T opologi al harge and sus eptibility

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 r 0 M

(a/r ₀ ) ²

MTM Nucleon Overlap Nucleon MTM Delta Overlap Delta

Figure 5.8: Continuum limit s aling of the MTM and overlap light baryon

masses (nu leon, delta). The overlap masses are omputed at the mat hing

mass. The MTMvales are slightly shiftedtothe leftand theoverlap ones to

the right, for learerpresentation.

In parti ular, even if the baryon massesare not onsiderably ae ted by the

zero modes, the matrix elements of the baryoni orrelation fun tions still

might hange signi antly, as is anti ipated in [124℄. Therefore, this issue

will be investigated further in the future [112℄. In parti ular, the role of

the zero modes an be assessed by usingdierentinterpolatingoperatorsfor

baryoni orrelationfun tions, sin e dierentoperators ouple ina dierent

waytozeromodes[124℄. Moreover,theexpli itsubtra tionpro eduremaybe

followedandtheoverlapofthesour esandthezeromodesmaybe omputed.

However,thepresentanalysisalreadyallowsusto on ludethatthe

mag-nitudeof thezero modesee ts indierentobservables maybedierentand

that some observables maybe mu hmore vulnerabletothe zeromodes

on-tribution(e.g.

f π

^{)than some other (e.g. the baryon masses).}

5.3 Topologi al harge and sus eptibility

Inthisse tion,wereporttheresultsofinvestigationofsometopologi alissues

relatedtogaugeeld ongurationsthatwehaveused. Forsomeofthem,we

have omputedthezeromodes, whi hallowsusto al ulatetheirtopologi al

-10

β=3.9, L/a=16, aµ=0.004, 544 confs -10 -5 β=3.9, L/a=16, aµ=0.0074, 260 confs

-10

β=4.05, L/a=20, aµ=0.003, 300 confs -10 -5 β=4.2, L/a=24, aµ=0.002, 396 confs

Figure 5.9: Monte Carlo history of the index of the overlap operator for

dierent ensembles. The verti al axiss ale is the same for allplots.

harge as the dieren e of the numbers of zero modes in the negative and

positive hirality se tors (eq. (1.79)), i.e. the index of the overlap Dira

operator. Inpra ti e, zero modes ona given ongurationo uronly inone

hirality se tor or, in other words, the probability of having zero modes in

both se torsfor the same ongurationiszero [122℄.

Fig. 5.9 shows the Monte Carlo histories of the index of the overlap

op-eratorforfourdierentensemblesof ongurations:

β = 3.9, L/a = 16, aµ = 0.004

^;

β = 3.9, L/a = 16, aµ = 0.0074

^;

β = 4.05, L/a = 20, aµ = 0.003

^;

β = 4.2, L/a = 24, aµ = 0.002

^{; all of them} orresponding to linear latti e extent of

L ≈ 1.3

^{fm. The plots indi atethat the} auto orrelations inMonte Carlotime arerathernot largeanddierenttopologi alse torsare sampled.

The verti al s aleon ea hof the plots is the same and hen eit isnoti eable

that topologi al harge u tuationsare the largestfor

β = 3.9

^and

onsider-ably smallerand omparable to ea h other for

β = 4.05

^and

β = 4.2

^.

The histograms of the index are shown in Fig. 5.10. To allow for

om-parison, the number of ongurations whi h orresponds to the given index

N(index)

^{hasbeennormalizedby thetotalnumberof} ongurationsforea h

0 β=3.9, L/a=16, aµ=0.004, 544 confs

0 β=3.9, L/a=16, aµ=0.0074, 260 confs

0 β=4.2, L/a=24, aµ=0.002, 396 confs

Figure 5.10: Histograms of the index of the overlap operator for dierent

ensembles. Also shown are Gaussian ts of the distributions. The axes

s ales are the same for allplots.

ensemble

N(total)

^{and the axesare plotted inthe same s ale. We alsoshow}

ts tothe Gaussian probabilitydistribution:

p(Q) = 1

√ 2πσ e ^{− Q}

2

2σ2 ,

^(5.5)

where

Q

^{isthe index and}

σ

^{isthe onlytting oe ient, i.e. we enfor e the}

mean ofthedistribution tobe0,sin ethe probabilitydistributionshouldbe

symmetri withrespe tto ongurations withnegativeandpositive

topolog-i al harge.

