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Physics Reports 389 (2004) 61–117 www.elsevier.com/locate/physrep

Mesons beyond the naive model

Claude Amslera;∗, Nils A. T,ornqvistb

aPhysik-Institut der Universitat Zurich, Winterthurerstrasse 190, CH-8057 Zurich, Switzerland bDepartment of Physical Sciences, University of Helsinki, P.O.Box 64, Fin-00014, Helsinki, Finland Accepted 1 September 2003 editor: J.V. Allaby

Abstract

We discuss theoretical predictions for the existence of exotic (non-quark-model) and review promi- nent experimental candidates. These are especially the f0(1500) and f0(1710) mesons for the scalar , fJ (2220) for the tensor glueball, Á(1410) for the pseudoscalar glueball, f0(600);f0(980);a0(980), the still 2 2 to be ÿrmly established Ä(800) and the f2(1565) for q q8 or two- states, and 1(1400);1(1600) for hybrid states. We conclude that some of these states exist, o9er our views and discuss crucial issues that need to be investigated both theoretically and experimentally. c 2003 Elsevier B.V. All rights reserved.

PACS: 12.39.Mk; 12.39.Jh; 13.25.Jx; 14.40.Cs Keywords: ; QCD; Scalar mesons; 4-quark states; Deuteronlike states; Gluonium; Hybride

Contents

1. Introduction ...... 62 1.1. The light meson spectrum ...... 63 2. Four-quark mesons ...... 66 2.1. Ja9e’s four-quark states ...... 66 2.2. Deuteronlike meson–meson bound states (or deusons) ...... 70 2.2.1. One- exchange ...... 70 2.2.2. Predictions for deuteronlike meson–meson bound states ...... 72 3. Are the scalars below 1 GeV non-qq8 states? ...... 74

3.1. The hadronic widths of the a0(980) and f0(980) mesons ...... 75 3.1.1. widths of the a0(980) and f0(980) mesons ...... 76

∗ Corresponding author. E-mail addresses: claude.amsler@.ch (C. Amsler), nils.tornqvist@helsinki.ÿ (N.A. T,ornqvist). URL: http://unizh.web.cern.ch/unizh/

0370-1573/$ - see front c 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physrep.2003.09.003 62 C. Amsler, N.A. Tornqvist / Reports 389 (2004) 61–117

3.1.2. Radiative widths of the (1020) to a0(980) and f0(980) ...... 76 3.1.3. The f0(980) produced in Ds → 3 ...... 79 3.2. A possible interpretation of the of a0(980) and f0(980) ...... 80 3.3. Is the f0(600) a non-qq8 state and does the Ä(800) exist?...... 81 3.3.1. The f0(600) (or ) ...... 81 3.3.2. The Ä(800) ...... 84 ∗ 3.4. Observation of a -strange state DsJ (2317) ...... 85 3.5. Do we have a complete scalar nonet below 1 GeV? ...... 87 4. ...... 88 4.1. Theoretical predictions ...... 88

4.2. Is the f0(1500) meson the scalar glueball? ...... 90 4.2.1. Hadronic decay width ...... 92 4.2.2. 2 -decay width ...... 95 4.2.3. Mixing with qq8 states ...... 96 4.3. The tensor glueball ...... 97 4.4. The pseudoscalar glueball ...... 99 5. Hybrid mesons ...... 102 5.1. Theoretical predictions ...... 102 −+ 5.2. A 1 , the 1(1400) ...... 104 −+ 5.3. Another 1 exotic meson, the 1(1600) ...... 108 5.3.1. Other hybrid candidates ...... 110 6. Conclusions and outlook ...... 111 Acknowledgements ...... 113 References ...... 113

1. Introduction

The nearly 40 years old naive or model (NQM), including many generalizations, has been since the pioneering work of Gell-Mann and Zweig [1,2] the basic framework within which most of the hadronic states could be understood, at least qualitatively. The NQMwas very successful in describing the observed spectrum, especially for the heavy (c and b) Navour sector. As expected, 3 1 there are very well established heavy quark–antiquark S-wave vector ( S1) and pseudoscalar ( S0) 3 3 3 1 mesons, as well as P-wave states ( P2; P1; P0 and P1) which can be identiÿed in the observed spectrum without ambiguities. No clearly superNuous and well established heavy meson state has been reported. The success of the NQMcan be understood within QCD from the fact that the bound system is approximately non-relativistic for heavy constituents, and from the fact that the e9ective couplings become suOciently small, so that higher order or non-perturbative e9ects can be neglected as a ÿrst approximation. 8 3 In particular, the scalar cc8 and bb states behave as expected for P0 states, whose axial and tensor 3 3 siblings are the heavy P1;2 mesons. Their production in radiative transitions from the 2 S1 states 3 and decays into 1 S1 or light are as expected. Nothing appears to be “exotic” (suggesting a composition di9erent from qq8) for these heavy scalar mesons. However, the situation is quite di9erent both theoretically and experimentally for the light meson spectrum. Since the e9ective within QCD becomes large, higher order graphs cannot be C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 63 neglected and there may be non-perturbative e9ects which cannot be described within the NQMnor by tree graphs within a phenomenological e9ective Lagrangian. The quark model has to be unitarized, requiring from the formalism the right analytic properties. Also, crossing should be at least approximately imposed. In addition, the qq8 system becomes inherently non-relativistic and one should allow the amplitudes and the spectrum to be consistent with the (almost exact) chiral symmetry of QCD for the light u and d . All this clashes with the NQMassumptions, and one should not a priori believe any simple results from the NQMwithout criticism. From the experimental side one should devote most e9orts to look for states that cannot be described within the NQM, but which are consistent with QCD and conÿnement, such as gluonium (composed of only glue), multiquark states (such as qqq8 q8), hybrid states (gqq8, composed of qq8 and a constituent ), or meson–meson bound states. Such states are expected especially among the light hadrons, where the NQMmust eventually break down. For most of the ground state light NQM qq8 nonets one can, with reasonable Navour symmetry breaking and binding assumptions, easily associate well established experimental candidates [3]. This 3 is remarkable indeed. The main exception is the scalar ( P0) nonet, for which there are too many observed candidates. On the other hand, if mesons with “exotic” quantum numbers (that do not couple to qq8 and therefore cannot appear in the qq8 NQM) were observed, this would give clues as to how the NQMshould be generalized. The light scalar mesons stand out as singular and their nature has been controversial for over thirty years. There is still no universal agreement as to which states are mainly qq8,astohowa glueball would appear among the light scalars, and whether some of the too numerous scalars are multiquark, or meson–meson states, such as KK8 bound states. Since the NQMperforms rather well for heavy constituents, the predicted spectrum of heavy mesons might be more reliable, hence non-qq8 states are easier to identify. For example, the recently ∗ discovered (presumably scalar) DsJ (2317) [4], the mass of which lies far below predictions, is likely to throw new light also on the light scalar sector. These are fundamental questions of great importance in physics. In particular, the scalar mesons have vacuum quantum numbers and are crucial for a full understanding of the symmetry breaking mechanisms in QCD, and presumably also for conÿnement. The structure of this review is as follows. In the next section we brieNy review the current status of the qq8 spectrum. For a recent comprehensive review on light quark spectroscopy we refer to Ref. [5]. We then discuss the theoretical predictions for the existence of –antidiquark states and qq8 − qq8 meson–meson bound states (Section 2). In Section 4 we present the predictions for the existence of mesons without quark constituents, the glueballs, and discuss current candidates for the scalar (4.2), the tensor (4.3) and the pseudoscalar (4.4) states. Section 5 is entirely devoted to hybrid mesons which are made of qq8 pairs with vibrating . In the last section we summarize the status of non-qq8 mesons and o9er our views on critical issues that need to be investigated theoretically and experimentally.

1.1. The light meson spectrum

In the NQMmesons made of the light quarks u; d; s are classiÿed in qq8 nonets of SU(3)-Navour. Theoretical predictions for their mass spectrum can be found in Ref. [6]. Fig. 1 shows the current 64 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

ν ≈ 2S+1 PC m[GeV] L =J n J ++ ++ 3 3 3 3 =1 D =3-- D =2-- 2P2=2 2P1 1 3 1 2

a2(1700) a1(1640) ρ3(1690) K *(1980) * 2 K 3(1780) K2(1820) f (2010) ω π(1800) 2 3(1670) f (1950) φ K(1830) 2 3(1850) η(1760) π ρ 2(1670) (1700) * 2 K2(1770) K (1680) η ω(1650) 1 -+ 3 -- 2(1645) S =0 S=1 η 3 0 3 1 2(1870) 1 -- D =2-+ 3D =1 3 ++ 311++ 2 1 1P2=2 1P1 =1

a2(1320) a1(1260) * K K2 (1430) 1a π(1300) ρ f (1285) (1450) f2(1270) 1 K(1460) K*(1410) f (1420) 1 η f2'(1525) 1 (1295) ω(1420) η(1440) φ n (1680) a0(1450) b1(1235) K *(1430) K -- 0 1b 1 -+ 3 f (1370) h (1170) 2S0=0 2S=11 0 1 f0(1710) (1380) 3 ++ 1 +- P =0 P =1 π ρ(770) 1 0 1 1 K * η K (892) 0 ω(782) η' L φ(1020)

-- 1 -+ 3S =1 1S0=0 1 1 0 1 2 L

Fig. 1. Tentative quark–antiquark mass spectrum for the three light quarks, according to SU(3). The states are classiÿed according to their total J, relative L, spin multiplicity 2S + 1 and radial excitation n. The vertical scale gives the radial number =n+L−1, the horizontal scale the orbital excitation L. Each box represents a Navour nonet containing the isovector meson, the two strange isodoublets, and the two isoscalar states. The mass scale is approximate. The shaded assignments are clear and deÿnitive.

experimental status of light quark mesons. The ground state (angular momentum L=0) pseudoscalars (J PC =0−+) and vectors (1−−) are well established, but many of the predicted radial excitations (n¿1) and orbital excitations (L¿0) are still missing. Among the ÿrst orbital excitations (L = 1), consisting of the four nonets 0++; 1++; 2++; 1+−, only the tensor (2++) nonet is complete and unambiguous. A nearby additional tensor meson, the f2(1565) [7] could be the ÿrst radial excitation (n =2)ofthef2(1270). However, this state is ob- served in - only, which suggests a rather di9erent nature, a four-quark state or app 8 (baryonium) state [8]. In the 1++ nonet two states compete for the ss8 assignment, C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 65

the f1(1420) shown in Fig. 1 and the f1(1510), which is not well established [9]. There are ++ too many scalar (0 ) mesons to ÿt in the ground state nonet: the f0(600) (or );a0(980); ∗ f0(980) and K0 (1430) are well established, but the former three are generally believed not to be qq8 states. This issue will be discussed in detail below, as well as the nature of the three iso- scalar states, f0(1370);f0(1500) and f0(1710), for which signiÿcant progress was made recently in pp8 annihilation and in pp central collisions. These states are believed to mix with the ground state +− scalar glueball. In the 1 nonet the ss8 meson is not established, although a candidate, h1(1380), has been reported [3]. The identiÿcation of the ÿrst radial excited pseudoscalars (n = 2) would be crucial to resolve the now 20 years old controversy on the existence of a pseudoscalar glueball around 1440 MeV. The Á(1440) meson (previously called E=–) is produced in radiative J= decay, a channel traditionally believed to enhance gluonic excitations. As we shall discuss below, there is now evidence for two pseudoscalar mesons in the 1400 MeV region, one of which could be non-qq8. Only overall colour-neutral qq8 conÿgurations are allowed by QCD. However, additional colourless states are possible, among them multiquark mesons such as q2q82 or q3q83 states. Bag model predictions for 0+; 1+ and 2+ q2q82 states have been presented in Ref. [10]. For q2q82 mesons one predicts a rich spectrum of 0, 1 and 2 states in the 1–2 GeV region, most of which have not been observed so far. This casts doubt on whether multiquark states really bind or are suOciently narrow to be observed. However, we shall show that the low lying scalar states a0(980);f0(980) and f0(600) are prime candidates for such states. Four-quark states were also searched for, in particular inpp 8 annihilation. The so-called “baryonia” [8] are bound states or resonances of the antiproton–proton system. The short range –nucleon interaction is repulsive, presumably due to heavy meson t-channel exchanges (e.g. ! exchange). Through G- transformation the interaction becomes attractive for various partial waves of the proton–antiproton system, and a rich spectrum of bound states and resonances was predicted [11]. With the possible exception of the f2(1565) candidate [7], none of these states was actually observed, perhaps because they easily decay into two mesons and are therefore very broad. Also, the predictions for bound states relies on the short range attraction of the nucleon–nucleon interaction which may instead be mediated by one-gluon exchange spin–spin contact interaction, in which casepp 8 and pp are not related by G-parity transformation. A remarkable prediction of QCD is the existence of isoscalar mesons which contain only gluons, the glueballs (to be discussed in Section 4). They are a consequence of the non-abelian structure of QCD which requires that gluons couple to themselves and hence may bind. Lattice gauge calculations predict the existence of the ground state glueball, a scalar, at a mass between 1500 and 1700 MeV [12]. The ÿrst excited state is a tensor and has a mass of about 2200 MeV. One expects that glueballs will mix with nearby qq8 states of the same quantum numbers [13]. We shall show below that the f0(1500) is a prime candidate for the ground state scalar glueball, possibly mixed with nearby qq8 states. Mesons made of qq8 pairs bound by an excited gluon g, the hybrid states, are also predicted [14]. We shall show in the section on hybrids below that the quantum numbers 1−+ do not couple to qq8. We shall refer to meson states with these quantum numbers as exotic states. Hence the discovery of mesons with such quantum numbers would prove unambiguously the existence of exotic (non-qq8) mesons. There are so far two prominent candidates for exotic states with quantum numbers 1−+, the 1(1400) and 1(1600). 66 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

In contrast with glueballs, exotic hybrids do not mix with qq8 states.

2. Four-quark mesons

2.1. Ja@e’s four-quark states

Already back in 1977 Ja9e [10], using the bag model [15] in which conÿned coloured quarks and gluons interact as in perturbative QCD, suggested the existence of a light nonet composed of four quarks with below 1 GeV. The essential result from that work was more recently reformulated by Ja9e in Ref. [16] which we follow here. To lowest order, the dominant graph is that of one gluon exchange in Fig. 2. The e9ective Hamiltonian is ˜ ˜ He9 ˙ − %i · %j˜ i · ˜ j ; (1) i=j ˜ where ˜ i and %i are the 2 × 2 Pauli spin and 3 × 3 Gell–Mann colour operators for the ith quark normalized in the usual way, Tr( k )2 = 2 for all three spin components k, and Tr(%a)2 = 2 for all eight gluons a. This is a simple generalization of the Breit spin–spin interaction to include a similar colour–colour piece. It is also known as the “colour-spin” or “colour-magnetic” interaction of QCD, and was ÿrst discussed in the pioneering work of De Rujula et al. [18]. The sum runs over all pairs of quarks in the state. For the light spectrum one takes only into account the light quarks u; d; s for which the masses are small or comparable to the QCD scale, and looks for states with the lowest . Of course conÿnement, strong renormalization (higher twist), ÿnite width e9ects, etc., are assumed not to completely distort the ÿrst order results obtained from the e9ective Hamiltonian in Eq. (1). Most of the results follow from ÿrst applying Eq. (1) to the simple case of a diquark q1q2. The spins can be combined either to a singlet antisymmetric spin S = 0 state |0, or a triplet symmetric spin S = 1 state |1. The eigenvalues of ˜ 1 · ˜ 2 are

˜ 1 · ˜ 2 = 1 for S =1 ;

= −3 for S =0 (2)

Fig. 2. One gluon exchange between two quarks following Ja9e [16,17]. C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 67

states. Thus deÿning a spin exchange operator 1 1 PS = + ˜ · ˜ ; (3) 12 2 2 1 2 one has

S S P12|0 = −|0;P12|1 =+|1 ; (4) S i.e., P12 tells whether the spin part of the state is antisymmetric or symmetric under the exchange of the quarks. One can deÿne a similar operator for the Navour and colour SU(3) degrees of freedom. For colour this is 1 1 PC = + %˜ · %˜ ; (5) 12 3 2 1 2 ˜ and for Navour one obtains a likewise expression, but the Gell–Mann matrices (here denoted ÿi) operate in Navour instead of colour: 1 1 PF = + ÿ˜ · ÿ˜ : (6) 12 3 2 1 2 C F S These operators P12 and P12 for colour and Navour have similar properties as P12 for spin in that their eigenvalues and eigenvectors tell whether the state is symmetric or antisymmetric. Now, coupling two triplets (u; d; s for Navour) one obtains 3×3=9 states, of which 6 (uu; dd; ss; ud + du; us + su; ds + sd) are symmetric and form the six dimensional representation 6F , while three are antisymmetric (ud − du; us − su; ds − sd) and form the three dimensional representation 8 3F of SU(3)F (see Fig. 3).

