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1
9
0
Scalar and Pseudoscalar Glueballs
0
2
Hai-YangCheng∗
n
a Institute of Physics, Academia Sinica,
J Taipei, Taiwan 115, ROC
∗E-mail: [email protected]
7
] Weemploytwosimpleandrobustresultstoconstrainthemixingmatrixoftheisosinglet
h
p scalar mesons f0(1710), f0(1500), f0(1370): one is the approximate SU(3) symmetry
empirically observed in the scalar sector above 1 GeV and confirmed by lattice QCD,
-
p andtheotheristhescalarglueballmassat1710MeVinthequenchedapproximation.In
e theSU(3)symmetrylimit,f0(1500)becomesapureSU(3)octetandisdegeneratewith
h a0(1450),whilef0(1370)ismainlyanSU(3)singletwithaslightmixingwiththescalar
[ glueball whichistheprimarycomponent off0(1710). Thesefeatures remainessentially
unchangedevenwhenSU(3)breakingistakenintoaccount.Theobservedenhancement
1
ofωf0(1710)productionoverφf0(1710)inhadronicJ/ψdecaysandthecopiousf0(1710)
v
productioninradiativeJ/ψdecayslendfurthersupporttotheprominentglueballnature
1 off0(1710). Wededuce themassofthepseudoscalar glueball Gfromanη-η′-Gmixing
4
formalism based on the anomalous Ward identity for transition matrix elements. With
7
the inputs from the recent KLOE experiment, we find a solution for the pseudoscalar
0
glueballmassaround(1.4±0.1)GeV,whichisfairlyinsensitivetoarangeofinputswith
.
1 orwithoutOkubo-Zweig-Iizuka-ruleviolatingeffects.Thisaffirmsthatη(1405),havinga
0 largeproductionrateintheradiativeJ/ψ decayandnotseeninγγ reactions,isindeed
9 aleadingcandidateforthepseudoscalarglueball.Itismuchlowerthantheresultsfrom
0 quenched latticeQCD(>2.0GeV)duetothedynamicfermioneffect.Itisthusurgent
: tohaveafullQCDlatticecalculationofpseudoscalarglueballmasses.
v
i
X
1. Introduction
r
a
TheexistenceofglueballsisanarchetypalpredictionofQCDasa confiningtheory.
Despite a great deal of experimental and theoretical efforts in the last 3 decades,
there is still no any compelling evidence for their existence. Surely, there are good
glueball candidates, such as f (1710) and η(1405) which are seen copioursly in
0
J/ψ radiative decays and yet not observed in γγ reactions.1 However, definitive
identification has been plagued by the complication that the branching ratios of
these candidates in the radiative decay of J/ψ, once thought to be the defining
channel to probe the glue-rich content of the mesons, are not orders of magnitude
large than the other known qq¯ mesons, that the glueball can mix with ordinary
mesons, and the fact that the statistics in experiments are neither high enough to
confidently detect all the major decay channels nor precise enough to disentangle
fromthenear-bystatessuchasf (1790)andη(1475).Inthiscase,thesupplemental
0
information on the quark content (or rather the lack of) via the γγ coupling and
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leptonicdecaysprovetobeimportanttorevealthenatureoftheglueballcandidates.
2. Scalar glueball
It is generally believed that the scalar glueball is hidden itself somewhere in the
isosinglet scalar mesons with masses above 1 GeV. The argument goes as follows.
Many scalar mesons with masses lower than 2 GeV have been observed and they
can be classified into two nonets: one nonet with mass below or close to 1 GeV,
such as σ, κ, f (980) and a (980) that are generally believed to be composed of
0 0
fourquarksandtheothernonetwithmassabove1GeVsuchasK∗(1430),a (1450)
0 0
and two isosinglet scalar mesons. This means that not all three isosinglet scalars
f (1710),f (1500),f (1370)can be accommodatedin the qq¯nonet picture. One of
0 0 0
them should be primarily a scalar glueball.
