77
Quark Confinement Arthur Corstanje Supervised by prof. J. Smit December, 2005
Quark Confinement
Arthur Corstanje
Supervised by prof. J. Smit
December, 2005
Quark confinement
Arthur Corstanje
December 14, 2005
Abstract
Quark confinement corresponds to a linear potential (i.e. a nonzero string tension) for a quark-antiquark pair. To investigate whether this linear potential is actually found in QCD, we needto calculate the potential at relatively long distances. Perturbative techniques like an expansionin terms of Feynman diagrams cannot be used. Numerical simulations of gauge theory on aspacetime lattice are an important tool for nonperturbative calculations. Further techniquesinclude the construction of an effective action for special field configurations such as monopolesand center vortices. These play an important role in establishing quark confinement.
We review simulation techniques for SU(2) gauge theory, including the Luscher-Weisz variance
reduction algorithm, as well as abelian projection and center projection methods. The latter are
used to locate monopoles and vortices in lattice simulations, and have established that monopoles
and vortices can both account for 94 % of the string tension. We also study an effective action
for monopoles in SU(2) theory, extending the analysis to gauge theory at high temperature.
Although the effective action tends to overestimate the string tension in the zero-temperature
case, it underestimates the high-temperature spatial string tension by a factor about 6.
Contents
1 Introduction 4
2 Theory 62.1 Continuum formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Definition of confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Lattice gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 The static quark potential from Wilson loops . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Polyakov line and Polyakov line correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 The quark potential at weak coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.7 Gauge theory at finite temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Simulating lattice gauge theory 153.1 Variance reduction in Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Luscher-Weisz method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 Topological excitations 254.1 Center vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2 The ’t Hooft-Polyakov monopole in SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5 Effective monopole action for SU(2) theory 335.1 Multimonopole configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2 Effective action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.3 String tension estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.4 String tension at high temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6 Simulation results for U(1) theory 426.1 Four-dimensional U(1) theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.2 Three-dimensional U(1) theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.3 Probability distribution for Wilson loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.4 Four-dimensional U(1) theory at finite temperature . . . . . . . . . . . . . . . . . . . . . . . 47
7 Abelian projection methods 517.1 Abelian projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.2 Center projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557.3 Higher representations of SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567.4 Higher representations of U(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587.5 String tension in higher representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597.6 Casimir scaling in abelian-projected SU(2) theory . . . . . . . . . . . . . . . . . . . . . . . . 617.7 Observation of string breaking in SU(2) theory . . . . . . . . . . . . . . . . . . . . . . . . . 637.8 Center projection and thick center vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2
7.9 Correlations between monopoles and vortices . . . . . . . . . . . . . . . . . . . . . . . . . . 66
8 The dual Abelian Higgs model 688.1 Comparison of SU(2) flux tubes with the DAH model . . . . . . . . . . . . . . . . . . . . . 68
9 Conclusion 72
10 Acknowledgements 74
3
Chapter 1
Introduction
The theory of the strong nuclear force, proposed about 30 years ago, is quantum chromo-dynamics (QCD). It states that hadronic matter, such as protons and neutrons, is builtup out of quarks, which are held together by gluons, the gauge bosons which mediate thestrong force.
However, no free quarks have been observed directly in the laboratory. There havebeen experiments in which electrons collide with very energetic protons, the so-called deepinelastic scattering experiments. The protons were found to have pointlike constituents,which have been identified as the quarks of QCD.
The experimental results can be accounted for by QCD if we can show that the potentialbetween quarks rises with distance, such that they are effectively confined to bound states.This is known as the ’confinement problem’, to be defined in more detail in due course.
Much theoretical work has been done since QCD was established. Studying quarkconfinement through perturbation theory, i.e. as an expansion in Feynman diagrams, wasfound to be insufficient, as the gauge interactions are too strong. In 1974 Wilson [5] useda lattice regularization to show that gauge theories are confining for large bare couplings.However, the continuum limit of the theory is at weak (bare) couplings, for which thestrong coupling expansions used in [5] do not apply.
Another approach, pioneered by Creutz [1] and Wilson [2] in 1980, is to simulate thegauge fields of QCD (and nowadays also the fermions) on a computer. This is done byMonte Carlo analysis, on a four-dimensional spacetime lattice. The simulations have givengood evidence of quark confinement, and the potential between quarks can be studied indetail.
Both analytical and numerical studies use the path integral formalism. The path in-tegral involves an average over all field configurations, with appropriate weight factorsdetermined by the action S. A field configuration is a particular set of values for the gaugepotential denoted by Aµ as a function of x. There is a special class of topological fieldconfigurations that we will study, such as monopoles and vortices, which play an importantrole in establishing a linear potential and confinement.
The outline is as follows: in chapter 2 we will give a short review of gauge theoryon a spacetime lattice, especially how to calculate the potential energy V (r) for a quark-
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antiquark pair at separation distance r. We will review simulation techniques for latticegauge theory (chapter 3), including recent improvements that allow for calculating thepotential at longer distance.
After reviewing center vortices and monopoles (chapter 4) we investigate the relevanceof monopoles for confinement by constructing an effective action (chapter 5). We therebymake use of the simulation data we produced (chapter 6).
In chapter 7 we review additional techniques to isolate monopoles and vortices in latticesimulations, including their results for the inter-quark potential. Chapter 8, is a review ofa numerical study relating gauge theory to the dual abelian Higgs model.
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Chapter 2
Theory
The following will be a brief introduction to lattice gauge theory. It is by no means intendedas a full introduction; we will assume the reader to be familiar with the subject, but wewill review some basic aspects (found in [3], [4]). We focus on the quantities we will needin this study, and fix some notations.
2.1 Continuum formulation
In a continuum spacetime, the action of QCD, which is SU(N) Yang-Mills theory, is givenby
S =∫
d4x
(
1
2g2Tr (FµνFµν) + ψ(γµDµ + m)ψ
)
(2.1)
Fµν = DµAν − DνAµ = ∂µAν − ∂νAµ − i [Aµ, Aν ] . (2.2)
The gauge field is Aµ(x), and can be parametrized by
Aµ = Akµtk, (2.3)
with tk denoting the N 2 − 1 generators of the SU(N) group. The presence of the gaugefields ensures gauge invariance of the action. That is, one can choose any smooth functionΩ(x) ∈ SU(N) such that the transformation
ψ′(x) = Ω†(x)ψ(x) (2.4)
leaves the value of the action unchanged, if it is countered by a transformation on theAµ-field:
A′µ(x) = Ω†AµΩ + iΩ†∂µΩ. (2.5)
Quantum chromodynamics is defined using the SU(3) group, but the formulation isessentially independent of the number of colors N . In some lattice studies one uses SU(2)theory. The main reason is the lower complexity of the group, making computer simulationsmore tractable. Moreover, the SU(2) and SU(3) theory give similar results, especially forquark confinement and string tension.
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2.2 Definition of confinement
In order to define the confinement problem in a useful way, we follow Greensite, [4]. Asmentioned, confinement should imply that there are no free quarks to be found in nature.Slightly more general is the statement that there exists no free color charge in nature. Thisrules out the possibility of a single quark bound to (for example) a massive scalar particle.It is usually assumed (supported by lattice simulations) that the color-electric field linesbetween a quark-antiquark pair will be collimated into a ’flux tube’ (fig. 2.1).
q q
q q
q q q q q q
q q
Figure 2.1: String breaking by quark-antiquark pair production. An artist impression isshown at the front cover.
This resembles the phenomenological model in which a meson is represented as a line-like object with an energy E = σL, where σ is the string tension and L is the length.Therefore the confinement problem will be to show that the quark-antiquark potential willapproach σL for large distances. An immediate problem is that the potential energy willnot rise indefinitely with quark separation. Above V ' 2mq additional quark-antiquarkpairs will be produced, leading to ’string breaking’, as shown in fig. 2.1. To circumventthis possibility we take the limit of quark masses to infinity. A linearly rising potentialbetween infinitely heavy quarks will be referred to as ’static quark confinement’.
2.3 Lattice gauge theory
When describing field theory on a spacetime lattice, a different set of variables is chosen.The gauge fields are described by link variables:
Uµ(x) = eiaAkµ(x)tk . (2.6)
The lattice constant is a. Most of the time, we will work in lattice units, i.e. we take a = 1.The link variables are defined on the links between lattice points; in this case between x
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and x + µ (see fig. 2.2). Fermions cannot be described immediately on a lattice. Naiveattempts to do this will result in a theory that describes too many fermions. This is knownas fermion doubling. Specific countermeasures exist to circumvent this problem, but wewill not consider them in this study, as a pure gauge theory (without dynamical fermions)will suffice for studying confinement.
y µ
ν
µU (y)x
Figure 2.2: The link variable Uµ(y) on the lattice, together with a plaquette at position x.
The action on the lattice is not uniquely defined; the criterion is that any valid latticeaction must lead to the continuum action (2.1) when taking the continuum limit a → 0.Usually one takes the Wilson action, which is
S = β∑
p
(
1 − 1
NReTr Up
)
, (2.7)
where β = 2Ng2 is the coupling strength for SU(N) theory. It is a sum over plaquettes,
elementary squares on the lattice, as indicated in fig. 2.2. The plaquette variable Up is aproduct of four link variables:
Up = Uµ(x)Uν(x + µ)U †µ(x + µ + ν)U †
ν(x + ν). (2.8)
The definition of a derivative on the lattice is as follows:
∂µUν = Uν(x + µ) − Uν(x) (2.9)
∂′µUν = Uν(x) − Uν(x − µ). (2.10)
These are labeled forward and backward derivative. For the plaquette variables Up oneobtains using these definitions
Up = exp(
ia2FµνFµν + O(a3))
, (2.11)
with no summation over µ and ν, and the continuum limit of the lattice action follows.For U(1) theory the action is slightly different:
S = β∑
p
(1 − ReUp) . (2.12)
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This is the Wilson action, with β = 1g2 ; further on we also make use of the Villain action,
which is
exp(βS) =∏
x,µ<ν
∞∑
nµν(x)=−∞exp
(
− 1
2g2(Fµν − 2πnµν(x))2
)
. (2.13)
2.4 The static quark potential from Wilson loops
In this section and the next we will give the operators needed to calculate the static quarkpotential. As we have taken the quark masses to infinity we can assume them to be static,so dynamical quarks are absent in the theory. A pure gauge theory is appropriate forstudying static quark confinement.
In lattice (pure) gauge theory a useful quantity is the Wilson loop. Imagine creating aquark-antiquark pair at a (spatial) distance R, at t = 0. Their masses are taken to infinity,so we may expect them to be static (no kinetic energy!). After a large time T we let themrejoin and annihilate. This is described in gauge theory as a source term in the action:
exp(−S) → exp(−S)P exp(i∮
CdxµAµ). (2.14)
As indicated, C is a rectangular contour with spatial length R and timelike length T . Thisis called the Wilson loop. The P symbol indicates path ordering of the integrand. This isimportant since we are dealing with nonabelian gauge theory. Moreover, it ensures gaugeinvariance of the Wilson loop. The definition of path ordering is as follows:
UC = P exp(i∮
CdxµAµ) = lim
dxi→0(1 + igAµ(x0)dx1,µ)..(1 + igAµ(xn−1)dxn,µ). (2.15)
It is a formal definition which is in general hard to evaluate. In the lattice theory theWilson loop has a simpler form - it is the path-ordered product of link variables along theloop C:
UC =∏
l∈C
Ul. (2.16)
In general, the real part of the trace (or group character) is taken over UC to obtain ascalar quantity, which is
WC =1
NTr UC . (2.17)
In the text we will often refer to WC as ’the Wilson loop’, and to UC as ’Wilson loopmatrix’. A gauge transformation Ω(x) acts on Uµ(x) as
Uµ(x) → Ω(x) Uµ(x) Ω†(x + µ). (2.18)
Inserting this in WC leads to an expression of the form
WC = Tr(
Ω(x)Uµ(x)Ω†(x + µ)Ω(x + µUµ(x + µ)...U †ν(x)Ω†(x)
)
. (2.19)
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In between link variables, the Ω and Ω† cancel; by using the cyclicity property of the tracewe can cancel Ω(x) against Ω†(x), and we see that WC is gauge invariant.
The path integral for the Wilson loop expectation value is written as
〈WC〉 =1
Z
∫
DUe−βSReTr P exp(
i∮
CdxµAµ
)
. (2.20)
Given that it describes creation and annihilation of a quark-antiquark pair, we can use theformulation of quantum mechanics in imaginary time to give a decomposition in terms ofenergy eigenstates. For this we introduce creation and annihilation operators Q†(t) andQ(t) resp. to get:
⟨
Q(T )Q†(0)⟩
=
∑
nm 〈0|Q|n〉⟨
n|e−HT |m⟩ ⟨
m|Q†|0⟩
〈0|e−HT |0〉 (2.21)
=∑
n
|cn|2e−(V (R)+∆En)T . (2.22)
The lowest energy eigenstate has energy V (R) by definition, which is referred to as thestatic quark potential. This is the quantity we are interested in, as static quark con-finement is equivalent to V (R) ∼ σR for large R. However, the existence of higher-energyeigenstates, denoted by their excess energy ∆En, means that we need to take T sufficientlylarge. The higher-energy terms in the expansion are then sufficiently suppressed. Therelation between the potential and the Wilson loop expectation value follows as
V (R) = − limT→∞
log〈W (R, T + 1)〉〈W (R, T )〉 . (2.23)
It is a priori not known how many of these higher-energy states there are (i.e. what thespectrum is), and what their overlap with the Wilson loop is compared to the ground state.Therefore, in lattice simulations, one has to find out by experiment which values of T arelarge enough at each R.
2.5 Polyakov line and Polyakov line correlators
A quantity related to the Wilson loop is the Polyakov line. Instead of a closed contourin spacetime, we now take a single line along the time direction (fig. 2.3). It stretchesout to t = ±∞ when an infinite spacetime is taken. On a finite lattice it is effectivelya loop, closed by periodicity in the time direction. In fact, the same holds for a theoryon an infinite spacetime, but at finite temperature T , where the timelike direction is alsocompactified, to a length Lt = 1
T.
The expectation value for the Polyakov line is written as
〈P (~x)〉 =1
Z
∫
DUe−βSP exp(i∮
CdxµAµ). (2.24)
10
t
x
Figure 2.3: The single Polyakov line on the lattice.
On the lattice, the Polyakov line is
P (~x) = Tr (U4(~x, 0) U4(~x, 1).. U4(~x, Lt − 1)) . (2.25)
We can interpret the Polyakov line as a single-quark state. When spacetime is infinite,it corresponds (formally) to a creation operator for a single quark at t = −∞, and acorresponding annihilation operator at t = +∞. The free energy Fq of a single quarkfollows from the expectation value, as
〈P (~x)〉 = e−FqLt . (2.26)
As mentioned before, free quarks cannot occur whenever confinement holds. This meansthat confinement implies Fq → ∞ and 〈P (~x)〉 = 0. A nonzero expectation value means noquark confinement. Therefore, the Polyakov line is an order parameter for confinement.
In finite-temperature gauge theory, a phase transition will occur when the temperatureT is raised above a critical temperature Tc. For T > Tc, or Lt < 1
Tc, one will find 〈P (~x)〉 6= 0
instead of zero for T < Tc. This is known as the deconfinement transition.
t
x
Figure 2.4: The Polyakov-line correlator (or double Polyakov line) on the lattice.
