Scalar Mesons in Radiative Decays and π-π Scattering ∗
Masayasu Harada Department of Physics, Nagoya University, Nagoya, 464-8602, Japan
In this write-up, I summarize the analyses on the low-lying scalar mesons I have done recently with my collaborators. I first briefly review the previous analyses on the hadronic processes related to the scalar mesons, which shows that the scalar nonet takes dominantly the qqq¯q¯ structure. Next, I summarize our analysis on the radiative decays involving the scalar mesons, which indicates that it is difficult to distinguish qqq¯q¯ picture and qq¯ picture just from radiative decays. Finally, I summarize our recent analysis on the π-π scattering in the large Nc QCD, which indicates that the σ meson is likely the qqq¯q¯ state.
I. Introduction This write-up is organized as follows: In section II, following Refs. [2, 5], I will briefly review the analyses on the hadronic processes related to the scalar mesons. According to recent theoretical and experimental anal- Next, in section III, I will briefly summarize the analysis yses, there is a possibility that nine light scalar mesons on the radiative decays involving the scalar mesons based exist below 1 GeV, and they form a scalar nonet [1]. In on the 4-quark picture [6]. I also present a new result on addition to the well established f (980) and a (980) ev- 0 0 the analysis on the decay processes based on the 2-quark idence of both experimental and theoretical nature for a picture [8]. In section IV, I will summarize our analysis very broad σ ( 560) and a very broad κ ( 900) has on the π-π scattering in the large N QCD [7]. Finally, been presented.' ' c in section V, I will give a brief summary. As is stressed in Ref. [2], the masses of the above low- lying scalar mesons do not obey the “ideal mixing” pat- tern which nicely explains the masses of mesons made II. Effective Lagrangian for Scalar Mesons from a quark and an anti-quark such as vector mesons [3]. As is shown in Ref. [2], the “ideal mixing” pattern qual- In this section I briefly review previous analyses [2, 5] itatively explains the mass hierarchy of the scalar nonet on the masses of scalar mesons and hadronic processes when the members of the nonet have a qqq¯q¯ quark struc- related to the scalar mesons. ture proposed in Ref. [4]. In this 4-quark picture, two In Ref. [2], the scalar meson nonet is embedded into quarks are combined to make a diquark which together the 3 3 matrix field N as with an anti-diquark forms a scalar meson. The resultant × scalar mesons have the same quantum numbers as the 0 + + NT + a0 /√2 a0 κ ordinary scalar mesons made from the quark and anti- − 0 0 N = a0 NT a0 /√2 κ , (2.1) quark (2-quark picture). It is difficult to clarify the quark − − 0 κ κ¯ NS structure of the low-lying scalar mesons just from their quantum numbers. The patterns of the interactions of where NT and NS represents the “ideally mixed” fields. the scalar mesons to other mesons made from qq¯, on the The physical σ(560) and f0(980) fields are expressed by other hand, depend on the quark structure of the scalar the linear combinations of these NT and NS as mesons. I expect that the analysis on the interactions of the scalar mesons will shed some lights on the quark σ cos θS sin θS NS = − , (2.2) structure of the scalar