Hamiltonian approach to Yang-Mills Theory in Coulomb gauge

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,

1

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

2

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

3

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]

5

References:

related work: SwiftSzczepanik & Swanson

Zwanziger

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

6

Coulomb gaugeCoulomb gauge

mD Gauß law:

|| 1m( D ) , ( A )

resolution of Gauß´ law

)()(*)(| AAAJDAcurved space

Faddeev-Popov )()( DDetAJ

A 0, A A

|| , / i A

7

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

8

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

9

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

10

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

12

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

0

Ak A k

k

Gluon energyGluon energy

13

gluon confinement

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

14

( ) ( ) 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

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)

15

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

16

running couplingrunning coupling

17

W. Schleifenbaum, M. Leder, H.R. PRD73(2006)

Comparison with lattice Comparison with lattice datadata

18

comparison with lattice D=2+1comparison with lattice D=2+1

19

lattice: L. Moyarts, dissertation continuum: C. Feuchter & H. Reinhardt

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

ghost propagator D=3+1ghost propagator D=3+1

22

424 - lattice Burgio, Quandt, Chimchinda, H. R.,PoS LAT2007:325,2007

Gluon propagator in Gluon propagator in D=3+1D=3+1

23

3/2( )D p p

( 0)D p const

K. Langfeld, L. Moyarts, 2004

1( ) (2 ( ))D p p

recent lattice calculations of recent lattice calculations of D=3+1 gluon propaD=3+1 gluon propagatorgator

gauge fixinggauge fixing renormalizationrenormalization

24


Static gluon propagator in Static gluon propagator in D=3+1D=3+1

25

4

2

1

2

( ) (2 ( ))

( )

0.8

´

6

Mk

Gribov s formu

D k k

k

V

la

k

M Ge


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

27

: ( ) : ( ) (log )IR D k k UV D k k k

The color electric fieldThe color electric field

ED:ED:

28

1, , =( )E E


ED:ED:

QCD:QCD:

29

1, , =( )E E

1

1

, ( )

, ( )

D D

E D

external static color sources

electric field

ghost propagator

1 DE

30

The color electric flux tube

missing: back reaction of the vacuum to the external sources

31


ED:ED:

QCD:QCD:

33

1, , =( )E E

1

1

, ( )

, ( )

D D

E D


ED:ED: mediummedium

QCD:QCD:

34

1, , =( )E E 11 = ( ) , dielectric constant

1

1

, ( )

, ( )

D D

E D


ED:ED: mediummedium

QCD:QCD:

ghost propagatorghost propagator

35

1, , =( )E E

1( ) / ( )D d

11 = ( ) , dielectric constant

1

1

, ( )

, ( )

D D

E D

The color dielectric The color dielectric „constant“ of the QCD „constant“ of the QCD

vacuumvacuum ED:ED:

mediummedium

QCD:QCD:


36

1, , =( )E E

1( ) / ( )D d


1

1

, ( )

, ( )

D D

E D

1 = ( ) ,d

The color dielectric The color dielectric „constant“ of the QCD „constant“ of the QCD

vacuumvacuum ED:ED:

mediummedium

QCD:QCD:


37

1, , =( )E E

1( ) / ( )D d


1

1

, ( )

, ( )

D D

E D

1 = ( ) ,d 1d H. Reinhardt, PhysRevLett.101.061602(2008)

The color dielectric fuction The color dielectric fuction of the QCD vacuumof the QCD vacuum 1d

38

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

39

1( 0) 0d k

40

0 1 1

D E

no free color charges in the vacuum: confinement

freeD

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

41

-magnetic pe rmeabili ty B H 0

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:

42

-magnetic pe rmeabili ty B H 0

B E

( 0) 0k

Confinement scenariosConfinement scenarios Gribov-Zwanziger: ≈Gribov-Zwanziger: ≈

(Kugo-Ojima)(Kugo-Ojima)

43

dual superconductor:dual superconductor:

magnetic monopole magnetic monopole condensationcondensation

1( 0) 0d k ( 0) 0k



lattice evidence:lattice evidence:

monopole condensation ≈monopole condensation ≈

vortex condensation ≈vortex condensation ≈

44



center vortex condensationcenter vortex condensation

Gribov-ZwanzigerGribov-Zwanziger

1( 0) 0d k ( 0) 0k

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

45

1( 0) 0d k

Coulomb potentialCoulomb potential

46J. Greensite, S. Olejnik , 2003



lattice evidence:lattice evidence:

monopole condensation ≈monopole condensation ≈

vortex condensation ≈vortex condensation ≈

47



center vortex condensationcenter vortex condensation

Gribov-ZwanzigerGribov-Zwanziger

1( 0) 0d k ( 0) 0k

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

48

( ) ( ) (1) (1)V f A f V ASU N SU N U U

( )V fSU N

( )A fSU N

(1)AU

qq

Witten-Veneziano-FormulaWitten-Veneziano-Formula

topological susceptibilitytopological susceptibility

topological charge densitytopological charge density

50

, 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

-vacuum in the Hamiltonian -vacuum in the Hamiltonian approachapproach

LagrangianLagrangian

canonical momentumcanonical momentum

hamiltonianhamiltonian


54

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

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

Numerical calculationsNumerical calculations

parametrizations:parametrizations:

56

4 2

2 2 2 2

2 1( ) ( ) 1m Mk k g k

k k G k

Numerical calculationsNumerical calculations

IR dominance of the integralsIR dominance of the integrals running coupling:running coupling:

IR limit:IR limit:

57

2 (0) 16(0)

4 3sC

g

N

Numerical ResultsNumerical Results

58

41.5 =(240Mev)C

4lattice: =(200-230Mev)

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


59

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

60

Comments on Gribov Comments on Gribov copiescopies

61

H. Reinhardt & W.SchleifenbaumarXiv:0809.1764[hep-th]

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

62

0 ( )exp( )Gribov

FPD Det M S J

Gribov

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

63

H. Reinhardt and W. Schleifenbaum, in preparation

64

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

PropagatorsPropagators

Gluon propagatorGluon propagator

Ghost propagatorGhost propagator

Ghost-gluon vertexGhost-gluon vertex

Coulomb form factorCoulomb form factor

65

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

Gribov copiesGribov copies

N-copiesN-copies

Gaussian distribution of copiesGaussian distribution of copies

66

1

0 0

... ...N

Nd d

212

0 0

... exp[ ( ) ]...N

d d

N

( 1):n n n

12 g L A

N-Gribov copiesN-Gribov copies

ghost form factorghost form factor

Ghost-gluon vertexGhost-gluon vertex

(dressing function)(dressing function)

67

Gaussian distributed Gribov Gaussian distributed Gribov copiescopies


Coulomb form factorCoulomb form factor

68

Effect of Gribov copies Effect of Gribov copies on ghoston ghost

69

3 1 1 1 D lattice D analytic

. , . , . ..G Burgio M Quandt H R

Effect of Gribov copies on Effect of Gribov copies on Coulomb form factorCoulomb form factor

70

3 1 1 1 D lattice D analytic

. .AVoigt et al

DSE with bare ghost-DSE with bare ghost-gluon vertexgluon vertex

gluon propagatorgluon propagator constant in D=1+1constant in D=1+1


71

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

72

Thanks for your Thanks for your attentionattention

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Hamiltonian approach to Yang-Mills Theory in Coulomb gauge