Lattice QCD Calculations of Hadron Spectra and Spectral Functions In

Home , Lambda baryon, List of quasiparticles, Quasiparticle

Lattice QCD Calculations

of Hadron Sp ectra and Sp ectral Functions

in the Vacuum and in a Thermal Heat Bath

Dissertation

zur Erlangung des Doktorgrades

at fur Physik an der Fakult

at Bielefeld der Universit

vorgelegt von

Ines Wetzorke

Septemb er

Contents

Intro duction

QCD and the Phase Diagram

Quantum Chromo Dynamics

Phenomena at Finite Temp erature

Phases at Finite Density

Strange Matter and the HDibaryon

Lattice Gauge Theory

Lattice Regularization

Gluonic Part of the Action

Fermionic Part of the Action

Continuum Limit

Monte Carlo Integration

Inversion of the Fermion Matrix

Gauge Fixing

Error Analysis

Contents

Strange Hadron Sp ectrum

Hadron Multiplets

Hadron Op erators and Correlation Functions

Extracting Masses from Correlation Functions

Extrap olation to the Physical Point

Improvement through the Fuzzing Technique

Hadron Sp ectrum

Strange Particle Masses

HDibaryon Stability

Hadron Sp ectral Functions

Sp ectral Functions in QCD

Principles of the Maximum Entropy Metho d

Details of the Algorithm

Dep endence on Input Parameters

Meson Sp ectral Functions

Diquark Sp ectral Functions

Thermal Sp ectral Functions

Free Meson Sp ectral Functions in the Continuum

Free Meson Sp ectral Functions on the Lattice

Meson Sp ectral Functions at Finite Temp erature

Conclusions

A Conventions

B Numerical Results

List of Figures

Schematic QCD phase diagram

U and chiral symmetry restoration of pseudoscalar mesons

Theoretical predictions of the Hdibaryon mass

Multiplet structure for pseudoscalar vector mesons and spin baryons

Multiplet structure for antitriplet and sextet diquark states

Fuzzed fermion op erator with radius R

Fuzzing for mesons baryons and dibaryons

Inuence of the fuzzing technique on the Hdibaryon correlation function

and the eective masses

Strange hadron and Hdibaryon masses at physical mass

Dierence of Hdibaryon and *lamb da masses for all lattice sizes

Dierence of Hdibaryon and *lamb da masses for dierent input

Relative error of Hdibaryon and lamb da correlation functions

Sharply p eaked p osterior distribution P jD m

Inuence of the input parameters on the sp ectral function at T

Correlation functions and corresp onding sp ectral functions for the zero tem

p erature pseudoscalar and vector meson at

List of Figures

Sp ectral functions for the pseudoscalar and the vector meson at dierent

values

Comparison of the MEM results with conventional twoexp onential t results

Inuence of the fuzzing technique on the sp ectral functions

Correlation functions and corresp onding sp ectral functions for the zero tem

p erature diquarks

Inuence of the covariance matrix smo othing pro cedure

Comparison of the MEM results with exp ts for the color antitriplet

and sextet diquarks

Discretized free thermal meson correlation function and reconstructed sp ec

tral functions

Cuto and nite volume eects on the free meson correlation function

Reconstructed free lattice sp ectral functions

Fuzzing of the free sp ectral function

Meson sp ectral functions at T

Meson masses at T obtained with MEM and twoexp onential ts the

band indicates the range obtained for the dierent data sets

Finite size eects on the pseudoscalar sp ectral function b elow the critical

temp erature

Sp ectral functions of the pseudoscalar and vector meson for several tem

p eratures

Meson correlation and sp ectral functions ab ove T

Ratio of meson and free thermal meson correlator on the lattice

Restauration of the chiral symmetry b etween the scalar and pseudoscalar

meson

Intro duction

Two asp ects of the strong interaction b etween quarks and gluons are investigated in the

present thesis On the one hand it covers the exploration of the Hdibaryon stability

a six quark state with equal content of the light u d and s quarks In cold and dense

matter this particle may b e formed as a condensate of two baryons due to strong

attractive interactions b etween quarks with maximal symmetry in the avor spin and color

quantum numb ers On the other hand changes of meson prop erties in a thermal medium

are investigated for a large temp erature range This constitutes the rst application of

the Maximum Entropy Metho d in the sp ectral analysis of thermal meson correlators

The phase diagram of QCD at vanishing baryon density is already well explored by lattice

simulations Starting at low temp erature in the conned hadronic phase and increasing

the temp erature a phase transition to the deconned quarkgluon plasma QGP phase

can b e observed Recent exp eriments at CERN SPS gave rst evidence that such a new

state of almost freely propagating quarks and gluons might exist In the near future the

colliders RHIC in Bro okhaven and LHC at CERN will hop efully op en the p ossibility to

explore the features of this plasma phase in more detail Exp erimental signatures at high

temp erature like dilepton pro duction rates are directly related to meson sp ectral functions

in this case to the vector channel The sp ectral analysis of meson correlation functions

b ecame recently accessible with the Maximum Entropy Metho d MEM which p ermits to

study the temp erature dep endent mo dications of the sp ectral shap e from rst principles

A similarly detailed theoretical understanding of the high density region of the QCD

phase diagram at low temp erature is of physical relevance in heavy ion collisions as well

as in astrophysics An interesting phase structure is exp ected to arise in this case from

quantum statistic eects which eg favor b osonic forms of matter over fermionic states

At high densities the SU color symmetry might b e sp ontaneously broken giving rise

to a color sup erconducting phase characterized by the formation of diquark condensates

Furthermore dibaryons or even larger quark clusters may play an imp ortant role as Bose

condensates in hyp ernuclear matter at increasing density In the current analysis previ

ously obtained diquark correlators are explored more precisely with the Maximum Entropy

Metho d whereas the stability of larger quark clusters is examined in a detailed analysis

of the smallest ob ject of this kind the Hdibaryon

Introduction

The investigation of these phenomena in lattice QCD is at present restricted to simulations

at an average baryon density of zero since the probabilistic interpretation of the QCD

partition function in the path integral representation breaks down for nonzero chemical

p otential Such calculations provide nevertheless an insight to the dominant quarkquark

interactions in this region

In the rst chapter QCD is describ ed as SU gauge theory in the Euclidean path integral

formalism The current knowledge ab out the QCD phase diagram at nite temp erature

and density is summarized with the fo cus on the changes of hadronic prop erties at the

dierent phase b oundaries After a short intro duction to the relevant asp ects of strange

matter in the cold and dense region of the phase diagram the previous exp erimental and

theoretical searches for a stable Hdibaryon are reviewed

The second chapter describ es the lattice regularization of the Euclidean path integral

which includes the sp ecication of the improved discretizations of the gauge and fermion

action used for the Monte Carlo simulation in the present thesis In the following some

numerical details of the lattice calculation are explained which includes the generation of

gauge eld congurations the inversion of the fermion matrix the gauge xing pro cedure

for gaugevariant observables as well as the error analysis for correlated data sets

