Scalar Mesons and the Fragmented Glueball

Scalar mesons and the fragmented glueball

Eberhard Klempta

aHelmholtz–Institut f¨urStrahlen– und Kernphysik, Universit¨atNußallee 14-16, Bonn, Germany

Abstract 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 a0(980) could be the spin-partner of a2(1320), a1(1260), 0 and b1(1235); f0(500) the spin-partner of f2(1270), f1(1285), and h1(1170); and f0(980) the spin-partner of f2(1525), f1(1420), and h1(1415). From the decay and the production of light scalar mesons we ﬁnd a consistent mixing angle s ◦ θ = (14 ± 4) . We conclude that f0(980) is likely “octet-like” in SU(3) with a slightly larger ss¯ content and f0(500) ∗ is SU(3) “singlet-like” with a larger nn¯ component. The a0(1450), K0 (1430), f0(1500) and f0(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.

1. Introduction established in the Review of Particle Physics (RPP) [15]. In this paper we study the possibility that all scalar Quantumchromodynamics (QCD) allows for the exis- mesons below 2.5 GeV have a qq¯ seed. They may acquire tence of a large variety of different states. SU(3) symme- large tetraquark, molecular or glueball components but all try [1] led to the interpretation of mesons and baryons as scalar mesons can be placed into spin-multiplets contain- composed of constituent quarks [2], as qq¯ and qqq states in ing tensor and axial-vector mesons with spin-parity J P = which a colored quark and an antiquark with anticolor or 2++ or 1±. For the light scalar mesons we apply the three colored quarks make up a color-neutral hadron [3]. center-of-gravity (c.o.g.) rule to the light tensor and axial Color-neutral objects can be formed as well as (qqq¯q¯) tetra- vector mesons to calculate the “expected” mass of scalar quarks [4]; the string between two quarks or a quark and mesons. Surprisingly we find, that the predicted masses an antiquark can be excited forming mesonic (qqg¯ ) [5] fall (slightly) below the mass of the light scalar mesons. We or baryonic (qqqg) [6] hybrids. Glueballs with two or speculate that small qq¯ components could be the seed of more constituent gluons may exist (gg, ggg) [3] as well as the light scalar mesons. Further, we investigate the possi- molecules generated by meson-meson [7], meson-baryon [8] bility to determine the mixing angle of the light scalar me- or baryon-baryon [9] interactions. These objects are all ∗ son nonet. We argue that a0(1450), K (1430), f0(1500), color-neutral, all may exist as ground states, all may have 0 and f0(1370) can be accommodated as radial excitations. orbital excitations. A recent review of meson-meson and At the end, we discuss the glueball. It seems not to in- meson-baryon molecules can be found in [10] and of further vade the spectrum as additional resonance but rather as non-qq¯ candidates in [11]. component in the wave function of regular scalar mesons. 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 heavy- 2. The nonet of light scalar mesons quark mesons - quarkonia exist only below the first rel- The σ meson is now firmly established as f0(500). In arXiv:2104.09922v1 [hep-ph] 20 Apr 2021 evant S-wave threshold for a two-particle decay or if all quarkonia have at least a QQ¯ seed. In this letter we re- the chiral limit, the pion is massless and QCD is controlled strict ourselves to the discussion of scalar mesons in the by a single parameter αs or ΛQCD. This is sufficient to gen- light-quark sector. Here, all mesons fall above their thresh- erate a pole dynamically, the f0(500) [16]. QCD dynam- old for S-wave two-particle decays. Indeed, the mostly ac- ics, with only few free parameters, control the spontaneous cepted view is that we have a nonet of mesonic molecules breaking of chiral symmetry, confinement and the f0(500). ∗ There is no need for any qq¯ or tetraquark component in a0(980), K0 (700), f0(980), f0(500) – also interpreted as ∗ its wave function. Inspite of this success, its nature as qq¯ tetraquarks. But the two states a0(1450) and K0 (1430) – certainly above their threshold for S-wave two-particle meson, tetraquarks or mesonic molecule is still a topic of a controversial discussion. A survey of interpretations of the decays – are usually interpreted as qq¯ states. The two ad- ∗ ditionally expected scalar isoscalar qq¯ states are supposed f0(500) can be found in [17]. The existence of κ or K0 (700) to mix with the scalar glueball thus forming the three is now certain as well [18–20]. Apparently the two reso- observed mesons f (1370), f (1500), f (1710) [13, 14]. nances form, jointly with f0(980) and a0(980), a nonet of 0 0 0 meson resonances. Here, we recall a few different views. Above f0(1710), no scalar isoscalar mesons are accepted as

Preprint submitted to Elsevier April 21, 2021 Jaﬀe calculated the spectrum of light tetraquarks in the Table 1: The center-of-gravity rule for heavy quarkonia MIT bag mode [4]. A nonet of light scalar mesons composed of two quarks and two antiquarks emerged while qq¯ measured - predicted scalar mesons were found at masses well above 1 GeV. This δ M : (0.08 ± 0.61) MeV view was supported later by lattice calculations [21]. Van hc(1P )

