Hamilton Approach to Yang-Mills Theory Confinement of Quarks and Gluons

Transcription

1 Hamilton Approach to Yang-Mills Theory Confinement of Quarks and Gluons H. Reinhardt Tübingen Collaborators: G. Burgio, M. Quandt, P. Watson D. Epple, C. Feuchter, W. Schleifenbaum, D. Campagnari, J. Heffner, M. Leder, W. Lutz, M. Pak, C. Popovici, J. Pawlowski, A.Szczepaniak, A. Weber 1

2 Deep inelastic e-p-scattering meson production

3 Confinement quarks and gluons do not exist as free particles but only in the interior of hadrons: Mesons Baryons Why are quarks und gluons confined? What is the confinement mechanism? 3

4 Clay Mathematics Institue Millenium Problem A. Yaffe & E. Witten: mass gap nuclear force is short ranged quark confinement no individual quarks chiral symmetry breaking current algebra theory of soft pions 4

5 aim of the talk microscopic description of infrared properties like confinement Hamilton approach to YMT Heidelberg 2008 H.Reinhardt 5

6 Plan of Talk Confinement Hamilton approach to Yang-Mills theory Variational solution of the YM-Schrödinger equation Field configurations relevant for confinement: Magnetic monopoles Center vortices Topological aspects 6

7 Classical Yang-Mills theory action 1 4g 2 4 S d x( F ( x)) 2 field strength tensor F ( x) A A A, A 7

8 Canonical Quantization of Yang-Mills theory cartesian coordinate s A a ( x) momenta a ( x) S / A a ( x) E a ( x) i i i a 0 ( x) 0 Weyl gauge : A a 0 ( x) 0 H d x( ( x) B ( x )) 2 a a quantization: ( x) / ia ( x) k k Gauß law: D m residual gauge invariance U(x) : ( A U ) ( A) 8

9 Coulomb gauge curved space A 0, A A * DA J( A ) ( A ) ( A ) Faddeev-Popov J( A ) Det( D), / ia Gauß law: D m resolution of Gauß law 1 m ( D ), ( A ) 9

10 YM Hamiltonian in Coulomb gauge H Coulomb term ( J J J J B HC 2 J J J ( D ) ( )( D ) J color density: A 2 m ) Christ and Lee -arises from Gauß law =neccessary to maintain gauge invariance -provides the confining potential 10

11 Perturbation theory D. Campagnari, H. R., A. Weber, Phys. Rev D(2009) Rayleigh-Schrödinger PT vacuum (QED) ß-function N C 11

12 aim: solving the Yang-Mills Schrödinger eq. H E for the vacuum by the variational principle H DAJ(A) (A)H(,A) (A) min with suitable ansätze for metric of the space of gauge orbits: J Det( D) 12

13 Vacuum wave functional QM: particle in a L=0-state YMT () r, r J ( r ) drr dr r 2 * * A exp 2 dxdy A(x) ( x, y) A(y) Det D gluon propagator A( x) A( y) (2 ( x, y) ) 1 variational kernel x, x determined from H min gap equation C. Feuchter & H. R. PRD70(2004) pure Gaussian: Szczepaniak, Swanson 13

14 gluon energy k 1/ k, k 0 k k, k gluon confinement 14

15 gluon propagator ω(k)-gluon energy ghost propagator ghost formfactor d(k): deviations from QED: QED: dk ( ) 1 Coulomb potential A( x) A( y) 1/ 2 ( x y) ( D) 1 dk ( ) 2 k V x y g x D D y V ( k) ( d( k)) / k

16 ghost formfactor gost propagator 1 ( D ) d / ( ) ghost form factor dk ( ) D. Epple, H. Reinhardt, W.Schleifenbaum, PRD 75 (2007) IR : d k 1 / k horizon condition d 1 ( k 0) 0 16

17 Coulomb potential V x y g x D D y V(k) (d(k)) / k 1/ k, d(k) 1/ k k0 k0 17

18 non-perturbative running coupling W. Schleifenbaum, M. Leder, H.R. PRD73(2006) 18

19 Comparison with lattice data 19

20 Static gluon propagator in D=3+1 A( k) A(0) (2 ( k)) Gribov s formula ( k 2 M ) k k M 0.88GeV deviations in the mid momentum regime: neglect of gluon loop G. Burgio, M.Quandt, H.R., PRL102(2009) 20

21 ghost formfactor: 1 ( D ) d / ( ) dk ( ) lattice G. Burgio, M.Quandt, H.R. in preparation d(k) 2 2 m k [ln(e )] 2 2 k m ~ 1 k, k 0 1 lattice data compatible with horizon condition d 1 ( k 0) 0 21

22 The color dielectric function of the QCD vacuum ghost propagator dielectric constant d 1 H.Reinhardt,PRL101 (2008) 1 ( D ) d / ( ) 22

23 The color dielectric function of the ghost propagator dielectric constant horizon condition: : d 1 H.Reinhardt,PRL101 (2008) d 1 ( k 0) 0 QCD vacuum QCD vacuum-perfect color dia-electricum ( k)<1 anti-screening 1 ( D ) d / ( ) ( k 0) 0 23

24 The color dielectric function of the QCD vacuum ghost propagator dielectric constant d 1 1 ( D ) d / ( ) H.Reinhardt,PRL101 (2008) horizon condition: : d QCD vacuum-perfect color dia-electricum ( k)<1 anti-screening QED: 1 ( k 0) 0 1 screening ( k 0) 0 24

