Lattice QCD with chirally invariant fermions

Adam Mi kiewi z University

Pozna«, Poland

Ph.D. Thesis

Latti e QCD with

hirally invariant fermions

Krzysztof Ci hy

Supervisors

Dr hab. Piotr Tom zak, Prof. UAM

Adam Mi kiewi z University

Quantum Physi s Division

and

Dr. habil. Karl Jansen

NIC, DESY Zeuthen

Introdu tion 7

1 Theoreti al prin iples

of Latti e QCD 11

1.1 The QCD Lagrangian. . . 11

1.2 Dis retizing gaugeelds . . . 20

1.3 Dis retizing fermions . . . 22

1.3.1 Naive dis retization . . . 22

1.3.2 Wilson fermions . . . 23

1.3.3 Wilson twistedmass fermions . . . 26

1.4 Chiral symmetry onthe latti e . . . 29

1.4.1 Ginsparg-Wilson relation . . . 29

1.4.2 Overlapfermions . . . 30

1.4.3 Other kindsof hiral fermions . . . 32

1.4.3.1 Domain wall fermions . . . 32

1.4.3.2 Creutz fermions. . . 33

1.4.4 Topology onthe latti e . . . 34

1.5 Observables in Latti e QCD . . . 35

2 Tree-level s aling test 41 2.1 Fermion propagators . . . 41

2.1.1 Overlapfermions . . . 41

2.1.2 Wilson twistedmass fermions . . . 42

2.1.3 Creutz fermions . . . 43

2.2 Observables . . . 43

2.3 Test setup . . . 45

2.4 Comparison of overlap,twistedmass and Creutz fermions . . . 45

2.5 Mat hing twisted mass and overlap fermions . . . 52

2.5.1 Unmat hed quarkmasses. . . 54

3.1 SimulatingQCD . . . 61

3.1.1 General idea . . . 61

3.1.2 Hybrid MonteCarlo . . . 62

3.2 Computationof the overlap operator . . . 64

3.3 Redu ing the ondition numberof

A

†

_A

. . . 66

3.3.1 Eigenvalue deation . . . 66

3.3.2 HYP smearingof gaugeelds . . . 68

3.4 Inverting the Dira operator . . . 69

3.4.1 Sto hasti sour es . . . 70

3.4.2 The SUMR solver . . . 72

4 Investigations of the ontinuum limit s aling properties of the mixed a tion setup 75 4.1 Mixed a tionapproa h . . . 75

4.2 S aling test light sea quark mass . . . 78

4.2.1 Simulationparameters . . . 79

4.2.2 Lo ality . . . 80

4.2.3 Mat hing the pion mass . . . 83

4.2.4 Pion de ay onstant  s alingtest . . . 85

4.3 Chiralzero modes and their ontributionto mesoni orrelators 88 4.3.1 Chiral zero modes . . . 88

4.3.2 The ontributionofthezeromodestomesoni orrelators 90 4.3.3 Comparison of orrelation fun tions . . . 93

4.4 The roleof the zeromodes smallvolume, lightseaquark mass 96 4.5 The role of the zero modes nite volume ee ts analysis. . . 101

4.5.1 Simulationparameters . . . 101

4.5.2 Mat hing the pion mass PP orrelator . . . 101

4.5.3 Pion de ay onstant  PP orrelator . . . 102

4.6 The role of the zero modes smallvolume, heavier sea quark mass . . . 104

4.6.1 Motivation and simulation setup. . . 104

4.6.2 Pion de ay onstant  s alingtest . . . 106

4.7 The role of the zero modes  on lusion . . . 109

4.8 Expli it subtra tion of zero modes . . . 111

4.8.1 Subtra tion pro edure . . . 111

4.8.2 Ee ts of expli it zero modes subtra tion . . . 112

5.1 Unitarity violations . . . 121

5.1.1 Motivation. . . 121

5.1.2 Small volumeanalysis . . . 124

5.2 Light baryonmasses . . . 128

5.3 Topologi al harge and sus eptibility . . . 131

Con lusions and prospe ts 135 A knowledgements 139 A Wilson gauge a tion 141 B Tree-level s aling test 143 B.1 Overlap fermions . . . 143

B.2 Creutz fermions . . . 145

B.3 Correlation fun tions . . . 147

C Improvements of the HMC algorithm 149 D Tree-level test of zero modes subtra tion 151 D.1 Analyti al formula . . . 152

D.2 GWC ode  point sour es . . . 153

D.3 GWC ode  sto hasti sour es . . . 155

The strongfor eplaysafundamentaland ru ialroleinnature. Itis

respon-sible for the formation of all hadrons, whi h an be lassied into mesons

and baryons. Examples of the former are the pion and the

ρ

meson and of the latter the proton and the neutron, whi h in turn form the nu lei of

all atoms. The theory of the strong intera tion is believed to be Quantum

ChromoDynami s (QCD). It postulatesthat allhadrons are not elementary

themselves but they have an inner stru ture and are built from onstituent

parti les. S attering experiments revealed that the onstituents are

point-like obje ts and we now have a large amount of eviden e that they an be

identied with quarks, whi h are spin-1/2 fermions and whose intera tion

is mediated by spin-1 bosons known as gluons. In order to understand the

intera tion amongquarksandgluonsand omprehendhowit an leadtothe

formation of hadrons anew quantum number, alled the olour harge, had

to beintrodu ed. However, allhadrons observed in experiment donot arry

this olour harge,butare olourless. Thismeansthatthequarksandgluons

an not beisolatedand donot existasfreeparti lesthey are onnedinto

olour-neutral omposite hadrons. This fundamental onnement property

of QCD resultsfrom the fa tthat at largedistan es (oratlow energies) the

QCD oupling onstant determining the intera tion strength between the

quarks and the gluons islarge. However, we know fromperturbation theory

analyses of QCDthat atsmalldistan es (orat highenergies) the QCD

ou-pling onstantbe omessmallandthe quarksbehaveasalmostfreeparti les.

This property of QCD is alled asymptoti freedom and has been tested by

onfronting experimental results with perturbative QCD predi tions. It is

one ofthe mostamazing hara teristi sof QCDthatitshoulddes ribeboth

phenomena, onnement and asymptoti freedom, simultaneously. Clearly,

inorder totest this theoreti alexpe tation, amethodisneeded whereQCD

an be evaluated both in the perturbative regime at small distan es and in

the non-perturbative regime at large distan es, where we enter the world of

the observed hadrons.

late the hadron spe trum and many stru tural properties of hadrons, like

form fa tors or parton distribution fun tions), non-perturbative methods

have to be employed. The only method whi h fullls the above riterion

and allows for pre ise quantitative predi tions is Latti e QCD (LQCD). It

onsists indis retizing spa e-time and formulatingQCD ona 4-dimensional

Eu lidean spa e-timegrid witha latti espa ing

a

. In thisway,the theoryis fullyregularizedandmathemati allywelldened, whi hledtomany

on ep-tualandtheoreti aldevelopmentsinourunderstandingofQCD.Ontheother

hand,byusingFeynman'spathintegralformulationofquantumeld theory,

LQCD anbeinterpreted asa kindof astatisti alme hani alsystem whi h

allows an evaluation with numeri al methods. LQCD was rst proposed in

a seminal paper by Wilson in 1974 [1℄ and shortly after Creutz indeed

per-formedsu hnumeri alsimulationsusingMarkov hainMonteCarlomethods

[2℄. IthastobesaidthatovermanyyearsLQCDsimulationswereperformed

in unphysi al setups with mu h too heavy and even innite quark masses.

