Light Scalar Meson sigma(600) in QCD Sum Rule with Continuum
aa r X i v : . [ h e p - ph ] J un Light Scalar Meson σ (600) in QCD Sum Rule with Continuum Hua-Xing Chen , , ∗ Atsushi Hosaka , † Hiroshi Toki , ‡ and Shi-Lin Zhu § Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka 567–0047, Japan
The light scalar meson σ (600) is known to appear at low excitation energy with very large widthon top of continuum states. We investigate it in the QCD sum rule as an example of resonancestructures appearing above the corresponding thresholds. We use all the possible local tetraquarkcurrents by taking linear combinations of five independent local ones. We ought to consider the π - π continuum contribution in the phenomenological side of the QCD sum rule in order to obtain agood sum rule signal. We study the stability of the extracted mass against the Borel mass and thethreshold value and find the σ (600) mass at 530 MeV ±
40 MeV. In addition we find the extractedmass has an increasing tendency with the Borel mass, which is interpreted as caused by the widthof the resonance.
PACS numbers: 12.39.Mk, 12.38.Lg, 12.40.YxKeywords: scalar meson, tetraquark, QCD sum rule
I. INTRODUCTION
The light scalar mesons, σ (600), κ (800), f (980) and a (980), have been intensively discussed for many years [1–3].However, their nature is still not fully understood [4–8]. They have the same quantum numbers J P C = 0 ++ as thevacuum, and hence the structure of these states is a very important subject in order to understand non-perturbativeproperties of the QCD vacuum such as spontaneous chiral symmetry breaking. They compose of the flavor SU (3) nonetwith the mass below 1 GeV, and have a mass ordering which is difficult to be explained by using a q ¯ q configurationin the conventional quark model [9–13]. Therefore, several different pictures have been proposed, such as tetraquarkstates and meson-meson bound states, etc. Here we note that hadrons with complex structures such as tetraquarksmay exist in the continuum above the threshold energy of two hadrons with simple quark structure.The tetraquark structure of the scalar mesons was proposed long time ago by Jaffe with an assumption of strongdiquark correlations [14, 15]. It can naturally explain their mass ordering and decay properties [16–18]. Yet the basicassumption of diquark correlation is not fully established. In this letter, we study σ (600) as a tetraquark state in theQCD sum rule approach as an example of resonances in the continuum states above the π - π threshold. In the QCDsum rule, we calculate matrix elements from the QCD (OPE) and relate them to observables by using dispersionrelations. Under suitable assumptions, the QCD sum rule has proven to be a very powerful and successful non-perturbative method in the past decades [19, 20]. Recently, this method has been applied to the study of tetraquarksby many authors [21–24].In our previous paper [24], we have found that the QCD sum rule analysis with tetraquark currents implies themasses of scalar mesons in the region of 600 – 1000 MeV with the ordering m σ < m κ < m f ,a , while the conventional¯ qq current is considerably heavier (larger than 1 GeV). To get this result, first we find there are five independent localtetraquark currents, and then we use one of these currents or linear combinations of two currents to perform the QCDsum rule analysis. But these interpolating currents do not describe the full space of tetraquark currents. In orderto complete our previous study, we use more general currents by taking linear combinations of all these