Beveren, Rupp and collaborators [22] developed a unita- δ Mhb(1P ): −(0.57 ± 1.08) MeV rized non-relativistic meson model. In the unitarization δ Mhb(2P ): −(0.4 ± 1.3) MeV scheme for S-wave scattering, a qq¯ resonance at 1300 MeV with J PC = 0++ quantum numbers originates from a con- fining potential. The resonance is coupled strongly to real The differences between the measured and the pre- and virtual meson-meson channels, creating a new pole ∗ dicted masses of the h masses are shown in Table 1. The in ππ scattering: the f0(500). Similarly, K0 (700), and masses were taken from the Review of Particle Physics a0(980) and f0(980) evolve from the unitarization scheme, (RPP) [15]. In the sector of heavy quarkonia, the c.o.g. no tetraquark configurations are required. A similar model rule is excellently satisfied. was suggested by Tornqvist and Roos [23]. In their model, ¯ We next test the rule for charmed mesons. However, the f0(980) and a0(980) are created by the KK → ss¯ → the mixing angle for the two charmed mesons D (2460) KK¯ interaction but owe their existence to qq¯ states. The s1 and Ds1(2536) is not known. In the heavy-quark limit, authors of Ref. [24] study in a Bethe-Salpeter approach we expect no mixing. We test the c.o.g. rule with this the dynamical generation of resonances in isospin singlet assumption: channels with mixing between two and four-quark states. 1 The authors conclude that the f0(500) wave function is M = 5M ∗ + Ds1(2536) 9 Ds2(2573) dominated by the ππ component; the tetraquark compo- 3M + M ∗ (2) nent is almost negligible, the quark-antiquark component Ds1(2460) Ds0(2317) contributes about 10%. The first radial excitation of the f (500) is tentatively identified with the f (1370). The The difference between the left-hand (2535.11±0.06) MeV 0 0 and right-hand (2504.6±0.7) MeV side is 30 MeV. A finite two states f0(980) and a0(980) are very close in mass to the KK¯ threshold. They are supposed to consist of two mixing angle would be needed to satisfy the c.o.g. rule. We note that according to the Weinberg criterion discussed uncolored qq¯ pairs and to be dynamically generated from ∗ KK¯ interactions [25]. Achasov et al. [26] provide argu- below, Ds0(2317) and Ds1(2536) have a large molecular ments in favor of the tetraquark picture of the light scalar component and could even be purely molecular states [34]. mesons. 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. The two resonances a0(980) and f0(980) are both seen in lattice calculations [27, 28]. Their unusual pole struc- rule. ture is argued to point at a strong KK¯ molecular compo- We now apply the c.o.g. rule for light mesons to predict nent. The f (500) is “stable” (the pion mass is 391 MeV). the mass of scalar mesons. We are aware of the fact that 0 this is dangerous and possibly misleading, in particular for Using the Weinberg criterium [29] discussed below, the ∗ authors find a molecular ππ component of about 70%. the calculation of the f0(500) and K0 (700) states. These The authors of ref. [30] performed lattice studies of the are very broad states, and the assumption that they have quark content of the a (980) in two-meson scattering. It similar internal dynamics as the tensor and axial vector 0 mesons must be wrong. Certainly, we do not know what is suggested that the a0(980) is a superposition of qq¯ and tetraquark. the impact is of meson-meson loops. Nevertheless, it seems Rupp, Beveren, and Scadron reject the hypothesis that legitimate to us to explore the borders of applicability of a strict distinction can be made between “intrinsic” and the c.o.g. rule. “dynamically generated” states [31], and Jaffe argued against We start with the equation the possibility to define a “clear distinction between a Mf0 = 9Mh1 − 5Mf2 − 3Mf1 , (3) meson-meson molecule and a qqq¯q¯ state” [32]. Such a dis- and use M =(1170±20) MeV, M =(1275.5±0.8), and M tinction is certainly possible only in the proximity of an h1 f2 f1 =(1281.9±0.5) MeV. We find a scalar mass of M =(307 S-wave threshold [10]. f0 ±180) MeV where the uncertainty is given by the uncertainty the h1(1170) mass. In the other cases, the er- 3. The center-of-gravity rule rors are added quadratically. For mesons with hidden strangeness, we have the masses Mh0 =(1416 ± 8) MeV, In the absence of tensor and spin-spin forces, the mass 1 Mf 0 =(1517.4±2.5), and Mf 0 =(1426.3±0.9) MeV, and we of the singlet heavy quarkonium states is given by the 2 1 find a scalar mass of Mf0 = (878±73) MeV. Likewise, we weighted average of the triplet states [33]: estimate the expected mass of the scalar isovector meson from 1 MhQ(nP ) = 5MχQ2(nP ) +3MχQ1(nP ) + MχQ0(nP ) (1) M = 9M − 5M − 3M . (4) 9 a0 b1 a2 a1 Table 2: Assignment of light scalar mesons to spin-multiplets. All masses are in MeV. The first line gives the predictions based on the center-of-gravity rule of light tensor, axial-vector and scalar mesons. The second line gives the name, the third one the best mass values of our assignment to be spin-partners of a2(1320), f2(1270), ∗ f2(1525), and K2 (1430). The assignments suggested by the Particle Data Group is given in the last line.