However, inthelastfewyearsatremendousprogresshas beena hievedwhen

new algorithmi developments provided a breakthrough in the performan e

oftheusedsimulationalgorithms. Atthesametime,thein reasing omputer

powermadeitpossibletosimulateonlargelatti eswithne latti espa ings

and pion masses approa hing the physi al pion mass. Latti e QCD

om-putations still require very large omputer resour es, parti ularly for fully

dynami al simulations, but its prospe ts are steadily improvingwith a new

generation of super omputers in the PetaFlop range. The algorithmi and

omputer improvementswere alsoa ompaniedby on eptualdevelopments

su hasonesleadingtoafasterapproa htothe ontinuumlimit(

a → 0

)and the formulationof non-perturbativerenormalizations hemes.

Anotherimportantaspe t of QCDis hiral symmetry,i.e. the invarian e

of the theory under the ex hange of massless left- and right-handed quarks.

Itisa ontinuoussymmetryandwebelievethatitisspontaneouslybroken in

nature, thusgiving rise to the appearan e of Goldstone bosons. In QCDwe

identify these Goldstone bosons with the pions, whosemass is mu hsmaller

than the mass of any other observed hadron. Assuming su h spontaneous

breaking of hiral symmetry inQCD, manyphenomenologi alinvestigations

an beperformed tointerpretexperimental data,the most notableof whi h

is hiral perturbationtheory.

Inprin iple, LQCDshouldbe abletodedu e the phenomenon of

sponta-neous hiral symmetry from the QCD Lagrangian itself and one would not

have to rely on assumptions. However, for many years, it seemed

impos-sible to preserve hiral symmetry on the latti e. Only in the late 1990s it

that a latti e fermion an be hiral, provided that we allow for a

latti e-modied version of hiral symmetry. This dis overy led to the introdu tion

of so- alled overlap fermions, a kind of latti e fermion whi h respe ts this

latti e modied hiral symmetry. Overlap fermions have many appealing

properties, but are mu h more omputationallydemanding than other

pop-ular fermion dis retizations, su h as Wilson fermions, modi ations thereof

or staggered fermions. This makes the use of overlap fermions still a

hal-lenge,espe iallyindynami alsimulations. Assu h,alternativestodynami al

overlap fermions are being looked for to keep hiral symmetry. The goal is

is to prot from the good hiral properties of overlap fermions, but at the

same time avoid the high omputational ost of generating dynami al

over-lapgaugeeld ongurations. One su happroa his alledmixed a tionand

it onsists in using overlap fermions only as valen e quarks and for the sea

se tor a heaperfermion dis retizationis used.

The aimof this thesis is toinvestigatethe mixed a tionsetup of overlap

valen e fermions and Wilson twisted mass sea quarks. One may suspe t

that usingdierentlatti efermionformulationsinthesea andinthevalen e

se tor leads tounphysi al ee ts. And, aswe willshowin this thesis,this is

indeedthe ase. Aswewilldemonstrate,inordertohaveasafe simulation,

where su hee ts an be avoided,a areful tuningof the physi alsetup has

to be performed. It is one of the main goals of this thesis to spe ify the

regime of parameter values (su h as the latti e volume and the pion mass)

that allows toperformsu hsafesimulations. Knowingthese parameterswill

then allow to address physi al questions and ompute physi al observables

without being ae ted by possible unphysi al ee ts. Therefore, providing

theparametersforsafesimulationsopens thewayforfuturesimulationswith

hirallyinvariantoverlapfermionsinthevalen ese torto omputeimportant

physi al quantities.

The outlineof the thesis is the following.

In Chapter 1, we review the theoreti al prin iples of Latti e QCD. We

startbyintrodu ingthe ontinuumQCDLagrangian anddis ussingits

sym-metries,parti ularlythe hiralsymmetry. Next,weshowhowthe ontinuum

theory is dis retized and we introdu e dierent fermion dis retizations,

in- luding the hirally-symmetri overlap formalism. We also shortly dis uss

the ways of extra tingphysi al observables from asimulation.

Chapter 2 presents the results of a latti e spa ing s aling test of

dif-ferent fermion dis retizations at tree-level of perturbation theory. For this

we use overlap, twisted mass and Creutz fermions. We also investigate the

ee ts of mat hing of twisted mass and overlap fermions, whi h is relevant

of QCD simulations. We review the HMC algorithm and the te hniques

used to ee tively deal with overlap fermions, in parti ular the method of

omputation of the overlap Dira operator, ways of redu ing the ondition

numberof its kernel and the use of sto hasti sour es.

The main results of the thesis are reported in Chapter 4. First, the

motivation and the general idea of a mixed a tionsimulationare dis ussed.

Then,a ontinuumlimits alingtestofthepionde ay onstantisperformed.

This test motivates the analysis of the role of hiral zero modes of the

over-lap operator. We show that this is a very important ee t in the ase of a

hirally-symmetri valen eandnon- hirally-symmetri seaquarks

dis retiza-tion. This hapter on ludes with the aforementioned range of parameter

values that are ne essary fora simulationsafe against these ee ts.

In Chapter 5 we dis uss some further results, in luding the unitarity

violationspresentinthemixeda tionsetup,lightbaryonmasses omputation

Theoreti al prin iples

of Latti e QCD

1.1 The QCD Lagrangian

Quantum ChromoDynami s (QCD) is a gauge theory of strong nu lear

in-tera tions between the onstituents of hadrons. The hadrons are a lass of

parti les in luding baryons (e.g. the nu leon) and mesons (e.g. the pion).

The theory is based on the prin iple of lo al gauge invarian e with a

non-Abelian SU(3) gauge group [3, 4℄. The fundamental degrees of freedom of

the theory are quarks and gluons. The Lagrangian density of QCD an be

written as:

L

QCD

= L

quark

+ L

gluon

+ L

int

,

(1.1)

where

L

quark

is the purely fermioni (quark) part,

L

gluon

the purely bosoni (gluon) part and

L

int

the intera tion part that ouples quarksand gluons.

Let us now onsider the dierent parts that onstitute the QCD

La-grangian. The quark term is 1 :

L

quark

=

N

f

X

f =1

¯

ψ

f

(x)(iγ

µ

∂

µ

− m

f

)ψ

f

(x),

(1.2)

where

N

f

is the number of avours 2

,

ψ

f

(x)

is the quark (spinor) eld or-responding to avour

f

and

m

f

is the

f

-avour bare quark mass and the

1

Throughoutthethesis,weemploytheEinsteinsummation onventionforDira indi es

(denoted byGreekletters)andSU(3)-groupgeneratorindi es(denotedbyLatinletters).

2

The Standard Model in orporates 6 avours of quarks (up, down, strange, harm,

bottom, top). However, investigating the low-energy properties of QCD with Latti e

{γ

µ

, γ

ν

_{} = 2η}

µν

,

(1.3) where

η

µν

₌

diag

(1, −1, −1, −1)

isthe metri tensor. The gluon part reads:

L

gluon

= −

1

4

F

a

µν

(x)F

a

µν

(x),

(1.4) where

F

a

µν

(x)

is the eld strength tensor, whi h is relatedto the gluon eld omponents

A

a

µ

(x)

:

F

_µν

a

(x) = ∂

µ

A

a

ν

(x) − ∂

ν

A

a

µ

(x) − gf

abc

A

b

µ

(x)A

c

ν

(x),

(1.5)

where

g

isthe bare oupling onstantand

f

abc

are the stru ture onstantsof

SU(3), satisfying the ommutation relations:

[t

a

, t

b

] = if

abc

t

c

,

(1.6)

where

t

a

are the generators of the group SU(3).

The purely bosoni part of the Lagrangian is invariant with respe t to

the lo al gauge transformation. If we want the fermioni part to obey the

lo algauge symmetry aswell, we haveto introdu e a term that ouples the

fermions and bosons, i.e. des ribes the intera tion between them. This is

the basi buildingprin ipleof alllo algaugetheories. It wasrst dis overed

in the ase of the ele tromagneti intera tion, where a term that ouples

ele trons and photons is ne essary to guarantee the lo al gauge invarian e.