currents. Itdescribes the full space of local tetraquark currents which can couple to σ (600). Since σ (600) meson is closely relatedto the π - π continuum and it has a wide decay width, we also consider the contribution of the π - π continuum as wellas the effect of the finite decay width.This paper is organized as follows. In Sec. II, we establish five independent local tetraquark currents, and performa QCD sum rule analysis by using linear combinations of five single currents. In Sec. III, we perform a numericalanalysis, and we also study the contribution of π - π continuum. In Sec. IV, we consider the effect of the finite decaywidth. Sec. V is devoted to summary. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected]
II. QCD SUM RULE
The local tetraquark currents for σ (600) have been worked out in Ref [24]. There are two types of currents:diquark-antidiquark currents ( qq )(¯ q ¯ q ) and meson-meson currents (¯ qq )(¯ qq ). These two constructions can be proved tobe equivalent, and they can both describe the full space of local tetraquark currents [24]. Therefore we shall just usethe first ones. Since we use their linear combinations to perform the QCD sum rule analysis, we can not distinguishwhether it is a diquark-antidiquark state or a meson-meson bound state. However, we find that tetraquark currentswith a single term do not lead to a reliable QCD sum rule result which means that σ (600) probably has a complicatedstructure. The five independent local currents are given by: S σ = ( u Ta Cγ d b )(¯ u a γ C ¯ d Tb − ¯ u b γ C ¯ d Ta ) ,V σ = ( u Ta Cγ µ γ d b )(¯ u a γ µ γ C ¯ d Tb − ¯ u b γ µ γ C ¯ d Ta ) ,T σ = ( u Ta Cσ µν d b )(¯ u a σ µν C ¯ d Tb + ¯ u b σ µν C ¯ d Ta ) , (1) A σ = ( u Ta Cγ µ d b )(¯ u a γ µ C ¯ d Tb + ¯ u b γ µ C ¯ d Ta ) ,P σ = ( u Ta Cd b )(¯ u a C ¯ d Tb − ¯ u b C ¯ d Ta ) . The summation is taken over repeated indices ( µ , ν, · · · for Dirac, and a, b, · · · for color indices). The currents S , V , T , A and P are constructed by scalar, vector, tensor, axial-vector, pseudoscalar diquark and antidiquark fields,respectively. The subscripts 3 and 6 show that the diquarks (antidiquarks) are combined into the color representations, ¯3 c and c ( c and ¯6 c ), respectively.These five diquark-antidiquark currents ( qq )(¯ q ¯ q ) are independent. In this work we use general currents by takinglinear combinations of these five currents: η = t e iθ S σ + t e iθ V σ + t e iθ T σ + t e iθ A σ + t e iθ P σ , (2)where t i and θ i are ten mixing parameters, whose linear combination describes the full space of local currents whichcan couple to σ (600). We can not determine them in advance and therefore we choose them randomly for the studyof the QCD sum rule.By using the current in Eq. (2), we calculate the OPE up to dimension eight. To simplify our calculation, weneglect several condensates, such as h g G i , etc., and we do not consider the α s correction, such as g h ¯ qq i , etc. Theobtained OPE are shown in the following. We find that most of the crossing terms are not important such as ρ ,and even more some of them disappear: ρ = 0, etc. For the most cases, we find that the OPE terms of Dim=6 andDim=8 give major contributions in the OPE series in our region of interest. This is because the condensates h ¯ qq i (D=6) and h ¯ qq ih g ¯ qσGq i (D=8) are much larger than others.Since the OPE series should be convergent to give a reliable QCD sum rule, we also calculate the OPE of Dim=10and Dim=12. However, we find that these terms are not important. Using the parameter set (2) and the the valuesof the condensates