The plots leadto the on lusion that the probabilitydistributions of the

index

Q

^areapproximatelyGaussian. However, thequalityofthetsisrather poor the most frequent value of the index is intwo ases at

Q 6= 0

^{and in}

general the symmetry with respe t to

Q = 0

^{is not very good. This results}

from the fa t that the total number of ongurations for ea h ensemble is

too smalland an order of magnitude in rease in statisti s would be needed

to obtaina reliable distribution.

0 0.02 0.04 0.06 0.08 0.1

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 r 0 4 χ top

(a/r ₀ ) ²

β=4.2 β=4.05 β=3.9

m _π ≈300 MeV m _π ≈450 MeV

Figure 5.11: Topologi al sus eptibilityfor 4 ensembles of gauge eld

ong-urations. The

m π ≈ 450

^{MeV ensemble is slightly shifted to the right for}

bettervisibility.

The u tuations of the topologi al harge determine the value of

topo-logi al sus eptibility

χ top

^{for ea hensemble. This quantity an be omputed}

from eq. (1.81), i.e. as the mean value of the topologi al harge squared,

normalized bythe volume. Alternatively, it an alsobe determinedfromthe

Gaussiant (5.5)as the varian e

σ ²

^{, againnormalizedby the volume. Even}

though the probability distributions are ratherfar away fromGaussian, the

statisti al errors of

hQ ² i

^{and of}

σ ²

^{are large and we have he ked that both}

methods lead to onsistent results. In Fig. 5.11 we plot the results of the

former method for three light-quark ensembles with sea quark masses

or-responding to the pion mass of around 300 MeV and one ensemble at the

heavier pion mass of approximately 450 MeV. However, sin e the statisti al

errors are large, a meaningful extrapolation to the ontinuum limit is not

possible one would learly need more statisti s. The same holds true with

regard tothe sea quark mass dependen e of the topologi alsus eptibility 

the expe ted in rease of

χ top

^{for larger quark mass is observed, but it is not}

statisti ally signi ant.

Therefore, the topologi alaspe ts alsoneed to beinvestigated further in

thefuturea onsiderablein reaseinpre isionisneededtodrawmeaningful

physi al on lusions.

Chiral symmetryis ofutmostimportan eforlow-energyproperties ofQCD.

Therefore, when dis retizingQCDona4-dimensionalspa e-timegrid to

ad-dress non-perturbativephenomena, retaining hiral symmetry inthis latti e

version of QCD (LQCD) is anessential element. Hen e, hirally-symmetri

fermion dis retizationsare needed to fully explore the lowenergy regime of

QCD. A very appealing kind of hiral fermions are the overlap fermions.

However, theiruse indynami alLatti eQCDsimulationsisstilla hallenge,

sin e they are very demandingfrom the omputationalpoint of view.

Chiralpropertiesof fermions are espe iallyimportantinthe valen e

se -tor. At the same time, the most expensive part of a simulation is the

gen-eration of gauge eld ongurations. Hen e, apossible way toover ome the

ost problemofdynami aloverlapsimulations,whi hatthesametimekeeps

theirgood hiralproperties,istofollowamixeda tionapproa hwheregauge

eld ongurations are generated using a omputationally heaper fermion

dis retization and the overlap operator isused only inthe valen e se tor.

The main aimof this thesis was to investigate a parti ular mixed a tion

setup of overlap valen e and maximally twisted mass (MTM) sea quarks.

In this way, we ould prot from a wide set of gauge eld ongurations

generated by the European Twisted MassCollaboration(ETMC).