In the ase ofQCD, the sum

L

quark

+ L

gluon

is not invariant withrespe t to the lo al SU(3) transformation and the way to guarantee this invarian e is

to introdu e the intera tion term

L

int

that ouples the quark elds

ψ

and gluon elds

A

µ

:

L

int

= g

N

f

X

f =1

¯

ψ

f

(x)γ

µ

A

µ

(x)ψ

f

(x),

(1.7)

where the gluon eld

A

µ

is related toits omponentsin the followingway:

A

µ

(x) = t

a

A

a

µ

(x).

(1.8)

Conventionally,onewritestheterms

L

quark

and

L

int

together,introdu ing the ovariant derivative

D

µ

:

L

QCD

=

N

f

X

f =1

¯

ψ

f

(x)(iγ

µ

D

µ

− m

f

)ψ

f

(x) −

1

4

F

a

µν

(x)F

a

µν

(x).

(1.10)

Letusalsodenethe ( lassi al)QCDa tion,whi his the integralof the

Lagrangian density overspa e-time:

S

QCD

=

Z

d

4

_{x L}

QCD

.

(1.11)

Anelegant(and relevantfrom the point of viewof Latti e QCD) way to

quantize a lassi al theory, like the one given by the lassi al QCD a tion

(1.11), is to use the Feynman path integral formalism [5℄. The expe tation

value of any observable

O

is given by:

hOi =

_Z

1

Z

D ¯

_{ψDψDA O[ψ, ¯}

ψ, A] e

iS

QCD

[ψ, ¯

ψ,A]

_,

(1.12)

with the partition fun tion:

Z =

Z

D ¯

ψDψDA e

iS

QCD

[ψ, ¯

ψ,A]

_.

(1.13)

Itisworthtoemphasizethatalleldsinthepathintegralare lassi al. Su h

path integral an not beevaluated analyti ally (ex ept for few spe ial ases

mu h simpler than QCD) and one has to swit h to approximate methods.

Formany theories, likeQuantumEle troDynami s (QED), averysu essful

method is perturbation theory. It onsists in expanding the path integral

with respe t to a small parameter (e.g. the ne stru ture onstant

α ≈

1/137.036

inQED)anddroppingtermsbeyondsomeorder. Forexample,the most re ent al ulation of the anomalous magneti moment of the ele tron

(usually parametrized in terms of the so- alled

g

-fa tor) up to fourth-order in

α

agrees with experiment up to 10 signi ant digits, making it one of the most pre isely veried predi tion of physi s  the ele tron

g

-fa tor is

g

e

= 2a

e

+2

,wherethetheoreti alvalue:

a

th

e

= 1 159 652 182.79(7.71)×10

−12

andtheexperimentalone:

a

exp

e

= 1 159 652 180.73(0.28)×10

−12

[6℄. However, for perturbative methods to work, there has to be a small parameter with

respe t to whi h one expands the path integral. In the ase of QCD, the

oupling onstant of the olour intera tion depends on energy and one has

to onsider two regimes. For high energy or large momentum transfer, the

QCD oupling onstantissmallenoughforperturbativemethodstowork. In

by Gross, Politzer and Wil zek. However, in the ase of low energy orsmall

momentum transfer, this oupling onstant be omes of the order of unity

and perturbation theory is bound to fail  the strong intera tions be ome

strongindeed. Quantitatively,theenergys alewhenithappens

Λ

strong

≈ 250

MeV, where the value is not pre isely dened and depends on the hosen

observable. Anyway, its approximate value implies that a vast number of

relevant phenomena in QCD, su h asthe onnement of quarks and gluons

into hadrons, happen in the non-perturbative regime. Thus, one needs

non-perturbativemethods,su hasLatti eQCD,whi histheonlyknownmethod

of extra ting quantitative predi tions about the low-energy regimeof QCD.

This approa h onsists in dis retizing the QCD path integral. In this way,

one obtainsafullyregularized andwell-dened theory, whi h an bestudied

numeri ally, but also analyti ally  the dis retized version of QCD enabled

manyrelevant on eptualdevelopmentsandledtoimportantinsightintothe

nature of strong intera tions.

However, the os illating exponential

e

iS

QCD

[ψ, ¯

ψ,A]

renders the numeri al

evaluation of the QCD path integral unfeasible from the pra ti al point of

view. Fortunately, integrals like (1.12) are tra table, if one swit hes from

Minkowski spa e-time with metri tensor

η

µν

with signature e.g.

(+ − −−)

toEu lideanspa e-timewithsignature

(++++)

. Thisisa hievedbyanalyti ontinuation (Wi k rotation of the time dire tion:

t → −iτ

). In order that the Eu lidean formulation an be ontinued ba k to physi al (Minkowski)

spa e,the Eu lidean orrelationfun tionshavetosatisfya ertain ondition,

alled the Osterwalder-S hrader ree tion positivity [7, 8℄. This ondition

ensures that the transition probabilities between gauge-invariant states are

non-negative and the quantum me hani al Hamiltonian has only real and

positiveeigenvalues [9℄.

The QCDLagrangian density inEu lidean spa e reads [10℄:

L

E

QCD

=

N

f

X

f =1

¯

ψ

f

(x)(γ

µ

E

D

µ

+ m

f

)ψ

f

(x) −

1

4

F

a

µν

(x)F

a

µν

(x)

(1.14)

and the Eu lidean gammamatri es satisfy:

{γ

µ

, γ

ν

_{} = 2δ}

µν

,

(1.15)

where

δ

µν

₌

diag

(1, 1, 1, 1)

isthe Eu lidean metri tensor. Theexpe tation value of any observable

O

is then given by:

hOi =

_Z

1

_E

Z

D ¯

_{ψDψDA O[ψ, ¯}

ψ, A] e

−S

QCD

E

[ψ, ¯

ψ,A]

,

where

S

E

QCD

=

R d

4

xL

E

QCD

is the Eu lidean a tion and the Eu lidean parti-tion fun tion reads:

Z

E

=

Z

D ¯

ψDψDA e

−S

QCD

E

[ψ, ¯

ψ,A]

.

(1.17)

The os illating exponential in (1.12) is repla ed by the well-behaved fa tor

e

−S

QCD

E

and thus the multi-dimensional integral (1.16) an beevaluated

nu-meri ally,atleastinprin iple,e.g. withMonteCarlomethods. Formally,the

quantum eld theory dened by the partition fun tion (1.17) an be

inter-preted as a statisti al me hani al system and the exponential

e

−S

E

QCD

plays

the role of aBoltzmann fa tor.

Fromnowon, wewillworkonlywiththe Eu lideanformulationof SU(3)

non-Abelian gauge theory (QCD) and hen e we drop the supers ript

E

and the subs ript

QCD

thatremind us of it.

Now,wewilldis ussafewimportantfeaturesof ontinuumQCDthatare

relevant from the point of view of further onsiderations, espe iallythe role

of hiral symmetry and spontaneous hiral symmetry breaking[10, 11,4℄.

Tobespe i , letusrestri tourselvestotwoavours ofquarks (

u

and

d

quarks). The lassi alQCD Lagrangian an be rewritten as:

L = ¯uγ

µ

D

µ

u + ¯

dγ

µ

D

µ

d + ¯

um

u

u + ¯

dm

d

d −

1

4

F

a

µν

F

a

µν

≡

≡ L

u

+ L

d

+ L

m

u

+ L

m

d

+ L

gluon

,

(1.18)

where

u ≡ ψ

u

and

d ≡ ψ

d

are the orresponding spinors and we have sep-arated the mass terms in the fermioni Lagrangian. We an de ompose the

quark Lagrangian further by dening left-handed and right-handed quark

spinor elds:

q

R

≡ P

+

q,

q

L

≡ P

−

q,

q = u, d,

(1.19) where:

P

±

=

1 ± γ

5

2

.

(1.20)

Eq. (1.19) impliesfor the onjugate spinor elds:

¯

q

R

= ¯

qP

−

,

q

¯

L

= ¯

qP

+

.

