The Physics of Glueball
Vincent Mathieu
Universidad de Valencia
February 2011
V. M., N. Kochelev, V. Vento,
“The Physics of Glueballs”,
Int. J. Mod. Phys. E18 (2009) 1
V. M., V. Vento,
“Pseudoscalar Glueball and η − η 0 Mixing”,
Phys. Rev. D81, 034004 (2010)
C. Degrande, J.-M. Gérard and V. M.,
in preparation,
Vincent Mathieu (Univ. Valencia)
Glueballs Spectroscopy
February 2011
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Introduction
QCD = gauge theory with the color group SU (3)
LQCD
=
Gµν
=
X
1
− Tr Gµν Gνµ +
q̄(γ µ Dµ − m)q
4
∂µ Aµ − ∂ν Aµ − ig[Aµ , Aν ]
Quark = fundamental representation 3
Gluon = Adjoint representation 8
Observable particles = color singlet 1
Mesons:
Baryons:
Glueballs:
3 ⊗ 3̄ = 1 ⊕ 8
3 ⊗ 3 ⊗ 3 = 1 ⊕ 8 ⊕ 8 ⊕ 10
8 ⊗ 8 = (1 ⊕ 8 ⊕ 27) ⊕ (8 ⊕ 10 ⊕ 10)
8 ⊗ ··· ⊗ 8 = 1 ⊕ 8 ⊕ ...
Three light quarks → nine 0± mesons : 3π (I = 1) ⊕ 4K (I = 1/2) ⊕ 2η (I = 0)
Glueball can mix with two isoscalars → glue content in η’s wave function and maybe a third
isoscalar
Vincent Mathieu (Univ. Valencia)
Glueballs Spectroscopy
February 2011
2 / 38
Lattice QCD
Quenched Results
Investigation of the glueball spectrum (pure
gluonic operators) on a lattice by
Morningstar and Peardon,
Phys. Rev. D60 (1999) 034509]
Identification of 15 glueballs below 4 GeV
M (0++ )
=
1.730 ± 0.130 GeV
M (0−+ )
=
2.590 ± 0.170 GeV
M (2++ )
=
2.400 ± 0.145 GeV
Quenched approximation (gluodynamics) →
mixing with quarks is neglected
Unquenched Results
Lattice studies with nf = 2 exist. The lightest scalar would be sensitive to the inclusion of
sea quarks but no definitive conclusion.
Theoretical status of glueballs :
V. M., N. Kochelev, V. Vento, “The Physics of Glueballs”, Int. J. Mod. Phys. E18 (2009) 1
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Glueballs Spectroscopy
February 2011
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Phenomenology - Strong Coupling Constant
Deur, arXiv:0901.2190
Saturation of αs (Q2 ) at large
distances
Gribov, Eur. Phys. J., C10,71 (1999)
Dokshitzer and Webber,
Phys. Lett., B352, 451 (’95)
Dasgupta and Salam,
J. Phys. G 30, R143 (2004)
Power correction in event shapes
(µI = 2 GeV)
α0 = µ−1
I
Z
µI
α(Q2 )dQ2 ∼ 0.5
0
Vincent Mathieu (Univ. Valencia)
Glueballs Spectroscopy
February 2011
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Landau Gauge Propagators
7
Gluon Propagator SU(3)
6
=5.7
L = 80 and
=5.7
Gluon Propagator SU(3)
10
L=64 and
=6.0
L=64 and
=6.2
L=80 and
=6.0
Fit
8
]
-2
4
(q )[GeV
]
-2
=5.7
L = 72 and
Fit
5
3
6
2
2
(q )[GeV
Gluon propagator from
lattice QCD in Landau
gauge
L = 64 and
4
2
2
1
Cucchieri and Mendes,
PoS (Lattice 2007) 297
0
1E-3
0
0,01
0,1
1
2
10
100
0,01
0,1
1
2
2
q [GeV ]
10
100
2
q [GeV ]
Gluon Propagator SU(2)
4
Bogolubsky et al.,
PoS (Lattice 2007) 290
L = 128 and
= 2.2
Fit
2
2
Oliveira and Silva,