In parti ular, we wanted to perform a ontinuum limit s aling test of

overlap fermions, a study that has not been done before. However, to

per-formsu h investigationwith atypi allinearlatti e extent of2fm verylarge

omputer resour es wouldberequired,even ifthe overlapoperator wasused

only in the valen e se tor. Therefore, we de ided to employ a small volume

with

L ≈ 1.3

^{fm. Su hvolumeissu ienttotestthe ontinuumlimits aling}

behaviour. Wede ided totake thepion de ay onstantas ourmain physi al

observabletostudythelatti eartefa tsoftheoverlapdis retization. Usinga

suitable mat hing ondition ofoverlap and twisted mass fermions,for whi h

we have taken the pion mass, and assuming universality, the same

ontin-uum limitvalue for

f π

^{shouldberea hedwithbothkindsoflatti efermions.}

Weexpli itly he ked inthe freetheory thatthis expe tationisfullled. We

learlyobservedthe

O(a ² )

^{leading ut-odependen eand}demonstratedthat

f π

^{agrees in the ontinuum limit.}

However, when moving to the intera ting ase, we en ountered a puzzle

in that the ontinuum limits of the two latti e fermions used ame out to

be in onsistent with ea h other. It is one of the main results of this thesis

that the solution of this puzzle ould be identied as the exa t hiral zero

modes of the overlap Dira operator. Being hiral, this operator admits

zero modes at any value of the latti e spa ing. This is in ontrast to the

non- hiral twisted mass Dira operator whi h does not admit su h hiral

zero modes, at least not at our urrent values of the latti e spa ing. In

order to demonstrate that the hiral zero modes are indeed the ause of

the mismat h of

f π

^{in the ontinuum limit, we used the fa t that the zero}

modes ouple inan identi al way tothe pseudos alar and s alar orrelation

fun tions. Hen e, in the dieren e of these orrelation fun tions (the

so- alled PP-SS orrelator), the zero modes ontribution is exa tly an elled.

Performing now a ontinuum limit s aling test of the pion de ay onstant

as omputed from the overlap PP-SS orrelator, whi h is not ae ted by

the zero modes, weobtained indeed onsistent ontinuum limitvalues for

f π

omputed from the two fermion dis retizations.

Wealso ross- he ked thisresult byexpli itlysubtra tingthezero modes

at the level of overlap propagators. This further onrmed the pi ture that

the hiral zeromodesneed tobetreatedspe ially,atleast inthe smallnite

volumeused here. However, the modi ationofpropagators byexpli it

sub-tra tionof a partof eigenmodes of theDira operatoris aeld-theoreti ally

not well dened pro edure and may lead to un ontrollable systemati

un- ertainties. Therefore, we interpret our ndings when subtra ting the zero

modes expli itly only as a plausibility he k, whi h however points in the

right dire tion.

The use of the PP-SS orrelator is, in ontrast, safe from the

eld-theoreti al point of view. However, it leads to another di ulty. The

sub-tra tion of the zero modes from the s alar orrelator introdu es signi ant

O(a ² )

^{ee ts related to the enhan ed doublepole} ontributionto the s alar orrelation fun tion, as suggested by results from hiral perturbation

the-ory. This ee t results from the fa t that the sea and valen e quarks are

dis retized in a dierent way and thus unitarity is violated at any non-zero

valueofthelatti espa ing. Althoughbeingadis retizationee t itvanishes

in the ontinuum limit,itmay render the approa h tothis limitdi ult.

Therefore,the on eptually leanestwaytota klethezeromodeproblem

is to avoid the region of parameters where the zero modes ontribution is

signi ant. To nd this region, we analyzed the dependen e of the zero

mode ee ts on the latti e volume and the sea quark mass. In this way, we

modes ontribution, a hazardous and a non-safe regime. We onsider

the identi ation of these regions to be the most important result of this

work. It allows to provide parameter values for future simulations where

problems with the zero modes will be ompletely absent. The situation is

best illustrated in Fig. 4.26. Letus givetwo expli itexamples of the values

of pion masses and latti e sizes for safe simulations:

•

^at

m _π ≈ 300

^{MeV, the safe linear latti eextent is}

L ≈ 2.6

^fm,

•

^at

m π ≈ 450

^{MeV, the safe linear latti e extent is down to}

L ≈ 2.0

fm.

