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(
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
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.
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