(1.21)

Thus, the rst twoterms in Lagrangian (1.18)be ome:

L

u

+ L

d

= ¯

u

L

γ

µ

D

µ

u

L

+ ¯

u

R

γ

µ

D

µ

u

R

+ ¯

d

L

γ

µ

D

µ

d

L

+ ¯

d

R

γ

µ

D

µ

d

R

=

(1.22)

=

u

¯

L

d

¯

L

γ

µ

D

µ

0

0

γ

µ

D

µ

 u

L

d

L



+ ¯

u

R

d

¯

R

γ

µ

D

µ

0

0

γ

µ

D

µ

 u

R

d

R



,

terms we obtain:

L

m

u

+ L

m

d

= m

u

(¯

u

L

u

R

+ ¯

u

R

u

L

) + m

d

( ¯

d

L

d

R

+ ¯

d

R

d

L

) =

(1.23)

=

u

¯

L

d

¯

L

m

u

0

0 m

d

 u

R

d

R



+ ¯

u

R

d

¯

R

m

u

0

0 m

d

 u

L

d

L



,

i.e. the mass terms oupleelds of opposite hiralities.

Let us now onsider the massless terms

L

u

and

L

d

in the Lagrangian. They are invariant with respe t to the following transformations,

respe -tively:

u

L

d

L



→ L

u

_d

L

L



,

u

R

d

R



→ R

u

_d

R

R



,

(1.24)

where

L

and

R

areunitary

2×2

matri es,i.e. elementsofthe(avour)group U(2). This means that the Lagrangian

L

u

+ L

d

is invariant with respe t to the group U(2)

L

×

U(2)

R

.

Let us take a loser look at the possible forms of transformations. The

masslessquark Lagrangianisinvariantunder fourSU(2)

×

U(1)ve tor trans-formations:

u

d



→ e

iαu

i

u

d



,

u ¯

¯

d

_{→ ¯u ¯}

d
e

−iαu

i

_,

(1.25)

wherethesubs ript

i = 0, 1, 2, 3

,

u

0

istheidentitymatrixinavourspa eand

u

i

(

i = 1, 2, 3

) are avour SU(2) group generators. There are 4 onserved (ve tor) Noether urrents

j

µ

i

asso iated with these 4 transformations and hen e 4 onserved harges

Q

i

=

R d

3

_xj

0

i

 the baryon number (

i = 0

) and the isospin (

i = 1, 2, 3

).

Inaddition,therearetransformationsinvolving

γ

5

, alled hiralrotations:

u

d



→ e

iαγ

5

u

i

u

d



,

u ¯

¯

d

_{→ ¯u ¯}

d
e

iαγ

5

u

i

_.

(1.26)

Togetherwithtransformations(1.25),themasslessquarkLagrangian

L

u

+L

d

is invariant underthe symmetry groupSU(2)

R

×

SU(2)

L

×

U(1)

V

×

U(1)

A

.

However, it an be shown that the fermion integration measure in the

quantized theory is not invariant under the transformation (1.26) for

i = 0

, whi hredu es the fullsymmetry to SU(2)

R

×

SU(2)

L

×

U(1)

V

. This isthe so- alled axial anomaly and it has important onsequen es e.g. for the meson

spe trum  the hiral avour singlet symmetry an not be broken

sponta-neously and hen e there isno Goldstone boson asso iatedwith spontaneous

breaking of this symmetry. This implies that the mass of the avour

sin-glet

η

′

opposed to the mass of the

η

meson, whi h is one of the pseudo-Goldstone bosons),but it isrelatedto topologi alu tuationsof the QCD va uumvia

the Witten-Veneziano formula[12, 13℄:

f

2

π

2N

f

m

2

_η

+ m

2

_η

′

− 2m

2

K

= χ

top

,

(1.27)

where

f

π

is the pion de ay onstant,

m

x

the mass of the

x

meson and

χ

top

the topologi al sus eptibility, whi h willbe dened later.

Letus now onsider the mass terms of the QCD Lagrangian

L

m

u

+ L

m

d

. They areinvariantwithrespe t tothe transformation(1.25)for

i = 0

,sothe baryon number is onserved also in the massive theory. For

i = 1, 2, 3

the transformation(1.25)isasymmetryonlyif thequarkmassesare equal

m

u

=

m

d

. Hen e, theisospinis onserved inthemassivetheory,butonlyfor mass-degenerate quarks. However, the mass terms

L

m

u

+ L

m

d

are not invariant under hiralrotations(1.26),whi his ausedbythefa tthattheexponential

in (1.26) is the same for the spinor

(u d)

T

and the onjugate spinor

u ¯

¯

d

,

whi h is, in turn, due to the anti ommutation relation

{γ

µ

, γ

5

} = 0

. Thus, the symmetry ofthe quantum QCDLagrangianisbroken toSU(2)

V

×

U(1)

V

in the mass-degenerate ase and to U(1)

V

×

U(1)

V

if

m

u

6= m

d

.

In the ase of arbitrary number

N

f

of quark avours, the analysis is easilygeneralized (the matri es

u

i

are nowthe

N

f

× N

f

identity matrix and

N

2

f

− 1

generatorsofthe avour groupSU(

N

f

))andthefullsymmetry ofthe quantized massless QCD Lagrangian is SU(

N

f

)

R

×

SU(

N

f

)

L

×

U(1)

V

, whi h is redu ed to SU(

N

f

)

V

×

U(1)

V

in the mass-degenerate ase and further to U(1)

V

× . . . ×

U(1)

V

(with

N

f

fa tors U(1)

V

) in the ase of dierent quark masses. Thus, in the latter ase, the only exa t symmetry is the baryon

number onservation.

However,sin etheisospinsymmetryisonlyslightlybrokenforthelightest

twoquarks, itisoftentreatedasexa t 3

,while theheavierquarksare treated

separately. Moreover, sin e the up and down quarks are so light, ompared

to the heavier quarks (

m

u

≈ m

d

≈

a few MeV, whereas already

m

s

≈ 100

MeV), the full symmetry of the massless Lagrangian with

N

f

= 2

avours SU(2)

R

×

SU(2)

L

×

U(1)

V

remains an important approximate symmetry and is the basis of

N

f

= 2

hiral perturbation theory (

χ

PT ). At low energy, the quarks and gluons are onned into hadrons and hen e one an dene

an ee tive eld theory, in whi h the fundamental degrees of freedom are

not quarksand gluons,but lighthadrons. Two-avour

χ

PT wasformulated by Gasser and Leutwyler [14℄. The Lagrangian of this theory is onstru ted

from elds des ribing the pions (

π

±

,

π

0

) in a way whi h is onsistent with

3

ganized in terms of expansion parameters

p/Λ

χ

and

m

π

/Λ

χ

, where

p

is the momentum,

m

π

thepion massand

Λ

χ

= (4πf )

2

thetypi alhadroni s ale

≈

1 GeV,with

f

the pion de ay onstantin the hirallimit. Thereare many appli ations of

χ

PT in the analysis of the low-energy regime of QCD, e.g. pion s attering experiments. Moreover, it is also essential in the analysis of

Latti eQCDdata, sin emost of ontemporaryLatti eQCDsimulationsare

performed at unphysi al values of the pion mass 4

 hen e an extrapolation

to the physi al point (physi al pion mass) is ne essary and is performed by

tting

χ

PT formulas. What is more, even thoughthe strange quarkmass is mu h larger than the mass of the up and down quarks, it is still relatively

small ompared to the typi al QCD s ale of

≈

1 GeV and the symmetry SU(3)

R

×

SU(3)

L

×

U(1)

V

of the massless

N

f

= 3

Lagrangian is also an ap-proximate symmetry and forms the basis of

N

f

= 3

hiral perturbation the-ory,whi hisalsoof use inthe analysis oflow-energyQCDexperiments,e.g.

in luding the kaons (also in kaon physi s from Latti e QCD). Three-avour

χ

PT was also introdu ed by Gasser and Leutwyler [15℄ as a generalization of the two-avour ase to in lude the strange quark. The three-avour

La-grangianin ludes,besidesthepionelds,alsootherlightpseudos alarmeson

elds (of the remaining pseudo-Goldstone bosons  the kaons

K

±

,

K

0

,

K

¯

0

and the

η

meson). Quantitatively, the expli it breaking of hiral symmetry by the quark masses an be expressed by the ratios

m

2

π

/(4πf )

2

≈ 0.007

and

m

2

K

/(4πf )

2

≈ 0.09

. In this sense, the expli itbreaking by the strangequark mass isroughlya10%ee t, whileforthe lightestquarksitisa<1% ee t.