25 D E D free no free color charges in the vacuum: confinement 25

26 Magnetic analog to the QCD vacuum : Superconductor 26

27 Magnetic analog to the QCD vacuum : Superconductor Magmetism in matter: B H -magnetic permeability 27

28 Magnetic analog to the QCD vacuum : Superconductor Magmetism in matter: B H -magnetic permeability perfect diamagneticum : superconductor 0 28

29 Magnetic analog to the QCD vacuum : Superconductor Magmetism in matter: B H -magnetic permeability perfect diamagneticum : superconductor QCD vacuum=dual superconductor: Duality: E B

30 Magnetic analog to the QCD vacuum : Superconductor Magmetism in matter: B H -magnetic permeability perfect diamagneticum : superconductor QCD vacuum=dual superconductor: Duality: E B 0 0 confinement=dual Meißner effect: t Hooft Mandelstam 30

31 (ordinary) Meißner effect 31

32 The dual Meißner-Effect 32

33 The flux tube Koma et al

34 Coulomb potential V x y g x D D y V(k) (d(k)) / k 1/ k, d(k) 1/ k k0 k0 34

35 Field configuration relevant for confinement Configurations on the Gribov horizon: singular ( D) 1 magnetic monopoles center vortices Lattice: monopoles & vortices are condensed in the YM vacuum 35

36 Von Karman-vortex Guadalupe Island,

37 Vortices fluid velocity vorticity v gauge theory gauge potential magnetic flux v A B A 37

38 vortices D=2 points D=3 loops D=4 closed surfaces self-intersect non-oriented 38

39 Center Vortices in Continuum Yang- Mills theory Wilson loop P exp C A( ) Z L( C; ) Gauss linking number center element ZZ(N) SU(N) C SU(2) : Z(2) non - trivial center element: Z 1 1,-1 39

40 Center vortices and magnetic monopoles W( C) 1 Dirac string not observable W( C) 1 1 center vortex 2 Dirac string magnetic monopoles change the orientation of the vortex flux

41 Center vortices & magnetic monopoles: instanton monopole loops are located on vortex sheets monopoles change the orientation of vortices 41

42 Q-Q-potential: SU(2) Del Debbio, Faber, Greensite, Olejnik Engelhardt, Langfeld, Reinhardt, Tennert 42

43 Coulomb potential V x y g x D D y V(k) (d(k)) / k 1/ k, d(k) 1/ k k0 k0 43

44 Kugo-Ojima confinement criteria: infrared divergent ghost form factor d 1 H.Reinhardt,PRL101 (2008) elimination of center vortices removes confinement: -string tension (Wilson s confinment criterium) -the infrared divergency from the ghost propagator (Kogu-Ojima confinement criterium) Gattnar, Langfeld, Reinhardt Nucl.Phys.B262 44

45 IR-dominant field configurations center vortices are responsible for confinement confined phase: percolated deconfined phase: small clusters Engelhardt, Langfeld, Reinhardt, Tennert 2000 percolation of center vortices cause magnetic monopoles to condense: dual Meißner effect magnetic monopoles make center vortices nonoriented and topologically non-trival Reinhardt 2002 Engelhardt & Reinhardt topolog. non-trivial field configurations are required for spont. breaking of chiral symm. 45

46 finite temperature QFT temporal extension of the lattice t=1/t T~0 T>Tc T>Tc: thick vortices align parallel to the time axis

47 Topological defects magnetic vortices quantized flux 1 ( SU (2) / Z (2)) Z 47

48 Topological charge A FF 4 EB Chern class i n m m SU U i i i 2, ( (2) / ( 1) ) magnetic monopole charge Reinhardt 1997 Quandt, Reinhardt, Schäfke

49 Center vortices in 3 closed surfaces of quantized flux magnetic vortices exist during a finite time period electric vortices exist at a single time instant 49

50 Topological charge of center vortices intersection points EB 0 topological charge 1 4 A 2 EB 50

51 Topological charge of center vortices intersection points EB 0 topological charge 1 4 A 2 EB self-intersection number vortex sheet 1 A I (, ) 4 Engelhardt, Reinhardt 2000 Reinhardt

52 The intersection number of oriented closed surfaces in D=4 vanishes time slice of D = 4 vortex sheet: S S 1 S 2 I( S S ) I( S S )

53 53

54 Topological charge of center vortices intersection points EB 0 topological charge index theorem 1 A EB n n 4 2 n Dq 0 # of L/R zero modes of Dirac operator 54

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56 Fermions in an intersecting center vortex field Vortices=fermion guides Reinhardt, Schröder, Tok, Zhukovsky, PRD66 56 >>spontaneous breaking of chiral symmetry?

57 Applications Topological susceptibility D. Campagnari & H. R., Phys. Rev. D (2008) t Hooft loop: perimeter law H. R. & D. Epple, Phys. Rev D(2007) Wilson loop: area law M. Pak & H. R., Phys. Rev.D(2009) FRG flow equation M. Leder, J. Pawlowski, H. R., A.Weber, arxiv

58 Summary & Conclusion Hamilton approach to YM gluon confinement quark confinement Infrared dominant field configurations: center vortices magnetic monopoles Topological properties of gauge fields magnetic charge of monopoles intersection and writhe of center vortices 58

59 Thanks for your attention 59