Hamiltonian approach to Yang-Mills 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 talk microscopic description of infrared properties like confinement microscopic description of infrared properties like confinement Hamiltonian approach to YMT Hamiltonian approach to YMT Coulomb gauge Coulomb gauge 2

Plan of Talk Hamiltonian approach to Yang-Mills theory in Coulomb gauge Hamiltonian approach to Yang-Mills theory in Coulomb gauge basic results: propagators basic results: propagators comparison with lattice comparison with lattice dielectric function of the Yang-Mills vacuum dielectric function of the Yang-Mills vacuum topological susceptibility topological susceptibility D=1+1: Gribov copies D=1+1: Gribov copies conclusions conclusions 3

Related work on Coulomb gauge QCD Swith Swith Szczepaniak & Swanson Szczepaniak & Swanson Zwanziger Zwanziger 4

C. Feuchter & H. R. hep-th/ , PRD70(2004) H. R. & C. Feuchter, hep-th/ , PRD71(2005) W. Schleifenbaum, M. Leder, H.R. PRD73(2006) D. Epple, H. R., W. Schleifenbaum, PRD75(2007) H. Reinhardt, D. Epple, Phys.Rev.D76:065015,2007 C. Feuchter & H. R,Phys.Rev.D77:085023,2008, D. Epple, H. R., W. Schleifenbaum, A. Szczepaniak, Phys.Rev.D77:085007,2008 H. Reinhardt, arXiv: [hep-th] PhysRevLett , D. Campangnari & H. R., arXiv: [hep-th], Phys.Rev.D, in press G. Burgio, M.Quandt, H.R., arXiv: [hep-lat] 5 References: related work:Swift Szczepanik & Swanson Zwanziger

Canonical Quantization of Yang-Mills theory Gauß law: 6

Coulomb gauge Gauß law: resolution of Gauß´ law curved space Faddeev-Popov 7

YM Hamiltonian in Coulomb gauge -arises from Gauß´law =neccessary to maintain gauge invariance -provides the confining potential Coulomb term Christ and Lee 8

aim: solving the Yang-Mills Schrödinger eq. for the vacuum by the variational principle for the vacuum by the variational principle with suitable ansätze for metric of the space of gauge orbits: 9

aim: solving the Yang-Mills Schrödinger eq. for the vacuum by the variational principle for the vacuum by the variational principle with suitable ansätze for reflects non-trivial metric of the space of gauge orbits: 10

Vacuum wave functional 11 QM: particle in a L=0-state YMT gluon propagator determined fromvariational kernel gap equation C. Feuchter, H.R, 2004

Vacuum wave functional 12 QM: particle in a L=0-state YMT gluon propagator determined fromvariational kernel gap equation converted into set of Dyson-Schwinger equations renormalization C. Feuchter, H.R, 2004

Gluon energy 13 gluon confinement

Propagators gluon propagator gluon propagator ω(k)-gluon energy ω(k)-gluon energy ghost propagator ghost propagator ghost formfactor d(k): deviations from QED: ghost formfactor d(k): deviations from QED: QED: QED: Coulomb potential Coulomb potential 14

numerical solution Confinement of gluons Confinement of gluons Excellent agreement with IR and UV analysis Excellent 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 16

running coupling 17 W. Schleifenbaum, M. Leder, H.R. PRD73(2006)

Comparison with lattice data 18

comparison with lattice D= lattice: L. Moyarts, dissertationcontinuum: C. Feuchter & H. Reinhardt

Lattice calculation in D=3+1 Cuccheri, Zwanziger Cuccheri, Zwanziger Langfeld, Moyarts, Langfeld, Moyarts, Cuccheri, Mendes Cuccheri, Mendes A. Voigt, M. Ilgenfritz, M. Muller-Preussker, A.Sternbeck A. Voigt, M. Ilgenfritz, M. Muller-Preussker, A.Sternbeck G.Burgio, M. Quandt, S. Chimchinda, H. R., G.Burgio, M. Quandt, S. Chimchinda, H. R., Dubna H.Reinhardt

ghost form factor 21 Burgio, Quandt, Chimchinda, H. R., PoS LAT2007:325,2007

ghost propagator D= Burgio, Quandt, Chimchinda, H. R., PoS LAT2007:325,2007

Gluon propagator in D= K. Langfeld, L. Moyarts, 2004

recent lattice calculations of D=3+1 gluon propa gator gauge fixing gauge fixing renormalization renormalization 24 G. Burgio, M.Quandt, H.R., arXiv: [hep-lat]

