Scalar mesons and the fragmented glueball

The center-of-gravity rule is tested for heavy and light-quark mesons. In the heavy-meson sector, the rule is excellently satisfied. In the light-quark sector, the rule suggests that the $a_0(980)$ could be the spin-partner of $a_2(1320)$, $a_1(1260)$, and $b_1(1235)$; $f_0(500)$ the spin-partner of $f_2(1270)$, $f_1(1285)$, and $h_1(1170)$; and $f_0(980)$ the spin-partner of $f_2'(1525)$, $f_1(1420)$, and $h_1(1415)$. From the decay and the production of light scalar mesons we find a consistent mixing angle $\theta^{\rm s}=(14\pm4)^\circ$. We conclude that $f_0(980)$ is likely"octet-like"in SU(3) with a slightly larger $s\bar s$ content and $f_0(500)$ is SU(3)"singlet-like"with a larger $n\bar n$ component. The $a_0(1450)$, $K^*_0(1430)$, $f_0(1500)$ and $f_0(1370)$ are suggested as nonet of radial excitations. The scalar glueball is discussed as part of the wave function of scalar isoscalar mesons and not as additional"intruder". It seems not to cause supernumerosity.


Introduction
Quantumchromodynamics (QCD) allows for the existence of a large variety of different states. SU(3) symmetry [1] led to the interpretation of mesons and baryons as composed of constituent quarks [2], as qq and qqq states in which a colored quark and an antiquark with anticolor or three colored quarks make up a color-neutral hadron [3]. Color-neutral objects can be formed as well as (qqqq) tetraquarks [4]; the string between two quarks or a quark and an antiquark can be excited forming mesonic (qqg) [5] or baryonic (qqqg) [6] hybrids. Glueballs with two or more constituent gluons may exist (gg, ggg) [3] as well as molecules generated by meson-meson [7], meson-baryon [8] or baryon-baryon [9] interactions. These objects are all color-neutral, all may exist as ground states, all may have orbital excitations. A recent review of meson-meson and meson-baryon molecules can be found in [10] and of further non-qq candidates in [11]. This is a large variety of predicted states, and we may ask if all these possibilities are realized independently. In Ref. [12] the question was raised, if -at least for heavyquark mesons -quarkonia exist only below the first relevant S-wave threshold for a two-particle decay or if all quarkonia have at least a QQ seed. In this letter we restrict ourselves to the discussion of scalar mesons in the light-quark sector. Here, all mesons fall above their threshold for S-wave two-particle decays. Indeed, the mostly accepted view is that we have a nonet of mesonic molecules a 0 (980), K * 0 (700), f 0 (980), f 0 (500) -also interpreted as tetraquarks. But the two states a 0 (1450) and K * 0 (1430) -certainly above their threshold for S-wave two-particle decays -are usually interpreted as qq states. The two additionally expected scalar isoscalar qq states are supposed to mix with the scalar glueball thus forming the three observed mesons f 0 (1370), f 0 (1500), f 0 (1710) [13,14]. Above f 0 (1710), no scalar isoscalar mesons are accepted as established in the Review of Particle Physics (RPP) [15].
In this paper we study the possibility that all scalar mesons below 2.5 GeV have a qq seed. They may acquire large tetraquark, molecular or glueball components but all scalar mesons can be placed into spin-multiplets containing tensor and axial-vector mesons with spin-parity J P = 2 ++ or 1 ± . For the light scalar mesons we apply the center-of-gravity (c.o.g.) rule to the light tensor and axial vector mesons to calculate the "expected" mass of scalar mesons. Surprisingly we find, that the predicted masses fall (slightly) below the mass of the light scalar mesons. We speculate that small qq components could be the seed of the light scalar mesons. Further, we investigate the possibility to determine the mixing angle of the light scalar meson nonet. We argue that a 0 (1450), K * 0 (1430), f 0 (1500), and f 0 (1370) can be accommodated as radial excitations. At the end, we discuss the glueball. It seems not to invade the spectrum as additional resonance but rather as component in the wave function of regular scalar mesons.

