Hamiltonian approach to Yang-Mills Theory in Coulomb gauge
description
Transcript of Hamiltonian approach to Yang-Mills Theory in Coulomb gauge
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Hamiltonian approach to Yang-Hamiltonian approach to Yang-Mills Theory in Coulomb gaugeMills Theory in Coulomb gauge
H. Reinhardt
Tübingen
Collaborators:
G. Burgio, M.Quandt, P. Watson D. Epple, C. Feuchter, W. Schleifenbaum,D. Campagnari, S. Chimchinda, M. Leder, W. Lutz, M. Pak, C. Popovici,
J. Pawlowski, A. Szczepaniak, A.Weber,
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aim of the talkaim of the talk
microscopic description of infrared microscopic description of infrared properties like confinement properties like confinement
Hamiltonian approach to YMTHamiltonian approach to YMT Coulomb gaugeCoulomb gauge
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Plan of TalkPlan of Talk Hamiltonian approach to Yang-Mills Hamiltonian approach to Yang-Mills
theory in Coulomb gaugetheory in Coulomb gauge basic results: propagatorsbasic results: propagators comparison with latticecomparison with lattice dielectric function of the Yang-Mills dielectric function of the Yang-Mills
vacuumvacuum topological susceptibilitytopological susceptibility D=1+1: Gribov copiesD=1+1: Gribov copies conclusionsconclusions
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C. Feuchter & H. R. hep-th/0402106, PRD70(2004) H. R. & C. Feuchter, hep-th/0408237, PRD71(2005)
W. Schleifenbaum, M. Leder, H.R. PRD73(2006)D. Epple, H. R., W. Schleifenbaum, PRD75(2007)
H. Reinhardt, D. Epple, Phys.Rev.D76:065015,2007C. Feuchter & H. R,Phys.Rev.D77:085023,2008,
D. Epple, H. R., W. Schleifenbaum, A. Szczepaniak, Phys.Rev.D77:085007,2008
H. Reinhardt, arXiv:0803.0504 [hep-th] PhysRevLett.101.061602,
D. Campangnari & H. R., arXiv:0807.1195 [hep-th], Phys.Rev.D, in press
G. Burgio, M.Quandt, H.R., arXiv:0807.3291 [hep-lat]
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References:
related work: SwiftSzczepanik & Swanson
Zwanziger
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Canonical Quantization of Yang-Canonical Quantization of Yang-Mills theoryMills theory
momenta ( ) / ( ) ( )a a ai i ix S A x E x
)( scoordinatecartesian xAa
0)( :gauge Weyl 0 xAa0)(0 xa
quantization: ( ) / ( )a ak kx i A x
))()(( 22321 xBxxdH
Gauß law: mD
)()( :)x U(invariance gauge residual AAU
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Coulomb gaugeCoulomb gauge
mD Gauß law:
|| 1m( D ) , ( A )
resolution of Gauß´ law
)()(*)(| AAAJDAcurved space
Faddeev-Popov )()( DDetAJ
A 0, A A
|| , / i A
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YM Hamiltonian in YM Hamiltonian in Coulomb gaugeCoulomb gauge
1 212 ( ) CH J J B H
-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
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aim: solving the Yang-Mills Schrödinger aim: solving the Yang-Mills Schrödinger eq.eq.
for the vacuum by the variational for the vacuum by the variational principleprinciple
with suitable ansätze for
H DAJ(A) (A)H( ,A) (A) min
metric of the space of gauge orbits: )( DDetJ
H E
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aim: solving the Yang-Mills Schrödinger aim: solving the Yang-Mills Schrödinger eq.eq.
