Yang-Mills Theory in Coulomb Gauge

H. Reinhardt

Tübingen

C. Feuchter & H. R. hep-th/0402106, PRD70

hep-th/0408237, PRD71

hep-th/0408236

D. Epple, C. Feuchter, H.R., hep-th/0412231

non-perturbative approach to continuum YMT

W. SchleifenbaumM. LederH. Turan

Previous work:

A.P. Szczepaniak, E. S. Swanson, Phys. Rev. 65 (2002) 025012

A.P. Szczepaniak, hep-ph/0306030

P.O. Bowman, A.P. Szczepaniak, hep-ph/0403074

Plan of the talk

• Basics of continuum Yang-Mills theory in Coulomb gauge

• Variational solution of the YM Schrödinger equation: Dyson- Schwinger equations

• Results:– Ghost and gluon propagators– Heavy quark potential– Color electric field of static sources

• YM wave functional• Finite temperatures• Connection to the center vortex picture of confinement

Classical Yang-Mills theory

24

41 ))((2 xFxdLg

AAAAxF ,)(

Lagrange function:

field strength tensor

Canonical Quantization of Yang-Mills theory

)()(/)( momenta xExALx ai

ai

ai

)( scoordinatecartesian xAa

0)( :gauge Weyl 0 xAa0)(0 xa

)(/)( :onquantizati xAix ak

ak

))()(( 22321 xBxxdH

Gauß law: mD

)()( :)x U(invariance gauge residual AAU

Coulomb gauge

mD Gauß law:

)( ,)( 1||mAD resolution of

Gauß´ law

)()(*)(| AAAJDAcurved space

Faddeev-Popov )()( DDetAJ

A 0, A A

|| , / i A

YM Hamiltonian in Coulomb gauge

)( 2||||1121 BJJJJH

-arises from Gauß´law =neccessary to maintain gauge invariance -provides the confining potential

Coulomb term11

C 2

1 1 2 112

m

H J J

J ( D ) ( )( D ) J

color density: A

Christ and Lee

Importance of the Faddeev-Popov determinant

ˆDet( D )

defines the metric in the space of gauge orbitsand hence reflects the gauge invariance

aim: solving the Yang-Mills Schrödinger eq.

for the vacuum by the variational principle

with suitable ansätze for

min)()()(

AHAADAJH

space of gauge orbits: metric )( DDetJ

Vacuum wave functional

determined from variational kernel

at the Gribov horizon:wave function is singular

-identifies all configurations on the Gribov horizon preserves gauge invariance -topolog. compactification of the Gribov region

FMR

QM: particle in a L=0-state

2*

12*

12

21 |

)(r , )(

drdrr

rJr

r

Minimization of the energy

set of Schwinger-Dyson equations for:

),...(k

Gluon propagator

2/ jiijijt

AAAAAAAAAAAAAAAA

transversal projector

Wick´s theorem:any vacuum expectation value of field operators can be expressed by the gluon propagator

Ghost propagator

ghost form factor d

Abelian case d=1

ghost self-energy

1 dG ( D )

Ghost-gluon vertex

rain-bow ladder approx: replace full vertex by bare one

bare vertex

constant structure )ˆ( abcbca fT

Curvature (ghost part of the gluon energy)

Coulomb form factor f

1f(k)order leading

Schwinger-Dyson eq.

)(),(),(),( fd

Regularization and renormalization:momentum subtraction scheme renormalization constants:

ultrviolet and infrared asymtotic behaviour of the solutions to the Schwinger Dyson equations is independent of the renormalization constants except for )(d

)0(

:)(

kd

dd critical

In D=2+1 is the only value for which the coupled Schwinger-Dyson equation have a self-consistent solution