Static gluon propagator in D= G. Burgio, M.Quandt, H.R., arXiv: [hep-lat]

Static gluon propagator in D= G. Burgio, M.Quandt, H.R., arXiv: [hep-lat]

Asymptotics lattice IR: α=0.98(2) UV: γ=1.005(10) δ=0.000(2) continuum IR: α=1 UV:γ=1.0 δ=0.0 27

The color electric field ED: ED: 28

The color electric field ED: ED: QCD: QCD: 29

external static color sources electric field ghost propagator 30

The color electric flux tube missing: back reaction of the vacuum to the external sources 31

The color electric flux tube 32

The color electric field ED: ED: QCD: QCD: 33

The color electric field ED: ED: medium medium QCD: QCD: 34

The color electric field ED: ED: medium medium QCD: QCD: ghost propagator ghost propagator 35

The color dielectric „constant“ of the QCD vacuum ED: ED: medium medium QCD: QCD: ghost propagator ghost propagator 36

The color dielectric „constant“ of the QCD vacuum ED: ED: medium medium QCD: QCD: ghost propagator ghost propagator 37 H. Reinhardt, PhysRevLett (2008)

The color dielectric fuction of the QCD vacuum 38

The color dielectric function of the QCD vacuum ghost propagator ghost propagator dielectric „constant“ dielectric „constant“ horizon condition: horizon condition: : QCD vacuum-perfect color dia-electricum QCD vacuum-perfect color dia-electricum QED: screening QED: screening k 39

40 no free color charges in the vacuum: confinement

magnetic analog to the QCD vacuum : superconductor magmetism in matter: magmetism in matter: perfect dia-magneticum : perfect dia-magneticum : Superconductor Superconductor 41

magnetic analog to the QCD vacuum : superconductor magmetism in matter: magmetism in matter: perfect dia-magneticum : perfect dia-magneticum : superconductor superconductor QCD vacuum:perfect dia-elektricum QCD vacuum:perfect dia-elektricum dual superconductor dual superconductor Duality: Duality: 42

Confinement scenarios Gribov-Zwanziger: ≈ Gribov-Zwanziger: ≈ (Kugo-Ojima) (Kugo-Ojima) 43 dual superconductor: magnetic monopole condensation

Confinement scenarios Gribov-Zwanziger: ≈ Gribov-Zwanziger: ≈ (Kugo-Ojima) (Kugo-Ojima) lattice evidence: monopole condensation ≈ monopole condensation ≈ vortex condensation ≈ vortex condensation ≈ 44 dual superconductor: magnetic monopole condensation center vortex condensation Gribov-Zwanziger Gribov-Zwanziger

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

Coulomb potential 46 J. Greensite, S. Olejnik, 2003

Confinement scenarios Gribov-Zwanziger: ≈ Gribov-Zwanziger: ≈ (Kugo-Ojima) (Kugo-Ojima) lattice evidence: monopole condensation ≈ monopole condensation ≈ vortex condensation ≈ vortex condensation ≈ 47 dual superconductor: magnetic monopole condensation center vortex condensation Gribov-Zwanziger Gribov-Zwanziger

Chiral symmery of QCD spontaneous breaking: spontaneous breaking: quark condensation quark condensation constituent quark mass constituent quark mass soft explicit breaking: soft explicit breaking: current massses current massses anomalous breaking: anomalous breaking: η´mass η´mass 48

Witten-Veneziano-Formula topological susceptibility topological susceptibility topological charge density topological charge density 49 in perturbation theory

Witten-Veneziano-Formula topological susceptibility topological susceptibility topological charge density topological charge density 50 in perturbation theory

-vacuum in the Hamiltonian approach -vacuum in the Hamiltonian approach vacuum wave functional vacuum wave functional winding number winding number 51

-vacuum in the Hamiltonian approach -vacuum in the Hamiltonian approach vacuum wave functional vacuum wave functional winding number winding number explicit realization explicit realization Chern-Simons action Chern-Simons action 52

