Theoretical Predictions for Exotic Hadrons T. Barnes

Computational and Theoretical Physics Group, Oak Ridge National Laboratory Oak Ridge, TN 37831-6373, USA

and E Q.

Department of Physics and Astronomy, University of Tennessee Knozville, TN 37996-1200, USA

Abstract. In this contribution we discuss current theoretical ex- pectations for the properties of light meson “exotica”, which are meson resonances outside the qf quark model. Specifically we discuss expecta- tions for gluonic hadrons (giueballs and hybrids) and muitiquark systems (molecules). Experimental candidates for these states are summarized, and the relevance of a TCF to these studies is stressed.

I. INTRODUCTION

The most exciting developments in QCD spectroscopy involve searches for resonances which are external to the conventional qp quark model of mesons. There are two general classes of such states, which are those with dominant gluonic excitations “gluonic hadrons” and states with more quarks and anti- quarks than the familiar qq states.

Since QCD is a theory which contains both quarks and gluons as dynamical degrees of freedom, we would expect to see evidence of both these building blocks in the spectrum of physical color-singlet hadrons. It is remarkable, however, that of the hundreds of hadronic states now known, most can be described as states made only of quarks and antiquarks in the nonrelativistic quark model, and none of the remaining problematic resonances have been established as having dominant gluonic valence components. The best evidence for the presence of gluons at low energies is indirect, for example in the Breit- Fermi one-gluon-exchange Hamiltonian used in potential models and in the qq t-) si$ configuration mixing evident in the q and q‘.

In addition to these gluonic states, one may also form color singlet combina- tions from multiquark systems of quarks and antiquarks, beginning with q2q2.

0 1996 American Institute of Physics 255

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Although these have been quite controversial, it now appears that light multi- quark resonances do exist i n nature, albeit as bound meson pairs “iriolecnles” rather than single four-quark clusters.

Experimental studies now in progress may alter the statns of hadronic exotica considerably, since there are now several resonances that, i f confirmed, appear to be likely candidates for gluehalls, hybrids and additional molecnles. As we shall see, these states share several common features with theoretical ex pec t,a t, ions for these unusn al hadronic states .

In this contribution we will review current theoretical expectations for glri- oriic hadrons and molecules, and briefly discuss some of the experimental can- didates for these states.

11. GLUEBALLS

A. Introduction

A priori one would expect glueballs to be the most attracbive glnonic hadrons experirnentally, since they might be expected to differ most notice- ably frorn qq. In practice this riaive expectation may not be realized; sttidies of the ligtti glueball spectrum using lattice gauge theory have found that the lowart.lyl~ig gluobrll Is (I mcnlsr, rrtd Its coirt~lltig to twa.pa~utltsncdbr nanl states suggests a typical hadronic width. The next glueballs encountered a t higher masses are predicted to he 0-+ and 2++, and states which couple to two transverse gluons (presumably the lightest glueballs) do not contain exotic J‘C .

Although there have been many studies of the spectrum and quantum num- hers expected for glueballs [ I ] , the results of lattice gauge theory should be treated as the most relevant to experiment, since they bear the closest resern- blance to full QCD. The assumptions of quenclied lattice gauge theory are that decay channels do not modify glueball masses significantly (since the neglect of quarks implies stable light glueballs) and that the extrapolations to small lattice spacing and large lattice volume do not introduce important biases. If glueballs are not very broad objects, the assumption of stable gluehalls should not introduce large mass errors.

There are lattice predictions for the masses of glueballs with varions J p c [2]; the most reliable is presumably for the scalar glueball ground state, which is predicted to have a mass of

1.550(50) CeV [3] hI(Ott) = ( 1.740(71) GeV [4] .

‘Hie corresponding mass estimate for the tensor glueballs is i n the 2.2-2.4 GeV range,

++ - 2.270(100) C C V I:$] 2.359(128) GeV (41 ; h!(2 ) - {

with the pseudoscalar glueball at a similar mass. There are obvious problems associated with tlie itlrntification of i~ S(.;I~;II.

state near 1.5 GeV. The fo sector is the most complicated of all nlesorl S( .C~,OI .~ . with a t least six problematical states, f0(980), f0(1300), f ~ ( 1368), j’,l( I5Oo) f ~ ( 1590) and fo(1710). Since this sector contains br0a.d and overlappirig ~ C S O . nances, the problem of identifying unusual st,ates against tlic: 91‘ and s s I ) ; I ( . ~ ground, and the related problems of separating individual resonailws KI,OIII interference arid threshold effects are dauiiting ones. If tlir scalar g111d~;Il! does have a typical hadronic width, as suggested by the work of Sext.oi~ ct u / . [5], it niay be quite difficult to identify t,liis state convincingly. A ~ n s l ~ r i 1 1 1 1 I Close [6] note that the near degeneracy of the pnrr ( q i ~ r i i d i d ) I , ( ; ‘ ] glild);l!t and the L = 1 qq and si multiplets may lead to complicated rnixingclfrc.t.s. S(I the physical states may be nontrivial combinations i n flavor space, as i l l I t i ( , q - ~ ‘ sector.