Amongtheisosingletscalarmesonsf (1710),f (1500)andf (1370),ithasbeen
0 0 0
quite controversialas to which of these is the dominant scalar glueball. It has been
suggested that f (1500) is primarily a scalar glueball,2 due partly to the fact that
0
f (1500), discovered in pp¯annihilation at LEAR, has decays to ηη and ηη′ which
0
are relatively large compared to that of ππ3 and that the earlier quenched lattice
calculations4 predictthescalarglueballmasstobearound1550MeV.Furthermore,
becauseofthesmallproductionofππ inf (1710)decaycomparedtothatofKK¯,it
0
has been thought that f (1710) is primarily ss¯dominated. In contrast, the smaller
0
productionrateofKK¯ relativetoππ inf (1370)decayleadstotheconjecturethat
0
f (1370) is governedby the non-strange light quark content.
0
Basedontheaboveobservations,aflavor-mixingschemeisproposed2toconsider
the glueball and qq¯ mixing in the neutral scalar mesons f (1710), f (1500) and
0 0
f (1370). Best χ2 fits to the measured scalar meson masses and their branching
0
ratios of strong decays have been performed in several references by Amsler, Close
and Kirk,2 Close andZhao,5 andHe et al.6 A typical mixing matrix in this scheme
is5
f (1370) 0.91 0.07 0.40 N
0
− − | i
f (1500) = 0.41 0.35 0.84 S .
0
− − | i
f (1710) 0.09 0.93 0.36 G
0
| i
Acommonfeatureoftheseanalysesisthat,beforemixing,thess¯massM islarger
S
than the glueball mass M which, in turn, is larger than the N( (uu¯+dd¯)/√2)
G
≡
mass M , with M close to 1500 MeV and M M of the order of 200 300
N G S N
− ∼
MeV.
Other scenarios also have been proposed. For example, based on their lattice
calculations of the quenched scalar glueball mass, Lee and Weingarten7,8 consid-
ered a mixing scheme where f (1500) is an almost pure ss¯ meson and f (1710)
0 0
and f (1370) are primarily the glueball and uu¯+dd¯meson respectively, but with
0
substantial mixing between the two ( 25% for the small component). With the
∼
effective chiral Lagrangian approach, Giacosa et al.9 performed a fit to the experi-
mentalmassesanddecaywidths off (1710),f (1500)andf (1370)andfoundfour
0 0 0
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possiblesolutions,dependingonwhetherthedirectdecayoftheglueballcomponent
is considered.In spit ofdifferent opinions in the communityonthe identification of
the scalar glueball, Particle Data Group10 tried to conclude the status as “Exper-
imental evidence is mounting that f (1500) has considerable affinity for glue and
0
thatthef (1370)andf (1710)havelargeuu¯+dd¯andss¯components,respectively”.
0 0
However, there are at least four serious problems with the above scenario: (i)
The isovector scalar meson a (1450) is now confirmed to be the qq¯ meson in the
0
lattice calculation.11 As such, the degeneracyof a (1450)andK∗(1430),whichhas
0 0
a strange quark, cannot be explained if M is larger than M by 250 MeV.
S N
∼
(ii) The most recent quenched lattice calculation with improved action and lat-
tice spacings extrapolated to the continuum favors a larger scalar glueball mass
close to 1700 MeV.12,13 (iii) If f (1710) is dominated by the ss¯content, the decay
0
J/ψ φf (1710) is expected to have a rate larger than that of J/ψ ωf (1710).
0 0
→ →
Experimentally, it is other way around: the rate for ωf (1710) production is about
0
6 times that of J/ψ φf (1710). (iv) It is well known that the radiative de-
0
→
cay J/ψ γf is an ideal place to test the glueball content of f . If f (1500)
0 0 0
→
has the largest scalar glueball component, one expects the Γ(J/ψ γf (1500))
0
→
decay rate to be substantially larger than that of Γ(J/ψ γf (1710)). Again,
0
→
experimentally, the opposite is true. Simply based on the above experimental ob-
servations, one will naively expect that γ α > β in the wave function of
≫
f (1710) =αN +β S +γ G .