The lattice study of the strange hadron sp ectrum and the investigation on the stability

of the Hdibaryon is the content of the third chapter After a few general considerations

ab out the hadron multiplet structure and the underlying interactions b etween quarks the

lattice setup is illustrated with the appropriate observables for the investigated hadron

channels Moreover the extrap olation of the particle masses to physical quark masses is

explained Finally results obtained on four dierent lattice sizes are presented for several

strange hadron masses as well as for the Hdibaryon mass

The fourth chapter gives an intro duction to the Maximum Entropy Metho d and describ es

the actual numerical implementation in the QCD context After the exploratory study of

the indep endence of the approach under changes of the input parameters the sp ectral

analysis is p erformed for meson and diquark correlation functions at zero temp erature

Furthermore the applicability of the approach at nite temp erature is tested with free

meson sp ectral functions Finally a lattice investigation of the scalar pseudoscalar and

vector meson correlators at nite temp erature and their sp ectral analysis with the Maxi

mum Entropy Metho d is presented

Chapter

QCD and the Phase Diagram

The starting p oint of the present investigation is the formulation of QCD as a nonab elian

SU gauge theory which is presented in the rst section The current knowledge

ab out the QCD phase diagram and the prop erties of hadronic matter at nite temp erature

are summarized in section while the rich phase structure at nite density is describ ed

in section This paragraph is followed by some considerations ab out the Hdibaryon in

the context of strange quark matter and a short review of previous investigations on the

stability of such a particle in section

Quantum Chromo Dynamics

In the framework of the Standard Mo del of elementary particle physics Quantum Chromo

Dynamics QCD is the fundamental theory of the strong interaction As a nonab elian

gauge theory it is lo cally gauge invariant under SU color transformations The basic

constituents namely the two light u and d quarks the s quark with intermediate mass

as well as the heavier c b and t quarks interact through the exchange of gluons the

gauge b osons of the theory This is reected in the fermionic and gluonic part of the

gaugeinvariant QCD Lagrangian for N quark avors with mass m

f f

L x L x L x

QC D F G

L x xi D m x

f =1

Tr F xF x L x

where the Greek letters denote the Dirac indices of the quark elds The covariant

derivative D as well as the eld strength tensor F involve the bare gauge coupling g

Chapter QCD and the Phase Diagram

D ig A x

F x A x A x ig A x A x

2 a

The gauge elds A are related to the N generators of the gauge group SU

A x A x

a=1

Due to the nonab elian character of QCD the gauge elds do not commute which is

manifest in the selfinteraction b etween the gluons

The quantization of the eld theory is realized in the Euclidean path integral formulation

The QCD partition function explicitly dep ends on the volume and the temp erature

(V T A ) S

QC D

Z V T D AD D e

while a p ossible dep endence on a nonzero chemical p otential is neglected herein The

thermal exp ectation value of an observable O in the Euclidean representation can b e

calculated as follows

(V T A ) S

QC D

D AD D O A e h O i

Z V T

The Euclidean action S at nite temp erature is dened through the integral over the

QC D

QCD Lagrangian in imaginary time

Z Z

3 E E

d d x L with S V T

QC D QC D

0 V

E f

L x x D m Tr F xF x

QC D

f =1

The Euclidean matrices which fulll the anticommutation relation f g

are given in app endix A Note that the index E will b e omitted from now on since only

quantities in the Euclidean metric are utilized

Apart from the lo cal SU color gauge symmetry the QCD Lagrangian p ossesses ad

ditional global symmetries In the massless continuum theory the chiral symmetry for

N quark avors is describ ed classically by SU N SU N U U The

L R V A

f f f

axial U symmetry is explicitly broken already at quantum level whereas the preserved

U symmetry implies the global baryon numb er conservation due to No ethers Theo

rem The chiral SU N SU N symmetry for the two lightest quark avors m

L R

f f ud

is sp ontaneously broken at zero temp erature which is asso ciated with the app earance of

2 + 0

N nearly massless Goldstone b osons the pseudoscalar isospin triplet f

Phenomena at Finite Temperature

The strong interaction is characterized by

two general features Quarks and gluons are

ery weakly coupled at small distances and

v quark gluon plasma

thus b ehave as freely propagating particles

asymptotic freedom whereas the cou

pling gets strong for large separations and

momenta As a consequence quarks small nuclear

color ve only b een observed in hadronic b ound

ha liquid superconducting

connement These prop erties states gas

hyper nuclear

reected in our basic picture of the QCD

are µ

phase diagram in gure When increas

Figure Schematic QCD phase diagram ing the temp erature T andor the baryon

chemical p otential the nonp erturbative

structure of the QCD vacuum which is characterized by connement and chiral symmetry

breaking gets lost and eventually will end in an asymptotically free quark gluon plasma

Phenomena at Finite Temp erature

In lattice simulations the phase diagram has quite successfully b een explored by varying

the temp erature T at vanishing baryon chemical p otential The chiral symmetry is

broken in the connement phase at low temp erature where the quarks and gluons form

b ound states within a hadron gas A phase transition to a plasma phase with restored

chiral symmetry is observed by increasing the temp erature Strictly sp eaking one has to

distinguish b etween the deconnement and the chiral symmetry restoring phase transition

In the quenched limit with innite quark masses the Polyakov lo op exp ectation value is

an order parameter for the deconnement transition whereas the chiral condensate is the

order parameter for the chiral symmetry restoration in the limit of vanishing quark masses

In QCD where the fermions b elong to the fundamental representation of the SU color

group these two phase transitions seem to o ccur simultaneously at a common critical

temp erature T However they are well separated in theories where the fermions

b elong to the adjoint representation

The critical temp erature as well as the order of the phase transition dep end on the numb er

of colors N and quark avors N resp ectively Furthermore the o ccurrence of a transition

dep ends on the magnitude of the quark masses In the quenched theory N N

with innitely heavy quarks the deconnement phase transition is rst order and o ccurs

at ab out T MeV For two massless quark avors the chiral phase transition is

second order at T MeV It is exp ected to change with increasing chemical

p otential to rst order at a tricritical p oint tc as indicated in gure Intro ducing a

small quark mass m the second order transition turns into a smo oth crossover and

the tricritical p oint b ecomes a critical endp oint which may b e determined quantitatively

in heavy ion collision exp eriments A more realistic strange quark mass m results

in a shift of the tricritical p oint towards the temp erature axis As a consequence the chiral

phase transition b ecomes rst order even at zero chemical p otential in the limit of three

Chapter QCD and the Phase Diagram

massless quark avors The critical temp erature T MeV in the case N was

obtained in a recent lattice simulation

light hadron sp ectrum is sensitive to 0.6 The

pion

changes of the chiral condensate and the a0 [δ] the 0.5

f0 [σ]

chiral symmetry restoration in the vicinity

the critical temp erature In the chirally

0.4 of

symmetric phase ab ove the critical temp era

0.3

ture the pseudoscalar meson is no longer a

0.2

Goldstone b oson For N the thermal

function in this quantum num 0.1 correlation m = 0.02

βc

channel should coincide with that of the L = 8 b er

0.0

singlet scalar meson Such a b e

5.24 5.26 5.28 5.3 5.32 5.34 isospin

havior is similarly anticipated for the axial

Figure U and chiral symmetry

vector mesons a and On the other hand

restoration of pseudoscalar mesons

an eective restoration of the U symme

try would involve a degeneracy of the avor

triplet pseudoscalar and as well as of the singlet and meson correlators Fig

ure shows results of pseudoscalar susceptibilities ie integrated meson correlation

functions in the Euclidean time interval T obtained in a lattice simulation with stag

gered fermions similar in The inverse of these susceptibilities may b e viewed as

the square of eective thermal masses The and meson masses b ecome degenerate near

