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閉じ込め四方山話 Much Ado about Confinement. Dec. 10 2008 @ Komaba Nuclear Theory Group Takuya Kanazawa Phys. Dep., Univ. of Tokyo. 1. Overview 1-1. Introduction: confinement. Emergence

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Much Ado about Confinement

Dec. 10 2008 @ Komaba Nuclear Theory Group

Takuya Kanazawa

Phys. Dep., Univ. of Tokyo


1. Overview

1-1. Introduction: confinement


“The way complex systems and patterns arise out of a multiplicity of relatively simple interactions.” (from Wikipedia)

Emergent phenomena in

Condensed Matter Physics:



quantum hall effect,

localization, …

Emergent phenomena in

Quantum Chromodynamics (QCD):

chiral symmetry breaking,

mass gap generation,

color confinement,

color superconductivity, …

Emergent phenomena in String Theory:

Please ask Yoneya sensei.


What is Confinement? Is it well-defined?

There is no gauge-invariant order parameter distinguishing

between the Higgs phase and the confining (symmetric) phase

when the system has a scalar in the faithful rep. of the gauge group.

(They are smoothly connected.)

T. Banks and E. Rabinovici, Nucl. Phys. B160, 349 (1979)

: Abelian Higgs model with Higgs minimally charged under U(1)

E. Fradkin and S.H. Shenker, Phys. Rev. D19, 3682 (1979)

: SU(N) gauge-Higgs model with Higgs in fundamental rep.



(composite particle description)


What is Confinement? Is it well-defined?

  • Electroweak theory also possesses a fundamental Higgs.
  • The thermal transition line may have an end point.
  • (K. Kajantie et al., Phys.Rev.Lett. 77 (1996) 2887-2890) 




“Confining picture of the Standard Model”

S. Dimopoulos et al., Nucl. Phys. B173, 208 (1980).

L. F. Abbott et al., Phys. Lett. 101B, 69 (1981); Nucl. Phys. B189, 547 (1981)

M. Claudson et al., Phys. Rev. D34, 873 (1986)

~80GeV ?

What about QCD?

No fundamental Higgs field … but a faithful scalar

can arise as a bound state particularly when





Global symmetry


What is Confinement? Is it well-defined?

Remarkably, the U(1)’ charges of all the elementary excitations in

CFL-phase are integers measured by the electron charge !

※U(1)’ differs from U(1)EM (which is broken by the diquark condensate).

The common criterion

“Absence of fractionally charged states in the physical spectrum”

does not distinguish CFL phase and hadron phase.

[More analogies between them, see T.Schafer and F. Wilczek, PRL 82 (1999) 3956]

Color-nonsinglet condensates might emerge even in zero-density QCD:

C. Wetterich, AIP Conf. Proc. 739, 123 (2005) [hep-ph/0410057]

J. Berges and C. Wetterich, PLB 512 (2001) 85 [hep-ph/0012311]

(Technicolor in QCD?)

This is a highly dynamical problem, worth a further study.

Only some restricted models (in addition to pure YM) have a clear,

kinematical distinction between confining phase and Higgs phase.

E.G.) SU(N)gauge-Higgs model with Higgs in the adjoint rep.

 Z(N) charge cannot be screened.

 Existence of a phase transition line.


What is Confinement? Is it well-defined?

  • In addition to the “confining phase” & “Higgs phase”,
  • non-Abelian gauge theories can be in a “massless phase”.
  • Case 1: Free phase (trivial IR fixed point, g=0)
  • Typically realized in theories with many flavors
  • Case 2: Non-abelian Coulomb phase (interacting IR fixed point, g≠0)
  • Envisaged by Banks and Zaks [NPB 196(1982)189]
  • and advertized by Seiberg

If Seiberg duality in SUSY is correct, then a confining theory of

electric d.o.f. and a Higgsed theory of magnetic d.o.f. is equivalent … !

(Far more nontrivial than the familiar complementality

of the Fradkin-Shenker type !)

Is the “confinement” a matter of the definition of variables ?

Is this ambiguity the reason why the Clay institute’s problem

centers on the mass gap, rather than confinement ? ---I don’t know.


