1. Isosinglet scalar mesons & glueballs
Hai-Yang Cheng
Academia Sinica, Taipei
August 28, 2011
Scalars 2011, Warsaw, Poland

2. Preamble
Why are scalar mesons so special among even-parity mesons ?
 model: pions are Goldstone bosons while  has nonzero v.e.v.
<> ⇒ spontaneous chiral symmetry breaking (SB)
In QCD, <qq>, <qGq> ⇒ dynamical SB, m2  mq<qq>
 Scalar mesons are the Higgs sector of strong interactions
can be very broad ⇒ difficult to identify (e.g. , )
possible tetraquark content
possible glueball content

3. Scalar Mesons (JPC=0++)

4.  and  are much broader than f0 and a0 in widths
It is commonly believed that the light scalars , , f0(980), a0(980) are composed mainly of four quarks forming an S-wave qqqq nonet, while scalars above 1 GeV: f0(1370), a0(1450), K*0(1430) and f0(1500)/f0(1710) form a P-wave qq nonet - - -
Close, Torqvist
inverted spectrum
widths
 and  are much broader than f0 and a0 in widths
(, )  MeV, (f0, a0)  MeV
  much faster than f0  f0 a0 f0    a0 
2-quark model expt

5. In 4-quark picture (Jaffe ’77)
Mass degeneracy of f0 and a0 is natural
 & K are OZI super-allowed (fall apart)  large width
f0, a0 KK are OZI super-allowed, but suppressed by p.s.
f0 & a0q are OZI allowed  small width
Supported by LQCD calculations:
1. Prelovsek et al. [ PR, D82, (2010) ]
 and  have sizable 4-quark components
2. Mathur et al. [ PR, D76, (2007)]
qq state for a0(1450) and K0*(1430)

6. mixing of  and f0(980)
 has a large glueball content ? [ Minkowski, Ochs (’99); Narison,…]
In 2-quark picture:
Ds+ f0(980)+,  f0(980)  large ss content of f0(980)
(i) J/ f0 comparable to J/ f0, (ii) f0 width dominated by 
 qq component of f0(980)
No universal mixing angle fit to all the data of J/, radiative  decay and f0 coupling to +- & K+K- ; it lies in the ranges
25o <  < 40o, 140o <  <165o
In 4-quark scenario
  175o based on mass spectrum Maiani et al. (’04)

7. 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

8. 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

9. 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

10. 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 (’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

11. 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.”

12. 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
glueball spectrum from quenched LQCD
MG before mixing should be close to 1700 MeV
Mathur et al.

13. 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 13 13

14. 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

15. 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)] 

16. 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

17. : 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)] 17

18. 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:
Using Br(f0(1710) KK)=  Br(J/f0(1710))= 2.410-3
Br(f0(1710) )=  Br(J/f0(1710))= 2.710-3

19. 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

20. 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