UNIVERSITA DEGLI STUDI DI TORINO

DIPARTIMENTO DI FISICA

SCUOLA DI SCIENZE DELLA NATURA

Corso di Laurea Magistrale in Astrofisica e Fisica Teorica

Tesi di Laurea Magistrale

RUNNING COUPLING IN YANG-MILLS THEORY

FROM THE SCHRODINGER FUNCTIONAL

Relatore: Candidato:

Prof. Marco Panero Olmo Francesconi

ANNO ACCADEMICO 2015-2016

ii

Acknowledgement

I would like to thank all the people with whom I shared the last five years

and that have made this thesis possible. In particular:

Thanks to my thesis supervisor prof. Marco Panero. The door to his office

has been always open whenever i ran into problems. Thanks also for all the

patience, encouragement and knowledge shared with me.

Thanks to my family. My father, who knows how to motivate me even in

front of the biggest challenges. My mother, who understands me like no

other can. My grandparents, aunts, uncles and cousins, with whom I shared

countless happy memories.

Thanks to Lorenza, my beautiful girlfriend, who has been beside me in the

most difficult periods, and with whom I shared the best moments of my life.

Thanks to all the friends that I have met along my way. You have made

every day of my life an unique experience and for that I have to be thankful.

Thanks to all the people with whom I shared moments of my life, however

small they may have been. If I am who I am now, it is also thanks to you.

Abstract

The Schrodinger functional provides a convenient way to define the running

coupling of non-Abelian gauge theories, in a wide energy range. In this

scheme, the coupling is extracted from the effective action that is induced on

a system of finite temporal extent, when some fixed boundary conditions are

imposed.

In this work, a novel and computationally efficient method is proposed,

to compute such effective action in lattice simulations of Yang-Mills theory,

by means of a statistical mechanics theorem due to C. Jarzynski.

iii

Contents

Abstract iii

1 Introduction 1

2 Yang-Mills theories 3

2.1 Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 The Yang-Mills Langrangian . . . . . . . . . . . . . . . . . . . 8

2.3 Quantization of non-Abelian gauge theories . . . . . . . . . . . 11

2.4 Key aspect: Asymptotic freedom . . . . . . . . . . . . . . . . 13

2.5 Key aspect: Confinement . . . . . . . . . . . . . . . . . . . . . 15

3 Lattice Regularization 17

3.1 Lattice scalar field theory . . . . . . . . . . . . . . . . . . . . 19

3.2 Lattice gauge field theory . . . . . . . . . . . . . . . . . . . . 21

3.2.1 Naive discretization of fermions . . . . . . . . . . . . . 22

3.2.2 Wilson gauge action . . . . . . . . . . . . . . . . . . . 28

3.3 Path integral formalism on the lattice . . . . . . . . . . . . . . 32

3.3.1 Fermion integration . . . . . . . . . . . . . . . . . . . . 33

3.3.2 Link variables integration . . . . . . . . . . . . . . . . 35

3.4 Scale setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4.1 Physical interpretation of the Wilson loop . . . . . . . 36

iv

CONTENTS v

3.4.2 The Sommer parameter and the lattice spacing . . . . 38

3.5 The continuum limit . . . . . . . . . . . . . . . . . . . . . . . 39

3.5.1 Running of the lattice coupling . . . . . . . . . . . . . 40

3.5.2 The true continuum limit . . . . . . . . . . . . . . . . 41

4 Schrodinger Functional 43

4.1 Lattice formulation . . . . . . . . . . . . . . . . . . . . . . . . 45

5 Jarzynski theorem 50

5.1 Jarzynski relation . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1.1 Application in the SF scheme . . . . . . . . . . . . . . 59

6 Numerical simulation 60

6.1 Monte Carlo simulation in lattice QCD . . . . . . . . . . . . . 63

6.1.1 Heat-bath algorithm . . . . . . . . . . . . . . . . . . . 64

6.1.2 Overrelaxation algorithm . . . . . . . . . . . . . . . . . 67

6.1.3 General workflow of a Monte Carlo simulation . . . . . 68

6.2 Implementation of Jarzynski’s algorithm . . . . . . . . . . . . 70

7 Conclusions 73

Bibliography 74

Chapter 1

Introduction

In quantum field theory the determination of the running coupling is of fun-

damental importance. It encodes how the coupling of the theory varies at

different momentum scale, thus it provides information about the behavior

of the theory at any energy scale.

In Quantum Chromodynamics, the theory that describes the strong nu-

clear, and sub-nuclear, interactions, the determination of the running cou-

pling has been a major issue. Weak-coupling analysis shows that the QCD

coupling tends to zero at high energies, while it becomes large at energy of

the order of a few hundreads of MeV. This implies that for high energy pro-

cesses the QCD perturbative computations are reliable, while they become

unreliable at the energy scale of the hadronic spectrum. A precise determina-

tion of the running coupling, using non-perturbative techniques, is therefore

very important for the study of processes mediated by strong interaction at

low energies.

We are going to propose a new method to evaluate the running cou-

pling using the Jarzinski theorem to compute the coupling defined in the

Schrodinger functional scheme.

1

CHAPTER 1. INTRODUCTION 2

In the present work we will describe the fundamental aspects of non-

Abelian quantum field theories, focusing our attention on some of the char-

acterizing properties of the theory, asymptotic freedom and confinement ; then

we will introduce an inherently non-perturbative approach to the theory, the

lattice regularization. Having done so we will define a renormalized coupling

by means of the Schrodinger functional ; and finally we will propose a new,

and more efficient, way to evaluate the aforementioned coupling using the

Jarzynski relation. Lastly we will describe the techniques that are currently

used in our lattice QCD simulations, and we will give a glimpse on our im-

plementation of the Jarzynski relation.

Chapter 2

Yang-Mills theories

Non-Abelian gauge theories are the core of the theoretical formulation of nu-

clear interactions. These theories are based on the assumption of invariance

under a local gauge transformation, that is more general than the simple

U(1) gauge invariance of quantum electrodynamics.

In early 1954 Yang and Mills extended the concept of gauge theory for

Abelian groups to non-Abelian groups to provide an explanation for strong

interactions. They studied the SU(2) group of isospin rotation, and inter-

preted the resulting vector field as the strongly-interacting vector meson in

analogy with the photons in quantum electrodynamics. The idea by Yang

and Mills was criticized by Pauli, as the quanta of the Yang and Mills field

must be massless in order to maintain gauge invariance, and it seemed that

any such particles should have been already detected.

The theory was set aside until the early 1960’s when the idea of particle

acquiring mas through a spontaneous symmetry breaking mechanism was

formalized, initially by Jeffrey Goldstone, Yoichiro Nambu, and Giovanni

Jona-Lasinio in regards of the global symmetry breaking, and then by Higgs,

3

CHAPTER 2. YANG-MILLS THEORIES 4

Englert and Brout, and Guralnik, Hagen and Kibble, for local symmetries.

The massless vector meson proposed by Yang and Mills could then acquire

mass and the theory finally found physical relevance. In the same years

non-Abelian gauge theories and their quantization continued to be studied

mathematically by Feynman, Fadeev and Popov and De Witt. The final

proof of physical relevance was obtained by ’t Hooft in 1971, as he proved

that the theory could be renormalized so the divergences could be eliminated

and actual physics could be extrapolated from the theory.

Since then Yang-Mills theories have been proven successful in the formu-

lation of electroweak unification and quantum chromodynamics, respectively

described by SU(2) × U(1) group and SU(3) group. Therefore such theo-

ries play a key role in the definition of the interactions of the Standard Model.

In this chapter a brief description of Yang-Mills theories will be given,

focusing on the fundamental concept of gauge invariance and the peculiar

aspects of these theories. Quantization and renormalization of Yang-Mills

theories in the continuum will be just mentioned since the attention will be

focused on the lattice regularization and the definition of a renormalizable

probe (the Schrodinger functional); appropriate references will be given in

the respective chapters.

2.1 Gauge Invariance

Let us start by studying the gauge invariance of QED; it is the fundamental

principle that determines the form of the Lagrangian.

The theory is required to be invariant under the following transformation

CHAPTER 2. YANG-MILLS THEORIES 5

of the Dirac field ψ(x)

ψ(x)→ eiα(x) ψ(x) (2.1)

This transformation is a simple phase rotation by an angle α(x) that varies

from point to point in spacetime.

The objective is to write the most general Lagrangian invariant under

this transformation. As long as we consider terms that have no derivatives

we can simply write the same terms that are invariant under a global phase

rotation, for example the fermion mass term mψψ is invariant under a global

transformation and also under local ones.

When we try to write terms including derivatives, problems arise. The

derivate of ψ(x) in the direction nµ is defined by

nµ∂µψ = limε→0

1

ε[ψ(x+ εn)− ψ(x)] (2.2)

This definition in a theory with local phase invariance is not very sensible,

since the fields ψ(x) and ψ(x+ εn) undergo different transformations under

the symmetry (2.1).

One must introduce a factor that compensates the difference in phase

transformation from one point to the next; the simplest way to do so is to

define a scalar quantity U(y, x) that has the transformation law

U(y, x)→ eiα(y)U(y, x)e−iα(x) (2.3)

We also assume that at zero separation this quantity takes the value U(x, x) =

1. With this definition the objects ψ(y) and U(y, x)ψ(x) have the same trans-

formation law and can be subtracted in a meaningful way. A proper covariant

CHAPTER 2. YANG-MILLS THEORIES 6

derivative can be defined as follows

nµDµψ = limε→0

1

ε[ψ(x+ εn)− U(x+ εn, x)ψ(x)]. (2.4)

To make this definition explicit, we need an expression of U(y, x) at infinites-

imally separated points:

U(x+ εn, x) = 1− i e ε nµAµ(x) +O(ε2). (2.5)

Where e is an arbitrary extracted constant, its role is to “tune” the intensity

of the interaction hence the obvious identification with the electric charge of

standard electromagnetism. Aµ is a new vector field that is the infinitesimal

limit of a comparator and is called a connection. The covariant derivative

then takes the form

Dµψ(x) = ∂µψ(x) + ieAµψ(x). (2.6)

By inserting (2.5) into (2.3) one finds the explicit transformation law for the

Aµ field

Aµ(x)→ Aµ(x)− 1

e∂µα(x). (2.7)

We have now recovered most the most familiar aspect of QED, namely

the gauge transformation law for the field Aµ that arises directly from the

postulate of local phase rotation symmetry.

To complete the construction of a locally invariant Lagrangian we must

find a kinetic-energy term for the field Aµ, a term that depends only on the

field itself and its derivatives. To do so we consider than if a field has the

local transformation law (2.1), then also its covariant derivative has the same

transformation law and the same conclusion holds for the commutator of the

CHAPTER 2. YANG-MILLS THEORIES 7

covariant derivative:

[Dµ, Dν ]ψ(x)→ eiα(x)[Dµ, Dν ]ψ(x). (2.8)

By inserting (2.6) in (2.8) one can give an explicit expression of the commu-

tator

[Dµ, Dν ]ψ = [∂µ, ∂ν ]ψ + ie([∂µ, Aν ]− [∂ν , Aµ])ψ − e2[Aµ, Aν ]ψ

= ie(∂µAν − ∂νAµ)ψ(2.9)

On the right-hand side of (2.8) the factor ψ(x) accounts for the entire

transformation law therefore the commutator acts on the field as a multi-

plicative constant rather than a differential operator and it can be rewritten

as:

[Dµ, Dν ] = ieFµν . (2.10)

As a result Fµν must be invariant and can be used to build the invariant

term of the Lagrangian that depends only on the field Aµ. Fµν physically

represents the electromagnetic field tensor from which one can derive the

values of classical electric and magnetic fields.

