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Coulomb energy, remnant symmetry in Coulomb gauge, and phases of non-abelian gauge theoriesPowerPoint Presentation

Coulomb energy, remnant symmetry in Coulomb gauge, and phases of non-abelian gauge theories

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### 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)

Confinement problem in QCD phases of non-abelian gauge theories

- 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.

Why Coulomb gauge? phases of non-abelian gauge theories

- 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.

Confinement scenario in Coulomb gauge phases of non-abelian gauge theories

- hA0A0i propagator:
- Classical Hamiltonian in CG:

Coulomb energy phases of non-abelian gauge theories

- Physical state in CG containing a static pair:
- Correlator of two Wilson lines:
- Then:

Measurement of the Coulomb energy on a lattice phases of non-abelian gauge theories

- Lattice Coulomb gauge: maximize
- Wilson-line correlator:
- Questions:
- Does V(R,0) rise linearly with R at large b?
- Does scoul match sasympt?

- Center vortices and Coulomb energy phases of non-abelian gauge theories

Scaling of the Coulomb string tension? phases of non-abelian gauge theories

- Saturation? No, overconfinement!

Center symmetry and confinement phases of non-abelian gauge theories

- 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, <P(x)>=0, but:

Coulomb energy and remnant symmetry phases of non-abelian gauge theories

- 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.

An order parameter for remnant symmetry in CG phases of non-abelian gauge theories

- Define
- Order parameter (Marinari et al., 1993):
- Relation to the Coulomb energy:

Massless phase phases of non-abelian gauge theories: field spherically symmetric

Compact QED, b>1

Confined phase: field collimated into a flux tube

Compact QED, b<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

Different phases of gauge theories(ZN center symmetric)

Compact QED phases of non-abelian gauge theories4

SU(2) gauge-adjoint Higgs theory phases of non-abelian gauge theories

A surprise: SU(2) in the deconfined phase phases of non-abelian gauge theories

- Does remnant and center symmetry breaking always go together? NO!

Center vortices and Coulomb energy phases of non-abelian gauge theories

- Center vortices are identified by fixing to an adjoint gauge, and then projecting link variables to the ZN 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

SU(2) in the deconfined phase: an explanation (?) phases of non-abelian gauge theories

- 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.)

SU(2) gauge-fundamental Higgs theory phases of non-abelian gauge theories

SU(2) with fundamental Higgs phases of non-abelian gauge theories

b phases of non-abelian gauge theories=0

Kertész line phases of non-abelian gauge theories?

Conclusions phases of non-abelian gauge theories

- 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).

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