PoS (QCD-TNT 2009) 033
(q )[GeV
- 2
]
3
1
0
1E-3
0,01
0,1
1
2
10
100
2
q [GeV ]
Predicted by Cornwall Phys. Rev. D26, 1453 (1982)
by introducing a dynamically generated gluon mass
Vincent Mathieu (Univ. Valencia)
Glueballs Spectroscopy
February 2011
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Dynamical Mass Generation
Not a violation of gauge invariance
Schwinger Mechanism like in
(1 + 1)−QED
Phys. Rev. 128 (1962) 2425
Self-energy gains a dynamical pole
−i
qµ qν
∆(q 2 )µν = gµν − 2
q
q 2 − q 2 Π(q 2 )
Mass is the residue
Π(q 2 )|pole =
m2 (q 2 )
q2
Composite bound state not present in
physical processes
R. Jackiw and K. Johnson, Phys. Rev. D 8, 2386 (1973)
D. Binosi and J. Papavassiliou, Phys. Rept. 479, 1 (2009)
A. C. Aguilar, D. Ibañez, V. M. and J. Papavassiliou, “The Schwinger Mechanism in QCD”,
in preparation
Vincent Mathieu (Univ. Valencia)
Glueballs Spectroscopy
February 2011
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Gluon Mass
Cornwall Phys. Rev. D26, 1453 (1982)
Gluons massless in the Lagrangian
Couplings and nonperturbative effects
→Dynamical mass
extracted from gauge-invariant propagator
ˆ 2) =
d(q

2
2
m (q ) =
m20



ḡ 2 (q 2 )
+ m2 (q 2 )
−12/11
q2
ln
q 2 +4m2
0
Λ2
ln
4m2
0
Λ2



How degrees of freedom has the gluon ?
On-shell decoupling of the pole → 2 d.o.f
Gluon massless in the Lagrangien
Vincent Mathieu (Univ. Valencia)
vs
Gluon mass generation → 3 d.o.f
Glueballs Spectroscopy
February 2011
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Two Models
Hgg = 2
p
p2 + 94 σr − 3 αrs + VOGE
Gluons spin-1 usual rules of spin couplings
J = L + S with
Gluons transverse Gluons helicity-1 particles
J 6= L + S
S = 0, 1, 2
OGE no needed VOGE = 0
No vector states
3 d.o.f. not compatible with J +−
Instanton contribution
V.M. , PoS [QCD-TNT09], 024 (2009)
Vincent Mathieu (Univ. Valencia)
Gluon has only 2 d.o.f in bound state w. f.
Glueballs Spectroscopy
February 2011
8 / 38
Bag Model
R. L. Jaffe and K. Johnson, Phys. Lett. B 60 (1976) 201
J. Kuti, Nucl. Phys. Proc. Suppl. 73 (1999) 72
Free Particles Confined in a Cavity
Gluonic Modes in a Cavity
TE mode
J P =1+
xTE = 2.74,
TE mode
J
−
xTE = 3.96,
TM mode
J P =1−
xTM = 4.49.
P
=2
Mass Spectrum
E
=
M2
=
X xi
4πBR3
α a a~ ~
+
ni
−
λ1 λ2 S1 · S2
3
R
4R
i
X xi 2
E2 −
ni
R
i
α = 0.5
B = (280 MeV)4
J. F. Donoghue, Phys. Rev. D 29 (1984) 2559
Gluon mass 740 ± 100 MeV in the bag model
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Glueballs Spectroscopy
February 2011
9 / 38
AdS/QCD
AdS/CFT correspondance:
Correspondance between conformal theories and string theories in AdS spacetime
QCD not conformal → breaking conformal invariance somehow
Introduction of a black hole in
AdS to break conformal
invariance
Parameter adjusted on 2++
Same hierarchy but
some states are missing
(spin 3,...)
