UNIVERSITÀ DELLA CALABRIA Dipartimento di Fisica

Corso di Studio Magistrale in

Fisica

Tesi di Laurea Magistrale

Three-quark potential from a 2D

effective SU(3) model

Relatore Candidato

Prof. Alessandro PAPA Emanuele MENDICELLI

Matr. 153388

______________________ ________________________

Anno Accademico 2016/2017

1

This page is intentionally left blank.

I dedicate this paperto individualswho exercisesound reasoningin all facets of life.

Contents

Abstract 8

Acknowledgements 9

Introduction 11

1 Basics of QCD and Lattice QCD 151.1 Lagrangian of Quantum Chromodynamics . . . . . . . . . . . . 16

1.1.1 Quantization in the Feynman-Integral Formalism . . . . 181.1.2 Asymptotic Freedom and Confinement . . . . . . . . . . 23

1.2 Lattice QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.2.1 Naive discretization of fermion . . . . . . . . . . . . . . . 251.2.2 Gauge action . . . . . . . . . . . . . . . . . . . . . . . . . 271.2.3 Wilson Loop . . . . . . . . . . . . . . . . . . . . . . . . . . 291.2.4 Finite Temperature . . . . . . . . . . . . . . . . . . . . . . 311.2.5 Polyakov Loop . . . . . . . . . . . . . . . . . . . . . . . . 331.2.6 The Continuum Limit . . . . . . . . . . . . . . . . . . . . 36

2 Monte Carlo Method 392.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.2 Markov chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.3 Update Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3.1 Metropolis Algorithm . . . . . . . . . . . . . . . . . . . . 442.3.2 Cluster Wolff Algorithm . . . . . . . . . . . . . . . . . . . 46

2.4 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.4.1 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . 472.4.2 Blocking Method . . . . . . . . . . . . . . . . . . . . . . . 492.4.3 Jackknife Method . . . . . . . . . . . . . . . . . . . . . . . 50

3 Effective Theory and the Proposed Model 523.1 Effective Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2 Proposed effective Theory . . . . . . . . . . . . . . . . . . . . . . 543.3 Definition of the Observables . . . . . . . . . . . . . . . . . . . . 56

3

CONTENTS 4

3.4 Phase transition determination . . . . . . . . . . . . . . . . . . . 583.5 Strong coupling test . . . . . . . . . . . . . . . . . . . . . . . . . 62

4 Numerical Study of the three quarks potential 654.1 State-of-the-Art in Three-quark potential . . . . . . . . . . . . . 654.2 Numerical determination of σqq and σqqq . . . . . . . . . . . . . 67

4.2.1 Extraction of σqq from Γ2 . . . . . . . . . . . . . . . . . . 674.2.2 Extraction of σqqq from Γ3 . . . . . . . . . . . . . . . . . . 75

4.3 Numerical study of the potential shape . . . . . . . . . . . . . . 76

Conclusions and Future Work 91

References 93

List of Figures

1 Proposed phase diagram for QCD. ALICE, RHIC,CBM, SPS and RHIC arethe names of relativistic heavy-ion collision experiments. . . . . . . . . . . 12

1.1 Cubic and quarting gluon self-interation Feynman diagram. Taken from [7]. 181.2 Representation of a lattice of size a. µ̂, ν̂ and ρ̂ are spacetime unit vector

useful to locate the fileds ψ(x) and Uµx . Taken from[15]. . . . . . . . . . . 251.3 Rapresentation of the two link variables Uµ(n) and U−µ(n) [7]. . . . . . 271.4 The four link variables which building up the plaquette Uµν(n). The circle

indicates the order followed in multiplying the link variables. Taken from [7]. 281.5 Two example of Wilson loop, the first planar and the second nonplanar. Taken

from [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.6 Diagram of a qq pair and its behaviour when the separation is increased. . . 311.7 Representation of a Polyakov loop. Taken from [9]. . . . . . . . . . . . . . 341.8 Representation of two Polyakov loop with opposite direction. Taken from [9]. 34

3.1 Representation of the 2D lattice used, where the lattice variables are tracesof SU(3) matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2 Graphic representation of the operator Γ2(R), where a and b are the positionsof the quark and antiquark sources and R is their distance. . . . . . . . . . 56

3.3 Graphic representation of the operator Γ2(R)Wall . The line connecting theblack balls forms a fermion wall while the other one, connecting the blueballs, forms the antifermion wall. . . . . . . . . . . . . . . . . . . . . . 57

3.4 Graphic representation of the operator Γ3, where a, b and c are the vertexesof a triangle where the three quarks are placed. . . . . . . . . . . . . . . . 57

3.5 Magnetization modulo versus β on 32x32 lattice. . . . . . . . . . . . . . 583.6 Scatter plots of the complex magnetization MP at β = 0.1, 0.42, 0.44

on a 32x32 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.7 Susceptibility versus β on a 300x300 lattice, and its fit with a Lorentzian

function. The fucntion is drawn only on the point actual used in the fit. . . 603.8 Pseudocritical β versus lattice size. . . . . . . . . . . . . . . . . . . . . 613.9 Two-point correlation function Vs β and strong coupling expansion in green. 633.10 Three-point correlation function Vs β and strong coupling expansion in green. 64

5

LIST OF FIGURES 6

4.1 Triangle spanned by the three quarks, where xi denotes the i thquark, F the Fermat-Torricelli point, hi the distance between Fand each side. So that ∆ and Y reads ∆ = (r1 + r2 + r3)/2 andY = l1 + l2 + l3. Taken from [30]. . . . . . . . . . . . . . . . . . . 66

4.2 Behaviour of σqq versus distance at different lattice sizes and forβ = 0.3, 0.41, 0.42. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3 Fit of σqq versus distance via σ + γ/2 ln [x/(x− 2)] at β = 0.42 . . . . . 714.4 σwall versus distance at β = 0.42 on 64x64 and 128x128 lattices. The green

line was used to help identifying the plateau region. . . . . . . . . . . . . 744.5 Behaviour of Γ3 versus triangle half perimeter (∆) for β =

0.3, 0.41, 0.42. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.6 Effective σqq verss R, σwall versus R and σqqq versus Y and ∆ at

β = 0.41 on a 64x64 lattice, for isosceles geometry. . . . . . . . . 794.7 Same as Fig 4.6 at β = 0.415. . . . . . . . . . . . . . . . . . . . . 804.8 Same as Fig 4.6 at β = 0.42. . . . . . . . . . . . . . . . . . . . . . 814.9 Effective σqq verss R, σwall versus R and σqqq versus Y and ∆ at

β = 0.41 on a 64x64 lattice, for right geometry. . . . . . . . . . . 824.10 Same as Fig 4.9 at β = 0.415. . . . . . . . . . . . . . . . . . . . . 834.11 Same as Fig 4.9 at β = 0.42. . . . . . . . . . . . . . . . . . . . . . 844.12 Effective σqq verss R, σwall versus R and σqqq versus Y and ∆ at

β = 0.41 on a 128x128 lattice, for isosceles geometry. . . . . . . 854.13 Same as Fig 4.12 at β = 0.415. . . . . . . . . . . . . . . . . . . . . 864.14 Same as Fig 4.12 at β = 0.42. . . . . . . . . . . . . . . . . . . . . 874.15 Effective σqq verss R, σwall versus R and σqqq versus Y and ∆ at

β = 0.41 on a 128x128 lattice, for right geometry. . . . . . . . . . 884.16 Same as Fig 4.15 at β = 0.415. . . . . . . . . . . . . . . . . . . . . 894.17 Same as Fig 4.15 at β = 0.42. . . . . . . . . . . . . . . . . . . . . 90

List of Tables

3.1 Fit parameters βpc obtained from fits of the susceptibility via a Lorentzian

function1c

(a

(x− b)2 + a2)

. . . . . . . . . . . . . . . . . . . . . . . . 61

3.2 Fit parameters obtained from the fit of βpc versus L via b +a

Lν. . . . . . . 62

4.1 Best-fit parameters b and a in lattice unit, obtained from fits of the σqq viaσ + γ/2 ln [x/(x− 2)] on a 64x64 lattice. . . . . . . . . . . . . . . . . 72

4.2 Best-fit parameters b and a in lattice unit, obtained from fits of the σqq viaσ + γ/2 ln [x/(x− 2)] on a 128x128 lattice. . . . . . . . . . . . . . . . 73

4.3 σwall extracted from Γwall2 (R) operator on a 64x64 and 128x128 lattice. . . 754.4 Numbers of σqqq values extracted for the Isosceles and Right geometries on

a 64x64 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.5 Numbers of σqqq values extracted for the Isosceles and Right geometries on

a 128x128 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

7

Abstract

The aim of the present thesis is to study the shape of the three-quark po-tential within a lattice effective model. The main tool of this study is ourproposed effective theory, which we candidate as a new means for studying,in a simplified manner, all the main aspects regarding the hadron systems.

Using a Monte Carlo code based on our model, we have numerically ad-dressed the shape of the three-quark potential. The result from all the numer-ical simulations, done almost all on the ReCaS cluster of INFN, are clearlysupporting the so called Y-ansatz, which describes the inner potential as lin-early increasing with the sum of distance of the three quarks from the Fermat-Torricelli point of the triangle spanned by the three quarks, suggesting theformation of a Y-shaped colour flux tube among the three quarks.

8

Acknowledgements

If you are here, full curious to read them, it means I have finally done it.

The journey was long and full of troubles, but I was lucky to be guided by mysupervisor Professor Alessandro Papa, to whom I express all my gratitude forall the times he with immense patience, has kept me back on the right way,always trying tirelessly to teach me the best way.

In this months I had the privilege to collaborate and mostly be helped by longand instructive discussions with Professor Oleg Borisenko and Dr. VolodyaChelnokov from the Ukrainian Bogolyubov Institute for Theoretical Physicsin Kiev, which I plentifully thank.

I wish to thank Dr. Leonardo Cosmai from INFN of Bari for some usefuldiscussion, during the first simulation period.

