Post on 17-Feb-2019
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
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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