The nonet of light scalar mesons
The σ meson is now firmly established as f 0 (500). In the chiral limit, the pion is massless and QCD is controlled by a single parameter α s or Λ QCD . This is sufficient to generate a pole dynamically, the f 0 (500) [16]. QCD dynamics, with only few free parameters, control the spontaneous breaking of chiral symmetry, confinement and the f 0 (500). There is no need for any qq or tetraquark component in its wave function. Inspite of this success, its nature as qq meson, tetraquarks or mesonic molecule is still a topic of a controversial discussion. A survey of interpretations of the f 0 (500) can be found in [17]. The existence of κ or K * 0 (700) is now certain as well [18][19][20]. Apparently the two resonances form, jointly with f 0 (980) and a 0 (980), a nonet of meson resonances. Here, we recall a few different views.
Jaffe calculated the spectrum of light tetraquarks in the MIT bag mode [4]. A nonet of light scalar mesons composed of two quarks and two antiquarks emerged while qq scalar mesons were found at masses well above 1 GeV. This view was supported later by lattice calculations [21]. Van Beveren, Rupp and collaborators [22] developed a unitarized non-relativistic meson model. In the unitarization scheme for S-wave scattering, a qq resonance at 1300 MeV with J P C = 0 ++ quantum numbers originates from a confining potential. The resonance is coupled strongly to real and virtual meson-meson channels, creating a new pole in ππ scattering: the f 0 (500). Similarly, K * 0 (700), and a 0 (980) and f 0 (980) evolve from the unitarization scheme, no tetraquark configurations are required. A similar model was suggested by Tornqvist and Roos [23]. In their model, the f 0 (980) and a 0 (980) are created by the KK → ss → KK interaction but owe their existence to qq states. The authors of Ref. [24] study in a Bethe-Salpeter approach the dynamical generation of resonances in isospin singlet channels with mixing between two and four-quark states. The authors conclude that the f 0 (500) wave function is dominated by the ππ component; the tetraquark component is almost negligible, the quark-antiquark component contributes about 10%. The first radial excitation of the f 0 (500) is tentatively identified with the f 0 (1370). The two states f 0 (980) and a 0 (980) are very close in mass to the KK threshold. They are supposed to consist of two uncolored qq pairs and to be dynamically generated from KK interactions [25]. Achasov et al. [26] provide arguments in favor of the tetraquark picture of the light scalar mesons.
The two resonances a 0 (980) and f 0 (980) are both seen in lattice calculations [27,28]. Their unusual pole structure is argued to point at a strong KK molecular component. The f 0 (500) is "stable" (the pion mass is 391 MeV). Using the Weinberg criterium [29] discussed below, the authors find a molecular ππ component of about 70%. The authors of ref. [30] performed lattice studies of the quark content of the a 0 (980) in two-meson scattering. It is suggested that the a 0 (980) is a superposition of qq and tetraquark.
Rupp, Beveren, and Scadron reject the hypothesis that a strict distinction can be made between "intrinsic" and "dynamically generated" states [31], and Jaffe argued against the possibility to define a "clear distinction between a meson-meson molecule and a qqqq state" [32]. Such a distinction is certainly possible only in the proximity of an S-wave threshold [10].

The center-of-gravity rule
In the absence of tensor and spin-spin forces, the mass of the singlet heavy quarkonium states is given by the weighted average of the triplet states [33]: The differences between the measured and the predicted masses of the h masses are shown in Table 1. The masses were taken from the Review of Particle Physics (RPP) [15]. In the sector of heavy quarkonia, the c.o.g. rule is excellently satisfied.
We next test the rule for charmed mesons. However, the mixing angle for the two charmed mesons D s1 (2460) and D s1 (2536) is not known. In the heavy-quark limit, we expect no mixing. We test the c.o.g. rule with this assumption: The difference between the left-hand (2535.11 ± 0.06) MeV and right-hand (2504.6 ± 0.7) MeV side is 30 MeV. A finite mixing angle would be needed to satisfy the c.o.g. rule. We note that according to the Weinberg criterion discussed below, D * s0 (2317) and D s1 (2536) have a large molecular component and could even be purely molecular states [34]. Here we assume that they have at least a cs seed. Too little is known for the D-meson excited states to test the c.o.g. rule.