of the next section as an example, we show the convergence of the two-point correlation functionΠ( M B , s ) ≡ R s ρ ( s ) e − s/M B ds in Fig. 1 as functions of M B . The threshold value is taken to be s = 1 GeV , andwe show its behavior up to certain dimensions. We find that the OPE up to Dim=0 and Dim=2 are very small; theOPE of Dim=4 gives a minor contribution; the OPE of Dim=6 and Dim=8 are both important; the OPE of Dim=10and Dim=12 are both small, and so we shall neglect them in the following analysis. Borel Mass [GeV ] -6-3036 B - G e V ] -6-3036Dim=0,2Dim=4Dim=6Dim=8,10,12 FIG. 1: The convergence of the two-point correlation function Π( M B , s ). The threshold value is taken to be s = 1 GeV ,and we show its behavior up to certain dimensions, as functions of M B . The solid line is for Π( M B , s ) up to Dim=8. Theshort-dashed line around it is for Π( M B , s ) up to Dim=10, and the long-dashed line around it is for Π( M B , s ) up to Dim=12. ρ ( s ) = t ρ ( s ) + t ρ ( s ) + t ρ ( s ) + t ρ ( s ) + t ρ ( s ) (3)+2 t t cos ( θ − θ ) ρ ( s ) + 2 t t cos ( θ − θ ) ρ ( s ) + 2 t t cos ( θ − θ ) ρ ( s )+2 t t cos ( θ − θ ) ρ ( s ) + 2 t t cos ( θ − θ ) ρ ( s ) + 2 t t cos ( θ − θ ) ρ ( s )+2 t t cos ( θ − θ ) ρ ( s ) + 2 t t cos ( θ − θ ) ρ ( s ) , where ρ ( s ) = s π + ( − m u π + m u m d π − m d π ) s + ( h g GG i π − m u h ¯ qq i π − m d h ¯ qq i π ) s +( − m u h g GG i π + m u m d h g GG i π − m d h g GG i π − m u h g ¯ qσGq i π − m d h g ¯ qσGq i π + h ¯ qq i π ) s (4) − m u h ¯ qq i π + m u m d h ¯ qq i π − m d h ¯ qq i π − m u h g GG ih ¯ qq i π − m d h g GG ih ¯ qq i π + h ¯ qq ih g ¯ qσGq i π ,ρ ( s ) = s π + ( − m u π − m u m d π − m d π ) s + ( h g GG i π + m u h ¯ qq i π + m d h ¯ qq i π ) s (5)+( − m u h g GG i π + m u m d h g GG i π − m d h g GG i π + m u h g ¯ qσGq i π + m d h g ¯ qσGq i π − h ¯ qq i π ) s (6)+ 11 m u h ¯ qq i π + 2 m u m d h ¯ qq i π + 11 m d h ¯ qq i π − m u h g GG ih ¯ qq i π − m d h g GG ih ¯ qq i π − h ¯ qq ih g ¯ qσGq i π ,ρ ( s ) = s π + ( − m u π − m d π ) s + ( 11 h g GG i π + m u h ¯ qq i π + m d h ¯ qq i π ) s (7)+( − m u h g GG i π − m d h g GG i π ) s + 5 m u h ¯ qq i π + 20 m u m d h ¯ qq i π + 5 m d h ¯ qq i π + 11 m u h g GG ih ¯ qq i π + 11 m d h g GG ih ¯ qq i π ,ρ ( s ) = s π + ( − m u π + m u m d π − m d π ) s + 5 h g GG i π s (8)+( − m u h g GG i π + 5 m u m d h g GG i π − m d h g GG i π − m u h g ¯ qσGq i π − m d h g ¯ qσGq i π + h ¯ qq i π ) s (9) − m u h ¯ qq i π + 8 m u m d h ¯ qq i π − m d h ¯ qq i π + m u h g GG ih ¯ qq i π + m d h g GG ih ¯ qq i π + h ¯ qq ih g ¯ qσGq i π ,ρ ( s ) = s π + ( − m u π − m u m d π − m d π ) s + ( h g GG i π + m u h ¯ qq i π + m d h ¯ qq i π ) s (10)+( − m u h g GG i π − m u m d h g GG i π − m d h g GG i π + m u h g ¯ qσGq i π + m d h g ¯ qσGq i π − h ¯ qq i π ) s (11)+ 17 m u h ¯ qq i π + 7 m u m d h ¯ qq i π + 17 m d h ¯ qq i π + m u h g GG ih ¯ qq i π + m d h g GG ih ¯ qq i π − h ¯ qq ih g ¯ qσGq i π ,ρ ( s ) = ( m u π + m u m d π + m d π ) s + ( − m u h ¯ qq i π − m d h ¯ qq i π ) s (12)+( − m u h g ¯ qσGq i π − m d h g ¯ qσGq i π + h ¯ qq i π ) s − m u h ¯ qq i π − m u m d h ¯ qq i π − m d h ¯ qq i π + h ¯ qq ih g ¯ qσGq i π ,ρ ( s ) = − h g GG i π s + ( 3 m u h g GG i π + 3 m d h g GG i π ) s − m u h g GG ih ¯ qq i π − m d h g GG ih ¯ qq i π , (13) ρ ( s ) = ( 3 m u h g GG i π + 3 m u m d h g GG i π + 3 m d h g GG i π ) s − m u h g GG ih ¯ qq i π − m d h g GG ih ¯ qq i π , (14) ρ ( s ) = ( − m u h g GG i π − m u m d h g GG i π − m d h g GG i π ) s + 3 m u h g GG ih ¯ qq i π + 3 m d h g GG ih ¯ qq i π , (15) ρ ( s ) = h g GG i π s + ( − m u h g GG i π − m d h g GG i π ) s + m u h g GG ih ¯ qq i π + m d h g GG ih ¯ qq i π , (16) ρ ( s ) = ( m u h g GG i π + m u m d h g GG i π + m d h g GG i π ) s − m u h g GG ih ¯ qq i π − m d h g GG ih ¯ qq i π , (17) ρ ( s ) = ( − m u π − m u m d π − m d π ) s + ( m u h ¯ qq i π + m d h ¯ qq i π ) s (18)+( − m u h g GG i π − m u m d h g GG i π − m d h g GG i π + 3 m u h g ¯ qσGq i π + 3 m d h g ¯ qσGq i π − h ¯ qq i π ) s + 5 m u h ¯ qq i π + 6 m u m d h ¯ qq i π + 5 m d h ¯ qq i π + 5 m u h g GG ih ¯ qq i π + 5 m d h g GG ih ¯ qq i π − h ¯ qq ih g ¯ qσGq i π ,ρ ( s ) = − h g GG i π s + ( 3 m u h g GG i π + 3 m d h g GG i π ) s − m u h g GG ih ¯ qq i π − m d h g GG ih ¯ qq i π . (19) III. NUMERICAL ANALYSIS
To perform the numerical analysis, we use the values for all the condensates from Refs. [25–30]: h ¯ qq i = − (0 .
240 GeV) , h ¯ ss i = − (0 . ± . × (0 .
240 GeV) , h g s GG i = (0 . ± .
14) GeV ,m u = 5 . , m d = 9 . ,m s (1 GeV) = 125 ±
20 MeV , (20) h g s ¯ qσGq i = − M × h ¯ qq i ,M = (0 . ± .
2) GeV . As usual we assume the vacuum saturation for higher dimensional operators such as h | ¯ qq ¯ qq | i ∼ h | ¯ qq | ih | ¯ qq | i .There is a minus sign in the definition of the mixed condensate h g s ¯ qσGq i , which is different with some other QCDsum rule calculation. This is just because the definition of coupling constant g s is different [25, 31].Altogether we took randomly chosen 50 sets of t i and θ i . Some of these sets of numbers lead to negative spectraldensities in the low energy region of interest, which should be, however, positive from their definition. This is dueto several reasons. One reason is that the convergence of OPE may not be achieved yet for those currents for thetetraquark state. Another reason is that some currents may not couple to the physical states properly. Except them,there are fifteen sets which lead to positive spectral densities. We show these fifteen sets of t i and θ i in Table I, andlabel them as (01), (02), · · · , (15). They are sorted by the fourth column “Pole Contribution” (PC):Pole Contribution ≡ R s e − s/M B ρ ( s )d s R ∞ e − s/M B ρ ( s )d s . (21)The pole contribution (PC) is an important quantity to check the validity of the QCD sum rule analysis. Here, ρ ( s )denotes the spectral function. It depends on the ten mixed parameters as well as M B and s . We note that π - π continuum which we shall study later is not included in the pole contribution. By fixing s = 1 GeV , we show thePC values in Table I for the fifteen sets. “PC(0.5)”, “PC(0.8)” and “PC(1.2)” denote pole contribution by setting M B = 0 . , 0 . and 1 . , respectively. We find that the pole contribution decreases very rapidly asthe Borel Mass increases. Since we have discussed the convergence of OPE in the previous section, and found thatthe Dim=10 and Dim=12 terms are much smaller than the Dim=6 and Dim=8 terms, and so it is only the polecontribution which gives a upper limitation on the Borel Mass. The Borel window is wider for the former parametersets (1), (2), · · · , and narrower for the latter ones. It almost disappears for the set (15), whose mass prediction is TABLE I: Values for parameters t i , θ i , the mass range M σ , the pole contribution (PC) and the continuum amplitude a ( t i , θ i ).The meaning of these quantities are given in the text. There are altogether fifteen sets, which are sorted by the fourth column“PC”. “PC(0.5)”, “PC(0.8)” and “PC(1.2)” denote pole contribution by setting M B = 0 . , 0 . and 1 . ,respectively.No t t t t t θ θ θ θ θ M σ (MeV) PC(0.5) PC(0.8) PC(1.2) a (GeV )(1) 0 .
