# An introduction to Lattice Gauge theories by liuhongmei

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```									          An introduction to Lattice Gauge theories

Antonio Rago

University of Wuppertal

CP3 -Origins Odense-17 August 2010

A. Rago (U. Wuppertal)               Lattice Introduction     17 August 2010   1 / 20
Overview of the Lattice Approach

LGT calculations are a non-perturbative implementation of ﬁeld theory using the Feynman
path integral approach.

A. Rago (U. Wuppertal)            Lattice Introduction               17 August 2010   2 / 20
Overview of the Lattice Approach

LGT calculations are a non-perturbative implementation of ﬁeld theory using the Feynman
path integral approach.
The calculations proceed exactly as if the ﬁeld theory was being solved analytically (had
we the ability to do the calculations).

A. Rago (U. Wuppertal)             Lattice Introduction                 17 August 2010   2 / 20
Overview of the Lattice Approach

LGT calculations are a non-perturbative implementation of ﬁeld theory using the Feynman
path integral approach.
The calculations proceed exactly as if the ﬁeld theory was being solved analytically (had
we the ability to do the calculations).
The starting point is the partition function in Euclidean space-time

Z=              ¯
DAµDψDψeS

where S is the gauge action, for example:

1           ¯
S=      d4 x       Fµν Fµν − ψM ψ .
4

and M is the Dirac operator. The fermions are represented by Grassmann variables ψ and
¯
ψ.

A. Rago (U. Wuppertal)              Lattice Introduction                17 August 2010   2 / 20
The ambitious plan

Deﬁning a ﬁnite dimensional system
The Yang-Mills action for gauge ﬁelds have to be transcribed onto a discrete space-time
lattice in such a way as to preserve
1   gauge invariance
2   chiral symmetry
3   topology,
4   and a one-to-one relation between continuum and lattice ﬁelds.
Discretisation Errors & Continuum limit
it is necessary to quantify and remove the discretisation errors by a reliable extrapolation
to a = 0.
Generating background gauge ﬁelds
the path integral is analogous to the Boltzmann factor in the partition function for
statistical mechanics systems. It can be regarded as a probability weight for generating
conﬁgurations.
Expectation values & Correlation functions

A. Rago (U. Wuppertal)              Lattice Introduction                  17 August 2010    3 / 20
Discretisation

There are a number of possible ways to discretized space-time in 4 Euclidean dimensions:
hypercubic, body-centered cubic, hexagonal, random lattices.

Of these the simplest is the isotropic hypercubic grid with spacing:
a = aS = aT and size NS × NS × NS × NT                               .

aT

aS
NT

NS

A. Rago (U. Wuppertal)              Lattice Introduction               17 August 2010   4 / 20
Discretisation
The lattice transcription of the matter ﬁelds ψ(x) → ψi is straightforward.

A. Rago (U. Wuppertal)             Lattice Introduction                 17 August 2010   5 / 20
Discretisation
The lattice transcription of the matter ﬁelds ψ(x) → ψi is straightforward.
The construction of gauge ﬁelds is less intuitive.
In the continuum: Aµ (x) carry 4-vector Lorentz indices
Wilson noticed that                        y
ψ(y) = Pe x igAµ (x)dxµ ψ(x)
This suggested that gauge ﬁelds be associated with links that connect sites on the lattice.

ˆ
µ
U (i, i + µ) ≡ Uµ (i) = eiagAµ (i+ 2 )

A. Rago (U. Wuppertal)             Lattice Introduction                 17 August 2010   5 / 20
Discretisation
The lattice transcription of the matter ﬁelds ψ(x) → ψi is straightforward.
The construction of gauge ﬁelds is less intuitive.
In the continuum: Aµ (x) carry 4-vector Lorentz indices
Wilson noticed that                        y
ψ(y) = Pe x igAµ (x)dxµ ψ(x)
This suggested that gauge ﬁelds be associated with links that connect sites on the lattice.