nonet. Actually, in Refs. [2, 5], f0 sin θS cos θS NT several hadronic processes related to the scalar mesons are studied. They concluded that the scalar nonet takes where θS is the scalar mixing angle. dominantly the qqq¯q¯ structure. The scalar mixing angle θS can parameterize the quark ◦ contents of the scalar nonet field: When θS = 90 , the Recently, for getting more informations on the struc- ± ture of the low-lying scalar mesons, we studied the ra- σ and f0 fields are embedded into the nonet field as diative decays involving scalar mesons [6] and the π-π 0 √ + + scattering in the large Nc QCD [7]. In this write-up I σ + a0 / 2 a0 κ − 0 0 will summarize these analyses, especially focusing on the N = a0 σ a0 /√2 κ . (2.3) − − 0 κ κ¯ f quark structure of the low-lying scalar nonet, and show 0 how these processes give a clue for understanding the structure of the scalar mesons. This is a natural assignment of scalar meson nonet based on the qq¯ picture:
uu¯ du¯ su¯ ∗ ud¯ dd¯ sd¯ . (2.4) Talk given at YITP workshop on “Multi-quark Hadrons; four, five ∼ and more ?”, February 17-19, 2004, Yukawa Insitute, Kyoto, Japan. us¯ us¯ ss¯ ◦ ◦ On the other hand, when θS = 0 or 180 , the scalar where ηT and ηS denote the ideally mixed fields. Based nonet field N becomes on the two-mixing-angle scheme introduced in Ref. [13] 0 0 √ + + the physical η and η fields are expressed by the linear f0 + a0 / 2 a0 κ combinations of η and η . − 0 √ 0 T S N = a0 f0 a0 / 2 κ , (2.5) By using the scalar meson nonet field N defined in − − 0 κ κ¯ σ Eq. (2.1) together with the above pseudoscalar nonet field which is a natural assignment of scalar meson nonet P , the general SU(3) flavor invariant scalar-pseudoscalar- based on the qqq¯q¯ picture: pseudoscalar interaction is written as [2] abc d e µ f ¯ ¯ ¯ NP P = A def Na ∂µPb ∂ Pc s¯dds s¯dus s¯dud −L µ µ s¯uds¯ s¯uus¯ s¯uud¯ . (2.6) + B tr[N] tr[∂µP ∂ P ]+ Ctr[N∂µP ] tr[∂ P ] ∼ µ u¯dds¯ u¯dus¯ u¯dud¯ + Dtr[N] tr[∂µP ] tr[∂ P ] , (2.14) Then, the present treatment of nonet field with the scalar where A, B, C and D are four real constants, and a,b,c = mixing angle can express both pictures for quark con- 1, 2, 3 denote flavor indices. The derivatives of the pseu- tents. doscalars were introduced in order that Eq. (2.14) prop- By using the scalar nonet field introduced above, the erly follows from a chiral invariant Lagrangian in which effective Lagrangian for the scalar meson masses are ex- the field P transforms non-linearly under chiral transfor- pressed as [2] mation. In Refs. [2, 5], four parameters A, B, C, D and the mass = a tr [NN] b tr [ NN] c tr [N] tr [N] scalar mixing angle are determined by fitting them to L − − M − d tr [ N] tr [N] , (2.7) the experimental data of the π-K scattering and the − M η0 ηππ decay together with the π-π scattering. The where a, b, c and d are real constants, and is the → M resultant best fitted values for A, B, C and D are “spurion matrix” expressing the explicit chiral symme- −1 −1 try breaking due to the current quark masses. This A 2.5 GeV , B 2.0 GeV , ' − '− − is defined by = diag(1, 1, x), where x is the ratioM of C 2.3 GeV 1 , D 2.3 GeV 1 . (2.15) strange to non-strangeM quark masses with the isospin in- '− '− The best fitted value of the scalar mixing angle is variance assumed. Note that the scalar mixing angle is ◦ expressed by a combination of the parameters a, b, c and θS 20 , (2.16) d. '− Here I use the following values of the masses of the which implies that the scalar meson takes dominantly the scalar nonet as inputs: qqq¯q¯ structure. It should be noticed that the coupling constant of the f0-π-π interaction determined from the