T while the approaches the other masses slowly with increasing temp erature which

implies that the U symmetry remains broken at T This b ehavior was conrmed by

A c

recent lattice simulations with overlap and domain wall fermions

So far information on thermal masses has either b een obtained from susceptibilities like

those in gure or from spatial correlators which dene screening masses Both ap

proaches do not yield direct information ab out thermal hadron masses In particular the

eective masses extracted from susceptibilities have no welldened counterpart in the

continuum limit In order to get access to the temp erature dep endence of p ole masses one

has to analyze thermal correlators in Euclidean time Information on masses is then en

co ded in the resp ective sp ectral functions where a dominant ground state mass is manifest

as a sharp function like p eak Moreover the sp ectral representation allows the distinc

tion b etween p ole and continuum contributions which b ecomes increasingly imp ortant at

nite temp erature A detailed investigation of hadronic correlators and sp ectral functions

at several temp eratures b elow and ab ove the phase transition is presented in chapter

Previous lattice simulations with staggered as well as Wilson fermions gave no

evidence for signicant changes of the meson masses b elow the critical temp erature Due

to the limited extent of the lattice in the Euclidean time direction T most of the

mentioned studies investigated only spatial correlation functions The screening and p ole

masses are exp ected to coincide b elow the critical temp erature if the T disp ersion

relation still holds true at T T At temp eratures well ab ove T the b ehavior of the

c c

spatial meson correlators can b e explained by the propagation of two weakly interacting

quarks in the plasma Thus the screening masses will approach the lowest Matsubara

Phases at Finite Density

frequency T induced by the antip erio dic b oundary conditions for fermions in the tem

p oral direction Nevertheless collective phenomena may still b e present in the plasma

phase for temp eratures close to T

In a recent simulation with Wilson fermions on anisotropic lattices it has b een observed

that the p ole masses are considerably smaller than the screening masses ab ove T which

led to the interpretation that mesonic excitations or metastable b ound states p ersist up

to at least T These conclusions however have b een obtained by using standard

exp onential tting pro cedures which are suitable at zero temp erature At T such an

approach should b e applied carefully b ecause the continuum contributions may have a

considerable inuence on the resulting p ole masses in the short temp oral direction Since

the sp ectral representation allows to distinguish b etween the dierent contributions the

analysis of correlation functions with the Maximum Entropy Metho d MEM will

provide new insight in the temp erature dep endence of p ole masses A rst application

of MEM for pseudoscalar scalar and vector meson correlators at nite temp erature is

presented in section

Phases at Finite Density

Now the fo cus is on the conjectured phase diagram g at low temp eratures and high

densities Moving away from the temp erature axis towards larger densities an additional

rst order phase transition line is observed separating the hadron gas with zero baryon

density from a nuclear liquid phase with density n f m At zero temp erature this

jump in the density is exp ected to o ccur at a baryon chemical p otential of the nucleon

mass minus its binding energy The endp oint of this rst order phase transition line

was estimated exp erimentally at GSI to lie in the region of times the nuclear

density and a temp erature of O MeV which is characteristic for the binding energy

of nuclei

At even higher densities an additional color sup erconducting phase is indicated in gure

In the theory of sup erconductivity the BCSmechanism leads to the formation of

a Bose condensate of electron Co op er pairs for an arbitrarily weak attractive interaction

A phononinduced interaction is needed in this case to reduce the repulsive electrostatic

force b etween the electrons In QCD the o ccurrence of a BCSlike mechanism is even more

straightforward Since quark momenta are large in the region of high density asymptotic

freedom implies that the coupling b etween the particles is weak Unlike in the case of

electrons the fundamental interaction b etween quarks is already attractive for u and d

quarks with dierent color and antiparallel spin This can b e deduced from the p er

turbative onegluonexchange as well as from the instanton liquid mo del

At high densities it is thus energetically favorable for the weakly coupled quarks to form a

diquark Bose condensate In this color sup erconducting phase the lo cal SU color gauge

symmetry is broken to SU Since the chiral symmetry is restored and none of the

global symmetries of QCD is broken no order parameter distinguishes the two avor color

sup erconducting phase from the quark gluon plasma

Chapter QCD and the Phase Diagram

In the case of three massless quarks such diquark condensates cannot b e avor singlets In

addition to a condensate of ud quark pairs also us and ds Co op er pairs may o ccur which

are only invariant under a correlated coloravor symmetry Such a sup erconducting

coloravor lo cked CFL phase is thus characterized by the broken baryon numb er

and chiral symmetry

SU SU SU U SU

C L R V C +L+R

This allows a distinction to the other phases since the breaking of global symmetries

provides an order parameter for the phase transition Such an order parameter for the

baryon numb er symmetry breaking would carry the quantum numb ers of the Hdibaryon

The symmetries of the CFL phase are the same as exp ected for hyp ernuclear matter

where hyp erons strange baryons are exp ected to pair into a SU avor singlet state

like the Hdibaryon or N Dierent from the phase transition at high

temp eratures and low baryon density where many new degrees of freedom arise their

numb er may remain the same with increasing baryon density at low temp erature The

hadronic degrees of freedom might b e followed continuously from hyp ernuclear matter to

the CFL phase of quark matter without a phase transition Such a b ehavior has b een

recently describ ed as quarkhadron continuity

Strange Matter and the HDibaryon

A somewhat dierent context in which the Hdibaryon might b e imp ortant is the physics

of strange quark matter At very high density and low temp erature the prop erties

of quarks and gluons in the deconned phase are exp ected to dier from those of the

quark gluon plasma at high temp erature therefore this state was given the name quark

matter Since the pressure gets higher at increasing density the quarks have to o ccupy

higher energy states due to the Pauli exclusion principle Then it could b e energetically

favorable for u and d quarks to convert into s quarks via weak interaction pro cesses After

such an equilibration pro cess of the quark avor content a nite numb er of strange quarks

is left which motivated the name strange quark matter or simply strange matter It

is susp ected that such a state of matter could exist in the core of neutron stars where

ordinary quark matter built from neutrons may b e converted to strange quark matter

under the inuence of the enormous pressure

It was suggested by Witten in that strange matter might b e more stable than

ordinary nuclei Strange matter could exist in small lumps called strangelets having lower

energy than a nucleus with the same amount of quarks This was the starting p oint for

the disaster scenarios for the new relativistic heavy ion collider RHIC in Bro okhaven

Concerns were raised that a transition to a lower vacuum state could b e initiated due to the

formation of stable strangelets The arguments against such a scenario were summarized

in a recent review article and were not proven false by the actual exp eriments at RHIC

in the meanwhile

Strange Matter and the HDibaryon

In the context of strange quark matter the Hdibaryon is the lightest p ossible strangelet

Such a six quark state udsuds is the lightest SU avor singlet state with

P +

spin zero strangeness and J It has the smallest energy p er baryon numb er

since b oth color and spins cancel pairwise to the greatest p ossible extent A stable H

dibaryon was rst predicted in by Jae in a bag mo del calculation He obtained

a mass of m MeV which is MeV b elow the threshold for strong decay

This observation had inspired many theoretical and exp erimental searches for a stable

