1. Scalar & Pseudoscalar Glueballs
Hai-Yang Cheng (鄭海揚)
Academia Sinica, Taipei
May 25, 2012
University of Science & Technology of China

2. What is the effect of quark degrees of freedom on glueballs ?
Glueball: color-singlet bound state of gluons as gluons have a self coupling
Y. Chen et al. PR, D73, (2006)
Lightest glueballs in pure YM theory:
JPG= 5080 MeV
JPG= 30120 MeV
JPG= ±35±120 MeV
What is the effect of quark degrees of freedom on glueballs ?
Glueball will mix with qq states so that a pure glueball does not exist in nature
Mass of 0-+ glueball could be drastically affected

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

4. Scalar Mesons (JPC=0++)

5. Scalar glueball
Three isosinglet scalars f0(1370), f0(1500), f0(1710) are observed, only
two of them can be accommodated in qq QM  glueball content
f0(1370) & f0(1500) decay mostly to 2 & 4, while f0(1710) mainly into
KK  f0(1370), f0(1500) are nn states, ss state for f0(1710)
Amsler & Close (’95) 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.
Amsler’s argument (’02) against a qq interpretation of f0(1500):
Let |f0(1500)> = |N>cos - |S>sin
Non-observation of f0(1500) in  reaction demands   75o and hence ss dominance. This contradicts nn picture  f0(1500) is not a qq state

6. f0(1710):  is suppressed relative to KK  primarily ss dominated
f0(1500): dominant scalar glueball  1550 MeV [Bali et al. ’93, Amsler et al. ’95]
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

7. Difficulties with this model:
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)] = (3.31.3) [J/ f0(1710)]
J/→gg and the two gluons couple strongly to glueball
⇒ If f0(1500) is primarily a glueball, it should be seen in “glue-rich”
radiative J/ decay, namely, J/  f0(1500) >> J/ f0(1710)
BES  [J/  f0(1710)] > 4 [J/ f0(1500)]
If f0(1500) is composed mainly of glueball, then the ratio
R=(f0(1500) )/(f0(1500) KK) = 3/4 flavor blind
< 3/4 for chiral suppression
Rexpt(f0(1500))=4.10.4, Rexpt(f0(1710))=
Tension with LQCD

8. Lattice calculations for scalar glueballs
Quenched LQCD: JPC=0++ glueball in pure YM theory
Morningstar, Peardon (’99): 1750 50 80 MeV
Lee, Weingarten (’00):  58 MeV
Y. Chen et al. (’06) :  50  80 MeV
Full QCD (unquenched): glueballs mix with quarks; no pure glueball
UKQCD (’10): ~ 1.83 GeV
Glueball spectrum is not significantly affected by quark degrees
of freedom

9. Other scenarios:
Lee, Weingarten (lattice): f0(1710) glueball ; f0(1500) ss ;
f0(1370) nn
Burakovsky, Page: f0(1710) glueball, but Ms-MN=250 MeV
Giacosa et al. ( Lagrangian): 4 allowed sloutions
PDG (2006): p.168
“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.”
PDG (2008), PDG (2010):
“The f0(1500), or alternatively, the f0(1710) have been proposed as candidates for the scalar glueball.”

10. approximate SU(3) symmetry in scalar meson sector (> 1GeV)
Based on two simple inputs for mass matrix, Keh-Fei Liu, Chun-Khiang Chua and I have studied the mixing with glueball :
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 and lattice QCD
glueball spectrum from quenched LQCD
MG before mixing should be close to 1700 MeV
Mathur et al.

11. 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 11 11

12. 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

13. 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
Carlson et al (’81);
Cornwall, Soni (’85);
Chanowitz (’07)
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 (’95)] 

14. 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 total 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

15. : tends to be an SU(3) octet
Amseler-Close-Kirk
: primarily a glueball
: tends to be an SU(3) octet
: near SU(3) singlet + glueball content ( 13%)
blue: nn
red: ss
green: G
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 3.31.3 (expt)
(J/ f0(1710)) >> (J/f0(1500))
in good agreement with expt.
[J/→f0(1710)] > 4 [J/→f0(1500)] 15

16. Scalar glueball in radiative J/ decays
LQCD:
Meyer ( ): Br(J/ G0)  910-3
Y. Chen et al. (lattice 2011): Br(J/ G0) = (3.60.7)10-3
BES measurements:
G.S. Huang (FPCP2012)
G.S. Huang (FPCP2012)
Using Br(f0(1710) KK)=  Br[J/f0(1710)]= 2.410-3
Br(f0(1710) )=  Br[J/f0(1710)]= 2.710-3

17. It is important to revisit & check chiral suppression effects
Chiral suppression in LQCD [Sexton, Vaccarino, Weingarten (’95)]:
(0++)=108  28 MeV
If chiral suppression in scalar glueball decays is confirmed, it will rule out f0(1500) and (600) as candidates of 0++ glueballs
(f0(1500) )/(f0(1500) KK)= 4.10.4 >> ¾
()  MeV: very broad

18. Pseudoscalar glueball
Hsiang-nan Li, Keh-Fei Liu, HYC
Phys. Rev. D 79, (2009)

19. 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) 19

20. 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;UKQCD]
QCDSR ⇒ mG > 1.8 GeV
QM ⇒ GeV > mG > 2.22 GeV
All yield m(0++) < m(0-+) 20

21. -’-G mixing
Consider flavor basis q=(uu+dd)/√2, s=ss. In absence of U(1) anomaly, q & s mix only through OZI-violating effects
Extend Feldmann-Kroll-Stech (FKS) mechanism of -’ mixing to include G,
where  =  o, G is the mixing angle between G and 1 (8 is assumed not to mix with glueball). Mixing matrix depends only on  and G 21

22. Applying EOM:
Six equations for many unknowns. We need to reply on large Nc rules (’t Hooft) 22

23. 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 analysis of  ’, 
⇒  = (40.4±0.6)o, G=(20 ±3)0 for fs/fq=1.3520.007
mG=(1.4±0.1) GeV
This result for mG is stable against OZI corrections

24. 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 with dynamic fermions is substantially lower than that in quenched approximation [Cabadadze (1998)]
Contrary to the mainstream, we conjecture that pseudoscalar glueball is lighter than the scalar one due to dynamic fermion or axial anomaly.
It is important to have a full lattice QCD calculation
quenched
unquenched

25. -’-G mixing yields Data can be accommodated with G  30o
supported by an analysis based on chiral Lagrangian with instanton effects, a solution for UA(1) problem
[Song He, Mei Huang, Qi-Shu Yan, arXiv: ]
Xin Liu, H.n. Li, Z.J. Xiao argued that B J/Ã(‘) decays imply large G content in ’ arXiv:
-’-G mixing yields
Data can be accommodated with G  30o

26. Conclusions 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
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 needed. 26