We now apply the c.o.g. rule for light mesons to predict the mass of scalar mesons. We are aware of the fact that this is dangerous and possibly misleading, in particular for the calculation of the f 0 (500) and K * 0 (700) states. These are very broad states, and the assumption that they have similar internal dynamics as the tensor and axial vector mesons must be wrong. Certainly, we do not know what the impact is of meson-meson loops. Nevertheless, it seems legitimate to us to explore the borders of applicability of the c.o.g. rule.
In the strange-quark sector, we have the well known K * 2 (1430) at (1427.3±1.5) MeV and two axial vector mesons, K 1 (1280) and K 1 (1400) with masses of (1253±7) and (1403 ±7) MeV, respectively. The scalar meson mass can be estimated to The two observed resonances K 1 (1270) and K 1 (1400) are mixtures of K 1A and K 1B . A recent analysis of the mixing parameters suggests a mixing angle of −(33. 6 With these values we derive a scalar mass M K * 0 = (574±40) MeV.
In Table 2 we compare the masses calculated using the c.o.g. rule with the masses of the lowest-mass scalar mesons. The comparison is very surprising: the predicted masses are rather close to the masses of the light scalar mesons. The comparison suggests that the light scalar mesons are not only dynamically generated. At the same time they might also play the role of ground-states of the scalar qq mesons. The Particle Data Group interprets the mesons a 0 (1450), f 0 (1710), f 0 (1370), and K * 0 (1430) as ground states of scalar mesons. This is clearly in conflict with the c.o.g. rule. Scalar mesons are subject to strong unitarization effects, hence significant mass shifts can be expected. But in the PDG interpretation, these effects would need to be extremely large. The masses calculated from the c.o.g. rule fall only a little below the actual lightscalar meson-masses. The masses of light mesons, given in the second line in Table 2, are certainly better in agreement with the values calculated using the c.o.g. rule than Table 3: Suggested spin-multiplet assignment for light mesons. with the assignment made by the Particle Data Group given in the last line of the Table. In Table 3 we show the suggested multiplet assignment. There is one remark to be made: the a 0 (980) is spinpartner of a 2 (1320), the f 0 (980) is not the spin-partner of f 2 (1270) but of f 2 (1525). In Jaffe's model, a 0 (980) and f 0 (980) are both nnss. Here, a 0 (980) has a nn seed, with nn = (uū + dd)/ √ 2, that acquires a large ss component while f 0 (980) has a ss seed acquiring a large nn component. Due to their mass very close to the KK threshold, both resonances develop a large KK molecular component and can thus be generated dynamically.
The assignment of, e.g., f 2 (1320), f 1 (1285), f 0 (500), and h 1 (1170), to one spectroscopic multiplet, to excitations with L=1, S=1 coupling to J=2,1,0 and L=1, S=0, J=1, is at variance with the Godfrey-Isgur model [37]: In this model, the lowest-mass isoscalar meson is predicted to have a mass of 1090 MeV, certainly below the f 2 (1270) mass but far above the f 0 (500) mass. The Bonn model [38], that is based on instanton-induced interactions instead of an effective one-gluon exchange, predicts 665 MeV (model B), in better agreement with the c.o.g. rule. Figure 1 shows a comparison of the spectrum of L = 1 excitations as listed in Table 3 with the results of the Godfrey-Isgur model [37]. The predicted masses of scalar mesons are all considerably above the scalar masses given in Table 3. However, all predicted scalar masses fall below their multiplet partners. The RPP assignments to the lowest-mass scalar qq mesons are all above their multiplet partners (except K * 0 (1430) that is degenerate in mass with K * 2 (1430)). Hence we believe that it is difficult to decide on the basis of a quark model which scalar mesons belong to the L = 1 multiplet.
Of course, the c.o.g. rule does not imply that the f 0 (500) is a pure qq state. It does not even need to have a large qq component. A small qq seed could acquire a strong nnnn component. Color exchange makes this indistinguishable from a ππ component, the leading term when the resonance is generated dynamically. Thus the light scalar mesons may have qq, tetraquark and molecular contributions [39].