We have seen that for extracting the static quark potential V (R), Wilson loops withtemporal size t as large as possible are optimal. The limiting case of large t is shown infig. 2.4. It corresponds to a double Polyakov line, or a Polyakov-line correlator. From the
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Wilson loop, its relation with the potential follows straightforwardly:
〈P (0)P ∗(R)〉 = e−V (R)Lt . (2.27)
This relation only holds if we can assume Lt to be large enough for the excited states tobe sufficiently suppressed. It should be no problem at zero temperature, or the practicalapproximation of it, i.e. an N 4 (hypercubic) lattice. However, at high temperatures suchthat Lt is small, we should regard the energy extracted from eq. (2.27) as the free energyof the static quark-antiquark pair instead of the static potential.
2.6 The quark potential at weak coupling
At weak couplings, corresponding to short distance scales, we can use perturbation theoryto extract the potential [3]. In this case, the Wilson loop expectation value, which iswritten as
Z(J)
Z(0)=
⟨
Tr P exp(ig∮
Aaµtkdxµ)
⟩
, (2.28)
is approximated using Feynman diagrams. The hermitian generators of the group are againwritten as tk. To lowest order in g2, we can write
Z(J)
Z(0)= exp
(
1
2g2Tr tktk
∫
d4xd4y Jµ(x)Dµν(x − y)Jν(y))
, (2.29)
in correspondence with the diagram shown in fig. 2.5. The diagrams represent a selfenergyand a one-gluon exchange, respectively. Using a lattice regularization, the potential V (r)becomes
V (r) = − limT→∞
log
(
Z(J)
Z(0)
)
(2.30)
= − 1
TTr tktk
1
2g2
∑
x,y
Jµ(x)Dµν(x − y)Jν(y) (T → ∞), (2.31)
with T the timelike distance. The propagator follows from the momentum-space represen-tation
Dµν(p) =δµνa
2
∑
µ (2 − 2 cos(apµ)), (2.32)
Figure 2.5: The diagrams showing one-gluon exchange in (a), and selfenergy in (b).
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which after Fourier transformation becomes
Dµν(x − y) =δµν
4π2(x − y)2(|x − y| À a). (2.33)
For the potential the summations lead to
V (r) = −g2Tr (tktk)
4πr+ g2Tr (tktk)v(0), (2.34)
which is the Coulomb potential plus a self-energy correction. The correction is finite be-cause of the lattice regularization. In particular, av(0) = 0.253 is a dimensionless number.
The Coulomb part is linear in Tr tktk, which is the quadratic Casimir of the group,denoted by C2. The value of C2 depends on the representation of the group SU(N), in whichthe quarks are defined. In QCD, quarks are defined in the fundamental representation ofSU(3). The Casimir scale factor turns out to be important for larger distances as well;further on we will see investigations of this scale factor at distances beyond the reach ofperturbation theory.
If corrections up to order g4 are taken into account, the potential acquires a logarithmiccorrection term:
V (r) = − C2
4πr
(
g2 +11N
48π2log(r2/a2) + O(g6)
)
, (2.35)
with N the number of colours, determining the group SU(N). From this form of the po-tential we can define a renormalised coupling gR in order to absorb the correction term:
V (r) ≡ C2g2R
4πr2; (2.36)
g2R = g2 + β0g
4 log(r2/a2); (2.37)
β0 =11N
48π2. (2.38)
The coupling constants can be expressed in terms of the distance scales r and a in termsof the renormalization group betafunctions:
βR(gR) = −r∂
∂rgR; (2.39)
β(g) = −a∂
∂ag. (2.40)
This way, we have defined gR = gR(r), and g = g(a). By definition, gR does not dependon a, and g is independent of r. The betafunctions are expanded in powers of the couplingconstant; it turns out that the first two terms are the same for both functions:
βR(gR) = −β0g3R − β1g
5R + O(g7); (2.41)
β(g) = −β0g3 − β1g
5 + O(g7); (2.42)
β1 =102
121β2
0 . (2.43)
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The most important feature of these functions is their sign, which implies that g and gR
will rise with the distance scales. In fact, it follows that
lima↓0
g(a) = 0, (2.44)
and the same for gR for R ↓ 0. This statement is known as asymptotic freedom. It isimportant, as it ensures the existence of a weak-coupling domain, i.e. for a small distancescale a perturbation theory can be applied, with accuracy improving when a is lowered.
When the lattice spacing a is increased, the coupling constant will rise, to such an extentthat an expansion in powers of g can no longer be used. Would we know the betafunctionaccurately up to large lattice spacings a and large g2, we would very likely enter the strong-coupling domain of large g(a). Static quark confinement is proven at strong coupling [3][6]. On the lattice, the theory is well accessible by (convergent!) analytic strong-couplingexpansions, for large values of the (bare) coupling constant g.
2.7 Gauge theory at finite temperature
So far, we have discussed lattice gauge theory at zero temperature. A generalization tononzero temperature is made by compactifying the timelike dimension. That is, we identifythe field configuration at t = 0 with the configuration at t = 1
T. T is the temperature,
having the dimension of energy, i.e. it is often written as kBT , with Boltzmann’s constantincluded.
In lattice simulations we have compactified all dimensions, so any finite lattice sim-ulation is a simulation at finite temperature (with additional finite-size effects from thespatial dimensions). If the (physical) length of the lattice is sufficiently large, the tem-perature is close to zero and the T = 0 case is well approximated. Higher temperaturesare described by a lattice of size N 3
s × Nt instead of N 4, with Nt < Ns. Let us examinethe static quark potential, calculated from Polyakov-line correlators. When temperatureis increased, the timelike length decreases. As we have described earlier this will implythat excited states in the energy expansion are no longer (fully) suppressed. Therefore theenergy we extract from the Polyakov correlators will change. Most importantly, there is acritical temperature Tc above which the string tension disappears. This is called the de-
confinement transition. The single Polyakov line acquires a nonzero expectation value forT > Tc, which is also a signal of deconfinement. The deconfinement transition is relevantfor high-energy experiments, aiming to create a quark-gluon plasma, a high-temperaturestate in which quarks and gluons appear in a dense, plasma-like state.
In the limit of high temperature, i.e. T À Tc, the timelike direction is so small thatthe theory is effectively reduced to 3 dimensions.
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Chapter 3
Simulating lattice gauge theory
To simulate lattice gauge theory on a computer one makes use of a Monte Carlo updateprocedure. As available computer time and memory space are finite, one has to approxi-mate the four-dimensional Euclidean space by a finite-sized lattice. By imposing periodicboundary conditions, one effectively removes boundary effects. However, one still has tobe aware of finite-size effects, especially when correlation lengths are similar to the latticesize. The average of observable (gauge-invariant) quantities is obtained in the path-integralrepresentation:
Z =∫
DU exp (−βS(U)) , (3.1)
〈X〉 =1
Z
∫
DUX(U) exp (−βS(U)) . (3.2)
In the simulation we estimate these path-integral averages by creating many configurationsof gauge fields. Configurations are created which are distributed according to exp(−βS).The Metropolis algorithm to do this is as follows:
• Start from an arbitrary configuration of link variables
• For link variable Uµ(x), propose a change Uµ(x) → U ′µ(x)
• Evaluate the change in the action ∆S
• If ∆S < 0, accept the change
• If ∆S > 0, accept with probability exp(−β∆S) (else reject the change and continue)
• Go through the lattice many times (called lattice sweeps) to reach equilibrium
After a large number of lattice sweeps, a situation of equilibrium is reached, analogous tothermal equilibrium in statistical mechanics. Then, one can extract the physical quantities(such as Wilson loops and Polyakov lines) from the (current) gauge configuration. Linkupdating is continued, and averaging of the quantities is done over many configurations,separated by n lattice sweeps. This measurement interval is useful, because measurements
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can then be taken from configurations which are less correlated. One may take intervalssuch as n = 20 or n = 100, depending on the parameters of the theory. The string tensioncan be estimated from averaged Wilson loops by using the Creutz ratio:
χ(R, T ) = − log
(
〈W (R, T )〉 〈W (R − 1, T − 1)〉〈W (R − 1, T )〉 〈W (R, T − 1)〉
)
. (3.3)
The rectangular Wilson loop contour is denoted here by the sides R and T of the rectangle.This quantity is constructed in such a way that only the area-law contribution to the falloffof Wilson loops with loop size is extracted. One thereby takes the following ansatz for theaverage Wilson loop at large R, T :
〈W (R, T )〉 ' exp(−σRT − 2µ(R + T ) + C), (3.4)
i.e. an area-law factor, a perimeter-law factor and a constant. Inserting this in the expres-sion for the Creutz ratio gives
χ(R, T ) = σ, (3.5)
so the perimeter-law factor and the constant cancel out. It should be noted, however, thatthe Creutz ratio does not have absolute accuracy in determining the string tension. Inreality, there may be other contributions present, which will only be of the constant orperimeter-law form in the limit of large loops.
For instance, one expects a correction term in the exponent of eq. (3.4), correspondingto the Coulomb-like potential derived at weak coupling in sect. 2.6. Using 〈W 〉 ' e−V (R)T ,this leads to
〈W 〉 ∝ exp(
−α(
T
R+
R
T
))
, (3.6)
with α = g2C2
4π. The presence of the R
T-term arises from 4D rotational symmetry of the
theory, as it is formulated in Euclidean space. Substituting this into the Creutz ratio gives
χCoulomb =−α
R(R − 1)+
−α
T (T − 1), (3.7)
which acts like an additive error in the relation between the Creutz ratio and the stringtension. The pure Coulomb potential is a good approximation only for short distances orsmall gauge couplings, but it gives an indication of the inaccuracy of Creutz ratios. Theerror terms in eq. (3.7) are optimal for R = T as large as possible, as the errors reduce
as O(
1R2
)
. Most authors use square Wilson loops for calulating Creutz ratios. For someapplications, studying only square Wilson loops is not sufficient. As indicated in sect. 2.4,one needs rectangular Wilson loops with a ’large’ size in the timelike direction in order toextract the ground state energy. In special cases, such as measurements of string breaking,the Wilson loop operator has such a small overlap with the ground state that one needsT ' 2R to obtain the ground state. For these cases, one determines the potential V (R)for the loops as defined in eq. (2.23), which can be fitted to a theoretical form, containinga Coulomb term, a string tension and a constant:
V (R) = − a
R+ bR + c, (3.8)
16
where b = σ is the string tension.
We determine string tensions using Creutz ratios, using R = T as large as possible. Asthe average Wilson loop 〈W (R,R)〉 ' e−σR2
it drops to zero exponentially with increasingarea. One needs correspondingly good statistics, which is the major (technical) difficultyin lattice simulations. The error analysis on Creutz ratios can, in principle, be done ana-lytically given the uncertainties in 〈W (R, T )〉. In practice one usually employs statisticalmethods such as the jackknife or bootstrap analysis. We choose the jackknife method, whichis explained in ref. [6]. The idea is to calculate N Creutz ratios from the set of N measuredWilson loop values. This is done by leaving out one of the measurements each time (no. 1,2, .., N), and calculating the Creutz ratio from the remaining N − 1 measurements. Thevariance in these Creutz ratios (when properly rescaled) gives the error estimate.
3.1 Variance reduction in Monte Carlo simulations
As mentioned in the previous section, statistical errors are an important limiting factor insimulating lattice gauge theory. In this section we will discuss this in further detail, alsoconsidering methods to improve accuracy.
-1,0 -0,5 0,0 0,5 1,0
1
10
Rel
ativ
e fr
eque
ncy
Wilson loop value W C
Histogram for 5x5 Wilson loops
Figure 3.1: Histogram for 5 × 5 Wilson loops in 3D U(1) theory, β = 1.
In the case of a nonzero string-tension, Wilson loop expectation values decrease ex-ponentially with loop area. This would not be a problem if the variance would decrease
17
rapidly as well, i.e. if the probability distribution were strongly peaked around the expec-tation value. However, a typical probability distribution for the Wilson loop is shown infig. 3.1, which concerns 5x5 loops in 3D U(1) theory, β = 1. The expectation value isless than 10−3, but the distribution has strong peaks near -1 and +1 (note the logarithmicscale). Therefore the variance is quite large, it is close to the maximum value of 1, whichwould occur if the Wilson loop takes values ±1 only. The uncertainty in the expectationvalue given N measurements is:
u =s√N
' 1√N
, (3.9)
when we write the variance as s2, and s is the standard deviation.For reasonably accurate measurements we need uncertainties smaller than a certain
fraction of the signal to measure; let us take 1% for example. Then, u < 1100
〈W 〉 isrequired. Using the area-law for a square Wilson loop this amounts to
〈W 〉 ∝ e−σR2
, (3.10)
u =1
100e−σR2
, (3.11)
N =1
u2= 10000e2σR2
. (3.12)
The result is a strong exponential dependence of the necessary computer time (here notedas the number of samples N which are required) on the distance R. As a consequenceone cannot stretch the range of R significantly by employing even an order of magnitudemore computer time. The achievable distance depends on the string tension of the theory:R ∝ 1√
s, with a proportionality constant determined by the order of magnitude of available
computer power.The limits of Monte Carlo computation for variables with small expectation values
call for additional methods to improve accuracy. Fortunately, there are various ways toaccomplish this. One method is to construct ’improved’ operators, e.g. Wilson loops with’thick links’, which have a larger overlap with the ground state of the theory. This methodis called link smearing, and it is now widely used. The potential can then be extracted forR×T loops with smaller T . However, the technique involves some parameters which haveto be tuned more or less by trial and error, i.e. there is no straightforward link with the(physical) theory. Therefore we will not use these methods in the following, and concentrateon measuring the ’pure’ Wilson loop. Improving accuracy in measuring pure Wilson loopsamounts to variance reduction.
The general idea is that, when calculating the expectation value of a (physical) quantity〈A〉, one may find another quantity A′ satisfying
〈A′〉 = 〈A〉 ,⟨
A′2⟩
<⟨
A2⟩
. (3.13)
This means that we can significantly reduce the uncertainty in calculations of 〈A〉 by using’improved estimators’, without introducing any systematic errors.
18
A first example is the multihit method, introduced in 1983 by [7]. In this method, onereplaces the link variables Uµ(x) by their ’thermal’ averages:
Uµ(x) =
∫
dUµ(x) Uµ(x) e−βS(U)
∫
dUµ(x) e−βS(U). (3.14)
One forms Wilson loops and Polyakov lines by multiplication of averaged link variablesUµ(x). The reason for this averaging to be allowed is nontrivial; we will check the requiredproperties, eq.(3.13). The averaging can in this case be done analytically for U(1) andSU(2) theory, but Monte Carlo averaging is also allowed as the expectation value is thesame. The latter method amounts to repeatedly updating a single link, hence the name’multihit’. Consider a Polyakov line, with expectation value given by
〈P 〉 =1
Z
∫
DU U4(x) U4(x + 4) .. U4(x + (L4 − 1)4) e−βS. (3.15)
Inserting averaged links Uµ(x), it is written as
〈P 〉 =
∫
D′U eβS(
∫
dU4(x) U4(x) e−βS)
..(
∫
dU4(x + (L4 − 1)4) U4(x + (L4 − 1)4) e−βS)
∫
D′U e−βS (∫
dU4(x) e−βS) ..(
∫
dU4(x + (L4 − 1)4) e−βS) .
(3.16)The path-integral measure D′U is the same as DU but with the averaged links U4(x)..U4(x + (L4 − 1)4)removed. The crucial condition for eq. (3.16) to equal eq. (3.15) is that the links in ques-tion can be integrated over independently, i.e. the integration can be factorized. This musthold for any pair of averaged links present in the integration. It can be expressed as thecondition:
∫ ∫
dUµ dUν Uµ Uν e−βS
∫ ∫
dUµ dUν e−βS=
(∫
dUµ Uµ e−βS
∫
dUµ e−βS
) (∫
dUν Uν e−βS
∫
dUν e−βS
)
. (3.17)
This condition is satisfied if the links Uµ(x) and Uν(y) have an independent effect onthe action. We therefore define two (different) links to be independent if the separatetransformations
Uµ(x) → Uµ(x) + ∆Uµ(x) giving S → S + ∆S1, (3.18)
Uν(y) → Uν(y) + ∆Uν(y) giving S → S + ∆S2 (3.19)
can be combined to get
Uµ(x) → Uµ(x) + ∆Uµ(x)
Uν(y) → Uν(y) + ∆Uν(y)giving S → S + ∆S1 + ∆S2.