3F 3F

6F 8 Fig. 3. Weight diagrams for the fundamental representation of SU(3)F (denoted 3F ), for the antisymmetric diquarks 3F and the symmetric diquarks 6F . 68 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

The same applies to colour SU(3)C with u; d; s replaced by red, blue, green. One then obtains instead of Eqs. (4): F 8 8 F P12|3F  = −|3F ;P12|6F  =+|6F  ; (7)

C 8 8 C P12|3C  = −|3C ;P12|6C  =+|6C  : (8) Hence, the antisymmetric combinations of the two quarks in a diquark (for Navour: ud − du; us − su; ds − sd) behave just as the anti-triplet composed of the anti-quarks (u; 8 d;8 s8). Now if the two quarks are in a relative S wave the spatial part of is symmetric. The remaining spin-Navour-colour part of the wave function must be antisymmetric for . This leads to a simple relation between the three exchange operators C F S P12P12P12 = −1 (9) or C S F P12P12 = −P12 : (10) Now using Eqs. (3) and (5) one can rewrite the colour-spin interaction (1)as 4 2 H ˙ − %˜ · %˜ ˜ · ˜ = −4PC PS + PS +2PC − : (11) e9 i j i j 12 12 3 12 12 3 i=j Inserting relation (10) the ÿrst term can be written in terms of the Navour exchange operator leading to 4 2 H ˙ − %˜ · %˜ ˜ · ˜ =4PF + PS +2PC − : (12) e9 i j i j 12 3 12 12 3 i=j Surprisingly, although the original relation does not depend on Navour, Cavour exchange has the largest weight (namely 4) in Eq. (12). Therefore the colour-spin interaction leads to large mass splitting between of di9erent Navours. It is now easy to evaluate Eq. (12) for the four possible totally antisymmetric systems of a diquark in the combined Navour-colour-spin variables. With obvious notation one gets 8 8 8 8 He9 |qq; 3F ; 3C ; 0 ˙ −8|qq; 3F ; 3C ; 0 ; (13) 8 8 He9 |qq; 3F ; 6C ; 1 ˙ −4=3|qq; 3F ; 6C ; 1 ; (14) 8 8 He9 |qq; 6F ; 3C ; 1 ˙ +8=3|qq; 6F ; 3C ; 1 ; (15)

He9 |qq; 6F ; 6C ; 0 ˙ +4|qq; 6F ; 6C ; 0 : (16) Hence the channel with the strongest attraction is in the conÿguration which is separately antisym- metric in all three variables, Navour, colour and spin. On the other hand, attraction or repulsion between the two quarks is weaker for symmetry in two variables and antisymmetry in the third. 8 8 This singles out the conÿguration |qq; 3F ; 3C ; 0, Eq. (13), as the lightest diquark conÿguration. It behaves much like an anti-quark under Navour and colour, but is a spin-. We shall de- note this state by the shorthand (qq)38. Thus in a multi-quark environment one expects large binding between two quarks in this particular Navour-colour-spin state. C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 69

This basic observation has several consequences for spectroscopy, which agree with experiment. The most obvious one is that the lightest composed of three quarks are in a Navour octet 8 with spin 1/2. Since baryons are colour singlets each quark must be coupled to a colour 3C diquark. 8 8 The octet can be obtained by adding a quark to the lightest diquark state |qq; 3F ; 3C ; 0. On the other hand, the decuplet 10F with total spin 3/2 can only be constructed by adding a quark to the 8 heavier |qq; 6F ; 3C ; 1. For qq8 mesons Eq. (1) reduces to the simple Breit spin–spin interaction and leads to pseudo- scalar mesons that are lighter than vectors, much like para- and orthopositronium. But Eq. (1) also leads to less obvious predictions such as the absence of light Navour-exotic states, e.g. ++ or K ++ resonances, or more generally higher Navour multiplets than octets and singlets not excluded by conÿnement. For Navour-exotic q2q82 states the colour-spin interaction Eq. (1) predicts that the colour-spin is repulsive. To build a colour singlet but Navour exotic q2q82 state, either the diquark or the anti-diquark (or both) must be a Navour sextet which is less tightly bound than the anti-triplet, see Eqs. (15), (16). Such exotic states should therefore be heavier and broader. However, Navour non-exotic light q2q82 states can be formed with both the diquark and the anti-diquark in spin singlets, colour triplets and Navour triplets (as in Eq. (13)). These states,

(qq)38(8qq8)3 would be light, and if they exist could be misinterpreted as or mixed with qq8 states, since they also form an SU(3)F nonet. Ja9e’s suggestion [10] was that the lightest scalar mesons (today named f0(600) or ; a0(980);f0(980), and the unconÿrmed Ä(800)) build up such a nonet. Then the qq8 0++ states would lie higher in the 1.2–1:7 GeV region, as shown in Fig. 1.

The most striking prediction of such a (qq)38(8qq8)3 model is the inverted mass spectrum shown in Fig. 4. This is simply obtained by letting the number of strange quarks determine the mass splitting. It is then tempting to identify the lightest state, an isospin singlet, with the f0(600), and the heaviest states which form a isospin triplet and a singlet, by the a0(980), and f0(980). Then the mesons with would lie in between, forming the isospin doublet Ä(800). Although such a Ä(800) was claimed earlier and more recently by the E791 experiment [19], its existence remains controversial [20,21]. This will be discussed in more detail in Section 3.3. Within the heavy meson sector Gelman and Nussinov [22] considered recently the possible 8 + existence of a (cc)38(8ud)3 four-quark state and argued that such a 1 isosinglet state may exist near the DD8 ∗ threshold, possibly mixed with a deuteronlike state of same quantum numbers (see Section 2.2).

Fig. 4. The inverted mass spectrum (left diagram) expected in Ja9e’s four-quark model when the s-quark is heavier than the u and d quarks, compared with a similar qq8 nonet (right diagram). 70 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

One can also raise objections to the four-quark scheme of Ja9e [10]. Obviously, an essentially nonrelativistic zero width model without chiral symmetry is used in a regime where the e9ective couplings to hadronic decay channels are very large. In such a case, mass shifts from e.g. → 8  → ; a0 → KK → a0 and Navour related loops should be large and distort any naive bare spectrum [23,24]. Furthermore, crossed channel e9ects can generate scalar bound states between two pseudoscalars [25], and even unitarization can generate new bound states in addition to those introduced, as we discuss in Section 3.3.

2.2. Deuteronlike meson–meson bound states (or deusons)

The deuteron and heavier nuclei being in fact multi-quark states, it is natural to ask whether a similar mechanism which binds the deuteron could also bind two mesons and produce four-quark states with given quantum numbers (Fig. 5). This question has been studied surprisingly sparsely in the literature. It is generally mentioned only in passing within general phenomenological models for meson–meson bound states (e.g. Refs. [27,28]). Some special attention to this problem was given in Refs. [29–31]. When assessing whether pion exchange binds two hadrons, the deuteron is certainly the prime reference state. There one knows that the dominant comes from pion exchange between two colourless qqq clusters—a proton and a (see Refs. [32–34]). For heavy enough constituents, one can follow the approach which was so successful for the deuteron, generalized to meson–meson states. Hence a rather simple nonrelativistic potential model can be used, and one looks for the bound states by solving the Schr,odinger equation. For states similar to the deuteron, i.e. with small binding and comparatively heavy constituents, a nonrelativistic treatment should already provide a very good approximation. The results are then easily understood without too many theoretical assumptions. In particular, if the constituent meson mass is assumed to be inÿnite and the interaction is attractive, the potential term dominates the kinetic energy term. Then a exists with a mass just below the sum of the two constituent masses. In other words one expects deuteronlike states to exist if pion exchange is attractive and strong enough.

2.2.1. One-pion exchange The broken chiral symmetry of QCD predicts that the pion is singled out as the by far lightest and one knows since long ago that the pion plays a crucial role in the dynamics of hadrons,

uu q q d Proton Meson Pion and meson cloud Pion and meson cloud Meson dd Neutron q q u

Fig. 5. The deuteron as neutron–proton bound state (left) and a loosely bound state of two mesons, called a deuson (right). C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 71 whenever long distance e9ects are important. In pion exchange was traditionally the ÿrst mechanism proposed for understanding nuclear binding. The ÿrst step in studying the e9ects of virtual is to ÿnd the one-pion exchange potential. The modern way of deriving this is from the QCD Lagrangian using chiral perturbation theory (see e.g. Ref. [31]). One can derive an e9ective quark pion interaction

g - Lint = q8() 5˜.q(x)9-(x) ; (17) F where F is the pion decay constant ≈ 132 MeV and g is an e9ective pion quark–pseudovector cou- pling constant. In a non-relativistic approximation, and using SU(6) wave functions for the hadrons, one solves the Schr,odinger equation with a potential that depends on the spin-isospin quantum num- bers of the constituents and the bound state. Deuteronlike states can be expected when the potential is attractive and strong enough to make a bound state like the deuteron. To describe the coupling between a qq8 pair and the pion (Fig. 6) we follow Ref. [29]. It is useful to introduce the constant m3 g2 V =  ≈ 1:3MeV; (18) 0 2 12 F the numerical value of which is ÿxed by the N (f2 =4 =0:08 = 25 (g2=F2)m2 N 9   from which g ≈ 0:6). The quantity V0 is a measure for the e9ective potential between the two quarks, when the total spin and the isospin are unity, Sqq = Iqq = 1 (see Fig. 6). This is a good approximation for heavy spectator quarks Q. A good test would be a measurement of the partial width of the D∗ meson to D. One of us [29] predicted 1 g2 p3 4(D∗+ → D0+)= p3 =2V  ; (19) 2  0 3 6 F m which gave 63:3 keV, in excellent agreement with the recent experimental result of 65 keV [35]. The same formula gives for the K∗ width 37 MeV, in reasonable agreement with the experimental result, 51 MeV. This agreement is also in accord with heavy quark symmetry, that the pion couples to the light quarks only, and the coupling does not depend on the heavy spectator quark mass. These relations then lead to a one-pion exchange potential between two hadronic constituents which can be written compactly in the form

V()=− V0[D · C(r)+S12(ˆr) · T(r)] ; (20)

d V0

Q g g Q

V0 u

Fig. 6. One-pion exchange coupling to qq8. The quark Q is assumed to be a spectator. 72 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

Here D is diagonal (often a unit ) and S12 is the tensor operator of matrix form which connects di9erent spin-orbitals. The quantity measures the strength of the one-pion exchange potential, as built up by pion exchange between the constituent quarks. For such a quark pair depends on a spin-isospin factor—( 1 · 2)(.1 · .2), where 1 · 2 (and .1 · .2) is +1 (for triplet) or −3 (for singlet) depending on the qq spin and isospin. Taking furthermore into account the internal symmetry wave functions of the two hadrons one ÿnds for example that is 25/3 for the deuteron and 6 for D∗D8 ∗ in the I =0;S= 0 state. The larger the , the stronger is the attraction, and for negative values of the interaction is repulsive. ∗ 8 ∗ In the equal mass case of the constituents (like D D ) the radial dependence of V(r) is given by the functions e−mr C(r)= ; (21) m r  3 3 T(r)=C(r) 1+ + ; (22) 2 mr (mr) where the tensor potential T(r) contains singular r−2 and r−3 terms. These make the latter dominate at small r like an axial dipole–dipole interaction. Therefore this singular behaviour of the tensor potential must be regularized at small distances, which introduces some model dependence.

2.2.2. Predictions for deuteronlike meson–meson bound states Since parity forbids three-pseudoscalar couplings, two pseudoscalar mesons cannot be bound by one-pion exchange. The lightest expected bound states are pseudoscalar (P)-vector (V) states. For V the pion is too light to be a constituent, because the small reduced mass of the V system would give a too large kinetic energy, which cannot be overcome by the potential. Thus the lightest bound states in which pion exchange can play a dominant role are KK8 ∗ systems which lie around 1400 MeV. For Navour exotic two-meson systems (I = 2, double strange, charm or bottom)—such as B∗B∗— pion exchange is always either weakly attractive or repulsive ( small or negative). Calculations do not support such bound states to exist from pion exchange alone, and shorter range are expected to be repulsive. Should they exist, however (cf. [31]), they would be quite narrow since they would be stable against strong decays. On the other hand, for non-exotic systems such calculations ÿnd that deuteronlike meson–meson bound states should exist [29]. The 12 expected states for D∗D8 ∗ and B∗B8∗ are given in Table 1. In the bottom sector these states are bound by about 50 MeV from one-pion exchange only. In the charm sector binding from pion exchange is weaker but states near threshold could also bind with small contributions from shorter range attraction. The widths are expected to be quite narrow, provided that heavy quark annihilation is not too strong. Such annihilation should be suppressed if the states are, like the deuteron, much larger in size than heavy quark qq8 states. A search for the heavy deuteronlike mesons predicted in Table 1 could be conducted with pp8 annihilation in Night, and possibly with 8 decay into open charm-anticharm states. Light mesons are much harder to bind since the attraction from one-pion exchange is not suOcient to overcome the large kinetic energy of the constituents. This is especially true for the pion itself as a constituent, but also for KK8 ∗ systems for which the potential term is only half as strong as C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 73

Table 1 Predicted heavy isoscalar loosely bound two-meson states, or deusons, with masses in MeV, close to the DD8 ∗ and the D∗D8 ∗ thresholds, and about 50 MeV below the BB8∗ and B∗B8∗ thresholds [29]

Composite J PC Mass (MeV) Composite J PC Mass (MeV)

DD8 ∗ 0−+ ≈ 3870 BB8∗ 0−+ ≈ 10545 DD8 ∗ 1++ ≈ 3870 BB8∗ 1++ ≈ 10562

D∗D8 ∗ 0++ ≈ 4015 B∗B8∗ 0++ ≈ 10582 D∗D8 ∗ 0−+ ≈ 4015 B∗B8∗ 0−+ ≈ 10590 D∗D8 ∗ 1+− ≈ 4015 B∗B8∗ 1+− ≈ 10608 D∗D8 ∗ 2++ ≈ 4015 B∗B8∗ 2++ ≈ 10602

As discussed in the text, the mass values are obtained from (a rather conservative) one-pion exchange contribution only.