Obviously,itisnot possibletotreat the

N

f

= 4

symmetryasapproximately valid,sin ethe harmquarkisalreadyheavy(

m

c

≈ 1.3

GeV)andthemesons ontaining itare mu hheavier than the s ale

Λ

χ

.

However, if hiralsymmetrywasbroken onlyexpli itly,wewouldobserve

degenerate multiplets of hadrons  e.g. there should be s alar mesons with

massesverysimilartothepseudos alarones. Also,inthis aseoneshouldnot

expe t su h big dieren e between the masses of the pions and kaons. The

explanation of these phenomena an beprovided by anassumption that the

hiral symmetry of QCD is not only expli itly broken by the quark masses,

butalsospontaneouslybroken. Wespeakofspontaneoussymmetrybreaking

if asymmetrywhi hispresentatthe Lagrangianlevelisabsentinthe

phys-i al ground state 5

. If a ontinuous symmetry is broken spontaneously, then

4

Some ollaborationshavere entlystartedorarepreparingsimulationsatthephysi al

pion mass.

5

A learexampleis provided byferromagnets. EventhoughtheHamiltonian ofsu h

system is invariantwith respe t to a simultaneous ip of allspins, in an experimentall

interpreted asthe wouldbe-Goldstonebosons of hiralsymmetrybreaking,

wheretheprex wouldbe-referstothefa tthattheyare notmassless,but

have asmallmass ( ompared tothe masses of other hadrons) that isdue to

(small) expli it breaking of hiral symmetry by the quark masses.

Also, spontaneous breaking of hiral symmetry an be observed in the

mass dieren e of parti les that are hiral partners and should have the

same mass,if hiralsymmetry wasexa t. Sin e hiralsymmetry isexpli itly

broken bythe quark masses,the experimental massvalues of hiral partners

should not be equal, but they should be lose to ea h other, be ause the

masses of the lightquarks are so small. This is not observed. Forexample,

the ve tor mesons

ρ

and

a

1

have massesequalto,respe tively, 770 and 1260 MeV,whi hisamu hlargerdieren ethanone wouldexpe tfromthesmall

expli itbreakingof hiralsymmetry[16℄. Anotherexampleisthenu leonand

itsnegative-paritypartner,usuallydenotedby

N

∗

[11,17℄. Theexperimental

value of the nu leon mass is

m

N

≈ 940

MeV, while

m

N

∗

≈ 1535

MeV. Spontaneous hiralsymmetrybreakingissignalledbyanon-zerovalueof

the hiral ondensate

h0|¯uu|0i

,where

|0i

is theva uumstate. This quantity emerges in hiral perturbation theory as an important low-energy onstant

B

0

:

B

0

= −f

−2

h0|¯uu|0i,

(1.28) where the tree-level pion de ay onstant

f

is another low-energy onstant. A well-known relationthat involves the hiral ondensate is the Gell-Mann,

Oakes, Renner (GMOR) relation[18℄:

f

2

m

2

_π

_{= −(m}

u

+ m

d

)h0|¯uu|0i,

(1.29)

whi h an be derived in

χP T

. As su h, it is desirable to assess the value of the hiral ondensate from experiment  thus the value of

B

0

would be known. It has been argued that the best estimate an be obtained from

the low-energypion-pions attering [19, 20℄. However, the al ulation of the

ondensatefromempiri aldatarequiressomemodelassumptions,i.e. one in

fa t has to assumethat spontaneous hiral symmetry breaking takes pla e.

Therefore,animportant he k would beto al ulatethe ondensate

non-perturbativelyfromrstprin iples,withoutanyadditionalassumptions. One

su hway isprovidedby Latti eQCD.Indeed, Latti eQCDsimulations

on-rmthatitisnon-zeroatzerotemperature(areviewofresultsonthistopi is

provided e.g. in. [21℄). However, thereexists atemperaturewherethe hiral

ondensate vanishes,thus signalling hiral symmetry restoration. Moreover,

ment temperature, i.e. the temperature at whi h the quark-gluon plasma

formsand quarksand gluons areno longer onned intohadrons. Up tothe

presentday,thisissuehasnotbeenresolved ompletely,butitisastronghint

that Latti eQCD al ulations point to the fa t that both temperatures are

equal, up tostatisti alerror. This strongly suggests that spontaneous hiral

symmetry breaking isrelated to onnement and onrms that

understand-ing hiral symmetry and spontaneous hiral symmetry breaking is essential

to fully omprehend QCD. However, mu h more pre ise results are needed

to unambiguously resolve this question. In Latti e QCD investigations of

these phenomena it is therefore essential to take hiral symmetry properly

into a ount,i.e. fermionswith good hiralpropertieshave tobeused. This

is one of the motivations for employing overlap fermions, whi h will be the

main subje t of this thesis.

1.2 Dis retizing gauge elds

In this se tionand the next one, we showhow QCD an be formulated ina

non-perturbative way on a Eu lidean 4-dimensional hyper ubi latti e with

latti e spa ing denoted by

a

[22℄.

The basi relationshipbetween the ontinuum and latti eformulationof

gauge elds is given by the followingequation:

U(x, x + aˆ

µ) = e

igaA

µ

(x)

_,

(1.30)

where

U(x, x + aˆ

µ)

represents the gauge eld on the latti e (itis a variable dened onthe link onne tingsites

x

and

x + aˆ

µ

, where

µ

ˆ

is the unitve tor in the

µ

-dire tion) and

A

µ

(x)

isthe ontinuum gauge eld. This expression also implies that the link variables are SU(3) matri es, sin e it involves the

generators of SU(3), a ording to eq. (1.8).

We nowdis uss the simplest gauge eld latti ea tion, alled the Wilson

a tion [1℄, and show that in the ontinuum limit itis equivalent tothe

on-tinuum gauge a tion. It is worth to emphasize that the hoi e of the latti e

a tion is non-unique. In prin iple, any latti e a tion an be used, provided

that it has the orre t ontinuum limit. The Wilson a tionreads:

S

Wilson

[U] =

β

3

X

x

X

1≤µ<ν≤4

(1 −

ReTr

U

P

(x, µ, ν)) ,

(1.31)

where

U

P

is alled the plaquette variableand is dened as:

U

P

(x, µ, ν) ≡ U(x, x + aˆµ)U(x + aˆµ, x + aˆµ + aˆν)

(1.32)

To simplify notation, one usually denes

U(x, x + aˆ

µ) ≡ U

x,µ

and

U(x, x −

aˆ

_{µ) ≡ U}

_x−aˆ

†

_µ,µ

. Theshort utnotationfortheplaquettevariableis:

U

P

(x, µ, ν)

≡ U

x,µν

, where

µν

identies the plane of the plaquette. In this way, the pla-quette an be writtenas:

U

x,µν

= U

x,µ

U

x+aˆ

µ,ν

U

x+aˆ

†

ν,µ

U

x,ν

†

.

(1.33)

Thegaugetransformationonthe latti eisasso iatedwithmultipli ation

of the fermion and gluon elds by a site-dependent SU(3) matrix

G(x)

. For the link matri esit an bewritten as:

U

x,µ

→ U

x,µ

′

= G(x)U

x,µ

G(x + aˆ

µ)

†

.