Note that in general the combined transformation gives S → S+∆S3 with ∆S3 6= ∆ S1 + ∆S2.The standard Wilson and Villain actions depend only on plaquette loops and therefore
are local. As a result, independence defined here holds for any two link variables which are
19
not on the same plaquette. It is easily seen that the condition (3.17) is satisfied for thePolyakov line (see also fig. 2.3).
For Wilson loops, there is a slight complication at the corners, as there are two linkson the same plaquette. The solution is either not to apply the multihit method on one ofthe links at each corner, or to average over both links in a corner simultaneously (e.g. byMonte Carlo updates).
Having established that the multihit-method can be applied correctly on Wilson loopsand Polyakov lines, it is also worthwile to examine the improvement in accuracy it brings.It is easy to see for U(1) theory that the one-link integral is strictly smaller than 1:
∣
∣
∣
∣
∣
∫
dU U e−βS
∫
dU e−βS
∣
∣
∣
∣
∣
< 1, (3.20)
as we can write U = eiθ which is integrated from θ = 0 to θ = 2π. In fact, it holds forSU(N) theory as well. For the U(1) case, the exact result is given by [18]:
∫
dU U e−βS
∫
dU e−βS=
I1(β |H|)I0(β |H|)
H
|H| . (3.21)
The sum of the 2(d − 1) staples surrounding the link U is denoted by H:
Hµ(x) =∑
ν 6=µ
(
U †ν(x + µ) Uµ(x + ν) Uν(x) + Uν(x + µ − ν) Uµ(x − ν) U †
ν(x))
. (3.22)
The Ik in eq. (3.21) is the k-th modified Bessel function. A useful way to compute Ik
is the (rapidly convergent) series representation:
Ik(x) =∞∑
n=0
1
(n + k)!n!
(
x
2
)2n+k
. (3.23)
A graph of I1(x)/I0(x) is shown in fig. 3.2. Every averaged link has a modulus r < 1,leading to a variance improvement following a perimeter-law:
V ar ∝ αP , P = Perimeter. (3.24)
The value of α is about 0.7 - 0.8, and varies with coupling constant β. For smaller β,the expectation values 〈W 〉 are smaller, and α is smaller as well. Therefore the varianceimprovement is better at strong couplings.
3.2 Luscher-Weisz method
Recently (in 2003), Luscher and Weisz [8] generalised the multihit method to more thanone link. The idea is again to factorize the path integral for the average of physicalquantities into separate sub-averages, like in eq. (3.16). The lattice is split up into time
slices, as shown in fig 3.3. Following our definition above, the link variables inside different
20
0 2 4 6 8 0,0
0,2
0,4
0,6
0,8
1,0
I 1 / I
0 (x)
x
Bessel function ratio I 1 (x) / I
0 (x)
Figure 3.2: Graph of the ratio of modified Besselfunctions, I1(x)/I0(x), which occurs inthe expression for thermally averaged links (multihit method).
1t
x
4
3
2
Figure 3.3: Schematic picture of the lattice with timeslices. Also present is a Polyakov-linecorrelator, which has parts lying in different timeslices.
time slices are independent. The boundaries between timeslices are kept fixed, and act asboundary conditions for the interior of the slices.
One first considers variables averaged over one timeslice (fixed boundary conditions).For a single Polyakov line, these variables are products of links as in fig. 3.3, denotedby P (x4). Timeslices have thickness d × a, i.e. d lattice units. Still, the variables P (x4)cannot be averaged analytically, so one uses Monte Carlo simulation within each timesliceto estimate the average. This ’subaverage’ is written as:
[P (x4)]d =1
Zsub
∫
DU |subP (x4)e−βS. (3.25)
21
The suffix ’sub’ refers to the timeslice in which P (x4) is located, and as a notation for thetimeslice average we use [...]d, following [8], but with timeslice thickness specified. If wenow look at a product of operators such as P (x4)P (x4 + d4) forming a Polyakov line wecan write its full average as follows:
⟨
P (x4)P (x4 + d4)⟩
=1
Z
∫
DU |boundary
(∫
DU |subP (x4)e−βS
Zsub
) (∫
DU |subP (x4 + d4)e−βS
Zsub
)
(3.26)
=⟨
[P (x4)]d
[
P (x4 + d4)]
d
⟩
. (3.27)
This is a ’two-step’ method:
• Calculate the sub-averages over timeslices with fixed boundary conditions.
• Take the full average by integrating these subaverages over all possible timesliceboundaries (as always, weighted by e−βS).
The two steps are easily implemented in a Monte Carlo simulation; by running a simulationon timeslices with fixed boundaries one can extract subaverages; by alternating timeslicesimulation with full lattice sweeps, averaging over boundaries is achieved.
So again we have a factorised expression for the path integral average, this time ina more general setting. Under some conditions, to be specified shortly, the benefit ofsuch factorized expressions in Monte Carlo estimates becomes clear. As we have seen insection 3.1, the inaccuracy of MC estimates reduce with computer time τ as 1√
τ. This can
now be applied to our subaverages. We have a product of two subaverages forming thePolyakov line; its inaccuracy decreases as 1
τwith computer time. If we had a product of
n subaverages, inaccuracy would scale as τ−n/2. Note that this is the situation before thefinal averaging over boundary conditions, which leads us to the necessary conditions forthis beneficial scenario to hold.
Fluctuations in the timeslice boundaries need to have a much smaller effect on mea-sured quantities than fluctuations inside timeslices. This is important, as the average overboundaries is not improved (i.e. inaccuracy scales as 1√
τ).
In particular, no ’phase transition’ must occur due to timeslices which are too small.This notion of phase transition is taken from finite-temperature field theory. Polyakovlines have nonzero expectation values when the time dimension is sufficiently compactified.The same is expected to hold for ’partial’ Polyakov lines in timeslices with small thickness.In this case the improved statistics will not yield a better estimate for the full Polyakovline, as the boundaries dominate the inaccuracy in this case. The similarity to the finite-temperature deconfinement transition suggests that also in this case there is a suddencrossover from zero to nonzero expectation values when decreasing slice thickness.
Finally, in this analysis it has been assumed that all measurements are statisticallyindependent. This is an issue in any MC simulation; statistical independence can always beapproximated by applying several ’interval sweeps’ of MC updates between measurements.
The algorithm as described here is suitable for Polyakov-line correlators (fig. 3.3). Inthe timeslices, one should store averaged products of two-link operators. These are tensor
22
products of time-like link variables (see L-W fig 1). Adopting the notation of [8] (butwriting L instead of T ), we can write such a tensor product as
L(x4)αβγδ = U †(x, x4)αβU(x + R1, x4)γδ. (3.28)
In SU(N) theory, these operators are represented by complex N 2×N2 matrices. Therefore,the method involves extra storage space for these ’intermediate’ variables. For U(1) theory,no tensor indices are present; as the theory is abelian, links can be multiplied in any order.Multiplication of these tensor products (to form Polyakov-line correlators, for instance) iswritten as
(L(x4)L(x4 + a))αβγδ = L(x4)αλγεL(x4 + a)λβεδ. (3.29)
t
x
4
3
2
1
Figure 3.4: In the original algorithm, Wilson loops are placed such that the spatial lineslie on timeslice boundaries.
Wilson loops can be calculated using the Luscher-Weisz method, but efficiency is some-what limited. In the original algorithm timeslices are proposed as in fig. 3.4. This meansthat the space-like lines of the Wilson loop are part of the timeslice boundaries. Theyare therefore kept fixed when taking sub-measurements, and no accuracy improvement ismade on these parts.
It is not possible to achieve a variance reduction similar to the timelike lines usingthis method. It is also not allowed to apply the multihit method on the spatial links.The averaging over the upper and lower timeslices in fig. 3.4 is done under fixed boundaryconditions; the spatial lines will affect the outcome of this average. By applying the multihitmethod afterwards, one would effectively take into account multiple boundary conditions.This would introduce false correlations (between temporal and spatial parts) in the Wilsonloop average.
A way to improve the situation is to take timeslices as in fig. 3.5. The spatial parts ofthe Wilson loop are now part of the timeslices (no. 1 and no. 4 in the figure) rather thanpart of the boundaries. Therefore, they are averaged over, and it is allowed to apply themultihit method on the spatial links. This will not affect the boundary conditions of the
23
t
x
4
3
2
1
Figure 3.5: The lattice setup with timeslices. The Wilson loop has been positioned suchthat the spatial lines are not located on timeslice boundaries.
subaverage in this case. The most common way to treat the spatial links, however, is theuse of link smearing techniques.
For large Wilson loops (large in lattice units) the Luscher-Weisz algorithm can befurther generalized. We have shown timeslices of thickness d = 2 in the figures. It isalso possible to split the averaging over the boundaries of d = 2 timeslices by introducinge.g. d = 4 timeslices. The procedure of averaging can then be described analogous to eq.(3.26):
⟨
P (x4)P (x4 + 2 × 4)⟩
=⟨
[
[P (x4)]2
[
P (x4 + 2 × 4]
2
]
4
⟩
. (3.30)
The algorithm is also named ’multilevel algorithm’, after the multiple levels of averagingthat can be introduced.
To which extent the multiple levels are useful is explored e.g. in [8] and [10].A practical example is reviewed in sect. 7.7. This simulation would not have been
possible without variance reduction; not even on a very large supercomputer. The calcula-tion would have involved rougly 1080 ordinary Monte Carlo updates. The development ofthis algorithm shows that improving the simulation method can be more rewarding thanrunning the simulation on a larger (super)computer.
24
Chapter 4
Topological excitations
We will now consider field configurations which have effect on Wilson loop values. Amongstthem, topological excitations are expected to be especially important, with monopoles andcenter vortices as typical examples.
To describe how quark confinement arises in SU(N) gauge theory, an old proposal is tomake an analogy with superconductivity [33].
In a superconductor, electric charges (electrons) are condensed into Cooper pairs, whichmakes the resistance for electric currents vanish. Magnetic flux cannot penetrate thematerial, as persistent electric currents will counter any magnetic flux applied from outside.In type-II superconductors, however, magnetic flux can penetrate; this can happen onlythrough the formation of Abrikosov flux tubes. These flux tubes are (more or less) one-dimensional, or string-like in nature, just like the flux tube in a confined quark-antiquarkpair. In a superconductor there would be an asymptotically linear potential betweenmagnetic charges (monopoles), i.e. confinement of magnetic charges.
In QCD, the quarks have (color-)electric charge instead of magnetic charge, so it shouldbe electric charge that is confined. This can be accomplished in a dual superconductor,in which the role of electric and magnetic fields is switched. Interchanging electric andmagnetic properties is possible because Maxwell’s equations, written as
∂µFµν = jeν , ∂ν∗Fµν = jm
ν (4.1)
are symmetric under exchange of fields and currents as follows:
~E → ~B, ~B → − ~E, jeµ ↔ jm
µ . (4.2)
Note that for the symmetry to hold one has to postulate the existence of elementarymagnetic charges (monopoles). In a dual superconductor, magnetic charges should becondensed, leading to a collimation of electric flux into flux-tubes, and confinement ofelectric charges.
Another proposal is the center-vortex picture. In this case one assumes that vorticesrather than monopoles are condensed. The vortices do not have a familiar physical inter-pretation such as electric or magnetic charges in superconductors. We will first discuss the
25
vortex model in more detail. After that, we will show an example of a monopole configu-ration, followed by a construction of multi-monopole configurations to study confinementproperties.
4.1 Center vortices
An example of a topological excitation affecting Wilson loops is the center vortex . Itsname refers to the center of the gauge group.
The center of a group is the set of group elements which commute with all otherelements of the group. For the group U(1) the center equals the group itself, as it is anabelian group. For SU(N) the center elements zn are all proportional to the unit matrix.Imposing the group condition that det zn = 1 gives:
zn = exp(
2πin
N
)
I (n = 0, 1, .., N − 1), (4.3)
with I the unit matrix. A center vortex is a (d − 2)-dimensional object in d spacetimedimensions. If we first consider 3-dimensional spacetime, a vortex is a linelike object.
An important concept is that a loop in 3-dimensional space can be topologically linked
to a line-like object, for instance another loop. That is, the line passes through the surfacespanned by the loop. One cannot deform the loop continuously such that the line no longerpasses through its surface (see fig. 4.1).
Figure 4.1: Schematic picture of a circular Wilson loop linked to a vortex, in 3 dimensions.
In two dimensions, a loop can be linked to a point, while in four dimensions, it canbe linked to a surface (see fig. 4.2). So it follows that a Wilson loop can be topologicallylinked to a center-vortex in d dimensions. The vortex has the effect of multiplying a Wilsonloop linked to it by zn:
W → znW. (4.4)
Therefore, for the group SU(N), there are N −1 varieties, not counting the identity matrixz0 = I.
26
(b)
x
y
z
t
xz
y
(c)
x
y
(a)
Figure 4.2: A loop topologically linked to: a) a point, d = 2; b) another loop or linelikeobject, d = 3; 4) a surface, d = 4.
A vortex is created from a vacuum configuration by a so-called singular gauge trans-
formation . This is not a true gauge transformation because it affects gauge-invariantobservables like the Wilson loop. On the lattice such a transformation would be
Uy(x, y0, ~x⊥) → znUy(x, y0, ~x⊥) (x > x0), (4.5)
which is shown in fig. 4.3.Here, ~x⊥ consists of all transversal directions: only z in d = 3, z and t in d = 4 etc.It is seen that any Wilson loop encircling (x0, y0, ~x⊥) for all ~x⊥ acquires a factor zn.
This holds for any spacetime dimension d ≥ 2.If we look at the plaquettes we see that at x = x0, y = y0 all plaquettes have been
multiplied by zn, while all others are unmodified. Let us take the group SU(2) as anexample, for which z1 = −I is the only nontrivial center element. All other plaquettes are
z z z zz z0
x
y
y
x0
z
Figure 4.3: Creation of a thin center vortex. The shaded plaquette, and all other x-yplaquettes at sites (x0, y0, ~x⊥) form the center vortex. The stack of vortex plaquettes liealong a line in d = 3 dimensions, or a surface in d = 4 dimensions.
27
for the moment assumed to equal I. The energy of such a configuration then follows fromthe lattice action:
S =1
g2
∑
p
(
1 − 1
2ReTr Up
)
= 2(
L
a
)d−2
(4.6)
L is the lattice length, i.e. the volume V = Ld, and a is the lattice constant. This meansthat the excess action of a single vortex is inversely proportional to (a power of) the latticeconstant, and rises with lattice size. In the continuum limit the vortex will be a line ofinfinitesimal thickness, with infinite field strength and an energy proportional to the lengthof the vortex. This gives rise to a divergent energy in the (physically relevant) continuumlimit. This means that these vortices are suppressed in the partition function due to theirlarge energy, and will not be relevant to continuum physics.
However, the vortex is defined here as a lattice object with size and energy dependingexplicitly on the lattice constant a. We should keep in mind that the finite lattice constanta only serves as a UV-regulator for our continuum gauge theory, and that its size can betuned independent from the physics the theory describes (of course, there are conditionson the parameters for this to be correct). To avoid this dependence we should think of thevortex ’line’ being a tube-shaped object with finite radius R. This object is then called athick center vortex, as opposed to the thin center vortex described above. When a Wilsonloop is linked to the entire cylinder, its effect is the same as for a thin center vortex.
