1. Scalar and Pseudoscalar Glueballs
Hai-Yang Cheng
Academia Sinica, Taipei
PAQFT08, Singapore
November 28, 2008

2. Glueball: color-singlet bound state of gluons as gluons have a self coupling
Y. Chen et al. PR, D73, (2006)
Lighest glueballs:
JPG= 5080 MeV
JPG= 30120 MeV
JPG= ±35±120 MeV
Glueball will mix with qq states so that a pure glueball does not exist in nature

3. Scalar glueball Chun-Khiang Chua, Keh-Fei Liu, HYC,
PR, D74, (2006)

4. Scalar Mesons (JP=0+)

5. Amsler & Close (1995) claimed f0(1500) discovered at LEAR as an evidence for a scalar glueball because its decay to ,KK,,’ is not compatible with a simple qq picture.
f0(1500): dominant scalar glueball  1550 MeV [Bali et al. ’93, Amsler et al. ’95]
KK/   small ss content
f0(1710):  is suppressed relative to KK  primarily ss dominated
f0(1370): KK is suppressed relative to   dominated by nn states
G ss nn
Amsler, Close, Kirk, Zhao, He, Li…
MS>MG>MN MG  1500 MeV, MS-MN  MeV

6. Other scenarios:
Lee, Weingarten (lattice): f0(1710) glueball ; f0(1500) nn ;
f0(1370) ss
Burakovsky, Page: f0(1710) glueball, but Ms-MN=250 MeV
Giacosa et al. ( Lagrangian): 4 allowed sloutions
PDG (2006): p.168 [PDG(2008), p.597]
“Experimental evidence is mounting that f0(1500) has considerable affinity for glue and that the f0(1370) and f0(1710) have large uu+dd and ss components, respectively.”

7. Difficulties:
Improved quenched LQCD  mG  1700 MeV rather than 
1500 MeV
Near degeneracy of a0(1450) and K0*(1430) cannot be
explained due to the mass difference between MS and Mn
If f0(1710) is ss dominated,
[J/f0(1710)]= 6 [J/f0(1710)] [Close,Zhao]
BES  [J/f0(1710)] = (6.62.7) [J/ f0(1710)]
J/→gg and glueballs couple strongly to gluons
⇒ If f0(1500) is primarily a glueball,
J/  f0(1500) >> J/ f0(1710)
Expt  [J/  f0(1710)]  5 [J/ f0(1500)]

8. In our work we reply on two simple inputs for mass matrix:
approximate SU(3) symmetry in scalar meson sector (> 1GeV)
a0(1450), K0*(1430), Ds0*(2317), D0*(2308)
 MS should be close to MN
a feature confirmed by LQCD
LQCD  K0*(1430)=1.41±0.12 GeV, a0*(1450)=1.42±0.13 GeV
 near degeneracy of K0*(1430) and a0(1450)
This unusual behavior is not understood and it serves as
a challenge to the existing QM
glueball spectrum from quenched LQCD
MG before mixing should be close to 1700 MeV

9. a0(1450) mass is independent of quark mass when mq ms
Mathur et al.
hep-ph/
1.420.13 GeV
a0(1450) mass is independent of quark mass when mq ms

10. x: quark-antiquark annihilation y: glueball-quarkonia mixing
uu dd ss G
x: quark-antiquark annihilation
y: glueball-quarkonia mixing
first order approximation: exact SU(3)  MU=MD=MS=M, x=xs=xss, ys=y
a0 octet singlet G
y=0, f0(1710) is a pure glueball, f0(1370) is a pure SU(3) singlet with mass = M+3x ⇒ x = -33 MeV
y 0, slight mixing between glueball & SU(3)-singlet qq. For |y| | x|, mass shift
of f0(1370) & f0(1710) due to mixing is only  10 MeV
 In SU(3) limit, MG is close to 1700 MeV

11. SU(3) breaking effect is treated perturbatively
Need SU(3) breaking in mass matrix to lift degeneracy of a0(1450) and f0(1500)
Need SU(3) breaking in decay amplitudes to accommodate observed strong decays
For SU(3) octet f0(1500),  = -2  R1=0.21 vs 0.026 (expt)
R2= vs 0.027 (expt)
LQCD [Lee, Weingarten]  y= 4331 MeV, y/ys=1.1980.072
y and x are of the same order of magnitude !
SU(3) breaking effect is treated perturbatively

12. Chiral suppression in scalar glueball decay
If f0(1710) is primarily a glueball, how to understand its decay to PP ?
If G→PP coupling is flavor blind,
chiral suppression: A(G→qq)  mq/mG in chiral limit [Chanowitz]
Chiral suppression at hadron level should be not so strong perhaps due to nonperturbative chiral symmetry breaking and hadronization
[Chao, He, Ma] : mq is interpreted as chiral symmetry breaking scale
[Zhang, Jin]: instanton effects may lift chiral suppression
LQCD [Sexton, Vaccarino, Weingarten] 

13. In absence of chiral suppression (i. e
In absence of chiral suppression (i.e. g=gKK=g), the predicted f0(1710) width is too small (< 1 MeV) compared to expt’l width of 13718 MeV  importance of chiral suppression in GPP decay
Consider two different cases of chiral suppression in G→PP:
(i)
(ii)
Scenario (ii) with larger chiral suppression is preferred