This work could not be done without the possibility of performing the simu-lations on the ReCaS Cluster present in the University of Calabria.

I owe a deep sense of gratitude to all the members of the administration ofthe Physics Department of University of Calabria, which have encouraged memany times during these years, with advices and a great friendly hospitality,that I will never forget.

***

The academic acknowledgements are written at the end of the thesis period,the following ones, one starts to think and write it long before the start of thethesis period and are almost made by the every day life with other people.

I am highly indebted to the friends and colleagues Eliana Lambrou, DafneBolognino, Giusy Monterosso, Francesca Cuteri and Mario Gravina, for theirdirect and indirect aid coming from the profitable reading, during these months,of their thesis work.

9

ACKNOWLEDGEMENTS 10

From world abroad, I express my hearty thanks to Vladimir Prochazka andAgnieszka Bomersbach which for their true friendship I always bring in mymind, and I hope to enjoy their company again, maybe in front of a goodplate of pierogi.

I have great pleasure to say thanks to friends whose good company full ofbig laughs I enjoyed during these months:

Jacopo Settino for our almost everyday discussions ranging from physics topolitics, luckily always ending in front of good food which was always wellchecked by our nutritionist, his partner Mariogiovanna Settino.

Our “priest”, Giuseppe Prete, which in this thesis period showed up veryadvanced cleaning skills that were previously unknown. . .

Federica Chiappetta, the Oracle, for her magnificent presence and constantsupport not only scientific but even with magnificent food, from parmigianato marvellous cakes.

Last but not the least the Unical “Papyrus Team” made by Sara Stabile andDavide Micieli, that is now fully equipped with a huge telescope to study thecosmos, which I hope to use one day.

Behind any success there is always an invisible and constant support, thatin my case is my family. I think nothing could be done without their aid.Word are inadequate to express my gratitude so I can only publicly thank mybrother Angelo, my mother Paola, my father Giuseppe and my grandfatherAngelo.

Introduction

Why does one write?To free oneself from anguish.Often writing represents theequivalent of a confession orFreud’s couch.

Other People’s TradesPrimo Levi

Everything started in 2013 when I decided to participate to the Erasmusplus project, thank to the advises of Professor Papa I decided to visit for fivemonths the Higgs Centre for Theoretical Physics, with the hope to understandunder the supervision of Professor Luigi Del Debbio the quantum chromody-namics (QCD) and its on lattice formulation (LQCD).

Before my departure, more or less one month, on the 6 March 2013 physi-cists announced at the Moriond conference in La Thuile, Italy, that the newparticle discovered at CERN last year is looking more and more like a Higgsboson. This news was spread out in all the media and induced a sense of vic-tory. Many "physics-for-all" events were organized and the main topic was:The Standard Model is now complete, the humankind has succeeded in un-derstand the particle physics word, we can now concentrate on other things.This sense of complete victory was of course a great beading of the true, be-cause the reality was far very different.

In the Higgs centre, where I met in months only one time Professor Higgs, hiscalmness was shared commonly by all the theoretical particle physics groupsall over the world, where their routine life was, as usual, full of troublingfor formulation of new theoretical ideas, i.e. analysing data and continuouscomparison among theory and experiments, guided by the constant hope tocomprehend something more on QCD that could let them be able to put anew tessera in the mosaic of phase diagram of QCD in Fig. 1.As we can see all the actual knowledge about QCD is condensated in thisphase diagram, where there are still many aspect as the transition phase from

11

INTRODUCTION 12

Figure 1: Proposed phase diagram for QCD. ALICE, RHIC,CBM, SPS and RHIC are thenames of relativistic heavy-ion collision experiments.

confined to deconfined phase that are only hypothetical. So as we have seenfrom the phase diagram, the truth that the media covered up, was that inparticle physics and in particular in QCD there were so many un-explainedbehaviours, many of them very conceptually complex and even some of themconnected to the mass-gap problem of Yang-Mills theory, for which the ClayMathematics Institute has proposed a millennium prize.

Among the unsolved problems there are ones that arise from simple ques-tions, easy to formulate which an entry level student could concentrate onand motivated by the Montessori idea of learning by doing, can try to discussand share ideas. One of these question is:

Which is the shape of the potential among three quarks inside a baryon, likea proton or a neutron?

The literature on this topic is not so plentiful and the numerical results are ac-tually very contradicting on the possible potential shapes that could describethe data coming from numerical results.

INTRODUCTION 13

The challenge resides in the fact that this topic is difficult to study in thefull QCD and even the numerical simulations are very complex and time con-suming, so that, researchers are forced to attack the problems with the for-mulations of simplified effective theories, that could be less complex to studyand more numerical feasible than the full one. The formulation of an effectivetheory is always a demanding task, because the physicist is completely free touse his imagination in the formulation and is provided by the only principleguide, that the more symmetries are reproduced from the original theory, themore the model will succeed in describing Nature.

The main aim that motivated this thesis, is proposing and using an effec-tive theory based on an evolution of a Potts model, whose building up sharesmany symmetries with the full QCD, that can hopefully give us the oppor-tunity to use the numerical results of the simulations for identifying the bestpossible candidate for the potential of the 3-quarks. So as in all the numericalsimulations study, the results always come after hard coding time, followedby long debugging analysis and finally impatient waiting for the runs end-ing. The possible success in describing the 3-points potential from numericalsimulations, will give us the opportunity to propose our effective model forthe preliminary study of any idea regarding the potential of the new exotichadron systems, like the pentaquarks that have started to appear in the re-cent experiments [1] or the mostly speculated tetraquarks systems for whichalready many possible experimental candidates exist [2, 3]

***

The present thesis is divided in four chapters structured as follows:

In Chapter 1 we introduce QCD, building up its Lagrangian from the gaugeprinciple and then focusing on the quantization via Feynman path integrals.After that we describe the two fundamental energy regimes, ruled by thestrong coupling constant so that we can present the asymptotic freedom andthe confinement. In the second part of the chapter, the lattice formulationof QCD is described, the fundamental observables commonly used are pre-sented, and it is shown how they can used to distinguish confined and decon-fined phase. Finally we discuss the delicate issue of the continuum limit, thathas got important implication on the numerical simulations.

In Chapter 2 we focus on the numerical techniques needed to study latticegauge theories (LGT), presenting the Monte Carlo methods and the standardtools to analyse data from simulations and how to estimate errors.

INTRODUCTION 14

In Chapter 3, we introduce the idea of an effective theory and then we presentour model, describing the effective theory from which the simulations are car-ried out. The numerical analysis carried out to characterize the model phasetransition is explained in details. We finally present the observables that areused in our study and how we have tested them.

In Chapter 4, a brief review of the state of the art on three-quark potentialstudies is delineated, then we extensively discuss the procedure used to studythe three-quark potential, describing how we analysed the numerical resultsproduced by simulations.

Chapter 1

Basics of QCD and Lattice QCD

...we see a number ofsophisticated, yet uneducated,theoreticians who are conversantin the LSZ formalism of theHeisenberg field operators, but donot know why an excited atomradiates, or are ignorant of thequantum theoretic derivation ofRayleigh’s law that accounts forthe blueness of the sky.

Modern Quantum MechanicsJun John Sakurai

Inside the Standard Model of particle physics (SM), Quantum Chromody-namics (QCD) is the theory introduced to describe the internal structure ofsubnuclear particles and their interaction and should be responsible of theexistence of simple matter form as nucleons, proton and neutron, and fromthem the nucleus inside atoms.Briefly from technical point of view, the QCD theory assumes the existence ofspinor field for quarks and vector fields for gluons, both these fields havingone internal degree of freedom called colour. The interaction among quarksand gluons is building up imposing the gauge invariance under local colourtransformation, and it is assumed to be able to explain all the known hadronphenomenology.The up-to-now, humankind knowledge supported by numerous experimentsgives us the possibility to state that QCD is the universally accepted theoryfor describing strong interaction. [4, 5]

15

1.1. LAGRANGIAN OF QUANTUM CHROMODYNAMICS 16

1.1 Lagrangian of Quantum Chromodynamics

Following the idea of Yang-Mills [6] and guided by the standard presen-tation in the books [4, 7] we build up the QCD Lagrangian and finally itsaction from the gauge invariance principle. To start, we have to identify anon commutative local transformation and then require, to the Lagrangianor the action of the theory, to be invariant under the chosen transformation.For QCD this transformation is the local rotations of quarks colour indices.For each space-time point x, we choose an independent complex 3x3 matrixΩ(x). The matrices are required to be unitary Ω(x)† = Ω(x)−1, and to havedet[Ω(x)] = 1, that is, they belong to the fundamental representation of thespecial unitary group group SU(3). This group is closed under matrix multi-plication that is not commutative, indeed the SU(3) is a non-abelian group.Next, we need to introduce a simple Lagrangian able to describe a free quark,and this is the Dirac fermion Lagrangian:

Lq = ψ(i γµ ∂µ −m)ψ , (1.1)

where the quark field ψ(x) is a 3-component column in colour space; eachcolor component is a 4-component spinor.We now require that the free quark Lagrangian is invariant under the trans-formation

ψ(x)→ ψ′(x) = Ω(x)ψ(x), ψ(x)→ ψ′(x) = ψ(x)Ω(x)†. (1.2)

Since Ω(x) belongs to the fundamental representation of SU(3), it can berewritten as

Ω(x) ≡ e−iTaθa(x),where Ta are eight generators of the Lie algebra corresponding to the SU(3)group and the θa are the gauge transformation parameters, with the index arunning from 1 to 8. The eight generators are subject to the following com-mutation relations:

[Ta, Tb] = i f abcTc,

where f abc are the structure constants characterizing the algebra of the group.The free quark Lagrangian given in (1.1) is invariant under the transforma-tion in equation (1.2) if the derivative is replaced by the so called covariantderivative, that reads

Dµ = ∂µ − igTa Aaµ, (1.3)where Aaµ are the boson gauge fields of the interaction and g is the constantrepresenting the coupling strength between the fields ψ and Aaµ.The new differential operator Dµ has the feature to make Dµψ transform un-der local gauge operator in the same way as field alone does.