If one assumes center vortices to be the dominant topological excitations present in theYang-Mills vacuum, there is a simplified scenario, which gives an idea how an area-law forWilson loops comes about [20] [11].
Assume statistical independence of the vortices, i.e. they do not interact. Consideragain the 3-dimensional case, in which vortices are linelike objects (fig. 4.1). The existenceof additional dimension(s) again does not change the argument. The probability that nvortices pierce a Wilson loop surface C (and therefore are topologically linked to the loop),is then given by the Poisson distribution:
Pn =nn
n!e−n; n = ρArea(C). (4.7)
The average number of linked vortices is n, and is proportional to the area of the loop. Theproportionality constant is a surface density, which we call ρ. Every vortex contributes afactor zk to the Wilson loop average. For the SU(2) theory with z1 = −1 this leads to:
〈W 〉 =∞∑
n=0
nn
n!e−n (−1)n (4.8)
= exp(−2ρArea(C)), (4.9)
which is the area-law, with string tension proportional to the vortex density. If one consid-ers thick center vortices, there will be corrections following a perimeter-law (see fig. 4.4).Inside the shaded region a center-vortex will not just lead to a factor z in the loop, butits effect will be different. There are suggestions for an effect which interpolates between a
28
Vortex diameter
Vortex diameter
Figure 4.4: Circular Wilson loop pierced by a thick vortex. Two cases are shown: onthe left-hand side the vortex diameter is small compared to the Wilson loop, and vorticesinside the shaded area will give a perimeter-law correction. On the right-hand side wehave large vortices; this means vortices give an area-law for Wilson loops smaller than thevortex diameter.
factor z when the vortex is linked entirely, and a factor 1 (i.e. no change) when it lies justoutside the loop.
The shaded region has an area proportional to the loop perimeter, and the expectednumber of vortices here is proportional to this area. One should be aware that the thicknessof the vortices is unknown; if the thickness turns out to be relatively large, one mayencounter a different area-law for Wilson loops of intermediate size (see right-hand side offig. 4.4), as the shaded region now fills the entire loop area.
4.2 The ’t Hooft-Polyakov monopole in SU(2)
In SU(2) gauge theory with a Higgs field, there exist magnetic monopoles, as was shownby ’t Hooft and Polyakov [16] [17]. These are field configurations which obey the classicalequations of motion, and therefore they are minima of the action. Here we consider gaugetheory (i.e. no Higgs field), which can be seen as a limiting case of the gauge-Higgs theory.
The existence of monopoles as solutions of the field equations does not depend on theHiggs parameters. Therefore they will appear in a limiting case of zero Higgs self-couplingas well, as shown by Bogomoln’yi, Prasad and Sommerfield (BPS), [21] [22]. The solutionfor pure SU(2) has Yang-Mills potentials as follows:
eAkm(~x, x4) = −εklm
xl
r(1 − K(µr)), (4.10)
Ak4(~x, x4) =
xk
rH(µr), (4.11)
29
K(µr) =µr
sinh µr, (4.12)
H(µr) = µr coth(µr) − 1. (4.13)
Here r = |x|, Akµ(x), k = 1, 2, 3 are the spatial potentials, while the role of the Higgs field
is taken over by the temporal component Ak4(x). This solution has a scale parameter µ
which is related to the energy by
E =4πµ
g2. (4.14)
It is convenient to perform a gauge transformation to ’static abelian gauge’, in whichA1
4 = A24 = 0, i.e. A4 is diagonal. The result is given by:
A34 =
H(µr)
r; A1
4 = A24 = 0, (4.15)
A3 = − sin θ
r(1 + cos θ)φ, (4.16)
Akm = −εkl3RlmK/r; k = 1, 2, (4.17)
Rlm = e−iφT3eiθT2eiφT3 . (4.18)
The potentials are given here in spherical coordinates (r, θ, φ). In this gauge the off-diagonal part given by (4.17) falls off exponentially for large r, while the diagonal part isdominant. The rotation matrix Rlm is calculated from eq. (4.18), with Ti the generators ofthe adjoint representation of SU(2) (or if you like, the generators of the 3D rotation groupSO(3)).
We now take a look at the effect of such a configuration on a large Wilson loop. Thisrequires calculating Ak
µ(x)tk to use in the definition of the Wilson loop:
UC = P exp(
i∮
CdxµA
kµ(x)tk
)
(4.19)
We consider a circular Wilson loop with radius R centered around the z-axis, as shown infig. 4.5. The monopole is placed at the origin. Now we calculate Aµdxµ in spherical polarcoordinates, using the potentials (4.15) - (4.17). For this we also need the rotation matrixRlm. We make use of the differential form of expressing (x, y, z) in terms of (r, θ, φ):
dx = sin θ cos φdr + r cos θ cos φdθ − r sin θ sin φdφ (4.20)
dy = sin θ sin φdr + r cos θ sin φdθ + r cos θ cos φdφ (4.21)
dz = cos θdr − r sin θdθ. (4.22)
In the case of our circular Wilson loop θ and r are constant, which means we only haveto take into account the dφ terms. This leads to:
Aµdxµ = −1
2
(
1 − cos θ K(r) sin θe−iφ
K(r) sin θeiφ −(1 − cos θ)
)
dφ.
30
When θ and r are variable, there are extra terms, both proportional to K(r), which gives
Aµdxµ = (...)dφ +K(r)
2ri
(
0 −e−iφ
eiφ 0
)
(cos θdr − r sin θdθ).
As we are interested in the long-distance behaviour of the Wilson loop we can simplifythis by neglecting all terms proportional to K(r). This is allowed since K(r) ∝ e−µr forlarge r. What remains is a diagonal matrix:
Akµtkdxµ = −1
2
(
1 − cos θ 00 −(1 − cos θ)
)
dφ.
This diagonal form can be integrated straightforwardly, which gives the resulting Wilsonloop matrix, with corresponding value for the trace WC :
UC = P exp(
i∫ 2π
0dφ [−(1 − cos θ)t3]
)
(4.23)
= exp (−2πi(1 − cos θ)t3) , (4.24)
WC =1
2Tr UC = cos (π(1 − cos θ)) . (4.25)
For cos θ = 0, corresponding to a monopole positioned in the plane of the Wilson loop,this gives UC = −I and WC = −1. This is the same as for a Wilson loop linked to a centervortex.
The Wilson loop value WC is plotted as a function of the elevation z of the Wilson loopin fig. 4.5. It varies on a length scale proportional to R, so for large loops it varies onlyslowly with z.
The Wilson loop result is independent of the Wilson loop radius R for large R, as theonly r-dependence was in the terms proportional to K(r) which are negligible at largedistances. It also holds for non-circular Wilson loops (for instance, rectangular loops asused in the lattice formulation) as long as the parametrized R(σ) stays in the long-distanceregime along the loop (such that K(R) can be neglected).
So far we have been dealing with a single classical, static monopole. For a realisticdescription we need to take into account multi-monopole configurations, as well as quantumfluctuations.
For the static monopole the potentials Akm, A3 have a correction to 1/r behaviour which
falls off exponentially with r (distance from monopole center). The A34 (also called the
’Higgs’ component) falls off as r−1. It is however suggested in [12] that in the quantumtheory this could become exponential as well, with a scale µ′ ∼ µ. This means A3
4 ∝ e−µ′r.It is then possible to build multi-monopole configurations, as long as they are separatedby a distance larger than αµ−1 with α a factor larger than 1.
Multiple monopoles cannot be simply superimposed if the ’exponential’ corrections nearthe center of the monopoles overlap. The larger their separation, the smaller the errorswe make by applying superposition. This can be reflected by the parameter α, which can
31
θ
z
x
y
0 1 2 3 4 -1,0
-0,5
0,0
0,5
1,0
Wils
on lo
op v
alue
W C
Wilson loop elevation z in units of R
Figure 4.5: The effect of a BPS monopole at the origin on a Wilson loop centered aroundthe z-axis, at elevation z, and at angle θ when expressed in spherical coordinates.
be taken larger or smaller, dependent on the tolerance for superposition errors one maychoose. Multi-monopole configurations will be approximate solutions of the field equations,which means that they are expected to be of physical importance in the quantum theory.In the next chapter we will discuss multi-monopole configurations in more detail, using aneffective action in field theory.
32
Chapter 5
Effective monopole action for SU(2)theory
In the next sections we will construct an effective action for multi-monopole configurations,based on ref. [12] (which has a follow-up in [13]). The purpose is to relate this action toperiodic U(1) theory, which is an abelian theory of monopoles with a Coulomb interaction.We therefore perform a (path) integration over a sub-class of field configurations, to becompared to the full path integration in SU(2) theory. The string tension of periodic U(1)theory can be analyzed using a strong-coupling expansion (large g2) and Monte Carlo data(smaller g2). The string tension resulting from our effective action can be related to thestring tension found in the full SU(2) theory. This will show to which extent the SU(2)string tension can be described by the effective action for monopoles.
5.1 Multimonopole configurations
We need to take into account non-static monopoles when describing multi-monopole con-figurations. A monopole trajectory in spacetime is described by kµ(x), which is a straight-forward generalization of the static case in which k4(x) = δ(3)(~x) for a monopole at ~x. Ona lattice this is written as kµ(x) = ∂νmνµ with mµν(x) an integer-valued plaquette field,antisymmetric in µ ↔ ν. Therefore kµ(x) is always an integer; anti-monopoles correspondto negative integers. Integers larger than ±1 represent multiple-charged monopoles.
As indicated, monopole trajectories will have to be restricted such that monopoles donot get too close to each other. For the moment this simply means that configurations inwhich monopoles exist at pairwise distance < O(µ−1) will not be allowed. This resemblesintroducing a hard-sphere potential for point particles in statistical mechanics. Smit & vdSijs use the notion of ’monopole containers’; a monopole can then be created by imposingappropriate boundary conditions on these regions in spacetime.
An important advantage of this restriction is the fact that monopoles retain their iden-tity; a pair of monopoles at distance closer than O(µ−1) cannot be described straightfor-wardly, due to the nonlinear nature of the theory. The approximate abelian description
33
(i.e. neglecting the nonabelian potentials after transforming to abelian gauge) can be car-ried over to the general case. This can be seen by interpreting the trajectories kµ(x) asbeing the local ’time’ direction of the monopole. The abelian gauge fixing can then bedone locally, in the same way as in the static case. The fact that this can be done is usedto make the abelian approximation for monopole configurations.
The interaction energy for a monopole-pair is given by a distance-dependent Coulombenergy:
Eint = ±(
4π
g(R)
)21
4πR; (µ−1 ¿ R ≤ dcl), (5.1)
where dcl is a maximum distance, to be defined shortly. Here, g(R) is the renormalizedgauge coupling which scales according to the betafunction, given by:
R∂g
∂R= −β(g), (5.2)
−β(g)
g' β0g
2 + β1g4, (5.3)
β0 =11
24π2; β1 =
102
121β2
0 . (5.4)
The semiclassical regime is defined by distance scales such that g2 is small, and O(g6)corrections to the betafunction can be neglected.
The asymptotic freedom result is
limR↓0
g(R) = 0 (5.5)
as noted before. For large R the approximation to order g4 breaks down, hence the re-striction R < dcl, which is a restriction to the semiclassical regime. The betafunction alsoholds for the potential between quarks, at least up to O(g4) as listed here.
The mass of a monopole changes somewhat as we assume exponential decay for the’Higgs’ potential A3
4. In the SO(3) gauge-Higgs theory, which was used to derive the HP-monopole [27], the mass changes by a factor C(µ′
µ) after introducing the decay scale µ′.
This factor ranges from C(0) = 1, C(1) = 1.24 to C(10) = 1.43; C(∞) = 1.79. With anassumption µ′ = O(µ) we can safely assume 1 < C ≤ 1.5. The monopole mass is then:
mM =4πµ
g2C (5.6)
34
5.2 Effective action
In the previous section we have described some properties of monopoles, making someapproximating assumptions. We will use these to write down an effective monopole action.For the action we make use of the fact that the monopoles have a mass (described inthe action by a mass-term), and an approximate Coulomb interaction, which leads to aninteraction term with a Coulomb propagator. The Coulomb propagator is modified for usein the Euclidean formulation of the theory we use throughout.
Seff = 2mM
∫
d4xkµ(x)kµ(x) +1
2
(
4π
g
)2∫
d4xd4ykµ(x)Dcont(x − y, x4 − y4)kµ(y) (5.7)
where L4 is the length of the system in the time-direction. It is related to the (physical)temperature as L4 = 1
T, and for the moment we take T = 0; L4 → ∞. This description is
given in continuous spacetime, for which the Euclidean propagator Dcont(x) is given by:
Dcont(x) =1
4π2x2. (5.8)
In this action we have included the diagrams in fig. 2.5, i.e. an interaction and aselfenergy. It is easily seen that the action is divergent for |~x− ~y| → 0. For the interactionthis is not too problematic, since we will exclude short distances. However, the selfenergy(affecting mM) is also divergent.
A way to proceed is to introduce a spacetime lattice, with lattice constant b ≥ µ−1. Thiswill provide the short-distance cutoff needed to keep the monopoles well-separated, and itwill fix the divergent selfenergy as well. However, the selfenergy now depends explicitly onthe cutoff size. Therefore it will be compensated by changing the mass term in the action,according to
m0 = mM − 1
2
(
4π
g
)2
VC(0)1
b, (5.9)
VC(x) =∑
n4
D(x). (5.10)
The effective action is now modified accordingly to get:
Seff = m0b∑
x
kµ(x)kµ(x) +1
2
(
4π
g
)2∑
x,y
kµ(x)D(x − y)kµ(y). (5.11)
The propagator used here is the lattice version of eq. (5.8):
D(x) = N−4b
∑
n6=(0,0,0,0)
e2πinµx/L 1∑
µ(2 − 2 cos(2πnµ/Nb))(nµ = 0, 1, .., Nb−1). (5.12)
In [12] it is shown by explicit calculation that the lattice propagator is a reasonableapproximation for distances x ≥ 2b. The deviations at smaller x (especially x = 0) can
35
be absorbed in the definition of m0, as we defined m0 in order to cancel cutoff-dependenteffects.
The coupling constant g in the effective action is taken independent of distance, asopposed to g = g(R) in eq. (5.1). This means one takes a single length scale, whichshould give a reasonably accurate description. The advantage of a single length scale isthat we do not need to adjust the interaction strength (through g = g(R)) as a function ofdistance. For this approximation to hold, the following conditions on the monopole currentdistribution kµ(x) should be satisfied:
• The distribution should be approximately uniform, at least on distance scales O(dcl)or smaller, i.e. the upper scale limit for the semiclassical approximation.
• The monopole currents should be densely packed, so Debije screening will take place,and pairwise interactions at large distances (d > dcl) will be suppressed.
We will see that at least the second condition is met, due to ’monopole condensation’. Inthe lattice action a natural choice is g = g(b), which is adopted here.
The effective action we have obtained now resembles closely the monopole action ofcompact U(1) theory, which is derived to give:
Seff = m0b∑
x
kµ(x)kµ(x) +1
2
(
2π
g
)2∑
x,y
kµ(x)D(x − y)kµ(y) (5.13)
The difference is that the bare-massterm proportional to m0 has been put in ’by hand’,and the monopole charge differs from the value for SU(2) theory by a factor four.
To study confinement properties of the effective theory we have to include a Wilsonloop (source) term, which is
SW = 2πi∑
x,y
kµ(x)D(x − y)Kµ(y) (5.14)
where Kµ(x) is defined as follows:
Kν(x) = ∂µMµν(x) (5.15)
Mµν =1
2εµνρσMρσ (5.16)
The plaquette variable Mµν is equal to 1 when a plaquette lies on the surface spanned bythe Wilson loop, and 0 otherwise.
