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Confinement mechanism and topological defects

Confinement mechanism and topological defects. Review + papers by A.V.Kovalenko, MIP, S.V. Syritsyn, V.I. Zakharov hep-lat/0408014 , hep-lat/0402017 , hep-lat/0212003. Dubna, 2 February 2005. Confinment in Abelian theories. Compact electrodynamics

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Confinement mechanism and topological defects

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  1. Confinement mechanism and topological defects Review + papers by A.V.Kovalenko, MIP, S.V. Syritsyn, V.I. Zakharov hep-lat/0408014, hep-lat/0402017, hep-lat/0212003 Dubna, 2 February 2005

  2. Confinment in Abelian theories • Compact electrodynamics Confinement is due to monopoles, which are condensed, and vacuum is a dual superconductor • Z(2) gauge theory Confinement is due to Z(2) vortices.

  3. Monopole confinement Compact Electrodynamics Vortex Confinement Z(2) gauge theory

  4. Confinement in compact electrodynamics Partition function:

  5. Confinement in compact electrodynamics Various representations of partition function Dual superconductor Action

  6. Vacuum of compact QED is dual to supercoductor Charge-anticharge in cQED vacuum Monopole-antimonopole in superconductor

  7. MONOPOLE CONFINEMENT hep-lat/0302006, Y. Koma, M. Koma, E.M. Ilgenfritz, T. Suzuki, M.I.P. Dual Abelian Higgs Model Abelian Projection of SU(2) gluodynamics

  8. Monopole is a topological defect EXAMPLE: 2D topological defect Topology: from real numbers we get integer, thus small, but finite variation of angles does not change topological number m !!!

  9. Monopole is a topological defect For each cube we have integer valued current j

  10. These monopoles are responsible for the formation of electric string

  11. Confinement in Z(2) Gauge Theory 2D example of “vortices” (they are points connected by “Dirac line”) In 3D vortices are closed lines, Dirac strings are 2d surfaces spanned on vortices In 4D vortices are closed surfaces, Dirac strings are 3d volumes spanned on vortices 1 -1

  12. Confinement in 3D Z(2) Gauge Theory If pisthe probability that the plaquette is pierced by vortex then the expectation value of the plaquette is: If vortices are uncorrelated (random) then expectation value of the Wilson loop is:

  13. Linking number 4D 3D

  14. Confinement in 4D Z(2) Gauge Theory 3D 4D Vortex in 4D is a closed surface, 3d Dirac volume is enclosed by the vortex. If vortices are random then the string tension is the same as in 3D:

  15. Confinement in 4D SU(2) Gauge Theory by Center Vortices 4D Center projection: partial gauge fixing Each piercing of the surface spanned on the Wilson loop by center vortex give (-1) to W.

  16. Monopole confinement Compact Electrodynamics Vortex Confinement Z(2) gauge theory

  17. In SU(2) gluodynamics we observe monopoles, center vortices and 3d Dirac volumes asphysical objects Physical object • Carry action (monopoles and vortices) • Scales (3d volumes, monopoles and vortices)

  18. The gauge fixing is a detector of gauge invariant objects Monopoles Center vortices 3d volumes (Dirac volume)

  19. Length of IR monopole cluster scales,

  20. P-VORTEX density, Area/(6*V4), scales:

  21. Minimal 3D Volumes bounded byP-vortices scale

  22. P-VORTEX has UV divergentaction density: (S-Svac)=Const./a2 In lattice units

  23. Monopoles have fine tunedaction density:

  24. “Usual” explanation of confinement Monopoles Center vortices

  25. Usually people are interestingin confinement mechanism (monopoles and vortices) Now inverse logic How to change gauge fields to get

  26. Field strength andgauge fields which areresponsible for the confinementare singular!

  27. Monopoles belong to surfaces (center vortices). Surfaces are bounds of minimal 3d volumes in Z(2) Landau gauge 3D analogue: 3d volume monopole Minimal 3d volume corresponds to minimal surface spanned on center vortex center vortex

  28. 1.Removing P-vortices (after Z(2) gauge fixing) we get zero string tension and zero quark condensate P. de Forcrand and M. D'Elia, Phys.Rev.Lett. 82 (1999) 4582 2.Removing P-vortices wealso remove 3d volumes 3.Wilson loop intersects with 3d volumes, not with P-vortices

  29. 3. Wilson loop intersects in points in 4D with 3d volumes, not with P-vortices 4. We get changing strong, fields in points, the average distance between these points is .

  30. 5. Note that we changed very small fraction of links to get , since we use Z(2) Landau gauge. De Forcrand and d’Elia had to change 50% of links 6. All information about confinement, quark condensate and any Wilson loop is encoded in 3d branes Holography

  31. Low dimensional structures in lattice gluodynamics • I. Horvathet al. hep-lat/0410046, hep-lat/0308029, Phys.Rev.D68:114505,2003 • MILCcollaboration hep-lat/0410024

  32. http:\www.lattice.itep.ru\seminars\ • e-mail: polykarp@itep.ru ИТЭФ, Б. Черемушкинская 25, Москва 117259

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