11

Scalar & Pseudoscalar Glueballs

Hai-Yang Cheng (鄭海揚 )

Academia Sinica, Taipei

May 25, 2012

University of Science & Technology of China

2

Glueball: color-singlet bound state of gluons as gluons have a self coupling

Lightest glueballs in pure YM theory:

JPG=0++ 17105080 MeV

JPG=2++ 239030120 MeV

JPG=0-+ 2560±35±120 MeV

Y. Chen et al. PR, D73, 014516 (2006)

Glueball will mix with qq states so that a pure glueball does not exist in nature

Mass of 0-+ glueball could be drastically affected

What is the effect of quark degrees of freedom on glueballs ?

3

Scalar glueballScalar glueball

Chun-Khiang Chua, Keh-Fei Liu, HYC,

PR, D74, 094005 (2006)

44

Scalar Mesons (JScalar Mesons (JPCPC=0=0++++))

)1710(

)1370(

0

0

f

f

)6 0 0(

qq

22qq

)1500(0f)1470(0

0a

)1430(*0K)1430(*

0K

)1430(*0K)1430(*

0K

)1470(0a )1470(0

a

)8 0 0(

)980(0f)980(0

a )980(0a

)980(00a

)8 0 0(

)8 0 0( )8 0 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

2

dduunnN

2

0 sin9

1cos

29

5)(

f

6

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

0 2

0

2

N

S

G

My

My

yyMG ss nn

SNGf

SNGf

SNGf

07.091.040.0)1370(

35.041.084.0)1500(

93.009.036.0)1710(

0

0

0

MS>MG>MN MG 1500 MeV, MS-MN 200-300 MeV

Amsler, Close, Kirk, Zhao, He, Li,…

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)] = (3.31.3) [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) ) = 3/4 flavor blind

< 3/4 for chiral suppression

Rexpt(f0(1500))=4.10.4, Rexpt(f0(1710))=0.41+0.11-0.17

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): 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

9

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.”

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 (2008), PDG (2010):

“The f0(1500), or alternatively, the f0(1710) have been proposed as candidates for the scalar glueball.”

10

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.

111111

Mathur et al.

hep-ph/0607110

a0(1450) mass is independent of quark mass when mq ms

1.420.13 GeV

12

0000

000

000

000

s

sssss

s

s

G

S

D

U

yyy

yxxx

yxxx

yxxx

M

M

M

M

M

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

GMy

yxM

M

M

300

3300

000

000

mixedslightly are glueball and )1370(

octet :14746/)2()1500(

MeV 191474 ,14742/)(

0

0

0

f

ssdduuf

MMdduua DU

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

a0 octet singlet G

13

Chiral suppression in scalar glueball decay

If G→PP coupling is flavor blind, 91.04

3)(/)( KKGG

PDG),,(/ from S BE0.41

),(/ from BES 0.13

WA102 04.020.0

))1710((

))1710((

0.110.17-

0

0

KKJ

KKJKKf

f

If f0(1710) is primarily a glueball, how to understand its decay to PP ?

chiral suppression: A(G→qq) mq/mG in chiral limit

364.0423.0

372.0402.0

603.0579.0 099.3:654.2:834.0::

ggg KK

LQCD [ Sexton, Vaccarino, Weingarten (’95)]

[Chao, He, Ma] : mq is interpreted as chiral symmetry breaking scale

[Zhang, Jin]: instanton effects may lift chiral suppression

? /)(/)( 22su mmKKGG

Chiral suppression at hadron level should be not so strong perhaps due to nonperturbative chiral symmetry breaking and hadronization

Carlson et al (’81); Cornwall, Soni (’85); Chanowitz (’07)

14

Consider two different cases of chiral suppression in G→PP:

(i)

(ii)

59.1:55.1:1:: ggg KK

74.4:15.3:1:: ggg KK

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 137eV importance of chiral suppression in GPP decay

Scenario (ii) with larger chiral suppression is preferred

1515

SNGf

SNGf

SNGf

52.078.036.0)1370(

84.054.003.0)1500(

17.032.093.0)1710(

0

0

0

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 3.31.3 (expt)

(J/ f0(1710)) >> (J/f0(1500))

in good agreement with expt.