ˆ
µ
U (i, i + µ) ≡ Uµ (i) = eiagAµ (i+ 2 )

aT
ψi
aS
NT

Aµ

NS

A. Rago (U. Wuppertal)              Lattice Introduction                17 August 2010   5 / 20
Symmetries
The SO(4) symmetric group is broken by the discretisation and replaced with the dihedral
group D3 .
Translations have to be by at least one lattice unit, so the allowed momenta are discrete
2πn
k=            n = 0, 1, ..., N
Na
A local gauge transformation Vi acts
ψi → Vi ψi                     ψi → ψi Vi†
¯    ¯
†
Uµ (i)    →     Vi Uµ (i)Vi+µ

The only gauge invariant operators must be written like:

¯
ψi U (Ci,j )ψj
U (Ci,i )

A. Rago (U. Wuppertal)                   Lattice Introduction                 17 August 2010   6 / 20
Quenched Approximation
In the rest of this lecture we will work in the “quenched approximation”.
All the dynamics of the fermions and the related problems will be presented in next lecture (A. Patella ... good luck)

A. Rago (U. Wuppertal)                          Lattice Introduction                             17 August 2010    7 / 20
Quenched Approximation
In the rest of this lecture we will work in the “quenched approximation”.
All the dynamics of the fermions and the related problems will be presented in next lecture (A. Patella ... good luck)

Quenched approximation:
Once integrated integrated out exactly the fermionic ﬁelds
4   1
Z=         DAµ det M e d x( 4 Fµν Fµν )

A. Rago (U. Wuppertal)                          Lattice Introduction                             17 August 2010    7 / 20
Quenched Approximation
In the rest of this lecture we will work in the “quenched approximation”.
All the dynamics of the fermions and the related problems will be presented in next lecture (A. Patella ... good luck)

Quenched approximation:
Once integrated integrated out exactly the fermionic ﬁelds
4   1
Z=         DAµ det M e d x( 4 Fµν Fµν )

det M is an highly non-local term

A. Rago (U. Wuppertal)                          Lattice Introduction                             17 August 2010    7 / 20
Quenched Approximation
In the rest of this lecture we will work in the “quenched approximation”.
All the dynamics of the fermions and the related problems will be presented in next lecture (A. Patella ... good luck)

Quenched approximation:
Once integrated integrated out exactly the fermionic ﬁelds
4   1
Z=         DAµ det M e d x( 4 Fµν Fµν )

det M is an highly non-local term
the partition function is an integral over only background gauge conﬁgurations

A. Rago (U. Wuppertal)                          Lattice Introduction                             17 August 2010    7 / 20
Quenched Approximation
In the rest of this lecture we will work in the “quenched approximation”.
All the dynamics of the fermions and the related problems will be presented in next lecture (A. Patella ... good luck)

Quenched approximation:
Once integrated integrated out exactly the fermionic ﬁelds
4   1
Z=         DAµ det M e d x( 4 Fµν Fµν )

det M is an highly non-local term
the partition function is an integral over only background gauge conﬁgurations
One can write the action, by re-exponentiating

1
S = Sgauge + Squarks =                 d4 x       Fµν Fµν − i           log (det Mi )
4                   i

where the sum is over the quark ﬂavors.

We deﬁne the quenched approximation setting

det M = constant

A. Rago (U. Wuppertal)                          Lattice Introduction                             17 August 2010    7 / 20
A pure gauge action

The simplest loop you can draw on a lattice is a square composed by the product of four links.

ˆ †       ˆ †
Uµν = Uµ (i)Uν (i + µ)Uµ (i + ν )Uν (i)

If you make an a expansion of such a term what you will ﬁnd is:

2                   a4
2
ReTr(1 − Uµν ) =    Fµν F µν + O(a6 )
g                    4

with
Fµν = ∂µ Aν − ∂ν Aµ + g [Aµ , Aν ]
with the derivative term obtained from

1
∂x f (x) → ∆x (f (x + a) − f (x))
a

A. Rago (U. Wuppertal)               Lattice Introduction            17 August 2010   8 / 20
The Haar Measure

The last ingredient needed to complete the formulation of LGT as a quantum ﬁeld theory via
the path integral is to deﬁne the measure of integration over the gauge degrees of freedom.
Unlike the continuum ﬁelds Aµ , lattice ﬁelds are SU(3) matrices with elements that are
bounded in the range [0, 1].
Wilson proposed an invariant group measure, the Haar measure.

