of Borel parameter in the two-resonance ansatz. Adding the lower resonance κ(900) in the sum rule analysis for the sq channel spoils the stability existing in the one-resonance ansatz, which is shown in Fig.4. From the requirement of numerical stability of QCD sum rule, the numerical analysis of the decay constant does not favor to include κ(900) in sq scalar channel. In addition, we can see from Fig.5a that the decay constant of the lower scalar resonance κ(900) tend to be zero at M 2 ∼ 1.01 and 1.05 GeV. This is consistent with the requirement that < 0|sq|κ(900) >∼ 0 in the one-resonance ansatz, where the stability window is located in the range 1.0 < M 2 < 1.2 GeV.
Therefore, both the analyses of the mass and decay constant of sq scalar meson from QCD sum rule imply that κ(900) is not dominated by quarkantiquark bound state, and the lowest sq scalar bound state is K * 0 (1430). The mass obtained from QCD sum rule is m(K * 0 (1430)) = 1.410 ± 0.049GeV (22) and the decay constant is In summary, we calculate the mass and decay constant of scalar meson sq in QCD sum rule. Our result favors that K * 0 (1430) is the ground state of sq scalar bound state. If this is correct, it would imply that scalar mesons below 1 GeV are not dominated by quark-antiquark pairs. We also predict that the mass of the first excited resonance of sq scalar bound state is larger than 2.0 GeV.
where the term with α s (µ) is the radiative correction to the perturbative contribution [20], and the scale is taken to be µ = M.