R. C. Brower et al., Nucl. Phys. B587 (2000) 249
Vincent Mathieu (Univ. Valencia)
Glueballs Spectroscopy
February 2011
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QCD Spectral Sum Rules
JS (x) = αS Tr Gµν Gµν
Gluonic currents:
Π(Q2 ) = i
Z
d4 x eiq·x h0|T JG (x)JG (0)|0i =
e µν
JP (x) = αS Tr Gµν G
1
π
∞
Z
0
ImΠ(s)
ds
s + Q2
Theoretical side (OPE):
µν
a
b
b
JG (x)JG (0) = C(a)+(b)+(e) 1 + C(c) Ga
µν Ga + C(d) fabc Gαβ Gβγ Gγα + · · ·
µν
Confinement parameterized with condensates h0|αs Ga
µν Ga |0i, . . .
Phenomenological side:
ImΠ(s) =
X
2
πfG
m4Gi δ(s − m2Gi ) + πθ(s − s0 )ImΠ(s)Cont
i
i
Vincent Mathieu (Univ. Valencia)
Glueballs Spectroscopy
February 2011
11 / 38
Correlators
Glueball masses extracted from correlators
0++
0−+
2++
β(g)
Tr Gµν Gµν
g
β(g)
e µν
JP (x) =
Tr Gµν G
g
1
JT,µν (x) = Tr Gµα Gαν − gµν JS
4
JS (x) =
Lattice: Euclidean Spacetime
Z
d3 xh0|T JG (x, t)JG (0)|0i ∝ e−MG t
Sum Rules: Spectral Decomposition
Z
Z
1 ∞ ρ(s)
i d4 x eiq·x h0|T JG (x)JG (0)|0i =
ds
π 0 s + Q2
2
ρ(s) = (2π)hG|JG |0i2 δ(s − MG
) + Cθ(s − s0 )
Vincent Mathieu (Univ. Valencia)
Glueballs Spectroscopy
February 2011
12 / 38
Low Energy Theorems
e µν
JS (x) = αS Tr Gµν Gµν
JP (x) = αS Tr Gµν G
Z
8πdO
ΠO (0) = lim i d4 x eiq·x h0|T JS (x)O(0)|0i =
h0|O|0i
b0
q 2 →0
Gluonic currents:
ΠS (Q2 = 0)
=
ΠP (Q2 = 0)
=
32π
µν
h0|αs Ga
µν Ga |0i
b0
mu md
h0|q̄q|0i
(8π)2
mu + md
Instantons contribution essential for LETs
Forkel, Phys. Rev. D71 (2005) 054008
M (0++ ) = 1.25 ± 0.20 GeV
Vincent Mathieu (Univ. Valencia)
M (0−+ ) = 2.20 ± 0.20 GeV
Glueballs Spectroscopy
February 2011
13 / 38
Padé Approximation and GZ propagators
Dudal et al, arXiv:1010.3638
Construction of spin-J glueball propagators
FJ (k2 ) =
Σ0
Z
0
X
σJ (s)
ds =
νn (−1)n (k2 )n
2
1 + sk
n
from GZ gluon propagator
D(k) =
k4
k2 + M 2
+ (m2 + M 2 )k2 + λ4
Lowest order Padé approximation p
after a subtraction analysis: mJ = ν0 /ν1
GZ Padé
m0++ = 1.96
m0−+ = 2.19
m2++ = 2.04
Models
m0++ = 1.98
m0−+ = 2.22
m2++ = 2.42
Lattice
m0++ = 1.73
m0−+ = 2.59
m2++ = 2.40
Coherent with Instanton contribution ∼ 300 MeV in 0±+
Problem induced by scale anomaly for the 2++
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Glueballs Spectroscopy
February 2011
14 / 38
Glueballs in the Real World
Crede and Meyer, The Experimental Status of Glueballs,
Prog. Part. Nucl. Phys. 63, 74 (2009)
Production in gluon rich processes (OZI forbidden,...)
Closely linked to the Pomeron:
J = 0.25M 2 + 1.08
Mixing between glueball 0++ and light mesons
0++
Scalar Candidates:
f0 (1370)
f0 (1500)
f0 (1710)
f0 (1810)
...
