-
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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DISCLAIMER
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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 / ’ : $q’)/(phasespace) = 3 : 4 :
I : O : I . ( : I )
Of conrse this simple pattern should a t least incorporat,e t .
l i e l j ’ I f iwii i phase space for an S-wave decay, and there
is in addition a decay forll1 l’ i l(’ t(D~’
286 287
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which depends on tlie unknowri scalar gliietxill wavefunction
and I.lie decay niecIianisIii. Experience wit,h the 31$-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 /
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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
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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
-
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