confinement mechanism and topological defects review+ papers by a.v.kovalenko, mip, s.v. syritsyn,...
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Confinement Confinement mechanism and mechanism and
topological defectstopological defects
Review Review + +
papers by A.V.Kovalenko, MIP, S.V. Syritsyn, V.I. Zakharovpapers by A.V.Kovalenko, MIP, S.V. Syritsyn, V.I. Zakharov
hep-lat/0408014hep-lat/0408014,, hep-lat/0402017 hep-lat/0402017, , hep-lat/0212003hep-lat/0212003
Dubna, 2 February 2005
Confinment in Abelian Confinment in Abelian theoriestheories
Compact electrodynamicsCompact electrodynamics
Confinement is due to monopoles, Confinement is due to monopoles, which are condensed, and vacuum is which are condensed, and vacuum is a dual superconductora dual superconductor
Z(2) gauge theoryZ(2) gauge theory
Confinement is due to Z(2) vortices.Confinement is due to Z(2) vortices.
Confinement in compact Confinement in compact electrodynamicselectrodynamics
Partition function:Partition function:
PP
eDZ l
cos
l4321 P
PP
eDZ l
cos
1 34
2
Confinement in compact Confinement in compact electrodynamicselectrodynamics
]},[exp{.
)},(4{exp.
lim
12
0
cos
BSDDBconst
jjconsteDZ
AHMdual
jl
PP
Various representations of partition function
BBG
igBGg
xdS AHMdual
])1|(||)(|1
[ 22222
4
Dual superconductor Action
Vacuum of compact QED is Vacuum of compact QED is dual to supercoductordual to supercoductor
Monopole-antimonopole in superconductor
Charge-anticharge in cQED vacuum
MONOPOLE CONFINEMENT hep-lat/0302006, Y. Koma, M. Koma, E.M. Ilgenfritz, T. Suzuki, M.I.P.
Dual Abelian Dual Abelian
Higgs ModelHiggs Model
Abelian ProjectionAbelian Projection
of SU(2)of SU(2)
gluodynamics gluodynamics
)2(%90 SUmon
Monopole is a topological Monopole is a topological defectdefect
EXAMPLE: 2D topological defect
1 2
34
i
}][
][
][
]{[21
254
243
232
221
m
1,0 m
Topology: from real numbers we get integer, thus small, but finite variation of angles does not change topological
number m !!!
Monopole is a topological Monopole is a topological defectdefect
dcdap
][][][
][][][2
654
321
PPP
PPPmonj
For each cube we have integer valued current j
These monopoles are These monopoles are responsible for the formation responsible for the formation
of electric stringof electric string
Confinement in Z(2) Gauge Confinement in Z(2) Gauge TheoryTheory
2D example of “vortices” (they are points2D example of “vortices” (they are pointsconnected by “Dirac line”)connected by “Dirac line”)
In 3D vortices are closed lines, Dirac stringsIn 3D vortices are closed lines, Dirac stringsare 2d surfaces spanned on vorticesare 2d surfaces spanned on vortices
In 4D vortices are closed surfaces, Dirac stringsIn 4D vortices are closed surfaces, Dirac stringsare 3d volumes spanned on vorticesare 3d volumes spanned on vortices
-1 1
4321}1{
;; ssssssSez ppP
S
s
Confinement in 3D Z(2) Gauge Confinement in 3D Z(2) Gauge TheoryTheory
If If pp isis the probability that the the probability that the plaquette is pierced by vortex then plaquette is pierced by vortex then the expectation value of the the expectation value of the plaquette is:plaquette is: pppP 21)1()1)(1(
If vortices are uncorrelated (random) then expectation value of the Wilson loop is:
)21ln(;)21( pepW AreaArea
Confinement in 4D Z(2) Gauge Confinement in 4D Z(2) Gauge TheoryTheory
3D 4D
Vortex in 4D is a closed surface, 3d Dirac volume isenclosed by the vortex. If vortices are random thenthe string tension is the same as in 3D:
)21ln( p
Confinement in 4D SU(2) Confinement in 4D SU(2) Gauge Theory by Center Gauge Theory by Center
VorticesVortices4D
Each piercing of the surface spanned on the Wilson loop by center vortex give (-1) to W.