We have now all the invariant terms that can appear in an invariant

Lagrangian under local gauge transformation:

L = ψ(i /D)ψ − 1

4(Fµν)

2 −mψψ. (2.11)

We have obtained some remarkable results just from postulating a local

gauge invariance. There must be a vector field in the theory, that is the

electromagnetic vector potential and its existence arises from the need of

defining a proper derivative, the covariant derivative. And the most general

CHAPTER 2. YANG-MILLS THEORIES 8

Lagrangian that is invariant under local gauge transformation is the Maxwell-

Dirac Lagrangian that is the basis of quantum electrodynamics.

2.2 The Yang-Mills Langrangian

Following the line traced in the previous section, we will show now how to

extend the concept of gauge invariance to invariance under any continuous

symmetry group, as proposed by Yang and Mills. The discussion will be

focused on the SU(2) symmetry group since it is the simplest non-Abelian

group and at the end a brief generalization to any arbitrary local symmetry

will be given.

We should now consider a doublet of Dirac fields,

ψ =

ψ1(x)

ψ2(x)

, (2.12)

which transform as a two-component spinor:

ψ → ψ′ = exp

(iαi

σi

2

)ψ. (2.13)

Here σi are the usual Pauli matrices and the sum over i is implied.

In analogy with the previous section we postulate that the theory must be

invariant under local transformations:

ψ(x)→ ψ′(x) = exp

(iαi(x)

σi

2

)ψ = V (x)ψ(x). (2.14)

And again following the path we define a comparator that has to be a 2× 2

unitary matrix since ψ is now a 2 component object and we set U(x, x) = 1.

CHAPTER 2. YANG-MILLS THEORIES 9

The transformation law of U(x, y) follows from (2.14),

U(y, x)→ V (y)U(y, x)V †(x). (2.15)

Near U = 1 we can give an expansion of U in terms of the Hermitian gener-

ators of SU(2), therefore we can write for an infinitesimal separation

U(x+ εn, x) = 1+ igεnµAiµσi

2+O(ε2). (2.16)

Here g is an arbitrary extracted constant that has the same role as e, the

electrical charge, has in QED, it is the coupling of the interaction.

Inserting this expansion in the covariant derivative definition (2.4) we find

the covariant derivative associated with local SU(2) symmetry:

Dµ = ∂µ − igAiµσi

2. (2.17)

Having defined a covariant derivative all the terms in the Lagrnagian con-

taining derivative of the field ψ and obviously the mass term are invariant

under local transformation of the SU(2) symmetry.

We still have to construct the analogue of the electromagnetic field tensor,

the gauge invariant term that depend only on Aiµ, to write the most general

Lagrangian. To do so, again, we use the transformation law of covariant

derivatives to express the transformation law of the derivative commutator

[Dµ, Dν ]ψ(x)→ V (x)[Dµ, Dν ]ψ(x) (2.18)

And again the commutator is not a differential operator but rather a multi-

CHAPTER 2. YANG-MILLS THEORIES 10

plicative factor, this time in the form of a 2× 2 matrix acting on ψ

[Dµ, Dν ] = −igF iµν

σi

2(2.19)

This time however due to the non-Abelian structure of the group the term

[Aµ, Aν ] no longer vanishes and the field tensor acquires a more complex

structure

F iµν

σi

2= ∂µA

iν

σi

2− ∂νAiµ

σi

2− ig

[Aiµ

σi

2, Ajν

σj

2

](2.20)

This definition of the field tensor is no longer gauge invariant since there are

three field strengths, however a gauge-invariant kinetic energy term for Aiµ

can be easily formed using a combination of the field strengths. For example,

L = −1

2Tr

[(F iµν

σi

2

)2]

= −1

4(F i

µν)2. (2.21)

We can now write the famous Yang-Mills Lagrangian by adding this

gauge-field Lagrangian to the Dirac Lagrangian with the derivative of ψ

replaced by the covariant derivative

L = ψ(i /D)ψ − 1

4(F i

µν)2 −mψψ. (2.22)

This Lagrangian despite being very similar to the QED Lagrangian describes

a wider and more complex set of interactions due to the non-Abelian nature

of the SU(2) group.

It is clear that if the particular behavior of the theory relies on the group’s

algebra to extend the discussion to any local symmetry one has to simply

express the fermionic field in a proper representation of the group and define

CHAPTER 2. YANG-MILLS THEORIES 11

the gauge field tensor, acording to (2.20), as:

F aµν = ∂µA

aν − ∂νAaµ + gfabcAbµA

cν , (2.23)

where the commutation law [Aµ, Aν ] has been generalized by replacing the

Pauli matrices with the group generator of a general group of symmetry

σi

2→ ta (2.24)

and the group commutation relation

[ta, tb] = ifabctc (2.25)

has been used to simplify the expression.

Again the form of the interaction is strictly related to the specified group

of symmetry; in the covariant derivative non-linear terms are proportional

to the ta generators and in the field tensor term they are proportional to the

group structure constants fabc.

2.3 Quantization of non-Abelian gauge theo-

ries

We have now defined a gauge-invariant Lagrangian for non-Abelian sym-

metries, this is however only the first step to define a proper quantum filed

theory describing real interactions between particles. We will not give a com-

plete description of the quantization process but rather an overview, focusing

on the challenges and limits of the analytical techniques.

The quantization of non-Abelian field theory is obtained using the formal-

CHAPTER 2. YANG-MILLS THEORIES 12

ism of Feynman path integrals. In this formalism, the fundamental object of

discussion is the generating function (the analogue of the partition function

of statistical mechanics) that can be expressed as

Z =

∫D[ψ, ψ] D[A] ei

∫d4x L. (2.26)

The computation of any n-point function reduces to the evaluation of

derivatives of the generating function, with respect to appropriate source

terms (omitted in the previous expression).

A correct evaluation of the functional integration of eq. (2.26) is not triv-

ial. There are two important aspects to take under consideration: the first

is that, due to the invariance of the Lagrangian under local gauge transfor-

mation, the integration over the gauge field degrees of freedom is not well

defined as the path integrals overcounts field configurations corresponding

to the same physical state. A solution has been proposed by Fadeev and

Popov by fixing the gauge, effectively evaluating the integral over one gauge

field configuration, and adding an additional term to the action to compen-

sate; this leads to the introduction of nonphysical fields in the discussion the

so-called ghost fields, they do not affect the final result as their effects get

removed along the way, but must be carefully considered. The other aspect

is that the effective evaluation of the integration can be achieved only in the

perturbative expansion of the exponential where the coupling g is taken to

be small, this leads to the representation of the theory in terms of the famous

Feynman diagrams; as we will see this assumption limits the effectiveness of

the analytical discussion to the description of high energy phenomenology.

In the next chapter we will discuss the lattice regularization of gauge

theories that, due to its nature, is inherently free of this problems. But

CHAPTER 2. YANG-MILLS THEORIES 13

before, let us discuss two of the characterizing aspects of of non-Abelian

gauge theories.

2.4 Key aspect: Asymptotic freedom

Asymptotic freedom is probably the most important aspect of non-Abelian

gauge theories, it is in fact an exclusive property of such theories (in four

dimensions) and has been the key to show how these theories could be the

right theoretical tool to explain the strong nuclear interaction.

The first appearance of asymptotic freedom is unrelated to the current

discussion, in fact in 1968 Bjorken proposed a new idea regarding the study

of deep inelastic scattering of light on strong interacting particles. This idea,

the effects of which are referred to as the Bjorken scaling, consists in the

assumption that for high enough momenta the hadron constituents behave

as point-like free objects. At the time there was however no theory that

showed such behavior but the first experimental data confirmed the Bjorken

prediction. This set off an urgent search in the theoretical physics community

for asymptotically free quantum field theories.

In 1973 David J. Gross, Frank Wilczek [1] and independently David

Politzer [2] published the discovery of the asymptotic freedom in non-Abelian

gauge theories; for the discovery they have been awarded with the Nobel Prize

in physics in 2004. The formalization of asymptotic freedom derives from the

study of the beta function in this class of theories. The beta function encodes

how, in a given physical process, the coupling (g) depends on the momentum

scale of the process (µ);

β(µ) =∂g(µ)

∂ log(µ). (2.27)

Its computation can be achieved in perturbation theory evaluating the multi-

CHAPTER 2. YANG-MILLS THEORIES 14

Figure 2.1: Experiemntal results of the evolution of the αS(Q) coupling. Imagetaken from ref. [3]; appropriate data reference also present in [3].

loop correction using Feynman diagrams. For a SU(N) gauge symmetry

group one finds that at one-loop correction the beta function is:

β(µ) = β0g3 +O(g5) = − g3

(4π)2

(11

3N − 2

3nf

)+O(g5) (2.28)

where nf is the number of fermion flavors in the chosen representation. It

is clear that this beta-function describes a theory whose coupling decreases

with increasing energy scale as shown in Figure 2.1, at least for nf < 11·N/2,

and therefore has the property of being asymptotically free.

There is then a direct connection between an observed phenomenon and

the theory, this leads to the modern formulation of QCD as a theory of quarks

and gluons with a fundamental symmetry, characterized by the non-Abelian

group SU(3), as the key to its asymptotically free nature. QCD is, by now,

well established as the fundamental theory of strong interactions for quarks,

gluons and the observed hadrons.

CHAPTER 2. YANG-MILLS THEORIES 15

2.5 Key aspect: Confinement

Confinement, or more correctly color confinement, is the property of the

particles with color charge of not existing as asymptotic, isolated states,

therefore such particles, quarks and gluons, cannot be observed directly. The

basic explanation of the phenomenon is that if two quarks are separated from

one another, the attractive force between the two tends to a constant value,

the so-called string tension.

This kind of behavior is incredibly different from any other force known in

nature. Let us take as an example two electrically reverse charged particles,

when held in a fixed position an electric field “fills” all the space around them

and if we start to separate the two charges the electric field diminishes quickly

and we are able to separate the charges, but if we take a quark-antiquark pair

with reverse color charge and start to take them apart the gluon field forms a

narrow tube between the two, and the more we separate the quark-antiquark

pair, the more energy the gluon field acquires and it would require an infinite

amount of energy to separate the two.

Furthermore, it is possible to separate the quark-antiquark pair enough

qq

q q

q q

q q q q q q

q q

Figure 2.2: Graphical schematization of the string breaking mechanism. Imagetaken from ref. [4].

CHAPTER 2. YANG-MILLS THEORIES 16

to make it more energetically favorable to generate another quark-antiquark

pair in between rather than to allow the tube to extend further, as graphi-

cally summarized in Figure 2.2. This scenario is what is thought to happen

in particle accelerator collisions: when a free quark should be produced, but

a jet of color neutral mesons and baryons clustered together is observed.

Another way of expressing this concept is that at large distances, or

equivalently low momenta, the coupling parameter grows. Having a large

coupling means that the usual perturbative approach cannot be used in this

region and this is the reason why there is no analytic proof of confinement.

An inherently non-perturbative approach to the theory must be taken and

that is what we are going to discuss in the next chapter.

Chapter 3

Lattice Regularization

The need to develop a non-perturbative tool to study strongly interacting

non-Abelian theories led to the formulation of QCD on a spacetime lat-

tice. Originally proposed by K.G. Wilson [5] in 1974, this formulation has

deep connection with statistical mechanics and statistical field theory, one

can therefore use a number of techniques borrowed from these fields such as

strong coupling expansions or numerical simulations.

Lattice field theory is today a well-established field of research with scien-

tist active worldwide; thanks to conceptual, algorithmic and computer-power

progress, numerical lattice QCD computations are now in a precision era.

The formulation of QCD on a lattice is not to be taken as a simple approx-

imation of the real phenomenon, it is instead the mathematically rigorous

non-perturbative definition of QCD. It provides a natural regularization of

the functional integral of QCD and a way to compute expectation values of

physical observables that does not require any assumption of perturbative

nature. The functional integrals appearing in the continuum formulation are

replaced by a discrete collection of ordinary integrals, that can be estimated

17

CHAPTER 3. LATTICE REGULARIZATION 18

numerically by means of Monte Carlo integration, and, once extrapolated

to the appropriate physical limit, these calculations provide a systematically

improvable estimate of expectation values of physical observables.