03 0 .
03 0 .
73 0 .
37 0 .
24 2 . . . . . ∼
580 92% 52% 13% 1 . × − (2) 0 .
03 0 .
92 0 .
75 0 .
70 0 .
03 5 . .
80 4 . . . ∼
590 90% 46% 11% 5 . × − (3) 0 .
25 0 .
79 0 .
16 0 .
95 0 .
22 1 . . . .
44 1 . ∼
600 87% 44% 11% 3 . × − (4) 0 .
53 0 .
26 0 .
93 0 .
24 0 .
76 2 . .
40 2 . . . ∼
610 85% 41% 10% 1 . × − (5) 0 .
74 0 .
54 0 .
74 0 .
65 0 .
67 0 .
15 3 . . . . ∼
640 81% 36% 8% 1 . × − (6) 0 .
98 0 .
50 0 .
12 0 .
33 0 .
03 2 . . . . . ∼
590 82% 32% 6% 5 . × − (7) 0 .
98 0 .
42 0 .
84 0 .
82 0 .
72 0 .
095 1 . . . . ∼
700 70% 26% 6% 4 . × − (8) 0 .
48 0 .
68 0 .
58 0 .
96 0 .
04 1 . . . . . ∼
690 70% 25% 6% 1 . × − (9) 0 .
53 1 . .
99 0 .
34 0 .
86 5 . . . . .
076 540 ∼
700 68% 24% 5% 4 . × − (10) 0 .
75 0 .
96 0 .
32 0 .
12 0 .
11 4 . . .
93 5 . . ∼
760 57% 17% 4% 9 . × − (11) 0 .
31 0 .
81 0 .
71 0 0 .
10 4 . . . . . ∼
780 55% 17% 4% 3 . × − (12) 0 .
47 0 .
40 0 0 .
46 0 .
91 0 .
18 1 . . .
091 0 .
94 540 ∼
730 58% 16% 3% 2 . × − (13) 0 .
60 0 .
26 0 .
44 0 .
27 0 .
24 3 . . .
92 5 . . ∼
850 43% 13% 3% 1 . × − (14) 0 .
74 0 .
73 0 .
73 0 .
32 0 .
28 1 . . . . . ∼
850 42% 12% 3% 4 . × − (15) 0 .
65 0 .
55 0 .
92 0 .
19 0 .