Pseudoscalar Candidates:
η(1295)
η(1405)
η(1475)
η(1760)
...
and
0−+
glueballs shared between those states
Three light quarks → 3 × 3 = 9 (pseudo)scalar mesons
A 10th light mesons would be the realization of the glueball
Vincent Mathieu (Univ. Valencia)
Glueballs Spectroscopy
February 2011
15 / 38
Scalar Glueball
Chiral Suppression
Chanowitz, Phys. Rev. Lett. 95 (1999) 172001
A 0++ −→ q̄q ∝ mq
Decay to K K̄ favoured over ππ
Controversy about mq (current or constituent mass ?)
A 0++ −→ K̄K
R=
>1
A (0++ −→ ππ)
f0 (1710) good glueball candidate
Results from Crystal Barrel (pp̄), OBELIX (pp̄), WA102 (pp), BES (J/ψ)
Name
f0 (1370)
Masse (MeV)
1200 − 1500
Width (MeV)
200 − 500
Decays
ππ, K K̄, ηη
f0 (1500)
1505 ± 6
109 ± 7
ππ, K K̄, ηη
f0 (1710)
1720 ± 6
135 ± 8
ππ, K K̄, ηη
Production
pp̄ → P P P , pp → pp(P P )
weak signal in J/ψ → γ(P P )
J/ψ → γ(P P ), pp → pp(P P )
pp̄ → P P P
J/ψ → γ(P P ), pp → pp(P P )
not seen in pp̄
Belle and BaBar puzzle : f0 (1500) strong coupling to K K̄ and weak to ππ
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Glueballs Spectroscopy
February 2011
16 / 38
Physical States
Pure states: |ggi, |nn̄i, |ss̄i
|Gi = |ggi +
hnn̄|ggi
hss̄|ggi
|nn̄i +
|ss̄i
Mgg − Mnn̄
Mgg − Mss̄
Analysis of
Production:
J/ψ → γf0 , ωf0 , φf0
Decay:
f0 → ππ, K K̄, ηη
Two mixing schemes
Cheng et al, Phys. Rev. D74 (2006) 094005
Close and Kirk, PLB483 (2000) 345
Mnn̄ < Mss̄ < Mgg
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Mnn̄ < Mgg < Mss̄
Glueballs Spectroscopy
February 2011
17 / 38
Pseudoscalar
Mark III : observation of two pseudoscalar in
J/ψ → γP P P
γγ fusion : η(1475) seen but not η(1405)
Results from Crystal Barrel (pp̄), OBELIX (pp̄)
Name
η(1295)
η(1405)
η(1475)
Masse (MeV)
1294 ± 4
1409.8 ± 2.5
1476 ± 4
Width (MeV)
55 ± 5
51.1 ± 3.4
87 ± 9
Decays
γγ, K K̄π, a0 π
K K̄π, a0 π, ηππ
γγ, K K̄ ∗ , K K̄π, a0 π
Production
not seen in pp̄
not seen in γγ
η(1205) and η(1475) radial excitations of η and η 0 .
η(1405) glueball candidate
Possible glue content in η 0
Γ(J/ψ → η 0 γ)
=
Γ(J/ψ → ηγ)
Vincent Mathieu (Univ. Valencia)
h0|GG̃|η 0 i
h0|GG̃|ηi
!2
2
MJ/ψ
− Mη20
2
MJ/ψ
− Mη2
Glueballs Spectroscopy
!3
= 4.81 ± 0.77
February 2011
18 / 38
Physical Pseudoscalar States
Axial Anomaly → mixing with η1
∂ µ (q̄γµ γ5 q) = 2imq̄γ5 q +
Mixing Scheme with pseudoscalar glueball

 
|ηi
cos ϕ − sin ϕ
 |η 0 i  =  sin ϕ
cos ϕ
|η 00 i
0
0
αS
1
Gµν G̃µν → gg|∂ µ Jµ5
|0 6= 0
4π

0
1
0 0
1
0
0
cos φG
sin φG


0
|η8 i
− sin φG  |η1 i
cos φG
|ggi
2 = hη 0 |ggi2
Measurement of ZG
V → P γ and P → V γ →
0.04 ± 0.09
Escribano and Nadal (2007)
Br(φ(1020) → η 0 γ)
→
Br(φ(1020) → ηγ)
0.12 ± 0.04
KLOE collaboration (2008)
J/ψ → V P →
0.28 ± 0.21
Escribano (2008)
Where is the physical pseudoscalar glueball |η 00 i ?
