Scalar & Pseudoscalar Glueballs
Hai-Yang Cheng (鄭海揚)
Academia Sinica, Taipei
May 25, 2012
University of Science & Technology of China
1
Glueball: color-singlet bound state of gluons as gluons have a
self coupling
Y. Chen et al. PR, D73, 014516 (2006)
Lightest glueballs in pure YM theory:
JPG=0++
17105080 MeV
JPG=2++
239030120 MeV
JPG=0-+
2560±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
2
Scalar glueball
Chun-Khiang Chua, Keh-Fei Liu, HYC,
PR, D74, 094005 (2006)
3
Scalar Mesons (JPC=0++)
K 0* (1430 )
K 0* (1430 )
qq
a 0 (1470 )
0
0
a (1470 )
f 0 (1500 )
K (1430 )
K 0* (1430 )
 (8 0 0 )
 (8 0 0 )
*
0
2
q q
2
a 0 (1470 )

0
a ( 980 )
 (8 0 0 )
a 00 ( 980 )
f 0 ( 980 )
a 0 ( 980 )
 (8 0 0 )

f 0 (1370)

 (600 )
f 0 (1710)
4
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
1
 5

 ( f 0   )  
cos   sin  
9
9 2

2
N  nn 
u u  dd
2
Non-observation of f0(1500) in  reaction demands   75o and hence
ss dominance. This contradicts nn picture  f0(1500) is not a qq state
5
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
 MG

 y

 2y
y
MS
0
nn
2y 

0 

MN 
f 0 (1710)  0.36 G  0.09 N  0.93 S
f 0 (1500)  0.84 G  0.41 N  0.35 S
f 0 (1370)  0.40 G  0.91 N  0.07 S
Amsler, Close, Kirk, Zhao, He, Li,…
MS>MG>MN
MG  1500 MeV,
MS-MN  200-300 MeV
6
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,
 Tension with LQCD
Rexpt(f0(1710))=0.41+0.11-0.17
7
Lattice calculations for scalar glueballs
Quenched LQCD: JPC=0++ glueball in pure YM theory
 Morningstar, Peardon (’99): 1750 50 80 MeV
 Lee, Weingarten (’00):
1648 58 MeV
 Y. Chen et al. (’06) :
1710 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
8
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.”
9
Based on two simple inputs for mass matrix, Keh-Fei Liu, ChunKhiang 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
Mathur et al.
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
10
Mathur et al.
hep-ph/0607110
1.420.13 GeV
a0(1450) mass is independent of quark mass when mq ms
11
uu
 MU

 0
M 
0

 0
dd
0
ss
0
MD
0
0
0
MS
0
G
0  x
 
0  x

0   xs
 
MG   y
x
x
xs
xs
xs
y
xss
ys
y

y
ys 

0
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
M

0
0

0

0
M
0
0
0
M  3x
0
3y
G
0 

0 
3y 

M G 
 a 0  (uu  dd ) / 2  1474, M U  M D  1474  19 MeV

  f 0 (1500 )  (uu  dd  2 ss ) / 6  1474 : octet
 f (1370 ) and glueball are slightly mixed
 0
 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
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, (G   ) / (G  KK ) 
3
 0.91
4
0.20  0.04 WA102
( f 0 (1710)   ) 
  0.13
BES from J /   ( KK ,  )
( f 0 (1710)  KK ) 
 0.11
BE S from J /   ( KK ,  ), PDG
0.41-0.17
chiral suppression: A(G→qq)  mq/mG in chiral limit
(G   ) / (G  KK )  mu2 / ms2 ?
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)] 
0.372
0.364
g : g KK : g  0.83400..603
:
2
.
654
:
3
.
099
579
0.402
0.423
13
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)
g  : g KK : g  1 : 1.55 : 1.59
(ii)
g  : g KK : g  1 : 3.15 : 4.74
Scenario (ii) with larger chiral
suppression is preferred
14
Amseler-Close-Kirk
f 0 (1710)  0.93 G  0.32 N  0.17 S
: primarily a glueball
blue: nn
: tends to be an SU(3) octet
red: ss
G
f 0 (1370)  0.36 G  0.78 N  0.52 green:
S
: near
SU(3) singlet + glueball content
f 0 (1500)  0.03 G  0.54 N  0.84 S
( 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 3.31.3 (expt)
 (J/ f0(1710)) >> (J/f0(1500))
in good agreement with expt.
[J/→f0(1710)] > 4 [J/→f0(1500)]
15
Scalar glueball in radiative J/ decays
LQCD:
 Meyer (0808.3151): Br(J/ G0)  910-3
 Y. Chen et al. (lattice 2011): Br(J/ G0) = (3.60.7)10-3
BES measurements:
J /  f 0 (1500)  
1.01  0.3210 4
J /  f 0 (1500)  
1.61 10
J /  f 0 (1710)  KK
8.5 10
J /  f 0 (1710)  
J /  f 0 (1710)  
 0.50
- 1.32
1.2
 0.9
5
 Br ( J /  f 0 (1500))  2.9  104
G.S. Huang (FPCP2012)
4
4.0  1.010 4
3.1  1.010 4
J /  f 0 (1710)  
2.35 10
J /  f 0 (1710)
> (1.8  0.3) 10 3
1.23
-0.72
4
G.S. Huang (FPCP2012)
Using Br(f0(1710) KK)=0.36  Br[J/f0(1710)]= 2.410-3
Br(f0(1710) )= 0.15
 Br[J/f0(1710)]= 2.710-3
16
It is important to revisit & check chiral suppression effects
 Chiral suppression in LQCD [Sexton, Vaccarino, Weingarten (’95)]:
0.372
0.364
g : g KK : g  0.83400..603
:
2
.
654
:
3
.
099
579
0.402
0.423
(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
 ()  600-1000 MeV: very broad
17
Pseudoscalar glueball
Hsiang-nan Li, Keh-Fei Liu, HYC
Phys. Rev. D 79, 014024 (2009)
18
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
1. Br[J/(1405)]  O(10-3) larger than J/(1295), (2225)
2.
 KK, : (1475) in KK was seen, but (1405) in  was not
Expt’l review: Masoni, Cicalo, Usai, J. Phys. G32, 293 (2006)
19
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
⇒ 2.62 GeV > m G > 2.22 GeV
All yield m(0++) < m(0-+)
20
-’-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
  2 / 3 sin(1 cos
G )  sin
  1/3 sin
 (1  cos G )  sin  sin G  q 
 q   cos
 