We emphasize that the interpretation of the light scalar meson-nonet as part of the qq spectrum is by no means in contradiction with chiral dynamics. This nonet is generated dynamically; but at the same time it could play its part in the qq spectrum. This is one version of the wellknown chicken-and-egg problem: Is the interaction first, that generates the pole, or is the pole first, that generates the interaction? QCD generates the interaction and the pole, and one aspect is not thinkable without the other one. The observation that the masses of the light scalar mesons are rather close to the predictions based on the c.o.g. rule opens the chance that these mesons could also be related the qq family. We are aware of the fact that this interpretation is at variance with the modern understanding of the spectrum of light scalar meson, but we have no other explanation why the c.o.g. rule "happens" to be satisfied for light mesons.

The flavor wave function
The mixing angle of pseudoscalar mesons. Mesonic mixing angles can be determined in the SU(3) singlet and octet basis or in a quark basis with nn and ss.
For the pseudoscalar mesons, the difference in the octet and singlet decay constants is not negligible, and the nonet is described by two mixing angles θ ps 1 and θ ps 8 [40]. Here, we neglect the difference and use the singlet/octet basis in the form and in the quark basis The ideal mixing angle θ ideal = 35.3 • is defined by tan θ ideal = 1/ √ 2 and leads to η qq = −ss and η qq = nn. In the quark basis the two mixing angles Φ ps n and Φ ps s are very similar in magnitude [41], we use ϕ ps = (39.3 ± 1.0) • [42]. In the singlet-octet basis, the unique mixing angle is given as θ ps = (39.
and for ϕ ps = 39.2 • we get The mixing angle of scalar mesons. We now discuss the possibility to define the mixing angle for scalar mesons. For scalar mesons, the refinement by two mixing angles exceeds the quality of data available at present, and we use a mixing scenario with one mixing angle. In formulas we use f 0 (500) = σ, K * 0 (700) = κ, a 0 (980) = a 0 , and f 0 (980) = f 0 as abbreviations.
We use the scalar mixing angles θ s and ϕ s defined by and This definition is used by Oller [43], and ϕ s = θ s + (90 The authors of Ref. [44] use a wave function in the qq basis: with Φ s = −ϕ s . Ochs [45] assigns f 0 (980) and f 0 (1500) to the same multiplet and determines the mixing angle. However, he uses only properties of f 0 (980) and the decomposition with ϕ s = Φ − 90 • . Most authors [48][49][50][51][52]56] use a definition of the mixing angle in the quark basis in the form with with ϕ s = φ − 90 • . If the light scalar mesons behave like ordinary qq mesons, they should have a unique mixing angle. Mixing angles can be derived from the production and decay of scalar mesons using SU(3) relations or from the masses exploiting the Gell-Mann-Okubo (GMO) formula. Most authors neglect the effects of the different densities at the origin of the f 0 (500) and f 0 (980) wave functions and assumed that these resonances are qq states: these are certainly two highly questionable assumptions. Nevertheless we will compare the mixing angles derived using different methods.
The mixing angle from decays of scalar mesons. Oller [43] determined the mixing angle in the singlet-octet basis of the light scalar mesons in a SU(3) analysis of the meson decay couplings. The coupling constants were obtained by determining the residues at the pole positions of the resonances. The analysis gave θ s = (19 ± 5) • or ϕ s = (74 ± 5) • . Many determinations of the mixing angle are ambiguous. In these cases, we choose the value that is closer to this value. The resulting mixing angles are collected in Table 4.
The authors of Ref. [44] study radiative decays of Φmesons and the two-photon width of scalar and tensor mesons. The calculated two-photon width of scalar and tensor mesons agree well with data even though approximately equal radial wave functions are used. For this reason, the authors argue that a 2 (1320), f 2 (1270) and f 2 (1525) might belong to the same P -wave multiplet as a 0 (980) and f 0 (980). For the scalar mixing angle, two solutions were found Φ s = −(48±6) • or (86±3) • . Equations (11) and (14) are related by Φ s = −ϕ s . The latter angle defines a mixing angle ϕ s = −(86 ± 3) • . The overall sign of the wave function is not important here, hence we quote ϕ s = (94 ± 3) • in Table 4.
