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Scalar Mesons and the Fragmented Glueball

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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 com- posed 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.
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