C.o.g. 791 ± 124 574 ± 40 878 ± 73 307 ± 180 ∗ Name a0(980) K0 (700) f0(980) f0(500) Mass 980 ± 20 [15] 648 ± 7 [19] 996 ± 7 [36] 457 ± 8 [36] ∗ RPP a0(1450) K0 (1430) f0(1710) f0(1370)

We use the masses Mb1 =(1229.5±3.2) MeV, Ma2 = (1316.9

±0.9) MeV, and Ma1 =(1230±40) MeV. Thus we obtain a scalar mass of Ma0 =(791±124) MeV. In the strange-quark sector, we have the well known ∗ K2 (1430) at (1427.3±1.5) MeV and two axial vector mesons, K1(1280) and K1(1400) with masses of (1253±7) and (1403 ±7) MeV, respectively. The scalar meson mass can be es- timated to Figure 1: (Color online) The spectrum of L = 1 excitations. Full symbols: Godfrey-Isgur model, open symbols: RPP masses of sug- M ∗ = 9M − 5M ∗ − 3M . (5) gested scalar spin-partners. The ﬁgure was suggested by W. Ochs. K0 K1B K2 K1A

The two observed resonances K1(1270) and K1(1400) are with the assignment made by the Particle Data Group mixtures of K1A and K1B. A recent analysis of the mixing parameters suggests a mixing angle of −(33.6 ± 4.3)◦ [35] given in the last line of the Table. In Table 3 we show the suggested multiplet assignment. and masses of MK1A =(1360±5) MeV and MK1B = (1310±5) MeV. With these values we derive a scalar mass M ∗ = There is one remark to be made: the a0(980) is spin- K0 (574±40) MeV. partner of a2(1320), the f0(980) is not the spin-partner of In Table 2 we compare the masses calculated using f2(1270) but of f2(1525). In Jaffe’s model, a0(980) and the c.o.g. rule with the masses of the lowest-mass scalar f0(980) are both √nns¯ s¯. Here, a0(980) has a nn¯ seed, with ¯ mesons. The comparison is very surprising: the predicted nn¯ = (uu¯ + dd)/ 2, that acquires a large ss¯ component masses are rather close to the masses of the light scalar while f0(980) has a ss¯ seed acquiring a large nn¯ compo- ¯ mesons. The comparison suggests that the light scalar nent. Due to their mass very close to the KK threshold, ¯ mesons are not only dynamically generated. At the same both resonances develop a large KK molecular component time they might also play the role of ground-states of the and can thus be generated dynamically. scalar qq¯ mesons. The Particle Data Group interprets The assignment of, e.g., f2(1320), f1(1285), f0(500), ∗ and h1(1170), to one spectroscopic multiplet, to excita- the mesons a0(1450), f0(1710), f0(1370), and K0 (1430) as ground states of scalar mesons. This is clearly in conflict tions with L=1, S=1 coupling to J=2,1,0 and L=1, S=0, with the c.o.g. rule. Scalar mesons are subject to strong J=1, is at variance with the Godfrey-Isgur model [37]: In unitarization effects, hence significant mass shifts can be this model, the lowest-mass isoscalar meson is predicted expected. But in the PDG interpretation, these effects to have a mass of 1090 MeV, certainly below the f2(1270) would need to be extremely large. The masses calculated mass but far above the f0(500) mass. The Bonn model [38], from the c.o.g. rule fall only a little below the actual light- that is based on instanton-induced interactions instead of scalar meson-masses. The masses of light mesons, given in an effective one-gluon exchange, predicts 665 MeV (model the second line in Table 2, are certainly better in agree- B), in better agreement with the c.o.g. rule. ment with the values calculated using the c.o.g. rule than 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 Table 3: Suggested spin-multiplet assignment for light mesons. mesons are all considerably above the scalar masses given in Table 3. However, all predicted scalar masses fall be- ∗ 0 a2(1320) K2 (1430) f2(1525) f2(1270) low their multiplet partners. The RPP assignments to the

a1(1260) K1A f1(1420) f1(1285) lowest-mass scalar qq¯ mesons are all above their multiplet partners (except K∗(1430) that is degenerate in mass with b (1235) K h (1415) h (1170) 0 1 1B 1 1 K∗(1430)). Hence we believe that it is difficult to decide ∗ 2 a0(980) K0 (700) f0(980) f0(500) 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 The mixing angle of scalar mesons. We now discuss f0(500) is a pure qq¯ state. It does not even need to have the possibility to define the mixing angle for scalar mesons. a large qq¯ component. A small qq¯ seed could acquire a For scalar mesons, the refinement by two mixing angles strong nnn¯ n¯ component. Color exchange makes this in- exceeds the quality of data available at present, and we distinguishable from a ππ component, the leading term use a mixing scenario with one mixing angle. In formulas ∗ when the resonance is generated dynamically. Thus the we use f0(500) = σ, K0 (700) = κ, a0(980) = a0, and light scalar mesons may have qq¯, tetraquark and molecu- f0(980) = f0 as abbreviations. lar contributions [39]. We use the scalar mixing angles θs and ϕs defined by