Very Useful criterion of confinement (at zero temperature)

  • = Area law of the Wilson loop
  • in a representation of non-zero N-ality
  • This is a statement in pure YM, but our world contains
  • nearly massless fermions. It might be that we are comparing
  • apples with oranges !
  • The relation between the area law and the absence of gluons
  • or the existence of mass gap is not clear to me …
  • String tension vanishes for but glueballs survive
  • the deconfinement transition, so there might be no relation.
  • ▲ according to
  • N. Ishii et al., PRD66 (2002) 094506

In strong-coupling LGT, area law is almost trivial:

it is simply a matter of plaquette disorder.

Random fluctuation of group elements leads to the area law with

But, are the holonomies

really fluctuating randomly,

independent of each other ?


(Figure is taken from

J. Ambjorn, J. Greensite, JHEP 9805 (1998) 004)



    • Gauge group is SU(2),
    • Each loop is sufficiently large,
    • space-time dimension >2
    • (strong coupling expansion is valid)

Then, for

we can show


What is fluctuating independently?

It turns out to be Z(2)⊂SU(2) at least at strong coupling, but

there is a debate at weak coupling:


Why center?

(2+1)-d gauge theory in Schroedinger picture

When the space is a punctured plane, 

Gauge transformation function may be


Such an “interesting” Ω(θ) will exist if


 Oh…! Nothing interesting happens!

If all fields (gauge fields and matter fields) are invariant under a subset of center symmetry, we still have a chance.

In pure SU(N) theory, the true gauge group = SU(N)/Z(N)

Let denote an operator implementing

such “interesting” gauge transformation Ω

n : linking number


‘t Hooft argued that

if there were no massless particles.

Generalization to 4d

“‘t Hooft loop operator”

n : linking number of C and C’

‘t Hooft argued that

if there were no massless particles.


for weak couplingin SU(2) LGT was

rigorously proved by E.T.Tomboulis. [ PRD23 (1981) 2371 ]

Awkward points:

  • Non-zero mass gap is assumed.
  • Validity of the cluster decomposition used by ‘t Hooft is subtle.
  • Two possibilities
  • exist for confinement. Hence
  • is not a necessary condition for confinement.

Attempts toward “surface operators”:

[S. Gukov and E. Witten, hep-th/0612073, 0804.1561]


‘t Hooft loop on the lattice: (2+1)d case

: ‘t Hooft’s disorder operator

: link variable multiplied by

center element

Arbitrary Wilson loop

non-trivially winding around

will be multiplied by z.

(This is the definition of ‘t Hooft


(snapshot in a fixed-time plane)


‘t Hooft `loop’ on the lattice: (2+1)-d case

An example of a Full 3d snapshot


: Twisted plaquettes

(unphysical Dirac string)

(snapshot in a fixed-time plane)


(It is a true`loop’ only in 4d.)


‘t Hooft `loop’ on the lattice: (2+1)-d case

‘t Hooft loop of maximal size

(with no source, closed by periodicity)

thin center vortex

A closed curve on the dual lattice.


‘t Hooft loop on the lattice: (3+1)-d case

Draw a closed curve (C) on the dual lattice.

Take a surface (S) on the dual lattice which is spanned by C.

Twist every plaquette on the original lattice which is dual to the plaquettes in S.

(snapshot in a fixed-time plane)

Taken from L. Yaffe, PRD21 (1980) 1574

Thermal behavior of ‘t Hooft and Wilson loops[de Forcrand et al., PRL86(2001)1438]

High temp.

Low temp.

temporal ‘t Hooft loop

spatial ‘t Hooft loop

temporal Wilson loop

spatial Wilson loop


area law


area law



area law

area law

dual behavior

Dual string tension: order parameter for deconfinement


‘t Hooft loop on the lattice: (3+1)-d case

‘t Hooft loop of maximal size

(with no source, closed by periodicity)

thin center vortex

A closed surface on the dual lattice.


Vortex surface


Based on an exact duality between electric and magnetic flux free energy

of 4dYM, ‘t Hooft argued that

must hold in the thermodynamic limit if the system is confining.

This behavior was verified for SU(2) in strong coupling

expansion by Munster [PLB95(1980)59] giving

to all orders of the expansion!