0
| i | i | i | i
In our recent work,14 we have employed two simple and robust results as the
inputforthemassmatrixwhichisessentiallythestartingpointforthemixingmodel
betweenscalarmesonsandtheglueball.Firstofall,weknowempiricallythatflavor
SU(3)isanapproximatesymmetryinthescalarmesonsectorabove1GeV.Thenear
degeneracy of K∗(1430), a (1470), and f (1500) has been observed. In the scalar
0 0 0
charmed meson sector, D∗ (2317) and D∗(2308) have similar masses even though
s0 0
the former contains a strange quark. It is most likely that the same phenomenon
also holds in the scalar bottom meson sector. This feature is also confirmed by the
quenchedlattice calculationofthe massesforthe isovectorscalarmesona andthe
0
axial-vectormesona .11 Itis foundthat, when the quark massis smallerthan that
1
ofthestrange,a massisalmostindependentofthequarkmass,incontrasttothose
0
of a and other hadrons that have been calculated on the lattice (see Fig. 1). This
1
explainsthefactthatK∗(1430)isbasicallydegeneratewitha (1450)despitehaving
0 0
one strange quark. This unusual behavior is not understood as far as we know and
it serves as a challenge to the existing hadronic models. In any case, these lattice
resultshint atanSU(3)symmetry in the scalarmesonsector.Second,animproved
quenched lattice calculation of the glueball spectrum at the infinite volume and
continuum limits based on much larger and finer lattices have been carried out.12
The mass of the scalar glueball is calculated to be m(0++)=1710 50 80 MeV.
± ±
This suggests that M should be close to 1700 MeV rather than 1500 MeV from
G
the earlier lattice calculations.4
We shall use U , D , S to denote the quarkonium states uu¯ , dd¯ and ss¯ ,
| i | i | i | i | i | i
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2
1.8
At strange mass
GeV) 1.6
Mass ( 1.4 a1 :
a0 :
1.2 a1(1223m0π) ::
a0(1450) :
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
mπ2 (GeV2)
Fig.1. Latticecalculationsofa0 anda1 massesasafunctionofm2π [6].
and G todenotethepurescalarglueballstate.Inthisbasis,themassmatrixreads
| i
M 0 0 0 x x x y
U s
0 M 0 0 x x x y
M= D + s , (1)
0 0 M 0 x x x y
S s s ss s
0 0 0 M y y y 0
G s
where the parameter x denotes the mixing between different qq¯ states through
quark-antiquark annihilation and y stands for the glueball-quarkonia mixing
strength. Possible SU(3) breaking effects are characterized by the subscripts “s”
and“ss”.Asnoticedinpassing,latticecalculations11 ofthea (1450)andK∗(1430)
0 0
masses indicate a good SU(3) symmetry for the scalar meson sector above 1 GeV.
This means that M should be close to M and M . Also the glueball mass M
S U D G
should be close to the scalar glueball mass 1710 50 80 MeV from the lattice
± ±
QCD calculation in the pure gauge sector.12
WebeginbyconsideringexactSU(3)symmetryasafirstapproximation,namely,
M =M =M =M and x =x =x and y =y. In this case, two of the mass
S U D s ss s
eigenstatesareidentifiedwitha (1450)andf (1500)whicharedegeneratewiththe
0 0
mass M. Taking M to be the experimental mass of 1474 19 MeV,1 it is a good
±
approximation for the mass of f (1500) at 1507 5 MeV.1 Thus, in the limit of
0
±
exact SU(3) symmetry, f (1500) is the SU(3) isosinglet octet state f and is
0 octet
| i
degeneratewitha (1450).Inthe absenceofglueball-quarkoniummixing,i.e.y =0,
0
f (1370)becomes apure SU(3) singlet f andf (1710)the pure glueball G .
0 singlet 0
| i | i
Thef (1370)massisgivenbym =M+3x.Takingtheexperimentalf (1370)
0 f0(1370) 0
mass to be 1370 MeV, x is found to be 33 MeV. When the glueball-quarkonium
−
mixing y is turned on, there will be some mixing between the glueball and the
SU(3)-singlet qq¯. If y has the same magnitude as x, the mass shift of f (1370)and
0
f (1710) due to mixing is only of order 10 MeV, a feature confirmed by the lattice
0
calculation.8
As discussed before, SU(3) symmetry leads naturally to the near degeneracy of
a (1450), K∗(1430) and f (1500).However,in order to accommodate the observed
0 0 0
branchingratiosofstrongdecays,SU(3)symmetrymustbebrokentocertaindegree
inthemassmatrixand/orinthedecayamplitudes.OnealsoneedsM >M =M
S U D
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in order to lift the degeneracy of a (1450) and f (1500). Since the SU(3) breaking
0 0
effect is expected to be weak, they will be treated perturbatively.