1.1. LAGRANGIAN OF QUANTUM CHROMODYNAMICS 17

With the introduction of the covariant derivative, the starting Lagrangian isnow endowed with a term describing the quark-gluon interactions,

Lq + Lqg = ψ(i γµ Dµ −m)ψ= Lq + gψγµTa Aaµψ.

(1.4)

The gauge invariance of the new Lagrangian is ensured by imposing the cor-rect transformation rule for the gauge fields Aaµ; to derive it, we simply requirethat

(Dµψ)′= Ω(x)(Dµψ),

holds, so from simple calculations we find that Aaµ obeys the transformationrule given by

Ta A′aµ = Ω(x)

[Ta Aaµ −

ig

Ω(x)−1∂µΩ(x)]

Ω(x)−1. (1.5)

The Lagrangian we have discussed so far describes only the dynamics offermion fields ψ(x) and its interaction with the gauge fields Aaµ. To completeour discussion we have to introduce a kinetic term describing the dynam-ics of free gauge fields Aaµ. This new term must be local gauge invariant.Here, instead of deducing it from infinitesimal transformation property andthe commutation rule between covariant derivatives,

[Dµ, Dν] = −igTa[ ∂µ Aaν − ∂ν Aaµ + g f abc Abµ Acν ]= −igTaGaµν.

(1.6)

that can be found in [4], we speed up the discussion following the constructionprocedure used in quantum electrodynamics (QED), so we propose a termand then we ensure that is local gauge invariant. Our request are satisfied bythe missing term

Lg = −14

GaµνGa µν, (1.7)

whereGaµν = ∂µ A

aν − ∂ν Aaµ + g f abc Abµ Acν (1.8)

represents eight strength tensors which are interpretable as the eight gluonsfields. Finally the QCD Lagrangian that is built from the invariance under thenon-abelian local gauge transformation has is final form:

LQCD = −14

GaµνGa µν + ∑

fψ f (i γµ Dµ −m f )ψ f (1.9)

where with the sun over f we explicit the sum over the know six flavour ofquarks (up, down, charm, strange, top and beauty).

1.1. LAGRANGIAN OF QUANTUM CHROMODYNAMICS 18

From a simple look of the previous equation we can notice the presence of theterm g f abc Abµ Acν in the field strength Gaµν which describe the self-interactionamong the gauge fields Aaµ. This interaction is one of the most interesting andpeculiar characteristic QCD, and is mainly responsible of asymptotic freedom[8], which we will try to explain briefly it in a further section.

32 2 QCD on the lattice

Fig. 2.1. Schematic picture of the cubic and quartic gluon self-interaction. Thewavy lines represent the gluon propagators and the dots are the interaction vertices

2.2 Naive discretization of fermions

In this section we introduce the so-called naive discretization of the fermionaction. Later, in Chap. 5, this discretization will be augmented with an ad-ditional term to remove lattice artifacts. Here, it serves to present the basicidea and, more importantly, to discuss the representation of the lattice gluonfield which differs from the continuum form. We show that on the lattice thegluon fields must be introduced as elements of the gauge group and not aselements of the algebra, as is done in the continuum formulation.

2.2.1 Discretization of free fermions

As discussed in Sect. 1.4, the first step in the lattice formulation is the intro-duction of the 4D lattice Λ:

Λ = {n = (n1, n2, n3, n4) |n1, n2, n3 = 0, 1, . . . , N − 1 ; n4 = 0, 1, . . . , NT − 1} . (2.25)

The vectors n ∈ Λ label points in space–time separated by a lattice constant a.In our lattice discretization of QCD we now place spinors at the lattice pointsonly, i.e., our fermionic degrees of freedom are

ψ(n) , ψ(n) , n ∈ Λ , (2.26)

where the spinors carry the same color, Dirac, and flavor indices as in thecontinuum (we suppress them in our notation). Note that for notational con-venience we only use the integer-valued 4-coordinate n to label the latticeposition of the quarks and not the actual physical space-time point x = an.

In the continuum the action S0F for a free fermion is given by the expression(set Aμ = 0 in (2.5))

S0F [ψ,ψ] =

∫d4xψ(x) (γμ∂μ +m)ψ(x) . (2.27)

Figure 1.1: Cubic and quarting gluon self-interation Feynman diagram. Taken from [7].

Now from the QCD density Lagrangian we can write the action,

SQCD =∫

dx0d3x

[−1

4Gaµν(x)G

a µν(x) + ∑f

ψ f (x)(i γµ Dµ −m f

)ψ f (x)

].

(1.10)It is useful to complete this discussion presenting the euclidean action ofQCD. The previous expression is written in the real time formulation, but inthe following section when we will discuss the path integral representation,we will use the QCD Euclidean action. To accomplish it, we perform the Wickrotation, changing from real to imaginary time, x0 → −ix4 and extend it tothe gauge fields A0 → iA4, then we introduce the euclidean gamma matricesγEµ which are defined by the anticommutation relation

{γEµ , γEν

}= 2δµν and

are related to the gamma matrices γµ according to γEµ = γ0 and γEi = −i γi.

Substituting in the previous equation we obtain the QCD Euclidean action:

SEQCD =∫

d4x

[14

GEaµνGEaµν + ∑

fψ f (x)

(γEµ D

Eµ + m f

)ψ f (x)

]. (1.11)

1.1.1 Quantization in the Feynman-Integral Formalism

Generally, in the standard course of quantum field theory the quantizationis normally carried on imposing relation of commutation or anticommutationamong the fields used in the definition of the Lagrangian of the theory. Theimportance of choosing and then imposing the right relations of quantization

1.1. LAGRANGIAN OF QUANTUM CHROMODYNAMICS 19

is that from them we can derive the expression of the propagator and of allGreen’s functions, i.e., or vacuum expectation values of the time-ordered fieldproducts, which finally enter the Feynman amplitudes inside the cross sectionof a particular process. Therefore the initial quantization is responsible of thecorrect agreement of the theory with the experiments.

The QCD, as any other gauge field theory, could be subjected to differentquantization schemes; among the realm of possible quantization, we choose topresent here the Functional-Integral Formalism. To accomplish it will guidedby the presentation made in[4, 9]. The motivation of our choice is twofold;first, Green functions are directly obtained by integrating the product of thefields over all of their possible functional forms with a suitable weight, andsecondly, this formulation clearly shows the structural equivalence existingbetween the statistical mechanics and the family of lattice gauge theories.

For a better introduction we prefer starting introducing the path integral inquantum mechanics and finally extending it to quantum field theory.In quantum mechanics, a time evolution of a physical system, whose statesare described by vectors in Hilbert space, can be written as

|ψ(t)〉 = e−iH(t−t0) |ψ(t0)〉 , (1.12)

where H is the Hamiltonian describing the system and it is represented in aHilbert space by a Hermitian operator.Let us define q = {qj} the coordinate degrees of freedom, and |q〉 the eigen-states of the relevant operators {Qj}, such that

Qj |q〉 = qj |q〉 , j = 1, 2, . . . , n (1.13)

Then, for the wave function ψ(q, t) = 〈q|ψ(t)〉 the subsequent relation can beeasily inferred:

ψ(q′, t′) =∫

dq G(q′, t′; q, t)ψ(q, t), dq =n

∑j=1

dqj (1.14)

The quantity in right hand of (1.14) is given by

G(q′, t′; q, t) = 〈q′| e−iH(t−t0) |q〉 , (1.15)

and it stands for the Green function describing the propagation of the wavefunction ψ(t) from |q〉 to |q′〉.In order to gain the path integral related to Green function just mentioned, wehave to notice that the matrix element in (1.15) is equal to 〈q′, t′|q, t〉, where|q, t〉 = eiHt |q〉 and it satisfies Qj(t) |q, t〉 = qj |q, t〉.

1.1. LAGRANGIAN OF QUANTUM CHROMODYNAMICS 20

To proceed further it is convenient to move the Euclidean space-time by per-forming of the Wick rotation, i.e. the analytic continuation of matrix elementin (1.15) to imaginary times (t → −iτ, t′ → −iτ′). This is done to avoid theappearance of strong oscillating imaginary exponential, which is not suitedfor numerical calculation. Making the replacement, we obtain the followingexpression

〈q′, t′|q, t〉∣∣∣∣t=−iτ

t′=−iτ′= 〈q′| e−H(t′−t) |q〉

∣∣∣∣t=−iτ

t′=−iτ′= 〈q′| e−H(τ′−τ) |q〉 . (1.16)

The imaginary-time interval can be divided into N infinitesimal segments,each one of width e = τ

′−τN . Accordingly, the Green function can be achieved

through infinitesimal time steps; and after that (N− 1) completeness relationsof the coordinate states |q〉 have been included, we obtain〈q′| e−H(τ′−τ) |q〉 = 〈q′| e−H(τ′−τN−1)e−H(τ′−τN−2) . . . e−H(τ′−τ) |q〉

=∫ N−1

∏l=1

dq(l) 〈q′| e−He |q(N−1)〉 . . . 〈q(1)| e−He |q〉 .(1.17)

In order to evaluate the matrix elements we need to explicit know the Hamil-tonian form, generally one can use,

H =12

n

∑j=1

P2j + V(Q), (1.18)

which can be approximate, in the limit of small e values as

e−eH ≈ e−e2 ∑j P

2j e−eV(Q) (1.19)

Therefore, making use of complete sets of momentum eigenstates |p〉, each ofmatrix elements in equation (1.17) results in

〈q(l+1)| e−He |q(l)〉 e→0= e−eV(q(l))∫ n

∏k=1

dp(l)k2π

n

∏j=1

e−e 1

2 p(l)2j −ip

(l)j

q

(l+1)j −q

(l)j

e

.