96 4 . . . . . ∼
930 25% 7% 2% 5 . × − also much different from others. The Borel window should be our working region. However, since the Borel stabilityis always very good when M B > , we shall keep the idea of Borel window in mind and work in the region0 . < M B < . On the other side, we shall care more about the threshold value s .By using these fifteen sets of numbers, we perform the QCD sum rule analysis. There are two parameters, the Borelmass M B and the threshold value s in the QCD sum rule analysis. We find that the Borel mass stability is usuallygood, but the threshold value stability is not always good. We show the mass range of σ (600), M σ , in Table I, wherethe working region is taken to be 0 . < s < . and 0 . < M B < . We find the mass rangeis small when the pole contribution (PC) is large.The parameter sets (01)-(06) lead to relatively good threshold value stability. Taking the set (02) as an example,we show its spectral density ρ ( s ) in Fig 2 as function of s . It is positive definite, and has a small value around s ∼ . . Therefore, the threshold value dependence is weak around this point, as shown in Fig. 3 for the extracted massas functions of both M B and s . We find all the curves are very stable in the region 0 . < M B < and0 . < s < . . From the set (02) we can extract the mass of σ (600) around 550 MeV. From other goodcases, we find that the mass of σ (600) is around 550 MeV as well. - O P E FIG. 2: The spectral density ρ ( s ) calculated by the mixed current η , as a function of s . We show the results of the parameterset (02) as an example. The parameter sets (07)-(15) lead to the threshold value stability, which is not good. Taking the set (13) as anexample, we show its spectral density in Fig. 4 as a function of s (left figure), and the extracted mass in Fig. 5 as afunction of s (upper three curves). The mass increases with s and we cannot extract the mass from this result.Many effects contribute to the mass dependence on the threshold value, but for σ (600) the π - π continuum contributionis probably the dominant one. Hence, we add a term ρ ππ ( s ) in the spectral function in the phenomenological side to M a ss [ G e V ] M a ss [ G e V ] FIG. 3: The extracted mass of σ (600) as a tetraquark state calculated by the mixed current η , as functions of the Borel mass M B and the threshold value s . We show the results of the parameter set (02) as an example. At the left panel, the solid,short-dashed and long-dashed curves are obtained by setting s = 0 . , . , respectively. At the right panel, thesolid and dashed curves are obtained by setting M B = 0 . , , respectively. - O P E - O P E FIG. 4: The spectral density ρ ( s ) calculated by the mixed current η , as a function of s . We show the results of the parameterset (13) as an example. The left figure shows the full spectral density as given on the left hand side of Eq. (22), while the rightfigure is the one with ρ ππ ( s ) subtracted. describe the π - π continuum: ρ ( s ) = f Y δ ( s − M Y ) + ρ ππ ( s ) + ρ cont . (22)where ρ cont is the standard expression of the continuum contribution except the π - π continuum. To find an expression M a ss [ G e V ] FIG. 5: The extracted mass of σ (600) as a tetraquark state calculated by the mixed current η , as functions of the thresholdvalue s . We choose the parameter set (13) as an example. The solid, short-dashed and long-dashed curves are obtained bysetting M B = 0 . , , respectively. The upper three curves are obtained without adding the contribution of the π - π continuum in the spectral density in the phenomenological side, while the lower three curves are obtained after adding thecontribution of the π - π continuum. for ρ ππ ( s ), we introduce a coupling λ ππ ≡ h | η | π + π − i . (23)The correlation function of the π - π continuum isΠ ππ ( p ) = i Z d q (2 π ) i ( p + q ) − m π + iǫ iq − m π + iǫ | λ ππ | , (24)and the spectral density of the π - π continuum is just its imaginary part ρ ππ ( s ) = ImΠ ππ ( s ) = 116 π r − m π s | λ ππ | . (25)We may calculate λ ππ by using the method of current algebra if we know the property of the resonance state. However,this is not the topic of this paper. Moreover, in this paper we use a general local tetraquark current to test the fullspace of local tetraquark currents, so we again make some try and error tests, and find that the following functionleads to a reasonable QCD sum rule result, λ ππ ∼ s . Hence, we take the spectral density of the π - π continuum as ρ ππ ( s ) = a ( t i , θ i ) s r − m π s . (26)We add the continuum contribution ρ ππ ( s ) in the phenomenological side and perform the QCD sum rule analysis.The values of parameter a ( t i , θ i ) are listed in Table I. After adding the continuum contribution, the threshold valuestability becomes much better. Still taking the set (13) as an example, we show its spectral density in Fig. 4 as afunction of s (right figure), and the extracted mass in Fig. 5 as functions of s (lower three curves). We see that nowthe spectral density has a small value around s ∼ . , and the stability of the threshold value is significantlyimproved.Hence, we made the same analysis for all the other cases. We found all the cases are good except one, which isthe case (15), where we are not able to get the desired stability as a function of s . The mass function has a smallstability region and increases rapidly with s . Hence, we consider this case is between the good case and bad case,and remove it from the further analysis in this paper. We show several results out of all the good cases in Fig. 6,which are obtained by using the parameter sets (01), (03), (06), (09), (12) and (14). We list the used a ( t i , θ i ) in Table1 for all the cases. All the masses behave very nicely as functions of the Borel mass and s as shown in Fig. 6. In ourworking region 0 . < s < . and 0 . < M B < , all the cases lead to a mass within theregion 495 MeV ∼