The continuum version of the theory is obtained in the limit in which the

intrinsic momentum cutoff of the lattice is taken to infinity, in other words

when the lattice spacing is sent to zero. In this limit the discretization arti-

facts decouple form the low-energy physics and the continuum theory arises

as a good low-energy effective description of the lattice theory.

The first step to discretize field theories on a lattice is obviously to de-

fine the lattice itself. Various choices can be made, from regular lattices to

completely random ones, but since the lattice topology has a direct influence

only on the discretization artifacts and those decouple from the theory in the

continuum limit, every choice of lattice is equivalent, therefore it is advisable

to choose a lattice on which the discretization procedure is the easiest. Such

lattice is a regular hypercubic grid Λ of spacing a in Euclidean spacetime

Λ = (x1, x2, x3, x4) = (an1, an2, an3, an4) : n1, . . . , n4 ∈ Z. (3.1)

This lattice definition has two direct consequences, the first one is that since

the lattice is a Euclidean one, we will discretize the Euclidean version of field

theories. Despite appearing as an odd choice, this enables us to interpret

our field theory as a statistical mechanics model, therefore we can use all

the methods developed in the latter field to study quantum field theories.

However we will always be just one Wick rotation away from the common

Minkowskian theory.

The other and most important consequence of the lattice discretization

is that the Fourier decomposition of any general function f defined on the

CHAPTER 3. LATTICE REGULARIZATION 19

lattice involves periodic momenta. If a function f is defined only on coor-

dinates that are integer multiples of the lattice spacing a it is trivial to see

that

f(an) =1

2π

∫dkf(k)eikan (3.2)

is invariant under the replacement: k → k + 2π/a. This applies to all

the directions, so in a four-dimensional hypercubic lattice the Brillouin zone

of the lattice is a periodic cell of size (2π/a)4. This means that the lattice

discretization of the theory automatically introduces a maximum momentum

in each direction, by doing so we are actually regularizing the theory in the

ultraviolet (UV).

3.1 Lattice scalar field theory

Having defined the lattice let us now use a simple scalar theory as an instruc-

tive example to show the basic aspects of discretization.

As a tradition in field theory we will discuss the massive φ4 scalar field

theory, described by the Euclidean action:

S =

∫d4x

[1

2(∂µϕ)2 +

1

2µ2ϕ2 +

1

4!λϕ4

](3.3)

In this basic example the discretization process consists in trading the con-

tinuum field φ with a lattice field defined on the lattice nodes, by doing so

the lattice action can be easily expressed as

SLatt =∑x∈Λ

[1

2m2φ2(x) +

1

4!λφ4(x)−

4∑µ=1

φ(x)φ(x+ aµ)

](3.4)

where we have defined the dimensionless lattice field φ(x) = aϕ(x), and the

CHAPTER 3. LATTICE REGULARIZATION 20

dimensionless mass parameter m2 = 4+a2µ2. By comparing the two actions,

it is clear that the integration over x has been mapped to a sum over all the

lattice nodes, and the kinetic term gets mapped to a sum of products of the

field over nearest-neighbor sites.

If the definition of a lattice action gives us the benefit of trading integrals

for sums, it is the study of the partition function in the quantum theory, using

the Feynmann path integral formalism, that gives us the biggest advantages

over the continuum theory.

The partition function is defined as

Z =

∫Dφ e−

∫d4x[ 1

2(∂µϕ)2+ 1

2µ2φ2+ 1

4!λφ4] (3.5)

here the integration is the common functional integration over all the field

configurations, which is a divergent expression if no regularization is imposed.

One can make the expression mathematically well-defined by using di-

mensional regularization of Feynman diagrams. This approach is, however,

limited to perturbation theory. The lattice allows us to formulate field the-

ory beyond perturbation theory, which is essential for strongly interacting

theories, therefore we can define a lattice partition function as

ZLatt =∏x

∫ +∞

−∞dφ(x)e−SLatt . (3.6)

Here the integration is done using an ordinary multiple integral, rather than

a functional one, over all values of the field at all lattice points, therefore it is

possible to evaluate the lattice partition function in a non-perturbative way.

With this definition of the partition function in mind, we can define the

CHAPTER 3. LATTICE REGULARIZATION 21

expectation value of a generic observable O as

〈O〉 =

∫ ∏x dφ(x) O exp(−Sφ)

ZLatt. (3.7)

The direct determination of expectation values on the lattice is still an

incredibly heavy computational task, but the lattice approach enables the

use of computational techniques developed in statistical mechanics, such as

the Monte Carlo method, which allows the evaluation of these observables

in a computationally efficient way. An overview of these techniques applied

to lattice gauge theories will be proposed in the last chapter of the present

work. We will now proceed to the formalization of gauge theories on the

lattice.

3.2 Lattice gauge field theory

Moving on to the lattice discretization of gauge fields theories we have to

focus our attention on the two fields necessary to describe these theories, the

fermionic field ψ and the gauge field Aµ. For the purpose of this work, only a

brief description of the discretization of the fermion field will be given, since

from this point forward we will discuss only pure gauge field theories, the

so-called Yang-Mills theories, therefore we will focus our attention on the

gauge field and the definition of a gauge invariant action. This decision is

mainly due to reasons of computational power: the simulation of pure gauge

theories is much more efficient; also, the Schrodinger functional places no

constraints on the presence of fermions which may be added in future works.

CHAPTER 3. LATTICE REGULARIZATION 22

3.2.1 Naive discretization of fermions

In this section we introduce the so-called naive discretization of the fermion

action. Despite not being a suitable discretization, since lattice artifacts are

present and can be eliminated using more advanced techniques, it serves to

present the basic idea and, more importantly, to discuss the representation

of the lattice gluon field which differs from the continuum form. We show

that on the lattice the gluon fields must be introduced as elements of the

gauge group and not as elements of the algebra, as is done in the continuum

formulation.

Discretization of free fermions

As already shown in the scalar field theory the translation of the continuum

matter field is straightforward, we simply associate a spinor to every node of

the lattice, hence our fermionic degrees of freedom are

ψ(n), ψ(n), n ∈ Λ. (3.8)

In the continuum the action SF for a free fermion is given by the expression

SF [ψ, ψ] =

∫d4x ψ(x)(γµ∂µ +m)ψ(x) (3.9)

When formulating this action on the lattice we have to discretize the integral

over space–time as well as the partial derivative. As we did with the scalar

theory, the integral discretization is implemented as a sum over all the lattice

nodes n, and the partial derivative is discretized as follows:

∂µψ(x) =1

2a(ψ(n+ µ)− ψ(n− µ)). (3.10)

CHAPTER 3. LATTICE REGULARIZATION 23

We can now give an expression of the naively discretized fermion action as

SF [ψ, ψ] = a4∑n∈Λ

ψ(n)

(4∑

µ=1

γµψ(n+ µ)− ψ(n− µ)

2a+mψ(n)

). (3.11)

This form is clearly not gauge invariant and is a good starting point for the

introduction of gauge fields.

Gauge fields as link variables

In the first chapter of this work we showed that requiring the invariance under

the action of a local symmetry group enforces the introduction of gauge fields.

Here we implement the same transformation on the lattice by choosing an

element Ω(n) of SU(N) for each lattice node n and transforming the fermion

field according to

ψ(n)→ ψ′(n) = Ω(n)ψ(n), ψ(n)→ ψ′(n) = ψ(n)Ω†(n). (3.12)

As in the continuum case, it is trivial to show that the mass term is invari-

ant under this transformation, for the discretized derivative term however it

is not. Considering the term

ψ(n)ψ(n+ µ)→ ψ′(n)ψ′(n+ µ) = ψ(n)Ω†(n)Ω(n+ µ)ψ(n+ µ), (3.13)

this is clearly not gauge-invariant, but if we introduce a field Uµ(n) with a

directional index µ we can build a gauge-invariant term

ψ′(n)U ′µ(n)ψ′(n+ µ) = ψ(n)Ω†(n)U ′µ(n)Ω(n+ µ)ψ′(n+ µ) (3.14)

CHAPTER 3. LATTICE REGULARIZATION 24

if we define the gauge transformation of the new field by

Uµ(n)→ U ′µ(n) = Ω(n)Uµ(n)Ω†(n+ µ). (3.15)

To make the fermionic action gauge-invariant we introduce the gauge

field Uµ(n) as an element of the group which transforms as given in (3.15).

These matrix-valued variables are oriented and are attached to the links of

the lattice and thus are often referred to as link variables.

Figure 3.1: Graphical representation of the Uµ(n) and U−µ(n) link variables.Image taken from ref. [6].

Uµ(n) lives on the link which connects the sites n and n+ µ. Since these

variables are oriented, we can also define link variables that point in the

negative µ direction. In particular, we can define a link variable that points

towards n− µ starting from n as

U−µ(n) ≡ U †µ(n− µ). (3.16)

These variables, however are not independent link variables, but are in-

troduced for notational convenience.

Having introduced these link variables and their properties under gauge

transformation we can give a gauge-invariant expression of the naive fermion

action for fermions in an external gauge field U as:

SF [ψ, ψ, U ] = a4∑n∈Λ

ψ(n)

4∑µ=1

γµUµ(n)ψ(n+ µ)− U−µ(n)ψ(n− µ)

2a+mψ(n)

.

(3.17)

CHAPTER 3. LATTICE REGULARIZATION 25

Link variables’ relation with continuum gauge field

Let us now discuss the link variables in more detail and see how they can

be related to the algebra-valued gauge fields of the continuum formulation.

We have introduced Uµ(n) as the link variable connecting the points n and

n+µ. The gauge transformation properties are consequently governed by the

two transformation matrices Ω(n) and Ω†(n + µ). In the continuum theory

an object with such transformation properties is known, the so called gauge

transporter. It is the path-ordered exponential integral of the gauge field Aµ

along some curve Cxy connecting two points x and y.

G(x, y) = P exp

(i

∫Cxy

A · ds

)(3.18)

These continuum gauge transporters transform under gauge transformation

as

G(x, y)→ Ω(x)G(x, y)Ω†(y), (3.19)

these transformation properties are the same as for the link variables if we

consider n and n + µ as the end points of a path. Therefore we interpret

the link variable as the lattice version of the gauge transporter, and we

introduce an algebra-valued lattice gauge field Aµ(n), a discretized version of

the continuum one, by which we can give an expression of the link variable

as:

Uµ(n) = exp(iaAµ(n)), (3.20)

where we have approximated the integral along the path by aAµ(n) as we

are implicitly considering an averaged valued field along all the path from

a node to the next one. This approximation is good to O(a) and no path

ordering is necessary.

CHAPTER 3. LATTICE REGULARIZATION 26

It is important to note that the group-valued link variables are not merely

an auxiliary construction to sneak the Lie algebra-valued fields of the contin-

uum into the lattice formulation. Instead, the group elements are considered

as the fundamental variables which are integrated over in the path integral.

This change from algebra-valued to group-valued fields has important conse-

quences. In particular, the role of gauge fixing changes considerably.

Continuum limit of the fermionic action

It is now possible to connect this lattice fermion action to the continuum

one, by taking the limit for a→ 0 of the lattice action. To do so we expand

the gauge transporters for small a,

Uµ(n) = 1+ iaAµ(n) +O(a2), (3.21)

U−µ(n) = 1− iaAµ(n− µ) +O(a2) (3.22)

and insert these expressions in the definition of the lattice action obtaining

SF [ψ, ψ, U ] = S0F [ψ, ψ] + SIF [ψ, ψ, A] (3.23)

where S0F denotes the free part of the action. The interaction part can be

written as

SIF [ψ, ψ, A] = ia4∑n∈Λ

4∑µ=1

ψ(n)γµAµ(n)ψ(n) +O(a). (3.24)

Taking now the limit for a→ 0 we recover the exact form of the continuum

action.