Iff (1710)isprimarilyapseudoscalarglueball,itisnaivelyexpectedthatΓ(G
0
→
ππ)/Γ(G KK¯) 0.9 after phase space correction due to the flavor independent
→ ≈
couplingofGtoPP.However,experimentallythereisarelativelylargesuppression
of ππ production relative to KK¯ in f (1710) decay. To explain the large disparity
0
between ππ and KK¯ production in scalar glueball decays, Chanowitz15 advocated
thatapurescalarglueballcannotdecayintoquark-antiquarkinthechirallimit,i.e.
A(G qq¯) m . Since the current strange quark mass is an order of magnitude
q
→ ∝
largerthanm andm ,decaytoKK¯ islargelyfavoredoverππ.Furthermore,ithas
u d
been pointed out that chiral suppression will manifest itself at the hadron level.16
To this end, it has been suggested16 that m should be interpreted as the scale of
q
chiral symmetry breaking since chiral symmetry is broken not only by finite quark
masses but is also broken spontaneously. Consequently, chiral suppression for the
ratio Γ(G ππ)/Γ(G KK¯) is not so strong as the current quark mass ratio
→ →
m /m .
u s
Guided by the lattice calculations for chiralsuppression in G PP,17 we have
→
performed a best χ2 fit to the measured masses and branching ratios. The mixing
matrix obtained in our model has the form:
f (1370) 0.78 0.51 0.36 N
0
− | i
f (1500) = 0.540.84 0.03 S . (2)
0
− | i
f (1710) 0.32 0.18 0.93 G
0
| i
It is evident that f (1710) is composed primarily of the scalar glueball, f (1500)
0 0
is close to an SU(3) octet, and f (1370) consists of an approximate SU(3) singlet
0
with some glueball component ( 10%). Unlike f (1370), the glueball content of
0
∼
f (1500)is very tiny because an SU(3) octet does not mix with the scalarglueball.
0
Because the nn¯ content is more copious than ss¯ in f (1710), it is natural that
0
J/ψ ωf (1710) has a rate larger than J/ψ φf (1710). Our prediction of
0 0
→ →
Γ(J/ψ ωf (1710))/Γ(J/ψ φf (1710)) = 4.1 is consistent with the observed
0 0
→ →
value of 6.6 2.7. Moreover, if f (1710) is composed mainly of the scalar glueball,
0
±
itshouldbethemostprominentscalarproducedintheradiativeJ/ψ decay.Hence,
it is expected that Γ(J/ψ γf (1710)) Γ(J/ψ γf (1500)), a relation borne
0 0
→ ≫ →
out by experiment. Finally, we remark that in our mixing model, the relative 2γ
coupling strength is f (1370):f (1500):f (1710)= 9.3:1.0:1.5. Hence f (1500)
0 0 0 0
has the smallest 2γ coupling of the three states even though it has the least glue
content in our model. Therefore, the fact that f (1500) has not been seen in γγ
0
reactions doesn’t necessarily imply its glueball content.
3. Pseudoscalar glueball
In 1980,Mark II observeda resonance with a mass around 1440 MeV in the radia-
tiveJ/ψdecay18 andidentifieditwiththeE(1420)mesonfirstdiscoveredatCERN
in 1963through pp¯interactions.19 It was then realized that the new state observed
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by Mark II was not E(1420) and was subsequently named ι(1440) by Mark II and
Crystal Ball Collaborations.20 Shortly after the Mark II experiment, ι(1440) now
knownasη(1405)wasproposedtobealeadingcandidateforthepseudoscalarglue-
ball.(ForanexcellentreviewoftheE andιmesons,see.21)Indeedη(1405)behaves
like a glueball in its productions and decays because it has a large production rate
in the radiative J/ψ decay and is not seen in γγ reactions. Besides η(1405), other
states with masses below 2 GeV have also been proposed as the candidates, such
as η(1760) and X(1835).