Hdibaryon during the following decades

Various QCD motivated mo dels were applied in the theoretical search for a stable H

dibaryon Calculations were p erformed in the bag mo del the nonrelativistic quark cluster

mo del the Skyrme mo del and with QCD sum rules References on all of these investiga

tions can b e found in a recent review article In general p erturbative calculations of

the hadron mass splittings are based on spindep endent quarkquark interactions which

includes colorspin coupling for the onegluonexchange OGE and avorspin coupling

for instanton induced interactions I I I and Goldstoneb osonexchange GBE This is

illustrated more explicitly in the rst section of chapter

ΛΛ (2.23)

ΝΝ (1.88) ΝΞ (2.26)

ΣΣ (2.38)

Mass ( GeV )

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

Figure Theoretical predictions of the Hdibaryon mass

Combining the previous theoretical investigations a slightly b ound or unb ound Hdibaryon

is predicted This is visualized in gure where the thresholds for strong N

and weak decay NN are indicated in addition In the case of a b ound state a denite

conrmation could b e exp ected from exp erimental searches in a range of a few hundred

MeV b elow the MeV threshold for strong decay The Hdibaryon may b e pro duced

in several pro cesses such as heavy ion collisions K K reaction or capture The

presence of a b ound state can only b e detected from the pro ducts of the weak decay

for example H p H n or H n The key reaction pro cesses for each

of the several exp eriments are listed in the review article of Sakai et al while the

details ab out the exp erimental setup can b e found in the references therein Most of the

rep orted investigations provided no evidence for a stable Hdibaryon Nevertheless a few

candidate events for a slightly b ound Hdibaryon were detected but it remained unclear

if these observations could b e explained by other than the exp ected pro cesses

Finally the previous attempts to calculate the Hdibaryon mass on the lattice should

b e reviewed The simulation parameters chosen by the dierent groups are summarized

Chapter QCD and the Phase Diagram

Ref GaugeFermion Action afm m MeV Lattice Size N m MeV

q C H

standard xx unb ound

RGimpstandard x

standard x larger

standard x

Table Summary of simulation parameters for the previous attempts to determine the

mass of the Hdibaryon on the lattice

in table The two earlier investigations of Mackenzie and Thacker and Iwasaki

et al gave contradicting results on the basis of limited statistics and relatively large

quark masses Moreover deviations could b e observed for dierent discretizations of the

gauge action A more precise study of Negele et al provided recently some insight

in the volume dep endence of the obtained results Considerable nite size eects could

not b e ruled out by the authors for the smaller lattice size The subsequent analysis on a

larger lattice led to the conclusion that the Hdibaryon is unb ound in the innite volume

limit

In the present study the volume dep endence of the Hdibaryon mass is investigated on

several lattices sizes with a relatively large lattice spacing which op ens the p ossibility to

examine a physical volume comparable and even larger than in the recent study of Negele

et al Preliminary results on two of the lattice sizes were already rep orted in

They have indicated an unb ound Hdibaryon state with the mass of two baryons The

detailed analysis of the strange hadron sp ectrum and in particular the Hdibaryon stability

is provided in chapter

Chapter

Lattice Gauge Theory

Having illustrated QCD in the Euclidean path integral formulation in the rst chapter

we can now pro ceed to the representation as a regularized gauge theory on the lattice

The basic asp ects of lattice QCD are intro duced in section including an overview of

improved discretizations of the gauge and fermion actions This section is completed by a

paragraph ab out the continuum limit of the lattice formulation For a general overview of

lattice gauge theories the interested reader is referred to the textb o oks of Rothe and

MontvayMunster

The following sections deal with the numerical details of the simulation First of all this is

the Monte Carlo integration metho d in section which enables us to carry out the high

dimensional QCD path integral An ecient inversion of the fermion matrix is imp ortant in

order to calculate the hadronic op erators see sec For some gaugevariant observables

a gauge xing of the gauge eld congurations is necessary This pro cedure is explained

in section The last section of this chapter contains a short description of the error

analysis for correlated data sets

Lattice Regularization

Since the Euclidean path integral intro duced in formula is mathematically not well

dened a regularization scheme has to b e implemented Perturbative metho ds usually

limit the magnitude of the momenta through the cuto parameter Another p ossibility

is to ensure the regularization in co ordinate space by intro ducing a four dimensional space

N with a small but nite lattice spacing a The volume as well time lattice of size N

as the temp erature on this hyp ercubic lattice are determined through its spatial and

temp oral extents in terms of the lattice cuto a

and T V N a

N a

The lattice regularization requires the discretization of integrals and derivatives which

transform into nite sums and dierences The fermion elds which are p ositioned on the

Chapter Lattice Gauge Theory

lattice p oints and observables like the string tension and particle masses can b e rescaled

to dimensionless quantities via the lattice spacing see for details The gauge elds

are identied with the link matrices U SU which are dened b etween neighb oring

lattice sites in the direction by the parallel transp orter

x+a ^

dy A y U x na exp ig

The discretized formulas for the gluon and fermion sector of the Euclidean action on the

lattice are provided in the following two sections Moreover the construction of improved

actions with reduced cuto eects will b e illustrated for b oth parts

Gluonic Part of the Action

The lattice gauge action should b e gaugeinvariant like the resp ective continuum expres

sion The simplest ob ject on the lattice which fullls this requirement is the plaquette

i e the closed lo op of four link variables around a square

y y

x U x U xU x U x U x

The gluonic part of the action can thus b e written in terms of U x

Re Tr U x S U

is related to the gauge coupling and the numb er of colors N where the coupling

In the limit of vanishing lattice spacing a the gluon action should repro duce the

continuum representation In order to show that this is indeed the case an expansion with

resp ect to the lattice spacing a is necessary After the application of the BakerHausdor

formula and a Taylor expansion around the center of the plaquette the following equation

is obtained

h i

2 2 2 4

Tr F F O a O g a S lim a

Z Z

3 2

d d x Tr F F O g

0 V

It can easily b e observed that this simple parametrization of the gluonic action repro duces

the continuum expression correctly up to O g but nevertheless yields cuto eects

of O a on the lattice Following the Symanzik scheme these deviations can b e

eliminated by additional gaugeinvariant terms For instance the improved lattice

gauge action is correct to O a

(12)

x Re Tr S

x Re Tr x

Lattice Regularization

The co ecients of the plaquette term and the additional and lo op

contribution are chosen such that the O a deviations eliminate each other By adding

further terms consisting of larger lo ops the cuto eects can b e reduced up to any

desirable order

Fermionic Part of the Action

The naive discretization of the fermionic part of the Euclidean action can b e obtained

simply by substituting the derivative with a nite dierence of the rescaled fermion elds

i ij

Moreover one intro duces the link matrices U instead of the gauge elds which

yields the following form of the fermion action

N i N

S y with the fermion matrix xM x y

F ij

xy ij

ij yij N

U x U x ma M x y

xy ij

x+ ^ y xy^

where i j refer to the color degrees of freedom By expanding this discretized expression

in p owers of the lattice spacing and the coupling constant it can b e veried that the naive

fermion action repro duces its continuum counterpart up to O a corrections

ob ey the anticommutation relation they are realized as Grass Since the fermion elds