This simple definition of the action has, however, a great inconvenience,

CHAPTER 3. LATTICE REGULARIZATION 27

known as fermion doubling. The doubling problem arises from our naive

lattice discretization of the Dirac operator and can be easily understood in

the free limit (Uµ(n) = 1 ∀µ,∀n), in which the momentum-space expression

of the lattice Dirac operator is

a4

∫B

d4k

(2π)4˜ψ(−k)

[m+

i

a

4∑µ=1

γµ sin(akµ)

]ψ(k). (3.25)

Of particular interest is the case of massless fermions, thus taking m = 0

in the previous expression one finds that the integrand is vanishing not only

when all components of k are 0, but also when any number of them is equal to

π/a. This means that the Dirac propagator, that in the continuum limit has

one pole at kµ = (0, 0, 0, 0) that represent the single fermion that is being

described, in the lattice discretization has 2d = 16 poles, of which 15 are

nonphysical ones representing the so-called doublers

kµ = (π/a, 0, 0, 0), (0, π/a, 0, 0), . . . , (π/a, π/a, π/a, π/a). (3.26)

Wilson fermions

Wilson fermions provide a lattice discretization of the continuum Dirac oper-

ator that gets rid of the doublers, by giving them a mass at the lattice cutoff

scale. By doing that, when taking the continuum limit, the doublers become

infinitely heavy, hence they decouple from the actual physics of the system.

The lattice discretization of the Dirac operator using the Wilson fermions

CHAPTER 3. LATTICE REGULARIZATION 28

reads as follows

a4∑n∈Λ

ψ(n)

[mψ(n) +

4∑µ=1

Uµ(n)ψ(n+ µ)− U †µ(n− µ)ψ(n− µ)

2a

− r4∑

ν=1

Uν(n)ψ(n+ ν)− 2ψ(n) + U †ν(n− ν)ψ(n− ν)

2a2

], (3.27)

The extra term, the so-called Wilson term, is exactly what we need. By

taking again the momentum-space expression of the lattice Dirac operator

one finds that for components with kµ = 0 it simply vanishes, while for each

component with kµ = π/a it provides an extra contribution 2/a to the mass

term, and the total mass of the doublers is given by

m+2l

a, (3.28)

where l is the number of momentum components with kµ = π/a. In the

limit a → 0 the doublers become very heavy and decouple from the theory,

as needed.

3.2.2 Wilson gauge action

We have introduced the link variables as the basic quantities for putting the

gluon field on the lattice. Now we construct a lattice gauge action in terms

of the link variables and show that in the a → 0 limit it approaches its

continuum counterpart.

Gauge-invariant objects built with link variables

Before defining an invariant action it is necessary to define which gauge-

invariant object can be built on the lattice using only link variables. We

CHAPTER 3. LATTICE REGULARIZATION 29

will start by considering a string of link variables along a path P connecting

points n0 and n1, and defining the ordered product

P [U ] = Uµ0(n0)Uµ1(n0 + µ0) . . . Uµk−1(n1 − µk−1) =

∏(n,µ)∈P

Uµ(n). (3.29)

Considering the transformation properties of the single link variable, it fol-

lows that for two subsequent link variables on the path, one ending at n

the other starting form n, the two transformation matrices Ω†(n) and Ω(n)

cancel each other at n. That happens for every node in the path that is

connected to two links, i.e. all but the ending ones, thus the product P [U ]

transforms according to

P [U ]→ P [U ′] = Ω(n0)P [U ]Ω†(n1). (3.30)

Like for the single link term, from such a product of link variables P [U ] a

gauge-invariant quantity can be constructed by attaching quark fields at the

starting point and at the end point,

ψ(n0)P [U ]ψ(n1). (3.31)

An alternative way of constructing a gauge-invariant product of link vari-

ables that does not need the presence of fermionic fields is to choose for the

path P a closed loop L and take the trace,

L[U ] = Tr

∏(n,µ)∈L

Uµ(n)

(3.32)

Under gauge transformation only the matrices at the end point n0 where

the loop is rooted remains, but these matrices cancel when taking the trace,

CHAPTER 3. LATTICE REGULARIZATION 30

because of the invariance of the trace of a product under cyclic permutations

of the factors.

L[U ′] = Tr

Ω(n0)∏

(n,µ)∈L

Uµ(n) Ω†(n0)

= Tr

∏(n,µ)∈L

Uµ(n)

= L[U ].

(3.33)

Thus the trace over any closed loop of link-variables is a gauge-invariant

object. We will now use these objects to build a gauge invariant action for

the gluon field and later we will also show how these objects serve as physical

observables.

The gauge action

To define the gluon action it is sufficient to use the smallest non trivial

closed loop on the lattice, the so-called palquette. This variable Uµν(n) is

the product of only four link variables defined as

Uµν(n) = Uµ(n)Uν(n+ µ)U †µ(n+ ν)U †ν(n). (3.34)

We can now define the Wilson gauge action as the sum over all plaquettes,

with each plaquette counted with only one orientation. This sum can be

realized by a sum over all the lattice nodes n, combined with a sum over the

Lorentz indices 1 ≤ µ < ν ≤ 4,

SG[U ] =2

g2

∑n∈Λ

∑µ<ν

Re Tr[1− Uµν(n)]. (3.35)

For later convenience, we also introduce the Wilson parameter β = 2Ng2 .

We will now show that in the limit of a → 0 the Wilson gauge action

CHAPTER 3. LATTICE REGULARIZATION 31

matches the continuum action. For establishing the correct limit we need

to expand the link variables for small a, this time however we have to deal

with products of link variables. It is useful to invoke the Baker-Campbell-

Hausdorff formula for the product of exponential matrices:

exp(A) exp(B) = exp

(A+B +

1

2[A,B] + . . .

), (3.36)

where A and B are arbitrary matrices and the orders larger than 2 are omit-

ted.

By expanding the product of link variables in the plaquette definition and

performing a Taylor expansion for the fields

Aν(n+ µ) = Aν(n) + a∂µAν(n) +O(a2) (3.37)

we obtain the plaquette expansion for small a that reads as follows

Uµν(n) = exp(ia2(∂µAν(n)− ∂νAµ(n) + i[Aµ(n), Aν(n)]) +O(a3))

= exp(ia2Fµν(n) +O(a3))(3.38)

where we have used the continuum definition of the field strength given in

eq. (2.23). This expansion can be inserted in the Wilson gauge action and

by expanding the exponential in (3.38) we find the expansion for small a of

the Wilson action

SG[U ] =2

g2

∑n∈Λ

∑µ<ν

Re Tr[1− Uµν(n)]

=a4

4g2

∑n∈Λ

4∑µ,ν=1

Tr[F 2µν(n)] [1 +O(a2)].

(3.39)

The Wilson action approximates the continuum action up to a correction

CHAPTER 3. LATTICE REGULARIZATION 32

of orderO(a2). Note that the factor a4, that derives from the expansion of the

exponential, together with the sum over the nodes n is just the discretization

of the spacetime integral, thus taking now the limit for a→ 0 we obtain

lima→0

SG[U ] =1

2g2

∫d4x Tr[F 2

µν(x)] = SG[A]. (3.40)

Up to this point we have taken the lattice spacing a to be a length defined

a priori, however, in view of a numerical simulation a more suitable approach

is to relate the lattice spacing to some physical observable easily computable

on the lattice, thus we will extrapolate the correct value of a by comparing

the lattice results with experimental ones, this procedure is known as scale

setting and will be discussed shortly; before doing that, let us see how the

path integral formalism of gauge field theory translates on the lattice.

3.3 Path integral formalism on the lattice

We have already introduced the path integral formalism in lattice field theory

in the case of a scalar field, now we are going to extend that definition to

lattice gauge field theory.

The starting point is always the partition function that in the path inte-

gral formalism is implemented as an integral over all field configurations

Z =

∫D[ψ, ψ] D[A] e−SF [ψ,ψ,A]−SG[A]. (3.41)

On the lattice the corresponding path integral measures are products of mea-

sures of all quark field components and products of measures for all link

CHAPTER 3. LATTICE REGULARIZATION 33

variables:

D[ψ, ψ] =∏n∈Λ

dψ(n)dψ(n), D[U ] =∏n∈Λ

4∏µ=1

dUµ(n). (3.42)

Thus the explicit definition of the partition function in the lattice formalism

reads as follows

Z =

∫ ∏n∈Λ

dψ(n)dψ(n)4∏

µ=1

dUµ(n) e−SF [ψ,ψ,U ]−SG[U ]. (3.43)

In the next sections both the fermionic integration and the link variables

integration will be discussed.

3.3.1 Fermion integration

As we have seen the lattice discretization of matter field ψ(x) is straightfor-

ward. However to recover the correct fermionic statistics, i.e. the Pauli ex-

clusion principle, in the path integral formalism the quantization of fermionic

field is based on Grassmann numbers which are variables formally defined by

classical anticommutation relations

ηi, ηj = ηiηj + ηjηi = 0. (3.44)

An element of the Grassman algebra is a polynomial in these generators

f(η) = f +∑i

fiηi +∑ij

fijηiηj +∑ijk

fijkηiηjηk + . . . (3.45)

One could now think that the obvious choice would be to simply associate

one of these variables for every node in the lattice, but if we consider that

CHAPTER 3. LATTICE REGULARIZATION 34

Grassamn numbers have also some notable integration rules

∫dηi = 0,

∫dηi ηi = 1, (3.46)

and that the fermion action is linear in both ψ and ψ, these rule can be

used to integrate over them. Thus the path integral reduces to one over

only the gauge degrees of freedom. It turns out that in practice one does

not have to worry about transcribing Grassmann variables on the lattice and

implementing the Pauli exclusion principle. However for a QCD-like theory

with six flavors of quarks the continuum euclidean action is given by

SQCD =

∫d4x

[1

2g2Tr[Fµν(x)Fµν(x)] +

6∑f=1

ψf (x)(mf + γαDα)ψf (x)

],

(3.47)

so using the Grassman integration rules the fermionic fields can be integrated

out, but the determinant of the Dirac operator appears as the fermionic

contribution to the quantum dynamics of the system

Z =

∫ ∏x,µ

dAµ(x)∏x,f

dψ(x)dψ(x) exp(−SQCD)

=

∫ ∏x,µ

dAµ(x) detD exp(−SG).

(3.48)

Note that, although the Dirac operator is local in the gauge fields, its de-

terminant is not. This is a source of significant computational overhead in

Monte Carlo simulations of lattice QCD.

Similarly, correlation functions involving fermionic fields can be expressed

in terms of matrix elements of the inverse of D. The real challenge of fermion

field discretization is now to give a good discretization of the Dirac matrix

D. The problem is however non trivial ad outside of the scope of the present

CHAPTER 3. LATTICE REGULARIZATION 35

work and will not be discussed.

3.3.2 Link variables integration

Having defined the link variables as elements of a Lie group, the integration

measure to be used is the normalized Haar measure for the gauge group. The

latter is such that

∫dgF (g) =

∫dgF (ug) =

∫dgF (gv) (3.49)

for any function F defined on the group and for any group elements u, and v.

This property ensures that the Haar measure is invariant under local gauge

transformations,

dUµ(n) = dU ′µ(n) = d(Ω(n) Uµ(n) Ω†(n+ µ)). (3.50)

As a result the expectation value of a generic quantity O, defined as

〈O〉 =1

Z

∫ ∏n,µ

dUµ(n)O exp(−SG[U ]) (3.51)

can be non vanishing only if O is gauge-invariant.

Taking into consideration the integration over a single link variable Uµ(n),

due to the invariance property discussed above, no gauge fixing is required,

hence there is no need to introduce ghost fields in the discussion. Moreover

for compact Lie groups, as the ones present in the Standard Model, the

integration reduces to integrating over the compact manifold defined by the

group parameters. Therefore the integration over the link variables poses no

further challenge.

