Scalar Glueball, Scalar Quarkonia, and their Mixing
Hai-Yang Cheng
Academia Sinica, Taipei
in collaboration with Chun-Khiang Chua & Keh-Fei Liu
NTHU, October 26, 2006
1
¾ Each quark has one of three different color charges and
antiquark carries an anti-color charge
q
q
q
¾ Quarks exchange gluons and create a very strong color force
¾ Gluons carry a color and an anti-color and hence can have
a self-coupling
¾ Single quark has not been observed yet. QCD tells it cannot
be observed (quark confinement). All naturally
occurring particles are colorless.
2
Glueball: color-singlet bound state of gluons as gluons
have a self coupling
Lighest glueballs:
JPG=0++
1710±50±80 MeV
JPG=2++
2390±30±120 MeV
Y. Chen et al. hep-lat/0510074
z ι(1440) [now η(1405)] as 0-+ glueball
z Amsler & Close (1995) claimed f0(1500)
discovered at LEAR as an evidence for a
scalar glueball because its decay to
ππ,KK,ηη,ηη’ is not compatible with a
simple qq picture.
z V.V. Anisovich et al. (2005) claimed
evidence of a tensor glueball, namely,
f2(2000)
3
Scalar Mesons (JP=0+)
K 0* (1430)
*
0
K (1430)
qq
a0− (1470)
0
0
a (1470)
f 0 (1500)
K (1430)
K 0* (800)
2
q q
2
a (980)
K 0* (800)
+
+
σ (600)
f 0 (1710)
K 0* (1430)
*
0
−
0
a0+ (1470)
f 0 (1370)
K 0* (800)
a00 (980)
f 0 (980)
a0+ (980)
K 0* (800)
4
f0(1500): dominant scalar glueball ∼ 1550 MeV [Bali et al. ’93, Amsler et al. ’95]
KK/ππ ∼ 0.25 ⇒ small ss content
f0(1710): ππ is suppressed relative to KK ⇒ primarily ss dominated
f0(1370): KK is suppressed relative to ππ ⇒ dominated by nn states
G
ss
⎛ MG
⎜
⎜ y
⎜⎜
⎝ 2y
y
MS
0
nn =
uu + dd
2
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
Further support:
„ f0(1710) is not seen in pp→ π0 f0 ⇒ f0(1710) dominated by ss
„ While f0(1710) is observed in γγ→KK, f0(1500) has not been seen in
γγ→KK or π+π- ⇒
⇒
absence of f0(1500) coupling to 2γ
glueball structure of f0(1500)
5
Other scenarios:
Lee, Weingarten (lattice): f0(1710) glueball ; f0(1500) nn ;
f0(1370) ss
Giacosa et al. (χ 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.
F.E. Close (Oxford): a leading figure in hadron spectroscopy
C. Amsler (Zurich): a dominant figure in searching for (exotic)
non-qq mesons
author of “Non-qq mesons”, Particle Data Group
6
7
Problems:
„ Near degeneracy of a0(1450) and K0*(1430) cannot be
explained due to the mass difference between MS and Mn
„ Improved LQCD ⇒ MG ∼ 1700 MeV rather than ∼ 1500 MeV
„ If f0(1710) is ss dominated, J/ψ→φf0(1710) >> J/ψ→ωf0(1710)
BES ⇒ Γ(J/ψ→ωf0(1710)) = (6.6±2.7) Γ( J/ψ →φf0(1710))
„ J/ψ→γgg and glueballs couple strongly to gluons
⇒
If f0(1500) is primarily a glueball,
J/ψ →γf0(1500) >> J/ψ→γf0(1710)
Expt ⇒
Γ(J/ψ →γf0(1710)) ∼ 5 Γ(J/ψ→γf0(1500))
8
In this work we consider two recent lattice inputs for mass matrix:
„ glueball spectrum from quenched LQCD
„ approximate SU(3) symmetry in scalar meson sector (> 1GeV)
Y. Chen et al. hep-lat/0510074
z improved quenched LQCD calculations
based on much larger & finer lattices
z infinite quark mass ⇒ no qq loops
M(0++)=1710 ± 50 ± 80 MeV
MG before mixing should be close to
1700 MeV
1650 MeV by Lee, Weingarten
1550 Mev by Bali
9
ψψ
IG (JPC) ≡ 1− (0+ + ), 1− (1+ + )
Mathur et al.
hep-ph/0607110
qq nature of a0(1450)
1.42±0.13 GeV
a0(980) is not seen ⇒ not a qq state !
a0(1450) mass is independent of quark mass when mq≤ ms
10
⇒ Flavor SU(3) is a good symmetry for scalar meson sector > 1 GeV
⇒ MS should be close to MN
LQCD ⇒
⇒
K0*(1430)=1.41±0.12 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
Near degeneracy also occurs in charm sector, Ds0*(2317) &
D0*(2308). This is the place where the conventional QM seems
not to work.