mann variables living on the lattice sites The calculation of exp ectation values of fermionic

observables involves the evaluation of the path integral over these Grassmann elds which

can b e easily p erformed by means of the Grassmann integration rules

M S

DUD D e h i

y x y x y x y x

n n n n

1 1 1 1

S 1 1 z z

DU det M e M M

z x z x y y

n n n

1 1 1

z z

where denotes the total antisymmetric tensor The integral of the nparticle observable

leads to a pro duct of the inverse fermion matrix comp onents while the

y x y x

n n

1 1

integration of the fermionic part of the action S M results in the determinant of

the fermion matrix M

Setting this determinant equal to one is a simplication commonly used for the com

putation of exp ectation values The socalled Quenched Approximation neglects the

contributions of internal quark lo ops and thus considers static quarks and dynamical gluon

background elds This pro cedure app ears to b e very restrictive at rst sight but it could

b e demonstrated that most of the prop erties of QCD can b e investigated qualitatively in

this approximation For instance the obtained hadron masses agree with the exp erimental

sp ectrum in an average range Since the path integral can now b e evaluated

in up dating only the dynamical gauge elds an enormous amount of computer time can

b e saved by the application of fast lo cal algorithms see section for details

Chapter Lattice Gauge Theory

A severe problem of the naive discretization b ecomes obvious in the momentum space

representation of the free fermion action for one quark avor

d p

S i sinp a m p p

The free quark propagator can thus b e obtained by employing the Grassmann integration

rules as the inverse of the ab ove fermion matrix

p q sinp a m i hpq i

p m i

sinp a p q with p

2 2

a p m

It can easily b e observed that this propagator has p oles not only at zero momentum but

also at all other corners of the Brillouin zone This phenomenon is referred to as fermion

doubler problem since a doubling of the particle content o ccurs in every dimension d In

the continuum limit this yields unwanted particles in addition to the physically

relevant one Hence the naive fermion discretization would pro duce N particles for

N quark avors which do not vanish in the limit a

In a NoGo theorem Nielsen and Ninomiya have shown that a lattice regularization

of QCD with exact chiral symmetry but without fermion doublers cannot b e achieved

The next few paragraphs deal with various metho ds which were develop ed to circumvent

or at least reduce the fermion doubler problem through the sacrice of the exact chiral

symmetry

Wilson Fermions

The aim of the Wilson fermion formulation is to increase the masses of the unwanted

doublers in a way which ensures that they diverge in the continuum limit This can b e

reached by adding a term prop ortional to the second derivative of the quark eld to the

naive fermion action

X X

i i i W W N

x2 x y xM x y S S

ij F F

xy ij

which breaks explicitly the chiral symmetry The parameter r controls the strength of this

auxiliary term In the following r will b e chosen for convenience since this yields the

pro jection op erators l in the Wilson fermion matrix

W ij

M x y ma l U x

xy ij

x+y ^

yij

x l U

xy^

which results in advantages in the explicit numerical implementation

Lattice Regularization

The free quark propagator in the momentum space can b e calculated similarly to

as inverse of the Wilson fermion matrix which leads to the momentum dep endent mass

dened b elow

p mp i

hpq i p q

2 2

p mp

mp m sin p a

This mass term ensures the divergence of the redundant particle masses in the limit a

for nonzero momenta For instance a fermion with p a would acquire a mass

while the mass of the physical particle at p remains the same mp m

In the Wilson formulation the bare quark mass exp eriences an additive renormalization

through the intro duction of the chiral symmetry breaking term Therefore the quark mass

This causes a severe tuning problem is controlled by the hopping parameter

2(ma+4)

since the quark mass now vanishes at varying values for dierent couplings In the case

of the free theory g this o ccurs at while the critical value is reached at

in the strong coupling limit

In the intermediate coupling range has to b e determined with the aid of other observ

ables in the numerical simulation In leading order p erturbation theory the quark mass

on the lattice

m m a

q q

by the PCAC hyp othesis in chiral can b e related to the square of the pion mass m m

p erturbation theory The value of at which the pion b ecomes the massless Goldstone

b oson thus denes the p oint of zero bare quark mass An alternative denition can b e

established on the basis of the axial Ward identity The quark mass is then obtained from

the ratio of the axial vector current combined with the pseudoscalar density and the pion

correlator M x see sec through

h A x M i m

hM x M i

where Z is a multiplicative renormalization factor for the axial current Having obtained

the bare quark masses for dierent values they have to b e extrap olated to zero again

Note that these two estimates of can dier by O a corrections

Finally the remark should b e added that the auxiliary Wilson term changes the dis

cretization errors from O a in the naive formulation to O a corrections for free Wilson

fermions The previous situation can b e restored by intro ducing the Clover fermion for

mulation

Chapter Lattice Gauge Theory

Clover Fermions

A further supplementary term prop ortional to the eld strength tensor was added to the

Wilson fermion action by Sheikholeslami and Wohlert in order to ensure that the

Clover action shows only O a discretization errors Rescaling the fermion elds with

x yields resp ect to the hopping parameter x

C W

S S ig c xF x x

F F i i

x y y xM

j i

xy ij

The Clover fermion matrix is dened through the Clover term A and the op erator

ij ij ij

x y x x y A M

F x A x l ig c

ij yij

x y l U x l U y

x+ ^ y xy^

j y j

where the eld strength tensor on the lattice is given by F x U x U x

8ig

The sum is dened over the four plaquettes in the plane around the site x the socalled

Clover term The numerical pro cessing of the fermion matrix is describ ed in detail

in section Altogether the O a improved Clover fermion action can b e visualized in

the following way

X X

Im x S x y l c x y

SW xy

The clover co ecient c in the action has to b e chosen appropriately For the treelevel

improved gauge action it is simply set to one but this parameter can also b e optimized

in a nonp erturbative way Simulations with dierent clover co ecients and couplings g

should ob ey the PCAC relation up to order a corrections This leads to the following

expression for c in the quenched approximation obtained with the Wilson plaquette

gauge action

2 4 6

g g g

for g c

An analogous result for simulations with dynamical fermions was rep orted in The

calculations for this thesis were p erformed with the improved gauge action and the

Clover fermion action with a treelevel clover co ecient at zero temp erature while the

simulations at nite temp erature were carried out with the plaquette gauge action and

the nonp erturbatively improved Clover fermion action

Lattice Regularization

Staggered Fermions

Another commonly used fermion discretization is the KogutSusskind or staggered

fermion formulation where the spinor degrees of freedom are distributed over hyp ercub es

of the lattice This reduces the fermion doubler problem but four degenerate quark avors

are still included in the continuum limit The advantage of staggered fermions is mani

fested in the preserved U U symmetry as remnant of the global chiral symmetry

Therefore the chiral condensate serves as an order parameter of the phase transition at

nite temp erature This is the reason why the staggered formulation is mostly used for

the calculation of thermo dynamic observables in the vicinity of the critical temp erature

Chiral Fermions

In recent years fermion discretizations with approximate chiral symmetry b ecame avail

able The starting p oint is the GinspargWilson relation D D aD D for

5 5 5

the Dirac op erator D which implies that the chiral symmetry as well as the axial U

anomaly are preserved for vanishing quark mass m on a lattice with nite lattice spacing