(1.34)

Thisformofthegaugetransformationimpliesthatthetra eoftheplaquette

(a tually, the tra e of any losed loopof link variables) is a gauge-invariant

quantity:

U

x,µν

→ U

x,µν

′

= G(x)U

x,µ

G(x + aˆ

µ)

†

G(x + aˆ

µ)U

x+aˆ

µ,ν

G(x + aˆ

µ + aˆ

ν)

†

× G(x + aˆµ + aˆν)U

x+aˆ

†

ν,µ

G(x + aˆ

ν)

†

G(x + aˆ

ν)U

x,ν

†

G(x)

†

=

= G(x)U

x,µ

U

x+aˆ

µ,ν

U

x+aˆ

†

ν,µ

U

x,ν

†

G(x)

†

,

(1.35)

Tr

U

′

x,µν

=

Tr

G(x)U

x,µ

U

x+aˆ

µ,ν

U

†

x+aˆ

ν,µ

U

†

x,ν

G(x)

†

=

(1.36)

=

Tr

U

x,µ

U

x+aˆ

µ,ν

U

†

x+aˆ

ν,µ

U

†

x,ν

=

Tr

U

x,µν

.

We will onsider the gauge transformationfor the fermionelds in the next

se tion.

InAppendix A,weshow thatthe Wilsongauge a tion anbewrittenas:

S

gauge

[U] = β

g

2

_a

4

6

X

x

X

µ,ν

 1

4

F

µν

(x)

2

+ O(a

2

)



.

(1.37)

Comparing this expression with the ontinuum gauge a tion

R d

4

_x

1

4

F

µν

(x)

2

,

we an immediatelysee that the ontinuumlimitof the dis retized a tionis

the ontinuum gauge a tion if we set:

β =

6

g

2

.

(1.38)

The leading dis retization ee ts are

O(a

2

₎

, sin e the fa tor

a

4

in front of

the sum omesjust from the dis retizationof the integral

R d

4

_{x → a}

4

P

improved a tion, whi h helps to de rease the size of latti e dis retization

ee ts. Su h a tions have the same ontinuum limit, but this limit is

ap-proa hed faster. One ofthe rst improved a tions wasderived by Weisz [23℄

and it is usually referred to as tree-level Symanzik improved gauge a tion.

The form of this a tion is:

S

tlSym

[U] =

β

3

X

x



b

0

X

µ,ν=1

1≤µ<ν

(1 −

ReTr

U

x,µν

) + b

1

X

µ,ν=1

µ6=ν

1 −

ReTr

U

re t

x,µν



,

(1.39)

where

b

0

,

b

1

areparameters 6

,

U

x,µν

isthe (denedabove)plaquettetermand

U

re t

x,µν

is the re tangle term:

U

re t

x,µν

= U

x,µ

U

x+aˆ

µ,µ

U

x+2aˆ

µ,ν

U

_x+aˆ

†

_ν+aˆ

_µ,µ

U

_x+aˆ

†

_ν,µ

U

x,ν

†

.

(1.40) If

b

1

= 0

,this a tionbe omesthe Wilson a tion.

1.3 Dis retizing fermions

1.3.1 Naive dis retization

Letusstartwithadis retizationofone-avour ontinuumfreefermiona tion

in Eu lidean spa e, given by:

S

free quark

=

Z

d

4

x ¯

ψ(x)Dψ(x),

(1.41)

where

D = γ

µ

∂

µ

+ m

is the Dira operator and

m

is the quark mass. The dis retization pro edure is not unique and we show here one of the hoi es

for the latti e derivative[25℄:

ˆ

∂

µ

ψ(x) =

1

2a

(ψ(x + aˆ

µ) − ψ(x − aˆµ)) .

(1.42) This an alsobe writtenas:

ˆ

∂

µ

ψ(x) =

1

2a

(ψ(x + aˆ

µ) − ψ(x) + ψ(x) − ψ(x − aˆµ)) ≡

1

2

∇

µ

+ ∇

∗

µ

ψ(x),

(1.43) 6

The omputationsrelevantforfurtherpartofthisworkusedgaugeeld ongurations

generatedbytheEuropeanTwistedMassCollaboration(ETMC),whousedthisa tionin

where we have dened the forward latti e derivative

∇

µ

and the ba kward latti e derivative

∇

∗

µ

. We also dis retize the spa e-time integral (

R d

4

_{x →}

a

4

P

x

),thus arrivingat:

ˆ

S

free quark

= a

4

X

x

X

µ

¯

ψ(x)(γ

µ

∂

ˆ

µ

+ m)ψ(x),

(1.44)

where the hat denotes latti e quantities.

ByFourier-transforming the latti e Dira operator (whi h is

onvention-ally alled the naive operator, sin e it orresponds to the simplest possible

dis retization)

D

ˆ

naive

= γ

µ

ˆ

∂

µ

+m

,one anobtaintheexpression forthe Dira operator in momentum spa e:

ˆ

D

naive

(p) = i˚

p

µ

γ

µ

+ m

1

,

(1.45)

where we havedened:

˚

p

µ

≡

1

a

sin(ap

µ

)

(1.46)

for later onvenien e and 1 isthe unit matrix inDira spa e.

The tree-level fermion propagator in momentum spa e is given by the

inverse of the Dira operator (1.45) and thusequals:

ˆ

D

−1

naive

(p) =

−i˚

p

µ

γ

µ

+ m

1

P

µ

˚

p

2

µ

+ m

2

.

(1.47)

Letus onsiderthe aseofmasslessfermions. One aneasilyobservethatthis

expression hasthe right ontinuumlimit

−ip

µ

γ

µ

/p

2

. However, italsoimplies

that the number of fermionsis doubled forea h spa e-timedimension, sin e

the poles of the fermion propagator are lo ated not onlyat zero momentum

(

ap

µ

= (0, 0, 0, 0)

), whi h orresponds to the single fermion given by the ontinuum Dira operator, but also whenever any momentum omponent

equals

π/a

. Thus, in4-dimensionalspa e-time, wehave

2

4

_{= 16}

fermions,of

whi h15areunphysi alandare alleddoublers. Thisistheso- alledfermion

doublingproblem.

1.3.2 Wilson fermions

The rst way to over ome the doubling problem onsists in treating

dier-entlythephysi alpoleandtheunphysi alonesandwasintrodu edbyWilson

[26℄, who suggested the following formof the latti eDira operator:

ˆ

D

Wilson

=

1

2

γ

µ

(∇

∗

µ

+ ∇

µ

) − ar∇

∗

µ

∇

µ

+ m,

(1.48)

where

r

is the Wilson parameter. The se ond-derivative term is now alled the Wilson term. In momentumspa e, this operatorreads:

ˆ

D

Wilson

(p) = i˚

p

µ

γ

µ

+

ar

2

p

ˆ

2

µ

1

+ m

1

,

(1.49)

where we havedened:

ˆ

p

µ

≡

2

a

sin



ap

_µ

2



(1.50)

and the tree-level fermion propagatoris:

ˆ

D

−1

Wilson

(p) =

−i˚

p

µ

γ

µ

+ (

ar

₂

P

µ

p

ˆ

2

µ

+ m)

1

P

µ

˚

p

2

µ

+ (

ar

2

P

µ

p

ˆ

2

µ

+ m)

2

.

(1.51)

The physi al pole at

ap

µ

= (0, 0, 0, 0)

gets no ontributionfrom the Wilson term, but the unphysi al ones a quire an additionalmass, whi h is

propor-tional to

a

−1

and hen e be omeinnitely heavy inthe ontinuum limitand

de ouple.

However, the pri e one has to pay for removing the doublers is twofold.

First, the Wilson term leads to an

O(a)

leading ut-o dependen e in ob-servables, whi h makes it, from the point of view of pra ti al simulations,

advantageous to introdu e further terms to the a tion, e.g. a twisted mass

term, whi hwillbedis ussed later,or ountertermswithin theframeworkof

the Symanzik improvement programme. The simplest way to obtain

O(a)

-improvement (the absen e of

O(a)

ut-o ee ts) is to add to the a tion a single term, alled the Sheikholeslami-Wohlert( lover) term [27℄.