We emphasize that the interpretation of the light scalar s s f0 cos θ − sin θ |8 > meson-nonet as part of the qq¯ spectrum is by no means in = s s (10) contradiction with chiral dynamics. This nonet is gener- σ sin θ cos θ |1 > ated dynamically; but at the same time it could play its and part in the qq¯ spectrum. This is one version of the well- s s known chicken-and-egg problem: Is the interaction first, f0 cos ϕ − sin ϕ |nn¯ > = s s . (11) that generates the pole, or is the pole first, that generates σ sin ϕ cos ϕ |ss¯ > the interaction? QCD generates the interaction and the This definition is used by Oller [43], and ϕs = θs + (90 − pole, and one aspect is not thinkable without the other 35.3)◦. one. The observation that the masses of the light scalar The authors of Ref. [44] use a wave function in the qq¯ mesons are rather close to the predictions based on the basis: c.o.g. rule opens the chance that these mesons could also s s be related the qq¯ family. We are aware of the fact that this f0 = sin Φ |ss¯ > + cos Φ |nn¯ > (12) interpretation is at variance with the modern understand- s s ing of the spectrum of light scalar meson, but we have with Φ = −ϕ . no other explanation why the c.o.g. rule “happens” to be Ochs [45] assigns f0(980) and f0(1500) to the same satisfied for light mesons. multiplet and determines the mixing angle. However, he uses only properties of f0(980) and the decomposition

0s 0s 4. The ﬂavor wave function f0 = cos Φ |ss¯ > + sin Φ |nn¯ > (13) The mixing angle of pseudoscalar mesons. Mesonic with ϕs = Φ0 − 90◦. mixing angles can be determined in the SU(3) singlet and Most authors [48–52, 56] use a deﬁnition of the mixing octet basis or in a quark basis with nn¯ and ss¯. angle in the quark basis in the form