This general idea led to two dierent formulations known as domain wall and

overlap discretization of the fermion matrix The domain wall approach is based

on the intro duction of a fth dimension in addition to the four dimensional spacetime

lattice Then the chiral symmetry breaking is suppressed exp onentially with the extent

L of this extra dimension Exact chiral symmetry can thus b e obtained theoretically

with an innite extent L The overlap formulation however works in an innite avor

space which is equivalent to the domain wall approach in the limit N Although

these realizations of chiral fermions in lattice simulations seem very promising the actual

calculations suer nevertheless from the increase in the required computer time which

rises linearly with the extent of the fth dimension

Continuum Limit

As already addressed b efore the observables on the lattice O g a a can b e rescaled

to dimensionless quantities O through the lattice spacing in the appropriate dimension

dimO They are related to their corresp onding physical value O in the continuum

phy s

limit via

dim(O )

O g a a a O g a a O

phy s

This relation is only valid if the coupling on the lattice g a approaches the critical

in analogy coupling g which is dened at the p oint of diverging correlation length

to statistical mo dels In QCD the limit is reached with increasing or equivalently

g due to the asymptotic freedom In this weak coupling regime p erturbative metho ds

provide the dep endence b etween the lattice spacing and the coupling

3 5 7

b g b g O g a

0 1 a

Chapter Lattice Gauge Theory

with the universal co ecients

b and b N N N N N

0 1 c c

f f

2 2 2

The integration of the function then yields directly the lattice spacing in terms of

the coupling

2 2b

ag b g exp

b g

L 0

Here the integration constant denes the invariant scale for the lattice theory which is

related by a multiplicative factor to the scale parameter in other regularization schemes

like or

MOM

Given the dimensionless results from lattice simulations they have to b e prop erly rescaled

in terms of the lattice spacing The physical scale can b e inferred from the determination

of the string tension or hadron masses on the lattice which are related to their wellknown

exp erimental values

1 1

a M eV m m a M eV or

Alternatively the ratio of two physical quantities can b e used to set the scale This

approach is employed in section in order to obtain the physical quark mass values

and in the Wilson quark formulation

Monte Carlo Integration

After the basic concepts of lattice QCD the fo cus is now on the numerical implementation

of the formerly describ ed metho ds The Euclidean path integral formalism was already

intro duced in section After the analytic integration over the fermion elds equation

the exp ectation value of an op erator O in the quenched approximation is given by

S [U ]

dU xO U e hO i

Nevertheless this leaves us with a high dimensional integral of order O which can

only b e evaluated approximately by Monte Carlo integration

In generating gauge eld congurations with the Boltzmann distribution in the thermal

equilibrium one can ensure to consider the most signicant contributions to the path inte

gral imp ortance sampling A new conguration U is pro duced from its predecessor U

in a socalled Markov chain with a certain transition probability The following approxi

mate expression for the exp ectation value is thus valid for a large numb er of congurations

O U hO i

i=1

Inversion of the Fermion Matrix

The detailed balance condition is required to obtain the equilibrium probability distri

bution

S [U ] S [U ]

e P U U e P U U

Furthermore ergo dicity is needed which means that any p ossible conguration U should

b e obtainable with nite probability from every conguration U

The rst and simplest algorithm which fullls these requirements was the one develop ed

by Metrop olis et al For the simulations rep orted in this thesis the pseudoheatbath

up date metho d was utilized improved by overrelaxation steps after

each heatbath step Both are lo cal up date routines which pro duce a new conguration

by a sweep over all lattice sites The pseudoheatbath algorithm consists of the successive

application of the heatbath algorithm on all SU subsets where the lo cal equilibration

resembles the contact with an innite heatbath

The overrelaxation up date is an ecient metho d to reduce the correlations b etween the

generated congurations Since it is desirable to analyze nearly uncorrelated gauge eld

congurations the observables are measured only on congurations separated by several

hundred sweeps

Inversion of the Fermion Matrix

The most fundamental fermionic observable in lattice QCD is the exp ectation value of the

quark propagator Gx y which can b e calculated using the analytic integration over the

fermion elds see formula

Gx y hx y i

M S

DUD D xy e

1 S

DU M x y det M e

The imp ortant quantity which is thus needed is the inverse of the fermion matrix M x y

evaluated on every gauge eld conguration in the quenched approximation det M

This requires the solution of the inhomogeneous equation

M x y y x

with p ointlike sources x for every colorspin combination of the quark elds

Since the fermion matrix is a very large but particularly sparse matrix the ab ove equation

can only b e solved approximately with an iterative metho d The rapidness of

convergence is determined by the distribution of the eigenvalues of this matrix The

numb er of iterations needed for the inversion is prop ortional to the ratio of the largest to

5 6

N N fermion matrix with N lattice size color spinor degrees of freedom O

Chapter Lattice Gauge Theory

the smallest eigenvalue which is divergent for vanishing quark mass m resp ectively

b elow the critical temp erature An optimized inversion of the fermion matrix

can b e reached in a twofold way On the one hand a fast and ecient algorithm should

b e used to solve the equation within a minimum of iteration steps on the other

hand a preconditioning pro cedure can further reduce the numb er of steps required for

convergence

Most of the common algorithms develop ed for the inversion of the fermion matrix stem

from the conjugate residual metho d For the simulations describ ed in this thesis

the BiCGstab algorithm was employed It was develop ed and improved from the

Conjugate Gradient CG metho d which guarantees convergence in N iteration steps

for a N N fermion matrix It could b e shown that the BiCGstab converges much

faster than other algorithms in particular in the region of small quark masses

The eveno dd preconditioning technique uses the prop erty of the Clover fermion matrix

that it only connects lattice sites of opp osite parity The equation M

decouples into separate parts for the even and the odd sites

M A

ee e e eo o e

o o oe e

2 1

with a dierent source vector and a mo died fermion matrix M A A

e ee ee eo oe

on the even sites of the lattice After the BiCGstab inversion on the even sites the

solution for the o dd sites can simply b e obtained by a back substitution in The

ma jor advantage of this preconditioning pro cedure is the new matrix M which is now

second order in Approaching the chiral limit the numb er of required iteration

steps is considerably lower since the dep endent smallest eigenvalue of M is ab out two

times larger than the one of the original fermion matrix M

Gauge Fixing

Some of the interesting op erators in lattice QCD like the quark or diquark propagators

are gaugevariant quantities They form color antitriplet or sextet states in contrast to

the color singlet mesons and baryons Therefore it is necessary to p erform gauge xing in

order to calculate quark and diquark correlation functions

GxU x The invariance of the gluonic action under lo cal gauge transformations U

G x with a matrix Gx SU is the basis for the gauge xing pro cedure in lattice

QCD After the generation of the gauge eld congurations with the Monte Carlo metho d

they can b e transformed separately into the desired gauge A whole bunch of gauges

is dened through the gauges The gauge condition in the continuum requires

4 3

X X

A x A x A x

4 4

=1 =1

Error Analysis

with the denition for and for The gauge indep endence

of the observables can thus b e studied by varying the parameter corresp onds to

the Landau gauge whereas yields the Coulomb gauge

Evaluating the gauge condition on the lattice corresp onds to minimizing the following

functional

X X

y G

GxU xG x U x Tr F G Tr

x x

After cho osing the gauge transformation matrix Gx appropriately the functional F can

b e driven to a minimum in an iterative pro cess Two common algorithms develop ed for

this purp ose are the Cornell and the Los Alamos metho d Both algorithms

suer from a considerable slowing down eect The more precisely the gauge condition is

fullled the nearer the transformation Gx resembles the unitary transformation which

makes the algorithm less and less ecient

Various metho ds have b een prop osed to sp eed up the convergence for example the overre

laxation pro cedure or the timeconsuming Fourier acceleration In contrast to all

other algorithms mentioned ab ove the Fourier acceleration is a nonlo cal pro cedure which

involves wide range communications on parallel machines Therefore the numerically op

timized fast Fourier transformation FFT is commonly used for the implementation

For the previous simulations at zero temp erature which were already rep orted in

the Landau gauge was used It was implemented with the Cornell metho d and acceler

ated by a FFT pro cedure For the new congurations ab ove the critical temp erature the