In Ref. [46], the use of the density of the wave function at the origin for hadronic molecules is critisized; instead, dressed meson propagators and photon emission vertices are used to calculate the two-photon widths of f 0 (980). The result is in excellent agreement with the experimental value. The radiative Φ decays into f 0 (980) and a 0 (980) are also consistent with the molecular view of these mesons [47]. The LHCb collaboration also studied the reactionB 0 s → J/ψπ + π − [50]. The π + π − invariant mass spectrum shows a large contribution from f 0 (980) and no sign for f 0 (500). From the upper limit, the scalar mixing angle is constrained to |φ s | < 7.7 • at 90% confidence level, or (3 ± 3) • . We use The authors of Ref. [51] study B 0 d,s → J/ψf 0 (500) [f 0 (980)] decays. In these decays, theb-quark in the B d,s converts into ac quark under emission of a W -boson. The W decays into a c quark plus ad -picking up the d-quark of the B d -or ans that combines with the s-quark of the B s . The squared transition amplitudes are thus proportional to the nn or ss content of the scalar wave functions. Table 4: Scalar mixing angle ϕ s from production experiments and from σ decay coupling constants [43]. The sign of the mixing angles is mostly undetermined. It is chosen to be compatible with Ref. [43].   [55]. The f 0 (500) contribution is proportional to cos 2 φ s , the f 0 (980) contribution to sin 2 φ s . Using the unweighted mean and spread of RPP results on Γ f0(980)→ππ /Γ tot = 0.72 ± 0.03 and after correction for the different phase spaces, mixing angles are determined that are listed in Table 4.
The eleven mixing angles are statistically not compatible. We assume the uncertainties are dominated by systematic uncertainties. Therefore, we take the mean value of the scalar mixing angles. The full spread is 11 • that corresponds to a statistical uncertainty of 4 • . The mixing angle θ s in the singlet/octet basis and the quark basis φ s are related by φ s = θ s + 90 − 35 • . Thus we have For θ s = 14.4 • , the octet contribution to the f 0 (500) and the singlet contribution to the f 0 (980) are small: In the quark basis and for φ s = 69.3 • , the wave functions can be cast into the form and f 0 (500) is mostly an nn state and f 0 (980) an ss state.
The flavor wave function from the GMO formula.
The mass pattern of the lightest scalar mesons differs decisively from the one observed for other mesons. There is an isovector meson a 0 (980) and an isoscalar meson f 0 (980)like ρ and ω -but in contrast to the vector mesons, these scalar mesons are heavier than their nonet partners. The other isoscalar meson, f 0 (500), has the lightest mass of the multiplet. The spectrum of scalar mesons resemble the pseudoscalar mesons but inverted: There are the lowmass π and η that correspond to a 0 (980) and f 0 (980), the K and K * 0 (700), and the high-mass η corresponds to the low-mass f 0 (500).
The sum rule contains f and f symmetrically. It is violated at the 6% level when the difference is compared to (8m 2 k + 3m 2 a ). We identify σ as f and f 0 as f .
For ideal mixing, we expect The expectation holds true at this 10% level. This is in line with the linear GMO formula that yields a negative mixing angle, θ s lin = −(55.7 ± 2.5) • , φ s lin = −(91.0 ± 2.5) • . Alternatively we use the mass formula in the form and obtain θ s lin = ±(67.  Table 5 we compare the mixing angles of scalar and pseudoscalar mesons. The GMO formula suggests that f 0 (500) is mostly an nn state and f 0 (980) an ss state. From the production and decay of light scalar mesons we conclude that f 0 (500) is mainly in the singlet, f 0 (980) mainly in the octet configuration, with a small mixing angle.
There is an important difference between the mixing angle derived from production and decay of mesons or from the GMO formula. Production and decay depend on the mesonic wave function, the GMO formula if sensitive to the mass content.
With the caveats expressed above (f 0 (500) and f 0 (980) may have different wave functions and are likely no qq states) we can still note that a mixing angle for the light scalar mesons can be defined. The f 0 (500) is a mainly nn state but has an additional singlet component, beyond the one expected from the decomposition of nn into singlet and octet. The f 0 (980) mainly ss with an additional octet component.
Size of light scalar mesons. Weinberg derived a criterium to decide if a bound state is elementary or compact or if it is an extended particle [29,34]. A quantity Z can be defined that is related to the scattering length a via where γ = √ 2µE B denotes the binding momentum. The quantity Z determines the compact fraction of the bound state, 1 − Z the "molecular" fraction. For Z = 0, the system is "molecular", for Z = 1 compact.