CHAPTER 3. LATTICE REGULARIZATION 36

3.4 Scale setting

Until now we have described the theory in terms of the lattice spacing a and

the lattice bare coupling g, but we have not given a proper way to relate

these parameters to the physical scales of the system. In this section we are

going to address this problem by comparing lattice results with experimental

data.

3.4.1 Physical interpretation of the Wilson loop

We have already introduced the Wilson loop as the trace of the ordered

product of link variables defined on a closed path, here to give a physical

interpretation of these objects we are going to study a particular class of

Wilson loops, the loops defined on a temporal plane (in fact, since we are on

a Euclidean lattice, all the planes can be treated equally, but we are giving

a different interpretation to the dimension we identify as the time).

These loops can be expressed as the product of four path transporters:

two spatial, from node n to node m, and two temporal, from time t = 0 to

t = nt,

WL[U ] = Tr[S(m,n, nt)T†(n, nt)S

†(m,n, 0)T (m,nt)], (3.52)

where S(m,n, nt) is the spatial transporter form n to m in the time slice

t = nt, and T (n, nt) is the temporal transporter from node n in the time slice

t = 0 to n in the time slice t = nt.

To allow a simple physical interpretation of these objects it is useful to

fix the gauge using the so-called temporal gauge,

A4(x) = 0. (3.53)

CHAPTER 3. LATTICE REGULARIZATION 37

We know that for a gauge theory a gauge fixing is always possible, and the

temporal gauge is a feasible choice because the field strength tensor present

in the action does not contain derivatives of A4 with respect to time. In the

following we use the temporal gauge only to find the physical interpretation

of the Wilson loop, for the actual computation of the expectation value we

do not need to fix the gauge since the result for the expectation value of the

Wilson loop is of course the same whether we fix the gauge or not.

As a direct effect of this choice the temporal transporters now become

trivial

T (n, nt) =nt−1∏j=0

U4(n, j) = 1. (3.54)

Thus the expectation value for the Wilson loop takes the form

〈WL〉 = 〈Tr[S(m,n, nt)S†(m,n, 0]〉. (3.55)

The temporal gauge makes it explicit that the Wilson loop is the corre-

lator of two Wilson lines S(m,n, nt) and S(m,n, 0) situated at time slices

nt and 0. The spatial Wilson lines are the representation on the lattice of a

quark-antiquark pair, thus the Wilson loop is related to the so-called static

quark potential

〈WL〉 ∝ e−tV (r). (3.56)

Here the potential V (r) describes the binding energy of two static (hence with

infinite mass) quarks at a distance r. By using strong coupling expansions

and perturbation theory, a parametrization of the static quark potential can

be given as

V (r) = A+B

r+ σr. (3.57)

The peculiar aspect of this potential is the linear term, it is in fact the

CHAPTER 3. LATTICE REGULARIZATION 38

one extracted from the strong coupling expansion and is characterized by

the presence of the real parameter σ, the string tension. From QCD phe-

nomenology, one expects a value of σ ≈ 900MeV/fm.

3.4.2 The Sommer parameter and the lattice spacing

Having now defined the static quark potential V (r), we can express the lattice

spacing in physical units by comparing the shape of the potential obtained in

lattice simulations with a certain physical distance r0, the so-called Sommer

parameter [7, 8]. Doing this allows us to determine the lattice spacing simply

by counting the number of lattice points between r = 0 and r = r0.

The Sommer scale definition is based on the force F (r) = dV (r)/dr be-

tween the two static quarks, rather than the potential V (r). For sufficiently

heavy quarks, quark–antiquark bound states can be described by an effective

non relativistic Schrodinger equation and the force F (r) can be studied.

From comparison with experimental data for the bb and cc spectra, one

finds that

F (r0)r20 = 1.65 corresponds to r0 ' 0.5fm. (3.58)

Using the definition of the static quark potential, we can now give an ex-

pression of the dimensionless term F (r)r2 in terms of the lattice spacing

as

F (r)r2 = r2 d

drV (r) = r2 d

dr

(A+

B

r+ σr

)= −B + σr2 (3.59)

Then using (3.58) we can express the Sommer parameter in terms of the

lattice spacing

X =r0

a=

√1.65 +B

σa2. (3.60)

CHAPTER 3. LATTICE REGULARIZATION 39

Figure 3.2: Static quark potential computed with the Wilson gauge action attwo different values of β. The dashed vertical lines are drawn at a distance thatcorresponds to the Sommer parameter. Image, and data reference, taken from ref.[6].

Having determined the parameters B and σ from numerical data the lattice

spacing in physical units is given by a = (0.5/X) fm.

As seen from the figure above the shape of the potential depends on the

lattice bare coupling, hence the lattice spacing itself and the bare coupling

are not independent parameters. In the next section we will analyze their

relation by means of the renormalization group.

3.5 The continuum limit

The basic idea now is to obtain a continuum limit of our theory keeping

the value of certain physical observables fixed, to do so we are going to use

the renormalization group to link the lattice spacing a to the lattice bare

coupling g.

CHAPTER 3. LATTICE REGULARIZATION 40

3.5.1 Running of the lattice coupling

Coupling constants like the gauge coupling g are usually called bare parame-

ters, these are not directly observable physical quantities. Only by computing

observables such as hadron masses, the string tension, or the Sommer param-

eter, and by identifying those with experimental values, one can find out the

values of the bare parameters of the action in physical units.

Lattice actions may differ in various aspects. They may use different

discretizations of derivatives or the lattice grid, which is usually taken to be

hypercubic, may vary in its structure. However, when sending the lattice

cutoff to infinity, i.e. sending a→ 0, physical observables should agree with

the experimental value and become independent of a. In general this will

imply that the bare parameters have a nontrivial dependence on the cutoff

a. As we send a → 0 the values of the bare parameters will have to be

changed in order to keep the physics constant.

Therefore we can use the same results obtained in the continuum theory

to relate the energy scale with the coupling. This time however we are not

doing this procedure to investigate the behavior of the physical coupling, but

to evaluate how the bare lattice coupling has to change in order to achieve

the continuum limit without changing the real physics of the system. We

define the beta function as

β(g) = − ∂g(a)

∂ log(a), (3.61)

that determines how the coupling g depends on the lattice spacing a. Using

the expression of the beta function in (2.28), we can solve the differential

CHAPTER 3. LATTICE REGULARIZATION 41

equation obtaining

a(g) =1

ΛL

exp

(− 1

2β0g2

)(1 +O(g2)). (3.62)

Inverting this relation one obtains the coupling g as a function of a

g(a)−2 = β0 ln(a−2Λ−2L ) +O(1/Ln(a−2Λ−2

L )) (3.63)

Changing a thus implies a corresponding change of g such that physical

observables remain independent of the scale-fixing procedure. The value of

ΛL depends on the regularization scheme, namely different lattice actions

have different values of ΛL.

3.5.2 The true continuum limit

We can now give a proper definition of the continuum limit, we have shown

that the lattice spacing a decreases with decreasing g (increasing β = 2N/g2),

hence we conclude that we simply have to study the limit

β →∞ (3.64)

to obtain the true continuum limit a → 0. There are, however, certain

caveats to be considered in this procedure. If one performs this limit, then

the physical volume of the box in which we study QCD is proportional to a4

and thus shrinks to zero, unless we also increase the numbers of lattice points

in the spatial (NS points) and temporal (NT points) directions of our lattice.

In an ideal world one would first perform the so-called thermodynamic limit

NS →∞, NT →∞ (3.65)

CHAPTER 3. LATTICE REGULARIZATION 42

and only after that step the continuum limit would be taken. However, since

in a numerical calculation this is not feasible, one is restricted to calculating

the physical observables for a few values of β, giving rise to different values

of a. The numbers of lattice points NS, NT are always chosen such that the

physical extension

L = aN, T = aNT (3.66)

of the box remains fixed for the different values of a. Studying the dependence

of the results at fixed physical volume allows one to analyze the dependence

on the scale a and to extrapolate the results to a→ 0. The extrapolation to

a = 0 can then be repeated for different physical sizes L, which in the end

allows one to extrapolate the data to infinite physical volume.

Before moving on we stress that the procedure that has been described

is not the running of the physical coupling of the theory in the continuum,

but just the running of the bare coupling of the lattice. Moreover having

used a result of a perturbative expansion, the beta function, we have to be

careful when planning a numerical simulation since the lattice spacing a must

be small enough to ensure that the continuum limit can be achieved while

maintaining the right scaling properties.

In general, however, it is possible to perform a non-perturbative scale-

setting by which one can extrpolate the relation between a and g for the

simulated values. This can be done, for example, by determining the Sommer

parameter, as previously discussed.

Chapter 4

Running coupling in the

Schrodinger Functional scheme

We have defined a proper non-perturbative, gauge-invariant way to discretize

gauge field theories on a lattice and the correct way to relate lattice observ-

ables with continuum ones. In this work, however, our main goal is to extract

the running-coupling of the continuum theory from the lattice formulation,

to do so we have to study how a lattice observable varies compared to the

energy scale.

The workflow we have to follow is the following:

1. Define on the lattice an object that has a non-trivial dependence on an

energy scale, in lattice-friendly terms a dependence on a linear exten-

sion L = an;

2. Based on this object, define a renormalized coupling g2(L) that does

not depend on any scale other than L and that will be considered our

running coupling;

3. Run the simulation at different values of L to obtain the running of

such renormalized coupling;

43

CHAPTER 4. SCHRODINGER FUNCTIONAL 44

4. Relate the behavior at small L to another more commonly used cou-

pling such as the coupling in the MS scheme (g2MS

) of dimensional

regularization;

5. Extract the scaling properties of our renormalized coupling in the non

perturbative region.

The definition of the running coupling g2(L) is arbitrary. The coupling

should however be accurately computable through numerical simulation and

its scaling properties should not be strongly influenced by the presence of

a non zero lattice spacing. We could now use the objects we have already

defined, such as the Wilson loops, to extract the running coupling.

Instead a renormalized coupling can be straightforwardly defined trough

the Schrodinger functional, as shown by Martin Martin Luscher in ref. [9].

With carefully chosen boundary values for the gauge field this coupling has

the desired technical properties, thus making a finite scaling study feasible.

Schrodinger functional

In the Schrodinger representation of quantum mechanics, in a system de-

scribed by the Hamiltonian H, the Schrodinger functional is the propagation

kernel for evolution from an initial state I at time t = 0, to a final state F at

time t = T , hence it represents the probability amplitude for the transition

from I to F and can be expressed as

Z[I,F ] = 〈F|e−HT |I〉. (4.1)

By inserting an orthonormal basis |ψn〉, n = 0, 1, 2, . . ., of gauge invariant

energy eigenstates we can formally express the Schrodinger functional in the

CHAPTER 4. SCHRODINGER FUNCTIONAL 45

spectral representation as

Z[I,F ] =∞∑n=0

e−EnTψn[F ]ψ†n[I], (4.2)

where the En are the energy eigenvalues. Being the matrix elements of the

Euclidean time evolution operator e−HT , they can be expressed through a

functional integral over all gauge field configurations Aµ(x) in four dimension

with 0 < t < T with fixed boundary conditions, specifying the I and F states

at times t = 0 and t = T , respectively, and periodic boundary conditions in

the spatial directions.

The functional integral representation of the Schrodinger functional thus

reads

Z[I,F ] =

∫I,FD[A]e−S[A], (4.3)

where the integration is done over all components of the Euclidean gauge

field compatible with the boundary conditions that define the states I and

F . The Euclidean action of pure gauge theories has already been defined as

S[A] =1

2g20

∫d4xTrFµνFµν. (4.4)

We have obtained an expression of the Schrodinger functional that is par-

ticularly suitable for lattice studies as we have already defined the Euclidean

lattice action and a proper way of integrating the discretized gauge field.