The tensor glueball may be an easitrr experirnrntal targcl., s i i w 1 . 1 ~ . 1’4 pected mass i s fat above the lowest-lying 2++ qua.rkoniurn st.at.rs. I I t w I I t ( < prohlern i s that the mass region above 2 GeV is poorly explored, so it is I I O I yet, possl\lla to cflst,lirgitlali (I tChOOr glucbnll frorn l l io Iinrk~rnctcirl ( J f , n c I i r t I 3P2 and 3F2 qq and sB states. This lack of adequate infornial.ion regardiiig the higher mass qnarkonium spectrum is even more of a probleni i n tlic. ( I - ’ sector.

B. Expectations for glueball properties

Since we have no confirnied glueballs and tlie states prc4ictcd arc- i i i cli i i i i nels with a complicated or poorly explored resonance spect,rriin, it. rvociltl I N . useful to have reliable theoretical predictiona of glueball propertice as $1 giiiilt.. The dsta we &re likely to have on gluoriic catcclidal,eo i n t.he n c ~ r fiitiirv t i n ’ their masses, widths and strong decay nmplitiides. Ilere a. very ~ l ~ i l r a ( ~ t , f * ~ i s t i,. naive glueball signature can be given, althongli it is easy to iniaginc: ways i i i which this signature might be violated.

As gluons at the bare lagrangian level have equa.1 sf.rength coiiplirigs t I ) quarks of all flavors, one can make the assumption t,hat flavor-syinnirt.ric mii- plings to hadron final states are approximately valid for physical gli ic4)aIl~. This gives a characteristic fla.vor-singIet, branching fraction to ~ ~ ~ ~ ? u t l o s c . a I i ~ ~ ~ pairs, which is (neglecting phase space differences)

r ( G -+ a*: ICrC : 9’1: i / i / ’ : $q’)/(phasespace) = 3 : 4 : I : O : I . ( : I )

Of conrse this simple pattern should a t least incorporat,e t . l i e l j ’ I f iwii i phase space for an S-wave decay, and there is in addition a decay forll1 l’ i l(’ t(D~’

286 287

which depends on tlie unknowri scalar gliietxill wavefunction and I.lie decay niecIianisIii. Experience wit,h the 31$-niodeI f O ( 9 9 ) clrcay itniplit i i d o K K , wliicli lias a node near tlie physical point [7], srigges1.s that. the naivr patkrii o f flavor-singlet decay aniplitucles may indeed be far froin the physical couplirigs.

'I'Jie accuracy of naive flavor-singlet couplings can he tested for a pure (qi1ericlied) scalar glueball in lattice gauge theory tlirough a tlrteririinat.ion of (.he glrieball-1's-Ps three point function. Preliminary results for this coupling [5) indicate that Havor-singlet syrnrnetry may indwd he I)a.tlly violated a t the ainplitude level, antl higher-mass Ps pairs are preferr(:d iri the ( h a y . I n view of the relatively large errors it is importarrt to improve tlit. statistics of this interesting lattice gauge theory nieasurement.. A n extrnsion of I.liis work to t.tie ttccay amplitudes of tensor 'and pseudoscalar gIiichaIIs woritd also I,e a very useful contribution.

111 future experirnerital work it may he possil~lc to clcterrriinc or liiiiit elrc- trornagrictic couplings of glriehall calididates. Mcasiireriwnts of one-plioton ( I 2 -+ yqq) and two-photon ( R + 77) transition rates o f t.\irse tesonacices A W cktrctrwly irriportnnt hscrtinc tlirorirt,n can rdc\iIatc (.lima for q i nt.nt,c>n with reasonably accuracy 181. The radiative transition rn.t.es of a. rclat,ivcly pure gluehall would clearly he anomalous relative to cxl)(:ct.ations for the cor- rrspoiiding f ~ ( q q ) state. If physical gliteballs are intlced strorigly mixed linear cor~il)i~iations of gluoriic. 9q and s.? basis states, a convincirig wily to identify tlre flavor corriponents of these mixed states would be tlirough a couiparison of the relative rates

1yn 3 ypo : yw : yo) since these act as flavor tags. Similarly, yy coiiplirigs can he usetl I,o locat,e the scalar rionstrarige fi) qq signal, since this stntk shorlltl have a, strong corlplitlg to yy. Itcsiilts on this reaction have already hreti obtained by tl iv Crystal 13all ill ~ , \ i e reactiorl yy -+ W ' K ~ [SI. Siric:c: a ~ I I I ( ~ I I sliolild liavci s~ippr~ss(*(I coriplings to yy, iiieasurt~tncnts of the yy wriplings of thc: various .f., sI,a.t.c:s arid other light resonances would be very important contributions to light meson spectroscopy a t a TCF.