The effective theory coincides with compact U(1) theory with Villain action in the casem0 = 0, and with the substitution:
g2U(1) =
1
4g2SU(2)(b) (5.17)
36
5.3 String tension estimate
To be able to estimate the string tension using the effective U(1) theory with Villain action,the bare-mass m0 has to be zero. Equation (5.9) can be rewritten as:
m0b =4π
g2(µ−1)µCb − 1
2
(
4π
g(b)
)2
Vc(0) (5.18)
Let us introduce a parameter ω to express m0 using only g(b) as the gauge coupling:
ω(b, µ) =g2(b)
g2(µ−1)(5.19)
m0b =4π
g2(b)µbωC − 1
2
(
4π
g(b)
)2
Vc(0) (5.20)
The condition for m0 = 0 follows as
µbωC = 1.59, (5.21)
suggesting that from now on we take µbωC fixed, with b, µ and C still variable under thiscondition.
Smit & vd Sijs choose C = 1.25. We consider configurations of monopoles with only afixed scale µ, and a fixed parameter C. As discussed before, this will lead to a lower stringtension than one finds if all possible configurations were considered.
Therefore we can proceed by varying the lattice spacing b such that the string tensionσ is maximized, giving an optimal lower bound. When this is done, the values of µ and ωfollow from the constraint (5.21).
We can evaluate the restriction b > µ−1 a priori: b > µ−1 implies µb > 1. For the gaugecoupling it implies g2(b) > g2(µ−1), i.e. ω > 1, as long as b and µ−1 are such that thebetafunction for g holds. It follows that the constraint can always be satisfied, but fromω > 1 and µbωC = 1.59 one can see that b ≈ 1.1µ−1, so the difference is small.
The string tension is found using the substitution (5.17):
g2U(1) =
1
4g2SU(2)(b), (5.22)
σ =1
b2F (g2
U(1)). (5.23)
The function F (g2) is the string tension in lattice units of compact U(1) theory withVillain action. It is characterized by a phase transition at g2/4 ≈ 1.553, with F (g2/4) = 0for g2/4 < 1.553. The function is well-known in terms of the strong coupling expansion,which is convergent for large g2, but no longer converges for g2 < 1.8, corresponding toa roughening transition as described in [14]. The function is also reasonably well-knownin terms of a scaling curve which holds close to the phase transition. The scaling curve,which is a fit to Monte-Carlo simulation results, is found in [26].
37
The maximization over b is done as follows:
∂σ
∂b= 0 ⇒ −β(g)
g=
F (g2)
g2F ′(g2)(5.24)
Smit & vd Sijs used the strong coupling expansion for F (g2):
F = − ln t −∑
k
cktk; t = exp(−g2
U(1)/2) (5.25)
The maximum is found at g2U(1) = 1.92, consistent with the use of the strong-coupling
expansion. The string tension of the SU(2) theory in physical units is calculated using arenormalization scheme, resulting in:
√σ = 44ΛL (5.26)
This is in qualitative agreement with the values√
σ ≈ 55ΛL found in Monte-Carloanalysis (cited in Smit & van der Sijs, [12]). Given the approximations made, an inaccuracyof at least 20− 30% is not unreasonable, and this qualitative agreement is the main resultof [12].
However, there have been more recent numerical studies, in which a lower string tensionhas been found. In particular, Teper [25], notes a value of
√σ = (37.3 ± 3.5)ΛL. This
was calculated from SU(3) theory, but it is indicated that the dependence on N in SU(N)theory is not very strong. This means that although the estimate using monopoles ismore accurate than earlier results suggested, it is too high, which disagrees with the ideaof calculating a lower bound. The simplifications made in the analysis are probably toostrong to obtain a more accurate quantitative result.
38
5.4 String tension at high temperature
We now consider the effective action for SU(2) at high temperature. This is an extension of[12] to the case of high temperature. When the temperature is raised, the time dimensionis compactified to L4 = 1
T. An important temperature scale is reached when L4 becomes
smaller than the dominating monopole scale µ−1. This means that we cannot have aLorentz rotated version of the static monopole, as its spatial size will not fit in the timedirection with L4 ¿ µ−1. Therefore only static monopoles remain in this limit.
Starting again from the continuum effective action (5.7), and taking into account onlystatic monopoles, we get:
Seff = mM
∫
d4xk4(x)k4(x) +1
2
(
4π
g
)2∫
d4xd4yk4(x)Dcont(x − y, x4 − y4)k4(y), (5.27)
which can be simplified to:
Seff = mML4
∫
d3xm(x)m(x) +1
2
(
4π
g
)2
L4
∫
d3xd3ym(x)VC(x − y)m(y). (5.28)
The monopole current k4(x) has been relabeled to m(x). In the second term, one integralover x4 has been used on the propagator, which becomes the 3D Coulomb propagator. Theother integral gives the overall factor L4. We see that the parameters in the action havebeen transformed according to:
mM → mM
T; g2 → g2T. (5.29)
As we already had mM ∝ 1g2 , the first transformation is consistent with the second.
The action is translated to a lattice version in the same way as for the case T = 0:
Seff = m0L4
∑
~x
m(x)m(x) +1
2
(
4π
g
)2
L4
∑
~x~y
m(x)VC(x − y)m(y), (5.30)
L4 = N4b =1
T. (5.31)
The gauge coupling now becomes g2 → g2bT . The effective action again coincides with theVillain form of (3D) compact U(1) theory for m0 = 0, now with the correspondence:
bg23,U(1) =
1
4g2SU(2)bT. (5.32)
The 3D U(1) theory is confining (i.e. σ > 0) for all values of the coupling constant. Infact, this has been proven [28]. The string tension in the weak coupling region is [30]:
F (bg23) = b2σ = A
√2
π
√bg3 exp
−1
4
(
2π√bg3
)2
bVC(0)
. (5.33)
39
The factor A is a scale factor put in to correct for disagreement with Monte Carlo data.In [28] it is 4.
From the simulations (section 6.2) one sees that the functional dependence on thegauge coupling agrees with the data, but a scale factor A ' 6.4 had to be introduced toget full agreement. For larger coupling, there is again a strong coupling series, again witha roughening transition near bg2
3 = 1.8, so it can only be used for larger bg23.
We employ the same maximization procedure over b, now using the weak couplingexpansion. We write
σ =1
b2F (g2bT/4), (5.34)
and introduce x = g2bT/4, so that the maximization can be written as
b2∂σ
∂b=
−2
bF (x) +
∂F
∂x
∂x
∂b. (5.35)
To proceed we need to make an additional assumption that
∂
∂b(g2bT/4) = g2T/4, (5.36)
i.e. we neglect any additional dependence of g2 on b at high temperature. The maximizationcan now easily be done analytically, and yields
g2bT/4 ≈ 1.66, (5.37)
for which the string tension is
σ =1
b20.129A = 2.93 10−3A g4T 2. (5.38)
This should be compared to the value given by [15] who find σ = 0.136 g4T 2 fromMonte Carlo computations of 4D SU(2) theory at high temperature. A more recent result[29] is σ = 0.104g4T 2. The factor A needed for equivalence is A ' 46 or A ' 35, while oursimulations only give A ' 6.4 (Villain action; see section 6.2 for simulation results). Sothe string tension is underestimated by about a factor 6.
We have taken the weak-coupling expansion as the function for the stringtension F (x).However, the value g2
3 = 1.66 may be too large for the expansion to be accurate. Thedetermination of the string tension through F (x) can be extended to three domains: theweak-coupling region which we have already done, the strong-coupling region for g2
3 > 1.8and a region in between, where Monte Carlo simulations can produce accurate results. Forthe latter two cases we maximize
σ =1
x2F (x) g4T 2, (5.39)
which is depicted in fig. 5.1.
40
0,5 1,0 1,5 2,0
0,010
0,015
0,020
0,025
0,030
F(x
) / x
2
x = g3
2
2x2 Creutz ratios 3x3 Creutz ratios Weak coupling expansion x 6.4 Strong coupling expansion, 12th order
Figure 5.1: The function F (x)/x2 from Monte Carlo data and from the weak-coupling andstrong-coupling expansions. The weak-coupling expansion has been multiplied by a factorA = 6.4; this factor was obtained when fitting to Creutz ratios obtained in simulations(sect. 6.2). The corresponding maxima of the string tension can be seen.
For the result we take the weak-coupling expansion up to the point where it crosses the2 × 2 Creutz ratios. The Creutz ratios (even for the small loops considered here) becomemore accurate for increasing g2. We then obtain a value for the string tension of
σ = 0.018 g4T 2, (5.40)
at x ≈ 1.4. The underestimation is now a factor 5.8. The strong-coupling expansion showsa lower maximum at a higher gauge coupling, g2 = 2.1. From the agreement of MonteCarlo data with the weak-coupling expansion we conclude that the region of validity forthe strong-coupling expansion is insufficient; it is accurate only at too high values of g2.Therefore we take the weak-coupling and Monte Carlo data as the main result.
41
Chapter 6
Simulation results for U(1) theory
6.1 Four-dimensional U(1) theory
The U(1) gauge theory is easy to simulate on a computer, compared to SU(N) theory. Inthe previous chapter it was found that there are semi-quantitative similarities of stringtension results for SU(2) theory and for U(1) theory with Villain action (four dimensions).Potentials are derived from Wilson loops; therefore we have calculated average Wilsonloops. Simulations were done at various values of the coupling constant β = 1
g2 . In fig.
Figure 6.1: Average Wilson loops as a function of inverse coupling strength β. The verticaldotted line indicates the phase transition point, at β ≈ 0.644.
6.1, results are shown for N × N Wilson loops, up to 4 × 4. It concerns 4-dimensionalU(1) theory, Villain action. Its main feature is the sudden ’jump’ of the loop values nearthe phase transition point, which is βc ≈ 0.644. For β < βc, or g2 > g2
c , the theory is
42
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5
0,01
0,1
1
Cre
utz
ratio
Beta
Creutz ratios: 2x2 3x3 4x4
Figure 6.2: String tension from Creutz ratios versus β, on a 124 lattice.
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5
0,01
0,1
1
Cre
utz
Rat
io
Beta
Creutz ratio 2x2; 84 lattice
idem, 124 lattice
Figure 6.3: Comparison between string tensions calculated on a 124 lattice and thosecalculated on an 84 lattice. Some deviations show up for β near and below the phasetransition point. However, no significant differences are found for higher β.
confining, i.e. the string tension is nonzero. At the phase transition, the string tensiondrops to zero. In the figure this corresponds to steeply rising values for larger Wilsonloops. The measurements of the string tension at the same values of β are plotted in fig.6.2. String tensions have been calculated using R × R Creutz ratios. The Creutz ratiosdrop, as expected, at the phase transition points. However, they do not reach zero. It is
43
clear that larger loops give lower values, and that the largest loops available still have notreached an asymptotic value (i.e. values still are not constant with increasing R).
If Creutz ratios vary with R, we should regard the value at the highest R as the bestestimate. We are interested in the asymptotic string tension, and spurious signals fromCoulomb-like potentials are smaller at large distance. At strong coupling (low β), only2×2 loops could be used for Creutz ratios, as the loop expectation values drop too rapidly.
At weak coupling and zero string tension, the correlation lengths on the lattice arelarger, as Wilson loops drop exponentially in the loop perimeter instead of the area. There-fore it is possible that finite lattice-size effects could play a role in the Creutz ratios wefind. However, simulations on a smaller (84) lattice show almost the same values (fig. 6.3),except for some deviations near the phase transition. It shows that the finite lattice sizeis not responsible for the nonzero string tensions we find. That is, other effects, caused byrelatively low values of R for the Creutz ratio, dominate.
6.2 Three-dimensional U(1) theory
In the three-dimensional theory we can measure Wilson loops and Creutz ratios the sameway as in the four-dimensional case. The values for the plaquette and Wilson loops up tosize 4x4 are shown in fig. 6.4. Results for the Wilson loops are in fig. 6.4. The difference
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5
0,0
0,2
0,4
0,6
0,8
1,0
Ave
rage
Wils
on lo
op
Beta
Loop size: 1x1 2x2 3x3 4x4
Figure 6.4: Average Wilson loops for 3D U(1) theory versus β.
with the four-dimensional theory is the absence of a phase transition. Instead, the theory
44
is confining for all β, a fact that has been proven rigorously in [28]. A weak-coupling
0,5 1,0 1,5 2,0 2,50,01
0,1
1
Str
ing
tens
ion
g2
U(1)
2x2 Creutz ratio 3x3 Creutz ratio 4x4 Creutz ratio Weak coupling x 6.4 Strong coupling 4th order Strong coupling 12th order
Figure 6.5: Creutz ratios of 3D U(1) theory combined with the weak-coupling and strong-coupling expansions, versus g2.
expansion [28], [30] gives for the string tension
a2σ = A
√2
πg exp
−1
4
(
2π
g
)2
VC(0)
, (6.1)
valid for small g2 (the coupling constant is taken such that ag is dimensionless). In theexponent is the lattice Coulomb potential at zero distance, VC(0) ≈ 0.2527 in latticeunits. The factor A, already introduced in section 5.4, turns out to be necessary to obtainagreement with simulation data. In [30] such a factor has been introduced for the theorywith Wilson action. As we have chosen the Villain action instead of the Wilson action,we obtain a different value for the constant. Data are shown in fig. 6.5. In this graph,the Creutz ratios from 2x2 to 4x4 loops are plotted versus g2 for clarity. The weak-coupling expansion, multiplied by A = 6.4, has beeen added, as well as the strong-couplingexpansion. It should be noted that the value of A is a manual fit, as there was no rigidcriterium determining the fit quality.
45
The reason is that systematic errors occur at low g2, which appear as a ’saturation’ ofCreutz ratios. Taking larger loop size leads to significantly lower Creutz ratios. Moreover,we may expect the weak-coupling approximation to break down above a certain value ofg2, which is a priori unknown.
The fit presented here is such that it matches the 4x4 Creutz ratios, and, for larger g2,the 3x3 Creutz ratios. It is consistent with the data out to g2 ≈ 1.5 in this case. However,the Creutz ratios may still be overestimating the string tension (certainly for g2 < 0.6,possibly for g2 < 0.8). Nevertheless, it follows that a significant factor A is necessary.
The strong-coupling expansions show an unexpected detail: while the 12th order ex-pansion no longer converges for g2 < 2.0, the 4th order expansion is still consistent withthe data out to g2 ≈ 1.0.
6.3 Probability distribution for Wilson loops
In lattice simulations, very many Wilson loops are sampled. This gives us the opportunityto gather statistics of the loops. In particular, we can plot histograms of Wilson loop values(WC = Re UC), and of Wilson loop phase angles as well. The phase angles θ are definedas UC = eiθ.
The probability distribution for Wilson loops is important whenever we examine latticegauge theory in higher group representations (see also sect. 7.4). For U(1) as well as SU(N),there is a 1-to-1 correspondence between (average) Wilson loops in higher representations,and higher moments in the distribution for WC . This was noted in [31] where an expansionof the probability distribution was given in terms of 〈Wq〉
p(θ) =1
2π
1 + 2∑
q>0
〈Wq〉 cos(qθ)
(6.2)
for Wilson loops. This expansion is a cosine transform, with the average Wilson loops ofcharge q as expansion coefficients.