570 MeV. From this mass range, the mass of σ (600) is extracted to be 530 MeV ±
40 MeV.
IV. THE EFFECT OF FINITE DECAY WIDTH
After the s stability has been improved, we notice now that the mass increases systematically with the Borel massas seen in Fig. 6 in all the cases. We therefore try to consider a possible reason of this systematic result. The σ (600)meson has a large decay width. We parametrize it by a Gaussian distribution instead of the δ -function for the σ (600). ρ F DW ( s ) = f X √ πσ X exp (cid:0) − ( √ s − M X ) σ X (cid:1) . (27)The Gaussian width σ X is related to the Breit-Wigner decay width Γ by σ X = Γ / .
4. We set σ X = 200 MeV, and M X = 550 MeV, and calculate the following “mass”: M ( M B , s ) = R s e − s/M B s exp (cid:0) − ( √ s − M X ) σ X (cid:1) ds √ s R s e − s/M B exp (cid:0) − ( √ s − M X ) σ X (cid:1) ds √ s . (28)We find that the obtained mass M is not just 550 MeV, but increases as M B increases as shown in Fig. 7. Hence, theextracted mass in the QCD sum rule analysis ought to depend on the Borel mass. The amount of the change of theextracted mass in the QCD sum rule analysis is similar to the one found in this model calculation. Moreover, we findthat the finite decay width does not change the final result significantly, which we have also noticed in our previouspaper [24]. M a ss [ G e V ] M a ss [ G e V ] M a ss [ G e V ] M a ss [ G e V ] M a ss [ G e V ] M a ss [ G e V ] FIG. 6: The extracted mass of σ (600) as a tetraquark state calculated by the mixed currents η , as functions of the thresholdvalue s . We choose the parameter sets (01), (03), (06), (09), (12) and (14). The results are shown in sequence. The solid,short-dashed and long-dashed curves are obtained by setting M B = 0 . , , respectively. M a ss [ G e V ] FIG. 7: The extracted “mass” considering a finite decay width. The solid, short-dashed and long-dashed curves are obtainedby setting M B = 0 . , , respectively. V. SUMMARY
In summary, we have studied the light scalar meson σ (600) in the QCD sum rule. We have used general localtetraquark currents which are linear combinations of five independent local ones. This describes the full space of localtetraquark currents which can couple to σ (600) either strongly or weakly. We find some cases where the stability ofthe Borel mass and threshold value is both good, while in some cases the threshold value stability is not so good.The resonance mass has an increasing trend with s , which indicates a continuum contribution. Hence, we haveintroduced a contribution from the π - π continuum, and obtained a good threshold value stability. The mass of σ (600)is extracted to be 530 MeV ± σ (600) meson and it is very important to consider thisfact in the QCD sum rule analysis for exotic states. We have seen clear tendency of the mass increase with the Borelmass after getting good signal of the threshold dependence. The decay width of σ (600) is related to this increasetendency. We are now trying to calculate this by using the three-point correlation function within the QCD sum ruleapproach. The present analysis is very encouraging to apply the QCD sum rule including the continuum states forother scalar mesons. Moreover, the continuum contribution should be important in many other resonances such asΛ(1405) etc, which lies in some continuum background. In the future, we will use the QCD sum rule analysis withcontinuum to study various resonances. Acknowledgments
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