Ma0 980 MeV ,Mf0 980 MeV , (2.8) π-π scattering [11] plays an important role to constrain ' ' the value of the mixing angle. listed in Particle Data Group (PDG) table [9, 10],
Mσ 560 MeV , (2.9) ' III. Radiative Decays Involving Scalar Mesons determined from the π-π scattering [11], and In the previous section, I briefly reviewed the analyses Mκ 900 MeV , (2.10) ' done in Refs. [2, 5], which shows that the experimen- tal data of the hadronic decay processes involving scalar determined from the π-K scattering [12]. The above ◦ mesons give θS 20 , i.e., the scalar meson is domi- choice yields the two possible solutions for the scalar mix- ' − ing angle [2] nantly made from qqq¯q¯. In this section, I show our anal- ◦ ysis on the radiative decays involving the scalar mesons θS 20 , (2.11) done in Ref. [6]. ∼− ◦ θS 90 . (2.12) In Ref. [6], the trilinear scalar-vector-vector terms were ∼− included into the effective Lagrangian as Solution in Eq. (2.11) corresponds to the case where the 0 0 0 a b c a b c scalar nonet is dominantly made from qqq¯q¯, while solu- SV V = βA abc [Fµν (ρ)] 0 [Fµν (ρ)] 0 Nc0 L a b tion in Eq. (2.12) to the case where it is from qq¯. + βB tr [N] tr [Fµν (ρ)Fµν (ρ)] For determining the scalar mixing angle, the authors + βC tr [NFµν (ρ)] tr [Fµν (ρ)] of Ref. [2] considered the tri-linear scalar-pseudoscalar- pseudoscalar interaction. There the pseudoscalar mesons + βD tr [N] tr [Fµν (ρ)] tr [Fµν (ρ)] . (3.1) are embedded into the nonet field as where N is the scalar nonet field defined in Eq. (2.1). 0 + + Fµν (ρ) is the field strength of the vector meson fields ηT + π /√2 π K − 0 0 defined as P = π ηT π /√2 K , (2.13) − − 0 K K¯ ηS Fµν (ρ)= ∂µρν ∂ν ρµ ig˜ [ρµ , ρν ] , (3.2) − − S 2 2 whereg ˜ 4.04 [14] is the coupling constant. (A term where kV = (mV mS)/(2mV ) is the photon momentum tr(FFN' ) is linearly dependent on the four shown). in the V rest frame.− For the energetically allowed V In∼ Ref. [6], the vector meson dominance is assumed to Sγ processes we have → be satisfied in the radiative decays involving the scalar 2√2 4 mesons. Then, the above Lagrangian (3.1) determines all f0 √ Cφ = βAc βB 2c + s the relevant interactions. Actually, the βD term will not 3 − 3 contribute so there are only three relevant parameters √2 √ β , β and β . Equation (3.1) is analogous to the P V V + βC c 2s , A B C 3 − interaction 1 which was originally introduced as a πρω σ 2√2 4 coupling a long time ago [16]. One can now compute the C = βAs βB c √2s φ − 3 − 3 − amplitudes for S γγ and V Sγ according to the → → 2 1 diagrams of Fig. 1. βC c + s , −3 √2 γ ( k , ε ) 1 1 a0 √ Cφ = 2 (βC 2βA) , ρ00 ,ω,φ − V(p, ε ) ρ 0,ω,φ γ( k ,ε) v σ 2√2 2√2 Cω = βA c + √2s + βB c √2s S 3 3 − ρ0 ,ω,φ S 2 γ(k , ε ) 2 2 βC √2c + s , −3 σ 0 √ √ √ (a) (b) Cρ = 2 2βAc +2 2βB c 2s . (3.6) − − FIG. 1: Feynman diagrams for (a) S → γγ and (b) V → Sγ. In addition, the same model predicts amplitudes for the energetically allowed S Vγ processes: f0 ωγ, 