(1.20)Hence, the entire transition amplitude, for small e values, has the followingform

〈q′| e−H(τ′−τ) |q〉 e→0=∫

Dq Dp eip(l)j

(q(l+1)j −q

(l)j

)e−eH(q

(l), p(l)), (1.21)

where q(0) = q, q(N) = q′, DqDp = ∏nk=1 ∏N−1l=1 dq

(l)k ∏

N−1l=0

dp(l)k2π and H

(q(l), p(l)

)=

12 ∑

nj=1 p

(l)2j + V(q

(l)). With these replacements it is simple to see that that, be-cause of quadratic kinetic term in H, the integral is solved carrying out the

1.1. LAGRANGIAN OF QUANTUM CHROMODYNAMICS 21

Gaussian integration.Now, again for small e, we obtain the coordinate space path integral repre-sentation

〈q′| e−He |q〉 e→0=∫ N−1

∏l′=1

n

∏j=1

dq(l′)

j e−e[

∑N−1l=0 LE(q(l), q̇(l))

], (1.22)

where LE(q(l), q̇(l)) = ∑nj=1q̇2(l)j2m +V(q

(l)) is the Lagrangian for imaginary timeand in the exponent there is the classical action SE[q].So the time Green function, can be write

〈q′| e−H(τ′−τ) |q〉 =∫ q′

qDq e−SE[q], (1.23)

where Dq is a collective short form for ∏nk=1 ∏N−1l=1 dq

(l)k and the explicit form

of the euclidean action reads

SE[q] =∫ τ′

τdτ′′LE(q(τ′′), q̇(τ′′)). (1.24)

We finally notice that, choosing the euclidean formulation, the main contribu-tions still come from the paths close to the one for which the action has gota minimum and the fluctuations around the classical path are exponentiallysuppressed, leaving wide the possibility for numerical determination of theintegral.

To extend the functional-integral formalism to the quantum fields theory, westart considering for simplicity a system characterized only by a neutral scalarfield φ(x) and its conjugate field π(x). The analogy is performed identifyingthe generalized set of coordinates {qj} seen before with the values of the fieldφ(x) at all points in space. For the two-point function we get

〈φ′, t′|φ, t〉 =∫ φ′

φDφ e−SE[q]

=∫ φ′

φDφ e−

∫ τ′τ dτ

′′LE(φ(τ′′), φ̇(τ′′)).(1.25)

As we have already pointed out at the begin, all physical information aboutthe system is stored in all the possible n-points Green’s functions, defined asvacuum expectation value of a time-ordered product of n field operators φ̂(x):

Gn(x1, x2, . . . , xn) = 〈0| T[φ̂(x1), φ̂(x2), . . . , φ̂(xn)] |0〉 . (1.26)

1.1. LAGRANGIAN OF QUANTUM CHROMODYNAMICS 22

Now, to accomplish our main task, we have to build up a path integral for-mulation for this object.Starting from the matrix element 〈φ′, τ′| T[φ̂(x1), . . . , φ̂(xn)] |φ, τ〉, and con-sider it for infinitely large values for τ and τ′, we obtain

〈0| T[φ̂(x1), . . . , φ̂(xn)] |0〉 =〈φ′, τ′| T[φ̂(x1), . . . , φ̂(xn)] |φ, τ〉

〈φ′, τ′|φ, τ〉 , (1.27)

Using again the analogy to the quantum mechanical case and consider thelimit (τ′ − τ) → ∞, we can easily rewrite the numerator and denominator interms of path integral representation, so that we can finally state the validityof the relation:

〈0| T[φ̂(x1), . . . , φ̂(xn)] |0〉 =∫

Dφ φ(x1) . . . φ(xn) e−SE[φ]∫Dφe−SE[φ]

. (1.28)

We can summarise the above discussion saying that in the Feynman functional-integral formalism, the fields are c-numbers and the Green function is de-duced by integrating the product of the fields over all possible functionalforms with proper weight, i.e. eSE[φ].

At this point the above formula is suitable to show the structural equiva-lence between field theory and statistical mechanics, the green function iscalculated as a statistical ensemble average with Boltzmann factor e−SE andpartition function as:

Z =∫

Dφe−SE[φ]. (1.29)

To conclude this section we need to underline that the previous discussion,dedicated only to bosonic fields, is not enough to the case of QCD, becausethe presence of fermions require the extension of the functional integral for-malism to anticommuting variables.

In the complete formulation, adapted completely to QCD, we can ensure thatafter few technical issue and many standard calculation, the Feynman pathintegral take a more complicate form, but at the end the partition functioncan be rewritten simply as:

Z =∫

DψDψDAe−SE[ψ,ψ,A]. (1.30)

The discussion about appearance of the "ghost" or "Fadden-Popov" fields inthe quantization formulation as an effect of gauge-fixing, has been omitted,since the Monte Carlo methods by which QCD is studied numerically or as aspace-time lattice do not need gauge fixing.

1.1. LAGRANGIAN OF QUANTUM CHROMODYNAMICS 23

1.1.2 Asymptotic Freedom and Confinement

After building up the mathematical framework of theory, i.e the Lagrangiandensity, the quantization of fields and the determination of the essential Green’sfunctions, one is interested to use the theory to describe Nature, or more sim-ply, to find a theoretical explanation of the experiments results. In accom-plishing this task one must solve the problem of divergences.

From the first appearance of divergences, theoretician have tried to overcamethis difficulty and they finally found a useful approach called renormalization.The renormalization theory is a very vast and complicate field; here we willpresent only the minimum knowledge useful to characterize the asymptoticfreedom and the confinement, so this section is a simply and short adaptationof what is present in [4, 10, 11].

The renormalization scheme is based on the idea that one can assume thatthe observables as the mass, the coupling and other parameters present in thetheory are not the physical one, but just numbers, called bare parameters, thatcould be redefined in order to absorb the divergences present in the theory.The bare parameters are dependent of the scale µ at which the subtraction ofdivergences take place, so they are not constant but their values change withµ so they are addressed as running constants. Once they are redefined, wecan calculate the elements of the scattering matrices and compare with the ex-periment measurements. From this comparison, we can deduce the runningconstant’s dependence from µ that renders the scheme meaningful and, if thetheory is renormalizable, when prediction are written in terms of physicalvariables instead of bare mass the divergences disappear.

The equations that are able to describe how the bare parameters change withthe scale µ are called renormalization group equations. The one relative tothe renormalized coupling constant given in the MS scheme is:

β(gr) = µ∂gr∂µ

∣∣∣∣gB,mB

, (1.31)

where gr is the renormalized coupling constant and gB and mB are the barecoupling and the bare mass. It can be shown that solving the previous equa-tion and using some standard field theory techniques, the strong couplingstrength is:

αs(µ) =12π

(33− 2N f ) ln(µ2/Λ2), (1.32)

where N f is the number of flavour and Λ is the QCD scale parameter, to bedeterminate experimentally, and the today accepted value is Λ ≈ 250MeV/c

1.2. LATTICE QCD 24

[12].

Now that we have fixed the value of Λ, we can state that at small distancesor high-energy regime (µ → ∞), the strong coupling becomes weaker andweaker as µ increases, and this is why quarks and gluons inside hadrons canbe described as a merely free particles. This QCD peculiar property is calledasymptotic freedom [13], and can be studied in the framework of perturbationtheory, which is reasonably correct only for αs(µ) � 1, and this is true whenQ2 � Λ2 ≈ 0.06(GeV/c)2.

On the contrary at large distances or at the low-energy regime (µ → 0),the strong coupling grows and finally diverge, which logically explain whycolour charged particles, as gluon and quark, cannot be isolated and thereforecannot be observed. This is the property of confinement. We like to stress that,up to now, the confinement is only a conjecture that still need a theoreticalproof, and this could be only arise from a non-perturbative approach as thelattice formulation of QCD.

1.2 Lattice QCD

To study the many non-perturbative aspects of QCD like confinement orphase transition, for which the strong coupling is not (αs � 1), therefore theperturbative theory is impracticable, we need a non-perturbative approachthat is gauge invariant and able to regularize the theory from UV divergences.

The solution was proposed by K. Wilson in 1974 with the introduction of anew approach, now called Lattice QCD [14]. The LQCD is based on the intro-duction of a 4D Euclidean spacetime lattice, on which a discrete formulationof QCD from first principle is established. The introduction of the lattice withconstant size a, provides a natural cut-off in the momentum space of order ofthe inverse of a, enough to remove the ultraviolet divergences present in thecontinuum formulation of QCD.The lattice could be defined as:

Λ = {n = (n1, n2, n3, n4)|n1, n2, n3 = 0, 1, ..., N − 1; n4 = 0, 1, ..., NT − 1}(1.33)

where N denotes the number of lattice sites in the three spatial directions,while NT is the number of lattice sites in the temporal direction. A schematicrepresentation could be seen in Fig. 1.2.The quark field ψ(x), ψ(x) are placed on the lattice points labelled by x = naand the gauge field Uµx on the edges among two consecutive lattice points.

1.2. LATTICE QCD 25

Figure 1.2: Representation of a lattice of size a. µ̂, ν̂ and ρ̂ are spacetime unit vector usefulto locate the fileds ψ(x) and Uµx . Taken from[15].

In the following sections, guided by the presentation in [7], we present andbuild up the main LQCD equations. These equations, which are not alwaysa direct discrete form of the continuum ones, can have different forms due tothe particular aspect we like to describe. This freedom in choosing their formis limited only by the fact that, to be meaningful, they have to reproduce thecontinuum equations when the spacetime lattice discretization is removed, inthe limit a → 0, the so called continuum limit. In this limit not only the dis-crete equations reproduce the continuum ones but all the symmetries brokenin the discrete formulations are restored. Finally we discuss how to extractphysics from this discrete formulation discussing the true continuum limit, inwhich the observables approach the physical ones.