The authors of Ref. [60] extended this relation and derived a probabilistic interpretation of the compositeness relation for resonances. Taking the effective range into account, the authors determined Z σ to 0.60 ± 0.02 where the error is of statistical nature only. The smallness of the molecular fraction in the f 0 (500) wave function is in line with a determination of the rms radius of the f 0 (500) from a Taylor expansion of the f 0 (500) form factor [61]: The f 0 (500) meson seems to be a compact object. A similar analysis was reported in Ref. [25] for a 0 (980) and f 0 (980). The authors concluded that the probability to find the a 0 (980) as qq state is about 25 to 50% and for the f 0 (980) meson this probability is even smaller, about 20% or less. In Ref. [60], the probability to find the f 0 (980) as qq state is Z f0 = 0.33 +0. 28 −0.28 . The compact fractions of f 0 (500) and f 0 (980) differ significantly, even though the large errors do not completely exclude similar spatial wave functions of f 0 (500) and f 0 (980). The K * 0 (700) has only a small molecular component, Z κ = 0.88.
Obviously, the Weinberg criterium does not exclude a possibly small qq component in the wave functions of light scalar mesons.

Excited scalar states
When a 0 (1450), K * 0 (1430), f 0 (1370), and f 0 (1710) are not elements of the lowest-mass qq multiplet with L = 1, do they fit into a radial excitation multiplet? Unfortunately, only few states are known to complete the qq multiplets with L = 1 and one unit of radial excitation. With the masses of a 2 (1700), a 1 (1640), and a 0 (1450) as given in the RPP, we predict -using the c.o.g. rule -a b 1 mass of M b1(xxx) = (1663 ± 23) MeV. Quark models predict higher masses for the radial excitations, hence we use mass differences relative to the a 2 (1700) mass. The Godfrey-Isgur [37] (Bonn [38]) model predicts the b 1 mass to be 40 (50) MeV below the a 2 (1700) mass; both values are in excellent agreement with the c.o.g. prediction. The f 2 (1640), h 1 (1595), f 0 (1370) can be used to predict the mass of a f 1 (xxx) to 1567 +60 −92 MeV. The Godfrey-Isgur [37] (Bonn [38]) model predicts the f 1 mass to be 40 ( ≈ 60) MeV below the f 2 mass, certainly compatible with the predicted mass. According to the c.o.g. rule, the a 0 (1450) resonance is a credible candidate to be the first radial excitation of the a 0 (980) and to be spin-partner of a 2 (1700), a 1 (1640) and an unobserved b 1 expected at 1655 MeV. The f 0 (1370) could be the radial excitation of f 0 (500) and spin-partner of f 2 (1640), h 1 (1595), and a missing h 1 at about 1590 MeV. There is neither a a 0 nor Table 6: Scalar mesons: ground states and radial excitations. Masses and the assignment to mainly-singlet and octet configurations stems from Ref. [62].
In Table 6 we collect the known scalar mesons. Three full nonets can be defined. The GMO formula suggest a mixing angle for the four mesons a 0 (1450), K * 0 (1430), f 0 (1500), f 0 (1370) that is nearly ideal. The mixing angle for a 0 (2020), K * 0 (1950), f 0 (2100), and f 0 (2020) vanishes in the singlet/octet basis. In both cases the errors are too large to make this a solid statement.

The fragmented glueball
QCD predicts the existence of glueballs, of states without constituent quarks. Here we quote a few calculations of the glueball spectrum [63][64][65][66][67][68][69]. The lowest-mass glueball is expected to have scalar quantum numbers, a mass in the 1500 to 2000 MeV range, to intrude the spectrum of scalar mesons and to mix with them. A large number of different mixing schemes were reported initiated by the work of Amsler and Close [13,14]. The proof for the presence of a glueball is supposed to be supernumerosity: it is expected that more mesons should be observed than predicted by quark models.