However, the pseudoscalar glueball interpretation for η(1405) is not favored by
most of the theoretical calculations. For example, quenched lattice gauge calcula-
tions predict the mass of the 0−+ state to be above 2 GeV in4 and around 2.6
GeV in.12,22 It is not favoredby the sum-rule analysiswith predictions higher than
1.8 GeV23,24 either. Hence, we are encountering an embarrassing situation that al-
thoughexperimentallyη(1405)is afavoredcandidate forthe pseudoscalarglueball,
theorists seem to prefer to have a 0−+ state heavier than the scalar glueball. The
motivationofourrecentwork25istoseeifwecanlearnsomethingabouttheglueball
mass by studying the η η′ G mixing.
− −
The η η′ mixing has been well studied by Feldmann, Kroll and Stech.26 We
−
extend the FKS formalism to include the pseudoscalar glueball G. In the FKS
scheme, the conventional singlet-octet basis and the quark-flavor basis have been
proposed. For the latter, the qq¯ (uu¯+dd¯)/√2 and ss¯ flavor states, labeled by
≡
the η and η mesons,respectively, are defined. The physicalstates η, η′ and G are
q s
related to the octet, singlet, and unmixed glueball states η , η and g, respectively,
8 1
through the combination of rotations
η cosφ+ 2/3sinθ∆ sinφ+ 1/3sinθ∆ sinθsinφ η
| i p G − p G − G| 8i
η′ = sinφ 2/3sinθ∆ cosφ 1/3cosθ∆ cosθsinφ η
G G G 1
| i −p −p | i
G 2/3sinφ 1/3sinφ cosφ g
G G G
| i −p −p | i
η
8
| i
U(φ,φ ) η , (3)
G 1
≡ | i
g
| i
where θ is the η η′ mixing angle in the octet-singlet basis, φ = θ+54.7◦, ∆ =
G
−
1 cosφ and φ is the mixing angle between G and η ; that is, we have assumed
G G 1
−
that η does not mix with the glueball.
8
We proceed to define decay constants for the physical and flavor states
i i
0q¯γµγ q η (P) = f Pµ, 0q¯γµγ q η (P),g(P) = fq Pµ ,
h | 5 | q i −√2 q h | 5 | s i −√2 s,g
0s¯γµγ sη (P) = if Pµ, 0s¯γµγ sη (P),g(P) = ifs Pµ . (4)
h | 5 | s i − s h | 5 | q i − q,g
The decay constants associatedwith the η meson, η′ meson,and the physical glue-
ballarerelatedtothoseassociatedwiththeη ,η ,andg statesviathesamemixing
q s
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matrix
fq fs f fs
η η q q
fq fs =U(φ,φ )fq f . (5)
η′ η′ G s s
fq fs fq fs
G G g g
Sandwiching the equations of motion for the anomalous Ward identity
α
∂ (q¯γµγ q)=2im q¯γ q+ s G Gµν , (6)
µ 5 q 5 µν
4π
e
between vacuum and η , η′ and G , we derive
| i | i | i
m2 +(√2/f ) 0q η m2 +(1/f ) 0q η 0
mq2q +(√2/fq)h0|q|ηqi ms2q+(1/fs)h0|q|ηqi0=U†(φ,φ )M2U(φ,φ )J˜,(7)
qs q h | | si ss s h | | si G G
m2 +(√2/f ) 0q g m2 +(1/f ) 0q g 0
qg q h | | i sg s h | | i
where q =α GG˜/(4π) and
s
m2 0 0 1 fs/f 0
η q s
M2 = 0 m2 0 , J˜=fq/f 1 0 , (8)
η′ s q
0 0 m2 fq/f fs/f 0
G g q g s
with the abbreviation
√2 2
m2 0m u¯iγ u+m d¯iγ dη ,η ,g ,m2 0m s¯iγ sη ,η ,g .
