Transcript Document

Confinement and screening
in SU(N) and G(2) gauge theories
Štefan Olejník
Institute of Physics
Slovak Academy of Sciences
Bratislava, Slovakia
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
1
The many faces…
… after seeing horrifying
faces of quantum fields.
J. Greensite, ŠO: Vortices, symmetry
breaking, and temporary confinement
in SU(2) gauge-Higgs theory, PR D74
(2006) 014502 [hep-lat/0603024].

J. Greensite, K. Langfeld, H. Reinhardt, T.
Tok, ŠO: Color screening, Casimir scaling, and domain structure in G(2) and
SU(N) gauge theories, PR D75 (2007)
034501 [hep-lat/0609050].
 …
J. Greensite, ŠO: Yang-Mills wave functDifferent reactions of people
ional in (2+1) dimensions, work in
progress.

13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
2
Pierre vs. Jeff and Casimir scaling
Subject: Lattice 97
Date: Mon, 28 Jul 1997 14:09:01 -0700 (PDT)
From: Jeff Greensite <[email protected]>
To: [email protected], [email protected]
[…] In the afternoon, Pierre van Baal gave a onehour, idiosyncratic plenary talk entitled “The QCD
Vacuum.” … [There] He told the audience that the
news here was that I had “changed my religion” to
vortices, […], and [he asked], hey, what about all
that Casimir scaling stuff?? […]
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
3
Outline

Introduction



Question 1: What if center symmetry is broken by matter
fields?





Temporary confinement in G(2) gauge theory
Casimir scaling
A simple (simplistic) model: Casimir scaling and color screening from domain structure
of the QCD vacuum
Question 3: Can we derive (at least some) elements of the
picture from first principles?


Permanent vs. temporary confinement
SU(2) gauge field coupled to fundamental Higgs fields
Question 2: What if center is trivial?


Roles of center symmetry
Center vortices and confinement in pure gauge theory
In search of the approximate Yang-Mills vacuum wave functional in 2+1 dimensions
Conclusions and open questions
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
4
Roles of center symmetry

Additional symmetry of pure-gauge SU(N) YM theory:

Polyakov loop not invariant:

On a finite lattice, below or above the transition, <P (x)>=0, but:
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
5

String tension depends on the representation class:

The asymptotic string tension depends only on the class or N-ality of the group
representation to which the charge belongs






Non-zero N-ality color charges are confined.
Zero N-ality color charges are screened.
Same N-ality means same transformation properties under the center
subgroup ZN.
Particle language: A flux tube, e.g., between adjoint color sources can crack
and break due to pair production of gluons.
The string tension of a Wilson loop, evaluated in an ensemble of
configurations from the pure YM action, depends on the N-ality of the loop
representation.
Large-scale vacuum fluctuations – ocurring in the absence of any external
source – must contrive to disorder only the center degrees of freedom of
Wilson loop holonomies.
Ambjørn, Greensite (1998)
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
6
Center vortices and confinement in pure gauge theory
Reviewed in Maxim Chernodub’s talk on Tuesday




The picture was proposed and elaborated at the end of 70’s and
beginning of 80’s by ‘t Hooft, Mack and Petkova, Ambjørn et al.,
Cornwall, Feynman and many others.
Some people helped to “bury” the model (incl. Jeff Greensite).
The model does not rely on any particular gauge, but …
… how to identify center vortices in vacuum configurations?
Del Debbio, Faber, Greensite, ŠO (1997)
Del Debbio, Faber, Giedt, Greensite, ŠO (1998)
many other groups joined our efforts
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
7


Center vortices are identified by fixing to an adjoint gauge, and
then projecting link variables to the ZN subgroup of SU(N). The
excitations of the projected theory are known as P-vortices.
Direct maximal center (or adjoint Lorenz) gauge in SU(2):
One fixes to the maximum of
and center projects

Fit of a real configuration by thin-center-vortex configuration.
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
8
Numerical evidence

Center dominance.
Correlation with gauge-invariant information: vortex-limited Wilson
loops.
Vortex removal: confinement also removed.
Scaling of the vortex density.
Finite-temperature: deconfinement as vortex depercolation.
Vortex removal: chiral condensate and topological charge vanish.