f (U )dU =      f (V U )dU =        f (U V )dU

As an example for SU(2) we have:

U   =      x0 Id + ix · σ
detU     =      x2 = (x0 )2 + x2 = 1
1
dU     =        δ(x2 − 1)d4 x
π

A. Rago (U. Wuppertal)                Lattice Introduction                17 August 2010   9 / 20
Continuum limit and critical behaviour

We just mentioned the possibility of taking the limit for a → 0.
It’s clear that at large distance correlator of two ﬁelds goes like

φ(x1 )φ(x2 )   c   ∝ e−m|x1 −x2 |

The correlation length which determines the rate of the exponential decay is therefore given by:

1
ξ=
am

If a continuum limit with a ﬁnite physical mass exists
By a suitable choice of bare parameters we can approach a limit where a → 0 while m stays
ﬁnite
ξ→∞
We are in presence of a critical point (second order phase transition)
Our concern is SU(N) gauge theories. Such theories are characterised by having an UV ﬁxed
point at g = 0.

A. Rago (U. Wuppertal)                Lattice Introduction              17 August 2010   10 / 20
The Wilson loop
Elitzur’s theorem:
In gauge theories it is impossible to ﬁnd a local quantity acting as an order parameter.

A. Rago (U. Wuppertal)               Lattice Introduction                17 August 2010   11 / 20
The Wilson loop
Elitzur’s theorem:
In gauge theories it is impossible to ﬁnd a local quantity acting as an order parameter.
Average values cannot distinguishes the Coulomb phase, in which charge states can be
produced, from the so called conﬁned phase.

A. Rago (U. Wuppertal)               Lattice Introduction                17 August 2010   11 / 20
The Wilson loop
Elitzur’s theorem:
In gauge theories it is impossible to ﬁnd a local quantity acting as an order parameter.
Average values cannot distinguishes the Coulomb phase, in which charge states can be
produced, from the so called conﬁned phase.
To study conﬁnement in pure gauge theories we need a gauge invariant non local quantity

A. Rago (U. Wuppertal)               Lattice Introduction                17 August 2010   11 / 20
The Wilson loop
Elitzur’s theorem:
In gauge theories it is impossible to ﬁnd a local quantity acting as an order parameter.
Average values cannot distinguishes the Coulomb phase, in which charge states can be
produced, from the so called conﬁned phase.
To study conﬁnement in pure gauge theories we need a gauge invariant non local quantity

Consider a rectangular loop CR,T of side lengths R and T

W (CR,T ) =               U
CR,T

The static quark potential V (R) is deﬁned by means of the large T behaviour of the
corresponding Wilson loop in the fundamental representation.

1
V (R) ≡ − lim        log W (CR,T )
T →∞     T

so that

W (CR,T )    ∼     Ce−T V (R)
T →∞

A. Rago (U. Wuppertal)               Lattice Introduction                17 August 2010   11 / 20
Strong coupling

One well-established nonperturbative method is strong coupling.

A. Rago (U. Wuppertal)            Lattice Introduction             17 August 2010   12 / 20
Strong coupling

One well-established nonperturbative method is strong coupling.
Strong coupling is tied to the lattice and cannot be derived directly for a continuum
theory.

A. Rago (U. Wuppertal)             Lattice Introduction                17 August 2010    12 / 20
Strong coupling

One well-established nonperturbative method is strong coupling.
Strong coupling is tied to the lattice and cannot be derived directly for a continuum
theory.
Once expanded each lattice integral in power of 1/g 2

A. Rago (U. Wuppertal)             Lattice Introduction                17 August 2010    12 / 20
Strong coupling

One well-established nonperturbative method is strong coupling.
Strong coupling is tied to the lattice and cannot be derived directly for a continuum
theory.
Once expanded each lattice integral in power of 1/g 2
It’s possible to perform the integrations that turns out to be only factorised group
integration

A. Rago (U. Wuppertal)              Lattice Introduction                17 August 2010   12 / 20
Strong coupling

One well-established nonperturbative method is strong coupling.
Strong coupling is tied to the lattice and cannot be derived directly for a continuum
theory.
Once expanded each lattice integral in power of 1/g 2
It’s possible to perform the integrations that turns out to be only factorised group
integration