η(1295),
η(1405),
η(1475),
...
Cheng et al., Phys. Rev D 79 (2008) 014024
Vincent Mathieu (Univ. Valencia)
Glueballs Spectroscopy
February 2011
19 / 38
Chiral Symmetry
QCD = gauge theory with the color group SU (3)
LQCD
=
Gµν
=
X
1
− Tr Gµν Gµν +
q̄(γ µ Dµ − m)q
4
∂µ Aµ − ∂ν Aµ − ig[Aµ , Aν ]
Degrees of freedom at High Energies: Quarks q and Gluons Aµ
Degrees of freedom at Low Energies: Pions, Kaons,...
Goldstone bosons of Chiral Symmetry Breaking
(Global) Chiral Symmetry: U (3)V ⊗ U (3)A
U (3)V :
q
→ exp(iθa λa )q
U (3)A :
q
→ exp(iγ5 θ5a λa )q
U (3)A broken spontaneously by quark condensate h0|q̄q|0i 6= 0
U (3)A broken explicitly by quark masses m
→ 9 Goldstone pseudoscalar bosons with a small mass ∝ mh0|q̄q|0i
3π, 4K and 2η
Vincent Mathieu (Univ. Valencia)
Glueballs Spectroscopy
February 2011
20 / 38
Chiral Anomaly - U (1) Problem
Observation of only 8 Goldstone bosons
Anomaly: Classical symmetry broken by quantum effects
Z
Z[J]
DqDq̄
=
U (1)A
−→
DqDq̄DA exp(LQCD + J · A + η̄q + η q̄)
DqDq̄ exp(θαβµν Gµν Gαβ )
Variation is a total derivative
αβµν Gµν Gαβ = ∂µ K µ
But non trivial gauge configuration (Instanton) with different winding number θ (topological
charge)
U (1)A is not a symmetry of QCD
→ η 0 is NOT a Goldstone boson Mη0 ∼ 958 MeV > Mη ∼ 548 MeV
But Anomaly vanishes for large N , for a gauge group SU (N → ∞)
Vincent Mathieu (Univ. Valencia)
Glueballs Spectroscopy
February 2011
21 / 38
Chiral Lagrangian
Lagrangian with 9 Golstone Boson π a in the large N limit
Nonlinear parametrization
 π0
√
√
U = exp(i 2π/f )
U ∈ U (3)
√ 
π = 2

UU† = 1
2
+
η8
√
6
π−
+
η0
√
3
π+
η8
√
6
−
K−
U → LU R†
π0
√
2
K̄ 0
K+
+
L ∈ U (3)L ,
η0
√
3
K0
q
− 23 η8 +

η0
√
3



R ∈ U (3)R
Effective Lagrangian based on Symmetry with a Momentum Expansion p2
Kinetic term at O(p2 )
E
f2 D
∂µ U † ∂ µ U
8
but not
1
ED
E
f2 D †
U ∂µ U
U ∂µU †
8
Symmetry breaking terms
Explicit breaking
U (1) Anomaly
Vincent Mathieu (Univ. Valencia)
E
f2 D
mU † + U m†
8
2
α0 f
det U
3
ln
∼ − α0 η02
2 4
det U †
2
B
Glueballs Spectroscopy
February 2011
22 / 38
Mass Matrix at Leading Order
Chiral Lagrangian at leading order (in p2 and 1/N )
L(p
Expand U = 1 +
√
2
)
=
E 1
f2 D
∂µ U † ∂ µ U + B(mU † + U m† ) − α0 η02
8
2
2πa λa /f − (πa λa )2 /f 2 + · · ·
L(p
2
)
=
1
1
3
∂µ π a ∂ µ π b δab − Bπ a π b hλa λb mi − α0 η02
2
2
2
Isospin Symmetry m = diag (m̃, m̃, m̃s ) → m2π = B m̃ and m2K = B(m̃ + ms )/2
Mass matrix in the octet-singlet (η8 − η0 )
1
4m2K − m2π