Relation to other scenarios:







13.4.2007
Monopole worldlines lie on vortex sheets.
Thin center vortices “live” on the Gribov horizon.
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
9
Question 1:
What if center symmetry is broken
by matter fields?
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
10
Permanent vs. temporary confinement

What is confinement?
The term is used, in the literature, in several inequivalent, related, ways:




Permanent confinement:




Electric flux-tube formation, and a linear static quark potential.
Absence of color-electrically charged particle states in the spectrum.
Existence of a mass gap.
Global center symmetry.
Flux tube never breaks.
Pure gauge theories.
Temporary confinement:



13.4.2007
Asymptotic string tension is zero.
At large scales the vacuum state is similar to the Higgs phase of gauge-Higgs
theory.
Static quark potential does rise linearly for some interval of color source
separations.
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
11

With temporary confinement:





The simple kinematical motivation
for the center-vortex mechanism is
lost.
Relevance (or irrelevance) of
vortices is a dynamical issue,
which can be investigated in
numerical simulations.
Real QCD with dynamical quarks.
G(2) pure-gauge theory.
SU(2)-gauge–fundamental-Higgs
theory.
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
12
SU(2) gauge field coupled to fundamental Higgs fields
Osterwalder, Seiler (1978); Fradkin, Shenker (1979); Lang, Rebbi, Virasoro (1981)
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
13
Center dominance
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
14
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
15
Correlation with gauge-invariant information

Wn(C) – a Wilson loop, computed from unprojected link variables,
with the restriction that the minimal area of loop C is pierced by n
P-vortices on the projected lattice.
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
16
Vortex removal

deForcrand-D’Elia procedure: fix to maximal center gauge, and
multiply each link variable by its center-projected value.
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
17
Kertész line?
?
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
18


Osterwalder-Seiler–Fradkin-Shenker theorem: no phase transition isolating
the temporary-confinement region from a Higgs-like phase; at least no
transition detected by any local order parameter.
Kertész line in the Ising model: With small external magnetic field the
global Z2 symmetry of the zero-field model is explicitly broken, there is no
thermodynamic transition between an ordered to a disordered state; still
there is a sharp depercolation transition; the line of such transitions in
the T-h plane is called Kertész line.
Kertész (1989); Chernodub, Gubarev, Ilgenfritz, Schiller (1998); Chernodub (2005)

Symmetry-breaking transition: A local gauge symmetry can never be
broken spontaneously (Elitzur). The local symmetry can be fixed by a gauge
choice. Certain gauges, such as Coulomb or Lorenz gauge, leave unfixed a
global remnant of the local symmetry, and this can be spontaneously
broken.
Greensite, Zwanziger, ŠO (2004)
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
19
Vortex percolation

Vortex-percolation order parameter:





f(p) – the fraction of the total number NP of P-plaquettes on the lattice,
carried by the P-vortex containing the P-plaquette p.
sw – the value of f(p) when averaged over all P-plaquettes.
sw – the fraction of the total number of P-plaquettes on the lattice,
contained in the “average” P-vortex.
sw=1 … all P-plaquettes belong to a single vortex.
sw→0 … in the absence of percolation, in the infinite-volume limit.
Bertle, Faber, Greensite, ŠO (2004)
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
20
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
21
Remnant-symmetry breaking

In the lattice Coulomb gauge, fixed by maximizing, at each time slice,
there remains “remnant” gauge freedom, local in time, global in space:


The color-Coulomb potential:
Asymptotically, this potential is an upper bound on the static quark
potential:
Zwanziger (2003)

Confining color-Coulomb potential: necessary, but not sufficient condition
for confinement.
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
22

Remnant-symmetry-breaking order parameter: define
then

Relation to Coulomb energy:
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
23
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
24

Similar situation with other order parameters (v.e.v. of the Higgs
field in Lorenz gauge).
Caudy, Greensite, work in progress
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
25
Question 2:
What if center is trivial?
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
26
Temporary confinement in G(2) gauge theory

Center-vortex confinement mechanism claims that the asymptotic
string tension of a pure non-Abelian gauge theory results from
random fluctuations in the number of center vortices.
No vortices implies no asymptotic string tension!

Is G(2) gauge theory a counterexample?