Table 2 Meson–meson channels in the light meson sector for which one-pion exchange is attractive. The nearby known mesons, some of which could be deuteron like (or mixed with deuteron-like states) are listed in the last column

Composite IJPC Threshold (MeV) Nearby states

KK8 ∗ 00−+ 1390 Á(1410)1 ∗ ++ KK8 01 1390 f1(1420) ∗ ∗ ++ K K8 00 1790 f0(1710) K ∗K8 ∗ 00−+ 1790 Á(1760) K ∗K8 ∗ 01+− 1790 ∗ 8 ∗ ++ K K √ 02 1790 −+ 2 (99 + !!)=√200 1540–1566 Á(1480) ++ (99 − !!)=√200 1540–1566 f0(1500) ++ (99 + !!)= 202√ 1540–1566 f2(1565) (K ∗9 − K ∗!)= 2 1 0++ 1665–1678 2 The Á(1410) (1) denotes the low mass region and the Á(1480) (2) the high mass region of the Á(1440).

needed (this conclusion depends somewhat on how the tensor potential is regularized). Table 2 shows the predicted states with the largest attractive channels and their quantum numbers, together with the nearby experimental candidates. We shall discuss below other more likely assignments for the f0(1500) and f0(1710) mesons. Further bound states could exist with additional attraction of shorter range. For Navour exotic systems or for states with exotic quantum numbers (such as 1−+; 0−−; 0+−) pion exchange is generally repulsive or very weakly attractive and hence these states do not bind. One pion exchange is generally a factor three weaker for I =1 systems than for I =0. Such states are therefore not expected within the light meson sector. Deuteron-like vector mesons are not expected either since the attraction from pion exchange is too small. Also, such states should have been seen in e+e− annihilation. 74 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

30 2

15 Events/5 MeV/c

0

3820 3860 3900

M(π+π-J/ψ) (MeV/c2)

Fig. 7. +−J= distribution in B± decays to K ±X (X → +−J= ) (from Ref. [36]).

A lower bound for the widths of the predicted light states in Table 2 is given by the widths of the constituents. For example, the width of a (KK8 ∗) deuteronlike meson should be at least that of the K∗ (51 MeV), while the width of a (K∗K8 ∗) state near threshold should be around 100 MeV. Annihilation amplitudes would of course increase these lower bounds. The BELLE Collaboration [36] reported a new narrow charmonium state in B± decay to K±+ −J= at 3871:8 ± 0:7 MeV with a width 4¡3:4 MeV, smaller than the experimental resolution. 3 This is 60-100 MeV above the expected spin 2 cc8( Dc2) state [37,38]. The new state (Fig. 7)is observed in the +−J= invariant mass distribution with a signiÿcance of 8:6 . This looks very much like one of the two deuteron-like DD8 ∗ states at 3870 MeV listed in Table 1, and as was predicted in Table 8 of Ref. [29] over 10 years ago. Its spin is, however, not determined yet. If deuteron-like, its spin-parity would be 0−+ or 1++ according to Table 1.It would be an isosinglet with a mass very close to the DD8 ∗ threshold. But, as the binding energy of such a deuteron-like state is of the same order as the isospin mass splittings one should expect large isospin breaking. In fact, the observed peak is almost exactly at the D0D8 ∗0 threshold (3871:2 MeV). The main decay mode of the new state, if deuteron-like, should thus be D0D8 00 since the charged modes lie about 2 MeV above resonance. One can then, using isospin and D∗ width measurements [35], estimate the width to be of the order of 50 keV. For further comments on this state see Ref. [39].

3. Are the scalars below 1 GeVnon- qq! states?

An essential experimental input for understanding the nature of the lightest scalar mesons comes from their couplings to two pseudoscalars, their -widths and their radiative widths which we now discuss in detail. C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 75

3.1. The hadronic widths of the a0(980) and f0(980) mesons

The a0(980) meson decays mainly into Á while f0(980) decays mainly into . However, their partial widths to KK8 are still rather large: 4[a (980) → Á] 0 =0:85 ± 0:02 ; (23) 8 4[a0(980) → Á + KK] 4[f (980) → ] 0 =0:78 ± 0:02 ; (24) 8 4[f0(980) →  + KK] where we have averaged over some listed data [3]. Indeed, one would naively expect the KK8 modes to be strongly suppressed as the nominal meson masses lie below KK8 threshold. Thus only the high end of the resonance peak can decay to KK8 . This is shown in Fig. 8 for couplings to Á 8 and KK derived frompp 8 annihilation at rest into a0(980) (see Ref. [40]). In fact, without phase 8 space correction the f0(980) couples much more strongly to KK than to . Models give e.g. 2 2 g − =g − =4:00 ± 0:14, see below. f0K+K f0+ √ 8 8 Now, if the isovector a0(980) is pure qq8 it must be nn8, i.e. ud; (uu8 + dd)= 2or8ud. Then the degeneracy√ of masses would suggest that the isosinglet f0(980) is also composed of nn8, i.e. (uu8 − dd8)= 2. But, this clearly contradicts the above large KK=8 coupling ratio of 4, since in that case that ratio would be 1/4 instead (assuming Navour symmetry and the OZI rule). The KK8 coupling can be large only if the f0(980) and a0(980) wave functions contain a signiÿcant fraction of ss8, either in the form of pure qq8, or in the form of KK8 , or within a multiquark structure. 8 Therefore other structures have been suggested such as four-quark states (a0(980) ≡ ss8(dd−uu8) and 8 8 f0(980) ≡ ss8(dd + uu8)) [10], or KK molecular states [41–43]. An issue which complicates things somewhat is that any unitarization, together with the fact that the KK8 threshold is nearby, always introduces a large KK8 component into the physical wave function if the coupling to KK8 is large.

14

12

10

8

6 Intensity

4

2 KK 0 0.8 1.0 1.2 1.4 m [GeV]

Fig. 8. A qualitative Á and KK8 mass distribution for the a0(980) resonance inpp 8 → ÁX and KKX8 (in arbitrary units, assuming that no other resonance is produced). The dashed line shows the Á intensity in the absence of KK8 decay mode (from Ref. [40]). 76 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

The bare states (“the seeds”), be they qq8 or q2q82, must thus be dressed with a cloud of KK8 around the core [23,26].

3.1.1. widths of the a0(980) and f0(980) mesons It is generally argued that molecular states should have much smaller widths than qq8 states [44]. The expected widths for qq8 mesons can be estimated using the measured widths of 2++ mesons and the (relativistic) formula [45] 3 m0 40 242 : (25) m2

One ÿnds ∼ 0:8 keV for the a0(980) and ∼ 0:1 keV for the f0(980), assuming a pure ss8 nature of the latter. More sophisticated calculations lead to 0:64 keV [46] and 0.3–0:5 keV [46,47], respectively, which are comparable to the widths of molecular states (∼ 0:6 keV for the a0(980) [48]). Hence measurements of the widths cannot distinguish between molecular and qq8 states. The experimental +0:10 values are 0:30 ± 0:10 keV for the a0(980) and 0:39−0:13 for the f0(980) [3], somewhat smaller than predicted. The comparison with theory and experiment is summarized in Table 3.

3.1.2. Radiative widths of the (1020) to a0(980) and f0(980) Radiative (1020) decay into a0(980) and f0(980) has been proposed as a sensitive reaction to distinguish between qq8 states, q2q82 and KK8 : KK8 molecules should be produced with a branching ratio of 10−5 while four-quark states (qqq8q8) would be produced with a much larger rate of 10−4 [44]. However, this has been criticized because of over-simpliÿed modelling of the decay process [49,50]. On the other hand, the a0(980) as an isovector qq8 state could not be produced in radiative decay (1020) → a0(980) in the limit of ideal mixing in the vector nonet. This is due to the OZI rule which prevents a pure ss 8 (1020) to decay into an nn8 state. Therefore one would expect the rate (driven by a KK8 loop) to be quite small, much smaller than the corresponding (1020) → f0(980) , assuming a dominant ss8 structure for the f0(980). Indeed, Achasov and Gubin −5 −5 [44] predict rates of about 10 for (1020) → a0(980) and 5 × 10 for (1020) → f0(980) , much smaller than for q2q82 states. The ÿrst measurements of these radiative decays were performed at the e+e− VEPP-2Mring at Novosibirsk by the SND and CMD-2 collaborations. The reactions (1020) → 00 and Á0 , 7 leading to ÿve ÿnal state , were reconstructed from a sample of 2×10 decays. The f0(980) 0 0 0 and a0(980) appear to dominate the   and Á mass spectra, respectively. The data samples were rather small (a few dozen a0(980) and a few hundred f0(980) events) and the expected contribution from the continuum  S-wave (e.g. in ) could not be determined precisely, leading to a large systematic error. SND reports the branching ratios given in Table 4. The corresponding results for

Table 3 Theoretical predictions of widths in KeV compared to experiment

State Eq. (25) Refs. [46,47] Experiment [3] a0(980) ∼ 0:8 0.64 0:3 ± 0:1 +0:10 f0(980) ∼ 0:1 0.3–0.5 0:39−0:13 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 77

Table 4 −4 0 0 Experimental and estimated theoretical branching ratios in units of 10 on → f0(980) →   and → a0(980) → 0Á

0 0 0 Experiment → f0 →   → a0 →  Á Ref.

+0:44 SND 1:17−0:58 0:88 ± 0:17 [51,52] CMD2 0:97 ± 0:68 0:90 ± 0:26 [53] KLOE 1:49 ± 0:07 0:74 ± 0:07 [54,55]

Theory qq8 state ∼ 0:5 ∼ 0:1[44] qqq8q8 state ∼ 1:0 ∼ 1:0[44] KK8 ∼ 0:1 ∼ 0:1[44]

Fig. 9. Left: Di9erential branching ratio for (1020) decay into 00 as a function of 00 mass (from Ref. [54]). Right: event distribution for (1020) decay into Á0 for Á → (top) and Á → +−0 (bottom) as a function of Á0 mass. The curve shows the theoretical prediction ÿtted to the data (from Ref. [55]).

CMD-2 agree, see Table 4. The errors for f0(980) are large, but at least the branching ratio for −4 2 2 a0(980) (∼ 10 ) appears to be consistent with expectations for q q8 states. More precise branching ratios were measured recently at the DAYNE factory with larger data samples (5 × 107 decays) and better reconstruction eOciency. The 00 mass distribution in 5 events from the KLOE collaboration [54] is shown in Fig. 9. The low mass tail is determined by the contribution to the  S-wave. For the the authors choose (somewhat arbitrarily) the results from D+ → ++− decay [60] in which a mass of 478 MeV and a width of 324 MeV were reported. The ÿnal state appears to interfere destructively with f0(980) . The contribution from → 900(90 → 0 ) is negligible. The net results are consistent with the Novosibirsk ones, but are more precise: multiplying the 00 rate by an isospin factor of three to take into account the 78 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

Table 5 2 0 0 Coupling constants in units of GeV or GeV extracted by KLOE (ÿt B with )[54,55] from → f0(980) →   0 and → a0(980) →  Á data

2 g − =(4)2:79 ± 0:12 f0K+K 2 2 g − =g − 4:00 ± 0:14 f0K+K f0+ g 0:060 ± 0:008 2 g + − =(4)0:40 ± 0:04 a0K K g =g + − 1:35 ± 0:09 a0Á a0K K 2 2 g 8 =g 8 7:0 ± 0:7 f0KK a0KK

Table 6 2 2 0 0 Experimental values of the coupling g + − =(4) in units of GeV extracted from → f0(980) →   assuming a f0K K virtual KK8 loop compared with some theoretical predictions

2 Experiment g + − =(4) f0K K KLOE ÿt A, (no )[54,55]1:29 ± 0:14 KLOE ÿt B (with )[54,55]2:79 ± 0:12 CMD-2 [53]1:48 ± 0:32 +0:73 SND [51,52]2:47−0:51

Theory Linear sigma model [62] ∼ 2:2 QCD sum rules [63] ∼ 4:0

+ − unobserved f0(980) →   decay mode, KLOE obtains the branching ratio −4 B[ (1020) → f0(980) ; (f0(980) → )]=(4:47 ± 0:21) × 10 ; (26) which is an order of magnitude larger than expected for a dominantly ssf8 0(980). The KLOE result for the a0(980) channel [55] also agrees with the Novosibirsk one. Here the Á + − 0 meson from a0(980) decay is detected in both its    and decay modes, leading to consistent results. This is an important check that systematical errors are under control. The contribution from → 900(90 → Á ) is negligible. The Á0 mass distribution is similar to the one for 00 in f0(980) decay and is also shown in Fig. 9. The branching ratio −4 B[ (1020) → a0(980) ; (a0(980) → Á)]=(0:74 ± 0:07) × 10 ; (27) is again much larger than expected for a qq8 state. The data are summarized in Table 4. The coupling 2 constants extracted from KLOE are given in Table 5. Their value for g + − =(4)=0:40±0:04 agree a0K K 2 well with a di9erent determination [56] in a coupled channel framework who get g + − =(4)= a0K K 2 0:356. Furthermore in Table 6 the values of g + − =(4) obtained by di9erent experiments are f0K K listed and compared with some theoretical predictions. 2 2 As q q8 states the f0(980) and a0(980) are assumed to be produced in radiative decays through the emission of a from a K +K− loop (Fig. 10). The radiative partial widths are therefore equal, C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 79

γ + φ K

K- π

a0, f0

η, π

Fig. 10. Radiative decay into a0(980) or f0(980). in the range of ∼ 2 × 10−4 [42,43,64]. From Eqs. (26) and (27) one obtains, however, the ratio (1020) → f (980) 0 =6:7 ± 0:7 ; (28) (1020) → a0(980) taking the KK8 decay modes into account according to Eqs. (23) and (24). An argument for the much larger f0(980) yield was presented in Refs. [57,220] as due to large isospin mixing arising + − 0 80 from the nondegenerate K K and K K loops, and the near degeneracy of a0(980) and f0(980). However, the fact that the decay channels f0 →  and a0 → Á are open complicates the issue. Since both f0 →  and a0 → Á are open one must use a coupled channel framework, and not only KK8 . This reduces the possible isospin breaking substantially [58]. (A large isospin breaking would in fact imply also large f0 → Á, which is not observed.) Achasov and Kiselev [59] have also done an independent analysis of the KLOE a0 data and (mainly by varying the a0 mass) found a considerably larger value than KLOE in Table 5 for 2 2 g + − =(4) between 0.55 and 0:82 GeV . a0K K From the large production rates one could argue that the a0(980) and f0(980) are four-quark states, although not everybody agrees as, for example, the authors of Ref. [47] using the linear sigma model (L M). However, the L Mfor the light scalars does not necessarily imply that the states need to be qq8. A four quark nonet as obtained from e.g. Ja9e’s model could equally well be used as scalars in the L M[65]. + − For a theoretical analysis of the → f0(980) using a model with the K K loop which includes both the L Mand chiral perturbation theory in a complementary way see Ref. [ 66]. The coupling constants and their ratios in Table 5 above are useful to understand the nature of the scalar mesons.

3.1.3. The f0(980) produced in Ds → 3 + + + − The meson f0(980) is strongly produced in Ds (cs8) →    decay [67,61]. Fig. 11 shows the +− mass distribution and D → 3 Dalitz plot in which a prominent peak is observed just below + 1 GeV. The f0(980) contributes about 50% to the Ds decay Dalitz plot. From the Cabibbo favoured c → s decay one would expect the f0 resonances such as f0(980) to be produced mainly as an ss8 state, which then decays to  through a virtual KK8 loop (Fig. 12), see Refs. [68,69]. Note that for the f0(980) only the  channel is open, hence the OZI rule must be violated in the decay even if it is produced as an ss8 pair at short distances. This 80 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

2 + − + + + − Fig. 11. (a) m (  ) invariant mass distribution in Ds →    decay which is dominated by the f0(980). The 2 broader peak around 1:9 GeV is due to the f0(1370). The hatched area is the background distribution; (b) Dalitz plot (from Ref. [61]).

π W c s K π

Ds f0(980) π

s K

Fig. 12. W -emission in the Cabibbo allowed c → s transition leading to the formation of the f0(980). explains its narrow width. However, the nearness of the KK8 threshold and the large g generates f0KK8 8 through unitarity large virtual KK clouds in the f0(980) wave function. + + + − However, for the f0(1370) which is also seen in Ds (cs8) →    decay (Fig. 11) the same argument poses a problem, since f0(1370) is known to be mainly an nn8 state. For instance, f0(1370) 8 is not observed in Ds → KK [70] which suggests that this state is an nn8 and not a naive ss8, since then one would expect the KK8 mode to be large. This indicates that the annihilation contribution Ds → W →  is also present, or more generally, that the production process is more complicated than for f0(980). 0 Another experimental fact is that the f0(980) is observed in hadronic Z decays with cross sections similar to those for nn8 states [71]. This argues in favour of a similar two-quark nature of the f0(980). However, the production cross section for four-quark states is not known.