√
M280 =
3 −2 2(m2K − m2π )
Mass matrix in the flavor basis (ηq − ηs )
2
mπ√+ 2α0
M2qs =
2α0
√
−2 2(m2K − m2π )
2
2
2mK + mπ + 9α0
√
2m2K
2α0
− m2π + α0
Anomaly only source of mixing
No anomaly for vector 1−− −→ ω and φ ideal mixed
Vincent Mathieu (Univ. Valencia)
Glueballs Spectroscopy
February 2011
23 / 38
Mass Matrix at Leading Order
Effective Lagrangian for vector (without anomaly)
2
L(p
)
=−
f2 (∂µ Vν − ∂µ Vµ )2 − 2B 0 mVµ V µ
8
Mass matrix in the flavor basis (ηq − ηs )
2
mρ
M2qs =
0
0
2m2K ∗
−
m2ρ
Theory at LO: Physical states are q q̄ and ss̄
Experiments: Decay Properties ω → πππ, φ → K + K −
ω(782) ∼ q q̄
φ(1020) ∼ ss̄
2 mass predictions at leading order:
m2ω = m2ρ
Satisfied at 3%
Gell-Mann−Okubo mass formula
m2φ = 2m2K ∗ − m2ρ
Satisfied at 8%
Vincent Mathieu (Univ. Valencia)
Glueballs Spectroscopy
February 2011
24 / 38
Mass Matrix at Leading Order
Mass matrix in the flavor basis (ηq − ηs )
M2qs =
2
mπ√+ 2α0
2α0
√
2m2K
2α0
− m2π + α0
Rotation to Physical States
R† (φ)M2qs R(φ) =
φ determines Decay Properties
η
cos φ
=
η0
sin φ
2
mη
0
− sin φ
cos φ
0
m2η0
ηq
ηs
.
Conservation of trace and determinant
m2η + m2η0
=
2m2K + 3α0
m2η m2η0
=
(4m2K − m2π )α0 + m2π (2m2K − m2π )
Vincent Mathieu (Univ. Valencia)
Glueballs Spectroscopy
February 2011
25 / 38
η − η 0 Mixing at Leading Order
Only 1 parameter α0 but 2 states (or 2
invariants)
Trace
α0 = (m2η + m2η0 − 2m2K )/3
Determinant
α0 =
m2η m2η0 − m2π (2m2K − m2π )
4m2K − m2π
! Not Equal !
2 photons decays : η → γγ and η 0 → γγ
Degrande and Gérard, JHEP 0905 (2009) 043
θ ∼ −(20 − 15)◦ in the U (3) basis (η8 − η0 )
φ ∼ (40 − 45)◦ in the flavor basis (ηq − ηs )
θ = φ − θi
with the ideal √
mixing angle
θi = arccos(1/ 3) ∼ 54.7◦
Vincent Mathieu (Univ. Valencia)
Glueballs Spectroscopy
February 2011
26 / 38
Decays with respect to θ
Γ(η 0 → ωγ)
Γ(ω → ηγ)
Γ(η 0 → ργ)
Γ(ρ → ηγ)
Γ(φ → η 0 γ)
Γ(φ → ηγ)
Γ(J/Ψ → η 0 γ)
Γ(J/Ψ → ηγ)
Γ(J/Ψ → ρη 0 )
Γ(J/Ψ → ρη)
Γ(J/Ψ → φη 0 )
Γ(J/Ψ → φη)
Glueballs Spectroscopy
February 2011
Vincent Mathieu (Univ. Valencia)
27 / 38
Improvements
1
Next to leading order O(p4 )
Different decay constants fπ 6= fK
J. Schechter, A. Subbaraman and H. Weigel, Phys. Rev. D 48, 339 (1993)
J. M. Gerard and E. Kou, Phys. Lett. B 616, 85 (2005)
V. Mathieu, V. Vento, Phys.Lett.B688:314-318,2010
2
Including a third gluonic state
√
2
2α0
mπ√+ 2α0
M2qs =
⇒ 3 × 3 Matrix M2qsg
2
2
2α0
2mK − mπ + α0
Description of the “glue” content in η 0
Γ(J/ψ → η 0 γ)
=