C . Summary of glueball candidates

At present the t.wo most proniincnt experinicnlal candidates for gliiebal\s a r c the scalar f o ( 1500) antl tlia ((2230), which is prohahly a t.ensor. 'I'ha scalar candidate has a mass arid width (as reported by Crystal Uarrel [ I O ] ) of

M ( J o ) = 1520 t20 MeV -55 (4)

r ( f o ) = 148 M e V . - 25

?'he fo(1500) seeriis rather too massive to be a nonstrange '1:) 94 $ta t ( . , t ) l l r 15 consistent with the lower mass estimates from LG'r for a scalar glllt+alI. '1'11(* widtli is also quite narrow for a 3P0 qq state a t this mass. ' h e &cay I",I I ( , I I I ( 0 pseudoscalar pairs is however inconsistent with Havor syrrirlietry; tlie S C ~ I I C I I invariant couplings cited by Arnsler [lo] are

r(f0(t500) -+ ?T?T : ftR : vll : v f ) l ) / ( p . s . ) =

1 : < 1/8.6(95%c.I.) : 0.24f0.12 : 0.05f0.15 A priori this argues against a pure glueball iIit~ri)r~!l,i~tioti, aiitl S I I ~ ) W < ~ I I ( ~ I I I work by Arnsler and Close [ G ] 1ia.s invcstigatcd the possil)ility t l l i i t t.l~c-sc* ( I ( % - cays may be consistent with a scalar glueball that has iinport,al1t pq ~ I I I ~ .e.4 COInpOnt3iltM, kllding Lo an titf motla arid bl ipprcaai l~g thc f i f i IIIO&!. 'i'110 l i l t l i t on the coupling to K K is actually inferred fro111 aiiotlier cxprrinicllt, ;lll(~ ;I more careful study of this coupling including iriterferenccs a.t tile (1ryst;tl 1ti1r- re1 appears to find a much larger K R coupling [ l I] . This state has also I ~ ~ Y ~ I I reported in a recent reanalysis of the Mark111 data on $ -+ y r + t r - ~ + s - I)y Bugg et a[ . [12]; in this channel the j o ( 1500) appears doniiiiantly i i i t,lic "rrn" mode of two S-wave *a pairs.

ported by Mark111 [13) i n $1 radiative decays, is rcyort,ctl hy I%ICS [ 1 . 1 1 to I I ; I \ x - very anomalous properties for a tensor above 2 GeV. 'l'lie inass and wit1I.h 131,:s cite for this state in /

such as 1(1(1270)1( are large, so r(fi(sS)) 2 400 MeV. Similarly for the 3F4 Ulundell and Godfrey now find a broader state given these additional modes, II(f4(sii)) 2 130 MeV. Thus the sB assignments now appear implausible if the ((2230) does indeed have an experimental width of < 50 MeV.

Several of the properties reported for this narrow ((2230) are disturbing. I t has surprisingly small branching fractions to pseudoscalar pairs in view of the available phase space 1141; branching fractions of only a few percent are implied by the PS185 limit on P P --+ 6 3 KI?. A more important concern is that t he rrported statistical significance in each of the four channels studied by BES is rather sniall, EJ 3a. A caution,ie appropriate because some previously reported narrow effects were subsequently found to be artifacts (for example the ((8.3)). In view of the remarkable properties reported for this state, measurement of tlicse channels with higher statistics is an extreniely important task for any e+e- facility operating a t the 1c, mass.

Although we have only discussed the fo(1500) and t(2230) glueball can- didates, this is largely because they have attracted considerable attention re- cently. Several other states with similar masses and the same quantum num- bers, notably the fo( 1710), should also be considered glueball candidates (51. Measurements of strong branching fractions and electromagnetic decays of this and other glueball candidates should be considered high priorities a t a TCF.

1

I

111. HYBRIDS

A. Introduction

Hybrid mesons may be defined as resonances in which the dominant valence basis state is qq combined with a gluonic excitation. Hybrids are attractive experimentally because, unlike glueballs, they span complete flavor nonets and hence provide many possibilities for experimental detection. In addition, the lightest hybrid multiplet is expected to include at lead one JPC-exotic (forbidden t o q j ) . In the bag model, for example, the lightest gluon mode has *Jp = 1+, so the lowest-lying q j g multiplet contains the quantum numbers

0-t 1 - t 2 - t , , (Sqq = (PQS) = { 1-- (Sqq = 0) . J PCn (9)

The flux tube model extends this bag niodel list by adding a degenerate set with reversed ( P , C) to the lowest hybrid multiplet. Constituent gluon models differ in that their lowest hybrid multiplet has P-wave q j quantum numbers [ 171 and so is nonexotic, although exotics appear i n excited hybrid multiplets. A I I investigatiorl of qijg interpolating fields [l8] shows that hybrids can have any . I " .

290

B. Hybr id masses.

Hybrids have been studied using a wide range of models and trctlnic~rlc~s:. These are the MIT bag model [19], constituent gluon models [17,20,21], t , \ \ ( x flux tube model [22-313, an adiabatic heavy-quark bag model [32], heavy-qrlarli lattice gauge theory [33] and QCD sum rules [34-381. There have been 1 1 0 published Monte Carlo lattice gauge theory studies of hybrid masses; a strldy of exotic hybrid masses would be an interesting application of this tecliniclrlct. In all the theoretical approaches employed to date the lightest hybrids ( I : , , involving u, d flavors) are predicted to have masses in the M 1 i-2 CeV rcgiou. A lrummary of hybrid maas predictions for the especially interretin5 I - + exoi i(. is given in the table below, taken from (28). A more detailed diacnssion o f these predictions plnd the literature on hybrids is given by Barnes, Close aut1 Swanson [28]; for other recent reviews of hybrids see [39].