Se ond, the Wilson term, being a mass term, expli itly breaks hiral

symmetryeveninthe hirallimit

m = 0

,i.e. eveninthislimit

{ ˆ

D

Wilson

, γ

5

} 6=

0

. Moreover, it has been proven by Nielsen and Ninomiya[28℄ that it is not possiblethatalatti eDira operator

D

ˆ

fulllsatthesametimethefollowing onditions

7

:

1. lo ality i.e. the norm of the Dira operator

D

ˆ

de ays exponentially, as afun tion of the distan e between latti epoints,

2. translational invarian e i.e. the Fouriertransform of the Dira

oper-ator exists and equals

D(p) = iγ

ˆ

µ

p

µ

+ O(ap

2

₎

for

p ≪ π/a

,

3. no fermion doublers  i.e.

D(p)

ˆ

is invertible everywhere, ex ept for

p

µ

= (0, 0, 0, 0)

, 7

Original formulation of the Nielsen-Ninomiya theorem is in fa t dierent. Here we

present anequivalent formulation(given e.g. in [29, 30℄), whi h stresses the important

{ ˆ

D, γ

5

} = 0.

(1.52)

Formanyyears,itseemedthatitwasnotpossibletohave hiral fermionson

the latti e without violating one of the other onditions. However, a great

progress has been made onthis topi when it wasrealized that(1.52) is not

the only possible form of latti e hiral symmetry. The impli ations of this

dis overy willbedis ussed inthe next se tion.

An important onsequen e of hiral symmetry breaking for the Wilson

a tion is that the quark mass

m

requires additive renormalization. Hen e, the massless ase does not orrespond to

m = 0

, but to

m = m

c

, where

m

c

is alled the riti alquark mass.

The quarkmass isoften expressed with the so- alledhoppingparameter

κ

, dened as:

κ =

1

8 + 2m

.

(1.53)

Now, we dis uss how to add gauge elds to the Wilson fermion a tion.

It is believed that in the intera ting ase the doubler modes also de ouple.

However, there isno rigorousproof of it.

Undergaugetransformation,thefermioneldstransforminthefollowing

way:

ψ(x) → ψ

′

(x) = G(x)ψ(x),

_{ψ(x) → ¯}

¯

ψ

′

(x) = ¯

ψ(x)G(x)

†

.

(1.54)

For onvenien e, weremind here that the gauge elds transform as:

U

x,µ

→ U

x,µ

′

= G(x)U

x,µ

G(x + aˆ

µ)

†

.

(1.55)

In this way, the fermion mass term is obviously gauge-invariant, but the

derivative terms, e.g.

ψ(x)γ

¯

µ

∇

µ

ψ(x) = ¯

ψ(x)γ

µ

(ψ(x + aˆ

µ) − ψ(x))

are not, sin e:

¯

ψ(x)ψ(x + aˆ

_{µ) → ¯}

ψ(x)G

†

(x)G(x + aˆ

µ)ψ(x + aˆ

µ).

(1.56) However, introdu ingthe ovariantderivative:

ˆ

D

µ

ψ(x) =

1

2a



U

x,µ

ψ(x + aˆ

µ) − U

x−aˆ

†

µ,µ

ψ(x − aˆµ)



,

(1.57)

one nds forthe derivativeterm (1.56):

¯

ψ(x)U

x,µ

ψ(x + aˆ

µ) → ¯

ψ(x)G

†

(x)G(x)U

x,µ

G(x + aˆ

µ)

†

G(x + aˆ

µ)ψ(x + aˆ

µ)

The gauge-invariantWilson-Dira operator an bewritten as:

ˆ

D

Wilson

(m) =

1

2

γ

µ

(∇

∗

µ

+ ∇

µ

) − ar∇

∗

µ

∇

µ

+ m,

(1.59)

whi his exa tlythe same formasineq. (1.48), but now

∇

µ

and

∇

∗

µ

are the forward and the ba kward ovariantderivatives

8 , dened by:

∇

µ

=

1

a

(U

x+aˆ

µ,µ

ψ(x + aˆ

µ) − ψ(x)) ,

(1.60)

∇

∗

µ

=

1

a



ψ(x) − U

x−aˆ

†

µ,µ

ψ(x − aˆµ)



.

(1.61)

1.3.3 Wilson twisted mass fermions

Intheremainderofthisse tion,wewilldis ussWilsontwistedmassfermions,

whi h are relevant from the point of view of further onsiderations.

Origi-nally, they were introdu ed to deal with the problem of unphysi ally small

eigenvalues (zero modes) of the Wilson-Dira operator [31℄, whi h is

an-other onsequen e of additive quark mass renormalization, whi h an bring

the renormalized quark mass to zero. In the quen hed approximation, the

ontribution of these modes is not balan ed by the fermioni determinant

and leads tolarge u tuations, whi hae t ensembleaverages in an

un on-trolled way. The gauge eld ongurations whi h ause this problem are

referred toas ex eptional ongurations. Thisis espe iallydangerousin the

ase ofsmallquark masses and makes the approa htowards the hiral limit

pra ti ally impossible with Wilson fermions. In dynami al simulationswith

Wilsonfermionstheproblemissuppressedbythefermioni determinant,but

it an still ause te hni al problems, su h as long auto orrelation times in

ertain observables, oming froma idental zero modes of the Wilson-Dira

operator. Moreover, it wasrealized that the twistedmass dis retization an

redu e the ee ts of expli it hiral symmetry breaking by the Wilson term

by suppressing the mixing problem of operators belonging to dierent

hi-ral representations. Finally,twisted mass a tion makes it possible toobtain

automati

O(a)

-improvement, by tuning just one parameter. This is an es-sentialadvantageoftwistedmassfermions,sin eotherimprovements hemes

make itne essary to ompute improvement oe ients for dierent

interpo-lating operators.

8

Wewillusethesamesymbols

∇

µ

and

∇

∗

µ

forthenon- ovariantand ovariant deriva-tivesandthemeaningofthesesymbolswillbedeterminedfromthe ontext.

rate quarks is given by:

ˆ

S

TM

= a

4

X

x

¯

χ(x) ˆ

D

TM

χ(x),

(1.62) with:

ˆ

D

TM

= ˆ

D

Wilson

(m) + iµγ

5

τ

3

,

(1.63)

where

µ

is an additionalmass parameter, alled the twisted mass,

τ

3

is the third Paulimatrix inavour spa e and

χ(x)

isthe quarkeld inthe twisted basis.

The physi aland twistedbases are relatedby an axialtransformation:

ψ(x) → χ(x) = e

iωγ

5

τ

3

/2

_ψ(x),

¯

ψ(x) → ¯

χ(x) = ¯

ψ(x)e

iωγ

5

τ

3

/2

_,

(1.64)

where

ω

is alledthe twist angle. Thistransformationleavesthe formof the a tion invariant,only transforming the mass parameters a ording to:

m → m cos(ω) + µ sin(ω),

(1.65)

µ → −m sin(ω) + µ cos(ω).

(1.66) A spe ial ase of this transformation, referred to as maximal twist, is

ω =

π/2

, whi h orresponds to sending the bare quark mass

m

to 0 or, taking additivemass renormalizationintoa ount,toits riti alvalue

m

c

. Conven-tionally, the value of the riti al bare quark mass is expressed in terms of

the parameter

κ

c

,given by eq. (1.53). This istheonlyparameter thatneeds to be tuned to obtain automati

O(a)

-improvement. The tuning is usually done by employingone of two methods. First, one an just nd the riti al

bare quark mass by looking for a quark mass value that gives a vanishing

pion mass. Alternatively, one an also tune the so- alled untwisted PCAC

mass:

m

P CAC

=

P

~

x

h∂

0

A

a

0

(~x, t)P

a

(0)i

2

P

~

x

hP

a

(~x, t)P

a

(0)i

,

a = 1, 2

(1.67)

to zero [24℄. The latter method seems towork very wellin pra ti al

simula-tions.