Clearly, the identi ation of safe simulation regions for valen e overlap

fer-mions is not onlyimportant for extensions of the present work, but alsofor

other ollaborationsworldwidewhoareusingoverlapfermionsinthevalen e

se tor.

Letus nish by givingsome dire tions for further work. We group these

intwoareas. The rstare possiblephysi stargetswith thesafe simulation

parameters. Withour knowledgeof these parameters, we plan to:

•

^omputeobservablesforwhi hgood hiralpropertiesofvalen efermions are essential e.g. the kaon bagparameter

B K

^{, orthe de ay}

K → ππ

^;

•

investigatequestionsthat arerelatedtotopology,i.e. the omputation oftopologi alsus eptibilityandthe determinationofthesingletmeson

mass

η ^′

^;

•

^{analyze in the mixed a tion setup unitarity violations in the s alar}

orrelatorandinmixed orrelationfun tions(withonevalen eandone

sea quark)this needsasetupwithnegligiblezero modes ontribution

to isolatethis ee t;

•

^{onfrontthesimulationresultswith(MixedA tion)PartiallyQuen hed}

Chiral Perturbation Theoryformulas toextra t the orrespondinglow

energy onstants;

•

^{perform a ontinuumlimits alingtestof thepion de ay onstant(and}

other observables) at larger volume in order to he k for the size of

quadrati latti espa ing dependen e.

Moreover, it would also be interesting to further investigate the role of the

zero modes torea ha better understanding. To this end, we plan to:

•

^test alternative mat hing onditions, dierent from the mat hing of the pion mass. In parti ular, weplan to ompute the ne essary

renor-malization onstants in order to use the mat hing ondition of equal

renormalized quarkmasses;

•

investigate the role of the zero modes in baryoni observables;

•

^{perform ananalysis of} topologi alaspe ts by expli itly omputing the zero modes.

Summarizing,webelievethattheresultsofthisworkprovideanessential

andsofarmissingbasisforfuturelarges alesimulationsusingmixeda tions.

Inparti ular,foroursetupofoverlapvalen eandmaximallytwistedmasssea

quarks we have determined simulation parameters for safe simulationson a

quantitativelevel. Thus, respe tingthe limitsontheparametersdetermined

here and performing simulations on large enough latti e volume at a given

pion mass, it will be possible to prot from the good hiral properties of

overlapfermionsandobtainpre isephysi alresultsforquantitiesthatwould

behardtoaddresswithnon hirally-symmetri versionsof latti efermions.

Firstofall,IwouldliketothankmysupervisorKarlJansen,whointrodu ed

metoLatti eQCDandwasalwayspatientinansweringallmyquestionsand

sharing his great experien e. Thank you for your onstant support, many

fruitfulandinspiringdis ussionsandthefriendlyatmospherethatyoualways

reate.

Ithank my supervisor PiotrTom zak whoalsosupported mewhile Iwas

working on this proje t and from whom I learned a lot over many years,

espe iallyabout s ienti programming and statisti alphysi s.

Very spe ial thanks go to Gregorio Herdoiza for many important and

insightful dis ussions, aswellas for numerous areful ross- he kings of the

results. Thank you for answering a lot of my naive questions and tea hing

me the right (patient)attitude tophysi s problems. Thank you alsofor the

very pleasant atmosphere ofour ommonwork.

Iwouldliketothankallthepeoplewith whomI haveworked ondierent

aspe ts related to this thesis: Vin ent Dra h, Elena Gar ia Ramos, Jenifer

Gonzalez Lopez, Agnieszka Kujawa (whois now my wife), Andrea Shindler.