This expectation was not met in a recent fit to a large body of reactions on radiative J/ψ decays, ππ elastic scattering, pion-induced reactions andpp annihilation [62]. The multi-channel fit to the data revealed the existence of ten scalar isoscalar resonances from f 0 (500) to f 0 (2330). Of particular importance for the interpretation are the data on radiative J/ψ decays to π 0 π 0 [70] and K S K S [71] shown in Fig. 2. In these reactions, the J/ψ is supposed to decay into one photon and two gluons (see Fig. 3, left). The two gluons interact and undergo hadronization. Figure 2 compares the invariant mass distributions resulting from radiative J/ψ decays with the pion and kaon form factors. Their square is proportional to the cross sections. The form factors were deduced by Ropertz, Hanhart and Kubis [72] exploiting the reactionsB 0 s → J/ψ π + π − [50] andB 0 s → J/ψK + K − [73] and imposing the S-wave phase shifts from ππ and KK. In theB 0 s decays, a primary ss pair converts into the final state mesons (see Fig. 3, right). The scale of the formfactor is chosen to match the intensity at high masses.
Both formfactors are dominated by the f 0 (980) resonance. The resonance connects the initial ss pair to π + π − in the final state: the f 0 (980) must have ss and nn components, and this statement holds true for the f 0 (1500) as well. The f 0 (980) is nearly absent in radiative J/ψ decays: this was explained by a dominance of the SU(3) octet component in f 0 (980). Most striking is the mountain landscape above 1500 MeV in the data on radiative J/ψ decays. The huge peak, e.g., at 1750 MeV in the KK mass spectrum and the smaller one at 2100 MeV decay prominently into KK but are produced with two gluons in the initial state and not -at least not significantly -with ss in the initial state. This is highly remarkable: the two gluons in the initial state must be responsible for the production gg → ππ ss → ππ gg → KK ss → KK Figure 2: (Color online) π 0 π 0 (a) [70] and K S K S (b) [71] invariant mass distributions from radiative J/ψ decays (histogramm) with fit [62] and the pion (a) and kaon (b) formfactors [72] from of resonances that decay strongly into KK but are nearly absent when ss pairs are in the initial state. Also the rich structure in the radiatively produced ππ mass spectrum is accompanied with little activity when the initial state is ss. The rich structure stems from gluon-gluon dynamics. In Ref. [62] it was shown that the total yield of scalar mesons as a function of their mass can be described by a Breit-Wigner function with mass M G = (1865±25 +10 −30 ) MeV and width Γ G = (370 ± 50 +30 −20 ) MeV. The peak is created by gluon-gluon interactions. It is the scalar glueball. The glueball peak extends over several scalar resonances, the glueball is fragmented. These scalar mesons can be grouped into a class of mesons with mainly octet and mainly singlet qq components. The two classes of mesons fall onto two linear (n, M 2 ) trajectories, there is no supernumerous state.
The classification into mainly singlet and mainly octet resonances is based on the opposite interference pattern in ππ and KK (see Ref. [62]). This observation seems to suggest that the scalar resonance in in the mass range from 1700 to 2100 MeV are produced largely due to their gg component in the wave function while they decay largely via their qq component.

Summary and conclusions
In this paper, an alternative view of scalar mesons was presented. The light scalar mesons are interpreted as members of a spin multiplet that includes the well known tensor and axial vector mesons. The scalar glueball of lowest mass is not seen as supernumerous state invading the spectrum of scalar isoscalar states and mixing with them but rather as component of the wave functions of scalar isoscalar mesons.
with possible contributions from higher terms. We thus could expect five (or more) different types of states. Since they all have the same quantum numbers, they can mix. But the number of states to be expected is large. We now make a conjecture: we assume that only one state exist and that the orthogonal states disappear in the continuum. Low-mass scalar mesons can be dynamically generated but still have a qq seed. Scalar mesons can have a large glueball component but still are part of the regular spectrum of scalar qq mesons. The situation can be compared to the one observed in case of the N (1535)1/2 − resonance. This resonance is generated dynamically [74] but still plays an important role in quark models as member of the nucleon's first excitation multiplet with orbital angular momentum L = 1. It may also be interpreted as pentaquark state with hidden strangeness [75]. These different views, to consider N (1535)1/2 − as dynamically generated from mesonbaryon interactions, its interpretation as pentaquark, and its assignment to the three-quark baryon spectrum are legitimate and provide additional insights. The light scalar mesons can be dynamically generated molecules but they could still have a qq seed and play their part in quark models.