In fig. 6.6, a histogram is shown for the phase angles of 2× 2 Wilson loops in 3D U(1)theory. We took 2000 measurements (of entire lattice configurations) at β = 1.0 on a 323
lattice. This gives 1.97 108 samples of Wilson loops. The histogram can be accuratelyfitted by a cosine. Histograms like this can always be expanded in a series of cosines, likeexplained in ref. [31]. It turns out that the first term in the expansion already gives a gooddescription of the probability distribution. Higher-order terms in the cosine transform,proportional to cos(qθ), correspond to the average Wilson loops at charge q. The stringtension rises quickly with charge, namely as [4]
σq = qσ1. (6.3)
The q dependence explains why the histogram is well-fitted by a single cosine. The higher-charge Wilson loops are much smaller. Figure 6.7 shows the histogram for the Wilson looptrace (which is the real part, by definition, in U(1) theory) corresponding to the phase
46
0 1 2 3 4 5 6 0,4
0,6
0,8
1,0
1,2
1,4
1,6
Fit: y = A cos(x) + 1
A = 0.4501 ± 0.00036
Rel
ativ
e fr
eque
ncy
Phase angle
Histogram of 2x2 Wilson loop phase angles Cosine fit
Figure 6.6: Histogram of phase angles of 2x2 Wilson loops. Relative frequencies are nor-malized such that a uniform distribution would have a relative frequency of 1 everywhere.The profile is well-fitted by a cosine.
angles shown in fig. 6.6. This distribution shows remarkably strong peaks at -1 and +1(note the logarithmic scale). The asymmetry in the distribution reflects the expectationvalue, which is 〈W2×2〉 = 0.2549.
6.4 Four-dimensional U(1) theory at finite tempera-
ture
In 4D U(1) theory at finite temperature, there are changes with respect to the zero-temperature case. To study these, we determine Creutz ratios and Wilson loops usingthe following lattice sizes: 124, 123 × 4, 123 × 2 and 123 (3D). In the finite-temperaturecases we only sampled spatial Wilson loops, as these are not directly affected by the shortertimelike dimension. From the spatial Wilson loops we can determine Creutz ratios, whichgive a spatial string tension. Note that this is not a physical string tension, but only anindicator of the strength of loop correlations on the lattice.
Creutz ratios (2 × 2) are shown as a function of β in fig. 6.8. The main result isthat the jump of the string tension at the phase transition becomes less strong. Thephase transition appears to remain at the same value of β. The data are consistent with
47
-1,0 -0,5 0,0 0,5 1,0
1
10
Rel
ativ
e fr
eque
ncy
WC
Histogram of 2x2 Wilson loops, WC = Tr U
C
Figure 6.7: Histogram of Wilson loop trace (2x2). Relative frequencies are again normal-ized. The asymmetry in the distribution yields the value for the average Wilson loop,which is 0.2549 in this case.
the theoretical prediction of [23], where it was proven that compact U(1) theory in fourdimensions acquires a string tension for all values of g2 when the temperature is finite(T > 0). This is again an indication that the phase transition becomes less strong.
We have also determined (timelike) Polyakov lines on a 123 × 4 lattice. Whenever thePolyakov line has a nonzero expectation value, it turns out that it can be either positiveor negative, when measured from one lattice configuration. This was also observed by [31],where absolute values of lattice averages were taken to avoid averaging over positive andnegative values. Instead of taking absolute values, we show in fig. 6.9 the consecutive latticeaverages, plotted versus simulation time (i.e. number of lattice sweeps). Measurementswere taken every 20 sweeps, while points are shown every 100 sweeps, averaging over5 measurements each. This way, the crossover (for increasing β) from zero to nonzeroexpectation value is illustrated. At β = 0.60, the Polyakov line P fluctuates closely aroundzero. Already at β = 0.63, fluctuations become slower, and have larger amplitude. Fromβ = 0.65 onwards, we can see that P is nonzero, and occasionally switches sign. Theprobability to switch sign is expected to drop further at higher values of β, as can be seenfrom the graph at β = 0.80.
48
0,0 0,5 1,0 1,5 2,0 2,5 3,0
0,1
1
Cre
utz
Rat
io
Beta
Creutz ratios for lattice sizes:
12 4
12 3 x4
12 3 x2
12 3
Figure 6.8: Creutz ratios for the lattice sizes as indicated. The phase transition at β = 0.644becomes weaker when the timelike direction is smaller.
49
0 5000 10000 15000 20000
-0,06
-0,04
-0,02
0,00
0,02
0,04
0,06
0,08
Pol
yako
v Li
ne
# lattice sweeps
Polyakov Line, 5-lattice average, beta = 0.60
0 5000 10000 15000 20000 -0,2
-0,1
0,0
0,1
0,2
0,3
Pol
yako
v Li
ne
# lattice sweeps
Polyakov Line, 5-lattice average, beta = 0.63
0 5000 10000 15000 20000
-0,4
-0,2
0,0
0,2
0,4
Pol
yako
v Li
ne
# lattice sweeps
Polyakov Line, 5-lattice average, beta = 0.65
0 5000 10000 15000 20000
0,0
0,1
0,2
0,3
0,4
0,5
0,6
Pol
yako
v Li
ne
# lattice sweeps
Polyakov Line, 5-lattice average, beta = 0.80
Figure 6.9: Polyakov lines on a 123 × 4 lattice, at four values of β. The Polyakov lineaveraged over 5 full lattices is shown as a function of ’simulation time’, i.e. the number oflattice sweeps.
50
Chapter 7
Abelian projection methods
The so-called abelian projection methods have been developed based on a paper by ’t Hooft[32]. In this paper he proposes to use the gauge freedom in the (continuum) theory to makethe relevant nonabelian variables (i.e. Aµ) diagonal. In the continuum SU(N) theory onecan make Aµ diagonal using an appropriate gauge transformation. This transformationcan be made such that in the Faddeev-Popov procedure, no massless ghost fields occur(i.e. no ghost fields with long-range interactions).
However, there are special points in space (or lines in 4-dimensional spacetime) wheretwo of the eigenvalues on the diagonal are equal. These are monopoles of the ’t Hooft-Polyakov type.
Later, the abelian projection has been accommodated for lattice calculations, in whichthe link variables Uµ(x) are made as diagonal as possible by gauge transformation. On thelattice we can also isolate monopoles, and investigate their relevance for the potential andthe string tension. In this chapter we review some methods and results.
7.1 Abelian projection
A method often used to investigate the configurations responsible for quark confinement,is the Abelian projection, sometimes followed by center projection [4]. These are techniquesused in computer simulations of lattice gauge theory. The idea is again to approximate anonabelian gauge theory by ’abelian configurations’, i.e. the potentials are elements of anabelian subgroup of the full gauge group. One can use the gauge freedom in the theoryto choose a gauge such that the nonabelian components of the potentials are minimized.This is called maximally abelian gauge (MAG).
In lattice SU(2) theory one chooses a gauge:
Ω(x) : Uµ(x) → Ω(x)Uµ(x)Ω†(x + µ), (7.1)
such that the following quantity is maximized:
R =∑
x
d∑
µ=1
Tr (Uµ(x)σ3U†µ(x)σ3). (7.2)
51
It is proportional to the 33-component of the adjoint representation for Uµ(x), as can beseen from the definition:
Rkl = 2Tr (U †tkUtl), (7.3)
where Rkl is the matrix for the adjoint representation. Maximizing R implies that the linkvariables become, on average, as diagonal as possible. This follows explicitly if we writeUµ as:
Uµ =
(
a bc d
)
,
and note that R then becomes:
R =∑
x
∑
µ
(a∗a + d∗d − b∗b − c∗c)µ(x). (7.4)
After R has been maximized, the link variables are projected onto the U(1) group space,as follows:
Uµ(x) = a0I + i~a · ~σ, (7.5)
uµ(x) =1
√
a20 + a2
3
(a0I + ia3σ3) , (7.6)
=
(
eiθµ(x) 00 e−iθµ(x)
)
.
Because the ’nonabelian components’ a1 and a2 are minimized, the error will be small.However, for a product of a large number of links (e.g. a large Wilson loop), the errors arenot necessarily small.
It is clear that this description of the gauge choice is algorithmic rather than analytic;the maximization has to be done numerically in Monte Carlo-simulated lattice configura-tions. It is, in general, intractable to find the global maximum of R; instead, one findslocal maxima using the following procedure:
• choose a position x and a direction µ
• propose a change Ω(x) → Ω′(x)
• evaluate the resulting change in R;
• if R increases: accept the change
• if R decreases: accept with probability exp(∆R/T ∗), else reject
• loop through the lattice multiple times, keeping track of the configuration with thehighest R, and gradually decreasing T ∗.
• take the best configuration found so far, and find the nearest local maximum of R:for each individual x, set Ω(x) to the value that maximizes R.
52
This procedure is an example of simulated annealing. Its main feature is the artificial’temperature’ T ∗, which allows the system to go (temporarily) to a state with lower R, inorder to find a better local maximum from there. The algorithm described here is a basicversion; many refinements are possible. For example, the way in which T ∗ evolves can betuned, and there are special algorithms for finding the nearest local maximum from a givenconfiguration. These go under the name of overrelaxation. The many local maxima for thegauge condition are known as Gribov copies, equivalent configurations that all satisfythe gauge condition.
Having defined the abelian projection, one can evaluate properties like the averageWilson loop, and the string tension, on the projected lattice as well as on the full lattice.It has been found in several studies (e.g. [34], [35]) that the full and abelian-projectedstring tensions agree up to (92 ± 4) % accuracy (the abelian-projected string tension is aslight underestimation). This property of the SU(2) theory is called Abelian dominance,i.e. the string tension can be reproduced by field configurations with only abelian links.
It should be noted that this does not imply that the relevant configurations are givenby U(1) theory. Indeed, they have only U(1) degrees of freedom, but they originate fromthe full SU(2) theory, and they have corresponding properties.
A particular way of isolating those field configurations leading to confinement is to iden-tify (lattice) monopoles in U(1)-projected configurations [36]. It is known as the monopole
dominance approximation. In U(1) lattice configurations we can write the plaquette vari-ables as
Uµν = eifµν , (7.7)
with fµν given by
fµν(x) = ∂µθν(x) − ∂νθµ(x), (7.8)
as explained in the introduction. The way to extract monopoles is known as the DeGrand-Toussaint criterion [37]. The value of fµν is split into fµν ∈ [−π, π] and 2πnµν with nµν
integer:
fµν(x) = fµν(x) + 2πnµν(x). (7.9)
Then, the monopole current kµ(x) like we used earlier in chapter 5 is given by
kµ(x) =1
4πεµαβγ∂αfβγ(x). (7.10)
Using the monopoles as sources for a (monopole) Coulomb gas, one obtains a lattice config-uration which can be considered as the ’monopole part’ of the original lattice configuration:
Umonµ (x) = exp
(
iθmonµ (x)
)
(7.11)
θmonµ = −
∑
y
D(x − y)∂ ′νnµν(y). (7.12)
The Coulomb propagator D(x − y) has been used for calculating the phase angle θmonµ
arising from all monopole sources (at positions y). A Coulomb gas with monopoles is
53
assumed for the monopole field configuration. The derivative ∂ ′ν is the backward lattice
derivative.The remaining part, called the ’photon’ contribution, is defined as the difference be-
tween the original, abelian-projected configuration and the monopole part defined here:
Uphµ (x) = exp
(
iθphµ (x)
)
(7.13)
θphµ (x) = θµ(x) − θmon
µ (x). (7.14)
We now have defined separate lattice configurations of the photon and monopole contri-butions, which are related to the original configuration according to
Umonµ (x) Uph
µ (x) = Uµ(x). (7.15)
The potential V (r) can be extracted in the usual way, for the monopole and photon parts.This has been done by Bali [36], with the results shown in fig. 7.1.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
2 4 6 8 10 12 14 16
aV(r
)
r/a
VAb
Vreg
Vsing
Vsing+Vreg
Figure 7.1: Static potentials computed from monopole fields only (V sing), and photon fields(V reg) only, in units of lattice spacing a = 0.081 fm. The potential computed from theabelian projected lattice is denoted V Ab. From Bali, ref. [36].
The photon part of the potential clearly saturates at a constant value, while themonopole part is indistinguishable from a straight line, right from the shortest distanceonwards.
54
The quality of the separation is striking; no residual linear behaviour is found in thephoton part (called V reg in the figure), while the monopole part V sing is purely linear (upto small measurement errors). The fact that V sing and V reg do not add up exactly tothe full potential V Ab is caused by the nonlinear dependence of the potential on the fieldconfigurations given by Uµ(x). This effect is not very large.
Quantitatively, it is found that the string tension of the monopole part shown here isabout 95 % of the string tension after abelian projection. The slight underestimation canbe ascribed to the nonlinear effects on V (r) of the splitting into monopole and photonparts. However, as the separation into linear and Coulomb parts is so good, it can also beargued that the string tension of the monopole part may be a better estimate. The stringtension of the abelian-projected theory is harder to measure, and might be an over-estimatedue to the Coulomb part which is still present. The abelian-projected string tension is inturn about 92 % of the full SU(2) string tension.
7.2 Center projection
The U(1)-projected configurations appear to describe the string tension of the full theoryrather well. However, the U(1) degrees of freedom still present on the lattice do notguarantee that U(1) monopoles as defined in eq. (7.10) are responsible for confinement.The confining fluctuations can still come in the form of, for example, Z2 vortices. Toinvestigate this possibility, one makes use of the center projection. The configurations inMAG possess a residual U(1) gauge freedom:
uµ(x) → eiφ(x)σ3uµ(x)e−iφ(x+µ)σ3 (7.16)
which leaves the quantity R invariant. This gauge freedom is used to select the gaugemaximizing:
R =∑
x,µ
cos2 θµ(x) for uµ(x) = diag(eiθµ(x), e−iθµ(x)) (7.17)
The result is that cos2 θµ will be, on average, close to 1, and uµ(x) will be close to a centerelement ∈ Z2 = −1, 1. Center projection is done by taking
Zµ(x) = sign cos θµ(x). (7.18)
There are other ways to obtain the center projection, i.e. without doing U(1) abelianprojection first. These are reviewed in [4], and references therein. In particular, the direct
Laplacian center gauge is used in some studies. Here we described the two-step method,also known as indirect maximal center gauge.
Again there are several studies of these Z2-projected configurations, and it has beenfound that also in this case the projected configurations have almost the same string tensionas the full configurations. This observation is known as center dominance. The accuracyis near 90% or higher, depending on the parameters of the theory. Results from [38] areshown in fig. 7.2.
55
0.01
0.1
2 4 6 8 10 12
χ cp(
R,R
)
R
Direct Laplacian Center Gauge
β=2.2
β=2.3
β=2.4
β=2.5
124
164
204
244
284
Michael, Teper; Bali et al.
Figure 7.2: Combined data, at β = 2.2 − 2.5, for center-projected Creutz ratios obtainedafter direct Laplacian center gauge fixing. Horizontal bands indicate the accepted valuesof asymptotic string tensions on the unprojected lattice, with the corresponding errorbars.From Faber et al., ref. [38].
7.3 Higher representations of SU(2)
The string tension of lattice SU(2) theory is reproduced by both the U(1) abelian projectionand the center projection, up to about 10% inaccuracy. This is a signal that these abeliansubgroups are important in describing the potential between static quarks. However, it isstill unclear whether the U(1) or the Z2 subgroup (or: the abelian monopole or the centervortex) is of fundamental importance.