[J/→f0(1710)] [J/→f0(1500)]

: 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

Amseler-Close-Kirk

blue: nnred: ssgreen: G

16

Scalar glueball in radiative J/ decays

LQCD: Meyer (0808.3151): Br(J/ G0) 910-3

Y. Chen et al. (lattice 2011): Br(J/ G0) = (3.60.7)10-3

30

41.230.72-0

40

40

42.10.90

50.501.32 - 0

40

100.3)(1.8 )1710(/

102.35 )1710(/

101.03.1 )1710(/

101.04.0 )1710(/

108.5 )1710(/

101.61 )1500(/

100.321.01 )1500(/

fJ

fJ

fJ

fJ

KKfJ

fJ

fJ

40 109.2))1500(/( fJBr

Using Br(f0(1710) KK)=0.36 Br[J/f0(1710)]= 2.410-3

Br(f0(1710) )= 0.15 Br[J/f0(1710)]= 2.710-3

BES measurements:

G.S. Huang (FPCP2012)

G.S. Huang (FPCP2012)

17

364.0423.0

372.0402.0

603.0579.0 099.3:654.2:834.0::

ggg KK

Chiral suppression in LQCD [Sexton, Vaccarino, Weingarten (’95)]:

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) )= 4.10.4 >> ¾

() 600-1000 MeV: very broad

(0++)=108 28 MeV

It is important to revisit & check chiral suppression effects

18

Pseudoscalar glueballPseudoscalar glueball

Hsiang-nan Li, Keh-Fei Liu, HYC

Phys. Rev. D 79, 014024 (2009)

1919

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)

2020

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 > mG > 2.22 GeV

All yield m(0++) < m(0-+)

2121

-’-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

cos sin3/1 sin3/2

sin cos )cos1(cos3/1cos )cos1(sin3/2sin

sinsin )cos1(sin3/1sin )cos1(sin3/2cos

),('

gg

U

Gs

q

GGG

GGG

GGG

s

q

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

PifGssPfi

Gqq

PifgssPfi

gqq

PifssPfi

qq

sG

qG

sgqq

sgqs

ssqq

,',5,',5

,5,5

55

,',||0 ,2

,',||0

,||0 ,2

,||0

||0 ,2

||0

gqg

sqs

sqq

G

sG

qG

sq

sq

ff

ff

ff

U

ff

ff

ff

),(

''

cos sin

sin cos)(

'

s

q

s

qU

Extend Feldmann-Kroll-Stech (FKS) mechanism of -’ mixing to include G,

2222

GGqqimqq sq

~

42)( 55

0 / /

0 1 /

0 / 1

),(

0 0

0 0

0 0

),(

0 ||0)/1( ||0)/2(

0 ||0)/1( ||0)/2(

0 ||0)/1( ||0)/2(

2

2'

2

22

22

22

ssgq

qg

qqs

ssq

G

G

G

ssgqqg

sssssqqs

qssqqqqq

ffff

ff

ff

U

m

m

m

U

gqfmgqfm

qfmqfm

qfmqfm

Applying EOM:

)4/(~

,,||02

,,,||02

52

,,552

,,

GGq

gsismf

mgdidmuiumf

m

s

sqss

sgsssqsqduq

qgqsqq

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

23

2222'

2222'

cos3/1))cos1(cos3/1(sinsin))cos1(cos3/1(sincos

cos3/2))cos1(cos3/2(cossin))cos1(cos3/2(sincos2

GGGG

GGGG

q

s

mmm

mmm

f

f

To leading order in 1/Nc, keep fq,s, neglect fq,sg, fs

q, fqs

keep mqq2 m

2, mss2 2mK

2-m2, neglect other mass terms

KLOE analysis of ’,

⇒ = (40.4±0.6)o, G=(20 ±3)0 for fs/fq=1.3520.007

This result for mG is stable against OZI corrections

mG=(1.4±0.1) GeV

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)]

4'

2

'4

2

4

0

0

mf

fm

f

fm

GG

G

GG

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

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/Ñ decays imply large G content in ’ arXiv:1205.1214

0.02)0.14(0.73 1cot)/(

)'/(

12.3

7.4 1tan

)/(

)'/(

2

1.91.8 -

2

JBBr

JBBrR

JBBr

JBBrR

s

ss

d

dd

-’-G mixing yields 22

)cos1(sin3/1sin

)cos1(cos3/1cos

)cos1(sin3/2cos

)cos1(cos3/2sin

G

Gs

G

Gd RR

Data can be accommodated with G 30o

2626

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

2626

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.