0 0 0 →0 0 → f0 ρ γ, a0 ωγ, a0 ρ γ and, if κ is sufficiently The decay matrix element for S γγ is written as → 0 →∗0 → 2 2 → heavy κ K γ. The corresponding width is (e /g˜ )XS k1 k2 1 2 k1 2 k2 1 where µ stands × · · − · · → for the photon polarization vector. It is related to the V 2 3 D width by Γ(S Vγ)= α kV S , (3.7) → S g˜ 2 π XS Γ (S γγ)= α2 m3 , (3.3) where kV = (m2 m2 )/(2m ) and → 4 S g˜2 S S V S − 2 2 ω √ √ where XS takes on the specific forms: Df0 = βA 2c + 2s + βB 2c + 2s 3 − 3 4 8 2 X = β √2s 4c + β c √2s , √ σ 9 A 3 B + βC c 2s , − − 3 − 4 8 ρ0 Xf0 = βA √2c +4s + βB √2c + s , D = 2√2βAs +2βB 2c + √2s , −9 3 f0 − ω 4√2 Da0 = 2βC , Xa0 = βA . (3.4) 3 ρ0 4 Da0 = βA , In the above expressions α = e2/(4π), s = sin θ and 3 S ∗ K 0 8 c = cos θS where the scalar mixing angle, θS is defined Dκ0 = βA . (3.8) in Eq. (2.2). Furthermore, ideal mixing for the vector −3 0 1 2 1 2 mesons, with ρ = (ρ1 ρ2)/√2, ω = (ρ1 + ρ2)/√2, Let me show the results obtained in Ref. [6] together 3 − φ = ρ3, was assumed for simplicity. with new results from a recent analysis [8]. I should Similarly to the one for S γγ, the decay ma- stress again that all the different decay amplitudes are → S trix element for V Sγ is written as (e/g˜)C described by only three parameters βA, βB and βC . → V × [p kV p k V ]. It is related to the width by In Ref. [6], the value of βA was determined from the · · − · · a0 γγ process. Substituting Γexp(a0 γγ)=(0.28 S 2 α 3 C → → ¯ ± Γ(V Sγ)= kS V , (3.5) 0.09) keV (obtained using [10] B(a0 KK)/B(a0 → 3 V g˜ ηπ)=0.177 0.024) into Eqs. (3.3) and→ (3.4) yields →
± βA = (0.72 0.12) GeV , (3.9) ± where positive in sign was assumed. By using this value, 1 It was shown [15] that the complete vector meson dominance → 0 + − the value of βC is determined from Γexp(φ a0γ) = (VMD) is violated in the ω π π π decay which is expressed → 0 by P V V interactions. However, since the VMD is satisfied in (0.47 0.07) keV (obtained by assuming φ ηπ γ is 0 ∗ ± → other processes such π → γγ as well as in the electromagnetic dominated by φ a0γ) and Eq. (3.6) as form factor of pion, it is assumed to be held in the processes → −1 related to SV V interactions in Ref. [6]. βC = (7.7 0.5 , 4.8 0.5) GeV . (3.10) ± − ± A ± ± It should be stressed that the values of βA and βC ob- β 0.72 0.12 0.72 0.12 βB 0.61 ± 0.10 −0.62 ± 0.10 tained above are independent of the mixing angle θS, βC 7.7 ± 0.52 7.7 ± 0.52 and that βA is almost an order of magnitude smaller | | Γ(σ → γγ) 0.024 ± 0.023 0.38 ± 0.09 than βC . As one can see from Eq. (3.8), the ampli- → ± ± | ω | ρ0 Γ(φ σγ) 137 19 33 9 tude Da0 is given by βC while Da0 is given by only βA. Γ(ω → σγ) 16 ± 3 33 ± 4 Then, the large hierarchy between βC and βA implies Γ(ρ → σγ) 0.23 ± 0.47 17 ± 4 that there is a large hierarchy between Γ(a0 ωγ) and Γ(f0 → ωγ) 126 ± 20 88 ± 17 → Γ(a0 ργ). Actually, by using the values of βA and βC Γ(f0 → ργ) 19 ± 5 3.3 ± 2.0 → given in Eqs. (3.9) and (3.10), they are estimated as Γ(a0 → ωγ) 641 ± 87 641 ± 87 Γ(a0 → ργ) 3.0 ± 1.0 3.0 ± 1.0 Γ(a0 ωγ) = (641 87 , 251 54) keV , → ± ± Γ(a0 ργ)=3.0 1.0 keV . (3.11) → ± TABLE I: Fitted values of βA, βB and βC together with the This implies that there is a large hierarchy between predicted values of the decay widths of V → S + γ and S → '− ◦ Γ(a0 ωγ) and Γ(a0 ργ) which