A way to gain further information is to consider higher representations of the gaugegroup, and/or a larger number of colors N (i.e. SU(N) theory for N > 3). So far, we haveonly discussed Wilson loops in the fundamental representation. Given a lattice configura-tion of link variables Uµ(x) = exp(αµ(x)tfµ) one can generalize to any representation r byreplacing the generators:
U rµ(x) = exp(αµ(x)trµ) (7.19)
56
The Wilson loop values change accordingly:
WC =1
χr(I)χr
∏
l∈C
U rl
, (7.20)
i.e. the trace in the fundamental representation has been changed into the group character
χr of representation r. Now one can examine how the potential (in particular the stringtension) will behave in higher representations. In the case of SU(2), the higher represen-tations are classified by the (half)-integers j (j = 1/2, 1, 3/2, ..). If we take the followingparametrization of the link variables:
U = exp(iωktk), (7.21)
ω1 = ω cos θ; ω2 = ω sin θ cos φ; ω3 = ω sin θ sin φ, (7.22)
then the group character (trace) depends only on
ω =√
ω21 + ω2
2 + ω23. In the fundamental representation we get
χ1/2(U) = 2 cos(ω/2), (7.23)
while for representation j the following relation is satisfied:
χj =sin
(
(2j + 1)12ω
)
sin(
12ω
) ; (7.24)
In fact, it turns out that χj(U) can be expressed in a polynomial in χ1/2(U). The Chebyshevpolynomials of the second kind are denoted by Un(x) (in standard notation; not to beconfused with the Wilson loop matrix or lattice link variables), and satisfy
Un(x) =sin(n + 1)θ
sin θ, for x = cos θ. (7.25)
Setting θ = 12ω and n = 2j relates it to eq. (7.24) and to χ1/2(U) = 2x. The Wilson loop
value Wj in representation j will be normalized to have a maximum value 1. Therefore onedivides by the dimensionality of representation j, as was shown in the definition (7.20). Theexplicit form of the Chebyshev polynomials is easily evaluated using the defining recursionrelation:
U0(x) = 1 (7.26)
U1(x) = 2x (7.27)
Un+1(x) = 2xUn(x) − Un−1(x). (7.28)
Noting that the dimensionality of representation j is d = 2j + 1 we express the Wilsonloop values as
Wj =1
2j + 1U2j(W1/2), (7.29)
57
with for example
W1 =1
3
(
4W 21/2 − 1
)
, (7.30)
W3/2 =1
4
(
8W 31/2 − 4W1/2
)
. (7.31)
It should be noted that these Wilson loop values Wj refer to single (local) Wilson loops, i.e.no averaging is done. The average Wilson loop, relevant for the potential, is denoted by〈Wj〉. If one considers the probability distribution for Wilson loop values p(W1/2), one canregard the Wilson loop values for higher representations as functions of higher momentsof p(W1/2). This probability distribution is accessible (albeit to finite accuracy) in latticeMonte Carlo computations, as very many Wilson loops are sampled (often up to 107 or108). An example for U(1) theory is described in section 6.3.
7.4 Higher representations of U(1)
In compact U(1) theory we also have higher representations. These can be labeled byintegers q. We define the link variables as
Uµ(x) = exp(iω(x)), (7.32)
which in representation q becomes:
U qµ(x) = exp(iqω(x)), (q ∈ Z). (7.33)
Going to representation q coincides with taking the q-th power of the link variable. Aphysical interpretation of q is the charge of the particles described. Higher-charged Wilsonloops and Polyakov lines are defined straightforwardly, analogous to eq. (7.20). The groupcharacter χq is in this case defined to be the real part of the exponential, for all q. Again,we can express Wq in terms of W1, using
Wq = cos(qω). (7.34)
We make use of the formula
cos(qω) = Tq(cos ω), (7.35)
with Tq(x) the q-th Chebyshev polynomial of the first kind. These polynomials are definedby:
T0(x) = 1, (7.36)
T1(x) = x, (7.37)
Tn+1(x) = 2xTn(x) − Tn−1(x). (7.38)
58
They satisfy the same recursion relation as the polynomials of the second kind, but with adifferent ’initial condition’ T1(x) = 1
2U1(x). The first few higher-charged Wilson loops are:
W2 = 2W 21 − 1, (7.39)
W3 = 4W 31 − 3W1, (7.40)
W4 = 8W 41 − 8W 2
1 + 1. (7.41)
7.5 String tension in higher representations
The string tension as evaluated from Wilson loops in higher group representations generallydiffers from the value in the fundamental representation. In 3D U(1) theory the stringtension is proportional to the charge (or representation number) q:
σq = qσ1. (7.42)
This follows from a semiclassical calculation of a monopole Coulomb gas.The SU(2) theory features an important distinction between half-integer and integer-
valued representations. For the integer-valued reps one expects, on general grounds, nostring tension:
σj = 0 (j = 1, 2, ..) (7.43)
In an abelian-projected theory, only the third generator T3 plays a role, and it has a zeroeigenvalue (for j=integer representations). For a Wilson loop one obtains:
W ∝ Tr exp(i∮
A3µT3dxµ) (7.44)
=j
∑
m=−j
exp(im∮
A3µdxµ) (7.45)
= 1 + 2j
∑
m=1
cos(m∮
A3µdxµ), (j = integer). (7.46)
The constant term effectively prevents an area-law. The off-diagonal components propor-tional to T1 and T2, which were neglected here, vanish exponentially with distance, andtherefore should give a perimeter-law correction.
By the same argument, one finds for the half-integer representations
σj = σf (j =1
2,3
2, ..), (7.47)
as this will lead to
〈W 〉 ∝ exp(−σ 1
2
Area) + exp(−3σ 1
2
Area) + .. (7.48)
The higher-order terms are strongly suppressed, and the fundamental string tension willdominate.
59
However, in simulations of (unprojected) SU(2) theory one finds in the adjoint repre-sentation (j = 1) a nontrivial string tension. It scales approximately according to Casimirscaling, which gives
σadj 'Cadj
Cf
σf ,Cadj
Cf
=8
3. (7.49)
Casimir scaling of the string tension is exact for d = 2 dimensions, as well as for SU(N)theory at N → ∞, as argued in [4]. The hypothesis is that it holds approximately in thecase of more spacetime dimensions, and at finite N .
At large distances, the potential saturates and the string tension vanishes, in agreementwith the earlier argument. The sudden vanishing of the string tension is called ’stringbreaking’. For a long time this has been a hypothesis, and it was not observed in latticesimulations. It has been observed first by [24], but it appears only at R = 10 lattice spacings(at β = 6.0). It is quite nontrivial to extract the potential at these large distances. Theoriginal result was questioned, as
non-standard simulation techniques (e.g. operator mixing) were required to get thedesired accuracy). However, the analysis has been redone using Wilson loops and Polyakov-line correlators, in an uncontroversial way; see sect. 7.7 for the results.
It is clear that one should be aware that the true long-distance behaviour may startat rather large distance R (large compared to the loop sizes attained in most computersimulations). Otherwise one could mistake (essentially) short-distance behaviour for long-distance behaviour.
In U(1) theory, long-distance calculations show that the string tension is proportionalto the charge q; we note that this is different from Casimir scaling, which would give
σq = q2σ1 (Casimir scaling). (7.50)
In SU(N) theory for N ≥ 3 the situation is similar; at intermediate distances Casimirscaling is expected to hold approximately. Then, at a crossover to large distances, color-screening by gluons takes place, and the string tension will change abruptly to a lowervalue, or disappear completely. This is known as string-breaking.
It should be noted that string breaking is a ’non-planar’ process when described interms of Feynman diagrams (weak coupling) or lattice strong coupling expansion diagrams(the diagrams cannot be drawn in a plane). This gives a suppression by a factor 1
N2 . Thestring-breaking distance will therefore rise with increasing N , and for the limit N → ∞,the intermediate-distance regime stretches out to infinite distance. In [4] it is argued thatCasimir scaling will hold exactly, due to factorization in the large-N limit.
In the adjoint representation, having zero N-ality, one expects
σadj ≈2N2
N2 − 1σf (7.51)
out to the string-breaking distance (Casimir scaling), and σadj = 0 thereafter.If we consider representation r with N-ality k, this screening will lead to a string-tension
corresponding to the lowest-dimensional representation with the same N-ality. Screening
60
does not lead to zero string tension, as gluons are defined in the adjoint representation,which has zero N-ality. Therefore they cannot change the N-ality of sources (quarks). Asa consequence, the asymptotic string tension is a function of N-ality only, i.e.
σr = f(k)σf . (7.52)
There are two common suggestions for the function f(k). The first one is Casimir scaling,which has a different meaning here. In this case it refers to the Casimir ratio Cr
Cfof the
lowest-dimensional representation of N-ality k:
σ(k) =k(N − k)
N − 1σf (7.53)
Another scenario is inspired by MQCD and supersymmetric Yang-Mills theory (notdiscussed here, see [39]), and is known as sine-law scaling:
σ(k) =sin πk
N
sin πN
σf . (7.54)
Two cases are special: for the physically relevant case of N = 3 the two functions are trivial,and there is only one string tension σf in the asymptotic regime; in the limit N → ∞ thetwo relations are identical for a given value k.
7.6 Casimir scaling in abelian-projected SU(2) theory
The properties of abelian dominance and monopole dominance have been verified up to(only) small deviations. Remarkably, this holds for center dominance as well.
One can perform additional tests to see the limits of these approximations. A commonway is to compare the SU(2) string tension in higher group representations to the abelian/ center-projected versions. There should be no string tension for j =integer representa-tions, when only diagonal links are considered. In [19] this has been investigated throughsimulations. The j = 1 representation of SU(2) therefore has to be projected onto U(1)variables.
In the fundamental representation, link variables are made as diagonal as possible, anda projection leads to:
uµ(x) =
(
eiθµ(x) 00 e−iθµ(x)
)
→ u′µ(x) = eiθµ(x).
In the j = 1 representation, Poulis distinguishes two types of abelian projection oftenconsidered in the literature.
In the first method, one only takes the T3 generator into account (after gauge fixing),i.e. one forms SU(2) variables like:
61
u′µ(x) ∝
ei2θµ 0 00 1 00 0 e−i2θµ
.
Note the constant term ’1’ in the trace over products of these variables. This method hasbeen called diagonal approximation.
The second method is to extract abelian link variables uµ(x) = eiθµ(x), and createhigher-representation values from them. One takes the double-charged version to comparewith the adjoint SU(2) case:
u2µ(x) = ei2θµ(x) (7.55)
. The difference is simply neglecting the constant term in the trace. The double-chargedabelian links correspond to the q = 2 representation of U(1).
The string tension has been determined (through Creutz ratios), comparing betweenj = 1 SU(2) and q = 2 projected theory, giving results as in fig. 7.3. This shows areasonable agreement. However, the string tension could only be measured up to a distanceof R = 3 lattice spacings (at β = 2.4). This is small compared to the string-breakingdistance (about 1 fm), as a = 0.12fm
Figure 7.3: Creutz ratios from unprojected adjoint SU(2) Wilson loops Wj=1 (o) versusCreutz ratios from doubly charged abelian Wilson loops Wn=2 in MA projection from the164 run (squares) and the 124 run (triangles).
From the way the abelian projection is defined, one can argue that the first methodis the correct one. There is no justification for removing the constant to consider higherrepresentations of U(1) to compare with j = 1, 3/2, .. SU(2) theory. Of course, one canwrite down an effective theory containing U(1) variables (with additional terms in theaction, for instance), in which one can analyze variables in various U(1) representations.However, there is no relation with SU(2) representations here.
62
But following the definition of abelian projection, one cannot simply remove the con-stant, even though it is known to be unphysical. In the full theory, the constant term isconverted into a perimeter-law contribution by the ’off-diagonal’ components.
7.7 Observation of string breaking in SU(2) theory
From considerations as in the previous section it follows that the static potential V (r)for the adjoint (or j = 1) representation of SU(2) should have zero string tension forr → ∞. However, for short and intermediate distances one finds a nonzero stringtension,approximately satisfying Casimir scaling. Only in recent simulations [24], the transitionto zero string tension has been found. The potential V (r) has been measured in SU(2)theory in 2+1 dimensions. Three different methods were used to extract the potential:Wilson loops, Polyakov line correlators and a so-called multi-channel ansatz. For thelatter, operators for the ’string-like’ state and for the broken-string state are introduced.The multi-channel method has been subject to criticism, which is discussed by the authorsthemselves. Therefore we will discuss the Wilson loop and Polyakov line results.
As was shown in sect. 2.4, it is necessary to consider R × T Wilson loops for largeenough T at each R, to suppress excited states.
Simulations have been done on a 482 × 64 lattice, at coupling constant β = 6.0. Atthis coupling, the lattice constant a is given as a ≈ 0.10(1) fm (note that this theory is3-dimensional; in 4 dimensions β and a have a different relation). Extensive use of theLuscher-Weisz method has been made for the necessary variance reduction. A three-levelscheme with timeslices of thickness 4a, 2a and 1a (multihit) has been employed. Togetherwith link-smearing techniques on the spatial links, this allows for signals as small as 10−40
to be detected. Without the variance reduction, this would take roughly 1080 Monte Carlomeasurements, which explains why this level of accuracy has never been reached before.
The results for the adjoint potential using Wilson loops are shown in fig. 7.4. Stringbreaking as well as deviation from Casimir scaling is demonstrated. The string-breakingdistance is about 10 lattice spacings, or 1.0−1.1 fm. For R = 12a, a temporal size T ≥ 22awas needed to extract the ground state. Therefore the Wilson loop operator has only a(very) small overlap with the ’broken-string’ state, which is the ground state at R = 12a.
7.8 Center projection and thick center vortices
It is clear that in lattice SU(2) theory, one cannot have a string tension in higher represen-tations which arises purely from thin center vortices. Since in the adjoint representation,all center elements are mapped onto the identity, so thin center vortices have no effect onWilson loops.
The only way vortices can give a good description is to consider thick center vortices. Asargued before, this is more reasonable anyway, as one expects the mechanism of confinementto be independent of lattice artifacts, which have an arbitrarily chosen size instead of a
63
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
0 2 4 6 8 10 12
V(R
) a
R/a
adjoint static potential8/3 fundamental static potentialadjoint unbroken string energy
2 M(Qg)
Figure 7.4: The adjoint and fundamental static potentials V (R) (the latter multipliedby the Casimir factor 8
3) versus R using Wilson loops only. The adjoint static potential
remains approximately constant for R ≥ Rb ≈ 10a proving string breaking. The unbroken-string state energy is also drawn. Apart from string breaking we see a clear deviation fromCasimir scaling.
fixed physical size.
String-breaking in the adjoint representation can be accounted for by thick vortices,as these have effects following a perimeter-law for Wilson loops larger than the vortexthickness (this holds only for j =integer representations). For smaller Wilson loops, thickvortices give an area-law (cf. sect. 4.1). This implies that string-breaking occurs at adistance equal to the vortex diameter.
From the results of Kratochvila & de Forcrand in the previous section we see thatquite a large thickness is required; in fact a diameter of ≈ 1 fm implies that it is notwell approximated by the thin center vortex picture at commonly used values of the gaugecoupling. It corresponds to the order of 10 lattice spacings, compared to 1 lattice spacingfor a thin vortex.
There is another indication that one should regard vortices as thick objects. Considera lattice (simulation) study as follows [38]. First create a number of SU(2) lattice configu-rations (ensemble), and store them. On each configuration, one applies center projection,locating thin center vortices.
Now we can use this information to distinguish, in the original ensemble, Wilson loopswhich are linked to n center vortices. Therefore we obtain sub-ensembles of Wilson loopswith 0, 1, 2, .. vortices linked. Note that we can take any size for the loop. The averageWilson loop linked to n vortices is denoted by Wn(C).
64
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
1 2 3 4 5 6
χ(R
,R)
R
β=2.3, 164-lattice
Bali et al.full theory
DLCGvortices removed
Figure 7.5: Creutz ratios in the unprojected configuration (open circles), in the center pro-jected configurations in DLC gauge (full circles), and in the unprojected configurations withvortices removed (open squares). The horizontal band indicates the accepted asymptoticstring tension of the unprojected theory. From Faber et al., [38].
