the physics of glueballs
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The Physics of Glueballs
Vincent Mathieu
Universite de Mons, Belgique
Zakopane, February 2009
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Outline
1 Introduction
2 Lattice QCD
3 AdS/QCD
4 QCD Spectral Sum Rules
5 Gluon mass
6 Two-gluon glueballs C= +
7 Three-gluon glueballs C=
8 Helicity formalism for transverse gluons
9 Conclusion
10 References
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References
V. Mathieu, N. Kochelev, V. VentoThe Physics of Glueballsinvited review for Int. J. Mod. Phys. EarXiv:0810.4453 [hep-ph]
V. Mathieu, F. Buisseret and C. Semay
Gluons in Glueballs: Spin or Helicity ? Phys. Rev. D 77, 114022 (2008), [arXiv:0802.0088 [hep-ph]]
V. Mathieu, C. Semay and B. Silvestre-BracSemirelativistic Potential Model for Three-gluon Glueballs.Phys. Rev. D77094009 (2008), [arXiv:0803.0815 [hep-ph]]
N. Boulanger, F. Buisseret,V. Mathieu, C. SemayConstituent Gluon Interpretation of Glueballs and Gluelumps.Eur. Phys. J. A38317 (2008) [arXiv:0806.3174 [hep-ph]]
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Introduction - QCD
QCD = gauge theory with thecolor group SU(3)
LQCD = 1
4Tr GG
+
q(D m)q
G = A A ig[A, A ]
Quark = fundamental representation 3Gluon = Adjoint representation 8Observable particles = color singlet 1
Mesons: 3 3= 1 8
Baryons: 3 3 3= 1 8 8 10
Glueballs: 8 8= (1 8 27) (8 10 10)
8 8= 1 8 . . .
Colored gluons color singlet with only gluons
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Introduction - Glueballs
Predictionof the QCDProduction ingluon rich processes(OZI forbidden,...)
Closely linked to thePomeron:
J= 0.25M2 + 1.08
Mixing between glueball 0++ and light mesons
Candidates: f0(1370) f0(1500) f0(1710)
One scalar glueball between those states [Klempt, Phys. Rep. 454].
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Physical States
Pure states: |gg, |nn, |ss
|G= |gg + nn|ggMgg Mnn
|nn + ss|ggMgg Mss
|ss
Analysis of
Production: J/ f0, f0, f0 Decay: f0 , KK,
Two mixing schemesCheng et al[PRD74, 094005 (2006)]
Mnn < Mss < Mgg
Close and Kirk [PLB483, 345 (2000)]
Mnn < Mgg < Mss
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Lattice QCD
Investigation of the glueball spectrum(pure gluonic operators) on a lattice byMorningstar and Peardon[Phys. Rev. D60, 034509 (1999)]Identification of15 glueballsbelow 4 GeV
M(0++) = 1.730 0.130 GeV
M(0+) = 2.590 0.170 GeV
M(2++) = 2.400 0.145 GeV
Quenchedapproximation (gluodynamics)
mixing with quarks is neglected
Lattice studies with nf= 2 exist. The lightest scalar would be sensitive to theinclusion of sea quarks butno definitive conclusion.
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AdS/QCD
AdS/CFT correspondance:
Correspondance between conformal theories and string theories in AdS spacetimeQCD not conformal breaking conformal invariance somehow
Introduction of a black hole
in AdS to break conformalinvariance
Parameter adjusted on 2++
Same hierarchybutsome states are missing
(spin 3,...)
[R. C. Brower et al., Nucl. Phys. B587, 249 (2000)]
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QCD Spectral Sum Rules
Gluonic currents: JS(x) =sGa(x)G
a(x) JP(x) =sG
a(x) G
a(x)
(Q2) =i
d4x eiqx0|T JG(x)JG(0)|0= 1
0
Im(s)
s + Q2ds
Theoreticalside (OPE):
JG(x)JG(0) =C(a)+(b)+(e)1 + C(c)GaG
a + C(d)fabcG
aG
bG
b +
Confinement parameterized with condensates 0|sGaG
a |0, . . .
Phenomenologicalside:
Im(s) =
i
f2Gim4Gi(s m
2Gi) + (s s0)Im(s)
Cont
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Sum Rules Constituent models
Sum rulescalculation by Forkel [Phys. Rev. D71, 054008 (2005)]
M(0++) = 1.25 0.20 GeV M(0+) = 2.20 0.20 GeV (1)
Two gluonsininteraction
Two gluons C= + Three gluons C=
Works on constituent models by Bicudo,Llanes-Estrada, Szczepaniak, Simonov,...
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Two-gluon glueballs C= +
Brau and Semay [Phys. Rev. D70, 014017(2004)]Two-gluon glueballs C= + and P = (1)L
J=L + Swith S= 0, 1, 2.
H0 = 2
p2 +9
4r 3
sr
.
