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} _{2} ^{D} ”

### ψ, ¯ ψ → ¯ ψe ^{iθγ} ^{5}

### “ 1− ^{a ˆ} _{2} ^{D} ”

### .

^{(1.71)}

Inthe ontinuumlimitthistransformationis(1.26)with

### u i = 1

^{.}

^{T}

^{o}

^{prove}

^{the}

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

intera tionsandtheexpli itviolationofisospinsymmetrybydierentupanddownquark

masses. InLatti eQCDwithmass-degeneratequarksthe hargedandneutralpionmasses

should beequal.

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)

^{:}

### P ˆ ± = 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 ^{±}

^{into}

^{a ount,}

^{one}

^{nds}

^{that}

^{the}

^{mass}

^{term}

^{is}

of the form

### m ¯ ψ

### 1 − ^{a ˆ} _{2} ^{D}

### ψ

^{,}

^{whi h}

^{means}

^{that}

^{the}

^{massive}Ginsparg-Wilson Dira operator

### D(m) ˆ

^{that}orresponds tothe massless operator

### D ˆ

^{reads:}

### D(m) = ˆ 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

^{T}

^{r}

### (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

Ginsparg-Wilsonrelationprovidesauseful (fromthesimulationalviewpoint)

tions of this equationhave been known. In 1997, Neuberger[37, 38℄found a

parti ularlysimpleformofalatti eDira operatorthatobeysthe

Ginsparg-Wilson relation. It is now usually referred to as overlap fermions and the

massless overlap Dira operatoris given by:

### D ˆ

^{ov}

### (0) = 1 a

### 1 − A(A ^{†} A) ^{−1/2}

### ,

^{(1.74)}

where:

### A = 1 + s − a ˆ D

^{Wilson}

### (0)

^{(1.75)}

and

### s

^{is}

^{a}

^{parameter}

^{whi h}

^{satises}

### |s| < 1

^{and}

^{ an}

^{be}

^{used}

^{to}

^{optimize}

lo ality properties. Note that instead of

### D ˆ

^{Wilson}

### (0)

^{,}

^{one}

^{ ould}

^{use}

^{in}

^{the}

kernel operator

### A

^{any}

^{massless}

^{latti e}

^{Dira }

^{operator}

^{that}

^{is}

^{lo al}

^{and}

^{has}

nodoublermodes[30℄. Moreover, ifthe operatorusedin

### A

^{itself}

^{satises}

^{the}

Ginsparg-Wilsonrelation,itwillbejustreprodu edbyeq. (1.74),sin ethen

### A ^{†} A = 1

^{.}

The massiveoperator is given, a ording to (1.73),by:

### D ov (m) =

### 1 + s − am 2

### D ov (0) + m,

^{(1.76)}

where

### m

^{is}

^{the}

^{bare}

^{overlap}

^{quark}

^{mass.}

Afterthe overlap operatorwas proposed,it was essential toshowthat it

is lo al. Else, this attitude would lead to ausality violations and render it

useless. The denition (1.74) in ludesthe highly non-lo al term

### (A ^{†} A) ^{−1/2}

^{,}

whi h raises doubts about lo ality. Algebrai ally, stri t lo ality (or

ultra-lo ality) would mean that the Dira operator matrix element

### D(x, y) ˆ

^{}

on-ne ting sites

### x

^{and}

### y

^{of}

^{the}

^{latti e}

^{is}

^{non-zero}

^{only}

^{if}

^{the}

^{distan e}

^{between}

### x

^{and}

### y

^{is}

^{smaller}

^{than}

^{some}

^{spe ied}

^{small}

^{value}

^{and}

^{also}

^{that}

^{this}

^{matrix}

element dependsongauge linksonlyin somesmall neighbourhoodofsites

### x

and

### y

^{[41℄.}

^{These}

^{properties}

^{are}

^{true}

^{for}

^{the}Wilson-Dira operator (hen e, itisasparsematrix),but theoverlapDira operatorhasnon-zero entriesfor

all pairs of latti esites and thusit is lear that itis not stri tly lo al.

However, stri t lo ality for a Dira operator is not really needed. It is

enoughthattheDira operatorfallsoexponentially,i.e. wehave

(suppress-ing the Dira and olorindi es):

### || ˆ D(x, y)|| ≤ Ce ^{−ρ||x−y||} ,

^{(1.77)}

for some onstants

### C

^{and}

### ρ

^{,}

^{where}

### || · ||

^{is}

^{the}

^{distan e}

^{between}

^{sites}

### x

^{and}

### y

^{,}

^{e.g.}

^{the}taxi-driver distan e

### ||x − y|| = P

### µ |x ^{i} − y ^{i} |

^{.}

^{If}

^{su h}

^{inequality}

holds, it means that the intera tion range in physi al units

### 1/ρ

^{tends}

^{to}

^{0}

(the de ay rate in latti e units

### aρ

^{does}

^{not}

^{depend}

^{on}

^{the}

^{latti e}

^{spa ing)}

asone approa hes the ontinuumlimitand inthe ontinuum onehas alo al

eld theory, as desired[11℄.