4.1 Lattice formulation

Let us give a proper lattice formulation of the Schrodinger functional and let

us show how to extract the running coupling from it, following refs. [9, 10].

CHAPTER 4. SCHRODINGER FUNCTIONAL 46

We are describing the theory in a volume V = L4, with L = na (n being

the number of nodes and a the usual lattice spacing). In this formulation we

identify the initial and final states as the field configurations C, C ′ respectively

in the first and last time slice of the lattice. The lattice gauge field is now

represented by the link variables Uµ and the functional integral over the

gauge field can be expressed as shown in (3.42). A discretized version of the

Schrodinger functional thus takes the form

Z[C, C ′] =

∫C,C′D[U ]e−S[U ] = e−Γ[C,C′], (4.5)

where Γ[C, C ′] is the effective action of the system with the specified bound-

ary conditions. The relation between the Schrodinger functional and the

effective action, and the crucial role that plays in the definition of the run-

ning coupling, will be discussed shortly, before doing that let us focus on the

definition of the boundary conditions.

Boudary conditions

As we already said the continuum gauge field (Aµ) configuration at t = 0 is

translated in the lattice formulation as a collection of discretized (algebra-

valued) variables that we identify collectively as C, in the same way the

configuration at t = L is identified as C ′. We can define the boundary lattice

variables W like

Wk(nx, ny, nz)|nt=0 = exp( ia C(nx, ny, nz)), (4.6)

Wk(nx, ny, nz)|nt=L = exp( ia C ′(nx, ny, nz)), (4.7)

where the index k identifies only the spacial links, i.e. a link that stars from a

node in a certain time slice and points to another node in the same time slice,

CHAPTER 4. SCHRODINGER FUNCTIONAL 47

thus only these links are affected by the boundary conditions. The time-like

ones starting from the first time slice and the ones pointing to nodes in the

last time slice are free to fluctuate and must be integrated over.

As it has been shown in ref. [9] a proper definition of these bound-

ary conditions ensures that the artifact effects that arise from the lattice

discretization can be minimized. The optimal choice are constant Abelian

fields, thus we can express the boundary condition in terms of the C field

configuration collectively as

Ck =1

L

φ1 0 . . . 0

0 φ2 . . . 0...

.... . .

...

0 0 . . . φN

, C ′k =1

L

φ′1 0 . . . 0

0 φ′2 . . . 0...

.... . .

...

0 0 . . . φ′N

. (4.8)

Unitarity and stability considerations constrain the angles

N∑i=0

φi = 0, φ1 < φ2 < . . . < φN , |φi − φj| < 2π. (4.9)

In practical terms this choice means that the lattice link variables (Uµ) that

we are going to set as boundary conditions have to be SU(N) group-valued

matrices with different than zero elements only on the diagonal, i.e. elements

of U(1)N−1.

Renormalized coupling definition

The relation (4.5) between the Schrodinger functional and the effective action

is clear if we note that the definition we have given of the former object is

exactly the definition of the partition function of a system with boundary

conditions C and C ′. Thus the relation arises from the quantum mechanical

CHAPTER 4. SCHRODINGER FUNCTIONAL 48

definition of the effective action as

Γeff = − ln[Z]. (4.10)

This object is called the effective (rather than the classical) action since it

contains all the quantum effects related to the path integral formulation of

the theory. The effective action can be expressed as an asymptotic series

Γeff =1

g20

∞∑n=0

(g20)nΓn, (4.11)

where Γ0 is the action of the classical configuration interpolating between

the I and F states, that can be evaluated analytically.

This equation can be used to directly define a renormalized coupling,

however, since in Monte Carlo simulations all the configurations are generated

with a statistical weight normalized by dividing for Z, it will be impossible

to evaluate such object, to get around this difficulty one generally measures

derivatives of the effective action. To evaluate such derivatives a dependence

of the boundary condition on a real dimensionless parameter χ is introduced

and a renormalized coupling can be defined as

g2 =Γ′0Γ′, Γ′ =

∂

∂χΓ. (4.12)

The initial and final states of the system can be defined in such a way that the

derivatives of the effective action correspond to certain well-defined operators,

therefore measuring those on the lattice gives us a good way to compute the

renormalized coupling.

A dependence on the lattice size L has been implied during all the dis-

cussion. It is now clear that running the simulation with different values of

CHAPTER 4. SCHRODINGER FUNCTIONAL 49

Figure 4.1: Running of the α = g2/(4 · π) coupling parameter in a SU(3) nu-merical simulation using the SF scheme. The full line is a fit of the data points,while the dotted and dashed lines are the theoretical extrapolation, starting fromthe right-most point, using respectively the 1-loop and 2-loop β-function. Imagetaken from ref. [11].

L and obtaining the correct continuum result for each of those simulations

will result in a precise evaluation of the running coupling.

The obtained results will then be compared to the analytical (and ex-

perimental) ones obtained in the perturbative region of the theory, as shown

in Figure 4.1. The Schrodinger functional scheme has already proven very

useful in many publications in a wide range of energies; particularly in ref.

[12] it has been shown that the region where the theory can be considered in

the perturbative regime could be narrower than what previously thought.

In this work instead of following the standard approach we will propose

a new way of measuring these derivatives of the effective action by means of

the Jarzinsky theorem.

Chapter 5

Jarzynski theorem

As we have shown the problem of extracting the renormalized coupling re-

duces to the evaluation of the derivative of the effective action of the system.

In our numerical simulations a discretized derivative of the effective action

will be evaluated as a finite difference between two slightly different config-

urations of the boundary conditions

Γ′ =∂

∂χΓ ∝ Γ[C,C ′′]− Γ[C,C ′] ∝ g−2, (5.1)

here we are assuming that the boundary condition C at one end of the lattice

(T = 0) does not vary, while at the opposite end (T = L) we are modifying

the field configuration from C ′ to C ′′.

Recalling again the definition of the effective action in quantum mechan-

ics, one can express the exponential difference of the effective action between

the two different configurations as the quotient between the respective par-

tition functions,

exp− (Γ[C,C ′′]− Γ[C,C ′])

=Z[C,C′′]

Z[C,C′]. (5.2)

50

CHAPTER 5. JARZYNSKI THEOREM 51

To evaluate this quantity we are going to use the Jarzynski’s relation

that states the equality between the exponential average of the work done on

the system in non-equilibrium processes and the ratio between the partition

function of the initial and final ensemble. Hence, we are going to evaluate

the exponential average of the action change of the system during a non-

equilibrium process in which the boundary conditions at T = L are switched

from C ′ to C ′′.

5.1 Jarzynski relation

Let us start by summarizing the original derivation, present in refs. [13, 14],

using natural units (~ = c = kB = 1) and focusing on a statistical-mechanics

system. As we will show below, the generalization to lattice gauge theories,

and in particular the use of the Jarzynski relation to compute the effective

action differences which are the basis to define a renormalized coupling in

the Schrodinger functional scheme, is straightforward.

Consider a system, whose microscopic degrees of freedom are collectively

denoted as φ. For instance, in our picture φ represents the configuration

of all the link variables Uµ(n) defined on the lattice; for simplicity we will

assume these parameters to be discrete; to obtain a proof for continuous pa-

rameters one should carefully replace the sums with integrals in the following

definitions. Let the dynamics of the system be described by the Hamiltonian

H, which is a function of the degrees of freedom φ, and depends on a set of

parameters. When the system is in thermal equilibrium with a large heat

CHAPTER 5. JARZYNSKI THEOREM 52

reservoir at temperature T , the partition function of the system is

Z =∑φ

exp

(−HT

), (5.3)

where∑

φ represents the multiple sum over the values that each microscopic

degree of freedom can take. The statistical distribution of φ configurations

in thermodynamic equilibrium is given by the Boltzmann distribution:

π[φ] =1

Zexp

(−HT

), (5.4)

which, in view of eq. (5.3), is normalized to 1:

∑φ

π[φ] = 1. (5.5)

Let us denote the conditional probability that the system undergoes a tran-

sition from a configuration φ to a configuration φ′ as P [φ→ φ′]. The sum of

such probability densities over all possible final configurations is one,

∑φ′

P [φ→ φ′] = 1, (5.6)

because the system must evolve to some final configuration. Since the Boltz-

mann distribution is an equilibrium thermal distribution, it satisfies the prop-

erty ∑φ

π[φ]P [φ→ φ′] = π[φ′]. (5.7)

In the following, we will assume that the system satisfies the stronger, detailed-

CHAPTER 5. JARZYNSKI THEOREM 53

balance condition:

π[φ]P [φ→ φ′] = π[φ′]P [φ′ → φ]. (5.8)

The Boltzmann distribution π, and consequently Z and P , will depend on

the couplings appearing in the Hamiltonian. Denoting them collectively as λ,

one can emphasize such dependence by writing the configuration distribution

as πλ, and the partition function and transition probabilities as Zλ and Pλ,

respectively.

We will now introduce a time dependence for the λ parameters. Starting

from a configuration, at the Monte Carlo time τ = τin, in which the system is

in thermal equilibrium and the λ parameters of the Hamiltonian take certain

values, the parameters of the system are modified as a function of time λ(τ)

until a final configuration at τ = τfin. λ(τ) is assumed to be a continuous

function, for simplicity we take it to interpolate linearly in (τfin−τin) between

the initial, λ(τin), and final, λ(τfin), values. During the transformation from

τin to τfin the system is, in general, out of equilibrium.

Now discretizing the ∆τ = τfin−τin interval in N sub-intervals of the same

length δt = ∆τ/N we define τn = τin + nδt for integer values of n ranging

from 0 to N , so that τ0 = τin and τN = τfin. With this discretization the linear

λ(τ) function mentioned above is now a piecewise-constant function, taking

the value λ(τn) for τn ≤ τ < τn+1. Furthermore let us define φ(t) as a pos-

sible trajectory in the space of field configurations, i.e. a mapping between

the time interval [τin, τfin] and the configuration space of the system. By dis-

cretizing the time interval [τin, τfin] we can associate φ(τ) with the discretized

path in the configuration space where the field configuration takes the values

φ(τin) → φ(τ1) → φ(τ2) → · · · → φ(τN−1) → φ(τfin). Since the parameter

values λ(τ) at fixed Monte Carlo time are common between all the variables

CHAPTER 5. JARZYNSKI THEOREM 54

under consideration, to ease the notation, we will use a single index n to

define the field configuration as φ(τn) = φn and the Boltzmann probability,

the Hamiltonian and the partition function as πλ(τn) = πn, Hλ(τn) = Hn and

Zλ(τn) = Zn, respectively.

Having defined these objects we can properly introduce the quantity

RN [φ] defined as

RN [φ] = exp

(−

N−1∑n=0

Hn+1[φn]

Tn+1

− Hn[φn]

Tn

), (5.9)

where each element of the sum appearing on the right-hand side is the work

(in units of the temperature) done on the system during a time interval

δt, by switching the couplings from their values at τ = τin + nδt to those at

τ = τin+(n+1)δt. Thus,RN [φ] provides a discretization of the exponentiated

work done on the system in the time interval from τ = τin to τ = τfin, during

which the parameters are switched as a function of time, λ(τ), and the fields

trace out the trajectory φ(τ) in configuration space. This discretization gets

more accurate for larger values of N , and becomes exact in the N →∞ limit.

Using eq. (5.4), eq. (5.9) can then be expressed in the form

RN [φ] =N−1∏n=0

Zn+1 · πn+1[φn]

Zn · πn[φn]. (5.10)

We will now take the average of eq. (5.10) over all possible field-configuration

trajectories realizing an evolution of the system from one of the configura-

tions of the initial ensemble (at t = τin, when the parameters of the system

take the values λ(τin)) to a configuration of the final ensemble (at t = τfin,

when the parameters of the system take the values λ(τfin)).