Γ(J/ψ → ηγ)
h0|GG̃|η 0 i
h0|GG̃|ηi
!2
2
MJ/ψ
− Mη20
2
MJ/ψ
− Mη2
!3
= 4.81 ± 0.77
V. Mathieu, V. Vento, Phys. Rev. D81, 034004 (2010)
Degrande, V.M., Gérard, in preparation
Vincent Mathieu (Univ. Valencia)
Glueballs Spectroscopy
February 2011
28 / 38
Third Gluonic State
Possible small “glue” content in η 0
Γ(J/ψ → η 0 γ)
Γ(J/ψ → ηγ)
=
=
h0|GG̃|η 0 i
h0|GG̃|ηi
!2
2
MJ/ψ
− Mη20
!3
2
MJ/ψ
− Mη2
4.81 ± 0.77
θ̂Gµν G̃µν →Glueball ∼ massive axion in the chiral Lagrangian
L(p
2
)
=
E α
1
1
f2 D
∂µ U † ∂ µ U + B(mU † + U m† ) − (η0 + θ)2 − m2θ θ2 + ∂µ θ∂ µ θ
8
2
2
2
Inclusion of a gluonic state via the θ−term and the anomaly
√
 2
mπ√+ 2α
2α
2
Mqsg =  √2α
2m2K − m2π + α
2β
β
√

2β
β 
γ
Rosenzweig, Salomone, and Schechter, Phys. Rev. D24 (1981) 2545
Degrande, V.M., Gérard, in preparation
Vincent Mathieu (Univ. Valencia)
Glueballs Spectroscopy
February 2011
29 / 38
Third Gluonic State
What are the theoretical constraints on R ?




|ηi
|ηq i
|η 0 i = R |ηs i
|Gi
|ggi
Inclusion of a gluonic state via the θ−term and the anomaly
√
 2
mπ√+ 2α
2α
2
Mqsg =  √2α
2m2K − m2π + α
2β
β
√

2β
β 
γ
Diagonalization
R† M2qsg R = D
2 ) with M 2 unknown
D = diag (Mη2 , Mη20 , MG
G
Three unknowns α, β, γ but three rotation invariants
α, β, γ ≡ F (Mη2 , Mη20 , Mη2θ )
2
β 2 ≥ 0 −→ MG
≥
Vincent Mathieu (Univ. Valencia)
2
2
⇒ R(Mη2 , Mη20 , MG
) ⇒ Decays(MG
)
2 − M 2 )2
(MK
8
1
π
2
(4MK
− Mπ2 ) +
∼ (1.5 GeV)2
2
3
3 4MK − Mπ2 − 3Mη2
Glueballs Spectroscopy
February 2011
30 / 38
Decay with Glueball for Mη = 530 MeV and Mη0 = 1030 MeV
Γ(η 0 → ωγ)
Γ(ω → ηγ)
Γ(η 0 → ργ)
Γ(ρ → ηγ)
Γ(η(0 ) → γγ)
Γ(π → γγ)
Γ(J/Ψ → η 0 γ)
Γ(J/Ψ → ηγ)
Γ(J/Ψ → ρη 0 )
Γ(J/Ψ → ρη)
Γ(J/Ψ → φη 0 )
Γ(J/Ψ → φη)
Glueballs Spectroscopy
February 2011
Vincent Mathieu (Univ. Valencia)
31 / 38
η 00 = η(1405)? decays for Mη = 530 MeV and Mη0 = 1030 MeV
Γ(η 00 → ωγ)
Γ(ω → ηγ)
Γ(η 00 → ργ)
Γ(ρ → ηγ)
Γ(η 00 → φγ)
Γ(φ → ηγ)
Γ(J/Ψ → η 00 γ)
Γ(J/Ψ → ηγ)
Γ(J/Ψ → ρη 00 )
Γ(J/Ψ → ρη)
Γ(J/Ψ → φη 00 )
Γ(J/Ψ → φη)
Glueballs Spectroscopy
February 2011
Vincent Mathieu (Univ. Valencia)
32 / 38
Preliminary Results
Degrande, V.M., Gérard, in preparation
Glueball ∼ massive axion in the chiral Lagrangian
L(p
2
)
=
E α
f2 D
1
1
∂µ U † ∂ µ U + B(mU † + U m† ) − (η0 + θ)2 − m2θ θ2 + ∂µ θ∂ µ θ
8
2
2
2
Decays in better agreement with data but very sensitive to Mη
θ = −11.4◦
ϕG = (4.7 ± 0.3)◦
ϕ = (46.8 ± 1.8)◦
Mη = 530 MeV
Mη0 = 1030 MeV
MG = 1400 − 1500 MeV
η(1405) would be the third state, mainly gluonic !