For the pseudoscalar mesons, the difference in the octet s s f0 cos φ sin φ |ss¯ > and singlet decay constants is not negligible, and the nonet = s s . (14) ps ps σ − sin φ cos φ |nn¯ > is described by two mixing angles θ1 and θ8 [40]. Here, we neglect the difference and use the singlet/octet basis in with with ϕs = φ − 90◦. the form If the light scalar mesons behave like ordinary qq¯ mesons, η cos θps − sin θps |8 > they should have a unique mixing angle. Mixing angles = (6) η0 sin θps cos θps |1 > can be derived from the production and decay of scalar mesons using SU(3) relations or from the masses exploit- and in the quark basis ing the Gell-Mann–Okubo (GMO) formula. Most authors ps ps η cos ϕ − sin ϕ |nn¯ > neglect the effects of the different densities at the origin 0 = ps ps . (7) η sin ϕ cos ϕ |ss¯ > of the f0(500) and f0(980) wave functions and assumed ◦ that these resonances are qq¯ states: these are certainly The ideal√ mixing angle θideal = 35.3 is defined by tan θideal 0 two highly questionable assumptions. Nevertheless we will = 1/ 2 and leads to ηqq¯ = −ss¯ and ηqq¯ = nn¯. In the ps ps compare the mixing angles derived using different meth- quark basis the two mixing angles Φn and Φs are very similar in magnitude [41], we use ϕps = (39.3 ± 1.0)◦ [42]. ods. In the singlet-octet basis, the unique mixing angle is given The mixing angle from decays of scalar mesons. as θps = (39.3 + 35.3 − 90)◦ = −(15.4 ± 1.0)◦, and the Oller [43] determined the mixing angle in the singlet-octet η0 is dominantly in the SU(3) singlet configuration. For basis of the light scalar mesons in a SU(3) analysis of the θps = −15.5◦, meson decay couplings. The coupling constants were ob- η p13/14 +p1/14 |8 > tained by determining the residues at the pole positions = , (8) η0 −p1/14 p13/14 |1 > of the resonances. The analysis gave θs = (19 ± 5)◦ or ϕs = (74 ± 5)◦. Many determinations of the mixing angle and for ϕps = 39.2◦ we get are ambiguous. In these cases, we choose the value that η p3/5 −p2/5 |nn¯ > is closer to this value. The resulting mixing angles are = . (9) η0 p2/5 p3/5 |ss¯ > collected in Table 4. The authors of Ref. [44] study radiative decays of Φ- Table 4: Scalar mixing angle ϕs from production experiments and mesons and the two-photon width of scalar and tensor from σ decay coupling constants [43]. The sign of the mixing angles mesons. The calculated two-photon width of scalar and is mostly undetermined. It is chosen to be compatible with Ref. [43]. tensor mesons agree well with data even though approxi- mately equal radial wave functions are used. For this rea- ◦ ◦ ◦ +15 ◦ ◦ ◦ (74 ± 5) (94±6) (72 ± 3) (56− 9 ) (67±4) (87±3) son, the authors argue that a2(1320), f2(1270) and f2(1525) [43] [44] [45] [48] [49] [50] might belong to the same P -wave multiplet as a0(980) and f (980). For the scalar mixing angle, two solutions were 0 ≈65◦ (58±8)◦ (67 ± 3)◦ (55 ± 5)◦ (63 ± 2)◦ found Φs = −(48±6)◦ or (86±3)◦. Equations (11) and (14) + + + − 0 0 + − are related by Φs = −ϕs. The latter angle defines a mixing [51] [52] D →π π π D →π π π J/ψ→γππ angle ϕs = −(86 ± 3)◦. The overall sign of the wave function is not important here, hence we quote ϕs = (94 ± 3)◦ in Table 4. Assuming similar hadronizations from the primary qq¯ pair s Ochs [45] determined φ from the couplings of f0(980) into the two scalar mesons, a scalar mixing angle in the ¯ to ππ and KK, the two-photon widths of f0(980) and nn¯-ss¯ basis of |φs| ≈ 25◦ or ϕs = 65◦ is deduced. + a0(980), from the production rates for D(s) → f0π and The authors of Ref. [52] compared the leptonic decays + ∗ + + + D → K0 (1430)π , and from J/ψ → f0(980)ω(Φ). He of the charmed mesons D into f0(980)/a0(980) + e νe, s ◦ s ◦ 0 + + + found two solutions φ = (30 ± 3) and φ = (162 ± 3) . D into a0(980) + e νe, and Ds into f0(980) + e νe. The Only the latter one is compatible with [43] and corresponds scalar mixing angle was |φs| shown to be compatible with to ϕ = (72 ± 3)◦. values in the (25 - 40)◦ range, a result that we use in the In Ref. [46], the use of the density of the wave func- form ϕs = (58 ± 8)◦. tion at the origin for hadronic molecules is critisized; in- We add a few further determinations of the scalar mix- stead, dressed meson propagators and photon emission ing angle. The errors are of statistical nature only. Dalitz- vertices are used to calculate the two-photon widths of plot analyses to D+ [53, 54] and D0 [55] decays into three f0(980). The result is in excellent agreement with the ex- pions reveal the fractional contributions of f0(500) and + perimental value. The radiative Φ decays into f0(980) and f0(980). The mean values for the D contributions from a0(980) are also consistent with the molecular view of these Refs. [53, 54] are (0.422±0.027)% for f0(500) and (0.048± 0 mesons [47]. 0.010)% for f0(980). For D decays the contributions (0.82±0.10 ±0.10)% and (0.25±0.04±0.04)% are reported The mixing angle from the production of scalar 2 s [55]. The f0(500) contribution is proportional to cos φ , mesons. Li, Du, and Lü[48] deduce the mixing angle 2 s the f0(980) contribution to sin φ . Using the unweighted from the rate (1.2±0.3)·10−4 for B0 → J/ψf (980) decays s 0 mean and spread of RPP results on Γ /Γtot = s + 9 ◦ s +15 ◦ f0(980)→ππ and find φ = (34−15) or φ = (146− 9 ) . The latter value 0.72 ± 0.03 and after correction for the different phase +15 ◦ yields ϕ = (56− 9 ) . spaces, mixing angles are determined that are listed in 0 In a study of the reaction B0 → D π+π− [49], the Table 4. collaboration found significant contributions from f0(500) From the ratio of f0(500) and f0(980) production in the 0 + − and f0(980). In this process, scalar mesons are produced reaction B → J/ψπ π , the LHCb collaboration con- (and decay) due to their nn¯ component. The produc- strained the mixing angle |φs| to be < 31◦ [56] at 90% con- tion ratio is interpreted within a qq¯ and tetraquark mixing fidence level. With increased statistics [57], the LHCb col- scheme. Assuming identical form factors for both mesons, laboration confirmed the absence of f0(980) production in 2 s +0.066 s the qq¯ model gives tan φ = 0.177−0.062 with φ = ±(23± this reaction. A reanalysis of the data exploiting a disper- ◦ s ◦ 4) . We choose the negative sign and get ϕ = −(113±4) sive framework demonstrated a significant role of f0(980) s ◦ 0 or, changing the overall sign, ϕ = (67 ± 4) . in the B → J/ψπ+π− reaction [58]. Therefore we exclude ¯0 The LHCb collaboration also studied the reaction Bs → this reaction in the present discussion. The ratio of the two J/ψπ+π− [50]. The π+π− invariant mass spectrum shows −4 frequencies f for J/ψ → ωf0(980), f=(1.4±0.5) 10 , and a large contribution from f (980) and no sign for f (500). −4 0 0 J/ψ → φf0(980), f=(3.2±0.9) 10 [15] gives a compati- From the upper limit, the scalar mixing angle is constrained ble mixing angle that is not used since this ratio is included s ◦ ◦ to |φ | < 7.7 at 90% confidence level, or (3 ± 3) . We use in the evaluation by Ochs [45]. ϕs = −(87 ± 3)◦. The frequencies for J/ψ → γf0(500), with f=(11.4±2.1) The authors of Ref. [51] study B0 → J/ψf (500) −4 −4 d,s 0 10 , and J/ψ → γf0(980), with f=(0.21±0.04) 10 [62], ¯ [f0(980)] decays. In these decays, the b-quark in the Bd,s can be used to determine the mixing angle θs=±(8.0±1.2)◦ converts into ac ¯ quark under emission of a W -boson. The that leads to φs = −(27.3 ± 1.2)◦ or −(43.3 ± 1.2)◦. The W decays into a c quark plus a d¯ – picking up the d-quark former value translates into ϕs = −(62.7 ± 1.2)◦ that is of the Bd – or ans ¯ that combines with the s-quark of the compatible the value obtained in Ref. [43]. Here it is as- Bs. The squared transition amplitudes are thus propor- sumed, that the radiative J/ψ decay couples only to the tional to the nn¯ or ss¯ content of the scalar wave functions. SU(3) singlet components. The eleven mixing angles are statistically not compat- Table 5: Pseudoscalar and scalar mixing angles from the GMO for- ible. We assume the uncertainties are dominated by sys- mula. The pseudoscalar mixing angles are from Ref. [15]. tematic uncertainties. Therefore, we take the mean value of the scalar mixing angles. The full spread is 11◦ that θ ps = −24.5◦ φs = +(91.0 ± 2.5)◦ corresponds to a statistical uncertainty of 4◦. The mixing lin lin ps s s ◦ s ◦ angle θ in the singlet/octet basis and the quark basis φ θquad = −11.3 φquad = +(91.2 ± 2.5) are related by φs = θs + 90 − 35◦. Thus we have