Belle:
2308±17±32 MeV for D0*
FOCUS: 2407±21±35 MeV
11
Lattice calculations of a0
Bardeen et al
Wilson fermion
ma0=1.326±0.086 GeV
RBC
domain wall fermion
ma0=1.43±0.10 GeV
ma0=1.58±0.34 GeV
SCALAR
Kunihiro et al
dynamical fermion
ma0∼ 1.8GeV
for mπ/mρ∼ 0.7
UKQCD(03’)
UKQCD(06’)
dynamical fermion
ma0=1.0±0.2 GeV
ma0=1.01±0.04 GeV
ma0 ∼ 1.45 GeV
BGR
MILC
Alford & Jaffe
stagged fermion
indication for a0
indication of bound qqqq
states
12
Prelovsek, Dawson, Izubuchi, Orginos, Soni PR, D70, 094503 (2004);
hep-ph/0511110: domain wall fermions
Mass for lightest nonsinglet two-quark a0 = 1.58 ± 0.34 GeV, which can be
identified with a0(1450)
13
ψ γ 5ψ ψ γ 5ψ [ππ , I (J
G
PC
+
) ≡ 0 (0
++
)]
Four quark spectrum (GeV)
2
1.8
1.6
Scattering states
1.4
m2
1.2
1
p=1
0.8
0.6
Eπ ( p = 1) + Eπ ( p = 1)
m1
p=0
0.4
m0
0.2
0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07
2
mπ (GeV2)
Possible BOUND state
σ(600)?
Eπ ( p = 0) + Eπ ( p = 0)
Scattering states
(Negative scattering
length)
In chiral limit, mσ=540±170 MeV
Need to check the volume dependence of the observed states
14
Two lattice sizes : 123x28, 163x28
For two-particle state, W12/W16=V3(16)/V3(12)=163/123=2.37
For one-particle state, W12/W16=1
scattering states
σ state
15
♦ Scalar qq meson has a unit of orbital angular momentum
) a higher mass above 1 GeV
f0(1370), a0(1450), K*0(1430) and f0(1500)/f0(1710) form a P-wave qq
nonet with some possible mixing with glueballs, supported by lattice.
♦ Four-quark qqq q scalar meson can be lighter due to
(i) absence of the orbital angular momentum barrier in S-wave 4-quark state.
(ii) a strong attraction between (qq ) 3 and (q q ) 3
) a mass near or below 1 GeV
Light scalar mesons σ, κ, f0(980), a0(980), form an S-wave nonet
16
Scalar Mesons and Glueball
f 0 (1710)
glueball
K 0* (1430)
*
0
K (1430)
qq
−
0
a (1470)
a00 (1470)
f 0 (1500)
K (1430)
K 0* (800)
2
q q
2
a (980)
K 0* (800)
+
f 0 (1370)
+
σ (600)
K 0* (1430)
*
0
−
0
a0+ (1470)
K 0* (800)
a00 (980)
f 0 (980)
a0+ (980)
K 0* (800)
17
uu
dd
⎛ MU
⎜
⎜ 0
M =⎜
0
⎜⎜
⎝ 0
0
MD
0
0
ss
0
0
MS
0
G
0 ⎞ ⎛x
⎟ ⎜
0 ⎟ ⎜x
+
0 ⎟ ⎜ xs
⎟⎟ ⎜⎜
MG ⎠ ⎝ y
x
x
xs
y
xs
xs
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
0
M + 3x
3y
G
0 ⎞
⎧a0 = (uu − dd ) / 2 = 1474, M U = M D = 1474 ± 19 MeV
⎟
⎪⎪
0 ⎟
⇒ ⎨ f 0 (1500) = (uu + dd − 2 ss ) / 6 = 1474
3y ⎟
⎪ f (1370) and glueball are slightly mixed
⎟
⎟
⎪⎩ 0
MG ⎠
„ 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
18
SU(3) Breaking
„ Need SU(3) breaking in mass matrix to lift degeneracy of a0(1450) and
f0(1500)
„ Need SU(3) breaking in decay amplitudes to accommodate observed
strong decays
e.g.