3.2. A possible interpretation of the nature of a0(980) and f0(980)

Since the radiative widths of both a0(980) and f0(980) are large and agree with theoretical expec- tations for compact four-quark states (Table 4), this suggests that they have substantial four-quark components. C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 81

On the other hand, these states lie just below the KK8 threshold and the resonance couplings to KK8 are therefore very large (see Tables 5 and 6). Hence their cores must by unitarity be surrounded by a rather large cloud of virtual KK8 pairs [23], which extends further out from the central four-quark core. From Section 2.1 one expects from QCD the strongest bound four-quark states to be those

(qq)38(8qq8)3 where the two quarks (and the two anti-quarks) are in a triplet state of both colour and Navour. Thus the following picture of the a0(980) and f0(980) emerges, which was discussed previously by Close and T,ornqvist [72] and is consistent with present data and theory: The central core is composed predominantly of a four-quark (qq)38(8qq8)3 statea  la Ja9e, whose quarks recombine at larger radii to two colour singlet qq8’s, and then form a standing wave of virtual KK8 at the periphery of the state. 8 The KK component also partly explains the narrowness of the a0(980) and f0(980) mesons: in order to decay the KK8 component must ÿrst annihilate near the origin to Á, respectively .

3.3. Is the f0(600) a non-qq8 state and does the Ä(800) exist?

Apart from the f0(980) and a0(980) there are two additional light candidates below 1 GeV, the f0(600) and the Ä(800). It is natural to ask whether these altogether four scalars could be related and form an SU(3)F nonet made of two- or four-quark mesons, or meson–meson bound states. Let us ÿrst discuss the experimental evidence for the f0(600) and the Ä(800) mesons.

3.3.1. The f0(600) (or ) There is now a rather widespread agreement that a light and very broad f0(600) or f0(600) pole exists in the  scattering data, although di9erent views prevail on its nature and its importance as a physical state in the nonperturbative regime of QCD. The PDG [3] cites numerous determinations of the pole mass in the neighbourhood of 600 MeV. One of the perhaps most precise determinations of the pole position was achieved using chiral perturbation theory together with constraints of analyticity, unitarity, crossing symmetry and the Roy equations [73], leading to the result m − i4=2 = (470 ± 30) − i(295 ± 20). Many analyses (see e.g. [25,74]) also generate the f0(600) from crossed channel exchanges. Such results do certainly not disprove the f0(600) as a true resonance as it is well known from the duality arguments of the 1970s that s and t channel resonances generally come together in hadronic amplitudes. The E791 Collaboration [60] observes in D+ → ++− a rather clear  resonance bump which dominates their 3 Dalitz plot (Fig. 13). They quote a (Breit–Wigner) resonance of mass +24 +43 478−23 ± 17 MeV and width 324−40 ± 21 MeV, which they interpret as the light f0(600). In their ÿt they, however, assume a Breit–Wigner shape, which means that the phase of the S-wave ampli- tude should reach 90◦ at the peak mass. This is not easily compatible with the known  phases (Fig. 14) and the Watson ÿnal state theorem, which states that the phase should be (up to some pos- sible constant production phase) the same as the  S-wave phase, which is only 45◦ at 600 MeV. This problem certainly needs further study. In Fig. 14 we show as an example the results of a recent new ÿt to the  →  elastic data in the scalar–isoscalar channel below 1 GeV by the Krakow group [75]. This group uses all well known theoretical constraints (e.g. crossing symmetry and Roy’s equations) in their ÿt, and ÿnd the unique ‘down-Nat’ solution shown in Fig. 14a. Together with chiral symmetry constraints, 82 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

Fig. 13. (a) m2(+−) invariant mass distribution in D+ → ++− decay, which is dominated by a broad low-mass f0(600). The hatched area is the background distribution; (b) Dalitz plot (from Ref. [60]).

(a) (b)

Fig. 14. Isoscalar S-wave phase shifts from Refs. [75,76]. The left ÿgure (a) shows the preferred ‘down-Nat’ solution (full circles) while (b) shows the ‘up-Nat’ data (open circles) [77]. The low-energy diamonds show the Ke4 data from Ref. [78]. The solid lines represent ÿts to Roy’s equations and to data. the slow increase of the  phase shift can be interpreted as due to the presence of a very broad f0(600) meson pole. 2 Chiral symmetry requires an Adler zero for the f0(600) in the  →  amplitude near s = m=2. This suppresses the low energy tail of the f0(600) as a naive Breit–Wigner resonance. Without that proper low energy behaviour one may easily miss the pole in the data analysis. A simple way to see this is to note [73] that current algebra predicts that (in the chiral limit, when the pion mass 0 2 vanishes, and s is small) the scalar–isoscalar amplitude, t0 = s=(16F ). Although this amplitude vanishes at the two-pion threshold, one soon reaches a very strong ÿnal state interaction which violates unitarity. An easy way to unitarize t0 is to write instead t0 =s=(16F2 −is). This expression √ 0 0  contains a pole at s = 463 − i 463 MeV, which is not far from the region where most f0(600) pole determinations are. Shifting the Adler zero sA from 0 and taking into account the ÿnite pion mass one can slightly improve the expression to t0 = −Im @(s)=[16F2 +Re@(s)+iIm @(s)], where Im @(s)= 0  2 −(s − sA) 1 − 4m=s and Re @(s) is determined from Im @(s) using a dispersion relation C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 83

2 (subtracted at s ≈ 0 to agree with the previous expression near s = 0). But since |spole|m the pole position is not changed signiÿcantly. It should, however, be emphasized that such unitarized ex- pressions, although utterly simple, cannot be approximated by a single Breit–Wigner amplitude. One must include other terms, or at least a speciÿc constant “background”, as e.g. the four-point contact term in the linear sigma model. This background then interferes destructively with the resonance at the low energy side to give the Adler zero. Another way to see that for a very broad resonance (say of width 4 ≈ 500 MeV) the pole parameters (M − i4=2) are quite di9erent from the Breit–Wigner mass MBW and width 4BW is to consider the following simple but instructive nonrelativistic (FlattÃe) form for the inverse propagator: g2 √ P−1(s)=M 2 − s − iM 4 (s)=M 2 − s − i M s − s (29) BW BW BW BW 8 BW th

◦ 2 where sth is the threshold. For this propagator the phase shift passes 90 at s = MBW, and the 2 Breit–Wigner width would be 4BW(MBW). Note that s appears twice in the above expression. On the other hand the pole position which is obtained from a zero in the inverse propagator

−1 P (spole) = 0 (30) is very di9erent from the narrow width approximation M − i 4 (M ), when g2=(4) is large BW 2 BW BW and the resonance is above threshold. In general M is much larger than the pole mass M obtained √ BW from s =(M − i 4)2 or M = Re( s ). pole 2 pole For a broad resonance it is important to give the pole position rather than the Breit–Wigner values, since the pole is independent of the reaction under study. Only at the pole does the amplitude factorize, and the pole is independent of the “background”. Therefore a pole can lead to resonances with di9erent Breit–Wigner masses and widths in di9erent reactions. On the other hand, determining the pole position requires a reliable theory for the amplitudes and this has unfortunately been for a long time a source of much confusion, especially when broad resonances were involved. We have shown in Section 3.1.2 above that for the f0(980) and a0(980) mesons the couplings + 8 − 0 8 0 2 2 to the channels K K and K K were very large (g + − =4 ∼ 1–4 GeV , cf. Table 6). Yet f0K K the resonances appear narrow in the  and Á channels, respectively. This is due to the KK8 threshold opening at the resonance masses. If the latter were increased far above decay threshold 2 (i.e. 1 − sth=M of order 1) the widths would become very large, easily reaching 500 MeV [79]. Now, if the f0(600) and Ä(800) indeed belong to the same family as the f0(980) and a0(980) mesons (say if the f0(600) were composed of 2 or 4 u and d type quarks) then no such mechanism would suppress the decay f0(600) →  or Ä(800) → K. Thus if f0(600) and Ä(800) belong to the same as a0(980) and f0(980) one would expect their widths to be very large. Therefore the broad f0(600) could well belong to the same family of light mesons as the narrow a0(980) and f0(980), as for example in the Ja9e four-quark model or in the U(3) × U(3) linear sigma model [65,80]. Suggestions that the f0(600) could be a glueball have been made [81,82]. However, as we shall show later, the best estimates from lattice QCD locate the glueball in the 1500–1700 MeV region. The partial width (4 =3:8±1:5 keV [83]) of the f0(600) can be understood from general gauge invariance requirements [84]. We shall deal with widths of glueballs in Section 4. 84 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

180 170 1.0 160 150 140 0.8 130 120 110

parameter 100 0.6 η | 90 π π K phase shift K 80 70 |A 0.4 60 50 K η threshold 40

Phase shift , 0.2 30 20 Kπ absorption 10 parameter*100 K η' threshold 0 0.0 0.6 0.8 1.0 1.2 1.4 1.6 0.6 0.8 1.0 1.2 1.4 1.6 (a) √s = m(K π ) GeV (b) √ s = m(K π ) GeV

Fig. 15. K S-wave phase shift (a) and magnitude of the S-wave partial wave amplitude (b) measured by LASS [85] (dots) and ÿtted in the unitarized quark model of T,ornqvist [23,24].

Fig. 16. The D+ → K −++ Dalitz plot. A broad Ä is reported under the dominating K ∗(892) bands (from Ref. [19]).

3.3.2. The Ä(800) Fig. 15 shows the LASS elastic K S-wave phase shifts [85]. The phase shift does not pass through 90◦ until 1350 MeV and hence there is no Breit–Wigner resonance behaviour below 1 GeV. Nonetheless several theoretical models arguing in favour of a light Ä around 800 MeV [10,47,65, 86–89] have been presented. However, in some of the analyses (especially the experimental analy- ses), one does not make a clear distinction between pole and Breit–Wigner mass. For instance, no distinction is made if only tree level graphs are included without the loops required by unitarity. +19 +43 The E791 Collaboration reported a light Ä with mass 797−43 MeV and width 410−87 MeV, but uses a Breit–Wigner amplitude [19]. Their Dalitz plot is shown in Fig. 16. This claim was, however, not conÿrmed by the CLEO Collaboration [20,90]. In fact, Cherry and Pennington [21] argued that C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 85

2 2 Fig. 17. (a) K S-wave squared running mass m (s)=m0 +Re@(s) and the corresponding width function m(s)4(s)=−Im @(s) which ÿts the phase shift in Fig. 15 (from Ref. [23]); (b) Running mass2 =Re@(s) and Im @(s) when the overall coupling of the model in (a) is increased from its physical value (from Ref. [26], see also [24]).

the Ä mass cannot exceed ∼ 825 MeV assuming the LASS phase shifts (Fig. 15) to be correct. A lighter and very broad Ä pole is nonetheless possible and should be looked for in future data analyses. In unitarized models with correct analytic behaviour at thresholds one must add a running mass 2 Re @(s) to the constant mass term m0 in the Breit–Wigner resonance amplitude. This is illustrated in Fig. 17a. When the running mass (top curve in Fig. 17a) crosses the K mass curve (s=m(K)2) the phase shift passes through 90◦. Note in particular the strong cusps at the K and KÁ thresholds. For a naive Breit–Wigner resonance the Re @(s) with cusp behaviour would be replaced by a constant. The nonlinear form of @(s) can produce two poles in the amplitude, although only one seed state (qq8 or 4-quark state) is introduced. In order to clarify this point one can, within a model, increase the e9ective coupling from its physical value. Then with increasing coupling one ÿnds a virtual bound state near the K threshold. For suOciently large coupling even a bound state in K would appear (for more details see Refs. [24,26]). An example is shown in Fig. 17b, where the ÿrst crossing between Re @(s) and s would be ∗ a bound state and the third crossing a resonance like the K0 (1430). (The second crossing corresponds to a slow anti-clockwise movement in the Argand plot which is not a resonance.) In conclusion, there are theoretical arguments for why a light and broad Ä pole can exist near the K threshold and many phenomenological papers support its existence [10,47,65,86–89]. But the question of whether a Ä(800) exists near the K threshold is not yet conclusive, since Breit–Wigner ÿts have been used. We believe that experimental groups should look for pole positions in their data analysis, which also include the aforementioned nonlinear e9ects from S-wave thresholds.

∗ 3.4. Observation of a charm-strange state DsJ (2317)

∗ The BABAR Collaboration has recently reported the observation of a very narrow meson DsJ (2317) (4¡10 MeV, smaller than the experimental resolution) in the heavy-light sector, which apparently + 0 decays through isospin violation to Ds  (Fig. 18)[4]. This state was soon conÿrmed by CLEO 86 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

± 0 ± + − ± ± + − ± 0 ∗ Fig. 18. Ds  mass distribution for Ds → K K  (a) and Ds → K K   (b) showing the DsJ (2317) (after Ref. [4]).

2.8 - 1 2.73 - 0 2.67 2.6 + + + 2 2.59 D (2573) 1 2.55, 2.56 sJ M + + Ds1 (2536) [GeV] 0 2.48

+ o 2.4 DK D K o + 2.32 D K

2.2 - 1 2.13 *+ Ds (2112)

2.0 - 0 1.98 + Ds (1969)

1.8

Fig. 19. Experimental (solid) and theoretical (dashed) [91] cs8 mass spectrum. The long horizontal dotted lines show the D+K 0 and D0K + thresholds, below which the BABAR state [4] is shown (after Ref. [92]).

[93]. Its spin is still uncertain, but the J P =0+ are preferred. This discovery may well turn out to be crucial also for the light scalars. 3 It is not easy to identify this state with the P0cs8 state because its mass is about 160 MeV below the expected value near 2480 MeV [91] (see also Fig. 19). Such a large deviation from C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 87

3 these predictions seems at ÿrst unusual. This expected P0cs8 state would lie above the DK threshold ∗ and would therefore [91] be several hundred MeV broad. However, the observed DsJ (2317) lies ∼ 45 MeV below this threshold (in fact the average of two isospin related thresholds, 2358 MeV for D0K + and 2367 MeV for D+K 0). The only open charm and strangeness conserving threshold is + 0 ∗ Ds  where it was observed. If the DJs(2317) had isospin 0 the narrow width could be understood, ∗ + 0 at least qualitatively, since DJs(2317) → D  would violate isospin. A small isospin breaking is expected from the rather large 9 MeV di9erence in the D0K + and D+K 0 thresholds, and from  − Á mixing. Educated guesses of the total width are of the order 10 keV [94,95]. Numerous discussions on this state can already be found in the literature with various interpre- ∗ tations [92,94–98]. We ÿnd the possibility especially intriguing that the DsJ (2317) could be a state related by Navour symmetry to the light scalars below 1 GeV. It is unavoidable that this state (be it composed of two quarks, four quarks or just a DK molecule) should couple strongly to the closed S-wave threshold DK. Then the hadronic mass shift (or running mass) due to DK loop would be important and could lead to a 160 MeV downward mass shift. The e9ect is similar to the Ä(800) running mass of the previous section. Remember also that two states are often generated by one input seed state, one near the strong threshold and the other higher in energy. In fact Ref. [97] supports such a picture with a model rather similar to the one used in the discussion above [23,24]. CLEO also found another new, also very narrow, charm-strange state at 2460 MeV [93]. This ∗ 0 ∗ peak is seen in the Ds  channel and lies about 40 MeV below the D K threshold, which would be the nearest strong (but closed) S-wave threshold, assuming that this state is an axial charm-strange meson, for which DK is forbidden by parity. Similar strong cusp e9ects in the running mass would also here be expected, which could explain the low mass, 100 MeV below predictions [91]. In fact, even among the heavy 8 states similar threshold e9ects (there due to the opening of the strong BB8 thresholds) explain why the 8(4S)–8(5S) mass splitting is so large, about 80 MeV larger than in naive potential models [99]. A better understanding of these two narrow states is thus likely to throw new light also on the enigmatic light scalars.