for the vacuum by the variational for the vacuum by the variational principleprinciple
with suitable ansätze for
H DAJ(A) (A)H( ,A) (A) min
reflects non-trivial metric of the space of gauge orbits:
( )J Det D
H E
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Vacuum wave functionalVacuum wave functional
11
2 * *1 2 1 2 1 2
( ) , r ( ) |
rJ r drr dr
r
QM: particle in a L=0-state
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1A exp dxdy (A(x) A(y)
Det Dx, y)
YMT
1( ) ( ) (2 ( ), )xA x y yA gluon propagator
determined from
H min
variational kernel
gap equation
x, x´
C. Feuchter, H.R, 2004
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0
Ak A k
k
Gluon energyGluon energy
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gluon confinement
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PropagatorsPropagators gluon propagatorgluon propagator
ωω(k)-gluon energy(k)-gluon energy ghost propagatorghost propagator
ghost formfactor d(k): ghost formfactor d(k): deviations from deviations from QED:QED:
QED: QED: Coulomb potentialCoulomb potential
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( ) ( ) 1/ 2 ( )yA x A y x
12
( )( )
Gk
kD
d
( ) 1d k
1 12 2V x y g x D D y
2 2( ) ( ( )) /V k d k k
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numerical solutionnumerical solution
• Confinement of gluonsConfinement of gluons• Excellent agreement with IR and UV analysisExcellent agreement with IR and UV analysis• (in)dependence on renormalization scale(in)dependence on renormalization scale
D. Epple, H. Reinhardt, W.Schleifenbaum, PRD 75 (2007)
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Coulomb potential
2 2 4k 0 k 0
V(k) (d(k)) / k 1/ k , d(k) 1/ k
1 12 2V x y g x D D y
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running couplingrunning coupling
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W. Schleifenbaum, M. Leder, H.R. PRD73(2006)
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Comparison with lattice Comparison with lattice datadata
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comparison with lattice D=2+1comparison with lattice D=2+1
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lattice: L. Moyarts, dissertation continuum: C. Feuchter & H. Reinhardt
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Lattice calculation in Lattice calculation in D=3+1D=3+1
Cuccheri, ZwanzigerCuccheri, Zwanziger Langfeld, Moyarts,Langfeld, Moyarts, Cuccheri, MendesCuccheri, Mendes A. Voigt, M. Ilgenfritz, M. Muller-A. Voigt, M. Ilgenfritz, M. Muller-
Preussker, A.SternbeckPreussker, A.Sternbeck G.Burgio, M. Quandt, S. Chimchinda, G.Burgio, M. Quandt, S. Chimchinda,
H. R.,H. R.,
Dubna 2008 20H.Reinhardt
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ghost propagator D=3+1ghost propagator D=3+1
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424 - lattice Burgio, Quandt, Chimchinda, H. R.,PoS LAT2007:325,2007
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Gluon propagator in Gluon propagator in D=3+1D=3+1
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3/2( )D p p
( 0)D p const
K. Langfeld, L. Moyarts, 2004
1( ) (2 ( ))D p p
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recent lattice calculations of recent lattice calculations of D=3+1 gluon propaD=3+1 gluon propagatorgator
gauge fixinggauge fixing renormalizationrenormalization
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G. Burgio, M.Quandt, H.R., arXiv:0807.3291 [hep-lat]
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Static gluon propagator in Static gluon propagator in D=3+1D=3+1
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4
2
1
2
( ) (2 ( ))
( )
0.8
´
6
Mk
Gribov s formu
D k k
k
V
la
k
M Ge
G. Burgio, M.Quandt, H.R., arXiv:0807.3291 [hep-lat]
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AsymptoticsAsymptotics
latticelattice IR: IR: αα=0.98(2)=0.98(2)
UV: UV: γγ=1.005(10)=1.005(10)
δδ=0.000(2)=0.000(2)
continuumcontinuum IR: IR: αα=1 =1
UV:UV:γγ=1.0=1.0
δδ=0.0=0.0
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: ( ) : ( ) (log )IR D k k UV D k k k
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The color electric fieldThe color electric field