s ◦ s ◦ ϕ = (69 ± 4) θ = (14 ± 4) . (15) Alternatively we use the mass formula in the form s ◦ For θ = 14.4 , the octet contribution to the f (500) and 4m − m − 3m 0 0 tan2 θ = K a f = 5.782 (19) the singlet contribution to the f0(980) are small: −4mK + ma + 3mf  q q  s ◦ 15 1 and obtain θlin = ±(67.4 ± 2.1) . For the negative sign we f0 16 + 16 |8 > ◦ ◦ = ·   . obtain φ = −(102.7 ± 2.1) . The value φ = −90 would σ q 1 q 15 |1 > − 16 16 indicate an inverted spectrum. Replacing the masses by squared masses in eqn. (18) yields θs = −(55.9 ± 2.5)◦ s ◦ quad In the quark basis and for φ = 69.3 , the wave functions s ◦ and φquad = −(91.2 ± 2.5) . In Table 5 we compare the can be cast into the form mixing angles of scalar and pseudoscalar mesons. p p f 1/8 − 7/8 |nn¯ > The GMO formula suggests that f0(500) is mostly an 0 = , σ p7/8 p1/8 |ss¯ > nn¯ state and f0(980) an ss¯ state. From the production and decay of light scalar mesons we conclude that f0(500) and f0(500) is mostly an nn¯ state and f0(980) an ss¯ state. is mainly in the singlet, f0(980) mainly in the octet configuration, with a small mixing angle. The flavor wave function from the GMO formula. There is an important difference between the mixing The mass pattern of the lightest scalar mesons differs deci- angle derived from production and decay of mesons or from sively from the one observed for other mesons. There is an the GMO formula. Production and decay depend on the isovector meson a0(980) and an isoscalar meson f0(980) – mesonic wave function, the GMO formula if sensitive to like ρ and ω – but in contrast to the vector mesons, these the mass content. scalar mesons are heavier than their nonet partners. The With the caveats expressed above (f0(500) and f0(980) other isoscalar meson, f0(500), has the lightest mass of may have different wave functions and are likely no qq¯ the multiplet. The spectrum of scalar mesons resemble states) we can still note that a mixing angle for the light the pseudoscalar mesons but inverted: There are the low- scalar mesons can be defined. The f0(500) is a mainly nn¯ mass π and η that correspond to a0(980) and f0(980), the state but has an additional singlet component, beyond the ∗ 0 K and K0 (700), and the high-mass η corresponds to the one expected from the decomposition of nn¯ into singlet low-mass f0(500). and octet. The f0(980) mainly ss¯ with an additional octet The sum rule component. 2 (m + m 0 )(4m − m ) − 3m m 0 (0.977 GeV ) f f K a f f Size of light scalar mesons. Weinberg derived a cri- 2 2 2 = 8mk − 8mK ma + 3ma (1.16 GeV ) terium to decide if a bound state is elementary or compact or if it is an extended particle [29, 34]. A quantity Z can contains f and f 0 symmetrically. It is violated at the 6% 2 2 be defined that is related to the scattering length a via level when the difference is compared to (8mk + 3ma). We identify σ as f and f as f 0. 1 − Z 1 0 a = −2 (20) Also the equal splitting law 2 − Z γ √ 2 2 2 2 2 mκ − mK = 0.172 = mσ − mπ = 0.191 GeV (16) where γ = 2µEB denotes the binding momentum. The quantity Z determines the compact fraction of the bound suggested by Scadron [59] is satisfied at the 10% level. state, 1 − Z the “molecular” fraction. For Z = 0, the For ideal mixing, we expect system is “molecular”, for Z = 1 compact. m + m The authors of Ref. [60] extended this relation and de- m = f0 σ . (17) κ 2 rived a probabilistic interpretation of the compositeness relation for resonances. Taking the effective range into ac- The expectation holds true at this 10% level. This is in count, the authors determined Z to 0.60±0.02 where the line with the linear GMO formula σ error is of statistical nature only. The smallness of the 4m − m − 3m 0 tan θ = √K a f = −1.465 (18) molecular fraction in the f0(500) wave function is in line 2 2(ma − mK ) with a determination of the rms radius of the f0(500) from s ◦ a Taylor expansion of the f0(500) form factor [61]: that yields a negative mixing angle, θlin = −(55.7 ± 2.5) , s ◦ σ φlin = −(91.0 ± 2.5) . rrms = [(0.44 ± 0.03) − i(0.07 ± 0.03)] fm. (21) ∗ The f0(500) meson seems to be a compact object. a K0 state that could be partner of f0(1770)/f0(1710) or A similar analysis was reported in Ref. [25] for a0(980) of f0(2330)/f0(2200). and f0(980). The authors concluded that the probability In Table 6 we collect the known scalar mesons. Three to find the a0(980) as qq¯ state is about 25 to 50% and for full nonets can be defined. The GMO formula suggest ∗ the f0(980) meson this probability is even smaller, about a mixing angle for the four mesons a0(1450), K0 (1430), 20% or less. In Ref. [60], the probability to find the f0(980) f0(1500), f0(1370) that is nearly ideal. The mixing angle +0.28 ∗ as qq¯ state is Zf0 = 0.33−0.28. The compact fractions of for a0(2020), K0 (1950), f0(2100), and f0(2020) vanishes f0(500) and f0(980) differ significantly, even though the in the singlet/octet basis. In both cases the errors are too large errors do not completely exclude similar spatial wave large to make this a solid statement. ∗ functions of f0(500) and f0(980). The K0 (700) has only a small molecular component, Zκ = 0.88. 