3.5. Do we have a complete scalar nonet below 1 GeV?

From the previous discussion, the a0(980);f0(980) and f0(600) could belong to the same Navour nonet, since the large di9erence in widths could be understood by the wide open phase space for the 8 , and by the KK threshold distortions for the a0(980) and f0(980) states. The conÿrmation of a light and equally broad Ä(800) would lead to a light scalar nonet below 1 GeV. The large radiative widths of the a0(980) and f0(980) mesons favour large four-quark components. This would not only agree with Ja9e’s original predictions for four-quark states, but would also be consistent with the conventional wisdom that meson–meson forces are attractive in octet and singlet channels, but repulsive for Navour exotic quantum numbers. Thus the formation of bound (or nearly bound) meson–meson bound states should be expected only in octet and singlet channels. Some years ago, one of us [23,24] was able to interpret many of the light scalars as originating from a nearly degenerate nonet of bare states which were strongly shifted in masses by unitarization. In that scheme a “resonance doubling” would appear, even though only one bare state was introduced. The lower state usually appeared near the ÿrst strongly coupled threshold. Flavour symmetry breaking 88 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 in the output spectrum came mainly from the large splitting between the two-pseudoscalar thresholds (i.e., for I =0:; Á; KK;8 ÁÁ; ÁÁ and ÁÁ). In an educated guess [72], the lightest scalars are composed of a central core with four quarks. Following Ja9e’s QCD arguments this central core would consist predominantly of a four-quark

(qq)38(8qq8)3 state. At larger distances the quarks would then recombine to a pair of colour singlet qq8’s, building two pseudoscalar mesons as a meson cloud at the periphery.

4. Glueballs

4.1. Theoretical predictions

QCD predicts the existence of isoscalar mesons which contain only gluons, the glueballs. These states are a consequence of the non-abelian structure of QCD which requires that gluons couple to themselves and hence may bind. Fig. 20 shows results from lattice gauge calculations as a function of lattice spacing a. When the scale parameter r0 (estimated from the string tension in heavy quark mesons) is taken to be about 0:5 fm, one ÿnds by extrapolation to a=0 a mass of 1611±30±160 MeV for the ground state glueball, a scalar (the ÿrst error is statistical while the second error reNects the uncertainty on r0). The ÿrst excited state is a tensor and has a mass of 2232 ± 220 ± 220 MeV [12]. Further mass predictions from the lattice can be found in Refs. [100,101]. Hence the low mass glueballs lie in the same mass region as ordinary isoscalar qq8 states, that is in 3 ++ 3 3 3 ++ the mass range of the 1 P0(0 ) and 2 P2; 3 P2; 1 F2(2 ) states, see Fig. 1. This is presumably the reason why they have not yet been identiÿed unambiguously.

Fig. 20. Predictions for the mass m of the ground state glueball (0++) and for the ÿrst excited state (2++); a is the lattice spacing and r0 is a scale parameter, see text (from Ref. [12]). C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 89

12 0+

+ 10 2 3 + 2 4 2' 1 3++ + 0' 3+ 8 2 + 1+ 3 ++ 0 + G 0

m 6 ++ [GeV]

0 2 r G

2 m ++ 4 0

1 2

0 +++ + 0 PC

Fig. 21. Predicted quenched glueball spectrum from the lattice (from Ref. [103]).

For pure gluonium one expects couplings of similar strengths to ss8 and uu8 + dd8 mesons because gluons are Navour-blind. This leads to the Navour “democracy” for the glueball decay rates into , KK;8 ÁÁ and ÁÁ of 3:4:1:0. In contrast, ss8 mesons decay mainly to , and uu8 + dd8 mesons mainly to pions. Hence decay rates to ; KK;8 ÁÁ and ÁÁ can be used to distinguish glueballs from ordinary mesons. Therefore a detailed understanding of the qq8 nonets is mandatory. Unfortunately, lattice calculations predict that glueballs with the exotic quantum numbers J PC = 0−−; 0+−; 1−+; 2+−, etc., lie far above 2 GeV [102,103]. This is in the diOcult region of radial and orbital excitations, where the states become increasingly broad and overlap. Fig. 21 shows the glueball spectrum from lattice QCD. The lightest glueball with exotic quantum numbers (2+−) has a mass of about 4 GeV. The lattice calculations assume that the quark masses are inÿnite and therefore neglect qq8 loops. Nonetheless, one expects that glueballs will mix with nearby qq8 states of the same quantum numbers [13]. There are indications that the predicted mass of the scalar glueballs decreases slightly in the unquenched approximation, at least with two quark Navours, while the mass of the tensor does not change signiÿcantly [104]. On the other hand, mixing with nearby qq8 states will modify the decay branching ratios and obscure the nature of the observed state. However, one would still ÿnd three isoscalar states in the regions of the 0++ and 2++ nonets, instead of only two. As we shall discuss below, signiÿcant progresses have been made recently to identify the 0++ glueball, while much uncertainty remains for the 2++ assignments. As discussed before, lattice gauge calculations place the ground state glueball, an isoscalar 0++ 3 ++ state, in the 1400 to 1800 MeV mass interval, that is in the mass region where the 1 P0(0 ) and 3 2 P0 isoscalar qq8 mesons are also expected. In the charmonium system, the Cc0(1P) cc8 meson lies 90 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

140 MeV below the Cc2(1P). A similar mass splitting is also predicted for light quark mesons in the 3 relativized quark model with chromodynamics of Ref. [6]. The 1 P0 isoscalar mesons are expected 3  somewhat below the corresponding 1 P2 mesons f2(1270) and f2(1525) while their ÿrst radial 3 excitations√ 2 P0 are predicted around 1900 MeV. The isoscalar qq8 mesons have the quark structure nn8 ≡ 1= 2(uu8 + dd8) and ss8, or a mixture thereof. One expects that glueballs will mix with nearby qq8 states of the same quantum numbers [13,105] but, nonetheless, one would still ÿnd three nearby isoscalar states, instead of only two.

4.2. Is the f0(1500) meson the ground state scalar glueball?

Five isoscalar resonances are well established: the very broad f0(600) (or ), the f0(980), the broad f0(1370), and the comparatively narrow f0(1500) and f0(1710). We have dealt with the f0(600) and the f0(980) in a previous section and discuss now the three upper mass states f0(1370);f0(1710);f0(1500). In the following we shall show that the data suggest that f0(1370) is largely nn;8 f0(1710) mainly ss8, and f0(1500) mainly glue. Experimental evidence for a 100 MeV broad isoscalar state at 1527 MeV, decaying into two pions, was ÿrst reported in pp8 annihilation at rest into three pions [106,107]. A spin 0 assignment was cautiously suggested and no KK8 decays were observed. A somewhat broader scalar meson at 1592 MeV, named G(1590) and decaying into ÁÁ [108], ÁÁ [109], but not  [108] was reported in high energy pion induced reactions. Clariÿcation and ÿrm evidence for the existence of a 100 MeV broad isoscalar scalar meson at 1500 MeV came with the high statistics data from Crystal Barrel in pp8 annihilation at rest (for a review see Ref. [40]). The newly baptized f0(1500) meson was reported to decay into  [110], ÁÁ [111], ÁÁ [112], KK8 [113] and 4 [114,115]. It was conÿrmed in many experiments, e.g. in pion induced reactions [116], by the Obelix collaboration also inpp 8 [117], in central collisions [118–120], in J= radiative decays [121], and perhaps in Ds decays [67]. It was, however, not observed in collisions [122,123]. A sketch of the Crystal Barrel apparatus, a large solid angle high granularity -detector, is shown in Fig. 22. Details can be found in Ref. [124]. The Dalitz plots forpp 8 annihilation at rest into 30;0ÁÁ and 00Á were analysed by replac- ing the usual Breit–Wigner amplitudes describing two-body 00 or Á0 resonances by T-matrices [125]. This is the recommended procedure for coupled channels and overlapping resonances of the same quantum numbers which ensures that unitarity is fulÿlled. Masses and widths were derived by searching for poles of the T-matrices in the complex energy plane. The  S-wave scattering data from the CERN-Munich collaboration [126] were included. 0 0 The 3 and  ÁÁ channels demand two high mass isoscalar scalar mesons, f0(1370) and f0(1500), decaying into 00 and ÁÁ, while annihilation into 00Á also requires a high mass isovector decay- 0 ing into Á , the a0(1450). Consistency between the three data sets was obtained by performing a simultaneous coupled channel ÿt [127]. Fig. 23 shows the resulting 00 and ÁÁ S-wave intensities for the three annihilation channels, apart from multiplicative phase space factors 2p=m (where p is the daughter momenta in the rest frame of a resonance with mass m). At low masses one observes a 0 strong contribution from f0(980) producing a dip in the 3 channel (due to destructive interferences in this channel) and a broad enhancement attributed to the f0(600) meson. C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 91

4

7 p 6 5 1

3 2

Fig. 22. Sketch of the crystal barrel detector at the low energy antiproton ring (LEAR) at CERN. 1,2–magnet yoke; 3–magnet coil providing a longitudinal ÿeld of 1.5 T; 4– detection barrel made of 1380 CsI(Tl) crystals with photodiode readout; 5– drift chamber; 6–proportional wire chambers; 7–4 cm long hydrogen target.

8 The KK decays of the f0(1370) and f0(1500) mesons were also observed by Crystal Barrel 0 inpp 8 annihilation at rest into  KLKL [113], where one of the two KL’s interacted in the CsI barrel and the other one was missing. The decay kinematics was reconstructed from the measured 0 directions and energies of the two ’s from  decay and from the direction of the interacting KL. 0 The contributions from f0(1370) and f0(1500) to the  KLKL Dalitz plot were found to be small, although they could not be determined precisely as a0(1450) also decays into KLKL since KLKL has both isospin 0 and 1. Therefore one must subtract the a0(1450) amplitude from the f0(1370) and f0(1500) amplitudes. The a0(1450) contribution was determined by using isospin conservation and ± ∓ analysing the reactionpp 8 → KLK  in which the isoscalar S-wave is absent [129]. ± ∓ The a0(1450) mass and width, determined from the annihilation channelpp 8 → KLK  are M = 1480 ± 30 MeV and 4 = 265 ± 15 MeV, respectively, in excellent agreement with the result from the annihilation channelpp 8 → Á00;M=1450±40 and 4=270±40 MeV [130], respectively. This argues against the low mass (M 1300 MeV) and narrow (4 80 MeV) a0 reported by the ± ∓ Obelix collaboration in the annihilation channel KS K  [131]. A high mass but somewhat narrower + − 0 a0(1450) is also found in the annihilation channelpp 8 → !   , where a0(1450) decays to !9 [132]. The f0(1370) and f0(1500) mesons were also observed by the WA102 collaboration in pp central + − collisions at 450 GeV. Signals for these states and the f0(1710) were observed in the   and K +K− S-waves [119]. Fig. 24 shows the K +K− S-wave from a coupled channel analysis of +− + − and K K data, using the T-matrix formalism. Signals from the f0(1500) and f0(1710) are clearly seen and pole positions for the f0(1370) and f0(1500) mesons are in excellent agreement with  Crystal Barrel data. The f0(1500) was also observed by WA102 in its ÁÁ [133], ÁÁ [134] and 4 [137] decay modes. 92 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

1600

f0(1500)

1200

f0(1370)

Intensity 800

400 f0(600)

f0(980) 0 400 800 1200 1600 m [MeV]

Fig. 23. Isoscalar 00 S-wave intensities in 30 (solid curve) and 00Á (dashed curve) and ÁÁ S-wave intensity in 0ÁÁ (dotted curve), apart from phase space factors. The vertical scale is arbitrary (from Ref. [128]).

3000

2000

f0 (1500) Events / 0.04GeV 1000 f0 (1710)

0 1 1.5 2.0 2.5 M (K+K ) [GeV]

Fig. 24. K +K − S-wave in central production (from Ref. [119]).

4.2.1. Hadronic decay width For the f0(1500) meson the crystal barrel and WA102 ratios of measured decay branching ratios into two pseudoscalar mesons are listed in Table 7. They are in good agreement. Ref. [135] quotes somewhat smaller ÁÁ= and ÁÁ= ratios but data from older less precise experiments are ÿtted C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 93

Table 7 Ratios of decay branching ratios into pairs of pseudoscalar mesons for the f0(1500)

Ratiopp 8 annihilation Ref. Central production Ref.

4(ÁÁ)=4()0:226 ± 0:095a [110,111]0:18 ± 0:03 [133] 0:157 ± 0:062b [127] 4(ÁÁ)=4()0:066 ± 0:028a [110,112]0:095 ± 0:026 [134] 0:042 ± 0:015b [127] 4(KK8)=4()0:186 ± 0:066a [110,113]0:33 ± 0:08 [119] 0:119 ± 0:032b [127]

aMeasured inpp 8 annihilation and in central production from the single channel analyses. bMeasured from the coupled channel analysis of 30; 20Á and 02Á.

Table 8

Ratios of decay branching ratios into 4 for the f0(1500) measured inpp 8 annihilation and in central production

Ratiopp 8 annihilation Ref. Central production Ref.

4(2[]S )=4(4)0:26 ± 0:07 [115]— 4(99)=4(4)0:13 ± 0:08 [115]— 4((1300))=4(4)0:50 ± 0:25 [115]—

4(a1(1260))=4(4)0:12 ± 0:05 [115]— a 4(99)=4(2[]S )0:50 ± 0:34 [115]2:6 ± 0:4 [137] 3:3 ± 0:5b [137]

aFrom 2+2−. bFrom +−20.

8 simultaneously. We note that the KK signal for the f0(1500) is much larger than for the f0(1710) (see Fig. 24), even though the latter couples more strongly to KK8 [119]. Hence the production of f0(1500) appears to be enhanced in central collisions, in accord with the conventional wisdom that gluonic states should be enhanced in exchange reactions. 8 For the f0(1500) the ratio KK= is much smaller than one. We recall that a pure ss8 meson does not decay to pions and, therefore, the f0(1500), if interpreted as qq8 state, cannot have a large ss8 content. A more quantitative statement will be given below. 0 The 4 decay modes of the f0(1500) were observed inpp 8 annihilation at rest into 5 [136], in pn8 annihilation into −40 [114,115], +2−20 [115] and in central collisions [137]. The partial width for annihilation into 4 is about half of the total width (44 =0:55 ± 0:05 4tot, following Ref. [115]). The ratios of decay branching ratios are given in Table 8. The 2[]S mode refers to the decay into two S-wave pion pairs. The S-wave parameterization was taken from the phase shift

analyses of Ref. [126]. The relative strengths of 99 decay to 2[]S decay is of interest to understand the internal structure of the f0(1500) meson. In the Nux tube simulation of lattice QCD one expects a glueball to decay in leading order into gluon pairs [13]. On the other hand, if the f0(1500) is a mixture of the ground state glueball with nearby qq8 scalars, 99 decay dominates 2[]S , at least in 3 the framework of the P0 model [138]. However, the experimental situation is still unclear, since the 99=2[]S ratio from crystal barrel and WA102 disagree (Table 8). 94 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

8 For the f0(1370) the KK= ratio is diOcult to determine precisely due to the large width of this state which cannot be easily disentangled from the f0(600). Ref. [115] quotes values between 0.2 and 1.4 while Ref. [119] reports 0:46 ± 0:19. The 4 decay mode of the f0(1370) is dominant [114] which also indicates that this meson cannot have a large ss8 content. Hence we are left with two nearby isoscalar mesons, both with a dominant nn8 structure. The 2002 issue of the Review of [3] quotes for the masses and widths

f0(1370) : M = 1200 − 1500 MeV;4= 300 − 500 MeV ; (31)

f0(1500) : M = 1507 ± 5MeV;4= 109 ± 7MeV: (32)

