the physics of glueballs

8/12/2019 The Physics of Glueballs

1/21

The Physics of Glueballs

Vincent Mathieu

Universite de Mons, Belgique

Zakopane, February 2009

Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 1 / 21
http://find/http://goback/


2/21

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

http://find/


3/21

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]]



4/21

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

http://find/


5/21

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].

http://find/


7/21

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.

http://find/


8/21

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)]



9/21

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

http://find/


10/21

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,...

http://find/


11/21


12/21

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
http://find/


13/21

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)

http://find/


14/21

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


15/21

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


H F
http://find/


16/21

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 ,


H F
http://find/


17/21

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

.


A ca o
http://find/


18/21

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

Extension for three-body systems ?Vincent Mathieu (UMons) The Physics of Glueballs Zakopane, February 2009 18 / 21

Three gluon glueballs with transverse gluons
http://find/


19/21

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


Conclusion
http://find/


20/21

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


References
http://find/


21/21

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]]


the physics of glueballs

Documents