-vacuum in the Hamiltonian approach -vacuum in the Hamiltonian approach vacuum wave functional vacuum wave functional winding number winding number explicit realization explicit realization Chern-Simons action Chern-Simons action topological susceptibility topological susceptibility 53

-vacuum in the Hamiltonian approach -vacuum in the Hamiltonian approach Lagrangian Lagrangian canonical momentum canonical momentum hamiltonian hamiltonian topological susceptibility topological susceptibility 54

Topological susceptibility in the Hamilton approach exact cancellation of Abelian part of BB exact cancellation of Abelian part of BB 2-and 3-quasi-gluons on top of the vacuum 2-and 3-quasi-gluons on top of the vacuum renormalization renormalization 55 D. Campangnari & H. R, Phys.Rev.D, in press

Numerical calculations parametrizations: parametrizations: 56

Numerical calculations IR dominance of the integrals IR dominance of the integrals running coupling: running coupling: IR limit: IR limit: 57

Numerical Results 58

Summary & Conclusion Hamiltonian approach to YMT in Coulomb gauge Hamiltonian approach to YMT in Coulomb gauge Variational solution of the YM Schrödinger eq. Variational solution of the YM Schrödinger eq. gluon confinement gluon confinement quark confinement quark confinement satisfactory agreement with lattice data satisfactory agreement with lattice data dielectric function of the YM vacuum dielectric function of the YM vacuum ε(k)=inverse ghost form factor ε(k)=inverse ghost form factor YM vacuum=perfect dual superconductor YM vacuum=perfect dual superconductor Gribov-Zwanziger Conf.↔dual Meißner effect Gribov-Zwanziger Conf.↔dual Meißner effect topological susceptibility topological susceptibility 59

Work in progress DSE in Coulomb gaue (first order formalism) DSE in Coulomb gaue (first order formalism) P. Watson P. Watson Hamiltonian flow equation Hamiltonian flow equation M. Leder, J. Pawlowski, A. Weber M. Leder, J. Pawlowski, A. Weber 60

Comments on Gribov copies 61 H. Reinhardt & W.Schleifenbaum arXiv: [hep-th]

Dyson-Schwinger Equations Exact relations between propagators and vertices Exact relations between propagators and vertices Not full QFT Not full QFT Missing:“ boundary“ condition Missing:“ boundary“ condition No information on Gribov region No information on Gribov region DSEs are the same in all Gribov regions but propagators are not DSEs are the same in all Gribov regions but propagators are not 62

Yang-Mills theory in D=1+1 Exact solution available Exact solution available Full control of Gribov copies Full control of Gribov copies Test approximation schemes used in D=3+1 Test approximation schemes used in D= H. Reinhardt and W. Schleifenbaum, in preparation

64 YMT on L FP determinant n-th Gribov regime spatial Wilson loop Coulomb gauge exact vacuum wave function(al) infrared limit of in D=3,4

Propagators Gluon propagator Gluon propagator Ghost propagator Ghost propagator Ghost-gluon vertex Ghost-gluon vertex Coulomb form factor Coulomb form factor 65

Gribov copies N-copies N-copies Gaussian distribution of copies Gaussian distribution of copies 66

N-Gribov copies ghost form factor ghost form factor Ghost-gluon vertex Ghost-gluon vertex (dressing function) (dressing function) 67

Gaussian distributed Gribov copies ghost form factor ghost form factor Coulomb form factor Coulomb form factor 68

Effect of Gribov copies on ghost 69

Effect of Gribov copies on Coulomb form factor 70

DSE with bare ghost-gluon vertex gluon propagator gluon propagator constant in D=1+1 constant in D=1+1 ghost form factor ghost form factor 71

conclusions Gribov copies tend to Gribov copies tend to damp IR enhencement of the ghost form factor and produce spurious peaks at intermediate momenta damp IR enhencement of the ghost form factor and produce spurious peaks at intermediate momenta increase the Coulomb form factor in the IR increase the Coulomb form factor in the IR Approximating the ghost-gluon vertex by the bare one puts the propagators(solutions of DSE) into the first Gribov region Approximating the ghost-gluon vertex by the bare one puts the propagators(solutions of DSE) into the first Gribov region Genuine IR physics cannot be properly described on the lattice unless Gribov copies are excluded Genuine IR physics cannot be properly described on the lattice unless Gribov copies are excluded 72

Thanks for your attention 73