Let us now deal with the f0(1710) meson. This state was ÿrst observed by Crystal Ball in radiative J= decay into ÁÁ [139]. The spin of the f0(1710) (J = 0 or 2) remained controversial for many years. The issue was ÿnally settled in favour of 0++ by the new data from WA102 in central collisions at 450 GeV [140]. This f0(1710) meson was discovered long before the f0(1500) and the 2++ assignment arose from the assumption that the 1500 MeV region was dominated by the  8 f2(1525). As mentioned before, the KK coupling of the f0(1710) is much larger than the . WA102 reports the ratio of partial widths [119] 4(KK8 )=4()=5:0 ± 1:1 ; (33) which, assuming a qq8 state, clearly points to a dominant ss8 structure. Nonetheless no signal for this − state was reported earlier in the amplitude analysis of K p → KS KS D interactions [141]. However, the assumption was that its spin would be 2. As we have seen, scalar mesons are strongly produced inpp 8 annihilation but the OZI rule forbids 0 the production of pure ss8 states. The f0(1710) was searched for inpp 8 annihilation into  ÁÁ and 30 with 900 MeV=c [142]. Fig. 25 shows the Dalitz plot forpp 8 → 0ÁÁ and the  corresponding ÁÁ mass projection. The a0(980) and a2(1320) → Á and f0(1500)=f2(1525) → ÁÁ

2.5 120 700 2.25 D 100 2 600

]

2 1.75 80 500

1.5 400 60 ) [GeV 1.25

πη 300 ( 40 2 1

m 200 0.75 20 100

0.5 number of events / 13.3 MeV

0.25 0 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 1000 1200 1400 1600 1800 2000 πη m 2( πη ) [GeV 2] m( ) [MeV]

0  Fig. 25. Left: Dalitz plot forpp 8 →  ÁÁ with a0(980) (A), a2(1320) (B), f0(1500)/f2(1525) (C). The arrow (D) shows  the expected location of the f0(1710). Right: ÁÁ mass projection showing the f0(1500)/f2(1525). The shaded histogram is the ÿt. C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 95

102 R 100 0.020 2 f (1710) 80 101 0 70 100 60 140 k=15/4 0=180 WA102 -1 10 CB ) [MeV] 0

f0(1500) (f k=2 10-2 40 10 γγ Γ

10-3 0.000 R1 0 30 60 90 120 150 180 10-4 10-4 10-3 10-2 10-1 100 101 α [o]

Fig. 26. Left: relative branching ratio R2 = B(KK8)=B() vs. R1 = B(ÁÁ)=B() as a function of F (in deg.); right: predicted -width for the f0(1500). The experimental upper limit is shown by the box (from Ref. [143]).

are clearly seen. However, no signal for the f0(1710) → ÁÁ is observed. This is prima facie evidence that f0(1710) cannot have a large nn8 component. 8 Fig. 26 (left) shows the SU(3) predictions for the ratio of branching ratios R2 = B(KK)=B() vs. R1 = B(ÁÁ)=B() for scalar mesons, apart from phase space factors. Details can be found in Ref. [143]. The boxes show the data from Crystal Barrel and WA102 (2 boundaries) on the f0(1500) and f0(1710). The angle F describes the mixing of the two nonet isoscalar mesons, uu8 + dd8 |f0 = cos F|nn8−sin F|ss8 with |nn8≡ √ : (34) 2

◦ Hence for F =0, f0 is pure nn8 and for F =90 , pure ss8 (ideal mixing). Note that SU(3) predictions for two-body decay branching ratios of tensor mesons (F =82◦) are in excellent agreement with data [13]. If one would assume that f0(1500) and f0(1710) are the isoscalar qq8 states, one would conclude from Fig. 26 (left) that the former is mainly nn8 and the latter mainly ss8. However, for f0(1500) we shall see in the next section that this conclusion is at variance with its 2 width.

4.2.2. 2 -decay width Let us now deal with two- processes which are useful to probe the content of mesons through their electromagnetic couplings. Glueballs do not couple directly to photons and their production should therefore be suppressed in -processes. New data in -collisions have been presented by the LEP collaborations. L3 observes three peaks below 2 GeV in the KS KS mass  distribution [122] (Fig. 27, left): f2(1270) (interfering with a2(1320)) and f2(1525), but the spin 0 f0(1500) is not seen. The spin of the third peak, fJ (1710) around 1765 MeV, is determined to be mainly 2 but a large spin 0 component is also present [144]. Since f0(1500) does not couple strongly to KK8 , its absence in Fig. 27 (left) is perhaps not surprising. However, ALEPH studying + − the reaction →   , does not observe f0(1500) either [123] (see Fig. 27, right). An upper limit of 1:4 keV (95% CL) can be derived for its -width from the ALEPH result [123], using the known  decay branching ratio of the f0(1500) [3]. 96 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

+ − Fig. 27. Left: KS KS mass distribution in -collisions at LEP/L3 (from Ref. [122]); right:   mass distribution from LEP/ALEPH showing only the f2(1270) (from Ref. [123]).

The -width of a qq8 state can be predicted from SU(3). Apart from an unknown nonet constant C and for a meson of mass m: √ 2 3 4 = C(5cos F − 2 sin F) m : (35) The -width of a scalar meson is related to that of the corresponding tensor by 3 ++ m0 ++ 4 (0 )=k 4 (2 ) ; (36) m2 with obvious notations. Here the factor k =15=4 arises from spin multiplicities in a non-relativistic calculation, while relativistically k 2. Data on the charmonium states Cc2 and Cc0 are in excellent agreement with Eq. (36). The -width for scalar mesons can now be predicted as a function of F by ÿrst calculating the constant C in Eq. (35) for tensor mesons, using their measured partial widths [3] and then introducing into Eq. (36). Fig. 26 (right) shows the prediction for the partial width of the f0(1500) as a function of F, together with the ALEPH upper limit [143]. Assuming a qq8 structure, one concludes that f0(1500) is dominantly ss8, at variance with the hadronic results discussed above. This contradiction indicates that f0(1500) is not qq8 and the lack of -coupling points to a large gluonic content. For the f0(1710), the ALEPH data are consistent with an ss8 state, although its  decay branching ratio is not known. In Ref. [143] we argued that the spin 0 component in the fJ region of Fig. 27 (left) was consistent with an ssf8 0(1710), while the spin 2 contribution arose from the (isovector) a2(1700) radial excitation of the a2(1320).

4.2.3. Mixing with qq8 states The most natural explanation is that f0(1500) is the ground state glueball predicted in this mass range by lattice gauge theories. However, one would expect a pure glueball to decay into ; ÁÁ; ÁÁ and KK8 with relative ratios3:1:0:4,incontradiction with the ratios in Table 7. Mixing of the pure glueball G with the nearby two N = nn8 and S = ss8 isoscalar scalar mesons was ÿrst introduced  8 to explain the ÿnite ÁÁ rate and the small KK rates observed for the f0(1500) meson [13]. In ÿrst C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 97 order perturbation one ÿnds, assuming that the quark–gluon coupling is Navour blind, √ |G + G( 2|N + !|S) |f0(1500) = ; (37) 1+G2(2 + !2) where ! is the ratio of mass splittings m(G) − m(N) ! = : (38) m(G) − m(S) If G lies between the two qq8 states, ! is negative and the decay to KK8 is hindered by negative interference between the decay amplitudes of the nn8 and ss8 components in Eq. (37). Conversely, the two isoscalars in the qq8 nonet acquire a gluonic admixture. In Ref. [13] G is at most 0.5 so that the nn8 state is essentially the f0(1370) and the ss8 state the (then not yet well established spin 0) f0(1710), both with a small glue admixture, while f0(1500) is dominantly glue. This model was extended in Ref. [105] and applied to both Crystal Barrel and WA102 data. The mass matrix  √  MG f 2f      fMS 0  (39) √ 2f 0 MN was diagonalized and the eigenstates f0(1710);f0(1500) and f0(1370) expressed as superposition of the G; N and S bare states. Here f = G|M|S and we have assumed Navour independence for simplicity (see Ref. [105] for a generalization). The best ÿt to the two-pseudoscalar decay branching ratios led to the dominantly ssf8 0(1710), while f0(1500) and f0(1370) share roughly equal amounts of glue ( 40%). The pure glueball was found at a mass M(G) 1440 MeV, while for pure nn8 and ssM8 (N) 1380 MeV and M(S) 1670 MeV, respectively. Mixing with nearby qq8 isoscalar 0++ states is hence probable but not necessarily required. In fact, as much as 60% of qq8 admixture in the f0(1500) wave function can hardly be accommodated by the ALEPH upper limit. Judging from Fig. 26 (right) one could tolerate an nn8 fraction of at most 25%. More accurate data in -collisions are needed for a more quantitative statement on mixing. Also, a systematic study of the so far not observed decay branching ratios of the f0(1710), in particular 4 or KK8 would have to be conducted, e.g. with the COMPASS experiment at CERN. We have discussed in Section 2 the nature of the low mass scalar mesons and have concluded that they are compatible with four-quark states and meson–meson resonances. From the present discussion we suggest that the ground state qq8 nonet lies in the 1200–1700 MeV range. Table 9 then shows the resulting classiÿcation scheme for scalar mesons.

4.3. The tensor glueball

++ 3 The ground state 2 nonet is well known. In this nonet the isoscalar 1 P2 mesons f2(1270)  and f2(1525) are well established. At higher masses three to four isoscalar states appear to be solid: (i) the f2(1565) (or AX ) observed at LEAR inpp 8 annihilation at rest [145] is perhaps the same state as f2(1640) also reported to decay into !! [146,147]; (ii) the rather broad f2(1950) decaying to 4 and ÁÁ is observed by several experiments, e.g. in central production [148] and in 98 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

Table 9 A likely classiÿcation of the low-mass scalar mesons showing the scattering resonances below 1 GeV and the ground 3 state qq8 nonet (1 P0). The supernumerary f0(1500) (not shown) is dominantly glue

State 4 (MeV) Isospin Dominant nature

2 2 a0(980) ∼ 50 1 KK;q8 q8 2 2 f0(980) ∼ 50 0 KK;q8 q8 2 2 f0(600) ∼ 800 0 ; q q8 Ä(800)? ∼ 600 1/2 K; q2q82

8 8 a0(1450) 265 1 ud; du;8 dd − uu8 8 f0(1370) ∼ 400 0 dd + uu8 f0(1710) 125 0 ss8 ∗ 8 K0 (1430) 294 1/2 us;8 ds;8 su;8 sd

pp8 annihilation at 900 MeV=c [142]; (iii) a broad structure (of perhaps several states) decaying to was reported around 2300 MeV in −N reactions [149,150] and in central collisions [151]. The JETSET Collaboration at LEAR measuring the cross section forpp 8 → also reported a broad enhancement at 2:2 GeV, just above the threshold [152], a channel that should be suppressed by the OZI rule. Since the 2++ glueball is expected around 2200 MeV we ÿrst discuss the experimental evidence for the narrow structure fJ (2220) (previously called G) reported around 2230 MeV. The observation of a ∼ 20 MeV broad state around 2230 MeV dates back to Mark III at SPEAR. It + − PC ++ was seen in radiative J= decay to K K and KS KS [153]. The latter implied that J = (even) . More recently, this state was reported by BES at the e+e− collider in Beijing with a mass of 8 + − 2231:1 ± 3:5 MeV and a width of 23 ± 7 MeV [3]. It was observed by BES to decay into   , + − 0 0 K K ;KS KS ; pp8 [154] and   [155] with statistical signiÿcance of about 4 in each decay mode. Several features made this state an attractive candidate for the 2++ glueball: (i) its mass which agrees with lattice predictions (although the 2++ assignment has not really been proven); (ii) its unusually narrow width for a qq8 excitation; (iii) its observation in the gluon rich environment of J= radiative decay; (iv) its comparable partial widths to  and KK8 , in line with Navour independence; (v) its non-observation in collisions [156]. According to BES the fJ (2220) meson decays topp 8 at BES and hence should be observed in pp8 formation experiments. However, all searches inpp 8 → G → 2; KK8 and have been negative so far. Crystal Barrel at LEAR has searched for narrow states decaying to 00 and ÁÁ (leading to 4 ) as a function ofp 8 momentum [157]. Fig. 28 shows the cross sections for nine momenta in the mass range of the G. The resolution was about ±0:6 MeV in the c.m.s. system. No structure was observed. Using the product of branching fractions B(J= → G)B(G → pp;8 00) measured by BES and the 95% CL upper limit of 6 × 10−5 for B(8pp → G)B(G → 00) measured by Crystal Barrel, one ÿnds that the observed decays amount to at most 4% of all G decays, hence most G decay channels have not been observed yet. Furthermore, B(J= → G) ¿ 3 × 10−3 which is comparable to the branching ratio for the known decay J= → Á. A striking Á is observed in the inclusive J= decay spectrum, while, however, G is not seen [158]. Hence the data are inconsistent: thepp 8 decay width measured at BES appears to be too large or the narrow G simply does not exist. C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 99

Fig. 28. Cross section forpp 8 → 20 (top) and 2Á (bottom) for cos ¡0:85). The curves are straight line ÿts (from Ref. [157]).

 Above the f2(1525), none of the nine reported isoscalars [3] can be deÿnitely assigned to the 3 3 3 expected radial or orbital excitations in the expected 2 P2; 3 P2 or 1 F2 nonets. Therefore, the identiÿcation of the tensor glueball is premature. A systematic study of the two-body channels ; KK;8 ÁÁ and ÁÁ, similar to the one performed for scalar mesons at lower energy, would have to be conducted.

4.4. The pseudoscalar glueball

The evidence for a 0−+ state around 1400 MeV dates back to the sixties. The then called E-meson was observed in the KK8 mass spectrum ofpp 8 annihilation at rest into KK8 3 [159]. This state ∗ 8 was reported to decay through a0(980) and K (892)K with roughly equal contributions. The quan- tum numbers of the E-meson (now called Á(1440)) remained controversial as the experimental − ++ evidence from  p peripheral reactions led to a 1 state, the f1(1420). The Á(1440) was later observed as a broad structure around 1400 MeV in radiative J= decay to KK8 [160]. Since radia- tive J= decay to light quark proceeds through an OZI forbidden process, namely the annihilation of both (cc8) quarks, the rather large production of Á(1440) (then called –) was indicative of a strong gluon–gluon interaction, presumably leading to the formation of a glueball. 8 In J= radiative decay the Á(1440) decays to KK through the intermediate a0(980) channel and hence a signal was also to be expected in the a0(980) → Á mass spectrum. This was indeed observed by Mark III, reporting a signal at 1400 ± 6 MeV (4 =47± 13 MeV) [161] and also in pp8 annihilation at rest. Crystal Barrel observed the Á(1440) in the reactionpp 8 → (Á+−)00 and (Á00)+− [162]. Fig. 29 shows the two mass distributions containing together roughly 9000 Á(1440) decays. The average mass between the neutral and charged channels was found to be 1409 ± 3 MeV and the width 4 =86± 10 MeV. The quantum numbers were determined to be 0−+ and the observation of the Á00 decay mode proved that the Á(1440) was indeed an isoscalar. 100 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

x 103 x 103 η η (1440) 12 ’ 6

η’ 8 4

η Events / 20 MeV 4 (1440) Events / 20 MeV 2

0 0 800 1200 1600 800 1200 1600 (a) m (π0π0η) [MeV] (b) m (π+π-η) [MeV]

Fig. 29. Á00 mass distribution (left) and Á+− (right) inpp 8 → 20+−Á. The dashed line shows the result of the partial wave analysis (from Ref. [162]).