1.2.1 Naive discretization of fermion

The naive form of the LQCD fermionic action, is built up from the dis-cretization of continuum one. To start, we place the spinor fields ψ(n), ψ(n)at the lattice points so that, n ∈ Λ.In the continuum, the action S0F for a free fermion field is given by

S0F[ψ, ψ

]=∫

d4x ψ(x)(γµ Dµ + m

)ψ(x). (1.34)

1.2. LATTICE QCD 26

Both the integral over space-time and the partial derivative, have to be dis-cretized, the integral as ∫

d4x → a4 ∑n∈Λ

and the partial derivative, with the symmetric expression as

∂ψ(x)→ 12a

(ψ(n + µ̂)− ψ(n− µ̂))

Finally we have the naive free fermionic action

S0F[ψ, ψ

]= a4 ∑

n∈Λψ(n)

(4

∑µ=1

γµψ(n + µ̂)− ψ(n− µ̂)

2a+ mψ(n)

). (1.35)

In analogy to the continuum formulation, we introduce the gauge fields, re-quiring the gauge invariance of the lattice action under an SU(3) transfor-mation, so choosing an element Ω(n) ∈ SU(3) for each lattice site n andtransforming the fermion fields according to

ψ(n)→ ψ′(n) = Ω(n)ψ(n), ψ(n)→ ψ′(n) = ψ(n)Ω(n)† (1.36)

But under these transformation, the naive fermionic action is not invariantbecause of the term with the discretized derivative which, under this trans-formation becomes

ψ(n)ψ(n + µ̂)→ ψ′(n)ψ′(n + µ̂) = ψ′(n)Ω(n)†Ω(n + µ̂)ψ′(n + µ̂) (1.37)

In order to achieve the gauge invariance we are forced to introduce thefield Uµx , which their matrix representation are elements of SU(3). We placethem on the link which connect the lattice sites n and (n + µ̂), therefore arecalled link variables.

It is useful to define the gauge transformation of the new field as

Uµ(n)→ U′µ(n) = Ω(n)Uµ(n)Ω(n + µ̂)† (1.38)

so that the quantity ψ(n)Uµ(n)ψ(n + µ̂) is gauge invariant.

For notational convenience we introduce U−µ(n) as the link variable point-ing in the negative µ direction, linking n to (n − µ̂), such as the relationU−µ(n) ≡ Uµ(n− µ̂)† hold, and transform as:

U−µ(n)→ U′−µ(n) = Ω(n)U−µ(n)Ω(n− µ̂)† (1.39)

1.2. LATTICE QCD 27

34 2 QCD on the lattice

U− μ(n) ≡ Uμ(n− μ̂)†n n+μ̂

Uμ(n)

nn− μ̂

Fig. 2.2. The link variables Uμ(n) and U−μ(n)

points from n to n − μ̂ and is related to the positively oriented link variableUμ(n− μ̂) via the definition

U−μ(n) ≡ Uμ(n− μ̂)† . (2.34)

In Fig. 2.2 we illustrate the geometrical setting of the link variables on thelattice. From the definitions (2.34) and (2.33) we obtain the transformationproperties of the link in negative direction

U−μ(n) → U ′−μ(n) = Ω(n)U−μ(n)Ω(n− μ̂)† . (2.35)

Note that we have introduced the gluon fields Uμ(n) as elements of the gaugegroup SU(3), not as elements of the Lie algebra which were used in the con-tinuum. According to the gauge transformations (2.33) and (2.35) also thetransformed link variables are elements of the group SU(3).

Having introduced the link variables and their properties under gaugetransformations, we can now generalize the free fermion action (2.29) to theso-called naive fermion action for fermions in an external gauge field U :

SF [ψ,ψ, U ] = a4∑

n∈Λψ(n)

(4∑

μ=1

γμUμ(n)ψ(n+μ̂) − U−μ(n)ψ(n−μ̂)

2a+mψ(n)

).

(2.36)

Using (2.30), (2.33), and (2.35) for the gauge transformation properties offermions and link variables, one readily sees the gauge invariance of thefermion action (2.36), SF [ψ,ψ, U ] = SF [ψ

′, ψ′, U ′].

2.2.3 Relating the link variables to the continuum gauge fields

Let us now discuss the link variables in more detail and see how they can berelated to the algebra-valued gauge fields of the continuum formulation. Wehave introduced Uμ(n) as the link variable connecting the points n and n+ μ̂.The gauge transformation properties (2.33) are consequently governed by thetwo transformation matrices Ω(n) and Ω(n + μ̂)†. Also in the continuum anobject with such transformation properties is known: It is the path-orderedexponential integral of the gauge field Aμ along some curve Cxy connectingtwo points x and y, the so-called gauge transporter:

Figure 1.3: Rapresentation of the two link variables Uµ(n) and U−µ(n) [7].

With the help of the link variables and their property under gauge trans-formation we can finally write the naive fermionic action for fermions in anexternal gauge field U as:

SF[ψ, ψ, U

]= a4 ∑

n∈Λψ(n)

(4

∑µ=1

γµUµ(n)ψ(n + µ̂)−U−µ(n)ψ(n− µ̂)

2a+ mψ(n)

).

(1.40)that is gauge invariant such that SF

[ψ, ψ, U

]= SF

[ψ′, ψ′, U′

]holds. This is

the so called naive fermionic action because its form is not definitive, but doto the presence of the doubling problem should be replaced with a form thatbetter suit to the system to study. We will not enter in this delicate topic be-cause in this thesis we will principally interested in the gluon action.

1.2.2 Gauge action

Differently from the fermion action, the bosonic or gauge action can bewrite directly using gauge invariant object made by link variables. These ob-jects ensure the gauge invariance and removing the lattice, in the limit a→ 0,they transform so that the continuum form is recovered.

The simplest gauge invariant object built with link variable is a closed loopcalled plaquette, and it is defined as the product of four link variables:

Uµν(n) = Uµ(n)Uν(n + µ̂)U−ν(n + µ̂ + ν̂)U−ν(n + ν̂)

= Uµ(n)Uν(n + µ̂)Uµ(n + ν̂)†Uν(n)†.(1.41)

The Wilson gauge action is a sum over all plaquettes, with each plaquettecounted with only one orientation. This sum is realized as a sum over alllattice points n, combined with a sum over the directional Lorentz indices1 ≤ µ < ν ≤ 4, so that

SG[U] =β

3 ∑n∈Λ∑

µ

1.2. LATTICE QCD 2838 2 QCD on the lattice

Fig. 2.3. The four link variables which build up the plaquette Uμν(n). The circleindicates the order that the links are run through in the plaquette

SG[U ] =2

g2

∑

n∈Λ

∑

μ

1.2. LATTICE QCD 29

Baker-Campbell-Hausdorff formula, we obtain

SG[U] =2g2 ∑n∈Λ

∑µ

1.2. LATTICE QCD 303.3 Wilson and Polyakov loops 55

xy

t

xy

t

Fig. 3.3. Examples for a planar (left-hand side plot) and a nonplanar (right-handside) Wilson loop. The horizontal direction is time

3.3.2 Temporal gauge

Before we can discuss the physical interpretation of the Wilson loop we mustdiscuss a peculiarity of gauge theories. If one wants to evaluate the canonicalmomentum (1.45) for the gauge field action (2.17) one finds that for thetemporal component A4 the canonical momentum vanishes. The reason isthat the field strength tensor Fμν(x) = −i[Dμ(x), Dν(x)] does not containderivatives of A4 with respect to time. A possible way out of this problem isto use a gauge where

A4(x) = 0 , (3.51)

i.e., the aforementioned temporal gauge. We remark that simply stating (3.51)does not by far do justice to the subtleties involved in the quantization of gaugetheories, and we refer the reader to field theory books such as [6–8] on thisissue. On the lattice, temporal gauge corresponds to the condition (3.36).

We stress that in the following we use the temporal gauge only to findthe physical interpretation of the Wilson loop. For the actual computationof the expectation value we do not need to fix the gauge. The result for theexpectation value of the Wilson loop is of course the same whether we fix thegauge or not.

3.3.3 Physical interpretation of the Wilson loop

In the temporal gauge (3.36), discussed in the last paragraph, the temporaltransporters become trivial,

T (n, nt) =

nt−1∏

j=0

U4(n, j) = 1 , (3.52)

and we obtain the following chain of identities

〈WL〉 = 〈WL〉temp =〈tr

[S(m,n, nt)S(m,n, 0)

†]〉temp

, (3.53)

where in the first step we have used the fact that the expectation value of agauge-invariant observable remains unchanged when fixing the gauge. In the

Figure 1.5: Two example of Wilson loop, the first planar and the second nonplanar. Takenfrom [7].

where L is a rectangular shape path with corners in (m,0),The simple definition of this object is greatly compensated by its ground stateexpectation value, that is very important in LQCD. Its expression could becalculated as the ensemble mean value:

〈WL[U]〉 =∫

DUDψDψWL[U] e−SQCD[ψ,ψ,U]∫DUDψDψ e−SQCD[ψ,ψ,U]

. (1.51)

It can be shown [7] that the static potential V(R) binding a qq pair locatedas spatial point m and n is related to the ground state average of the Wilsonaccording to

〈WL[U]〉 ∼T→∞

Ce−TV(R), (1.52)

where R and T are the extension of the loop in space and time expressed inlattice spacing unit. Inverting this expression we obtain

V(R) = − limT→∞

1T

ln 〈WL[U]〉 . (1.53)

The actual knowledge regarding the qq potential could be summarized as:

V(r) = A +Br+ σr (1.54)

in the first term A is a constant taking track of the energy normalization, thesecond is Coulomb shape part and the last is a linear rising terms where σis the so-called string tension. The last terms is the one more studied, be-cause its growing with the distance is a direct manifestation of confinement,so proposing model describing the dynamic origin of this term, could in prin-ciple shed some light on the intimate mechanism of confinement.

From the potential V(R) we can define the string tension as

σ = limR→∞

1R

V(r), (1.55)

1.2. LATTICE QCD 31

which, in a numerical simulation can be estimate from the vacuum expecta-tion value of the Wilson loop as:

σ ∼ −ln( 〈WL[U]〉

RT

)(1.56)

Numerical simulations in LQCD has established that the chromoelectric fieldlines generated by the strong interaction are collimated into a tube-like shapecalled chromoelectri flux tube as shown Fig. 1.6.