)( criticaldd

horizon condition

Asymptotic behaviour D=3+1

-angular approximation

infrared behaviour

ultraviolet behaviour

kkfkkd

kkkkk

ln/1)(,ln/1)(

ln/1)(/)( ,)(

constkfkd

kk

k

k

)(,)(

)()(1

1

Numerical results (D=3+1)

ghost and Coulomb form factors gluon energy and curvature

GeVkGeVk 3)(,5.1 minmin mass gap:

c cinput: Coulomb string tension: 3 lattice: 1.5...3

Coulomb potential

)()())(()(0

02

21

2 krjkfkddkrV

3.7 4ck 0

V(k) / k (angular approximation: V(k) 1/ k )

external static color sources

electric field

ghost propagator

1 DE

The color electric flux tube

The flux between 3 static color charges

)()()( :chargecolor xqtxqx jaiji

a

1)ˆ( DE

a=3 a=8

The „baryon“= 3 static quarks in a color singlet

ijk i 1 j 2 k 3baryon q (x )q (x )q (x ) A

baryonxExEbaryon )()(

)()()(

chargecolor

xqtxqx jaiji

a

eliminating the self-energies

baryonxExEbaryon int))()((

The dielectric „constant“ of the Yang-Mills vacuum

)/()( 1 dD

EDDdE , , Maxwell´s displecement

dielectric „constant“

)(/1)( kdk

k

)(k

Importance of the curvature

Szczepaniak & SwansonPhys. Rev. D65 (2002)

• the = 0 solution does not produce a quasi-linear confinement potential

The vacuum wave functional & Fadeev-Popov determinant

kSzczepania &Swanson 0

rkpresent wo ansatz) (radial 21

0/ 0/ dHdHto 1-loop order:

oft independen is AA

AADDetA 21exp)(

Robustness of the infrared limit

Infrared limit = independent of

gauge fields at different points are completely uncorrelated

stochastic vacuum

exact in D=1+1

AADA 21exp)(

1)( A

3-gluon vertex

a b c abc abci 1 j 2 k 3 ijk 1 2 3 ijk 1 2 3

ijk 1 2 3

2 2 21 2 3

A (p ),A (p ),A (p ) : (p ,p ,p ) f (p ,p ,p )

(p ,p ,p ) (p)(Lorentz structure of the perturbative vertex) ...

at the symmetric point p p p

2

2.54p 0

0.29p

p

infrared divergent : (p) p

UV: (p) p

M.LederW.Schleifenbaum

2 6p 0 p 0

Landau gauge (Alkofer, Fischer, Llanes-Estrada) :

ghost : d(p) p , 3-gluon: (p) p

Coulomb gauge : 0.42

Finite temperature YMT

• ground state wave functional

• vacuum

• gas of quasi-gluons with energy

12(A) exp( A A) A 0

† 12

a 0 0 a,a 1 a ( A i / )

(k)

Energy density

Lattice: Karsch et al.

minimization of the free energy: (k,T)

Connection to the Center Vortex Picture

Center Vortices in Continuum Yang-Mills theory

);()(exp

CL

C

ZAP

SU(N)Z(N)Z

Wilson loop

Linking number

center element

C

1

:elementcenter trivial-non

1,-1 Z(2):)2(

Z

SU

Q-Q-potential: SU(2)

vortices

Confinement mechanism in Coulomb gauge

infrared dominant field configurations:

:

static quark potential

1 1V ( D ) ( )( D )

Gribov horizon

Det( D ) 0 ����������������������������

similar results in Coulomb gauge: Greensite, Olejnik, Zwanziger, hep-lat/0407032

Kugo-Ojima confinement criteria:

infrared divergent ghost propagator

center vortices

Suman &Schilling (1996)Nakajima,…Bloch et al.

Gattnar, Langfeld, Reinhardt, Phys. Rev.Lett.93(2004)061601,hep-lat/0403011

Ghost Propagator in Maximal Center Gauge (MCG)

fixes SU(2) / Z (2)

2

,x

(trU (x)) max

ghosts do not feel the center Z (2)

• no signal of confinement

in the ghost propagator

removal of center vortices does not

change the ghost propagator (analytic result!)

FP

Faddeev-Popov operator:

M (center vortex) center vortices

center vortex Z (x) vacuum U (x) 1

Landau(Coulomb)gauge maximum center gauge

center vortices

Gribov´s confinement criteria (infrared ghost propagator) is realized in gauges where the center vortices are on the Gribov horizon

Summary and Conclusion

• Hamilton approach to QCD in Coulomb gauge is very promising for non-perturbative studies

• Quark and gluon confinement• Curvature in gauge orbit space (Fadeev –Popov

determinant) is crucial for the confinement properties

• Center vortices are on the Gribov horizon and are the infrared dominant field configuratons, which give rise to an infrared diverging ghost propagator (Gribov´s confinement scenario)

Thanks to the organizers