Much of the recent interest in hybrids has derived from the flux tube ~ n o t k l , which gives rather precise predictions for masses and decay modes of hyhritls. The original flux tube references [23-251 cited masses of x 1.9 GeV for 1l1c lightest (u,d) hybrid multiplet, NN 4.3 GeV for ce hybrids and x 10.8 (h\' for b6 hybrids. There is an overall variation of about 0.2-0.3 GrV i n t l i c w predictions, as indicated in Table I. Multiplet splittings are usually ricglccld in the flux tube model. This approximation rnay not be justified; a large, inverted spin-orbit term was found for hybrids by Merlin and Paton [%I.

TABLE I. Predicted 1-+ Hybrid Masses. state mass (GeV) model Her. Hu.d 1.3-1.8 bag model (191

122 '~5.2~1 1.8-2.0 flux tube model 2.1-2.5 QCD sum rules (most after 1984) [X 371 2.1 constituent gluon model 1211

H, = 3.9 adiabatic bag model [:I21 [2:I- 25,'LXI 4.1-4.5 flux tube model

4.1-5.3 QCD sum rules (most after 1984) [35 - -3 i ] 4.19(3) f sys. HQLGT [33]

Hb 10.49(20) adiabatic bag model I321 10.8- 11.1 flux tube model 123 251

10.81(3) f sys. HQLGT [:n] 10.6-1 1.2 QCD sum rules (most after 1984) 135 371

29 1

A recent Hamiltonian Monte Carlo study (281 of the flux tube model de- termined hybrid masses without using the questionable approximations of the earlier flux tube model studies, such as an adiabatic separation of quark and flux-tube motion and a small oscillation approximation for the flux tube. This Monte Carlo study generally confirmed the accuracy of the earlier flux-tube model mass estimates, both for qQ and c? mesons (compared to experiment) and for hybrids (compared to the earlier approximate analytical calculations). These flux tube predictions are shown in Fig.1 below for light quarks and in Fig.2 in the discussion of charmonium hybrids.

3.0 1 I

[0.63]

0.0

Fig.1. The light qqbar (q=u,d) and hybrid spectrum in the flux tube model 1281.

By varying the model parameters over a plausible range, this study con- cluded that the lightest hybrid masses in the flux tube model were

M ( Hu,d) = 1.8 - 1.9 GeV (10)

for light quark hybrids and

& f ( H c ) = 4.1 - 4.2 GeV (11)

for charmonium hybrids. Excited hybrids were also considered, and the first hybrid orbital excitation ( A L = I D ) was found at about 2.3 GeV, 400 MeV

above the lightrst. ( 1 f’) I1yl)rids. The sanie n1itticricaI rc*siilt. w i t s Iotili(I c b ; i r I i ( x i . I)y Merlin [26] using tltc: atliii.hatic approximatioli. ‘IIiis 11) iiiiilt.ipI(*t c.oIiI,;iiiis the J p c states (1,2,3)** and 2**, which includes t.l~e fxotics I - + , %+- i l l l ( l 3-+. One way to test the experimental caiiditlates for groitntl-stilt.r I iyI ) i . i , Iy near 1.8 GeV [do] arid 1.6-2.2 GeV [ d l ] woiilcl be to smrcll for I I I V I I I I ) ( W 01 t Iiis excited ID hybrid niultiplct about 0.4 GeV h i g h i n IttiLss.

C. Light hybrid decay modes.

r . I Iteorrtical tnodrls predict. ra.tlier cliaractc~ristic two- I)otly clpci~y I I I O ( I ~ ~ S [()I. 1iyl)rkIs. 13otli const.itrient gliiori 1201 and f lux till)(: [2i] r~~otIc*ls fitici I I I ; I I , t I I ( ’

I meson “S+l”’, for exartiplc: *SI arid 7~61. ‘J’liosc iiiirisrial I I I O ~ ( ! S l ) I ( - \~ i~~~isI , , received little experiment at at teri t ion because they involv(’ cot I 1 plic:aI.cv I I i 1 1 a I states, which may explain why hybrids were not discovered previously. ‘ I ’ \ I ( S flux-tiibe decay predictions of Isgur, Kokoski a d I’aton (271 arc cliiittr i i i t ( % 1 . - estiiig because t.hey suggest that, tnaiiy Iiyl)ritls arr so ImacI t.Iiat, I , I I C ~ ~ wiII c4rectivrly invisible, whercw ii. fvw I i y h i t l s slioriltl IW t i : ~ I r o ~ ( - 1 i o i 1 ~ 1 1 to ( \ i t s iIy observable i n certain ctianriels. The I = I = I - + vxot ic. IiatI ; t ~ ~ . ( , i i ( ~ ! . been cited as an attractive experimental candidatc, arid this work sriggc-st ( , ( I that this state shorild be relatively narrow, l’tot M 200 MrV, i l l l t l t.11i11. t,li(% S+P modes ~ 6 1 and rf1 should be tlie dominant final st.il1.c~~. ‘I’ltcw slii(Ii(*s have motivated sevcml experimrntal invcstigations of 7r6, arid .JI, wliicli s l i o w possible indications of resonant amplitudes i n I -+.