BUT only leading order
Gérard and Kou, Phys.Rev.Lett.97 (2006) 261804
Other anomalous decays
Br(B 0 → K 0 η 0 ) = 65 10−6 ,
Br(B 0 → K 0 η) < 2 10−6 ,
Br(B 0 → K 0 π 0 ) = 10 10−6 .
η0 : η : π = 3 : 0 : 1
via Penguin diagram
Vincent Mathieu (Univ. Valencia)
Glueballs Spectroscopy
February 2011
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Summary
Pure Yang-Mills
Various models reproduce lattice spectrum
Dynamical gluon mass generation BUT only 2 d.o.f. in wave function
Instanton effects → mass difference between 0++ and 0−+
Scalar Mesons
Chiral suppression A 0++ −→ q̄q ∝ mq
→ f0 (1710) realization of the glueball ?
Not without controversy
A confirmation of three f0 is required
Accurate data about their productions and decays would improve the understanding of
their structure.
Pseudoscalar Mesons
Anomaly in pseudoscalar → no ideal mixing for η and η 0
Chiral Lagrangian at LO not enough to describe η and η 0
→ Inclusion of glueball in chiral Lagrangian
Preliminary results favor η(1405) realisation of the glueball
Prediction for η 00 decays → need experimental confirmation !
Vincent Mathieu (Univ. Valencia)
Glueballs Spectroscopy
February 2011
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Backup Slides
Backup Slides
Vincent Mathieu (Univ. Valencia)
Glueballs Spectroscopy
February 2011
35 / 38
Decays with respect to θ
Γ(η 0 → ωγ)
Γ(ω → ηγ)
Γ(η 0 → ργ)
Γ(ρ → ηγ)
Γ(φ → η 0 γ)
Γ(φ → ηγ)
Γ(J/Ψ → η 0 γ)
Γ(J/Ψ → ηγ)
Γ(J/Ψ → ρη 0 )
Γ(J/Ψ → ρη)
Γ(J/Ψ → φη 0 )
Γ(J/Ψ → φη)
Glueballs Spectroscopy
February 2011
Vincent Mathieu (Univ. Valencia)
36 / 38
Decay with Glueball
Γ(η 0 → ωγ)
Γ(ω → ηγ)
Γ(η 0 → ργ)
Γ(ρ → ηγ)
Γ(J/Ψ → ωη 0 )
Γ(J/Ψ → ωη)
Γ(J/Ψ → η 0 γ)
Γ(J/Ψ → ηγ)
Γ(J/Ψ → ρη 0 )
Γ(J/Ψ → ρη)
Γ(J/Ψ → φη 0 )
Γ(J/Ψ → φη)
Glueballs Spectroscopy
February 2011
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η 00 Decays
Γ(η 00 → ργ)
Γ(ρ → ηγ)
Γ(η 00 → ωγ)
Γ(ω → ηγ)
Γ(η 00 → φγ)
Γ(φ → ηγ)
Γ(J/Ψ → η 00 γ)
Γ(J/Ψ → ηγ)
Γ(J/Ψ → ωη 00 )
Γ(J/Ψ → ωη)
Γ(J/Ψ → φη 00 )
Γ(J/Ψ → φη)
Glueballs Spectroscopy
February 2011
Vincent Mathieu (Univ. Valencia)
38 / 38