Thus, one an writethe maximallytwistedmass(MTM) QCDa tion as:

ˆ

S

MTM

= a

4

X

x

¯

χ(x) ˆ

D

MTM

χ(x),

(1.68) with:

ˆ

D

MTM

= ˆ

D

Wilson

(m

c

) + iµγ

5

τ

3

.

(1.69)

r

χ

0

m

PS

= 0.614

r

χ

0

m

PS

= 0.900

r

χ

0

m

PS

= 1.100

r

χ

0

f

PS

(a/r

χ

0

)

2

0.06

0.04

0.02

0

0.42

0.38

0.34

0.30

0.26

= 0.045

= 0.090

r

χ

0

µ

R

= 0.130

(r

χ

0

m

PS

)

2

(a/r

χ

0

)

2

0.06

0.04

0.02

0

1.4

1.0

0.6

0.2

Figure 1.1: Continuum limits aling inxed nitevolume for

r

0

f

P S

at xed values of

r

0

m

P S

(a) and for

(r

0

m

P S

)

2

at xed values of renormalized quark

mass

r

0

µ

R

(b). In (b) data at

β = 4.2

(

(a/r

0

)

2

_{= 0.0144}

) are not in luded,

due tothe missing value of the renormalizationfa tor

Z

P

. Sour e: [33℄.

The spe ial meaningof the maximally twisted ase is that it guarantees an

automati

O(a)

-improvement,whi hwasproven in[32℄. Thismeansthat all terms of

O(a)

in the Symanzik expansion of parity even operators (whi h givee.g. the hadronmasses)are absent. This observation makesthe twisted

mass formulation (at maximal twist) very useful from the point of view of

pra ti alsimulations.

Anexample of

O(a)

-improvement isprovided by ETMC simulations[33℄ and isdepi ted in Fig. 1.1. The left plot shows the ontinuum limits aling

ofthe pseudos alarde ay onstant(inxed volume)atxedreferen evalues

of the pseudos alar mass

r

0

m

P S

. Four latti e spa ings are in luded, but the linear t does not in lude the data at the largest latti e spa ing. The

rightplotshows the s alingofthe pseudos alarmass(againinxed volume)

at xed values of the renormalized quark mass

r

0

µ

R

. Here, the data for only three latti e spa ings are presented (all of them in luded in the t),

sin e it was impossible to in lude the points at the nest latti e spa ing

(

(a/r

0

)

2

_{= 0.0144}

), due tothe missing value of the renormalizationfa tor of

the pseudos alar urrent

Z

P

. Bothplotsshowthatthe leading ut-o ee ts are indeed

O(a

2

₎

and their overall magnitude israther small.

However, one should mention here that the twisted mass term violates

parity and the isospin symmetry. This ee t e.g. makes the masses of the

harged and neutral pions dierent from ea h other 9

and in fa t this mass

9

Both parity and isospinbreaking are

O(a

2

₎

ee ts and hen e they vanish in

the ontinuum limit.

1.4 Chiral symmetry on the latti e

In this se tion we dis uss the great breakthrough asso iated with the fa t

that itwas realized that there is an alternative viewon hiral symmetry on

the latti e, i.e. that the Nielsen-Ninomiya theorem an be over ome in an

elegantway.

1.4.1 Ginsparg-Wilson relation

In 1982 (i.e. only one year after establishing the Nielsen-Ninomiya

theo-rem), Ginsparg and Wilson, basing on renormalization group

transforma-tions,showed [34℄thataremnantof hiralsymmetryispresentonthelatti e

without the doublermodes, if the orresponding Dira operator

D

ˆ

obeys an equation now alledthe Ginsparg-Wilsonrelation:

γ

5

D + ˆ

ˆ

Dγ

5

= a ˆ

Dγ

5

D.

ˆ

(1.70)

It is a modi ation of the anti ommutation relation(1.52) and the term on

the right-handside vanishesinthe ontinuum limithen e, inthislimitthe

standard hiral symmetry relation(1.52) is regained.

However, for many years it has not been realized that the

Ginsparg-Wilson relation allows one to dene hiral symmetry also on the latti e,

i.e. at non-vanishinglatti e spa ing. It lasted until around 1997 before the

Ginsparg-Wilson relation was redis overed. First, P. Hasenfratz realized

that a kind of latti e fermions alled the xed point fermions satises this

relation [35, 36℄ and shortly afterwards a similar observation was made by

Neuberger regarding the overlap formalism [37, 38℄. Moreover, Lüs her [29℄

found that the Ginsparg-Wilsonrelationleads toa non-standardrealization

oflatti e hiralsymmetry. Thefermiona tionisinvariantunderthefollowing

hiral rotation:

ψ → e

iθγ

5

“

1−

a ˆ

D

2

”

ψ, ¯

_{ψ → ¯}

ψe

iθγ

5

“

1−

a ˆ

D

2

”

.

(1.71)

Inthe ontinuumlimitthistransformationis(1.26)with

u

i

= 1

. Toprovethe invarian e of the massless a tion with respe t to the above transformation,

intera tionsandtheexpli itviolationofisospinsymmetrybydierentupanddownquark

masses. InLatti eQCDwithmass-degeneratequarksthe hargedandneutralpionmasses

the left- and right-handed proje tors of fermion elds [11℄ with a modied

γ

5

-matrix

γ

ˆ

5

= γ

5

(1 − a ˆ

D)

:

ˆ

P

±

=

1 ± ˆγ

5

2

.

(1.72)

Thusdened proje torshavethe sameproperties asthe standard ontinuum

proje tors andhen e one an de ompose the fermion part ofthe Lagrangian

into left- and right-handed massless parts and a symmetry breaking mass

term that mixes the left- and right-handed omponents. Taking the

proper-ties ofthe latti eproje tors

P

ˆ

±

intoa ount,onends thatthe masstermis of the form

m ¯

ψ



1 −

a ˆ

D

2



ψ

, whi h meansthat themassive Ginsparg-Wilson Dira operator

D(m)

ˆ

that orresponds tothe massless operator

D

ˆ

reads:

ˆ

D(m) = ˆ

D + m

_{1 −}

a ˆ

D

2

!

=



_{1 −}

am

2

 ˆ

D + m.

(1.73)

Sin etheGinsparg-Wilsonrelationisanon-standardrealizationof hiral

symmetry,the onditionsoftheNielsen-Ninomiyatheoremdonotapply and

one anhave hiralsymmetry withoutthe doublers,whi hwas onsidered to

beimpossible for many years.

Moreover, itwasalsoshownbyHasenfratz,Laliena,Niedermayer[39℄and

in adierentway by Lüs her [29℄ thatthe Ginsparg-Wilsonrelationimplies

thatthe axialanomalyis orre tlyreprodu ed. Thea tionisinvariantunder

the transformation (1.71), but the fermioni measure

D ¯

ψDψ

is not  its Ja obian

J

is non-trivial:

J = exp[−2iθ

Tr

(γ

5

(1 − a ˆ

D/2))]

and it an also be expressed as

J = exp[−2iθQ

top

]

, where

Q

top

is the topologi al harge, to be dis ussed later. This issue was further elu idated by Fujikawa [40℄, who

studied the ontinuum limitof this Ja obian.

Furthermore, a onsequen e of the Ginsparg-Wilson relation is that

fer-mionsare prote tedfromadditivemass renormalizationandmixingbetween

four-fermion operators in dierent hiral representations (Hasenfratz [36℄)

and there an beno

O(a)

latti e artefa ts (Niedermayer [30℄).

Inthenextse tionwedis ussoneofthesolutionsofthe Ginsparg-Wilson

equation, dening the so- alled overlap operator.

1.4.2 Overlap fermions

As wehave already stated, for many years it has not been realizedthat the