is caused by an order V +γ for θS 20 . Only two out of four sets of (βA, βB , βC ) → → −1 of magnitude difference between βC and βA . are listed here. Units of βA, βB and βC are GeV and those | | | | I next show how to determine the value of βB from the of the decay widths are keV. f0 γγ process. Xf0 in Eq. (3.4) depends on βB as well as on→ β and the scalar mixing angle θ . Here the scalar A S These predictions are very close to the ones in Eq. (3.14). mixing angle θ is taken as S This can be understood by the following consideration: ◦ ω θS 20 , (3.12) From the expression of Df0 in Eq. (3.8), one can see that '− it is dominated by the term including β which is pro- which is characteristic of qqq¯q¯ type scalars [2]. By using C portional to (cos θS √2 sin θS). Then, the approximate this and the value of βA in Eq. (3.9), Γexp(f0 γγ) = relation − 0.39 0.13 keV yields → ± ◦ ◦ − cos( 20 ) √2 sin( 20 ) β = (0.61 0.10 , 0.62 0.10) GeV 1 . (3.13) B − − ◦ − ◦ ± − ± cos( 90 ) √2 sin( 90 ) 1.4 (3.17) This implies that βB is on the order of βA , and almost ' − − − ' | | | | ω ◦ an order of magnitude smaller than βC . Equation (3.8) implies that the value of Df0 for θS = 90 is close to ω | ρ | ◦ − ◦ shows that Df0 includes βC while Df0 does not. Thus, that for θS = 20 , and thus Γ(f0 ωγ) for θS = 90 − ◦ → − there is a large hierarchy between decay widths of f0 to that for θS = 20 . As for Γ(f0 ργ) I should note → − → ωγ and f0 ργ: The typical predictions are given by that the following relation is satisfied for Xf0 in Eq. (3.4) → ρ0 Γ(f0 ωγ) = (88 17) keV , and Df0 in Eq. (3.8): → ± Γ(f0 ργ)=(3.3 2.0) keV . (3.14) 0 → ± ρ 4 3Xf0 2√2D = √2βA(c √2s) . (3.18) This implies that there is a large hierarchy between − f0 −3 − Γ(f0 ωγ) and Γ(f0 ργ) which is caused by the fact → → Since the experimental value of Γ(f0 γγ), i.e., Xf0 is that βC is an order of magnitude larger than βA and used as an input, this relation implies→ that the predicted β .| I summarize| the fitted values β , β and| β| to- ◦ B A B C value of Γ(f0 ργ) for θS = 90 is roughly equal to |gether| with several predicted values of the decay widths → ◦ − that for θS = 20 . Similarly, the predicted values of of V S + γ and S V + γ in Table I. − ◦ other radiative decay widths for θS 90 are also very Let→ me make an analysis→ when the scalar mixing angle ◦ '− ◦ 2 close to those for θS 20 as I list in Table II. is taken as θS 90 . As I stressed above, the values of '− '− The result here indicates that it is difficult to distin- βA and βC are independent of the scalar mixing angle θS. guish two pictures just from radiative decays. Of course, The value of βB determined from Γ(f0 γγ) becomes → other radiative decays should be studied to get more in- −1 βB = (1.1 0.1 , 0.12 0.13) GeV . (3.15) formations on the structure of the scalar mesons. Fur- ± ± thermore, inclusion of the loop corrections may be im- Then the typical predictions for Γ(f0 ωγ) and Γ(f0 portant [17]. However, there are still large uncertainties → → ργ) are given by in the experimental data which make the analysis harder. So instead of the analysis which can be compared with Γ(f0 ωγ) = (86 16) keV , → ± experiment, some theoretical analyses give a clue to get Γ(f0 ργ)=(3.4 3.2) keV . (3.16) → ± more informations on the structure of the scalar mesons.