cos

sin


q
q
 
 
 
   
 




U
(

)





  (1  cos G ) cos sin G  s 
 ' 2 / 3 sin(1  cos sin
  1/3 cos
 '   U ( , G ) s   sin
G )  cos
cos




 
s
s




 
G 
g

 g 
 2 / 3 sin G
 1 / 3 sin G
cos G
 
  

Extend Feldmann-Kroll-Stech (FKS) mechanism of -’ mixing to include G,
where  =  + 54.7o, G is the mixing angle between G and 1 (8 is assumed
not to mix with glueball). Mixing matrix depends only on  and G
i
f q P ,
0 | s    5 s |  s   if s P
2
i s
0 | q    5 q |  s , g  
f q , g P ,
0 | s    5 s |  q , g   if qs, g P
2
i q
0 | q    5 q |  , ' , G 
f , ',G P , 0 | s    5 s |  , ' , G  ifs, ',G P
2
0 | q    5 q |  q  
 f q f qs 
 fq fs 




q
s
q
  f ' f '   U ( , G ) f s f s 
 q



q
s
 fG fG 
 fg fg 




21
Applying EOM:
  (q    5 q )  2imq q  5 q 
s
~
G G 
4
2
s
2
 mqq
 ( 2 / f q )0 | q |  q  msq2  (1 / f s )0 | q |  q  0 



1
f
/ fs 0 
m
0
0
q







 m 2  ( 2 / f )0 | q |   m 2  (1 / f )0 | q |   0   U  ( ,  ) 0 m 2 0 U ( ,  ) f q / f
1 0 
qs
q
s
ss
s
s
G
'
G
s
q






2
q
s
2
2
 0 0 mG 
 m  ( 2 / f )0 | q | g  m  (1 / f )0 | q | g  0 
 f g / fq f g / fs 0
q
sg
s




 qg

2
mqq
, qs , qg 
2
fq
 0 | mu u i 5u  md d i 5 d |  q , s , g  , msq2 , ss, sg 
2
 0 | ms s i 5 s |  q , s , g 
fs
~
q   s GG /( 4 )
Six equations for many unknowns. We need to reply on large Nc rules (’t Hooft)
22
 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
2
2
2
2
2 f s cos  (sin   2 / 3 cos  (1  cos G )) m '  sin  (cos   2 / 3 cos  (1  cos G )) m  2 / 3 cos G m G

fq
cos  (sin   1 / 3 cos  (1  cos G )) m2 '  sin  (sin   1 / 3 cos  (1  cos G )) 2 m2  1 / 3 cos G m G2
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
23
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)]
2
2
 fG 
f 
mG4   0  mG4 0    '  m4'
 fG 
 fG 
quenched
Contrary to the mainstream, we
conjecture that pseudoscalar
glueball is lighter than the scalar
one due to dynamic fermion or
axial anomaly.
unquenched
It is important to have a full lattice
QCD calculation
24
 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:0903.5032]
 Xin Liu, H.n. Li, Z.J. Xiao argued that B J/Ã‘ decays imply
large G content in ’
arXiv:1205.1214
Rd 
Br ( Bd  J /' )
 tan 2   1
Br ( Bd  J /)

7.4 
 

1.9 
 12.3- 1.8 
Rs 
Br ( Bs  J /' )
 cot 2  > 1
Br ( Bs  J /)
(0.73  0.14  0.02)
-’-G mixing yields
 sin   2 / 3 cos  (1  cos G ) 

Rd  

 cos   2 / 3 sin  (1  cos G ) 
2
 cos   1 / 3 cos  (1  cos G ) 

Rs  

  sin   1 / 3 sin  (1  cos G ) 
Data can be accommodated with G  30o
2
25
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 needed.
26