Holland, Minkowski, Pepe, Wiese (2003), Pepe, Wiese (2006)




13.4.2007
No!
The asymptotic string tension of G(2) gauge theory is zero, in perfect
accord with the vortex proposal.
G(2) gauge theory however exhibits temporary confinement, i.e. the
potential between fundamental charges rises linearly at intermediate
distances. This can be qualitatively explained to be due to the group
center, albeit trivial! A model will be presented.
Prediction: Casimir scaling.
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
27
Linear potential at intermediate distances

Numerical simulations:



Potential computed from expectation
values of rectangular Wilson loops
W(r,t).




Metropolis algorithm with
microcanonical reflections, real
representation of G(2) matrices
(Langfeld et al.).
Cabibbo-Marinari method, complex
representation of G(2) matrices (Pepe
et al.; Greensite, ŠO).
Smeared spacelike links.
Unmodified timelike links.
Fit as usual: constant + Coulomb +
linear terms.
What should the linear rise be
attributed to?
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
28
Casimir scaling




At intermediate distances the string
tension between charges in
representation r is proportional to Cr.
Argument: Take a planar Wilson loop,
integrate out fields out of plane,
expand the resulting effective action:
Truncation to the first term gives
Casimir scaling automatically.
A challenge is to explain both Casimir
and N-ality dependence in terms of
vacuum fluctuations which dominate
the functional integral.
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
Bali, 2000
29
A simple (simplistic) model: Casimir scaling and color
screening from domain structure of the QCD vacuum



Casimir scaling results from uncorrelated (or short-range
correlated) fluctuations on a surface slice.
Color screening comes from center domain formation.
Idea: On a surface slice, YM vacuum is dominated by overlapping
center domains. Fluctuations within each domain are subject to the
weak constraint that the total magnetic flux adds up to an element
of the gauge-group center.
Faber, Greensite, ŠO (1998)
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
30

SU(2)
13.4.2007

G(2)
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
31



Consider a set of random numbers, whose probability distributions
are indep’t apart from the condition that their sum must equal K :
For nontrivial and trivial center domains, in SU(2):
Leads to (approximate) Casimir scaling at intermediate distances
and N-ality dependence at large distances.
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
32



The same would work for G(2) with only one type of “center
domain”, but the string tension is always asymptotically zero.
Prediction: Casimir scaling for potentials of various G(2)representation charges – needs to be verified in simulations!
How the domains arise and how to detect them? Why should the
string tension in SU(N) be the same at intermediate and large R?
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
33
Question 3:
Can we derive (at least some) elements of the
picture from first principles?
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
34
In search of the approximate Yang-Mills vacuum wave
functional in 2+1 dimensions


Confinement is the property of the vacuum of quantized non-abelian
gauge theories. In the hamiltonian formulation in D=d+1 dimensions and
temporal gauge:
Strong-coupling lattice-gauge theory – systematic expansion:
Greensite (1980)
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
35

At large distance scales one expects:
Greensite (1979)
Greensite, Iwasaki (1989)
Karabali, Kim, Nair (1998)

Property of dimensional reduction: Computation of a spacelike loop in
d+1 dimensions reduces to the calculation of a Wilson loop in Yang-Mills
theory in d Euclidean dimensions.
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
36

At weak couplings, one would like to similarly expand:

For g!0 one has simply:
Wheeler (1962)
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
37

A possibility to enforce gauge invariance:

No handle on how to choose f’s.
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
38
Our suggestion for the YM vacuum wave-functional in D=2+1
Samuel (1996)
Diakonov (unpublished)
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
39
Zero-mode, strong-field limit

Let’s assume we keep only the zero-mode of the A-field, i.e. fields
constant in space, varying in time. The lagrangian is
and the hamiltonian operator

The ground-state solution of the YM Schrödinger equation, up to
1/V corrections:
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
40


Now the proposed vacuum state coincides with this solution in the strongfield limit, assuming
The covariant laplacian is then
and it can be shown easily
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
41
Dimensional reduction


It was Samuel who suggested that the Ansatz
interpolates between perturbative vacuum at short wavelengths, and a
dimensional reduction form at large wavelengths.
However, what is short and long wavelength is ambiguous and gaugedependent. To make things better defined, we decompose:
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
42