These original flux tube tlecay calcrilatioiis w ( w ror ~ I i c . t1irc.c. (w)t ic. .I”“ qitantum numbers in the lowest flux-tube iniiltiplct,. S i n w I.liis i i i i i l t ip l c t , W I I tains a total of eight Jpc assignments, I** (for S,, = O ) and y * T ; I*T; ~ f l f (for Sqq = I ) , one might wonder whcxt.lier any of t l ic : tiotir:rolic Iiyhritls ; I I X > narrow enough to be observed. ‘t‘he decay arnplitutlcs of thcsr tioiicxot.ic 11s- britls were recently calculated by Close and Page [29], who also clirckr(l t I I V exotic decay amplitudes and fourid reasonable numerical agreernent with Isgiii., Kokoski and I’aton.

Close and Page predict t,liat many of tliesc noncxotic Iiybritls iirv iilso so broad as to be effectively unobservable. ’There are two striking excrpt,ioiis. One is a 1-- w-hybrid with a total width of only E 100 MeV, wliicli t l t w y s 1 0 KI( 127O)fi arid K1( I4OO)K; this should he searched for in lit I< final st , i l l~ (~s . perhaps in photoproduction. A second intcrcsting nonc.xot.ic Ityl)titl is ;I 7r). with rtOt w 170 MeV. This may he tlie Iiigli-mass sl,al.c: wl t id i liiis I ) c ~ v i I v ~ i o i ~ t ~ ~ I in several photoproduction experiments a mass ncar 1775 MrV [40]. 0 1 l i ( % i . notalde conclusions are that I ) several other hybrids, including cxxot.ic-s, llitv(% total widths near 300 MeV a.nd so should be observable, and 2) tlie I = 0 O+

lightest Iiylirids decay preferentially to pairs of ono f,,,=O ntttl O I I ( >

292 293

exotic found by Isgur et a/. to have rblr = 250 MeV actually has very large Kt K modes and so should be unobservable.

In addition Close and Page investigate the “forbidden” decay modes such as j(1900) 3 pn, and find that, due to differences in the p and A spatial wavefunctions, these S+S modes are present with partial widths of typically - 10 MeV. An important pn coupling was found earlier by deViron and Go- vaerts [35] using QCD sum rules. Thus it is interesting to search relatively straightforward modes such as P A for hybrids, in addition to the favored but more difficult S+P modes such as b l r , ~ f 1 and K l K .

D. Prospects for charmonium hybrids at a TCF.

The predictions of the recent flux tube model calculations ( [28], shown be- low) and heavy-quark LGT [33] that hybrid charmonium states should appear beginning at 4.1-4.2 GeV are especially relevant for the physics program of a Tkk42hhtm Fkebary,

5.5 1 4

5.0 - M (GeV) :

4.5 -

4.0 -

3.5 -

3.0 -

4.46

P - - - - - - - - - - 4.21 D - - - - - - - - - -

1

1 F 3.99 D 3.77

P [3.52]

S [3.07] 4

Fig.2. Charmonium and ccbar-hybrid masses in the flux tube model [28].

Charmonium spectroscopy is rather well understood up to about 3.8 GeV, so searches for unusual states should be straightforward near this mass. Since only a few open charm channels occur below 4.3 GeV, for a considerable range of masses one might anticipate rather narrow hybrid resonances. This pos-

sibility is supported by the theoretical prefcrencc of hybrids for S-t 1’ modes, which have thresholds of about 4.3 GeV for c? and 11.0 ( ~ c V IO, /,/) Calculations of the decay widths of charmonium hybrids have been carric(I OIII in the flux tube model by Close and Page [31], assuming niasses of zz I . I. 1.:’ GeV. The partial widths (to D*D) are found to be quite stiiall, tyl)icaIIy O I I ~ ! - 1 - 10 MeV. Thus if there are relatively unmixed charinoniurri hybritls, t 1 1 ( - 1-- vector hybrids should appear as narrow spikes in R in this Inass r;il lR(’. For this reason a detailed scan of R starting near the open charrn threslloIlI would be a first priority at a Tau-Charm Factory.

Close and Page subsequently speculate about a more coniplicatccl possiI) i I ity, which is that the $(4040) and $(4160) may be equal-weight l inear C O I I I binations of 3 s ICE) and 1-- ci:-hybrid basis states. (The usual assigniiic.iil i< that the G(4040) is a 3s CC? and the $(4160) is a 2L) cc (431.) The Close-I’;~g~~ linear combinations would explain why the e+e- widths are a.pproxiniat (4) equal and relatively large for both states, which is surprising if otie is a / I wave c t . The assignments for the II, states above open-charrn thresholds C ~ I I I be teeted by nieawrenianta et t,lieir bratictiing tracbloiia to D L J , L)*f>, ..., /I: />:. The branching fractions predicted by these models are very scwsit,ivt, to I 1 1 8 . initial state assignments [42]; unfortunately thcy have not yet. I)crn i i i ( ~ ; i s i i i accurately. Determination of these branching fractions would be aiiotht~r I i i r : l l priority a t a TCF.