Table 10

The Á(1440) splits into two pseudoscalar mesons, ÁL and ÁH . The slightly di9erent masses for two decay modes of the ÁL are not considered to be signiÿcant

State Mass (MeV) Width (MeV) Decays

ÁL 1405 ± 556± 6 Á (a0(980) and ()S ) ÁL 1418 ± 258± 4 KK8 (a0(980) dominant) ∗ ÁH 1475 ± 581± 11 KK8 (K K8 dominant)

There is now evidence for the existence of two pseudoscalars in the Á(1440) region which are called ÁL and ÁH by the [3]. The ÁL around 1410 MeV decays into Á (through a0(980) or Á()S , where ()S is an S-wave dipion). The ÁH around 1480 MeV decays mainly to ∗ 8 8 K (892)K. In addition, the axial f1(1420) also contributes to the KK ÿnal state. The simultaneous observation of the two pseudoscalars ÁL and ÁH is reported with three production mechanisms [3]: peripheral −p reactions, radiative J= (1S) decay, andpp 8 annihilation at rest. All of them give values for the masses, widths and decay modes in reasonable agreement, with the exception of DM2 8 ∗ 8 which ÿnds the ÁL → KK above the ÁH → K K [163]. The 1400 MeV region is extremely complicated, due to the presence of both the K∗K8 thresh- 8 8 old at 1390 MeV and the a0(980) at the KK threshold in the KK channel. The average of all measurements, following Ref. [3], is given in Table 10. Systematic e9ects and model dependence are probably important. Therefore the error scaling factors between di9erent experiments analysing di9erent reactions are large, which is presumably the reason why the two states are not yet explicitly divided into Á(1410) and Á(1480) in Ref. [3]. However, the presence of a pseudoscalar doublet is highly suggestive. ++ Axial (1 ) states, such as the f1(1420) meson, are diOcult to observe inpp 8 annihilation at rest, because annihilation proceeds mainly through the atomic S-states which are dominantly populated in liquid hydrogen targets. However, these states can be observed using gaseous targets in which annihilation from atomic P-states is enhanced [164]. The Obelix Collaboration at LEAR has analysed C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 101

-+ π 0 a0(980) 0-+KK* 1++KK* 400

300

200

100 Events / 0.03GeV

0 1.4 1.5 1.4 1.5 1.4 1.5 m (KKπ) [GeV]

Fig. 30. Partial wave analysis of the KK8 mass distribution forpp 8 annihilation into K ±K 0∓+− showing the contri- −+ ∗ ++ ∗ butions from the 0 wave to a0(980) and K K8, and from the 1 wave to K K8 (adapted from Ref. [167]). the KK8 mass spectrum inpp 8 annihilation at rest into KK8 3 in liquid and in gas. They observe the ÁL and ÁH [165] but also the f1(1420) recoiling into an S-wave dipion [166–168]. The three resonating contributions in the partial wave analysis are shown in Fig. 30. One of the two pseudoscalars could be the ÿrst radial excitation of the Á, with the Á(1295) being the ÿrst radial excitation of the Á. Ideal mixing is suggested by the nearly equal masses of the Á(1295) and (1300) which then implies that the high mass isoscalar in the nonet is mainly ss8 ∗ 8 and hence couples to K K, in agreement with the ÁH . Assuming that ÁH is mainly ss8 and Á(1295) mainly nn8, one furthermore predicts from the mass formula that the mass of the strange member in the nonet would be about 1400 MeV, in agreement with the mass of the so-called K(1460) [3]. The mass of the latter is, however, poorly established. Finally, we note that the ÁH width is in accord 3 with expectation from the P0 model for the radially excited ss8 state [169,170]. ± ∓ The 2 -width of the ÁH was observed by L3 at LEP in the reaction → KS K  [171]. Fig. 31 ± ∓ shows the KS K  mass distribution for various transverse momenta. At small transverse momenta the photons are quasi-real and therefore the production of spin 1 states are forbidden by the Yang theorem. A high mass (but not low mass) pseudoscalar is observed. The ÁH is observed with a mass of 1481 ± 12 MeV and a width of 48 ± 9 MeV. At high transverse momentum the photons become virtual and the distribution is dominated by the axial f1(1285) and f1(1420) mesons (see Ref. [171]). The partial width for 2 production and decay into KK8 (212 ± 60 eV) is in agreement  with ÁH being the ÿrst radially excited state of the Á (958) [172]. The ÁL state therefore appears to be supernumerary. An exotic interpretation was proposed, perhaps gluonium mixed with qq8 [169] or possibly a bound state of [173]. Note that the ÁL is also not observed in → Á+− [171]. This, however, does not argue in favour of a gluonium nature + − for the ÁL since the Á(1295) is not seen either. Also, the radiative decay partial width into   width has been measured to be rather large for a gluonium candidate: crystal barrel reports a ratio of +− to Á+− widths of 0:111 ± 0:064 [174]. Finally, the gluonium interpretation is also not favoured by lattice gauge theories, which predict the 0−+ ground state glueball to lie above 2 GeV. However, a low mass pseudoscalar glueball is possible in gluonic Nuxtubes [175]. To summarize this section, there is strong evidence for the presence of two overlapping pseu- doscalar isoscalar mesons around 1400 MeV, separated in mass by about 50 MeV. One of them, 102 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

± ∓ Fig. 31. KS K  mass distribution in 2 -collisions for di9erent intervals in transverse momentum PT .AtlowPT the −+ ++ 0 ÁH is produced while at large PT the 1 f1(1285) and f1(1420) dominate (from Ref. [171]). probably the low mass state, is of a di9erent nature than qq8, but the two states overlap and hence are likely to mix. This situation is reminiscent to that of the scalar spectrum around 1500 MeV, here with the additional complication from axial vector mesons contributing to the same ÿnal states. The lower mass pseudoscalar state around 1405 MeV decays into KK8 and Á, but its precise decay 8 branching ratios into a0(980); Á()S and direct KK have not been established unambiguously. The higher mass state around 1480 MeV decays into K∗K8 . A comprehensive study of this compli- cated spectrum will require large statistical samples in J= radiative decays such as those expected from CESR running at the (2S).

5. Hybrid mesons

5.1. Theoretical predictions

According to the Nux tube model, hybrid mesons should lie in the 1:9 GeV region. Eight nearly mass degenerated nonets with quantum numbers J PC =0±∓; 1±∓; 2±∓ and 1±± have been predicted [176–178]. Lattice QCD also predicts the lightest hybrid, an exotic 1−+, at a mass of 1:9 ± 0:2 GeV [179,180]. However, the bag model predicts the four nonets 0−+; 1−−; 2−+ and the exotic 1−+ at a much lower mass, around 1:4 GeV [14,181]. Hybrids have distinctive decay patterns. They are expected to decay mainly into pairs of S- and P-wave mesons (for example the 1−+ state into C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 103

m1

π, K JPC L X

m Y 2

pp

Fig. 32. Peripheral p or Kp reaction leading to the production of a resonance X with quantum numbers J PC which in turn decays into two mesons m1 and m2.

f1(1285); b1(1235)), while the decay into two S-wave mesons is suppressed [182]. Most hybrids are rather broad but some can be as narrow as 100 MeV [183]. In contrast to glueballs, hybrids can have isospin 0 and 1. A state with quantum numbers 1−+ does not couple to qq8: for J PC =1−+ the angular momentum ‘ between the quark and the anti-quark must be even, since P = −(−1)‘. The positive C-parity then requires the total quark spin s to be zero, since C =(−1)‘+s. This then implies J = ‘ and therefore excludes J = 1. Likewise, it is easy to show that the quantum numbers 0−−; 0+− and 2+− do not couple to qq8 either. The discovery of a state with such quantum numbers would prove unambiguously the existence of exotic (non-qq8) mesons. We brieNy describe the nomenclature used in the framework of the isobar model for the partial wave analysis of peripheral p or Kp reactions of the type ab → Xc, where a is the incident  or PC K and X a resonance with quantum numbers J decaying into two mesons m1 and m2 (Fig. 32). Details can be found in the literature [184,185]. Since parity P is conserved in physics, so is the reNection

R = P exp(−iJy) : (40) The quantization axis z is chosen in the direction of the incident particle a, seen from the rest frame of the decaying resonance X . The y-axis around which the rotation is performed in Eq. (40)is chosen orthogonal to the plane spanned by z and the direction of the resonance decay daughters in the rest frame of X , This is the so-called Gottfried–Jackson reference frame. To avoid negative values of the spin projection M, it is convenient to express the eigenstates of R in the so-called reNectivity basis. They are |jM = M(M)(|M−jP(−1)J −M |−M) ; (41) √ where P is the parity of the resonance X; M =1= 2 or 1/2 for M¿0orM = 0, respectively. The reNectivity j is +1 for natural parity and −1 for unnatural parity exchanges of the meson Y . 1 For M = 0 the reNectivity j = P(−1)J IS excluded, since the eigenstate (41) vanishes. For peripheral reactions of the type shown in Fig. 32 and for incident spin zero like  or K the projection M is 0 or 1. If one now assumes that, say m1, is a meson resonance, one can

1 For natural parity exchange the meson Y has quantum numbers J P =0+; 1−; 2+, etc., while for unnatural parity exchange J P is 0−; 1+; 2−, etc. 104 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

PC j characterize the partial waves by their quantum numbers J [m1]LM , where L is the relative angular momentum between m1 and m2. A particular case arises when both m1 and m2 are pseudoscalar mesons, in which case J(X )=L and P =(−1)L. The eigenstates of R are then |jM = M(M)(|M−j(−1)M |−M) : (42)

Obviously, for M =0 j = +1 is excluded. The contributing partial waves Lj are then

P+;D+;F+; etc:; for M = 1 and natural parity exchange ;

S0;P0;D0; etc:; for M = 0 and unnatural parity exchange ;

P−;D−;F−; etc:; for M = 1 and unnatural parity exchange : (43) For example, for an exotic 1−+ meson decaying into Á (L = 1), the contributing partial waves are P+ for natural parity exchange, P0 and P− for unnatural parity exchanges.

−+ 5.2. A 1 exotic meson, the 1(1400)

The decay channel Á is a favourable one to search for 1−+ hybrid, in which case the two pseudoscalar mesons would be in a relative P-wave. The Á state would be isovector and hence could not be confused with a glueball. Both neutral and charged decays (Á0 and Á±) should be observed. PC −+ − The J =1 exotic meson decaying to Á , called 1(1400), was reported in the reaction −p → Á−p at 18:3 GeV=c by the E852 collaboration using the Multi-Particle Spectrometer (MPS) at the AGS [186,187]. A sketch of the MPS, a large angle magnetic spectrometer, is shown in Fig. 33. The Á was detected in its decay mode (47,235 events). Results for the Á → +−0 decay mode (2,235 events) are statistically less signiÿcant but consistent [186]. The 1(1400) was observed

Fig. 33. Sketch of the MPS spectrometer used by the E852 Collaboration at the AGS (after Ref. [187]). 1 – cylindrical drift chambers surrounding the 30 cm long hydrogen target; 2 – array of 198 CsI(Tl) crystals; 3 – multiwire proportional chambers; 4 – drift chambers; 5 – array of 3054 lead-glass crystals; 6 – lead-scintillator veto; 7 – beam veto scintillation counters. C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 105

Fig. 34. Acceptance corrected forward–backward for Á emission in the Gottfried–Jackson frame as a function of Á mass (after Ref. [187]).

− − − Fig. 35. D+ (a) and P+ (b) intensities for the Á system in  p → Á p at 18 GeV (after Ref. [187]). as an interference between the angular momentum L = 1 and L =2 Á amplitudes, leading to a forward/backward asymmetry in the Á angular distribution (Fig. 34). The (natural parity exchange) D+ and the exotic P+ intensities are shown in Fig. 35. The peak in the D+ amplitude is due to the a2(1320) meson, the peak in the P+ amplitude is due to the exotic 1(1400). The exotic intensity is a small fraction (about 3%) of the dominating a2(1320) contribution. There is an eightfold ambiguity in the ÿt (central error bars in Fig. 36), which was already noticed earlier in the partial wave analysis of this reaction [188,189]. Contributions from unnatural parity exchanges were found to be small. Fig. 36 shows the phase movement as a function of Á mass. The mass and width of the 1(1400) are given in Table 11. The 1(1400) state (called9 ˆ(1405) previously), was reported earlier by the GAMS collaboration in −p reactions at 100 GeV=c [190]. Mass and width are given in Table 11. However, the enhancement was observed in the (unnatural parity exchange) P0 wave. Ambiguous solutions in the partial wave analysis were pointed out in Refs. [188,189]. Clear enhancements in the P+ wave were also reported at 6:3 GeV=c [191] and 37 GeV=c [192], although the evidence for an actual resonance was not deemed to be conclusive. 106 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

Fig. 36. Phase movements of the D+ and P+ waves, as a function of Á mass; 1–D+ phase, 2–P+ phase, 3–relative phase (after Ref. [187]).

Table 11 Mass and width (in MeV) of 1−+ exotic mesons. The ÿrst error is statistical, the second represents the systematic uncertainties

State Reaction Mass (MeV) Width (MeV) Ref.

− − +50 +65 1(1400)  p → Á p 1370 ± 16−30 385 ± 40−105 [186,187] −p → Á0n 1406 ± 20 180 ± 20 [190] 0 − +71 pn8 → Á  1400 ± 28, 310−58 [195] pp8 → Á00 1360 ± 25, 220 ± 90 [196]

− 0 − +29 +150 1(1600)  p → 9  p 1593 ± 8−47 168 ± 20−12 [198,199] −p → 90−p 1620 ± 20 240 ± 50 [201] −  − +45  p → Á  p 1597 ± 10−10 340 ± 40 ± 50 [202] −  −  p → [b1(1235);Á;9] p 1560 ± 60 340 ± 50 [200]

On the other hand, an analysis of the reaction −p → Á0n at 18:3 GeV=c was performed [193]. The data were also collected by the E852 collaboration at the MPS. The Á and 0 were both reconstructed from their observed decays to 2 (45,000 events). An exotic P-wave similar to the one reported for −p [186,187] was found. However, the resonance behaviour was not compelling. The authors of Ref. [193] pointed out that the inclusion of the M = 0 and 1 contribution, i.e. unnatural parity exchange, did not lead to a consistent set of Breit–Wigner parameters. The crystal barrel collaboration at LEAR also searched for a 1−+ resonance in the Á P-wave in low energypp 8 annihilation into Á. Forpp 8 → 00Á (isospin I =0) and with stopping antiprotons, annihilation proceeds through the initialpp 8 atomic states. In liquid hydrogen and for the intermediate 0 1 3 1(1400) →  Á this is mainly S0 with some, presumably small contribution from P1, since density e9ects enhance S-wave annihilation. Forpn 8 → −0Á (isospin I = 1) and with stopping antiprotons 3 3 in liquid deuterium, the dominating initial states are S1 and P1. Here the spectator proton may remove angular momentum and hence the inclusion of initial P-waves becomes mandatory. C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 107

Fig. 37. Dalitz plot ofpn 8 → −0Á (after Ref. [195]).

A weak 1−+ structure with poorly deÿned mass and width was ÿrst reported inpp 8 → 00Á 1 3 [194]. However, the production of 1(1400) could be suppressed from S0 but enhanced from S1. The reactionpn 8 → −0Á was therefore studied in deuterium [195]. The increasing complexity in the amplitude analysis due to P-waves is compensated by the absence of 0++ isocalars which do not contribute to −0Á. Events were selected with a single − and 0Á → 4 . Spectator of less than 100 MeV=c were required, hence not escaping from the deuterium target. The channel −0Á could thus be treated as quasifree, thereby avoiding ÿnal state rescattering with the proton. The Dalitz plot (52,567 events) is shown in Fig. 37. The accumulation of events in the Á mass regions around 1300 MeV above the 9 band indicates the presence of interferences between a2(1320) and some other amplitude. The ÿt could not describe the observed interference pattern without the inclusion of a resonant Á P-wave. The accumulation of events above the 9 (visible in Fig. 37) also leads to a forward=backward asymmetry in the Á rest frame along the a2(1320) band. Mass and widths given in Table 11 are in good agreement with the results from E852 [186,187]. The − 0 3 contribution of 1(1400) to the   Á channel was 34% of the dominating S1 contribution, hence much larger than for the −p → Á−p reaction of Refs. [186,187]. The channelpp 8 → 00Á was studied again with annihilation in liquid hydrogen (280,000 events) but now using also annihilation in high pressure gas (270,000 events), in which the contribution of P-waves was enhanced [196]. Both data sample were ÿtted simultaneously, but the relative con- tribution from S- and P-waves was ÿxed by atomic cascade calculations [197]. The 1(1400) was 3 1 observed dominantly from the P1 atomic state with a small contribution from S0 which explained the weak signal reported earlier in liquid [194]. The mass and width are given in Table 11. It appears 3 3 that 1(1400) is mostly produced frompp 8 spin triplet states ( S1 or P1). 108 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

+ + 1 [ρ ] P 0 ,1 1 [ρ ] P 1+ 4000 (a) (b) 1500

1000 2000 Intensity 500

0 0 1.0 1.5 2.0 1.0 1.5 2.0 Mass [GeV]

Fig. 38. Exotic 1−+ contribution to 9 in −p → −−+p at 18 GeV for (a) unnatural (M = 0 and 1) and (b) natural (M = 1) parity exchanges. The dark histogram shows the background contribution (after Ref. [199]).