14. Amseler-Close-Kirk
: primarily a glueball
: tends to be an SU(3) octet
: near SU(3) singlet + glueball content ( 13%)
MN=1474 MeV, MS=1498 MeV, MG=1666 MeV, MG>MS>MN
MS-MN  25 MeV is consistent with LQCD result
 near degeneracy of a0(1450), K0*(1430), f0(1500)
Because nn content is more copious than ss in f0(1710),
(J/f0(1710)) = 4.1 ( J/ f0(1710)) versus 6.62.7 (expt)
If f0(1710)=ss, one needs large doubly OZI  5 OZI [Close, Zhao]
(J/ f0(1710)) >> (J/f0(1500))
in good agreement with expt.
[J/→f0(1710)]  5 [J/→f0(1500)]
Our scenario for scalar glueball and its mixing with qq is consistent with all the existing data

15. Pseudoscalar glueball
Hsiang-nan Li, Keh-Fei Liu, HYC
arXiv: [hep-ph]

16. Br[J/(1405)]  O(10-3) larger than J/(1295), (2225)
1980: J/  + resonance (1.44 GeV) was seen by MarK II and identified as
E(1420) [now known as f1(1420)] first discovered at CERN in 1963.
Renamed the ¶(1440) by Crystal Ball & Mark II in 1982
1981: (1440) was proposed to be a pseudoscalar glueball by
Donoghue, Johnson; Chanowitz; Lacaze, Navelet,…
2005: BES found a resonance X(1835) in J/ +-’
2006: X(1835) as an 0-+ glueball by Kochelev, Min; Li; He, Li, Liu, Ma
It is commonly believed that (1440) now known as (1405) is a leading candidate of G
Br[J/(1405)]  O(10-3) larger than J/(1295), (2225)
 KK, : (1475) in KK was seen, but (1405) in  was not
Expt’l review: Masoni, Cicalo, Usai, J. Phys. G32, 293 (2006)

17. All yield m(0++) < m(0-+)
Pseudoscalar glueball interpretation of (1405) is also supported by the flux-tube model (Faddeev et al.), but not favored by most of other calculations:
LQCD ⇒  2.6 GeV [Chen et al; Morningstar, Peardon]
QCDSR ⇒ mG > 1.8 GeV
QM ⇒ GeV > mG > 2.22 GeV
All yield m(0++) < m(0-+)

18. -’-G mixing
Extend Feldmann-Kroll-Stech (FKS) mechanism of -’ mixing to include G.
Consider flavor basis q=(uu+dd)/√2, s=ss. In absence of U(1) anomaly, q & s mix only through OZI-violating effects
where  =  o, and 8 is assumed not to mix with glueball. Mixing matrix depends only on  and G

19. Applying EOM:
Six equations for many unknowns. We need to reply on large Nc rules (‘t Hooft)

20. Then turn on msg2 and neglect mqg, mqs, msq
To leading order in 1/Nc, keep fq,s, neglect fq,sg, fsq, fqs
keep mqq2  m2, mss2  2mK2-m2, neglect other mass terms
KLOE data on  ’,  ⇒  = (39.7±0.7)o, G = (22 ±3)o, fs/fq=1.40±0.02
Analysis of V P, P V by Escribano, Nadal ⇒  = (41.4±1.3)o, G=(12 ±13)0, for fs/fq=1.50±0.27
mG=(1.4±0.1) GeV
OZI correctios:
Next turn on fgq,s, fsq,fqs and assume flavor-independent couplings between g and qq ⇒ fgs=√2 fgq ⇒ mG is independent of fgs/fs
Then turn on msg2 and neglect mqg, mqs, msq

21. msg2=0, <0|q|q>=0.065, 0.050, 0.035 GeV3
G=22o
G=12o
msg2 0, <0|q|q>=0.050 GeV3
G=22o
G=12o
Large mG is not favored as msg2 is negative; to be checked by LQCD ⇒ mG=(1.4±0.1) MeV is stable

22. It is urgent to have a full lattice QCD calculation
Is this compatible with LQCD which implies mG > 2GeV ?
Lattice results are still quenched so far. It has been noticed that mG in full QCD is substantially lower than that in quenched approximation due to dynamic fermion effect or axial anomaly [Cabadadze (1998)]
Note that topological susceptibility is around (190 MeV)4 in quenched approximation, while it is very small ( 10-5 GeV4) in full QCD and becomes zero in chiral limit
It is urgent to have a full lattice QCD calculation
for pseudoscalar glueball !

23. Conclusions
We use two simple & robust results to constrain mixing matrix of f0(1370), f0(1500) and f0(1710): (i) empiric SU(3) symmetry in scalar meson sector > 1 GeV, (ii) scalar glueball mass  1700 MeV
Exact SU(3) ⇒ f0(1500) is an SU(3) octet, f0(1370) is an SU(3) singlet with small mixing with glueball. This feature remains to be true even when SU(3) breaking is considered
Analysis of -’-G mixing yields mG  1.4±0.1 GeV, suggesting that (1405) is a leading candidate of pseudoscalar glueball. A full lattice QCD calculation is urgent.

24. Spare slides

25. f0(1710) is not seen in pp→0f0 at LEAR (2002), but now
observed in pp→0 at Fermilab (2006)
⇒ f0(1710) is not dominated by ss
2-quarkonium coupling
f0(1370): f0(1500): f0(1710) = 9.3 : 1.0 : 1.5
2 coupling of f0(1500) is weak even if it has no glue content !
⇒ The fact that f0(1500) is not seen in  reaction doesn’t
necessarily imply its glue content

26. We predict (G)  1 keV, (G +-)  210-2 eV
Since glueball couples strongly to gg and weakly to , it is expected to be produced copiously in J/ radiative decays [J/ggG] and suppressed in  reactions.
typical anomaly matrix elements:
Br(J/ ’)=(4.31±0.30)10-3 could be comparable to Br(J/G)
In order to identify G, we need to combine information from J/ radiative decays, hadronic decays, pp and  collisions, leptonic decays
Chanowitz: “stickness” for identifying glue rich states