Cornell potentialdoes not lift the degeneracybetween states with different S
Corrections of order O(1/2), with
= |
p2|
Structures coming from theOGE
Voge=
14
+13S2
U(r) 2
(r)
52S2 4
3
22U(r)
r L S
1
62
U(r)
r U(r)
T
with = 3S and U(r) = exp(r)/r.Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 12 / 21
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Two-gluon glueballs C= +
Smearing of the attractive 3
(r) by agluon size
(r, ) = exp(r/)/r
Replacement of potentials byconvolutionswiththe size function
V(r) V(r, ) =
V(r+ r)(r, )dr
Parameters
= 0.21 GeV2 s = 0.50 = 0.5 GeV1
Good agreement with lattice QCDGluons =spin 1 spurious J= 1 statesandindetermination of statesFor instance, 0++ can be (L, S) = (0, 0) or (2, 2)
JPC (L, S)
0++
(0,0) (2,2) (0,0)
0+ (1,1) (1,1)
2++ (0,2) (2,0) (2,2)2+ (1,1) (1,1)
3++ (2,2)4++ (2,2)
1++ (2,2)1+ (1,1) (1,1)
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Three-gluon glueballs C=
The Model
Extension of the model to three-gluon systems
H=3
i=1
p2i +9
4
3
i=1
|ri Rcm| + VOGE
Same parameters(, S, ) as for two-gluonglueballs
VOGE = 3
2S
3
i
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Three-gluon glueballs C=
The resultsGluons with spin J=L + S
Good results for 1 and 3 buthigher 2 symmetry
Disagreement with lattice QCDforP C= +This model cannot explain the splitting
2 GeV between 1+
and 0+
0+, 1+, 2+, 3+ are L= 1 anddegenerate in a model with spin
JPC (L, S) JPC (L, S)
1 (0,1) 0+ (1,1)
2
(0,2) 1+
(1,1)3 (0,3) 2+ (1,1)0+ (1,1) 3+ (1,2)
Solution: Implementation of the helicityformalism for three transverse gluons.
dabcAaA
bA
c [(88)8s8]
1 C=
fabcAa
Ab
Ac
[(88)8a8]
1 C= +
S Sint Symmetry JPC
0 0 1 A 0+
1 0, 1, 2 1 S, 2 MS 1
2 1, 2 2 MS 2
3 2 1 S 3
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H F
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Helicity Formalism for two-gluon glueballs
Yangs theorem: gg J= 1
J =L + Scannot hold for relativisticsystem. Jis the only relevant quantumnumber.Solution: Helicity formalism for transversegluons with helicity si = 1. Only twoprojection,i =1.
Formalism to handle with massless particles[Jacob and Wick (1959)]State JPC in term of usual (L, S) states (with =1 2):
|J, M; 1, 2=
L,S
2L + 1
2J+ 1
1/2
L0S| J s11s2 2| S
2S+1LJ
.
Not eigenstates of theparityand of thepermutation operator.
P |J, M; 1, 2 = 12(1)J |J, M; 1, 2 ,
P12 |J, M; 1, 2 = (1)J2si |J, M; 2, 1 ,
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H F
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Helicity Formalism for two-gluon glueballs
Construction of states symmetric (bosons) and with a good parity selection rules.
S+; (2k)+ 0++, 2++, 4++, . . .
S; (2k)
0+, 2+, 4+, . . .
D+; (2k+ 2)+ 2++, 4++, . . .
D; (2k+ 3)+ 3++, 5++, . . .
States expressed in term of the usual basis (useful to compute matrix elements!).
Low-lying states:
S+; 0+ =
2
3
1S0
+
1
3
5D0
,
S; 0
=
3P0
,
D+; 2+
=
2
5
5
S2
+
4
7
5
D2
+
1
7
5
G2
,
D; 3+ =
5
7
5D3
+
2
7
5G3
,
S; 2
=
2
5
3P2
+
3
5
3F3
.
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A ca o
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Application
Same hierarchyas the lattice QCDApplication with a simple Cornellpotential
H0 = 2
p2 +9
4r 3
sr
.
Parameters:= 0.185 GeV2 s = 0.45
Addition of an instantoninduced interaction to split the degeneracy between the 0++
and 0+HI=PIJ,0 with I= 450 MeV.
Instanton attractive in the scalar channel and repulsive in the pseudoscalar and equalin magnitudeVery good agreementwithout spin-dependent potential
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Three gluon glueballs with transverse gluons
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Three-gluon glueballs with transverse gluons
SU(2) SU(3) decomposition for two gluons to the lowest J:
a
b=
(ab) (ab)
[ab]
LowestJallowed for 3 gluons with helicity:a
b
c
= (abc)
(abc)
[abc]
Low-lying states are J= 1 and J= 3 withsymmetric colourfunction and J= 0 withan antisymmetric colour function
Low-lying states are the1, 3 and the 0+
J= 0P are not allowed for three-gluon glueballs 0+ four transverse gluons
The result should be confirmed by a detailedanalysis of thethree-body helicity formalismThese developments are under construction
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Conclusion
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Conclusion
Model withspin-1 gluonsreproduces the puregauge spectrum butspin-dependentpotentialsshould be added and the spectrum is plagued with
unwanted statesThree-gluon glueballs withmassive gluon cannotreproduce the lattice data
With twotransverse gluons, a simplelinear+Coulomb potential reproduce the latticeQCD results.Implementation of thehelicity formalismfor threetransverse gluons should solve the hierarchyproblem
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References
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References
V. Mathieu, N. Kochelev, V. VentoThe Physics of Glueballsinvited review for Int. J. Mod. Phys. EarXiv:0810.4453 [hep-ph]
V. Mathieu, F. Buisseret and C. Semay
Gluons in Glueballs: Spin or Helicity ? Phys. Rev. D 77, 114022 (2008), [arXiv:0802.0088 [hep-ph]]
V. Mathieu, C. Semay and B. Silvestre-BracSemirelativistic Potential Model for Three-gluon Glueballs.Phys. Rev. D77094009 (2008), [arXiv:0803.0815 [hep-ph]]
N. Boulanger, F. Buisseret,V. Mathieu, C. SemayConstituent Gluon Interpretation of Glueballs and Gluelumps.Eur. Phys. J. A38317 (2008) [arXiv:0806.3174 [hep-ph]]
Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 21 / 21
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