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閉じ込め四方山話Much Ado about Confinement
Dec. 10 2008 @ Komaba Nuclear Theory GroupTakuya Kanazawa
Phys. Dep., Univ. of Tokyo
1.Overview1-1. Introduction: confinement
Emergence“The way complex systems and patterns arise out of a multiplicity of relatively simple interactions.” (from Wikipedia)
Emergent phenomena in Condensed Matter Physics:
superconductivity, superfluidity, 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.
continuously
connected
(composite particle description)
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)
What is Confinement? Is it well-defined?
~80GeV ?
1st
2nd?
?
“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)
What about QCD?
?
QGP
CSCHadron
No fundamental Higgs field … but a faithful scalar
can arise as a bound state particularly when
Global symmetry
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?
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]
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 complementalityof 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 loopin 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
(Figure is taken from J. Ambjorn, J. Greensite, JHEP 9805 (1998) 004)
But, are the holonomies
really fluctuating
randomly,
independent of each
other ?
No.
Assume
1. Gauge group is SU(2),
2. Each loop is sufficiently large,
3. space-time dimension >2
4. (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
multi-valued.
Such an “interesting” Ω(θ) will exist if
Let denote an operator implementing
such “interesting” gauge transformation Ω
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)
∴
n : linking number
‘t Hooft argued that
if there were no massless particles.
Generalization to 4d
n : linking number of C and C’
“‘t Hooft loop operator”
‘t Hooft argued that
if there were no massless particles.
for weak coupling in SU(2) LGT was
rigorously proved by E.T.Tomboulis. [ PRD23 (1981) 2371 ]
1. Non-zero mass gap is assumed.2. Validity of the cluster decomposition used by ‘t Hooft is
subtle.3. Two possibilities
exist for confinement. Hence is not a necessary condition for confinement.
Awkward points:
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
(snapshot in a fixed-time plane)
Arbitrary Wilson loop
non-trivially winding around
will be multiplied by z.
(This is the definition of ‘t Hooft
operator.)
‘t Hooft `loop’ on the lattice: (2+1)-d case
An example of a Full 3d snapshot
Monopole-pair
time
(snapshot in a fixed-time plane)
: Twisted plaquettes
(unphysical Dirac string)
(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
(snapshot in a fixed-time plane)
Taken from L. Yaffe, PRD21 (1980) 1574
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.
Thermal behavior of ‘t Hooft and Wilson loops [de Forcrand et al., PRL86(2001)1438]
temporal ‘t Hooft loop
spatial ‘t Hooft loop
temporal Wilson loop
spatial Wilson loop
screening
screening
area law
area law
screening
area law
screening
area law
High temp.
Low temp.
Dual string tension: order parameter for deconfinement
dual behavior
‘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.
But
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.
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 forawrd a proof of area law.
Outline of the Proposed Proof ●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.
[See T.Kanazawa, 0805.2742 for details. (To appear in PLB)]
We can very easily show
Tomboulis (in essence) claims that α can be equated to 1,
but his proof is mathematically incomplete.
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.
• Conversely, the removal of vortices cause both loss of confinement
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.
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!
: 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…
SU(N)×SU(N) Principal chiral model (PCM)
Extension of TY inequality to spin systems:
is it worth doing? Yes.
in 2D has Profound Similarity to SU(N) LGT in 4D:
plaquette, link ⇔ link, siteWilson loop ⇔ correlation function
area ⇔ lengthstring tension ⇔ mass gap
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,
‥‥
Confining phase ⇔ Disordered phaseHiggs phase ⇔ Ordered phaseCoulomb phase ⇔ KT phase
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 theorists1. 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.’’
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…
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.
Theorem [TY inequality in spin systems]
Both sides of inequalities are expected to decay exponentially as n
→ ∞
in the disorder phase.
Comments
● As long as the high-temperature expansion is
valid, this behavior can be seen. Groeneveld-Jurkiewicz-Korthals Altes (’81) All order proof is also possible.
But 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.
(with p.b.c.)
Twist
L
• 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??
Summary
• 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.