ED:ED:
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1, , =( )E E
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The color electric fieldThe color electric field
ED:ED:
QCD:QCD:
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1, , =( )E E
1
1
, ( )
, ( )
D D
E D
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external static color sources
electric field
ghost propagator
1 DE
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The color electric flux tube
missing: back reaction of the vacuum to the external sources
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The color electric fieldThe color electric field
ED:ED:
QCD:QCD:
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1, , =( )E E
1
1
, ( )
, ( )
D D
E D
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The color electric fieldThe color electric field
ED:ED: mediummedium
QCD:QCD:
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1, , =( )E E 11 = ( ) , dielectric constant
1
1
, ( )
, ( )
D D
E D
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The color electric fieldThe color electric field
ED:ED: mediummedium
QCD:QCD:
ghost propagatorghost propagator
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1, , =( )E E
1( ) / ( )D d
11 = ( ) , dielectric constant
1
1
, ( )
, ( )
D D
E D
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The color dielectric The color dielectric „constant“ of the QCD „constant“ of the QCD
vacuumvacuum ED:ED:
mediummedium
QCD:QCD:
ghost propagatorghost propagator
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1, , =( )E E
1( ) / ( )D d
11 = ( ) , dielectric constant
1
1
, ( )
, ( )
D D
E D
1 = ( ) ,d
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The color dielectric The color dielectric „constant“ of the QCD „constant“ of the QCD
vacuumvacuum ED:ED:
mediummedium
QCD:QCD:
ghost propagatorghost propagator
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1, , =( )E E
1( ) / ( )D d
11 = ( ) , dielectric constant
1
1
, ( )
, ( )
D D
E D
1 = ( ) ,d 1d H. Reinhardt, PhysRevLett.101.061602(2008)
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The color dielectric fuction The color dielectric fuction of the QCD vacuumof the QCD vacuum 1d
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The color dielectric The color dielectric function of the QCD function of the QCD
vacuumvacuum ghost propagatorghost propagator dielectric „constant“dielectric „constant“
horizon condition:horizon condition: ::
QCD vacuum-perfect color dia-QCD vacuum-perfect color dia-electricumelectricum
QED: screeningQED: screening
1( ) / ( )D d
1d
k
( )k
1
( )<1 anti-screeningk
, freeD E D
( 0) 0k
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1( 0) 0d k
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0 1 1
D E
no free color charges in the vacuum: confinement
freeD
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magnetic analog to the QCD magnetic analog to the QCD vacuum :vacuum : superconductor superconductor
magmetism in matter:magmetism in matter:
perfect dia-magneticum :perfect dia-magneticum :SuperconductorSuperconductor
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-magnetic pe rmeabili ty B H 0
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magnetic analog to the QCD magnetic analog to the QCD vacuum :vacuum : superconductor superconductor
magmetism in matter:magmetism in matter:
perfect dia-magneticum :perfect dia-magneticum :superconductorsuperconductor
QCD vacuum:perfect dia-elektricumQCD vacuum:perfect dia-elektricumdual superconductordual superconductor
Duality:Duality:
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-magnetic pe rmeabili ty B H 0
B E
( 0) 0k
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Confinement scenariosConfinement scenarios Gribov-Zwanziger: ≈Gribov-Zwanziger: ≈
(Kugo-Ojima)(Kugo-Ojima)
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dual superconductor:dual superconductor:
magnetic monopole magnetic monopole condensationcondensation
1( 0) 0d k ( 0) 0k
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Confinement scenariosConfinement scenarios Gribov-Zwanziger: ≈Gribov-Zwanziger: ≈
(Kugo-Ojima)(Kugo-Ojima)