Finally, we note that the non-vector hybrids can also Iw protl i ic:c~tl i l l ; I T C F through a “continuum cascade”, as suggested by 1).13iigg, and tlisciisscvl in references 143,441. In this approach one produces a high-mass ci: syst.riii iii the continuum, for example a t 5 GeV; this may tlien decay hatlroriically to hybrid charrnonium levels of various J P G acrornpanied hy it light I i i i ( l r o i i oi Iiadrons. The cc-hybrid ill t u rn decays hadronically to a cliaract.rtist,ic st ill ( , such as the $. Thus one can search for example for the decay chain

e’e- -+ c? -+ H , q ; H , -+ q$; d, -+ e’e-

in the final state v t le+e- , triggering on a lepton pair at. t,lie d, iiiass aid 3~ I m i i S from the t,wo 7s. The q$ invariant mass distribution can then be st,iitlic.tl 101 evidence of hybrids or c? states. Other quantum iiuinbers can IIC. iiivcst.igiil by replacing by other hadrons, for example ( m r ) ~ , in the hadronic caSci\(I(’\

E. Hybrid Experimental Candidates

There are several experimental candidates for Iiybrids, but just as for gliii. balls there are no generally accepted states at present.

In the exotic channels (which would provide the most convincing rvi t l r i i r 1 for hybrids), previous claims by GAMS that a resonant signal had 1)rvii ( 1 1 .

294 295

tcctc:tl i n tlie 1-+ wave of nq [45] have now been withdrawn. A K E K experi- inent (461 finds evidence for a resonant 1-+ ai] wave, but with I.lie ma.ss a.nd widt.li of the nz( 1320); this surprising result obviously must be checked care- fully for “feedthrough” of the a2 amplitude. VES (471 has studied r ~ ] arid r p f and report a broad, higher-mass effect in rq and especially in rtf, near 1.6 CeV. The phase motion of the 1-+ component ha.s not yet been determined. Studies of the nfl final state suggested by the flux tube model are underway [41,47], and preliminary evidence for a possible 1-+ signal has been reported by E818 a t BNL (411.

There have been several observations of a photoproduced I = 1 state i n p r and r j 2 a t about 1775 MeV [40], which is too heavy to be the r2( 1670) without coniplicated interference effects. Although the qrraiitriiri titimbers of this state have not been determined definitively, 1-+ is preferred over 2-+. A possible narrow 1-+ state has heen reported by GAMS i n pq’ a t a mass of 1910 MeV [48]; here there are rather few events, so it will be important to improve the statistics. Several experiments plan future studies of these channels, iiicluding E818 (tm atudy r-J1 ) [4S] ~ n d E852 ( to study nfi mid aq) [GO] Ht IINL.

I n addition to exotic hybrids there are several noriexotic candidates; recall for example the Close-Page result that a hybrid with ~2 quantum numbers is expected to be relatively narrow, and should be visible i n a.fz. One way t,o tlistinguish hybrids from q9 spin-singlet states is through their strong decay ani1)Iitudes; for example, in the T Z sector tlie relative F / P a n d D / S arnplittrtle ratios in ~ ~ ( 9 4 ) -+ p r and x f2 are reasonably well coiistrainctl i n the ‘1’0 aii t l fl i ix tube decay models [SI]. ‘l’liese decay niodels provide a.n interesting svlcction rule for 9q decays; t,liey forbid the decay of a spin-singlet qq state to two final spin-singlet quarkonia,

( v i ) s = o f , (rlil)s=o t (rlCl)s=o

1 1 1 1 . 1 1 ~ ~2 chaniicl this selcctiori rule forI)itIs tlic decay of a IDz 9q T plus a ‘1’1 61,

to a ‘So

712(94) f t K b l

I)ut allows it for a hybrid n2 wliicli does not ha.ve the 99 pair i n a.ri S’ = 0 configuration. Close and Page find the rbl mode of a T Z Iiybritl sliorilcl be rat her large, so it is especially important to search the r b l channel for evitlcrice of a 2-+ signal.

0 t h nonexotic hybrid candidates which have been siiggcst.rcl recently are a X( 1800) report.ed by VES [52] antl the nonstrange I-- stat,es near 1.1-1.7 GeV [MI. ’The T ( 1800) is cited as a possible hybrid because it has unusual branch- irig fractions. including a significant couplirig tlo T~,v/, apparently tliroctglr I,he glric4)all candidate .fo( 1500) --+ ?pi. This ?I( 1800) is also reported by VI% i n wp,

~]no(980), rfu(980) and rfo(1300) . l‘lie decay mode n( 1800) -+ p r is MI i ~ l ~ l , ~ absent, and n fi is also weak or absent.

brid, a a( 1800) second radial excitation is expected in quark potrtiI.iaI I I I O ( I ( ~ I S (Godfrey and lsgur [54] predict 1.88 GeV), so one stlorlltl corlsitler this i 1 ~ - signrnent as well. Radial quarkonia can have unusual branching fractions titi(, to nodes in their decay amplitudes, a.nd in tlie 31:, decay niotlrl w i t . I i Si[() wavefunctions the amplitude for 743s) -+ pn has a iiotle a t hf = 1.88 (;(,\’ for p = 0.35 GeV. Tlie weakness of tlie pr rriotle is I.h(~ref(,rc~ r ~ ~ ~ t l c ~ t ~ t . n ~ ~ t l i ~ l ~ l ~ ~ for a 3s state. Tlie same niodel however predict,s a weak rfo( 1 : ) ~ ) l l i o ~ l ( ~ . which disagrees wi1.h experiment. ‘I’lle decay atripl i tr i t lcr for n(3.Y) -+ r p ( ~ , < ) is predicted to be quite large [55 ] , so a search for a *p ( 1450) final stilt(, W O I I I ( I be useful.