3

2 [rad]

ϕ 1 1 2

0 1.5 1.6 1.7 1.8 Mass [GeV]

−+ −+ Fig. 39. Phase of the natural parity exchange 1 9(770) wave (curve 1) and 2 f2(1270) wave (curve2, after Ref. [199]).

−+ 5.3. Another 1 exotic meson, the 1(1600)

−+ Another 1 state, 1(1600), was reported to decay into 9 [198,199]. It was observed by the E852 collaboration in the peripheral reaction −p → −−+p at 18:3 GeV=c. Contaminating reac- tions involving excited (e.g. ++ → p0) could be removed with the arrays of CsI(Tl) and lead-glass calorimeters vetoing s from 0-decay. The partial wave analysis was based on 250,000 reconstructed events. Apart from the known mesons (f2(1270);a2(1320), 2(1670), etc.) a resonat- ing partial wave was found in the exotic waves 1−+[9(770)]P0− and 1−+[9(770)]P1− (unnatural parity exchange) and 1−+[9(770)]P1+ (natural parity exchange). This is shown in Fig. 38. Accord- ingly, this resonance was named 1(1600). No statement on the 1(1400) → 9 could be made below 1500 MeV due to leakage from other partial waves. A rapid phase movement was observed for the M j =1+ wave with respect to all other signif- icant natural parity exchange waves. Fig. 39 shows for example the 1−+ [9(770)]P1+ phase mo- −+ + tion, resonating around 1600 MeV, and that of the 2 [f2(1270)]S0 , resonating at the 2(1670). C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 109

No statement could be made on a phase advance of the unnatural parity contribution to the formation −+ ofa1 resonance since contributions from unnatural parity exchanges (f1 or b1) were small for all partial waves. The mass and width of the 1(1600) meson are given in Table 11. The VES collaboration at IHEP studied the same reaction −A → −−+A but at 36:6 GeV=c, using a beryllium target and a large aperture magnetic spectrometer [200]. Preliminary results on this channel were reported earlier by VES [201]. In the −90 channel they reported a broad shoulder around 1:6 GeV for the (natural parity) 1−+[9(770)]P1+ wave while the contributions from unnatural parity exchanges remained negligible. The mass and width from Ref. [201] are given in Table 11. However, the author of Ref. [200] warns that the intensity and width of the 1:6 GeV enhancement is quite sensitive to details of the partial wave analysis. Also, the 1−+ contribution is small, about 2% to the total intensity. Note that the 3 system is rather complicated with 27 contributing partial waves in the simplest ÿts [199]. A much simpler channel is the one for which 1(1600) decays into two pseudoscalar mesons. The  −  − 1(1600) decaying into Á  was reported by the E852 collaboration in the reaction  p → Á  p at 18:3 GeV=c [202]. The Á was reconstructed through its decay mode Á → Á+− with Á → . The photons were detected in the lead-glass array. A sample of 6,040 Á−p events were collected. The main contributing amplitudes were the (natural parity) P+;D+ and G+ waves. Fig. 40 shows the exotic P+ intensity and D+ contributions. The former intensity is dominant in the 1500–1800 MeV range, reaching a maximum around 1600 MeV, while the latter peaks at the a2(1320). There is a weak indication of the 1(1400) in the P+ intensity, although the ÿt does not require it. Table 11  also gives the reported mass and width of 1(1600) in the Á  decay mode. The VES collaboration observed earlier a broad enhancement in the exotic 1−+ Á wave at  37 GeV [192] but mass and width were not given. Fig. 41 shows the P+ wave for Á  and Á  (contributions from P− and P0 are negligible). The Á  contribution exceeds Á in the 1600 MeV region, although phase space favours the latter. This would favour hybrids over q2q82 states [203]in this mass region. A signal for 1(1600) → b1(1235) was also reported by VES [200] and a combined ÿt to  the b1(1235), Á  and 9(770) data was performed [200]. Mass and width of the 1(1600) are compatible with the results of E852 (Table 11). Furthermore, the 1(1600) decay branching ratios to  the three ÿnal states b1(1235); Á  and 9(770) are of compatible strength, 1: 1:0 ± 0:3:1:5 ± 0:5. The experimental errors are quite large, and the predicted dominance of b1(1235) for hybrid states cannot be excluded.

1500

P+ D+ 1000

500 Events / 0.05GeV

0 1.5 2.0 2.5 1.5 2.0 2.5 M (η'π ) [GeV]

−  − Fig. 40. Intensity of the P+ (left) and D+ (right) partial waves in the reaction  p → Á  p (after Ref. [202]). 110 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

6 η'π ηπ

2 4 |T|

2

0 1.00 1.25 1.50 1.75 2.00 Mass [GeV]

− − −  − Fig. 41. Intensity of the P+ wave in  A → Á A and  A → Á  A at 37 GeV (after Ref. [192]).

Evidence for the 1(1600) decaying to b1(1235) is also reported frompp 8 annihilation at rest into !+−0 [132]. −+ Summarizing this part, we now have evidence for two 1 exotics, 1(1400) and 1(1600) from peripheral reactions and antiproton annihilation. However, Á and Á rescattering e9ects appear to be large and a non-resonant interpretation for the 1−+ wave has been suggested [204]. This a9ects −+ especially 1(1400). The 1 signals are rather small in peripheral reactions and an imperfect description of the experimental acceptance, or of the dominating a2(1320) meson, could mimic a resonance at 1400 MeV. However, the signal for 1(1400) is rather strong inpn 8 andpp 8 annihilation and directly visible in the Dalitz plots. In Ref. [205] it is suggested that a Deck generated Á background from ÿnal state rescattering in 1(1600) decay could mimick 1(1400). However, this mechanism is absent inpp 8 annihilation. The Á data require 1(1400) and cannot accommodate a state at 1600 MeV [206]. Hence antiproton annihilation data argue for the existence of 1(1400). As isovectors, 1(1400) and 1(1600) cannot be glueballs. The coupling to Á of the former points to a four-quark state while the strong Á coupling of the latter is favored for hybrid states [207,208]. As mentioned already, the Nux tube model and lattice calculations concur to predict a −+ mass of about 1:9 GeV. The 1(1600) mass is not far below these predictions. Note that a 1 structure around 2 GeV decaying to f1(1285) was reported by one experiment [209].

5.3.1. Other hybrid candidates Hybrid candidates with quantum numbers 0−+; 1−−, and 2−+ have also been reported. The (1800) decays mostly to a pair of S- and P-wave mesons [210]), in line with expectations for a 0−+ hybrid meson, although recent data contradict this, indicating a strong 9! decay mode [211]. This meson is also rather narrow if interpreted as the second radial excitation of the pion. The evidence for 1−− hybrids required in e+e− annihilation and in . decays was discussed in −+ Ref. [212]. A candidate for the 2 hybrid, the Á2(1870), was reported in interactions [213], inpp 8 annihilation [214] and in central production [215]. The near degeneracy of Á2(1645) and −+ 2(1670) suggests ideal mixing in the 2 qq8 nonet and hence the second isoscalar should be mainly ss8. Data for K∗K8 decay are unfortunately not available for such high masses. However, Á2(1870) decays into a2(1320) and f2(1270)Á with a relative rate of 4:1 ± 2:3[214]or20:4 ± 6:6 [216]. These large numbers are compatible with a predicted ratio of 6 for a hybrid state [178]. C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 111

6. Conclusions and outlook

The light scalars below 1 GeV have for long been considered as good candidates for non-qq8 states. The measured radiative decay widths to  and Á from DA NE [55,54] are not compatible with f0(980) and a0(980) being qq8 states [44]. The f0(600) could belong to the same Navour nonet, since the large di9erence in width compared to f0(980) and a0(980) might be due to the wide open phase space for the former, and to the KK8 threshold distortions for the latter states. There are theoretical arguments in favour of a light and broad Ä(800) pole near the K threshold. However, the experimental evidence is not conclusive. A more detailed analysis of experimental data using the K-matrix formalism [125] rather than Breit–Wigner amplitudes will be required. This remark also applies to the f0(600) and, in general, to all overlapping broad resonances. In a recent topical review [72], it was suggested that the lightest scalars are at the central core composed of a four quarks. Following Ja9e’s QCD arguments this central core would consist pre- dominantly of a four-quark (qq)38(8qq8)3 state. At larger distances from the core the four quarks would then recombine to a pair of colour singlet qq8’s, building two pseudoscalar mesons as a meson cloud at the periphery. Better experiments with adequate theoretical treatments of the f0(600) pole and the likely Ä(800) pole in D and Ds decays are important. A better understanding of the nature of the recently discovered ∗ very narrow DsJ (2317) by BABAR [4] and the DsJ (2460) by CLEO [93], which lie slightly below the ÿrst allowed strong S-wave (DK, respectively D∗K) threshold, is likely to throw new light, also on the light scalar sector. 3 There is now strong experimental evidence that the lightest qq8 scalar nonet (1 P0) consists of ∗ the mesons a0(1450);K0 (1430);f0(1370) and f0(1710). From hadronic reactions the f0(1710) appears to be made dominantly of ss8 quarks [105]. The nonet mixing angle is not far from ideal mixing [143]. However, the decay branching ratios of the f0(1710) to two pseudoscalar mesons have been measured by one experiment only [119,133]. They should be checked, e.g. with the COMPASS experiment at CERN. Furthermore, the strongest decay channels, presumably into 4 and KK8 , have not been observed yet. Branching fractions are important to (i) determine the nonet mixing angle and (ii) to establish the partial width of the f0(1710). 8 The KK and the upper limit for the partial widths of the f0(1500) are not compatible with a qq8 state [143]. The absence of signal in suggests that f0(1500) contains a large fraction of glue. Mixing with qq8 is likely [13]. On the other hand, data in collision are statistically limited. Much better data in → KK8 and  are required to pin down the fraction of glue in the wave function. The a0(1450) was observed so far inpp 8 annihilation only, presumably because the production branching ratio is rather small. Experimental data [127,129] argue for a high mass a0(1450) and against a low mass state around 1300 MeV. The a0(1450) should also be observed in collisions, 8 in particular decaying to KS KS . However, the branching ratio for a0(1450) → KK is not known. An upper limit of 33% can be deduced from SU(3), in accord with data [129]. Data from LEP 8 (see Fig. 27) are consistent with a0(1450) for a branching ratio to KK of about 10% or less. Thus, a search for many body decays of the a0(1450) e.g. to Á3 or 5 would be useful. In fact, the decay branching ratio into !9 seems rather large [132]. These channels are diOcult but data in collision to Á would be very useful, since the Á decay branching ratio of a0(1450) is comparable to that for KK8 [129]. 112 C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117

The f0(1500) should be observed in J= radiative decay, which is traditionally believed to enhance gluonium production. A scalar state is indeed observed in this region [121,218] but the data are statistically limited and do not allow a proper treatment with the K-matrix formalism. Large statistical samples will hopefully become available with the commissioning of CLEO-III at Cornell. For a complete theoretical treatment of the light scalars one would need to include the qq8 states, the possible four-quark states, and a glueball as seed states, and treat all Navour related states si- multaneously in order to constrain the parameters. Nonperturbative e9ects from the strongly coupled two-pseudoscalar channels would be included through unitarization, in a way consistent with at least unitarity, analyticity, Navour and chiral symmetry. They would mix the bare resonance states (seeds) with themselves and with the meson–meson continuum and distort any naive mass spectrum and tree-level resonance widths. This would be clearly an ambitious program. Quoting Ja9e [217]: “It would be wonderful to have a uniÿed quark/hadron description of scattering with (i) open (un- conÿned), (ii) closed (kinematically forbidden), and (iii) conÿned channels”. Such a program should not be too diOcult to realize with modern computers. In the tensor sector, the identiÿcation of the 2++ glueball is premature. The ÿrst radially excited 3 nonet (2 P2) is not established, although candidates exist (see Fig. 1). A high statistics systematic investigation akin to the one performed inpp 8 annihilation at rest [40] is called for in high energy −p or K−p interactions between 1500 and 2300 MeV, e.g. with the COMPASS experiment at CERN. The nature of the f2(1565) is unclear. It could be a 99 + !! molecule (see Table 2). However, it is observed only inpp 8 annihilation [3] and as such could be a deeply bound nucleon– antinucleon state [11]. Here also, good data in radiative J= decays will hopefully settle the issue on the existence and quantum numbers of a narrow state around 2200 MeV. The nature of the Á(1440) is unclear. In the eighties, this state observed in the gluon rich radiative J= decay process was considered a prime candidate for the ground state scalar glueball. However, lattice gauge theories later predicted the 0−+ glueball to lie around 2:5 GeV (see Fig. 21). On the experimental side little progress was made due to the lack of good statistics data, but the experimental evidence now points to the existence of two pseudoscalar states in this mass region, one around 1410 MeV, the other around 1480 MeV [3,167]. This situation is likely to improve with the commissioning of CLEO-III. The high mass state in the Á(1440) mass region decays mainly to K∗K8 and is hence consistent with being the radially excited ss8 pseudoscalar state. On the other hand, the ÿrst radially excited 1 pseudoscalar nonet (2 S0) is not well established (see Fig. 1). The other isoscalar, the Á(1295) was reported so far only in peripheral −p reactions [3]. It is not observed inpp 8 annihilation, in contrast to Á(1440). Also the K(1460) → K is poorly established. This calls for new attempts to investigate the strange meson sector. The Á(1440) structure is furthermore complicated by the presence of an axial vector, the f1(1420). Whether this state is the ss8 state of the 1++ nonet or a KK8 ∗ molecular state (see Table 2) is unclear. In the latter case the elusive f1(1510) [9] could be the ss8 state. Experiments are called for in which both the f1(1420) and the f1(1510) are observed simultaneously.  In the hybrid sector, the exotic 1(1600) seems well established. Good data for the channel Á  in pd8 orpp 8 would be very valuable. The 1(1400) is strongly observed in antiproton annihilations but its resonant nature is currently being debated. On the other hand, according to theoretical predictions, 1−+ hybrids should be observed at a mass of about 1:9 GeV [176,179]. The issue of 1−+ exotics will be addressed further in photoproduction at CEBAF. C. Amsler, N.A. Tornqvist / Physics Reports 389 (2004) 61–117 113

In the non-exotic sector the nature of the Á2(1870) is unclear. Its decay fractions into a2(1320) −+ and f2(1270)Á are consistent with expectations for a 2 hybrid. However, it could be the ss8 member of the 2−+ nonet in which case a strong signal should appear in the KK∗ decay channel. On the other hand, the mass region above 1500 MeV is very complicated due to the overlapping of broad resonances and the opening of two-body thresholds. Charmed hybrids should be easier to identify since predictions for the qq8 charmonium spectrum are believed to be quite reliable. Furthermore, charmed hybrids are expected to be narrow if they lie below the DD1(2420) threshold, since decay into a pair of S-wave mesons is suppressed. For instance, in the bag model one expects the sequence m(0++) ¡m(1−+) ¡m(1−−) ¡m(2−+)[219]. However, complications due to threshold e9ects and surprises can be expected. Charmed hybrids will be investigated at the planned GSI facility.

Acknowledgements

N.A.T. acknowledges partial support from the EU grant HPRN-CT-2002-00311 (Eurodice). References

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