IV. π-π Scattering in Large Nc QCD 2 It should be noticed that the predicted value of f0-π-π coupling ◦ for θS '−90 is much larger than the value obtained in Ref. [11] by fitting to the ππ scattering amplitude [2] as I discussed in In this section, I briefly review our recent analysis [7] section II. on the π-π scattering in QCD with large Nc, where Nc is βA 0.72 ± 0.12 0.72 ± 0.12 plies that the unitarity is violated, this amplitude breaks − ± ± βB 0.12 0.13 1.1 0.1 the unitarity in the energy region around 500 MeV. The βC 7.7 ± 0.52 7.7 ± 0.52 → ± ± solid line in Fig. 2 shows the curve when the following Γ(σ γγ) 0.023 0.024 0.37 0.10 ρ-exchange contribution is inclded in addition: Γ(φ → σγ) 140 ± 22 35 ± 11 → ± ± Γ(ω σγ) 17 4 33 5 g2 Γ(ρ → σγ) 0.20 ± 0.43 17 ± 4 ρππ 2 Aρ(s,t,u)= 2 (4mπ 3s) Γ(f0 → ωγ) 125 ± 19 86 ± 16 2mρ − Γ(f0 → ργ) 18 ± 8 3.4 ± 3.2 2 gρππ u s Γ(a0 → ωγ) 641 ± 87 641 ± 87 2 − 2 → ± ± − 2 (m t) imρΓρθ(t 4m ) Γ(a0 ργ) 3.0 1.0 3.0 1.0 ρ − − − π t s + 2 − 2 . (4.2) (m u) imρΓρθ(u 4m ) TABLE II: Fitted values of βA, βB and βC together with ρ − − − π the predicted values of the decay widths of V → S + γ and ◦ Note that the appearance of the first term is required S → V + γ for θS ' −90 . Only two out of four sets of by the chiral symmetry. From Fig. 2, we can easily see (βA, βB , βC ) are listed here. Units of βA, βB and βC are − GeV 1 and those of the decay widths are keV. that a large cancellation occurs between the contribution from the pion self-interaction and that from the ρ-meson exchange. However, the unitarity is still violated around the number of colors. 550 MeV. First, let me briefly review the analyses done in To recover unitarity, we need negative contribution to Refs. [11, 18] which stressed that the scalar meson σ is the real part above the point where the solid line in Fig. 2 needed for satisfying the unitarity in the isospin I = 0, violate the unitarity. While below the point a positive S-wave π-π scattering amplitude in real-life QCD with contribution is preferred by the experiment. Such prop- erty matches with the real part of a resonance contribu- Nc = 3. First contribution included in the ππ scattering amplitude is the one from the pion self interaction given tion: The resonance contribution is positive in the en- by the current algebra, or equivalently, expressed by the ergy region below its mass, while it is negative in the leading order chiral Lagrangian: energy region above its mass. In other words, the unitar- ity requires the existence of the resonance in this energy 2 region. Then we have included a low mass broad scalar s mπ Aca(s,t,u)= − , (4.1) state, σ. The contribution of the σ to the real part of the F 2 π amplitude is given by where Fπ = 92.42 MeV is the pion decay constant. The γ2 (s 2m2 )2 (M 2 s) contribution from this to the real part of the I = 0, S- ReA (s,t,u)= σππ π σ , σ − 3 2 2 − 2 02 2 Mσ (s M ) + M G wave ππ scattering amplitude is shown by the dashed − σ σ line in Fig. 2. Since the amplitude greater than 0.5 im- (4.3) where G0 is a parameter corresponding to the width and γσππ is the σ-π-π coupling constant. This γσππ is related to the parameters A and B in the scalar-pseudoscalar- o R o pseudoscalar interaction Lagrangian given in Eq. (2.14) as [2]
γσππ =2B sin θs 2√2(B A)cos θs . (4.4) − − In Ref. [11], a best overall fit was obtained with the pa- rameter choices:
0 −1 Mσ = 559MeV , G = 370MeV γσππ =7.8 GeV . (4.5) 0 The result for the real part R0 due to the inclusion of the σ contribution along with π and ρ contributions is shown in Fig. 3. It is seen that the unitarity bound is
s (GeV) satisfied and there is a reasonable agreement with the experimental points up to about 800 MeV. 0 FIG. 2: Predicted curves for R0 [11]. The horizontal axis The above analysis on the π-π scattering in real-life shows the energy, and the vertical axis shows the magnitude QCD tells an important lesson: The mass of σ meson is of the real part of the amplitude. The dashed line shows the determined by the point where the amplitude constructed contribution from the pion self interaction given by the cur- from π + ρ contribution violates the unitarity. rent algebra. The solid line shows the result with ρ-exchange Now, let me show the results in Ref. [7], where the π-π contribution included in addition. scattering in the large Nc QCD was analyzed. o R o