A thorough analyti al and numeri al investigationof the lo ality of the

overlap Dira operator was performed by Hernandez, Jansen and Lüs her

[41℄, who showed that this operator is lo al under very general onditions,

i.e. for a wide range of bare oupling onstants.

1.4.3 Other kinds of hiral fermions

Apart from overlap fermions, there a few kinds of latti e fermions that also

preserve hiral symmetry. In this subse tion, we shortly dis uss a few of

them.

1.4.3.1 Domain wall fermions

Closely related (mathemati allyequivalent) to overlap fermions are domain

wall fermions, introdu edby Kaplan[42℄ and Shamir[43℄ in 1992 and 1993,

respe tively. The general idea of this approa h is to introdu e an auxiliary

(non-physi al) fth dimension and onsider massive Dira fermions with a

spa e-dependent mass in the shape of a domain wall. Kaplan showed that

su h theory has a zero mode with denite hirality lo alizedon the domain

wall and from the pointof view of the 4-dimensional theory this zero mode

is a hiral fermion. The way that this formulation ir umvents the

Nielsen-Ninomiya theorem is that translational invarian e in the 5-dimensional

sys-temisbroken(by thespa e-dependentmassterm),butitisstill onserved in

the 4-dimensionalphysi alworld [44℄. Ifthe fthdimension isinnite,there

does not exist a doubler mode. But in the ase of a nite fth dimension

(whi h is of ourse always true in a latti e simulation),an extra zero mode

of opposite hirality appears on a se ond domain wall. However, both zero

modes have an exponentially smalloverlap and hen e an not ommuni ate

if theirseparation islarge enough. What ismore,itwas alsoshown that the

anomaly stru ture is orre t both in the innite and nite fth dimension

ase. A rst investigationof these properties was performedby Jansen [45℄,

shortly after the birth of the idea of domain wallfermions.

Aftertheoverlapformalismwasinvented,Neubergeralsoshowed[46℄that

domain wall fermions with innite fthdimension are equivalent to overlap

fermions. Therefore, at nite fth dimension, they an be regarded as an

approximationto overlap fermions.

In pra ti al simulations, the domain wall formalism is now widely used

in a dynami al setup (e.g. by the RBC/UKQCD Collaboration [47℄) or in

staggered fermions in the sea se tor [48℄). However, the size of the fth

dimension is usually taken in the range 8-16, whi h means that the hiral

symmetryisonlyapproximateandthisentailsadditivemassrenormalization

of the quark mass, i.e. a shift away from zero of the bare quark mass for

whi h one has a vanishing pion mass [10℄. The value of this shift is usually

referred to asthe residualmass.

The mainadvantage of domainwall fermions withrespe t to the Wilson

fermions(and othernon- hirallysymmetri formulations)isthat hiral

sym-metry breaking by the domain wall fermions is rather mild and is believed

to be ontrollable. Their main disadvantage is that one needs to simulate

a 5-dimensional theory, instead of a 4-dimensional one, and hen e the

om-putational ost is higher by a fa tor of the order of the size of the fth

dimension.

Withrespe ttooverlapfermions,anadvantageisthatthis omputational

ost is still mu h smaller than the one for overlap, atthe pri e, however, of

not havingexa t hiral symmetry, but only anapproximationto it.

1.4.3.2 Creutz fermions

A dierent approa h to ir umvent the Nielsen-Ninomiya theorem is to

re-stri t oneself to the minimal number of doubler modes, i.e. to two modes

of opposite hirality. This was pointed out in the 1980s by Karsten [49℄

and Wil zek [50℄. Re ently, this idea reemerged in the work of Creutz [51℄,

who wasmotivatedbythe ele troni stru tureof graphene (whi hisbuilt of

two-dimensionallayers of graphite). The low-energyex itations ingraphene

are des ribed byatwo-dimensionalDira equationformasslessfermionsand

are hen e hiral. Furthermore, hirality is a hieved exa tly in the way that

involves the minimum number of fermion modes required by the

Nielsen-Ninomiya theorem, i.e. they are minimally-doubled. Creutz showed how

to generalize these properties to four dimensions. Creutz's idea was soon

elaboratedon by Bori i[52℄, who derived a moregeneralform ofthe a tion.