Thank youfor many stimulatingdis ussions and for the ni e working

atmo-sphere.

IamalsoindebtedtoKarolinaAdamiakthanks towhomI metKarl and

ame to Zeuthen for the rst time.

I thank allthe people who ontributed tothe omputer ode that I have

been using and who have helped me in the use of this ode, espe ially to:

Remi Baron, Vin ent Dra h, Luigi S orzato, Andrea Shindler, Carsten

Ur-ba h, Mar Wagner.

I a knowledge useful dis ussions with: Mariane Brinet, Maarten

Golter-man, Dru Renner, Luigi S orzato, Stefan S haefer, Carsten Urba h, Urs

Wenger.

I would also like to thank the Organizers of Les Hou hes 2009 Summer

S hool  Modern perspe tives in Latti e QCD: Quantum eld theory and

high performan e omputing Laurent Lellou h,Rainer Sommer,Benjamin

Svetitsky,AnastassiosVladikasformakingitpossibleformetoparti ipate

s hool for their very lear presentation of di ult topi s that allowed me to

learn many importantaspe ts for this work.

I have alsoproted alot from the Latti e Pra ti es workshop in 2008. I

thank the Organizers ofthis s hool Karl Jansen,Dirk Pleiterand Carsten

Urba h.

I a knowledge the use of omputer resour es of the Leibniz

Re henzen-trum inMuni hand Pozna«Super omputingandNetworking Centre. I also

thank the staof these institutions for te hni al support.

I thank DESY Zeuthen for hospitality and nan ial support during my

stays in Zeuthen.

Thiswork waspartly nan ed fromMinistryofS ien e and Higher

Edu- ation grant nr. N N202 237437. I alsoa knowledge nan ialsupportfrom

the Foundation for Polish S ien e who granted me the START s holarship

(2009, 2010).

Lastbut notleast, Ithankmyfamilywhosupported meovermanyyears

 espe iallymy wife Agnieszka and my parents.

Wilson gauge a tion

We showhere that the expression for the Wilson gauge a tion has the right

QCD ontinuum limit. We willuse the Baker-Campbell-Hausdorformula:

e ^aA e ^aB = e ^{aA+aB+ a}

2

2 [A,B]+O(a ³ ) ,

^(A.1)

generalized to:

e ^aA e ^aB e ^aC e ^aD = e a(A+B+C+D)+ ^a ₂ ² ([A,B]+[A,C]+[A,D]+[B,C]+[B,D]+[C,D])+O(a ³ ) ,

(A.2)

setting:

A = igA _µ (x)

^,

B = igA _ν (x+aˆ µ)

^,

C = igA _µ (x+aˆ ν)

^and

D = igA _ν (x)

^.

Now, inserting(1.30) in (1.33)and using (A.2), weobtain:

U x,µν = exp h

iga A µ (x) + A ν (x + aˆ µ) − A ^µ (x + aˆ ν) − A ^ν (x)
+

− g ² a ² 2



[A µ (x), A ν (x + aˆ µ)] − [A ^µ (x), A µ (x + aˆ ν)] +

−[A µ (x), A _ν (x)] − [A ν (x + aˆ µ), A _µ (x + aˆ ν)] +

−[A ^ν (x + aˆ µ), A ν (x)] + [A µ (x + aˆ ν), A ν (x)]  + + O(a ³ ) i

.

^(A.3)

WeTaylor-expand terms like:

A _µ (x + aˆ ν) ≈ A µ (x) + a∂ _ν A _µ (x)

^(A.4)

to order

a

^{and this implies:}

From (1.8) and (1.6), the ommutatorof the gauge elds an berearranged

as:

[A _µ (x), A _ν (x)] = A ^b _µ (x)A ^d _ν (x)[t _b , t _d ] = if ^bdc A ^b _µ (x)A ^d _ν (x)t _c ,

^(A.6)

nally yielding(negle ting

O(a ³ )

^terms):

U _x,µν = exp h

Expanding to

O(a ⁴ )