ll ZU
Center projection: partial gauge fixing
)2(%)70%60( SUvortex
In SU(2) gluodynamics we In SU(2) gluodynamics we observe monopoles, center observe monopoles, center
vortices and 3d Dirac volumes vortices and 3d Dirac volumes asas physical objectsphysical objects
Physical objectPhysical object1.1. Carry action Carry action (monopoles and (monopoles and
vortices)vortices)
2.2. Scales Scales (3d volumes, monopoles and vortices)(3d volumes, monopoles and vortices)
The gauge fixing is a detector The gauge fixing is a detector of gauge invariant objectsof gauge invariant objects
MonopolesMonopoles
Center vorticesCenter vortices
3d volumes3d volumes
(Dirac volume)(Dirac volume)
a
LS
fmV
L percmonmon 9.1,31 3
4
224 53.0,24
aA
SfmV
A vortvortvort
0,2 34
3 dd SfmV
V
UVa1
P-VORTEX has UV divergentP-VORTEX has UV divergent action density: action density:
(S-S(S-Svacvac)=Const./)=Const./2 2
In lattice units In lattice units
““Usual” explanation of Usual” explanation of confinement confinement
Monopoles Center Monopoles Center vortices vortices
Usually people are interestingUsually people are interesting in in confinement mechanism confinement mechanism
(monopoles and vortices)(monopoles and vortices)
Now inverse logicNow inverse logic
How to change gauge fields to get How to change gauge fields to get ?0fm2
C
CaPS
dxAi
e
PeTrW C
Field strength andField strength andgauge fields which aregauge fields which are
responsible for the responsible for the confinementconfinementare singular!are singular!
AF
UVUV
aa ,0;1
Monopoles belong to surfaces (center Monopoles belong to surfaces (center
vortices).vortices). Surfaces are bounds of Surfaces are bounds of minimalminimal 3d volumes 3d volumes
in Z(2) Landau gaugein Z(2) Landau gauge
3D analogue: 3D analogue:
center vortexcenter vortex
Minimal 3d volume corresponds to minimal surface spanned on center vortex
)1
(
)1
( 2
aOA
aOF
)
1(
)1(
aOA
OF
monopolemonopole
3d volume3d volume
1.Removing P-vortices1.Removing P-vortices
(( after Z(2) gauge after Z(2) gauge fixing)fixing)
we get zero string tension andwe get zero string tension and zero quark condensatezero quark condensate
P. de Forcrand and M. D'Elia, Phys.Rev.Lett. 82 (1999) 4582P. de Forcrand and M. D'Elia, Phys.Rev.Lett. 82 (1999) 4582
2.Removing P-vortices 2.Removing P-vortices wewe also remove 3d volumesalso remove 3d volumes
3.Wilson loop intersects with 3.Wilson loop intersects with 3d volumes3d volumes, , not with P-vorticesnot with P-vortices
xxx ZUU ~
xx TrUsignZ
fm2
C
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 .fm2
0
C
dxAi
PeTrW
,ac
A
fm2
C
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
0
ac
A
Holography
Low dimensional structures Low dimensional structures in lattice gluodynamicsin lattice gluodynamics
1.1. II. . HorvathHorvath et alet al. . hep-lat/0410046hep-lat/0410046, , hep-lat/030802hep-lat/0308029, 9, Phys.Rev.D68:114505,200Phys.Rev.D68:114505,20033
2.2. MILCMILC collaboration collaboration hep-lat/0410024 hep-lat/0410024
http:\www.lattice.itep.ru\http:\www.lattice.itep.ru\seminars\seminars\
e-mail: e-mail: polykarp@itep.rupolykarp@itep.ruИТЭФ, Б. Черемушкинская 25, Москва 117259ИТЭФ, Б. Черемушкинская 25, Москва 117259
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