FIG. 5: Real part of the I = 0 S-wave amplitude due to the current algebra +ρ terms, plotted for the following increasing values of Nc (from up to down), 3, 4, 5, 6, 7. The curve with s (GeV) largest violation of the unitarity corresponds to Nc = 3 while 0 the ones within the unitarity bound are for Nc = 6, 7. [7] FIG. 3: Predicted curves for R0 with the (current algebra) +ρ + σ contribution included [11]. As I showed in Fig. 3, the violation of the unitarity in the real-life QCD is recovered by the existence of the σ First one to be included in the amplitude is the current pole. The σ pole structure is such that the real part of algebra contribution given in Eq. (4.1). Note that the 2 its amplitude is positive for s
FIG. 4: Real part of the I = 0 S-wave amplitude due to the current algebra term plotted for the following increasing values of Nc (from left to right), 3, 5, 7, 9, 11, 13. [7]
Next, I show that this result is strongly modified by the presence of the well established qq¯ companion of the 2 pion – the ρ meson. The amplitude is obtained by adding FIG. 6: The value M (Nc) as a function of Nc and normal- to the current algebra contribution the ρ meson contri- ized to M 2(3)/3. [7] The dashed line corresponds to the pure 0 bution given in Eq. (4.2). In Fig. 5, I show the plot of R0 current algebra contribution while the solid line to current due to current algebra plus the ρ contribution for increas- algebra +ρ contribution. ing values of Nc. Here the scaling property of the ρ-π-π coupling is taken as gρππ(Nc)/gρππ(Nc = 3) = 3/Nc larger Nc, and when Nc 6, the σ is not needed in the ≥ with mρ kept fixed. This figure shows that the unitarityp energy region below 2 GeV. From this we concluded [7] 0 (i.e., R0 1/2) is satisfied for Nc 6 till well beyond that the σ meson is unlikely the 2-quark state and likely the 1| GeV| ≤ region. However unitarity≥ is still a problem the 4-quark state. This is similar to the conclusion ob- for 3, 4 and 5 colors. tained in Ref. [19]. V. Summary in Ref. [7], in which we studied the π-π scattering in large Nc QCD. Our analysis shows that the mass of the σ me- son becomes larger for larger Nc, and when Nc 6, the In this write-up, focusing on the structure of the low- ≥ lying scalar nonet, I summarized the analyses in two π-π scattering amplitude satisfies the unitarity without works [6, 7] which I have done recently. the σ meson. From this we concluded that the σ meson In section II, following Refs. [2, 5], I first briefly re- is unlikely the qq¯ state and likely the qqq¯q¯ state. viewed what the hadronic processes involving the scalar nonet tell about the quark structure of the low-lying scalar mesons. The analysis on the pattern of the Acknowledgments hadronic processes implies that the scalar nonet takes dominantly the qqq¯q¯ structure (or the diquark–anti- I would like to thank the organizers to give an opportu- diquark structure). Next, in section III, I summarized nity to present my talk. I am grateful to Deirdre Black, the work in Ref. [6], in which we analyzed the radiative Francesco Sannino and Joe Schechter for collaboration decays involving the scalar nonet based on the qqq¯q¯ pic- in Refs. [6, 7] on which this talk is based. My work is ture. I also presented a new result [8] on the radiative supported in part by the JSPS Grant-in-Aid for Scien- decays based on the qq¯ picture. Our result indicates that tific Research (c) (2) 16540241, and by the 21st Century it is difficult to distinguish two pictures just from radia- COE Program of Nagoya University provided by Japan tive decays. Finally, in section IV, I summarized the work Society for the Promotion of Science (15COEG01).
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