The unusual properties of the nonstrange I = 0 and I = I V C ~ I . O ~ S 1 1 ( b i l i . 1.5 GeV have led to suggestions that hybrid vetbor st,atea rnay Iw p r c ~ i i t near this mass (53,561. In I = 1, for cxattiplc, 1111: two s t . a k s p ( l , i . ~ ) iiii(l ~ ( 1 7 0 0 ) are iiaudly nsaigned to ‘Z3S1 and ‘Dl rcspcct,ivc*ly, 1 ~ 1 1 . t h v c ~ y I;trgr. p ( 1450) + 2 ( r + x - ) mode [SS] is in conflict with q u a r k 1iiotlc4 cxpvct iit.ioiis for a 23Sl state 196-581. A better understanding of t.liese vcct,or st.atc,s iiiii! require a detailed isobar analysis of their quasi t.wo-l)otly st.rong tl(’cii.y i i i o ~ l ( ~ s .

These comparisons of strong decay modes illiistratc the iii~port~aiiw of Ita\. ing an accurate understanding of the tlccays of iiidially cxcitctl qij st.iitcx ( ‘iii.i< f i l l studies of the strong decays o f radially cxcitctl 99 cniitlitlatc.s siic.li iis I 111. r(1300), p(1450), 4(1680), ~ ( 1 8 0 0 ) and so forbh will rcyiiirc~l i f W P a i ~ ’ to distinguish qq from non-9q states with identical quarlturil n ~ i i ~ i b c ~ r s .

Although the weakness of the p r S+S mode is indcetl suggestive ol‘ i l 11)

IV. MULTIQUARK SYSTEMS A N D MOLECULES

A. Introduction

Multiqriark systems have liad a coinplicat.r(l his(.ory, i l l i t1 ( . t irt(- i i t . I Iiwivt ii.iiI expectations for these states now differ radically fronl I , I I C rarlic,st. siiggc*st i o i i s . I n the pre-QCD quark model era it was thought, t1ia.t. niult,iqiiavk I i i i t l i ~ o ~ ~ s should exist as resonances i n the liaclron spcct,rirni. A h 1 . 1 1 ~ clisc0vc.i.y ( 1 1 QCD antl confinement it was still widcly cxprct.etl tliiit iiiiilt.i(liiarl< I i i i O i ~ i ~ i i ~ should exist (in color singlet scvdors), and rnotlrls tyl)ic.aIly ptwlictcd ii v r i ~ ~ ~ I i i .0 spectrum of states. I n the light 9’q2 scctor these “I)aryonitini” rc~soi i~ i i ic~~s \v i ’ i i. expected t,o appear beginning at ahout 1 G c V . It . was clcv~r l i o w ( ~ v w I . l i i 1 t t111, i . i . were problems wibli these predict.ions, l)rcaust, i i i I.lie icl i i t, ivvly i i i i (~(i i i i l ) l i t . ; i I i . 1 1 flavor-exotic channels srdi as I = 2 J r c = o++ no q2q2 r o s o i i i i i i ( ~ s i v i ~ 1 ’ observed [,59] whereas t.hey w w e prctlictctl t,o I)(- rc4ii1,ivc4y liglil (E I .2 (;(,\ i n the M I T bag model). Siniihrly, the cvitlciicc for ( I i l n i i iOOi i l i y l ) ~ * r i i i i ( ~ l ( ~ i [ ( i ( l l

296 297

makes the existence of an H six-quark resonance well below A A threshold (another bag model prediction) appear very unlikely.

The problem with these predictions of multiquark resonances such as q2q2 was that they were above (qq)(qq) thresholds, and could spontaneously disso- ciate “fall-apart” into two mesons [61]. Thus the mass predictions in models which assumed a priori that the q2q2 system existed as a single hadron were spurious, because the physical eigenstates were usually a continuum of scatter- ing states [62]. Whether single multiquark clusters exist as resonances under any conditions is a detailed dynamical question, which should be investigated using models that allow the system itself freedom to choose between a sin- gle cluster or separate color singlets. At present it appears that single q2q2 hadronic clusters may only exist m resonances in heavy-light syateme such M c2qa [SS].

More realistic models of multiquark systems were subsequently developed which gave the q2q2 system freedom to choose dynamically between a bound system and a two-meson scattering state. The variational calculations of We- instein and Isgur [64] are the best known of these studies; in this work it was found that most O+ sectors of the light qZq2 system had two free mesons as the ground state, but that the I = 0 and I = 1 qs@ sectors actually had a weakly bound, deuteronlike Itl? pair as the ground state. These states were obvious assignments for the problematical fo(980) and ~ ( 9 8 0 ) resonances, which were difficult to explain as 3Po qq states but could easily be understood as Kl? systems with nuclear binding energies of 10s of MeV. These states have been the “prototypes” for hadron molecules, although they remain somewhat con- troversial. We note in passing that molecule states as a general category are not a t all controversial, since the > lo4 known nuclear levels are all examples of hadronic molecules. Here we will discuss meson molecules; candidates dso exist in baryon nectora, for example the A(l406), which may be a /(N bound system [as].

Signatures for the a priori most likely molecular states [66] can be ab- stracted from our experience with short-ranged hadronic forces and the Weinstein-Isgur results:

1 ) J p c and flavor quantum numbers of an L=U hadron pair.

