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.

Monopole confinementCompact Electrodynamics

Vortex ConfinementZ(2) gauge theory

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

Linking numberLinking number3D 4D

||1

41

1 2yx

dydxL l

C C

iklki

||1

)(8

1

1 2

2 yxdyxdL

C C

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

Monopole confinementCompact Electrodynamics

Vortex ConfinementZ(2) gauge theory

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

Length of IR monopole cluster Length of IR monopole cluster

scales,scales,

4VL

IR a

constUV

P-VORTEX density, Area/(6*VP-VORTEX density, Area/(6*V44), ), scales: scales:

Minimal 3D Volumes bounded Minimal 3D Volumes bounded byby

P-vortices scale P-vortices scale

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

Monopoles have fine tunedMonopoles have fine tuned action density:action density:

““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