The largest contribution correspond to the minimal area bounded to the Wilson loop

2
W (CR,T ) ∼ e−RT ln g

If no phase transition occurs when g 2 varies from inﬁnity to zero the gauge theory underlying is
always in a conﬁnement phase
Ex. SU(2) SU(3) in 4d

On the other side is a phase transition exists, the phase in which we can demonstrate the
existence of a conﬁning behaviour is not connected to the continuum limit.
Ex. SU(N) with N> 3 in 4d

A. Rago (U. Wuppertal)               Lattice Introduction                17 August 2010   12 / 20
Thermodynamics of Quantum Field Theory and the Lattice

In order to describe a ﬁnite temperature theory we require that one dimension (the Time) is
compactiﬁed with periodic boundary conditions.

1
The compactiﬁcation length LT a =        T
, where T is the temperature.
Periodic boundary conditions implies a new global symmetry of the action.
The symmetry group is the center C of the gauge group (SU(N ) → ZN ).

This symmetry can be realised as follows.

UT (i) → W0 UT (i)
∀ i belonging to an hyperplane a ﬁxed time

A. Rago (U. Wuppertal)                Lattice Introduction                   17 August 2010   13 / 20
Thermodynamics of Quantum Field Theory and the Lattice

The space-like plaquettes are not aﬀected by the transformation
−1
In timelike plaquette two contribution appear: W0 and W0 . Since they belong to the
center they cancel each other.
The Wilson action is invariant under such transformation.
It is impossible to re-absorb this global twist by means of local gauge transformations.

It is possible to deﬁne gauge invariant observables which are topologically non-trivial: The
Polyakov loop

LT
P (x) ≡ Tr         UT (i).
i=1

A. Rago (U. Wuppertal)               Lattice Introduction                17 August 2010   14 / 20
Thermodynamics of Quantum Field Theory and the Lattice

The relevant feature of the Polyakov loop is that it transforms under the above discussed
symmetry as follows:

P (x) → W0 P (x);
thus it is a natural order parameter for this symmetry.

It will acquire a non zero expectation value if the center symmetry is spontaneously
broken.
Its expectation value is related to the free energy of a single isolated quark.
Signature of deconﬁnement: phase transition which separates the regime in which the
center symmetry is unbroken from the broken symmetric phase.

A. Rago (U. Wuppertal)                Lattice Introduction                 17 August 2010   15 / 20
A partial summary

What we have learned (or at least mentioned) so far
Discretized the theory on a lattice

A. Rago (U. Wuppertal)                 Lattice Introduction   17 August 2010   16 / 20
A partial summary

What we have learned (or at least mentioned) so far
Discretized the theory on a lattice
Deﬁne the Wilson action

A. Rago (U. Wuppertal)                 Lattice Introduction   17 August 2010   16 / 20
A partial summary

What we have learned (or at least mentioned) so far
Discretized the theory on a lattice
Deﬁne the Wilson action
Search for a continuum limit

A. Rago (U. Wuppertal)                 Lattice Introduction   17 August 2010   16 / 20
A partial summary

What we have learned (or at least mentioned) so far
Discretized the theory on a lattice
Deﬁne the Wilson action
Search for a continuum limit
Order parameter for conﬁnement at zero Temperature (Wilson Loop)

A. Rago (U. Wuppertal)                 Lattice Introduction         17 August 2010   16 / 20
A partial summary

What we have learned (or at least mentioned) so far
Discretized the theory on a lattice
Deﬁne the Wilson action
Search for a continuum limit
Order parameter for conﬁnement at zero Temperature (Wilson Loop)
Order parameter for conﬁnement at non-zero Temperature (Polyakov Loop)

A. Rago (U. Wuppertal)                 Lattice Introduction         17 August 2010   16 / 20
A partial summary

What we have learned (or at least mentioned) so far
Discretized the theory on a lattice
Deﬁne the Wilson action
Search for a continuum limit
Order parameter for conﬁnement at zero Temperature (Wilson Loop)
Order parameter for conﬁnement at non-zero Temperature (Polyakov Loop)
An analytical method to justify conﬁnement of the continuum theory if the theory does
not have any bulk PT

A. Rago (U. Wuppertal)                 Lattice Introduction          17 August 2010   16 / 20
A partial summary

What we have learned (or at least mentioned) so far
Discretized the theory on a lattice
Deﬁne the Wilson action
Search for a continuum limit
Order parameter for conﬁnement at zero Temperature (Wilson Loop)
Order parameter for conﬁnement at non-zero Temperature (Polyakov Loop)
An analytical method to justify conﬁnement of the continuum theory if the theory does
not have any bulk PT

What we miss now is to understand how to obtain results from our discretized model.