Figure 1.6: Diagram of a qq pair and its behaviour when the separation is increased.

The figure above shows that the potential increasing linearly with the separa-tion of the source until it produces through the vacuum polarization effects,a new qq pair, that screens the original interaction.

Knowing the behaviour of the potential for big enough qq separation we candeduce the so called area law

〈WL[U]〉 → e−σRT (1.57)

recognizing that R times T forms a spacetime area, from which the name.

Finally, this equation is the mathematical base for the Wilson loop criterion,that states that a static quark confinement holds if Wilson loops obey the arealaw and the associated string tension is not zero. The Wilson loop criterionis very important since it allows us to distinguish different phases, with andwithout static quark confinement, by means of this gauge invariant observ-able. Therefore the string tension has been used as an order parameter inmany numerical and analytical investigation of lattice gauge theories [16].

1.2.4 Finite Temperature

Up to the following section, all the discussion we have done is only relativeto the QCD at zero temperature, but there so many interesting QCD aspectsat high temperature that still needs an explanation, such as the condensationof quarks and gluons in hadronic system, due to the cooling of the universeafter the big bang. To accomplish this scope, physicist were forced to intro-duce the study of a field theory at finite temperature.

1.2. LATTICE QCD 32

Recalling that the thermodynamic of a quantum system in a heat bath withtemperature T in the continuum, is described by the following canonical en-semble partition function [7]:

Z(T) = Tr[e−βH

]. (1.58)

were H is the Hamiltonian and here β is not the inverse of the couplingconstant but is equal to 1/kbT, with kb the Boltzmann constant. The parti-tion function can be expanded using a base formed by the eigenstates of thecoordinate operator. The corresponding path integral representation in con-figuration space is:

Z =∫ q

qDq e−

∫ β0 dτ

∫d3x LE(q,q̇), (1.59)

working at finite temperature means that the temporal integration is nowrestricted to the interval [0, β].In the path integral formalism the analogue of the above partition function,for the case of a bosonic scalar field φ(~x, τ) with Lagrangian density L(φ, ∂µφ)is:

Z =∫

Dφ e−∫ β

0 dτ∫

d3x LE(φ,∂µφ), (1.60)

with the fields satisfying periodic boundary conditions φ(~x, 0) = φ(~x, β). Thepartition function is a weighted sum over all field configurations which liveon a Euclidean space-time surface compactified along the time direction. Intwo space-time dimensions this surface can be viewed as a cylinder with itsaxis along the spatial direction. The radius of this cylinder becomes infinitein the limit of vanishing temperature [9].

Getting close to our need, we consider a pure gauge theory on the lattice,where the partition function reads:

Z =∫

Uµ(n,0)=Uµ(n,β)

DU e−S[U], (1.61)

where S[U] is the Wilson gauge action (1.42), and the integration must nowdone on all the link variables Uµ(n) constrained to the periodic boundaryconditions:

Uµ(n, 0) = Uµ(n, β). (1.62)

The concept of temperature is here introduced by a compactification of a

1.2. LATTICE QCD 33

temporal direction and identifying the inverse of the temperature with thetemporal extension of the spacetime [9], so that

β = aNT =1T

(1.63)

holds, and in the limit β → ∞ corresponds to T → 0 reproducing the resultof zero temperature.

Since the lattice space a and the coupling constant g2 are related, the tem-perature can be expressed as [17]:

T =1

NT∆QCD exp

[∫ dg2B(g2)

](1.64)

where the integrand coming from renormalization group theory and B(g2) isthe Gell-Mann-Low function. Therefore one can change the temperature onthe lattice by varying either the size along the temporal axis Nt or the cou-pling constant g2.

Now that we are fully equipped with the partition function rewritten to finitetemperature, we can study the behaviour of the thermodynamic observableswith the temperature of the system and deduce the critical property of thetheory, such as the critic temperature and the order of the phase transition.

1.2.5 Polyakov Loop

To study the properties of a system on a lattice at finite temperature, weneed to introduce new observables that can substitute the ones used in theinfinite-volume limit, were the temperature is zero.A very important gauge invariant observable commonly used at finite tem-perature, and very useful to characterize a system phase transition is thePolyakov loop, which is defined as follows:

P(m) = Tr

[NT−1∏j=0

U4(m, j)

]. (1.65)

It is nothing more than the trace of a temporal transporter T(m, nt) with thetemporal extent equal to the maximum possible size NT, hence the name ther-mal Wilson line. A common representation is the black arrowed line windingaround the time direction, as we can see in Fig. 1.7.

Taking the product of two Polyakov loops having opposite orientation as isshown in the following figure, and considering its vacuum expectation value,

1.2. LATTICE QCD 34

Non-Perturbative QCD at Finite Temperature 491

the time direction, and the Wilson loop no longer plays this role. The questiontherefore arises which is the corresponding object to be considered when studyingQCD at finite temperature. To this effect we notice that, because of the periodicstructure of the lattice, we can also construct gauge-invariant quantities by takingthe trace of the product of link variables along topologically non-trivial loopswinding around the time direction. In fig. (20-1) we show a two-dimensionalpicture of the simplest loop we can construct, located at some spatial lattice siten. Consider the following expression, constructed from the link variables locatedon this loop,

PL(n) = Tr Yl UA{n,nA), (20.13)

This expression is invariant under periodic gauge transformations:

Ui{n,ni) —¥ G{n,ni)Ui{n,ni)G^l{fi,ni + 1)

G(n,l)=G(n,0+l).

L(n) is referred to in the literature as the Wilson line, or Polyakov loop. As we nowshow, its expectation value has a simple physical interpretation. The argumentgiven below is only qualitative. For a more detailed discussion we refer the readerto the work of McLerran and Svetitsky (1981). To keep the presentation as simpleas possible, we will consider the U(l) gauge theory in the absence of dynamicalfermions. Furthermore we shall use the continuum formulation.

Fig. 20-1 Two dimensional picture of a loop winding around the timedirection.

Consider the partition function of the system consisting of an infinitely heavyquark coupled to a fluctuating gauge potential.

Z = '$2(s\e-PH\s). (20.14)S

Figure 1.7: Representation of a Polyakov loop. Taken from [9].

Non-Periurbative QCD at Finite Temperature 493

L(x) is invariant under periodic gauge transformations. In the non-abelian case

the potential A4 is a Lie algebra-valued matrix, and (20.16b) must be replaced

by the trace in colour space of the corresponding time-ordered expression. From

(20.17) we conclude that the free energy of the system with a single heavy quark,

measured relative to that in the absence of the quark is given by

e~^ = (L) = i £(£(£)). (20.18)X

where V is the spatial volume of the lattice. In writing the last equality we have

made use of the translational invariance of the vacuum. On the lattice, the left-

hand side of (20.18) is replaced by exp(-(3Fq), where /3 and Fq are the inverse

temperature and the free energy measured in lattice units.

The Polyakov loop resembles the world line of a static quark in a Wilson

loop. This suggests that the free energy of a static quark and antiquark located

at x = na and y = ma, with a the lattice spacing, can be obtained from the

correlation function of two such loops with base at n and m, and having opposite

orientations (see fig. (20-2)):

T(n, m) = (L{n)L\m)). (20.19)

Fig. 20-2 Two Polyalov loops, winding around the time direction, used

to measure the gg-potential.

Indeed, by the same type of arguments which led to the connection between the

Wilson loop and the static gg-potential in chapter 7, but with T now replaced by

ft, one can show that (20.19) is related to the free energy Fqq(n,m) of a static

quark-antiquark pair, measured relative to that in the absence of the qq pair, as

follows:

T(n,rh) = e-/3^(".™). (20.20)

Figure 1.8: Representation of two Polyakov loop with opposite direction. Taken from [9].

one can prove that its relation with the free energy Fqq of a qq pair, is:

〈P(m)P(n)†〉 ∝ e−aNT Fqq(a|m−n|) (1.66)

The similarity of this equation with the one related to the Wilson loop (1.52)should not be surprising, We stress, that this is a similarity, indeed the iden-tity of the potential as defined via Wilson or Polyakov loop is not rigorouslyproven; there exists, however, a proof of the fact that the string tension de-rived from the Polyakov loop correlator is bounded from above by the oneobtained from the Wilson loop correlator [7].

If we now consider a large distances we have:

lim(a|m−n|)→∞

〈P(m)P(n)†〉 = 〈P(m)〉〈P(n)†〉 = |〈P(n)〉|2 (1.67)

where in the last we used the translation invariance, that means the spatialposition of each Poliakov loop is irrelevant.Using the previous equation we can state

|〈P(n)〉| =√

lim(a|m−n|)→∞

〈P(m)P(n)†〉 ≡√

lim(a|m−n|)

e−aNT Fqq(a|m−n|) (1.68)

Carefully analysing this equations chain, we can say that if 〈P〉 = 0 then thefree energy grows up with the quarks separation, and this can be interpretedas a confinement signal. While if 〈P〉 6= 0 the free energy will tend to aconstant for large quark separation, this is interpreted as a deconfinementsignal.This simple reasoning is the base for using the mean value of a single Poliakovloop to distinguish the phase of a pure gauge system, where only gluons are

1.2. LATTICE QCD 35

present:〈P(n)〉 = 0⇔ Confined Phase〈P(n)〉 6= 0⇔ Deconfined Phase (1.69)

Due to this behaviour, the Polyakov loop can be used as order parameter. Thiscan be cleared discussing the inner mechanism behind the phase transition.Commonly in statistical mechanics the phase transitions are associated to abreak down of a global symmetry.

The SU(3) pure gauge theory action is invariant not only for periodic gaugetransformation but for transformation of the center of SU(3), Z(3) the groupformed by the three complex root of the identity. So that each element ofSU(3), g, is invariant under the transformation:

zgz−1 = g (1.70)

The Polyakov loop is not invariant under that transformation, infact it trans-form as:

P(n)→ zP(n) (1.71)and this is can be used to understand its behaviour close to the transitionpoint. In our pure gauge theory the fundamental state follow the action sym-metry, so that we have exact the same number of configuration connected bythe center symmetry and because ∑k e(2πik/3) = 0, the vacuum expectationvalue of a Polyakov loop is zero, and this is the reason why we aspect that thecenter symmetry hold at temperature below the critical temperature at whichthe deconfinement take place.