6. The fragmented glueball Obviously, the Weinberg criterium does not exclude a possibly small qq¯ component in the wave functions of light QCD predicts the existence of glueballs, of states with- scalar mesons. out constituent quarks. Here we quote a few calculations of the glueball spectrum [63–69]. The lowest-mass glueball is expected to have scalar quantum numbers, a mass in the 5. Excited scalar states 1500 to 2000 MeV range, to intrude the spectrum of scalar ∗ mesons and to mix with them. A large number of differ- When a0(1450), K0 (1430), f0(1370), and f0(1710) are not elements of the lowest-mass qq¯ multiplet with L = 1, ent mixing schemes were reported initiated by the work of do they fit into a radial excitation multiplet? Unfortu- Amsler and Close [13, 14]. The proof for the presence of a nately, only few states are known to complete the qq¯ mul- glueball is supposed to be supernumerosity: it is expected tiplets with L = 1 and one unit of radial excitation. With that more mesons should be observed than predicted by the masses of a2(1700), a1(1640), and a0(1450) as given quark models. in the RPP, we predict – using the c.o.g. rule – a b1 This expectation was not met in a recent fit to a large body of reactions on radiative J/ψ decays, ππ elastic scat- mass of Mb1(xxx) = (1663 ± 23) MeV. Quark models predict higher masses for the radial excitations, hence we tering, pion-induced reactions andpp ¯ annihilation [62]. use mass differences relative to the a2(1700) mass. The The multi-channel fit to the data revealed the existence of Godfrey-Isgur [37] (Bonn [38]) model predicts the b1 mass ten scalar isoscalar resonances from f0(500) to f0(2330). to be 40 (50) MeV below the a2(1700) mass; both val- Of particular importance for the interpretation are the 0 0 ues are in excellent agreement with the c.o.g. predic- data on radiative J/ψ decays to π π [70] and KSKS [71] tion. The f2(1640), h1(1595), f0(1370) can be used to pre- shown in Fig. 2. In these reactions, the J/ψ is supposed +60 to decay into one photon and two gluons (see Fig. 3, left). dict the mass of a f1(xxx) to 1567−92 MeV. The Godfrey- Isgur [37] (Bonn [38]) model predicts the f1 mass to be The two gluons interact and undergo hadronization. 40 ( ≈ 60) MeV below the f2 mass, certainly compatible Figure 2 compares the invariant mass distributions re- with the predicted mass. According to the c.o.g. rule, sulting from radiative J/ψ decays with the pion and kaon the a0(1450) resonance is a credible candidate to be the form factors. Their square is proportional to the cross sec- first radial excitation of the a0(980) and to be spin-partner tions. The form factors were deduced by Ropertz, Han- ¯0 of a2(1700), a1(1640) and an unobserved b1 expected at hart and Kubis [72] exploiting the reactions Bs → J/ψ + − ¯0 + − 1655 MeV. The f0(1370) could be the radial excitation π π [50] and Bs → J/ψK K [73] and imposing the ¯ ¯0 of f0(500) and spin-partner of f2(1640), h1(1595), and a S-wave phase shifts from ππ and KK. In the Bs decays, missing h1 at about 1590 MeV. There is neither a a0 nor 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. Table 6: Scalar mesons: ground states and radial excitations. Masses Both formfactors are dominated by the f0(980) reso- and the assignment to mainly-singlet and octet configurations stems nance. The resonance connects the initial ss¯ pair to π+π− from Ref. [62]. in the final state: the f0(980) must have ss¯ and nn¯ components, and this statement holds true for the f0(1500) I = 1 = 1/2 I = 0 I = 0 as well. The f0(980) is nearly absent in radiative J/ψ “octet” “singlet” decays: this was explained by a dominance of the SU(3) ∗ octet component in f0(980). Most striking is the mountain a0(980) K (700) f0(980) f0(500) 0 landscape above 1500 MeV in the data on radiative J/ψ ∗ a0(1450) K0 (1430) f0(1500) f0(1370) decays. The huge peak, e.g., at 1750 MeV in the KK¯ mass f0(1770) f0(1710) spectrum and the smaller one at 2100 MeV decay promi- ∗ nently into KK¯ but are produced with two gluons in the a0(2020) K0 (1950) f0(2100) f0(2020) initial state and not – at least not significantly – with ss¯ in f (2330) f (2200) 0 0 the initial state. This is highly remarkable: the two gluons in the initial state must be responsible for the production 7. Summary and conclusions gg → ππ ss¯ → ππ In this paper, an alternative view of scalar mesons was presented. The light scalar mesons are interpreted as mem- bers 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 gg → KK¯ but rather as component of the wave functions of scalar ss¯ → KK¯ isoscalar mesons. These two conjectures are linked. The wave function of a scalar meson can be expanded