CHAPTER 5. JARZYNSKI THEOREM 55

The average of eq. (5.10) over all possible field-configuration trajectories

realizing an evolution of the system from τ = τin to τ = τfin can be written

as

∑φ(t)

RN [φ] =∑φ(t)

πin[φin]N−1∏n=0

Zn+1

Zn· πn+1[φn]

πn[φn]· Pn+1 [φn → φn+1]

,

(5.11)

where we used the fact that the system is initially in thermal equilibrium,

hence the probability distribution for the configurations at τ = τin is given by

eq. (5.4), and where∑φ(t) denotes the N + 1 sums over field configurations

at all discretized times from τin to τfin:

∑φ(t)

. . . =∑φ(τin)

∑φ(t1)

∑φ(t2)

· · ·∑

φ(τfin−δt)

∑φ(τfin)

. . . . (5.12)

Focusing now on the right-hand side term, the product of partition-function

ratios in eq. (5.11) simplifies, and the equation can be rewritten as

ZfinZin

∑φ(t)

πin[φin]N−1∏n=0

πn+1[φn]

πn[φn]· Pn+1 [φn → φn+1]

. (5.13)

Using eq. (5.8), this expression can be turned into

ZfinZin

∑φ(t)

πin[φin]N−1∏n=0

πn+1[φn+1]

πn[φn]· Pn+1 [φn+1 → φn]

. (5.14)

Here, also the product of ratios of Boltzmann distributions can be simplified,

reducing the latter expression to

ZfinZin

∑φ(t)

πfin[φfin]N−1∏n=0

Pn+1 [φn+1 → φn] . (5.15)

CHAPTER 5. JARZYNSKI THEOREM 56

Note that, in eq. (5.15), φin appears only in the P1 [φ1 → φin] term: thus,

one can use eq. (5.6) to carry out the sum over the φin configurations, and

eq. (5.15) reduces to

ZfinZin

∑φ1

· · ·∑φfin

πfin[φfin]N−1∏n=1

Pn+1 [φn+1 → φn] . (5.16)

Repeating the same procedure, eq. (5.16) can then be simplified using the

fact that the only remaining dependence on φ1 is in the P2 [φ2 → φ1] term,

and so on. One arrives at

ZfinZin

∑φ(τfin)

πfin[φfin]. (5.17)

Finally, eq. (5.5) implies that also the last sum yields one, so recalling the

full index notation, one gets,Zλ(τfin)

Zλ(τin)

. (5.18)

This result has been obtained with total generality in regards of the sys-

tem under consideration, hence the application to the lattice field is straight-

forward. The partition function Zλ(τfin) and Zλ(τin) can be directly related to

the partition function of our lattice field, with the boundary condition in the

initial and final states, namely Z[C,C′] and Z[C,C′′].

The left-hand side of eq. (5.11), instead, has to be taken with more

consideration. In statistical mechanics terms, the definition of RN [φ], given

in eq. (5.9), represents the discretized exponential work (in units of the

temperature) done on the system during the transformation from τin to τfin.

Hence the right-hand term of eq. (5.11) can be written, for N →∞, as

⟨exp

[−∫δW

T

]⟩. (5.19)

CHAPTER 5. JARZYNSKI THEOREM 57

However recalling that in the usual mapping between statistical mechanics

and lattice field theory one associates H/T with the Euclidean action of the

lattice theory, one easily realizes that, from the point of view of the lattice

theory, each term within the braces on the right-hand side of eq. (5.9) can be

interpreted as the difference in Euclidean action for the field configuration

denoted as φ (tn), which is induced when the parameters are changed from

λ (tn) to λ (tn+1). Thus, evaluating the work done on the system by changing

the boundary conditions corresponds to evaluating the variation in Euclidean

action in the lattice gauge theory. The Jarzynski relation in our picture thus

reads, ⟨exp

[−

N∑n=1

(SG(τn)− SG(τn−1))

]⟩N→∞−−−→

Z[C,C′′]

Z[C,C′], (5.20)

where SG(τn) denotes the Wilson action of the field configuration φ(τn).

We now point out some important remarks; let us start by noting that

the Jarzynki relation is an exact equality only when we consider the limit

for N →∞ and we evaluate the average on all the possible field trajectories

φ(t) starting from τin and ending in τfin. This means that in every simulation

we will have two sources of systematic errors:

Finite number of trajectories: this is a purely statistical effect, mean-

ing that increasing the sample size will reduce the uncertainty related to the

mean of the exponential work, hence giving a better approximation of the

free energy difference.

Finite N : the effect of a finite N is more complex. In the N →∞ limit

we can assume∫ τfin

τinδS = ∆S[τfin, τin] to be an exact relation, but for a finite N

the sum of the infinitesimal work will differ from the real value. In thermody-

CHAPTER 5. JARZYNSKI THEOREM 58

100 1000 10000 1e+05 1e+06N

6

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

7

7.1

7.2

7.3

7.4

7.5F

(1)

reference value, from JHEP 09 (2007) 117

reverse transformationdirect transformation

β = 0.223102, N0 = 96, N

1 = 24, N

2 = 64

Figure 5.1: Results of Monte Carlo simulations, using the Jarsynsky method,for the interface free energy of an SU(2) model at different intermediate steps N .Image taken from ref. [15]

namic terms this means that if the transformation is done ”smootly” enough

(N →∞) it is a reversible process and no energy is dissipated, instead if we

are doing finite steps in the transformation it is an irreversible process and

the system is dissipating energy. To prove this one can for example study

both the direct and inverse transformations at increasing N , as shown in

Figure 5.1. As expected increasing N leads to a better approximation of the

real value and the direct and inverse transformations approximate the correct

result from opposing sides, and for N →∞ since ∆S[τin, τfin] = −∆S[τfin, τin]

both transformation will correctly approximate the expected value.

CHAPTER 5. JARZYNSKI THEOREM 59

5.1.1 Application in the SF scheme

Recalling the definition of the running coupling in the Schrodinger functional

scheme, given in eq. (4.12), and the relation between the effective action and

the partition function of eq. (5.2), we can now associate, using the Jarzynski

relation, our definition of the running coupling with a quantity that is easily

evaluable in numerical simulations:

g2 ∝ log

[Z[C,C′′]

Z[C,C′]

]N→∞←−−− log

⟨exp

[−

N∑n=1

(SG(τn)− SG(τn−1))

]⟩.

(5.21)

The evaluation of the running coupling thus reduces to the computation

of the difference of the Wilson action between the initial and final configu-

ration, defined in eq. (5.1), over non-equilibrium trajectories in the space of

configurations. By running the simulation at different values of the physi-

cal lattice size L the scaling properties of the coupling can be obtained and

compared with theoretical predictions of other rinormalization schemes (and

possibly with experimental data, too).

In the next chapter we will discuss the techniques that are used in nu-

merical simulations of lattice field theory and we will propose a numerical

implementation of the Jarzynski relation.

Chapter 6

Numerical simulation

We have now given a proper discussion of the fundamental theoretical aspects

of the topic under consideration, we have shown how to study a gauge field

theory on a lattice, thus providing a non perturbative approach, we have then

defined a renormalized coupling and proposed a new method to evaluate such

observable. What remains to be done, to complete the present discussion, is

to define how such theoretical aspects can be numerically simulated, that is

what we are going to do in this chapter.

Let us start by analyzing in more detail the definition of the mean value

of an observable in our discretized approach. We have already shown in eqs.

(3.7) and (3.51) a proper definition of such object as

〈O〉 =1

Z

∫ ∏n,µ

dUµ(n)O exp(−SG[U ]). (6.1)

We have also briefly stated that a direct evaluation (i.e. a simple sampling) of

this object is an absurdly tough numerical task; to give the scale of such state-

ment let us consider the simple Ising gauge model, where the link variables

60

CHAPTER 6. NUMERICAL SIMULATION 61

are represented by spin variables that may assume only the values ±1: for a

four-dimensional lattice with N nodes in every dimension we have 4 ·N4 in-

dependent spin variables, that means that for a reasonably sized lattice with

N = 24 the total number of spin configurations is 24·N4= 21327104 ≈ 10399498,

the evaluation of (6.1) corresponds to summing over all these configurations,

it is clearly an unrealistic task.

However, in our path integral we have to take into account the Boltzmann

factor exp(−SG[U ]). Depending on the value of the action SG, it will give

largely different statistical weights to different field configurations. When

summing over the configurations it is therefore more important to consider

the configurations with the largest weights. The central idea of Monte Carlo

sampling is to approximate the huge sum by a comparatively small subset

of configurations, which are sampled according to the weight factor. An ap-

proximation of eq. (6.1) can be obtained if one generates field configurations

distributed with probability P (U) ∝ exp(−SG[U ]) as

〈O〉 ≈ 1

N

∑Uµ(n)

with probability

exp(−SG[U ])

O(Uµ(n)), (6.2)

where we are assuming to evaluate the observableO overN different field con-

figurations. The uncertainty in this estimate of the correct average behaves

like O(√

1/N)

, hence the precision can be made arbitrarily small, provided

enough computing power is available, by increasing the sample size.

The challenge is now to develop an algorithm that could generate these

configurations. This problem has been already studied in statistical mechan-

ics and the result has been the development of the so-called Monte Carlo

simulations. These simulations are a very powerful instrument that is widely

used in fundamental and applied science, hence we will present only the ba-

CHAPTER 6. NUMERICAL SIMULATION 62

sic aspects of a general Monte Carlo simulation and we will focus on the

characterizing aspects of its application in lattice field theory.

General Monte Carlo algorithm

The idea is to start from an arbitrary configuration and then to construct a

stochastic sequence of configurations that eventually follows the equilibrium

distribution. This is done with a Markov process

U0 −→ U1 −→ U2 −→ . . . . (6.3)

In this chain the configurations Un are generated subsequently and the index

n labels the configuration in the order they appear in the chain.

Markov processes are characterized by a conditional transition probabil-

ity, the probability to get Un+1 starting from Un, P (Un+1 = U ′|Un = U) =

T (U ′|U) that depends only on the configurations U and U ′, but not on the

index n.

The determination of T (U ′|U) is not trivial: in fact, it is the most chal-

lenging aspect of the development of a good algorithm. The oldest and most

famous of such algorithms is the Metropolis algorithm, which advances the

Markov chain by proposing a new configuration U ′ according to some a priori

selection probability T0(U ′|U), then by comparing the proposed configuration

U ′ and the most recent one U . By means of the acceptance function

TA(U ′|U) = min

(1,

T0(U |U ′)P (U ′)

T0(U ′|U)P (U)

)(6.4)

it determines whether or not the proposed step is a step towards the equi-

librium configuration or not. If it is, then the configuration U gets updated

to the new configuration U ′. The simulation repeats these steps until equi-

CHAPTER 6. NUMERICAL SIMULATION 63

librium is achieved. At that moment every new configuration is properly

distributed and can be part of the sampling ensemble used to evaluate some

observable.

The Metropolis algorithm is very simple to implement in a numerical

simulation, and for simulation with a simple and discrete symmetry group,

e.g. the Ising model, it can be sufficiently efficient. For our purposes however,

a simple Metropolis algorithm is not efficient enough. In the next section we

will illustrate a better alternative for simulating lattice gauge field theories.

6.1 Monte Carlo simulation in lattice QCD

For gauge field theories the most common update algorithms are local al-

gorithms, i.e. the update process changes one variable at a time. In this

section we will discuss the Heat-bath algorithm [16, 17], equivalent to an it-

erated Metropolis algorithm optimizing the local acceptance rate, and the

Overrelaxation algorithm [18, 19], a sometimes very efficient method to im-

prove the step size in the Markov chain.

For both of those an efficient implementation is known for SU(2) variables

and it is possible to update an SU(N) variable by updating a sequence of

different SU(2) subgroups of SU(N), as shown in ref. [20].

Before going in the details of the algorithms let us define the represen-

tation of SU(2) that will be used. For SU(2) the minimum number of pa-

rameters for the group is 3. Although one can think of representations of

group elements that have just these minimal sets of parameters, in practical

calculations it is often more convenient to use a redundant representation.