qq,qs,qg ≡ f h | u 5 d 5 | q s i sq,ss,sg ≡ f h | s 5 | q s i
q s
Nowwehavesixequationsformanyunknowns.Hencewehavetoreplyonthelarge
N counting rules27
c
f O( N ), fq,s O(1), fs fq O(1/ N ),
q,s ∼ p c g ∼ q ∼ s ∼ p c
m O(1), m2 O(1), m2 O(1)+O(1/N ),
G ∼ η8 ∼ η1 ∼ c
m2 O(1), m2 O(1), φ O(1/ N ), (9)
qq ∼ ss ∼ G ∼ p c
m2 m2 O(1/ N ), m2 m2 O(1/N ),
qg ∼ sg ∼ p c qs ∼ sq ∼ c
0q g O(1), 0q η 0q η O(1/ /N ),
h | | i∼ h | | qi∼h | | si∼ p c
to solve the equations step by step.
To the leading order of 1/N expansion, we shall keep the decay constants f
c q
and f and neglect fq,s, fs and fq as they are suppressed by 1/√N and 1/N ,
s g q s c c
respectively.Likewise,wecanjustretainthediagonalmasstermsm2 m2,m2
qq ≈ π ss ≈
2m2 m2 andneglectother off-diagonalmass terms.It turns outthat the ratioof
K− π
the last two equations in the third low of Eq. (7) yields
cθ(sφ−p2/3cθ∆G)m2η′ −sθ(cφ+p2/3sθ∆G)2m2η−p2/3cφGm2G = √2fs(10)
cθ(cφ−p1/3cθ∆G)m2η′ +sθ(sφ−p1/3sθ∆G)2m2η−p1/3cφGm2G fq
where cφ (sφ) is the shorthand notation for cosφ (sinφ) and similarly for others.
This simple equation tells us that the pseudoscalar glueball mass m can be deter-
G
mined provided that the mixing angle φ and the ratio f /f are known. Note that
G s q
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Table 1. Solutions for the input of h0|q|ηqi = 0.050 GeV3 and fs being fixed to trade for
m2sg as a free parameter. The upper (lower) table is for φG = 22◦ (φG = 12◦), r ≡ fgs/fs
andR≡fqs/fs.
fs r R m2sg(GeV2) mG(GeV) h0|q|ηsi(GeV3) h0|q|gi(GeV3)
1.24fπ 0.22 −0.001 −0.009 1.60 0.028 0.036
1.26fπ 0.22 −0.003 0.004 1.47 0.028 0.036
1.28fπ 0.23 −0.005 0.016 1.34 0.028 0.038
1.30fπ 0.24 −0.007 0.029 1.21 0.028 0.040
1.24fπ 0.12 0.001 −0.054 2.15 0.027 0.030
1.26fπ 0.13 −0.001 −0.029 1.84 0.027 0.031
1.28fπ 0.15 −0.003 −0.005 1.52 0.027 0.034
1.30fπ 0.24 −0.005 0.018 1.16 0.028 0.045
the φ dependence appears at order of ∆ φ2 for small φ . So the solution for
G G ≈ G G
m is stable against the most uncertain input φ .
G G
The mixing angles φ andφ have been measuredrecentlyfrom the φ γη,γη′
G
→
decays by KLOE.28 Using the decay constants f = (1 0.01)f and f = (1.4
q π s
± ±
0.014)f asinputs,29KLOEobtainedtheanglesφ=(39.7 0.7)◦andφ =(22 3)◦
π G
± ±
inferred from the relevant data. In30 the data of P γV and V γP were first
→ →
considered and the fit gave the outcomes φ = (41.4 1.3)◦ and φ = (12 13)◦.
G
± ±
Withoutpreciseinputsoff andf itisnotunexpectedtogetawiderangeforφ .
q s G
Sinceφ hasawiderange,theresultsf =(1.05 0.03)f andf =(1.57 0.28)f
G q π s π
± ±
alsohavelargererrors.Usingthecentralvaluesoff /f andφ from28,30 asinputs,
s q G
we derive the pseudoscalar glueball mass from Eq. (10) to be
m =(1.4 0.1) GeV. (11)
G
±
The proximity of the predicted m to the mass of η(1405) and other properties of
G
η(1405) make it a very strong candidate for the pseudoscalar glueball.
