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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, M F , JG, ŠO) paper in preparation (JG, ŠO, DZ).

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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

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

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.


Why coulomb gauge

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.


Confinement scenario in coulomb gauge

Confinement scenario in Coulomb gauge

  • hA0A0i propagator:

  • Classical Hamiltonian in CG:


Coulomb energy

Coulomb energy

  • Physical state in CG containing a static pair:

  • Correlator of two Wilson lines:

  • Then:


Measurement of the coulomb energy on a lattice

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 b?

    • Does scoul match sasympt?


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

  • Center vortices and Coulomb energy


Scaling of the coulomb string tension

Scaling of the Coulomb string tension?

  • Saturation? No, overconfinement!


Center symmetry and confinement

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


Coulomb energy and remnant symmetry

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.


An order parameter for remnant symmetry in cg

An order parameter for remnant symmetry in CG

  • Define

  • Order parameter (Marinari et al., 1993):

  • Relation to the Coulomb energy:


Different phases of gauge theories

Massless phase: 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 4

Compact QED4


Su 2 gauge adjoint higgs theory

SU(2) gauge-adjoint Higgs theory


A surprise su 2 in the deconfined phase

A surprise: SU(2) in the deconfined phase

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


Center vortices and coulomb energy

Center vortices and Coulomb energy

  • 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

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.)


Su 2 gauge fundamental higgs theory

SU(2) gauge-fundamental Higgs theory


Su 2 with fundamental higgs

SU(2) with fundamental Higgs


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

b=0


Kert sz line

Kertész line?


Conclusions

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).


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