1. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Coulomb energy, remnant symmetry in Coulomb gauge, and phases of non-abelian gauge theories with J. Greensite and D. Zwanziger (a part with R. Bertle and M. Faber) hep-lat/0302018 (JG, ŠO) hep-lat/0309172 (JG, ŠO) hep-lat/0310057 (RB, MF, JG, ŠO) paper in preparation (JG, ŠO, DZ)

2. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 20032 Confinement problem in QCD The problem remains unsolved and lucrative: The phenomenon attributed to field configurations with non-trivial topology: Instantons? Merons? Abelian monopoles? Center vortices? Their role can be (and has been) investigated in lattice simulations.

3. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 20033 Why Coulomb gauge? Two features of confinement: Long-range confining force between coloured quarks. Absence of gluons in the particle spectrum. Requirements on the gluon propagator at zero momentum: A strong singularity as a manifestation of the long-range force. Strongly suppressed because there are no massless gluons. Difficult to reach simultaneously in covariant gauges! In the Coulomb gauge: Long-range force due to instantaneous static colour- Coulomb field. The propagator of transverse, would-be physical gluons suppressed.

4. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 20034 Confinement scenario in Coulomb gauge h A 0 A 0 i propagator: Classical Hamiltonian in CG:

5. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 20035 Coulomb energy Physical state in CG containing a static pair: Correlator of two Wilson lines: Then:

6. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 20036 Measurement of the Coulomb energy on a lattice Lattice Coulomb gauge: maximize Wilson-line correlator: Questions: Does V(R,0) rise linearly with R at large  ? Does  coul match  asympt ?

7. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 20037

8. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 20038 Center vortices and Coulomb energy

9. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 20039

10. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200310 Scaling of the Coulomb string tension? Saturation? No, overconfinement!

11. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200311

12. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200312 Center symmetry and confinement Different phases of a stat. system are often characterized by the broken or unbroken realization of some global symmetry. Polyakov loop not invariant: On a finite lattice, below or above the transition, =0, but:

13. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200313 Coulomb energy and remnant symmetry Maximizing R does not fix the gauge completely: Under these transformations: Both L and Tr[L] are non-invariant, their expectation values must vanish in the unbroken symmetry regime. The confining phase is therefore a phase of unbroken remnant gauge symmetry; i.e. unbroken remnant symmetry is a necessary condition for confinement.

14. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200314 An order parameter for remnant symmetry in CG Define Order parameter (Marinari et al., 1993): Relation to the Coulomb energy:

15. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200315 Different phases of gauge theories Massless phase: field spherically symmetric Compact QED,  >1 Confined phase: field collimated into a flux tube Compact QED,  <1 Pure SU(N) at low T SU(N)+adjoint Higgs Screened phases: Yukawa-like falloff of the field Pure SU(N) at high T SU(N)+adjoint Higgs SU(N)+matter field in fund. representation (Z N center symmetric)

16. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200316 Compact QED 4

17. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200317 SU(2) gauge-adjoint Higgs theory

18. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200318

19. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200319

20. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200320

21. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200321 A surprise: SU(2) in the deconfined phase Does remnant and center symmetry breaking always go together? NO!

22. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200322

23. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200323 Center vortices and Coulomb energy Center vortices are identified by fixing to an adjoint gauge, and then projecting link variables to the Z N subgroup of SU(N). The excitations of the projected theory are known as P-vortices. Direct maximal center gauge: Vortex removal: What happens when “vortex-removed” configurations are brought to the Coulomb gauge? Coulomb energy

24. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200324 SU(2) in the deconfined phase: an explanation (?) Spacelike links are a confining ensemble even in the deconfinement phase: spacelike Wilson loops have an area law behaviour. Removing vortices removes the rise of the Coulomb potential. Thin vortices lie on the Gribov horizon! (A proof: D. Zwanziger.)

25. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200325 SU(2) gauge-fundamental Higgs theory

26. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200326 SU(2) with fundamental Higgs

27. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200327  =0

28. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200328

29. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200329

30. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200330

31. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200331 Kertész line?

32. Štefan Olejník Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia Tübingen, November 18, 200332 Conclusions The Coulomb string tension much larger than the true asymptotic string tension. Confining property of the color Coulomb potential is tied to the unbroken realization of the remnant gauge symmetry in CG. The deconfined phase in pure GT, and the “confinement” region of gauge-fundamental Higgs theory: color Coulomb potential is asymptotically linear, even though the static quark potential is screened. Center symmetry breaking, spontaneous or explicit, does not necessarily imply remnant symmetry breaking. Strong correlation between the presence of center vortices and the existence of a confining Coulomb potential. Thin center vortices lie on the Gribov horizon. The transition between regions of broken/unbroken remnant symmetry: percolation transition (Kertész line).