2 ) A binding energy of at most about 50 - 100 M e V . 3 ) Strong couplings to constituent channels.

4 ) Anomalous EM couplings relative to expectations for conventional quark model states.

B. Experimental molecule candidates

1 ) fo(975) and a0(980): The lLItk-molecules”.

Weinstein and Isgur [64] found an exception to the Call-apatt p l i c t i ~ t i i ~ ~ i ~ ~ ~ ~ ~ in the scalar sector, with parameters corresponding to the qsqs system. 1 1 m . weakly-bound deuteronlike states of kaon and antikaoti were found to Iw I I N . ground states of the four-quark system; Weinstein and Isgur refer to t.lic.sc. ;IS “KK molecules”. The scalars fo(975) and ao(980) were obvious candidates I’oI these states, having masses just below Kl? threshold and strong coiiplitigs I O strange final states. Subsequently the 77 couplings of the f0(975) arid no(!)80) were found to be anomalously small relative to expectations for tight ‘1;) states ( q = u,d ) , as discussed in references 167,681. The status of the J i ‘ K molecule assignment and the many points of evidence in its favor have Iw(*ii discussed recently by Weinstein and Isgur 169,701.

Morgan and Pennington have argued against a molecule interpretatioii 0 1 the fo(975) 1711. Their criticism however applies to a Itl? potential iiio(I(~1 in which the fo(975) is a single pole in the scattering amplitude. ’ l l i e I I I ( J I ( . recent work of Weinstein and Isgur [69,70] incorporates couplings to i i w s o i i meson channels and heavier 3P0 qq states, so the physical resonances arc. 1 1 0 1 only IItR). Since there has been much criticism of the idea of a. piirc / < I \ bound state, a direct quote from Weinstein and Isgcir (691 (regarding tlic I =.. 0 state) is appropriate:

“Despite its name and location, the “II’K molecde” 1s not n s m p k bound state, Its stability is dependent on its coupliitgs to the otltrt. I =; ( 1 channels and at E 3 Mp ihe coupled-channel wauejunctton has ~ ~ 6 d i o t t i 1 1 I compondnte of thd olkcr Itatea.”

Although the j o and a0 states remain dominantly Itk, tliese rnodilicat.ioiis may answer the objections of Morgan and Pennington. I’ennington siiggcst s that the term “deuteronlike” may be a misnomer, if couplings to other s t . i . t v s than Kl? play an important r6le in these states [67]. Thus it a.ppea.rs that. t l i c . important question regarding the fo and a0 may be one of detail, sp(x:ificiill\. how large the subdominant non-Itl? components are in these states a.nd l ion. they can be observed experimentally.

The experimental measurements which would be most iisefril for stit(1ic.s of these states at a TCF are 1) their 77 widths, which are as yet rat.licv poorly known, and 2) their cross sections in hadronic decays, in $ -+ L I , / ~ , and dfo. (The latter are flavor-tagging and in studies a t BES have s11ow11 that the f0(980) does appear to be a mixed flavor state.) Other intercst i r i l : measurements at low energies are the radiative transitions --+ rf,l and 3 1 ) ~ ~ ~

298 299

which drpend strongly oii the scalar assignment 1721 and may be measured a t DXI'IINE [i3] and CEUAI: (74).

2 ) J,(1420)

Siiice the fl( 1420) is above t.he K*/i threshold of 1390 MeV it. is acantlitlate for a nonresonant threshold enhancement, ( I i*k+/i .c . ) rather than a iiiolecular \)ound state. '!his possibilit,y was suggested by Chltlwcll 1751, and satisfies the criteria of lying jnst above the A'*K thresholtl (alltiparticle lahels are implicit,) and having qnantum numbers allowed for that pair i n S-wave. The apparent width of the enliancement should not be narrower than the intrinsic width of the I

M.Teper, Phys. Lett. R152, 107 (1985); G.Berg, Nucl. Phys. B221, 109 (1983); a.nd references cited therein.

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[5] .I. Sexton et nl. , I R M report IIIM-HET-94-5 (contribiition to 1,attice 94). (61 C.Anisler and F.E.Close, “Evidence for Glueballs”, Itntlierford Lahoratory and

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302 I 303

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304 ! 305

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I861 T.Barnes and E.S.Swanson, Phys. Rev. D46,131(1992); for closely related work on meson-meson scattering see B.Masud, J.Paton, A.M.Green and G.Q.Liu, N u d . Phys. A528,477 (1991); D.Blaschke and G.Ropke, Phys. Lett. B299,332 (1993); K. Martins, D. Blaschke and E. Quack, Phys. Rev. C51,2723 (1995). A.LeYaouanc, L.Oliver, 0.PCne and J.-C.Raynal, Phys. Rev. D42,3123 (1990).

(1992).

[87] E.S.Swanson, Ann. Phys. (NY) 220, 73 (1992). [88] T.Barnes, E.S.Swanson and J.Weinstein, Phys. Rev. D46,4868 (1992). (891 T.Barnes, S.Capstick, M.D.Kovarik and E.S.Swanson, Phys. Rev. C48, 539

[go] T.Barnes and E.S.Swanson, in preparation. (1993).

306