A. Rago (U. Wuppertal)                 Lattice Introduction          17 August 2010   16 / 20
Dynamics

Ergodic Hypotheses:
average on a statistical ensemble           time average

Problem: give a dynamic to the system

Fundamental request: at the equilibrium you must generate conﬁguration
according to the Boltzmann distribution

A. Rago (U. Wuppertal)     Lattice Introduction         17 August 2010   17 / 20
Markov Chain

n           t                     t−1
Series of conﬁgurations Cm for which Cm depends only on Cn according
to a probability distribution Pnm (upper index time, lower index
conﬁguration index)
Irreducible if for each Cj it’s possible to reach Cl l > j, or
equivalently it exists a time k for which
k
Pjl = i1 ...in Pji1 Pi1 i2 ...Pin l = 0 for every j, k
k
Aperiodic if Pii = 0 for each i
Ci state is said to be positive if it reappear in a ﬁnite time

A. Rago (U. Wuppertal)       Lattice Introduction         17 August 2010   18 / 20
Equilibrium distribution
Given a {Ci } Markov chain irreducible, aperiodic and with only positive
states
The equilibrium distribution exists and is unique
hence it doesn’t depend on the start state
N
lim Pij = Pi
N →∞

A. Rago (U. Wuppertal)      Lattice Introduction        17 August 2010   19 / 20
Equilibrium distribution
Given a {Ci } Markov chain irreducible, aperiodic and with only positive
states
The equilibrium distribution exists and is unique
hence it doesn’t depend on the start state
N
lim Pij = Pi
N →∞

The equilibrium distribution is stationary
1
Pj =            Pij Pi
i

A. Rago (U. Wuppertal)      Lattice Introduction        17 August 2010   19 / 20
Equilibrium distribution
Given a {Ci } Markov chain irreducible, aperiodic and with only positive
states
The equilibrium distribution exists and is unique
hence it doesn’t depend on the start state
N
lim Pij = Pi
N →∞

The equilibrium distribution is stationary
1
Pj =            Pij Pi
i

if the variance of the reappear time is ﬁnite
N
1
Pi O(Ci ) = O = lim                    O(Cj )
N →∞ N
i                                    j=1

A. Rago (U. Wuppertal)              Lattice Introduction                17 August 2010   19 / 20
Detailed balance

Montecarlo Dynamics: any Markov dynamics

A. Rago (U. Wuppertal)   Lattice Introduction   17 August 2010   20 / 20
Detailed balance

Montecarlo Dynamics: any Markov dynamics
Problem: Given an action, build a Markov dynamic

A. Rago (U. Wuppertal)    Lattice Introduction      17 August 2010   20 / 20
Detailed balance

Montecarlo Dynamics: any Markov dynamics
Problem: Given an action, build a Markov dynamic
Necessary condition: not known

A. Rago (U. Wuppertal)    Lattice Introduction      17 August 2010   20 / 20
Detailed balance

Montecarlo Dynamics: any Markov dynamics
Problem: Given an action, build a Markov dynamic
Necessary condition: not known
Suﬃcient condition: Detailed balance

e−βH(Ci ) Pij = e−βH(Cj ) Pji

A. Rago (U. Wuppertal)        Lattice Introduction       17 August 2010   20 / 20
Detailed balance

Montecarlo Dynamics: any Markov dynamics
Problem: Given an action, build a Markov dynamic
Necessary condition: not known
Suﬃcient condition: Detailed balance

e−βH(Ci ) Pij = e−βH(Cj ) Pji

and you are still free to choose Pij , decide you algorithm

A. Rago (U. Wuppertal)        Lattice Introduction       17 August 2010   20 / 20

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