(Their diagrams have a

one-to-one correspondence.)


This equality as well as ‘t Hooft’s conjecture is not rigorously proved yet.


was proved in SU(2) LGT by Tomboulis and Yaffe [CMP100(1985)313]

using reflection positivity.

 ’t Hooft’s conjecture is at least a sufficient condition

for the area law to hold.

  • Numerical Check at Weak coupling: Very hard task.
  • Kovacs and Tomboulis, PRL85(2000)704
  • ・・・Multihistogram method ( gradually)
  • de Forcrand et al., PRL86(2001)1438
  • ・・・Snake algorithm (# of twisted plaquettes grow gradually)

 Expected behavior was confirmed.

outline of the proposed proof
Outline of the Proposed Proof

Extension of Tomboulis-Yaffe’s inequality to SU(N) LGT:

[T. Kanazawa, 0808.3442]

This inequality can be verified in 2D SU(N) LGT explicitly.

Tomboulis [0707.2179] has put forawrda proof of area law.

●Estimate Z-ratio in order to bound <W(C)> from above,

●Represent Z^(-) and Z on a coarser lattice

by the Migdal-Kadanoff transformation with a larger β,

●Make the β’s of Z and Z^(-) coincide, [Incorrect proof]

●Apply strong-coupling expansion. The End.


The Migdal-Kadanoff (MK) transformation is proved to

flow into the strong coupling fixed point for arbitrary

initial coupling, for arbitrary b, for arbitrary compact

gauge group including U(1), SU(N), …

(Muller-Schiemann, ’88).

b ∈ Z・・・ decimation parameter.


We can very easily show

Tomboulis (in essence) claims that α can be equated to 1,

but his proof is mathematically incomplete.

[See T.Kanazawa, 0805.2742 for details. (To appear in PLB)]

To prove the area law, we need to show

It must be exponentially close to 1. This still remains unsolved.


Abelian dominance Take SU(2).

Fix the gauge on the lattice nonperturbatively by maximizing

  • which leaves a residual U(1) symmetry. Project all link variables
  • onto the diagonal part.  Nearly full string tension is recovered!
  • Off-diagonal gluons somehow acquire dynamically generated mass,
  • thus decoupling in the IR?
  • Was it proved? [K.-I. Kondo and A. Shiba, 0801.4203]
  • Monopole density ρ in projected configuration seems to have a
  • well-behaved continuum limit (a 0).
  • The action measured on the plaquettes bounding the monopole is
  • greater than the average plaquette action on the lattice.
  • (▲Gauge invariant check!! Is monopole not gauge artefact…?)
  • However it does not explain the breaking of
  • adjoint string.

Center dominance Take SU(2).

Fix the gauge on the lattice nonperturbatively by maximizing

which leaves Z(2) (=center) symmetry. Project all link variables

onto the center element (DMCG)

 Nearly full string tension is recovered!

  • Conversely, the removal of vortices cause both loss ofconfinement
  • and chiral symmetry breaking (for SU(2).) Why chiral symmetry????
  • [de Forcrand and D’Elia, PRL82 (1999) 4582]
  • Recently the survival of χSB in SU(3) in vortex-removed ensemble was
  • reported !! [Leinweber et al., NPB(Proc.Suppl.)161(2006)130] Not yet settled…
  • Using IDMCG, most monopoles are found to lie on P-vortices.
  • P-vortices really capture the core of physical thick vortices
  • in the vacuum: correlation with plaquette action and Wilson loops.

: exceptional Lie group with trivial center and

trivial first homotopy group

  • Fundamental rep. can be screened by adjoint rep. No area law
  • This gauge theory in 4d shows a 1st order deconfinement transition,
  • but there is NO order parameter! Can we understand it?
  • There is a chiral condensate! [J. Danzer et al., 0810.3973]
  • It vanishes exactly at the transition temperature.
  • [J. Greensite et al., PRD75(2007)034501] discovered that
  • the string tension is recovered under the projection onto SU(3)
  • subgroup, but with no gauge invariant support.
  • Very strange…

Extension of TY inequality to spin systems:

is it worth doing?  Yes.