If we keep nmax fixed for V!1, the eigenvalues can be approximated by 0 and
So the part of the wave-functional that depends on “slowly-varying” B has the
dimensional reduction form, i.e. the probability distribution of the D=2 YM theory.
One can then compute the string tension analytically and gets
An experiment: take m=(4/3) and compute the mass gap with the full proposed
wave-functional.
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
43
Numerical simulation of |0|2

To extract the mass gap, one would like to compute
in the probability distribution:


Looks hopeless, K[A] is highly non-local, not even known for arbitrary fields.
But if - after choosing a gauge - K[A] does not vary a lot among
thermalized configurations… then something can be done.
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
44

Define:

Hypothesis:

Iterative procedure:
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
45

Practical implementation:
choose e.g. axial A1=0 gauge, change variables from A2 to B. Then
1.
2.
3.
4.


All this is done on a lattice.
Of interest:



given A2, set A2’=A2,
P[A;A’] is gaussian in B, diagonalize K[A’] and generate new B-field (set of Bs)
stochastically;
from B, calculate A2 in axial gauge, and compute everything of interest;
go back to the first step, repeat as many times as necessary.
Eigenspectrum of the adjoint covariant laplacian.
Connected field-strength correlator, to get the mass gap:
For comparison the same computed on 2D slices of 3D lattices generated by
Monte Carlo.
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
46
Eigenspectrum (=18, 402 lattice)


Very close to the spectrum of the free laplacian.
Very little variation among lattices.
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
47
Mass gap
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
48
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
49




M(fit) is our result, obtained as described earlier.
M(MT) is result of the computation of the O+ glueball mass by Meyer and
Teper.
M=2m is the naïve estimate using the mass parameter entering the
approximate vacuum wave-functional.
Optimistic message: The glueball mass comes out quite accurately.
β
13.4.2007
M (fit)
M (MT)
2m
6
1.152
1.198
1.031
9
0.740
0.765
0.627
12
0.540
0.570
0.445
18
0.404
0.397
0.349
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
50
Why should m be different from 0?
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
51
What about N-ality?


This cannot be the whole story – with the simple wave-functional one
would expect Casimir scaling even of higher-representation asymptotic
string tensions, while asymptotic string tensions should in fact depend only
on the N-ality of representations. The simple zeroth-order state is clearly
missing the center-vortex or domain structure which has to dominate the
vacuum at sufficiently large scales.
Add a gluon mass term. Then |0|2 has local maxima at center vortex
configurations (seems a bit ad hoc, though).
Cornwall (2007)

Another possibility: include the leading correction to dimensional reduction.
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
52
Conclusions and open questions

Part I:





The vortex mechanism for producing linear potential can work even when the
gauge action does not possess global center symmetry; global center symmetry
is not necessarily essential to the vortex mechanism.
No single Kertész line separating the “confinement phase” and “Higgs phase” in
gauge-Higgs theory.
Do vortices have a branch polymer structure at large scales?
What is the situation in QCD with dynamical quarks?
Part II:



13.4.2007
G(2) gauge theory is not an exception to the claim that there is no asymptotic
string tension without center vortices.
Temporary confinement and screening may be explained by domain structure of
the vacuum.
It is crucial to verify the prediction of Casimir scaling for G(2) gauge theory.
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
53

Part III:

The proposed approximate vacuum wave-functional








13.4.2007
is a solution of the YM Schrödinger equation in the g→0 limit;
solves the YM Schrödinger equation in the strong field, zero-mode limit;
confines if m>0, and m>0 seems energetically preferred;
results in the numerically correct relationship between the string tension and mass gap.
But: Does the variational solution satisfy m ~ 1/ ?
Validity of approximations ?
Will corrections to dimensional reduction give the right N-ality dependence ?
D=3+1 ?
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
54
G(2), some mathematical details


G(2) is the smallest among the exceptional Lie groups G(2), F(4), E(6), E(7), and E(8). It has a
trivial center and contains the group SU(3) as a subgroup.
G(2) has rank 2, 14 generators, and the fundamental representation is 7-dimensional. It is a
subgroup of the rank 3 group SO(7) which has 21 generators.

With respect to the SU(3) subgroup:

3 G(2) gluons can screen a G(2) quark:
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
55
Dimensions
13.4.2007
The Many Faces of Quantum Fields, Lorentz Center, Leiden, April 10-13, 2007
56