f 0 (1500) = α (| uu ⟩+ | dd ⟩ ) + β | ss ⟩
2
Γ( f 0 (1500) → KK ) 1 ⎛ β ⎞ p K
,
R1 =
= ⎜1 + ⎟
Γ( f 0 (1500) → ππ ) 3 ⎝ α ⎠ pπ
2
β ⎞ pη
Γ( f 0 (1500) → ηη ) 1 ⎛
R2 =
=
⎜2 + ⎟
α ⎠ pπ
Γ( f 0 (1500) → ππ ) 27 ⎝
For SU(3) octet f0(1500), β = -2α ⇒ R1=0.21 vs. 0.246±0.026 (expt)
R2=0
LQCD [Lee, Weingarten] ⇒
y= 43±31 MeV,
vs. 0.145±0.027 (expt)
y/ys=1.198±0.072
y and x are of the same order of magnitude !
SU(3) breaking effect is treated perturbatively
19
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, Γ(G → ππ ) / Γ(G → KK ) = 0.91
⎧0.20 ± 0.04 WA102
Γ( f 0 (1710) → ππ ) ⎪
= ⎨< 0.13
BES from J /ψ → ω ( KK , ππ )
Γ( f 0 (1710) → KK ) ⎪
+ 0.11
BES from J /ψ → γ ( KK , ππ )
⎩0.41-0.17
chiral suppression: A(G→qq) ∝ mq/mG in chiral limit
[Chanowitz]
Γ(G → ππ ) / Γ(G → KK ) ∝ mu2 / ms2 ?
Chiral suppression at hadron level is probably not so strong 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] ⇒
+0.372
+0.364
g ππ : g KK : gηη = 0.834 +−00..603
:
2
.
654
:
3
.
099
579
−0.402
−0.423
20
In absence of chiral suppression (i.e.
gππ=gKK=gηη), the predicted f0(1710)
width is too small (< 1 MeV) ⇒
importance of chiral suppression in
G→PP decay
Consider two different cases
of chiral suppression in G→PP:
(i)
g ππ : g KK : gηη = 1 : 1.55 : 1.59
(ii)
g ππ : g KK : gηη = 1 : 3.15 : 4.74
Scenario (ii) with larger chiral
suppression is preferred
21
Amseler-Close-Kirk
f 0 (1710) = 0.93 G + 0.32 N + 0.17 S
: primarily a glueball
f 0 (1500) = 0.03 G − 0.54 N + 0.84 S
: tend to be an SU(3) octet
f 0 (1370) = −0.36 G + 0.78 N + 0.52 S
: near SU(3) singlet + glueball content
(∼ 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),
Γ(J/ψ→ωf0(1710)) = 4.1 Γ( J/ψ →φf0(1710))
versus 6.6±2.7 (expt)
If f0(1710)=ss, one needs large doubly OZI ∼ 5 OZI
[Close, Zhao]
„ Γ(J/ψ →γf0(1710)) >> Γ(J/ψ→γf0(1500))
in good agreement with expt.
Γ(J/ψ→γf0(1710)) ∼ 5 Γ(J/ψ→γf0(1500))
22
„ f0(1710) is not seen in pp→π0f0 at LEAR (2002), but now
observed in pp→π0ηη at Fermilab (2006)
„ 2γ-quarkonium coupling
f0(1370): f0(1500): f0(1710) = 9.3 : 1.0 : 1.5
2γ coupling of f0(1500) is weak even if it has no glue content
23
Conclusions
„ We use two recent lattice results to constrain mixing matrix
of f0(1370), f0(1500) and f0(1710): (i) scalar glueball mass ∼ 1700
MeV, (ii) SU(3) symmetry in scalar meson sector > 1 GeV
„ 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
„ Chiral suppression in G→PP decays is essential. Hadronic
and radiative J/ψ decays all indicate prominent glueball nature
of f0(1710)
24