lattice evidence:lattice evidence:
monopole condensation ≈monopole condensation ≈
vortex condensation ≈vortex condensation ≈
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dual superconductor:dual superconductor:
magnetic monopole magnetic monopole condensationcondensation
center vortex condensationcenter vortex condensation
Gribov-ZwanzigerGribov-Zwanziger
1( 0) 0d k ( 0) 0k
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elimination of center vortices removes:-string tension (Wilson´s confinment criterium)
-the infrared divergency from the ghost propagator (Kogu-Ojima confinement criterium)
Gattnar, Langfeld, Reinhardt NPB262(2002)131
Kugo-Ojima confinement criteria:
infrared divergent ghost form factor
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1( 0) 0d k
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Coulomb potentialCoulomb potential
46J. Greensite, S. Olejnik , 2003
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Confinement scenariosConfinement scenarios Gribov-Zwanziger: ≈Gribov-Zwanziger: ≈
(Kugo-Ojima)(Kugo-Ojima)
lattice evidence:lattice evidence:
monopole condensation ≈monopole condensation ≈
vortex condensation ≈vortex condensation ≈
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dual superconductor:dual superconductor:
magnetic monopole magnetic monopole condensationcondensation
center vortex condensationcenter vortex condensation
Gribov-ZwanzigerGribov-Zwanziger
1( 0) 0d k ( 0) 0k
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Chiral symmery of QCDChiral symmery of QCD
spontaneous breaking:spontaneous breaking: quark condensationquark condensation constituent quark massconstituent quark mass
soft explicit breaking: soft explicit breaking: current masssescurrent massses
anomalous breaking: anomalous breaking: ηη´mass´mass
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( ) ( ) (1) (1)V f A f V ASU N SU N U U
( )V fSU N
( )A fSU N
(1)AU
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Witten-Veneziano-FormulaWitten-Veneziano-Formula
topological susceptibilitytopological susceptibility
topological charge densitytopological charge density
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, 2
22 2 22 fN
K Fm m m
0 ( ) (0) 0q x q
2
28( ) ( ) ( )g a a
i iq x E x B x
in perturbation theory0
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-vacuum in the Hamiltonian -vacuum in the Hamiltonian approachapproach
LagrangianLagrangian
canonical momentumcanonical momentum
hamiltonianhamiltonian
topological susceptibilitytopological susceptibility
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2
2
d HV
d
2
28( ) ( ( )) ) (gx x E x B x
L L
2
28( ) ( ) ( )ga a ax xx B
2
2
2 21 12 28
( )g BH B
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Topological susceptibility in Topological susceptibility in the Hamilton approachthe Hamilton approach
exact cancellation of Abelian part of exact cancellation of Abelian part of BBBB 2-and 3-quasi-gluons on top of the 2-and 3-quasi-gluons on top of the
vacuumvacuum
renormalizationrenormalization55
2
2
2
g 2 2
8n n
n B 0V ( ) 0 B (x) 0 2
E
† † † † †( ) 0 0, { ( ) ( ) 0 , ( ) ( ) ( ) 0 }a k n a k a k a k a k a k
0, G(k) G(k)-G(k)PT PT
D. Campangnari & H. R, Phys.Rev.D, in press
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Numerical calculationsNumerical calculations
parametrizations:parametrizations:
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4 2
2 2 2 2
2 1( ) ( ) 1m Mk k g k
k k G k
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Numerical calculationsNumerical calculations
IR dominance of the integralsIR dominance of the integrals running coupling:running coupling:
IR limit:IR limit:
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2 (0) 16(0)
4 3sC
g
N
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Numerical ResultsNumerical Results
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41.5 =(240Mev)C
4lattice: =(200-230Mev)
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Summary & ConclusionSummary & Conclusion Hamiltonian approach to YMT in Coulomb gaugeHamiltonian approach to YMT in Coulomb gauge Variational solution of the YM Schrödinger eq.Variational solution of the YM Schrödinger eq.