SU(N)×SU(N) Principal chiral model (PCM)

in 2D has Profound Similarity to SU(N) LGT in 4D:

plaquette, link ⇔ link, site

Wilson loop ⇔ correlation function

area ⇔ length

string tension ⇔ mass gap


Confining phase ⇔ Disordered phase

Higgs phase ⇔ Ordered phase

Coulomb phase ⇔ KT phase

  • Asymptotic freedom,
  • Nonperturbatively generated mass gap,
  • Identical behavior under Migdal-Kadanoff transformation,
  • Bound state spectrum of PCM in 1D

= k-string spectrum of 2D SU(N) LGT (EXACT),

  • Durhuus-Olesen phase transition at N=∞,
  • Large N expansion in planar graphs,


Ref: L. Del Debbio, H. Panagopoulos, P. Rossi and

E. Vicari, JHEP 01, 009 (2002)


Integrability of PCM

becomes flat for certain f and g;

is invariant under continuous change of C

→ Conserved quantity

→ Taylor expansion

→ Infinitely many conserved quantities

→ Factorization of S-matrix

(Symmetry transformation: non-local)


Hasenfratz, Niedermayer (’90), Hollowood (’94), …

  • calculated exactly,
  • by combining
  • perturbation theory based on asymptotic freedom
  • and
  • Thermodynamic Bethe Ansatz based on factorizability.
  • The mass gap was assumed from the beginning
  • (Highly nontrivial) Consistency Check ; not Proof
  • Mechanism of mass generation is still unclear

Attempts of gauge theorists

1. Kovacs (’96), Kovacs-Tomboulis (’96)

extended TY inequality to SU(2) PCM; then

``SO(3) vortex is responsible for the mass gap.’’

2. Borisenko-Chernodub-Gubarev (’98)

used Abelian projection and observed

Abelian dominance & vortex percolation.

3. Borisenko-Skala (’00)

extended the Mack-Petkova inequality to SU(N) PCM;

then observed Center (Z2) dominance in SU(2) PCM.

``Center vortex is responsible for the mass gap.’’


Center vortex is induced by a center twist.

…effectively changing the boundary condition.


Fat vortices diminish the cost of the twist,

disordering the system quite efficiently.

Ex: (-1)-twist in 2D XY model


Why is center special?

In spin systems,

there’s no restriction for twist to center;

Arbitrary element of the symmetry group

could be used…


3 pi/4 twist

pi twist

pi/2 twist


Configuration induced by a twist

effectively contains effects of other twists.

In (abelian or non-abelian) spin systems,

twists may mix with each other.

Perceiving any subgroup as special

does not seem very natural physically.

In addition, if it were not for the center ?


Extending the TY’s inequality in SU(N) LGT

to SU(N) PCM is straightforward,

but more general formulation is preferable.



Definition of the correlation function


We require

as an important assumption.



Theorem [TY inequality in spin systems]

Both sides of inequalities are expected to decay exponentially as n → ∞

in the disorder phase.



● As long as the high-temperature expansion is valid,

this behavior can be seen.

Groeneveld-Jurkiewicz-KorthalsAltes (’81)

All order proof is also possible.


what is truly mysterious is

the mechanism of mass generation

at arbitrarily weak coupling.

● The interpretation of the reported

``Center dominance’’ in SU(2) PCM should be

carefully reexamined…


● As a testing ground, let’s consider the

planar antiferromagnetic isotropic triangular Ising model.

 Twist


(with p.b.c.)


In disordered phase for all T >0 ; (Houtappel, ’50)

  • no thermodynamic singularity.
  • was obtained analytically by Wu-Hu (’02).
  • We found

This is the first non-analyticity (or crossover)

found in the model.

Physical interpretation remains to be clarified.

My guess:

commensurate-incommensurate transition

Is it related to the breakdown of high-temperature

expansion for the mass gap??



  • The confinement is physically understood
  • in terms of vortex percolation in QCD vacuum.
  • The recovery of the string tension is still mysterious.
  • The exceptional deconfinement is mysterious.
  • Magnetic degrees of freedom at high
  • temperature is poorly understood.
  • The relation between non-abelian spin models
  • and non-abelian gauge theories seems to be
  • not yet fully clarified.
  • Relations to recent approaches should be
  • explored also.