Meson = α1|qq¯ > +α2|qqq¯q¯ > +α3|gg > (22)

+α4|meson−meson > +α5|qqg¯ > + ··· . with possible contributions from higher terms. We thus 0 0 Figure 2: (Color online) π π (a) [70] and KS KS (b) [71] in- could expect five (or more) different types of states. Since variant mass distributions from radiative J/ψ decays (histogramm) they all have the same quantum numbers, they can mix. with fit [62] and the pion (a) and kaon (b) formfactors [72] from But the number of states to be expected is large. We now ¯0 + − ¯0 + − Bs → J/ψπ π [50] and Bs → J/ψK K [73]. make a conjecture: we assume that only one state exist and that the orthogonal states disappear in the continuum. γ c¯ J/ψ Low-mass scalar mesons can be dynamically generated but c¯ ¯b still have a qq¯ seed. Scalar mesons can have a large glueball + c J/ψ B¯0 W component but still are part of the regular spectrum of s s¯ scalar qq¯ mesons. c s The situation can be compared to the one observed in case of the N(1535)1/2− resonance. This resonance f0 f0 is generated dynamically [74] but still plays an important role in quark models as member of the nucleon’s Figure 3: (Color online) In radiative J/ψ decays, two gluons, in first excitation multiplet with orbital angular momentum B¯0 → J/ψ + ss¯, a ss¯ pair may convert into a scalar meson. s L = 1. It may also be interpreted as pentaquark state with hidden strangeness [75]. These different views, to con- of resonances that decay strongly into KK¯ but are nearly sider N(1535)1/2− as dynamically generated from meson- absent when ss¯ pairs are in the initial state. Also the rich baryon interactions, its interpretation as pentaquark, and structure in the radiatively produced ππ mass spectrum is its assignment to the three-quark baryon spectrum are le- accompanied with little activity when the initial state is gitimate and provide additional insights. The light scalar ss¯. The rich structure stems from gluon-gluon dynamics. mesons can be dynamically generated molecules but they In Ref. [62] it was shown that the total yield of scalar could still have a qq¯ seed and play their part in quark mod- mesons as a function of their mass can be described by a els. +10 Breit-Wigner function with mass MG = (1865±25 −30) MeV +30 and width ΓG = (370 ± 50 −20) MeV. The peak is created by gluon-gluon interactions. It is the scalar glue- I would like to thank Chr. Hanhart, W. Ochs, J. Oller, ball. The glueball peak extends over several scalar reso- and U. Thoma for a careful reading of the manuscript and nances, the glueball is fragmented. These scalar mesons helpful discussions. Funded by the NSFC and the Deutsche can be grouped into a class of mesons with mainly octet Forschungsgemeinschaft (DFG, German Research Foun- and mainly singlet qq¯ components. 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