gluon confinementgluon confinement quark confinementquark confinement
satisfactory agreement with lattice datasatisfactory agreement with lattice data dielectric function of the YM vacuumdielectric function of the YM vacuum
εε(k)=inverse ghost form factor (k)=inverse ghost form factor YM vacuum=perfect dual superconductorYM vacuum=perfect dual superconductor Gribov-Zwanziger Conf.↔dual Meißner effectGribov-Zwanziger Conf.↔dual Meißner effect
topological susceptibilitytopological susceptibility
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Work in progressWork in progress
DSE in Coulomb gaue (first order formalism)DSE in Coulomb gaue (first order formalism)
P. WatsonP. Watson Hamiltonian flow equationHamiltonian flow equation
M. Leder, J. Pawlowski, A. WeberM. Leder, J. Pawlowski, A. Weber
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Comments on Gribov Comments on Gribov copiescopies
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H. Reinhardt & W.SchleifenbaumarXiv:0809.1764[hep-th]
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Dyson-Schwinger Dyson-Schwinger EquationsEquations
Exact relations between propagators Exact relations between propagators and verticesand vertices
Not full QFTNot full QFT Missing:“ boundary“ conditionMissing:“ boundary“ condition No information on Gribov regionNo information on Gribov region DSEs are the same in all Gribov DSEs are the same in all Gribov
regions but propagators are notregions but propagators are not
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0 ( )exp( )Gribov
FPD Det M S J
Gribov
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Yang-Mills theory in Yang-Mills theory in D=1+1D=1+1
Exact solution availableExact solution available Full control of Gribov copiesFull control of Gribov copies Test approximation schemes used in Test approximation schemes used in
D=3+1D=3+1
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H. Reinhardt and W. Schleifenbaum, in preparation
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YMT onYMT on 1 1( ) ( )S space R time L
2
2
sin( ) ( )J A Det D
FP determinant
( 1)n n n-th Gribov regime
1
12 exp( ) cos
S
tr A spatial Wilson loop 0
Coulomb gauge 0A
exact vacuum wave function(al) 1
( 0)( )A kA infrared limit of in D=3,4
12 g L A
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PropagatorsPropagators
Gluon propagatorGluon propagator
Ghost propagatorGhost propagator
Ghost-gluon vertexGhost-gluon vertex
Coulomb form factorCoulomb form factor
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213
a b abA A const
12
( )( )
d kD
k
1 1 01( ) ( ) ( ) , A D AA D D
1 2 1 1 2 1( ) ( )( ) ( ) ( ) ( )D DfD D
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Gribov copiesGribov copies
N-copiesN-copies
Gaussian distribution of copiesGaussian distribution of copies
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1
0 0
... ...N
Nd d
212
0 0
... exp[ ( ) ]...N
d d
N
( 1):n n n
12 g L A
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N-Gribov copiesN-Gribov copies
ghost form factorghost form factor
Ghost-gluon vertexGhost-gluon vertex
(dressing function)(dressing function)
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Gaussian distributed Gribov Gaussian distributed Gribov copiescopies
ghost form factorghost form factor
Coulomb form factorCoulomb form factor
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Effect of Gribov copies Effect of Gribov copies on ghoston ghost
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3 1 1 1 D lattice D analytic
. , . , . ..G Burgio M Quandt H R
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Effect of Gribov copies on Effect of Gribov copies on Coulomb form factorCoulomb form factor
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3 1 1 1 D lattice D analytic
. .AVoigt et al
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DSE with bare ghost-DSE with bare ghost-gluon vertexgluon vertex
gluon propagatorgluon propagator constant in D=1+1constant in D=1+1
ghost form factorghost form factor
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conclusionsconclusions
Gribov copies tend toGribov copies tend to damp IR enhencement of the ghost form factor damp IR enhencement of the ghost form factor
and produce spurious peaks at intermediate and produce spurious peaks at intermediate momentamomenta
increase the Coulomb form factor in the IRincrease the Coulomb form factor in the IR Approximating the ghost-gluon vertex by the Approximating the ghost-gluon vertex by the
bare one puts the propagators(solutions of bare one puts the propagators(solutions of DSE) into the first Gribov regionDSE) into the first Gribov region
Genuine IR physics cannot be properly Genuine IR physics cannot be properly described on the lattice unless Gribov copies described on the lattice unless Gribov copies are excludedare excluded
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Thanks for your Thanks for your attentionattention
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