^{, we obtainfor the Wilson a tion (1.31):}

S gauge [U] = β X

sin e

O(a ² )

^{terms are purely imaginary. Finally, we use}

P

x

Tree-level s aling test

B.1 Overlap fermions

We show here expli itly the omputation of the overlap Dira operator in

momentum spa e, whi h was given by Lüs her [29℄. We begin with the

derivation of the kernel operator  the massless Wilson-Dira operator in

momentumspa e. In position spa e, this operator isgiven by:

D ˆ

^Wilson

= 1

2 γ µ (∇ ^∗ µ + ∇ ^µ ) − ar∇ ^∗ µ ∇ ^µ
,

^(B.1)

where we use the notation introdu ed in Chapter 1. The Eu lidean a tion

an bewritten as:

S = X

x,y

ψ(x)K ¯ xy ψ(y),

^(B.2)

with:

K xy = 1 2

X

µ

 γ µ (δ x+ˆ µ,y − δ ^{x−ˆ µ,y} ) − r (δ ^{x+ˆ µ,y} + δ x−ˆ µ,y − 2δ ^x,y ) 

.

^(B.3)

Using the integral representation of the Krone ker delta:

δ _x,y = Z π

−π

d ⁴ p

(2π) ⁴ e ^ip(x−y)a ,

^(B.4)

we obtain:

K xy = Z π

−π

d ⁴ p

(2π) ⁴ e ^ip(x−y)a

"

X

µ

1

2 γ µ e ^ipµa − e ^−ipµa
+

^(B.5)

− r

2 e ^{ipˆ µa} + e ^{−ipˆ µa} − 2

!#

.

e ^{ipˆ µa} − e ^{−ipˆ µa} = 2i sin ap µ ,

^(B.6)

The expression inbra kets isthe Fouriertransform of

K mn

^{, i.e. the}

Wilson-Dira operator inmomentum spa e

D ˆ

^Wilson

(p)

^{. Hen e, adding expli itlythe}

identity matrix in Dira spa e, we have:

D ˆ

^Wilson

(p) = iγ µ ˚ p µ + ar

2 p ˆ ²

¹

,

^(B.9)

whi hisexpression (1.49)for theWilson-Dira operatorat

m = 0

^or

expres-sion (1.45)for the naiveDira operator,ifthe Wilson parameter

r = 0

^{. One}

an alsonoti ethattheinverseofthis formulaimmediatelygivesexpressions

for the fermion propagator (1.47)in the naive ase and (1.51)in the Wilson

ase.

We now set the Wilson parameter

r = 1

^{and then the denition of the}

operator

A

^{for overlap fermions (1.75)(with}

s = 0

^)yields:

A = 1 − aD ^W (p) = 1 − iaγ ^µ ˚ p µ − a ²

2 p ˆ ² .

^(B.10)

Sin e the massless overlap Dira operatoris given by:

D ˆ

^ov

(0) = 1

Using nowthe following rearrangements:



a ² X

The nal form of the tree-level overlap Dira operator in momentum spa e

is:

One an show [55℄ that the free Dira operator for Creutz fermions an be

written inmomentum spa e as:

D Creutz (p) = i X

¯b = 1

where the onstants

C

^,

R

^and

S

^{are parameters hosen} appropriately to ensurethe orre t ontinuumlimitofthe fermionpropagator. One anshow

that these values are:

C = 3/ √

with the fun tions

s

^and

c

^{given by}

s 1 (p) = [˚ p 1 + ˚ p 2 − ˚ p 3 − ˚ p 4 ] ,

^(B.25)

The freeDira operator forthe modi ation suggested by Bori i is:

D Borici (p) = i X

S Borici (p) = −i P

The starting point for the derivation of the expression for the pseudos alar

orrelation fun tion asa sum over momentais eq. (2.15):

orrelation fun tion asa sum over momentais eq. (2.15):

W dokumencie Uniwersytet im. Adama Mickiewicza Adam Mickiewicz University (Stron 132-148)