Coulomb gauge condition was applied realized through the Los Alamos algorithm com

bined with the overrelaxation metho d The Coulomb gauge was chosen with regard to

the maximum entropy metho d MEM which is describ ed in detail in chapter The aim

was to yield a p ositive denite sp ectral function for the quark propagator resp ectively a

plateau for the eective quark mass which is reached from ab ove see section This is

denitely not the case for the Landau gauge where no transfer matrix can b e constructed

Error Analysis

The gauge eld congurations generated by Monte Carlo integration sec are not

completely uncorrelated The Jackknife pro cedure takes the auto correlation b etween

the observables on dierent congurations into account and provides an improved estimate

of the mean values and errors

For this purp ose the complete data set is divided into N subsamples of equal length

Leaving out one subsample resp ectively the mean value on each reduced sample D can k

Chapter Lattice Gauge Theory

b e calculated The improved mean value and the corresp onding error are then obtained

by the statistical average

O J J with J ND N D

k k k

k =1

J J

k =1

N N

where D denotes the mean value on the whole sample

Furthermore strong correlations b etween dierent time slices are observed for hadronic

correlation functions The standard t for a function F p which dep ends on the

parameters p

F p D

i i

includes only the standard deviation for each time slice indep endently and neglects

the correlations among them completely Therefore it is essential to take the inverse of

the covariance matrix C into consideration which results in a mo died t ansatz

1 2

F p D F p D C

j j i i

where the symmetric covariance matrix is given by

m m

D D D D C

i i j j ij

N N

C C

m=1

This approach is contained in the Likeliho o d function of the Maximum Entropy Metho d

which will b e describ ed in chapter Setting all odiagonal comp onents of the covariance

matrix to zero then corresp onds to the uncorrelated t in

In the case of a limited numb er of congurations one is sometimes confronted with very

small or nearly zero eigenvalues of the covariance matrix which results in unreasonable

large eigenvalues of the inverse One solution is to calculate the inverse with a singular

value decomp osition SVD which corresp onds to omitting the smallest eigenvalues A

less restrictive pro cedure the eigenvalue smo othing was prop osed by Michael and

McKerrel Thereby the smaller eigenvalues are replaced by the average of them

whereas the larger ones remain the same The eigenvectors of the covariance matrix and

thus of its inverse are kept unchanged

Considering N eigenvalues of the covariance matrix with the smo othing

i i i+1

metho d can b e implemented as follows

max with

i i min min

N N

i=N +1

where N denotes the numb er of retained large eigenvalues This pro cedure often provides

a more stable mo del of the correlation matrix from the sample data see section

Chapter

Strange Hadron Sp ectrum

After a short intro duction to the symmetries of the particle sp ectrum and the underlying

quark interactions in section the appropriate op erators and correlation functions for

the simulations are summarized in The subsequent paragraph explains how the

particle masses are extracted from the correlation function while section describ es the

extrap olation to physical quark masses The following section illustrates the improve

ment through the application of the fuzzing technique The last part of this chapter

describ es the present lattice simulation and provides the results obtained for the strange

hadron masses and the Hdibaryon stability

Hadron Multiplets

Considering the three lightest quark avors to b e mass degenerate m m m

u s

the SU isospin symmetry accounts for the multiplet structure of the observed light

hadron sp ectrum The three antiquarks u d and s form a avor antitriplet in the

framework of the SU symmetry group which is generated by the eight GellMann

matrices a

In order to construct a meson q q the pro duct of a SU triplet and an antitriplet has

to b e evaluated It separates into irreducible singlet and o ctet representation

In gure the o ctets of the pseudoscalar and vector mesons are visualized where I

denotes the third comp onent of the isospin and S is the strangeness quantum numb er

The pseudoscalar singlet is approximately the meson whereas a mixing of singlet and

o ctet states contributes to the vector mesons and

Chapter Strange Hadron Spectrum

S S Κo Κ*o Κ+ Κ*+ n p +1 0 ds- us- ddu uud - ρ- uu- dd ρ+ uds 0 - πo ρo - -1 Σ- dds uus Σ+ π- du ud π+ Λ o η ss- ω/φ Σ

su- sd- ssd ssu -1 _o _ o -2 Ξ- Ξo Κ- Κ*- Κ Κ* I3 I3

-1 0 +1 -1 0 +1

Figure Multiplet structure for the pseudoscalar and vector mesons left as well as

baryons right for the spin

Alternatively two quarks can b e assembled in a diquark state The nine p ossible qq

combinations group themselves into an antitriplet and a sextet according to

This leads to the antisymmetric antitriplet states and the symmetric sextet states shown

in gure The splitting in irreducible representations works similarly in the SU color

group Combining two color triplets results likewise in a color antitriplet and a sextet

Note that in contrast to the ordinary mesons and baryons no color singlet is obtained for

such a two quark state Therefore the diquarks are color carrying ob jects which may only

o ccur in the color sup erconducting phase of QCD at high densities see g

In order to construct a total antisymmetric wave function for the diquark state the

color and avor representations have to b e combined appropriately with a total spin

zero or one of the two quarks The p ossible diquark states are summarized in table

The last two columns give the relative

of the qq interaction due to a

S strength

vorspin or a colorspin coupling which dd[ud] {ud} uu a

can b e calculated via twob o dy matrix ele

ments with the p otential

a a

s s

[ds] [us] V

i j j -1 i

{ds} {us}

denote the generators of the SU The

avor or color symmetry group resp ectively

qq interaction is attractive for the avor -2 The

titriplet diquarks considering the color

ss 3 an

coupling which arises in p erturbative

-1 0 +1 spin

QCD from onegluonexchange OGE

Figure Multiplet structure for the

In contrast to that the instanton liquid mo del

antitriplet and sextet diquarks avor predicts an attractive interaction in the color