-1.5
-1
-0.5
0
0.5
1
1.5
0 5 10 15 20
Wn/
W0
Loop Area
Ratios of Vortex-Limited Wilson Loops, β=2.3, 164 Lattice
W1/W0W2/W0
Figure 7.6: Wn/W0 ratios at β = 2.3. From Faber et al., ref. [38].
65
In this way, it is found that:
• The Wilson loops without vortices do not give a string tension (see fig. 7.5). At leastthe string tension is consistent with zero rather than with the accepted value for thefull theory.
• The ratio W1(C)/W0(C) tends to -1 at moderate distances, while the ratio W2(C)/W0(C)tends to +1. See fig. 7.6.
Both results are consistent with the vortex picture for a vortex thickness (diameter) ofabout 4-5 lattice spacings, at coupling strength β = 2.3 in 4D. The value for β correspondsto a ' 0.17 fm, giving a vortex diameter of about 0.8 fm. This relatively large distance iscomparable to the string-breaking distance in the adjoint representation of 3-dimensionalSU(2) theory. We may conclude that the thick center vortex picture, which leads tostring-breaking in the adjoint representation at a distance equal to the vortex diameter, isconsistent with these results.
7.9 Correlations between monopoles and vortices
Ambjorn et al. [31] studied correlations between monopoles and center vortices. Monopolesand vortices can be located, albeit not entirely uniquely, using the indirect maximal centergauge described in sect. 7.1, 7.2. That is, one first applies U(1) abelian projection, forwhich monopoles can be located according to eq. (7.10). After this, the residual gaugefreedom is used to apply center projection, for which one can locate vortices.
At fixed time x4 on the lattice, vortices are linelike objects, as they form surfaces ind = 4. Monopoles are then pointlike, so they are localised in space at every fixed time.Of course, in the Euclidean formulation uses throughout, monopoles can move in purelyspatial directions as well. However, points in space can still be associated to monopoles.
For every monopole position found on the lattice, the authors have checked whether avortex line runs through it. For 93% of the monopoles, a single vortex line passes through,while for 3% of the monopoles no vortex passes through. The remaining 4% has more than1 vortex. This result clearly demonstrates that monopoles and vortices are correlated.
Alternatively, one can look at all vortex lines (again, at fixed x4), and see if monopoles oranti-monopoles lie on it. In 61% of the cases no monopoles were found; 31% of the vorticeshad one monopole-antimonopole pair on it. On 8% of the vortices, multiple monopole-antimonopole pairs were present. Moreover, as one traces a path along the vortex onefinds monopoles and antimonopoles in an alternating fashion. There was a fraction of only1.2% which broke this rule of alternation. The picture that arises from these results isshown below (fig. 7.7)
We may conclude from this analysis that (on the lattice) there is a strong correlation inlocation between U(1) monopoles and center vortices. Therefore, monopoles and vorticesappear to be related, which would be consistent with the results for the potential and thestring tension. These are described rather well using monopoles and using vortices.
66
+
+
-
+
-
+
-+
-
+
-
+
-
+
-
Figure 7.7: The picture of collimation of monopole/antimonopole flux into center vortextubes on the abelian-projected lattice.
As a comment on correlating monopoles and vortices, we note that the locations ofthese objects are not gauge invariant, which means that different choices of abelian/centergauge may lead to different results for the correlation.
If we consider, for example, the U(1) projection as described here compared to the centerprojection using Direct Laplacian Center gauge (DLC, ref. [38]), we may expect somewhatlower correlations. The DLC gauge does not involve performing a U(1) projection first.This gives the possibility that vortex locations are less related to monopole locations foundafter U(1) projection.
The strength of this gauge dependence is unknown; however, both monopoles andvortices found using these projection methods are ’thin’ objects. This means that they(both) have the effect of generating plaquettes (i.e. small Wilson loops) that have valueclose to −1 rather than close to +1. The plaquette value, which is ReTr Up, is gaugeinvariant and is directly included in the action. The location of vortices and monopolesmust be somehow tied to the positions of plaquettes with large action (values near −1).Therefore, the correlation result is not expected to depend strongly on the particular choiceof abelian and center gauge.
67
Chapter 8
The dual Abelian Higgs model
8.1 Comparison of SU(2) flux tubes with the DAH
model
Within the formalism of abelian-projected SU(2) theory, the shape of the potential hasbeen investigated in detail [40], as well as the ’flux tube’ between the static quarks. Thisis done by numerical simulation. These results are used to test the dual superconductor
model of the QCD vacuum. In ordinary superconductivity, electrical charges (in Cooperpairs) are condensed, while electric flux is collimated into a (quasi)-onedimensional fluxtube. The same happens in dual superconductivity, but then for magnetic charges, andmagnetic flux. The dual abelian Higgs model happens to have classical flux-tube solutions,to which the SU(2) simulation results are compared.
The analysis has been carried out on a relatively large (324) lattice, on which theSU(2) gauge configurations were generated. The abelian projection has been applied, tobe able to separate monopole and ’photon’ contributions according to eq. (7.10). Theflux tube profile is measured as follows: with the quark-antiquark system positioned alongthe z-direction, at a distance of R lattice spacings, one chooses the x − y midplane formeasurements. The electric field Ez along the z-axis is measured, as well as the monopolecurrent Kφ around the z-axis. This is then considered as a function of the off-axis distanceρ, in fig. 8.1. The upper figures refer to the full (abelian-projected) theory, the middle andlower figures show monopole and photon contributions, respectively.
The different types of measurement indicated in the figures refer to different choicesof the MAG gauge fixing procedure. The authors used an algorithm using only overrelax-ation (OR) as well as a combined overrelaxation-simulated annealing algorithm (simulatedannealing was described in sect. 7.1). The latter uses Ng different initial (random) gaugecopies. The comparison is done to investigate the lattice Gribov copy problem. The OR-SA algorithm, which finds better local maxima, produces different results for some of themeasured values. The Gribov ambiguity therefore plays a significant role, especially atshort distances. The effect appears mainly in the monopole current densities Kφ, withbetter gauge fixing leading to lower monopole currents. This is consistent with the notion
68
0.005
0.004
0.003
0.002
0.001
0.000
-0.001
k ϕ
14121086420
ρ/a
OR OR-SA (Ng=5) OR-SA (Ng=10) OR-SA (Ng=20)
0.05
0.04
0.03
0.02
0.01
0.00
Ez
14121086420
ρ/a
OR OR-SA (Ng=5) OR-SA (Ng=10) OR-SA (Ng=20)
0.025
0.020
0.015
0.010
0.005
0.000
-0.005
Ez
14121086420
ρ/a
OR OR-SA (Ng=5) OR-SA (Ng=10) OR-SA (Ng=20)
0.005
0.004
0.003
0.002
0.001
0.000
-0.001
k ϕ
14121086420
ρ/a
OR OR-SA (Ng=5) OR-SA (Ng=10) OR-SA (Ng=20)
0.025
0.020
0.015
0.010
0.005
0.000
-0.005
Ez
14121086420
ρ/a
OR OR-SA (Ng=5) OR-SA (Ng=10) OR-SA (Ng=20)
0.005
0.004
0.003
0.002
0.001
0.000
-0.001
k ϕ
14121086420
ρ/a
OR OR-SA (Ng=5) OR-SA (Ng=10) OR-SA (Ng=20)
Figure 8.1: Flux tube profiles at β = 2.6 for 8 × 8 Wilson loops.
of monopoles being ’defects’ in the MAG fixing [32].
For different numbers of initial gauge copies Ng the results do not change significantly.This would suggest that the gauge fixing of this algorithm is ’good enough’ to produceunambiguous results. However, the authors have not made a quantitative comparison ofthe gauge-fixing quality defined by R, eq. (7.2). It is possible that R remains nearlyconstant when increasing Ng, which would explain the similarity of the profiles withoutthe conclusion that the Gribov ambiguities are under control.
The simulated SU(2) flux tubes have been compared quantitatively to classical flux-tube solutions in the 3D dual Abelian Higgs model. The model is studied on the (dual)lattice, to have discretization effects similar to the lattice SU(2) case. In this formulation,the action is given by:
69
SDAH = βg
∑
m
1
2
∑
i<j
Fij(m)2 +m2
B
2
3∑
i=1
∣
∣
∣Φ(m) − eiBi(m)Φ(m + i)∣
∣
∣
2+
m2Bm2
χ
8
(
|Φ(m)|2 − 1)2
(8.1)The dual field strength Fij is given by
Fij(m) = Bi(m) + Bj(m + i) − Bi(m + j) − Bj(m) − 2πnij(m), (8.2)
with nij(m) a ’Dirac-string’ variable. Bi is the dual gauge field, and Φ(m) is a complexscalar monopole field. These fields are defined on the dual lattice, and positions are labeledby m instead of ~x.
There are 3 parameters present, which are free to choose in order to fit to the SU(2)results: the dual gauge coupling βg = 1
g2 , the dual gauge boson mass mB and the monopolemass mχ. The mass parameters are related to the type of superconductivity by theGinzburg-Landau parameter κ = mχ/mB. The case κ < 1 corresponds to type-I, whileκ > 1 implies type-II superconductivity. Given the parameter values, one obtains flux-tube solutions by solving the field equations. There are 3 equations, which follow from thevariation of the action, eq. (8.1) with respect to Bi(m), ReΦ(m) and ImΦ(m) respectively.
The field equations are solved numerically using a relaxation algorithm employing thesecond variation of the action with respect to each field variable. This is similar to Newton’smethod. The relation between this solution and the 4D SU(2) flux tube is established bytaking the DAH electric field E and monopole current Ki and by comparing with the AP-SU(2) electric field. The latter is decomposed into monopole and photon parts as describedin sect. 7.1. The explicit form of the field equations and the monopole current Ki are givenin [40].
The 3 parameters of the theory are optimized as to fit to the SU(2) flux tube. Resultsof the fit are shown in fig. 8.2, where the lower two figures correspond to the profiles infig. 8.1
The quality of the fit is remarkably good; especially when one again notes that one fitsa classical abelian flux tube solution to a profile arising from a full quantum theory. Thefit is (nearly) consistent with the data points over the full range. The fit values are:
mB = 1091(7) MeV, (8.3)
mχ = 953(20) MeV, (8.4)
leading to κ = mχ
mB= 0.87(2) < 1, which would be classified as type I (dual) superconduc-
tivity.
70
0.010
0.008
0.006
0.004
0.002
0.000
k ϕ
1086420
ρ/a
β=2.5115: W(6,6) AP (Abelian) DAH (Abelian)
0.08
0.06
0.04
0.02
0.00
Ez
1086420
ρ/a
β=2.5115: W(6,6) AP (Abelian) AP (Monopole) AP (Photon) DAH (Abelian) DAH (Monopole) DAH (Photon)
0.05
0.04
0.03
0.02
0.01
0.00
Ez
14121086420
ρ/a
β=2.6: W(8,8) AP (Abelian) AP (Monopole) AP (Photon) DAH (Abelian) DAH (Monopole) DAH (Photon)
0.005
0.004
0.003
0.002
0.001
0.000
k ϕ
14121086420
ρ/a
β=2.6: W(8,8) AP (Abelian) DAH (Abelian)
Figure 8.2: Some examples of the fitting at β = 2.5115 for R = 6 (upper row), and β = 2.6for R = 8 (lower row). The solid line is the DAH flux-tube profile (obtained by the fit).The dotted and dashed lines correspond to its monopole and photon parts (as predictedusing the fit parameters).
71
Chapter 9
Conclusion
We have reviewed techniques for simulating lattice SU(N) theory, including the recentLuscher-Weisz variance reduction algorithm. The variance, or signal-to-noise ratio, inMonte Carlo estimates is a serious limitation to the distance at which the quark potentialcan be calculated. The necessary computer time was shown to rise exponentially in theinter-quark distance for the potential. That is, to achieve significantly longer distances,one may quickly need an order of magnitude more computer power. The recent variancereduction method has improved the situation.
We have reviewed topological field configurations that are relevant for quark confine-ment. For SU(2) theory we have studied monopoles of the ’t Hooft-Polyakov type, inthe BPS limit. Based on a number of general assumptions, an effective action for thesemonopoles has been constructed ([12]). In the four-dimensional theory, at zero tempera-ture, the string tension agrees with simulations of the full theory, given the approximationsmade. The effective theory gives about 118 % for
√σ relative to the full theory as found in
[25]. However, we should remark that the string tension is overestimated by the effectivetheory, while several approximations have been set up in order to obtain a lower bound onthe string tension.
When we extended the effective theory to finite temperature (the high-temperaturelimit), we found that the string tension was underestimated by about a factor 6. Thereforethe approach does not appear to work in explaining the string tension in this case.
From our review of abelian and center projection methods, we conclude that abelianmonopoles and center vortices both successfully reproduce the string tension in SU(2)theory with 90 to 95 % accuracy. To see significant differences (as far as the potential V (r)is concerned) between abelian-projected and center-projected theories we have to look athigher group representations. The potential has three distance regimes: short-distance(Coulomb-like potential satisfying Casimir scaling), intermediate distance (approximateCasimir scaling) and long distance. From the study of string-breaking in SU(2) theory, itwas found that the intermediate-distance regime is rather large (out to 1.1 fm). Casimirscaling of the string tension was shown to hold approximately, with deviations rising withdistance up to 10%. The abelian-projected theory still reproduces Casimir scaling atintermediate distances, and it is expected to reproduce the long-distance behaviour. That
72
is, zero string tension for integer representations, and the fundamental string tension forhalf-integer representations. However, the long-distance regime has never been reached forabelian / center-projected theories as the modern variance reduction methods are (yet)incompatible with abelian gauge fixing and projection.
The center-projected theory has a trivial representation dependence: all half-integerrepresentations are mapped onto the fundamental representation, while all integer rep-resentations are mapped onto the identity. This means that at least the long-distancephysics is preserved, but the intermediate and short-distance regimes are not described bythe projected theory.
When thick vortices are introduced, there can be a string tension again for intermediatedistances, in integer representations. String breaking happens at a distance equal to thevortex thickness (diameter). Numerical studies indicate that the vortex thickness shouldbe about 0.8 fm, which is similar to the string-breaking distance, which was found to be1.1 fm.
Based on this review, it is impossible to tell whether we should take monopoles asthe relevant objects causing confinement, or center vortices. A way to continue wouldbe, for example, to study SU(N) gauge theory for larger N . This will bring new grouprepresentations, and increasing relevance of the large-N limit. This subject is also reviewedin [4]. Both approaches ’work’ as advertised: they reproduce the linear part of the inter-quark potential, which can be separated from the Coulomb-like (saturating) part. Thenumerical projection methods give a clear demonstration of this separation.
However, the advantage of the monopole approach is that it has a direct and well-studiedinterpretation in terms of (dual) superconductivity. The relevance of dual superconduc-tivity has been further demonstrated by the flux tubes of the dual abelian Higgs model.These flux tubes, which are solutions of the field equations, have been fitted to SU(2) fluxtubes with high accuracy. The dual superconductivity picture is therefore (still) relevant.
73
Chapter 10
Acknowledgements
I wish to thank prof. J. Smit for supervising this thesis, and for many discussions on quarkconfinement, and lattice gauge theory in general. This way, I learned about the manysubtleties that are natural to this subject. I also thank my parents and my brother fortheir continuous support throughout my academic career.
Some words of thank also go to the ITF for starting the master’s program in theoreticalphysics before the bachelor/master system was generally introduced in the Netherlands.The thorough introduction to the various topics in theoretical physics has been the reasonfor me to come to Utrecht for the master’s program.
74
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