This leads to faster evaluation of the multiplication of group elements. This

operation is the most time-consuming part of the calculation, because it has

CHAPTER 6. NUMERICAL SIMULATION 64

to be done so frequently.

Therefore we define a general link variable U as

U = x01+ i~x · ~σ, (6.5)

where σ are the usual Pauli matrices. The request for U to be unitary leads

to the condition for the parameters (x0, ~x):

det[U ] = x20 + |~x|2 =

3∑i=0

x2i = 1. (6.6)

6.1.1 Heat-bath algorithm

In the heat-bath method one combines the proposing and the accepting steps

of the Metropolis update into a single step and chooses the new value U ′µ(n)

according to the local probability distribution defined by the surrounding

“staples”,

dP (U) = dU exp

(β

NRe Tr[U A]

). (6.7)

The staple term A is defined as the sum of the open strings of three links

starting from the node n and ending in n + µ, the simplest choice is to use

only the 6 square staples built using the closest variables to Uµ(n). Improve-

ments can be made by using a combination of rectangular and square staples

of different sized. In this definition the Boltzmann weight is given by the

exponential of the local action in the bulk around the link Uµ(n).

If an efficient heat-bath algorithm exists only for SU(2), it is because any

sum SU(2) elements is proportional to an SU(2) matrix. Thanks to this

unique property, we can give a useful definition of the staple A in the form

A = aV with a =√

det[A], and V ∈ SU(2). (6.8)

CHAPTER 6. NUMERICAL SIMULATION 65

We can now write our probability distribution as

dP (U) = dU exp

(1

2aβ Re Tr[U V ]

). (6.9)

Using the invariance of the Haar measure ( dU = d(UV ) ), we can define a

new matrix X = UV , with the local probability distribution

dP (X) = dX exp

(1

2aβ Re Tr[X]

). (6.10)

If we generate a matrix X distributed accordingly, the candidate link is

obtained by

U ′µ(n) = XV † = XA†1

a. (6.11)

We have reduced the problem to generating matrices X distributed ac-

cording to eq. (6.10). The Haar measure in that equation may be written

in terms of the real parameters used in the representation (6.5) of the group

elements as

dX =1

π2d4x δ(x2

0 + |~x|2 − 1)

=1

π2d4x

θ(1− x20)

2√

1− x20

(δ

(|~x| −

√1− x2

0

)+ δ

(|~x|+

√1− x2

0

)),

(6.12)

where the common property of the Dirac delta-distribution has been used.

We can now rewrite the volume element in terms of spherical angles and use

the Dirac delta-distribution to carry out the integration over the modulus of

~x, the Haar measure thus takes the form

dX =1

2π2d2Ω dx0

√1− x2

0 (6.13)

CHAPTER 6. NUMERICAL SIMULATION 66

In the chosen matrix representation we have Tr[X] = 2x0, therefore we

end up with the distribution for X in the form (using d2Ω = d cos θ dφ )

dP (X) =1

2π2d cos θ dφ dx0

√1− x2

0 eaβx0 , (6.14)

with x0 ∈ [−1, 1], cos θ ∈ [−1, 1] and φ ∈ [0, 2π). In order to find a random

matrix X we have to determine random variables x0, θ, and φ according to

this distribution.

For x0 the task is to find values distributed according to√

1− x20 e

aβx0 ,

without going in the details of the algorithm that generates this variable it

is important to note that this step is the only one that requires the presence

of an accept/reject step, but the acceptance rate is very high. The cos θ and

φ variables are instead uniformly distributed in their domains.

We can see a great difference between the heat-bath algorithm and a local

Metropolis one: in the former a new variable is generated at every update

while in the latter there is always the chance of a no update result. As

a consequence, a simulation carried out with the heat-bath method should

generally be able to get to the equilibrium in less Monte Carlo time and once

achieved should “move” through the configurations in a more efficient way,

meaning that less updates are required to find an uncorrelated configuration.

We can summarize the steps to update a SU(2) link variable as follows:

• Evaluate the sum of staples A, compute a =√

det[A] and set V = A/a;

• Generate a group element X as described above;

• Compute the new link variable as U = XV †.

This process changes the value of a single variable at a time hence we have

to apply it on every link variable present in our lattice to generate a new

CHAPTER 6. NUMERICAL SIMULATION 67

field configuration. We will therefore define a Heat bath update as the sweep

across the lattice of the single link updater.

6.1.2 Overrelaxation algorithm

We have now found an efficient way of updating our lattice variables, however

it still is a local method and as all the local ones has the problem of generat-

ing always highly correlated configurations. A way to mitigate this problem

is by means of the overrelaxation algorithm; in a nutshell this algorithm

generates a new variable U ′, that has the same probability as U and that is

as far as possible from U , to speed up the motion through configuration space.

The defining term in the probability weight is the local action Re Tr[U A],

hence we have to find a new variable that gives to the system the same action

content. One suggests a change according to the Ansatz

U −→ U ′ = V †U †V †, (6.15)

with a gauge group element V chosen such that the local action is invariant.

In the general case the choice of V is non-trivial, however as we have

already discussed SU(2) has an interesting property. We consider again that

A, the sum of the staples built around Uµ(n), is proportional to a group

matrix V = A/a with a =√

det[A]. With this choice one finds that

Tr[U ′A] = Tr[V †U †V †A] = aTr[V †U †] = Tr[A†U †] = Tr[UA] (6.16)

In the last step we have used the reality of the trace for SU(2) matrices.

This choice for U ′ indeed leaves the action invariant.

The overrelaxation algorithm alone is not ergodic. It samples the configu-

CHAPTER 6. NUMERICAL SIMULATION 68

ration space on the subspace of constant action. Since one wants to determine

configurations according to the canonical ensemble, i.e., distributed accord-

ing to the Boltzmann weight, one has to combine the overrelaxation steps

with other updating algorithms, such as heat bath steps.

6.1.3 General workflow of a Monte Carlo simulation

After having introduced algorithms for the configuration update we can now

discuss how to organize an actual simulation. A Monte Carlo simulation of

a lattice gauge theory consists of the following basic steps:

Initialization: in this phase all the variables of the simulation are ini-

tialized, the actual implementation of data structures is obviously at the

discretion of the programmer. Moreover the first field configuration has to

be chosen. Two typical start configurations are the so-called cold start, where

all the link variables are set to the unit element (Uµ(n) = 1 ∀µ,∀n), this

situation corresponds to minimal gauge action, and the hot start, where the

gauge field matrices are generated randomly. Since the equilibrium config-

uration is not affected by this choice, the starting configuration is purely

arbitrary.

Thermalization: In this step the update algorithms described above are

used to generate new field configurations. Usually one combines the two al-

gorithms, heat-bath and overrelaxation, to optimize performance, i.e to have

less autocorrelation in the Markov chain. A common choice is to loop over a

subroutine composed of a heat bath sweep and some overrelaxation sweeps.

To evaluate when the system is correctly thermalized one can compare how

CHAPTER 6. NUMERICAL SIMULATION 69

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

0 5 10 15 20

⟨(

) ⟩

β = 1.00 β = 1.20

β = 1.40 β = 1.60

β = 1.80 β = 2.00

Figure 6.1: Results of our simulation of an SU(2) lattice of size 244. The data-points represent the mean value of the real part of the plaquette trace at increasingMonte Carlo time for different values of the Wilson couping parameter β; the redlines are the termalization time, estimated with the 5τ rule, for β = 1.00 andβ = 2.00. Note that, due to its locality, the plaquette does not properly representthe thermalization status of the entire lattice.

a set of observables changes every time the gauge field is updated. Generally

the speed at which the system approaches the equilibrium will depend on

the size of the lattice and the gauge coupling β as a larger system and large

β require more update steps than small lattices and small β, as shown in

Figure 6.1

Evaluation of the observables: once the lattice is correctly thermal-

ized the configurations can be used for the evaluation of the observables.

Generally a few update steps are necessary between subsequent evaluations

of the observables to ensure that the configurations are not correlated. After

a large enough sample has been evaluated, standard statistical analysis can

be used to determine mean values and fluctuations.

CHAPTER 6. NUMERICAL SIMULATION 70

6.2 Implementation of Jarzynski’s algorithm

To implement the Jarzynski theorem we have to slightly modify the general

workflow of the simulation. Let us start with the first two steps, initialization

and thermalization; here we have to set the boundary conditions [C,C ′] at

the starting point of the simulation and then we must thermalize the system

taking care not to vary them. In practical term it means that the update

algorithm must sweep over all the lattice except for the spacial links in the

first and last time slices. By doing that, we generate thermalized configu-

rations with the correct boundary conditions that are the starting point for

the evaluation of the effective energy difference. The third step is where

the Jarzynski relation is effectively implemented, we have already discussed

in section 5.1 the limitations of a numerical implementation, namely the fi-

nite number of intermediate steps N , so if we assume that the evolution of

the boudary condition from C ′ to C ′′ has been formalized as a linear func-

tion in some adimensional parameter η ∈ [0, 1] so that B(η = 0) = C ′ and

B(η = 1) = C ′′, we can describe the basic steps of the simulation as:

• Evaluation of the Euclidean action in the starting configuration

• Loop over the N steps (for n = 1 to n = N)

– Evolve the boundary conditions to B(η = n/N)

– Update the lattice

– Evaluate the action in the new configuration

– Compute the action difference done during the transformation and

add it to the total action difference

• Compute the average of the exponential of the action difference over

many realization of this transformation.

CHAPTER 6. NUMERICAL SIMULATION 71

0,6

0,62

0,64

0,66

0,68

0,7

0,72

0,74

0 1 2 3 4 5

⟨(

)⟩

Euclidean time

[C,C'] [C,C''] No boundary condions

Figure 6.2: Results of our preliminary simulation with fixed boundary conditionson a 5 · 243 lattice with β = 2.3. Values on integer euclidean time are the averageof the space-like plaquette in the corresponding time-slice, while on half-integerare represented the average of the time-like plaquette connecting two nearby time-slices. The three series of data are obtained with the following boundary condi-tions: triangles have no boundary conditions; dots have C = C ′ = 1; rhombi haveC = 1, C ′′ = −1. The difference between the last two datasets is what we willevaluate using non equilibrium methods. Note that in this simulation the effectis magnified, having used a small NT , a rather large β and antipodal boundaryconditions.

To conclude we have to point out that the characterizing aspect of this

approach is that only the first configuration is a correctly thermalized one,

all the subsequent ones, generated during the evolution of the system, are

not. We are updating the lattice only once for every step in the evolution

of the boundary conditions, hence we can say that “we are not giving the

system enough time to thermalize”. Hence by repeating the routine we are

sampling the action difference between the initial and final configuration over

trajectories in the field configurations space that are out of equilibrium, that

is exactly what the right-hand side of eq. (5.21) requires us to evaluate.

As a second effect, the absence of a thermalization requirement at every

CHAPTER 6. NUMERICAL SIMULATION 72

step is an enormous advantage in terms of efficiency of the algorithm. That

means that increasing the number of degrees of freedom will increase the

computational time only polynomially, since the number of lattice updates

will not vary, while evaluating the same quantity on thermalized configura-

tions would require an increasing number of updates in correspondence of an

increase of the number of degrees of freedom, thus leading to an exponential

increase of the required computational time.

Chapter 7

Conclusions

In this thesis we have proposed a new way to evaluate the difference of the

effective action (from which one can obtain the renormalized coupling g2) in

the Schrodinger functional scheme. We have done that by means of a non

equilibrium relation: the Jarzynski theorem.

Large part of the time spent working on this thesis has been devoted

towards the development of a Monte Carlo code that implements this new

method for Yang-Mills theories on the lattice. The main advantage of our

non equilibrium approach to the problem is the increase in the computational

efficiency.

In the future, we are going to run the aforementioned code on a super-

computer to obtain high precision results, that we expect to publish in a

scientific article.

73

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