In the deconfined phase, where 〈P(n)〉 6= 0, because the central symmetryis spontaneously broken, we aspect to see the Polyakov loop values clusteringaround any of the Z(3) elements.

Finally for locating the exact point at which the transition phase accours, oneusually adopts the so called Polyakov loop susceptibility, defined as

χP = K(〈P2〉 − 〈P〉2

)(1.72)

which, as it is known from statistical mechanics, peaks in proximity of thetransition point. Studying how its position changes on the different latticesizes, one is able to find its finite size scaling law, which functional formdepends of the particular order of the phase transition, so that one can extractthe infinite volume values from finite value measurements [18]. In chapter 3we will use exactly this technique to extract the phase transition point of ourmodel.

1.2. LATTICE QCD 36

1.2.6 The Continuum Limit

We cannot conclude this chapter without discuss how is conceptually pos-sible extracting physics from numerical simulations on discretized theory, andwhich condition should exist in the system that guarantee that this is possi-ble. The topic is rather technical and conceptually complex so we more orless adhere to the presentation done in [9]. The introduction of the lattice isnecessary to remove the theory UV divergences, but with its introduction, theobservables become inevitably dependent of the lattice spacing a, so beforeeliminating it with the so called continuum limit (a → 0), we have to findthe conditions that ensure that the system and the observables approach theircontinuum corrispectives, the so called true continuum limit.

To understand the nature of these conditions, lets suppose that we studieda lattice theory that could posses them in principle, and we like to determinethe physical value of an observables as the mass from its lattice counterpartm̂(a). The relation existing between them is simply m̂(a) = m · a, so that tobe the mass finite when (a → 0) the m̂(a) has to vanish too. This has got avery important consequence, because the lattice mass is not a common ob-servables, it is intimately connect to the system correlation lengths ξ̂, whichmeasure how far the fundamental objects on the lattices, as the fields, arecorrelated, therefore due to their existing relation (ξ̂ = 1/m̂), we can say thatwhen the lattice mass goes to zero the lattice correlations lengths diverge. Themaster condition for a lattice field theory to be able to describe a continuumsystem, is the presence of a critical region in parameter space where the cor-relation lengths diverge. In the case of a gauge field theory as QCD, there isonly one parameter that we can tune to bring the system close to the criticalpoints, where ξ diverge, that is the bare coupling g0. Hence the critical regionis the one around g0 → gc where

ξ̂(g0) →g0→gc ∞ (1.73)

holds, so we can read a divergent correlations lengths as a sign that the sys-tem is close to the critical region and it is approaching the continuum theoryloosing its memory of the original lattice structure.

Doing a step forward in our discussion we notice that if considering differentlattices the physics of our theory should not change, that means that the bareparameters as g0 must change with a in a way depending only of the dynam-ics of the theory. In a more concrete way, let consider a physical observablesO and Ô its lattice version, with mass dimension d, and suppose one is inter-ested in study it on the lattice. So if the continuum limit exist, the relations

1.2. LATTICE QCD 37

among them:

O(g0, a) =(

1a

)dÔ(g0) (1.74)

has got a finite limit when a → 0 only if ”g0” is tuned with a in the rightway, that it can approach the critical coupling gc for which the equation (1.73)holds, so that

O (g0(a), a) →a→0

Ophys. (1.75)

Hence fixing the value of Ophys. and knowing the functional dependence ofÔ(g0) from g0 we can deduce the correct form of g0(a) at lest for small a value.We finally notice that, choosing a small enough lattice spacing, the functionaldependence of g0 from a is independent of the particular observable chosento extract it.

The delineated method is realized using the renormalization group equationfor a generic and well behaving observable. Imposing that its physical valueis independent of lattice spacing when a→ 0, so that

[a

∂

∂a− β(g0)

∂

∂g0

]O(g0, a) = 0 (1.76)

hold, where β(g0) = −a∂g0∂a

is the Callan-Symanzik β−function. The value ofβ(g0) cannot be calculate exactly, but is commonly deduced from the pertur-bation theory after solving the above equation for a chose observable in thecontinuum theory. Going directly at the hearth of the problem, in the case ofSU(3) with NF flavours, we have:

β(g0) = −β0g30 − β1g50 + O(g70),

β0 =1

16π2

(11− 2

3NF

),

β1 =1

(16π2)2

(102− 38

3NF

),

(1.77)

where β0 and β1 are universal coefficients, independent of the regularizationscheme instead the β(g0) depends on the details of the regularization.

Finally, integrating β(g) we obtain

a =1

ΛLR(g0), R(g0) =

(β0g20

)−β12β0 e

−1

2β0g20(

1 + O(g20))

(1.78)

that express the relation existing between g0 and a. The constant ΛL is notsimply a constant with mass dimension, but should be interpretate as the

1.2. LATTICE QCD 38

physical scale in term of which dimensioned quantities can be measured.

To answer to the question with which we opened the chapter, we need tofind the coupling constant interval of values for which the continuum physicscould be extracted while we perform numerical calculations on finite lattices.From the equations (1.78) we see how the bare coupling constant controlsthe lattice spacing, so that, when g → 0, a → 0 and when g → ∞, a → ∞.Inserting (1.78) into (1.74) the requirements (1.75) reads:

Ô(g0) ≈g→0

OR(g0)d

Λd= Ĉ(R(g0))d (1.79)

and, by studying the ration Ô(g0)/ (R(g0))d as a function of g0 one finds the

constant Ĉ. In a common numerical calculation on a finite size lattice therewill exist only a narrow region in coupling constant where observables be-have according to (1.79) and hence physics could be extract.

This region is called scaling window, and its extension is limited by the finitesize effect when g0 is too small, that means lattice too fine, so that the physicscould not be seen because is on a bigger scale. On the contrary, when g0 istoo big, lattice is too coarse, physics is on a small scale, hence its fluctuationsare not seen. In a concrete simulation one has to choose from the observablesbehaviour the extension of the scaling window, choosing the good compro-mise between rejecting un-physical signals and taking reasonable ones.

Finally, knowing the constant Ĉ, inside the Scaling Window, we can extractphysical value from numerical calculations as

Ophys. = Ĉ(ΛL)d (1.80)

From this equation we finally notice that in lattice calculation all the physicalquantities are in units of the undetermined mass scale ΛL, due to one is com-monly forced to express lattice results as dimensionless rations of physicalquantities.

Chapter 2

Monte Carlo Method

Some calculus-tricks are quiteeasy. Some are enormouslydifficult. The fools who write thetextbooks of advancedmathematics—and they aremostly clever fools—seldom takethe trouble to show you how easythe easy calculations are. On thecontrary, they seem to desire toimpress you with theirtremendous cleverness by goingabout it in the most difficult way.

Calculus Made EasySilvanus P. Thompson

The origin of the Monte Carlo Method can be traced back to the deter-mination of π in the 18th century, by the algorithm Buffon’s needle, whichconsist in dropping n needless on a set on parallel strips, and from the eval-uation of probability that the needle will lie across a line between two stripsone can determine the value of π. This historical example clarify the belongof the methods to the fields of experimental math [15].

The modern idea of Monte Carlo Simulations that uses calculator was firstintroduced by Neumann, Ulam and Fermi in the period of the Manhattanproject. The key of their introduction was the idea that by generating pseu-dorandom numbers for a long time the desired result can be reached.

Since then, Monte Carlo simulations have become a standard tool in matterphysics and successively the main tool in lattice gauge theories as lattice QCD.

39

2.1. INTRODUCTION 40

In this chapter we describe firstly the general property of the Monte Carlomethod, describing the necessity to move from simple sampling to impor-tance sampling for the determination of the mean value of an observable. Wedescribe the main property of the Markov chains necessary to generate thesystem configurations that goes toward equilibrium and the two updatingalgorithms used in our code, the Metropolis and Wolff algorithm. The MCsimulations are not free from errors, therefore in the last section we discussone of the main sources of error and the statistical analysis necessary to cor-rectly estimate the experimental errors.

In presenting this chapter we will follow the presentation done in the stan-dard books [7, 9, 19], where we remand the interested reader.

2.1 Introduction

To study a system one normally has to calculate the expectation value ofmany interesting physical observables such O, whose value can be calculatedin the Feynman path integral approach of lattice theory as:

〈O〉 =∫

DU O[U] e−S[U]∫DU e−S[U]

. (2.1)

This is an ensemble average, which means that all the elements of the en-semble contribute to the determination of the final mean value, even the oneswith high action value S[U], whose weight is close to zero due to the multiply-ing factor e−S[U]. Calculating this integral analytically is always not possibledue to its high dimensionality and the standard numerical way to calculatethe integral, using mesh technique, are impractical [20].

To understand how expensive would be, let’s consider a lattice with N4 sites,whose has N4 variables. Counting all possible combinations of values in Z(3),

(1, 1e2πi/3, 1e−2πi/3)

we have 3N4

configurations. For a common lattice we may have N = 16 andtherefore 365536 ≈ 1031269 possible configurations. So that the exact evaluationof (2.1) corresponds to summing over all these configurations, clearly an im-possible task.

A way to provide an estimate for this sum can be deduced noticing that whensumming over the configurations it is therefore more important to consider

2.1. INTRODUCTION 41

the configurations with larger weight than those with smaller weight, this isthe fundamental idea behind the so-called importance sampling.

In the importance sampling Monte Carlo method, the huge sum is approx-imate by a comparatively small subset of configurations, which have moreimpact on the observable being estimated, and this can be done by an ade-quate choice of a probability distribution in which larger weight is attributedto more important configurations.

Commonly speaking, given a probability distribution with density ρ(x), theexpectation value of a generic function f (x) is expressed as

〈 f 〉ρ =∫ b

a dx ρ(x) f (x)∫ ba dxρ(x)

(2.2)

with the assumption that x is a random variable defined over the interval(a, b).