Our next task is to check the stability and robustness of our prediction when
higher order effects in 1/N are included. We first turn on the decay constants fq,s
c g
and fs, fq. Assuming flavor-independent couplings between the glueball g and the
q s
pseudoscalar uu¯,dd¯and ss¯states, we then have the relations
fq =√2fs, fq =fs. (12)
g g s q
It turns out that the above simple formula Eq. (10) still holds even after keeping
the OZI-correcting decay constants, as long as they obey Eq. (12).
Finally we turn on the mass term m2 and neglect m2 , m2 and m2 . It is
sg qg qs sq
justifiedtodosobecausem2 isproportionaltothelightu/dquarkmass,whilethe
qg
last two mass terms are 1/√N suppressed relative to m2 . To explore the impact
c sg
of m2 on our solutions, we add f as an input so that m2 can be introduced as
sg s sg
an unknown. The results for the various inputs of f = (1.24-1.30)f , φ = 22◦
s π G
and 12◦, and 0q η =0.050 GeV3 are listed in Table 1. We see that m2 and m
h | | qi sg G
do depend on f sensitively. In some cases, we have m as large as 1.84 GeV and
s G
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9
2.15 GeV, for which m2 are negative and large.We cannot discard these solutions
sg
of m2 a priori, but they are not favored due to their negative values. This issue
sg
can be sorted out, when lattice calculations of m2 with dynamical fermions are
sg
available. Therefore, if excluding the solutions with large and negative m2 , the
sg
range (1.4 0.1) GeV of the pseudoscalar glueball mass obtained in Eq. (11) will
±
be more or less respected.
Onemayfeelratheruncomfortablewithoursolutionform asbothlatticeQCD
G
and QCD sum rules indicate a pseudoscalar glueball heavier than the scalar one.
The point is that lattice calculations so far were performed under the quenched
approximation without the fermion determinants. It is believed that dynamical
fermions will have a significant effect in the pseudoscalar channel, because they
raise the singlet would-be-Goldstone boson mass from that of the pion to η and
η′. Indeed, it has been argued that the pseudoscalar glueball mass in full QCD is
substantially lower than that in the quenched approximation.24 In view of the fact
that the topological susceptibility is large (of order 10−3GeV4) in the quenched
approximation, and yet is of order 10−5GeV4 in full QCD and zero in the chiral
limit, it is conceivable that full QCD has a largeeffect onthe pseudoscalarglueball
as it does on η and η′.
According to our analysis, the η(1405) γγ decay width is 0.6-3 keV, and
→
the leptonic decays η(1405) ℓ+ℓ− are very small.25 Both predictions can be
→
confronted with future experiments. There may not exist a unique feature which
tellsaglueballapartfromaquark-antiquarkstate.Weneedtocombineinformation
from J/ψ radiative decays, hadronic decays, as well as γγ and leptonic decays as
advocated in.31
4. Conclusions
We have employedtwo simple and robust results to constrainthe mixing matrix of
theisosingletscalarmesonsf (1710),f (1500),f (1370):(i)empiricSU(3)symme-
0 0 0
try in the scalarsector above1 GeV, and(ii) the scalarglueballmass at1710MeV
in the quenched approximation. In the SU(3) symmetry limit, f (1500) becomes a
0
pureSU(3)octetandisdegeneratewitha (1450),whilef (1370)ismainlyanSU(3)
0 0
singletwithaslightmixingwiththescalarglueballwhichistheprimarycomponent
off (1710).ThesefeaturesremainessentiallyunchangedevenwhenSU(3)breaking
0
is taken into account.
From the analysis of the η η′ G mixing together with the inputs from the
− −
recent KLOE experiment, we find a solution for the pseudoscalar glueball mass
around (1.4 0.1) GeV, suggesting that η(1405) is indeed a leading candidate for
±
the pseudoscalarglueball.Itisthus urgenttohaveafull QCDlattice calculationof
pseudoscalar glueball masses.
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10
Acknowledgments
I’m very grateful to Chun-Khiang Chua, Hsiang-nan Li and Keh-Fei Liu for the
fruitful collaboration on glueballs and to K K Phua for organizing this wonderful
conference.
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