Creutzfermionsexhibitanexa t

### N _{f} = 2

^{avour}

^{ ontinuum}

^{ hiral}

^{}

symme-try, whi h implies also that the leadingdis retization errors are of

### O(a ^{2} )

^{10}

^{,}

and they are stri tly lo al. These are very appealing properties, sin e they

imply that one ould simulate hiral fermions without the high

omputa-tional ost ofoverlap fermions. However, Creutz fermionsbreak anumberof

dis rete symmetries, su h as parity, harge onjugation and time ree tion

[53℄. Therefore, toapproa h the ontinuum limitinthe intera ting ase one

10

An expli ittestof thispropertywillbedis ussedin Chapter2.

in the Symanzik ee tive a tion and this would make the pra ti al

simula-tions with Creutz fermions very di ult. However, a preliminary quen hed

test wasperformedby Bori i[54℄, who omputed thepion mass andfound a

behaviour onsistentwiththe predi tionsof hiral perturbationtheory. This

ledhimtoa on lusionthat Creutz fermionsare stillworth exploringin the

future, despitethefa tthattheybreakimportantdis retesymmetries. Also,

the on lusion by Bedaque et al. [53℄ was that for ertain values of the

pa-rameters, the minimally-doubled fermion a tions may exhibit non-standard

symmetries, that ouldeliminaterelevantoperatorsoftheSymanzikee tive

theory and hen emoderate the problemof dis rete symmetriesbreaking.

The expressions for the Dira operator for Creutz fermions (by whi h

we will mean both fermions related to Creutz's original idea and Bori i's

generalization) will be dis ussed in Chapter 2, only in the ontext of a test

of their ontinuum-limits alingat tree-level of perturbation theory [55, 56℄.

1.4.4 Topology on the latti e

TheQCDva uumhasanon-trivialtopologi alstru ture,whi hhasmany

im-portant impli ations for hadron properties. Forexample, we have remarked

in Se tion 1.1 that the mass of the avour singlet

### η ^{′}

^{meson}

^{is}

^{related}

^{to}

the topologi alu tuations of the QCD va uum. This is aninherently

non-perturbativephenomenon andhen eseems tobewell-suitedtobeaddressed

by Latti eQCD al ulations.

Letusstart withthe eld-theoreti aldenitionof thetopologi al harge:

### Q top = 1 32π ^{2}

### Z

### d ^{4} x ǫ µνρσ

^{Tr}

### (F µν (x), F ρσ (x)) ≡ Z

### d ^{4} x q(x),

^{(1.78)}

where

### q(x)

^{is}

^{ alled}

^{the}topologi al harge density [57℄. Gauge eld ong-urations that have a non-zero and integer topologi al harge are e.g.

super-positionsof instantons [58℄ and anti-instantons,whi hare lassi al solutions

of the Eu lidean eld equations.

Thetopologi al harge an berelatedtothenumberof hiralzero modes

of the massless Dira operatorvia the Atiyah-Singer index theorem [59℄:

### Q top = Q index ≡ N ^{−} − N ^{+} ,

^{(1.79)}

where

### N _{±}

^{denotes}

^{the}

^{number}

^{of}

^{zero}

^{modes}

^{in}

^{the}positive/negative hirality se tor and

### Q index

^{is}

^{ alled}

^{the}

^{index}

^{of}

^{the}

^{Dira }

^{operator.}

If one wants to ompute the topologi al harge of a given gauge eld

ongurationonthe latti e,one an,in prin iple,use the dis retizedversion

[60℄. This an be over ome by applying smearing on gauge ongurations,

e.g. APE smearing[61℄, whi hmovesthe topologi al harge losertointeger

values, but it an also destroy small topologi al obje ts and thus lead to

in orre t values of the harge.

Su h problems are avoided if one uses the index theorem and omputes

the topologi al harge as the index of the massless Dira operator. For this

to be possible, one has to employ a Dira operator that an have hiral

zero modes (at any value of the latti e spa ing), i.e. eigenstates with zero

eigenvalue,whi hhavedenite hirality(thatistheyarealsoeigenstatesof

### γ 5

witheigenvalue

### ±1

^{).}

^{This}

^{means}

^{that}

^{the}

^{massless}

^{Dira }

^{operator}

^{must}

^{obey}

(latti e) hiral symmetry, e.g. it an be the overlap Dira operator, whi h

will beused to ompute topologi al harge infurther part of this work.

Sin ethe QCDpath integralissymmetri with respe t tothesign of the

topologi al harge, wehave

### hQ ^{top} i = 0

^{.}

^{However,}

^{a}non-trivialquantity that one an ompute is relatedto the u tuations of the topologi al harge and

is alled topologi alsus eptibility. In the ontinuum, itis dened by:

### χ top = Z

### d ^{4} xhq(x)q(0)i,

^{(1.80)}

whi hon the latti e be omes

### χ _{top} = hQ ^{2} index i

### V ,

^{(1.81)}

where

### V

^{is}

^{the}

^{latti e}

^{volume.}

Ithasbeenmentionedbeforethatthetopologi alsus eptibility isrelated

via the Witten-Veneziano formula(1.27) to the mass of the

### η ^{′}

^{meson.}

^{}

Phe-nomenologi ally, this formula implies a value of

### χ top ≈ (180

^{Me}

^{V}

### ) ^{4}

^{.}

^{It}

^{is}

worth to emphasize that this value agrees rather wellwith several quen hed

latti e omputations,e.g. [62℄.