In the importance sampling Monte Carlo integration, this expectation value isapproximated by an average over N values

〈 f 〉ρ = limN→∞

1N

N

∑n=1

f (xn), (2.3)

where each variable xn ∈ (a, b) is randomly sampled according to the nor-malized probability density

dP(x) =ρ(x)dx∫ b

a dx ρ(x). (2.4)

This reasoning can be applied to evaluate the ensemble average in (2.1) lead-ing to the expression

〈O〉ρ = limN→∞

1N

N

∑n=1

O[Un], (2.5)

where each variable Un is randomly sampled according to the normalizedprobability distribution density

dP(U) =e−S[U]D[U]∫D[U]e−S[U]

, (2.6)

the so-called Gibbs measure.

Finally, this justify why using a Monte Carlo integration method we can ap-proximate the ensemble average with a sample average.

2.2. MARKOV CHAINS 42

2.2 Markov chains

The field configurations [Un] needed to calculate the ensemble averageshould be generated numerically in a reasonable way, so that they can fol-low the Gibbs measure probability distribution. A common way is using aMarkov process, which consists in starting from a random configuration andgenerating a new one according to some transition probability, so that theconfiguration with larger probability are more often visited. The result, aftermany update or Monte Carlo steps is a memoryless sequence configurations,that is called Markov chain.

C0 → C1 → C2 → . . . . (2.7)

The Markov chains useful to our aim have to satisfy three fundamental re-quirements: irreducibility, aperiodicity and positiveness.

1. irreducible if, starting from an arbitrary configuration Ci, there existsa finite probability of reaching any other configuration Cj after a finitenumber of Markov steps. There exist a finite N such that

P(N)ii = P(N) ([Ci]→ [Cj]

)= ∑

ik

Pii1Pi1i2 . . . PiN−1 j 6= 0 (2.8)

2. aperiodic if, P(N)ii 6= 0 for any N

3. composed of positive states if the mean recurrence time of each stateis finite. If P(n)ii is the probability to get from Ci to Cj in n-steps of theMarkov chain, without reaching this configuration at any intermediatestep, then the mean recurrence time of Cj is

τi =∞

∑n=1

np(n)ii (2.9)

For the Markov chains that satisfying the above requirements, there are twotheorems of fundamental importance:

Theorem 1. If a Markov chain is irreducible and the states are positive andaperiodic, then the limit N → ∞ of (2.8) exists, and is unique; in particularone can show that

limN→∞

p(N)ij = πj (2.10)

2.2. MARKOV CHAINS 43

where πj are numbers which satisfy the following equation:

π > 0,

∑j

πj = 1,

πj = ∑i

πi Pij.

(2.11)

The equation (2.10) states that the limit N → ∞ of PNij is independent ofthe configuration used to start the Markov process. Furthermore the last twoequations express the condition that the system is in equilibrium, with πi theprobability of finding the configuration Ci.

Theorem 2. If a Markov chain is irreducible and its states are positive andif

τ(2)i ≡

∞

∑n=1

n2p(n)ii < ∞ (2.12)

then, the time average

〈O〉N =1N

N

∑i=1

O(Ci) (2.13)

approaches the ensemble average

〈O〉 = 1N ∑i

πi O(Ci) (2.14)

with a statistical uncertainty of order O(1/√

N)

In short, if the three fundamental requirements are satisfied the Markov pro-cess reaches an equilibrium configuration, unique for every start point. Aftera certain number of steps we are sure to get the thermalized configuration tobe used to measure the observables.

The equilibrium is achieved if the transition respect the balance equation:

∑C

P(C → C′)P(C) = ∑C

P(C′ → C)P(C′) (2.15)

which means that the probability for our system to reach the configurationC′ at the step Cn−1 → Cn has to be equal to the probability for being in thatconfiguration and move to any other one at the same step.

2.3. UPDATE ALGORITHM 44

2.3 Update Algorithm

There are a lot of algorithms which are based on Markov process and canbe used to implement a Monte Carlo simulation. However, to be sure that thealgorithms are generating the correct system and they work properly, theyhave to satisfies two conditions: ergodicity and detailed balance.

By ergodicity it is meant that in a Markov process starting from an initialconfiguration it is possible, in principle, to reach any other configuration.

The detailed balance condition is expressed by the fact that the transitionprobability rate in and out of a configuration must be equal.

P(C → C′)P(C) = P(C′ → C)P(C′) (2.16)

For a Markov process to sample the distribution e−S(C), the condition to befulfilled for every pair C and C′ is

P(C → C′)e−S(C) = P(C′ → C)e−S(C′) (2.17)Then, if we consider the probability density

Peq =e−S(C)

∑C e−S(C). (2.18)

.

Using the normalization condition ∑C P(C′ → C) = 1, we see that the de-tailed balance condition implies the validity of the balance condition.

The detailed balance condition dose not determine the transition probabil-ity uniquely, so this freedom has been used to propose different updatingalgorithms. Two of them are presented in the following sections.

2.3.1 Metropolis Algorithm

The Metropolis algorithm was proposed in 1953 by Metropolis et al. [21]and its diffusion is justified by the fact that can be applicable to any system,whose evolution can be reproduced by updating a single variable at time.

The algorithm for generating the sequence of configurations, given the start-ing one C is:

1. Choose some candidate configuration U’ according to some a prioriselection probability P0(U → U′), which has to satisfy the conditionP0(U→ U′) = P0(U′ → U).

2.3. UPDATE ALGORITHM 45

2. Accept the candidate configuration C’ as the new configuration, if thefollowing Metropolis acceptance test is satisfied:

PA(U→ U′) = min(

1,P0(U′ → U)e−S[U′]P0(U→ U′)e−S[U]

)=(

1, e−∆S)

(2.19)

where ∆S = S[U′] − S[U]. Once the new configuration is accepted itsprobability is exactly PA(U→ U′). If a suggested change is not accepted,the unchanged configuration is considered again in the Markov chain.

3. Repeat these previous steps.

It is interesting to note that the possibility to accept configurations thatincrease the system action is the key to reproduce quantum fluctuation.

The reason why the Metropolis algorithm is not suited for the update of morethan a variable at once is that such update produces a large change in theaction, leading to a very small acceptance rate, hence the system would movevery slowly through configuration space.

It is easy to show that the Metropolis algorithm satisfy the detailed balancecondition:

P(U→ U′)e−S[U] = P0(U→ U′)PA(U→ U′)e−S[U]

= P0(U→ U′)min(

1,P0(U′ → U)e−S[U′]P0(U→ U′)e−S[U]

)e−S[U]

= min(

P0(U→ U′)e−S[U], P0(U′ → U)e−S[U′])

= P0(U′ → U)min(

P0(U→ U′)e−S[U]P0(U′ → U)e−S[U′]

, 1

)e−S[U

′]

= P(U′ → U)e−S[U′]

(2.20)

Commonly to describe how the algorithm is able to update the configura-tion, one defines a parameters called acceptance that is the ration between thenumber of accepted variables Nacc and the total number of proposed variablesNacc. This parameter give an estimation of how much change was done bythe algorithm, in formula reads:

Macc =NaccN

(2.21)

In the adaptation of the algorithm for a specific system one has to try toincrease the acceptance without disturbing the thermalisation toward the sys-tem equilibrium. The Metropolis algorithm can be more computationally eco-nomic if the the updating step of a single variable is repeated few times. This

2.3. UPDATE ALGORITHM 46

modification is called multi-hit Metropolis algorithm and was used in ourcode.

2.3.2 Cluster Wolff Algorithm

For many physical system the single flip algorithm, such as the Metropo-lis, are not efficient to study the system close to the transition point, becausethey are affected by the critical slowing down.

To understand what it is, let’s consider a discrete system on a lattice, wereat each node there is a variables as a spin. Starting from an initial config-uration and updating it, we can get close to the transition point, where thevariables present on the lattice, the spins, start to correlate, forming large lat-tice region where all the variable have got the same value. The size of theseregion, so called domains or clusters, is measured by the correlation length ξ.For a single flip algorithm, proposing a new configuration close to the transi-tion point is very difficult, because changing one single element of a domainhas got a high probability of being rejected, and this is more true as strong isthe local interaction.

A solution to this problem was proposed in 1989 by Ulli Wolff [22] usingprevious work made by Swendsen and Wang [23]. Their idea is to look forclusters on the lattice and try to flip all the cluster components all in one step;with this idea they pose the base for a new class of update algorithm, now socalled cluster algorithm.Generally, the algorithm for generating the sequence of configurations, giventhe starting one C is:

1. In the configuration C choose a single variable at random from the lat-tice.

2. Look in turn at each of the neighbours of that variable. If they have thesame value as the initial variable, add them to the cluster with probabil-ity Padd.

3. For each variable that was added in the last step, examine each of itsneighbours to find the ones which have the same value and add each ofthem to the cluster with the same probability Padd. This step should berepeated as many times as necessary until there are no variable left inthe cluster whose neighbours have not been considered for inclusion inthe cluster.

4. Flip the cluster.

2.4. ERROR ANALYSIS 47

The Padd should be choose in relation to the system. It can be proven thatWolff algorithm satisfies the detailed balance condition and the criterion ofergodicity [19].

In our code the Wolff algorithm was used to update only the variable be-longing to Z(3).

2.4 Error Analysis

As in all process of measurements there are inevitably sources of errorssuch as the common physical fluctuation of the observable due to its nature,whose can be addressed by the common error statistical analysis. And it iswell known that, if N is the number of measurements statistically indepen-dent, then the error coming out from from the statistical fluctuations in themean value will be of the order 1/

√N, and can be reduced simply increasing

the statistics.

But in the MC methods there is a characteristic source of statistical errorsthe autocorrelation. When the system is updated from an old configurationto a new one, the new configurations are not all statistically independent, sothat the this is the problems now as autocorrelation.

Other important source of error is coming from the use of finite lattice size,that is manifested by the fact that the ob