Hadron Multiplets

FSC diquark state FS coupling CS coupling

d C u

abc

u u

abc

y y

u d

c c

u C u

Table Diquark states in the color antitriplet and sextet representation

antitriplet channel Such a spin and avor dep endent coupling b etween the con

stituent quarks is needed to describ e the ne structure of the exp erimentally observed

hadron sp ectrum correctly

Our previous analysis of diquark correlation functions in lattice QCD provided

evidence that the diquark splitting at zero temp erature and density follows the order given

by the avorspin interaction For the lightest state a mass compatible with twice the

constituent quark mass was found A more sophisticated investigation of the diquark

correlators will b e given in section by applying the Maximum Entropy Metho d Since

the onegluonexchange CS coupling gains in imp ortance at increasing baryon density

a reversal of the order for the spin states and a slightly b ound diquark state seems

p ossible which would supp ort the existence of a color sup erconducting diquark phase at

high density see section

Adding a further quark q to the diquark state qq leads to the formation of the baryons

qqq which b elong to the following multiplets

A M M S

1 3

baryons build the symmetric decuplet while the spin baryon o ctet see The spin

2 2

gure is describ ed by a mixed representation The antisymmetric singlet state can

b e identied with the heavier baryon In the lattice simulation rep orted in this

thesis the correlation functions for the nucleon sigma and lamb da baryons as well as

for the pion rho K and K mesons were calculated The particles in the same isospin

multiplet are mass degenerate since the states carry no charge quantum numb er on the

lattice

As already discussed in section the main purp ose of the analysis of the strange hadron

sp ectrum is to examine the stability of the six quark state called Hdibaryon This particle

is built of two quarks of each light avor u d and s and forms the antisymmetric SU

avor singlet state In order to obtain a total antisymmetric wave function color and

spin have to b e symmetrized in the combined SU group The Hdibaryon b elongs

to the representation of this group which contains the antisymmetric color and spin

singlets This is the most attractive channel for the qq interaction in formula arising

from onegluonexchange

Chapter Strange Hadron Spectrum

Hadron Op erators and Correlation Functions

In lattice calculations the particle masses can b e obtained from the exp onential decay of

the twop oint correlation functions This paragraph provides the appropriate op erators

and corresp onding correlators for mesons diquarks baryons and nally the Hdibaryon

A general meson op erator is constructed from the quark and antiquark elds with the

desired quantum numb ers combined with one of the matrices l or

5 5

x x Mx

Given the op erator the connected avor nonsinglet part of the twop oint correlation

function Gx hMx M i can b e evaluated according to in analogy to

the quark propagator in formula Moreover the anticommutation relations of the

fermion elds and the matrices have to b e taken into account The sp ecic op erators and

resulting correlators for the scalar pseudoscalar and vector mesons are summarized b elow

PC ++ cf

x x Scalar Meson J M x

c c c c

1 2 1 2

G x h U D i

S 5 5

1 1

where U M x and D M x denote the inverse fermion matrix for

the quark avors f u and f d which are usually chosen to b e degenerate The

1 2

expression h i should b e understo o d as average over the gauge eld congurations U

according to resp ectively

PC + cf

x x Pseudoscalar Meson J M x

5 PS

Here the additional matrix is eliminated in the correlator via the relation D D

5 5 5

which was already used for the scalar meson

c c c c

1 2 1 2

G x h U D i

c c c c

1 2 1 2

G x h U S i

Hence the choice of the quark avors f determines the particle describ ed with the ab ove

correlation function which is the pion for u and d and the K meson for u and s quark

avors where S M x

PC cf

x x Vector Meson J M x

For u and d quark avors this op erator characterizes the meson while the K meson is

given by ud and s quark content Since the op erator includes the matrices or

1 2 3

the total correlator is obtained by summing over all three combinations

c c c c

1 2 1 2

h U D i G x

5 5

k k

k =1

c c c c

1 2 1 2

G x h U S i

K 5 5

k k

k =1

Hadron Operators and Correlation Functions

The four p ossible diquark states with a total antisymmetric wave function were already

intro duced in table Instead of the matrices app earing in the meson op erators the

spin zero diquark states involve the combination of the charge conjugation matrix C with

which ensures the appropriate combination of the spinor indices The tensor provides

the antisymmetric color distribution for two quarks in the antitriplet representation In

the following the fermion eld notation is abbreviated according to the avor content

au a

x The diquark correlators can b e obtained in analogy to those for x as u

mesons but unlike in the case of color singlet states they are gaugevariant observables

Therefore these correlation functions have to b e calculated on gauge xed congurations

see section Note that for the avor sextet diquarks with two u quarks two p ossible

combinations for the quark propagator arise This leads to a second term with negative

sign due to the anticommutation relation for the quark elds The resulting correlation

functions are listed b elow

a b

Diquark Flavor Spin Color D x C u xd x

abc

3 03

ad be

G x h C C U D i

5 5

abc def

303

a b

Diquark Flavor Spin Color D x u xu x

abc

613

s s

ad be ae bd

G x h U U U U i

abc def

613

s s s s s s s s

1 2 1 2 1 2 1 2

c c

x xd Diquark Flavor Spin Color D x u

316

s s

c c c c

1 2 1 2

G x h U D i

316

s s s s

1 2 1 2

c c

Diquark Flavor Spin Color D x C u xu x

606 5

c c c c c c

c c

1 2 1 2 1 2

1 2

G x h C C U U U U i

606 5 5

Combining the diquark op erator with an additional quark eld yields the general form

baryon op erator for the spin

af cf

1 2

B x x x C x

abc

As describ ed b efore the antisymmetric tensor takes care of the color neutrality of the

baryon while the C matrix ensures the correct spinor distribution of the quarks elds

Accordingly the op erators of the nucleon and the strange sigma baryon together with the

corresp onding correlators can b e written as

b c a

x xu xC d Nucleon B x u

5 N

abc

cf cf

ad be ae bd

G x h Tr C C U U D U U D i

N 5 5

abc def

Chapter Strange Hadron Spectrum

a b c

Sigma B x u xu xC s x

abc

cf cf

ad be ae bd

G x h Tr C C U U S U U S i

5 5

abc def

The lamb da baryon consists of the three light quark avors u d and s Hence the op erator

contains the three p ossibilities to combine the spin of a quark pair to zero

a b c a b c a b c

Lamb da B x C u xd xs xd xs xu xs xu xd x

abc

G x h Tr C C

5 5

abc def

cf cf cf

ad be ad be ad be

S U D D U S U D S

cf cf cf

ad be ad be ad be

U D S D U S U D S

cf cf cf

ad be ad be ad be

i S U D S D U D U S

cf cf

ad be ad be

h Tr C C U U S S U U

5 5

abc def

cf cf cf

ad be ad be ad be

i S U U U U S U U S

It is obvious from the ab ove correlation function that the nine original comp onents reduce

to only ve if the masses of the light quark avors u and d are degenerate

Finally the Hdibaryon correlator is needed for the investigation of this particle on the

lattice As SU avor singlet this six quark state contains two of each light quark

avors u d and s The explicit construction of the op erator requires the symmetrization

of the color and spinor indices of two triplets of quarks in order to obtain color and spin

singlets as describ ed in the last section This pro cedure can b e illustrated most suitable

with the following op erator notation for the dierent six quark combinations

a b c d e

x ab cdef C C C a xb xc xd xe xf

5 5 5

abc def

The quark pairs ab de and cf couple to spin zero while the rst three quarks ab c and

the last three ones def form each a colorless state Using this notation the Hdibaryon

op erator can b e written as

H x udsuds usdusd dsudsu ussudd dssduu usudsd

udsuds ussudd dssduu

where the additional relations

usdusd udsuds dssduu usudsd

dsudsu udsuds ussudd usudsd

were employed to reduce the numb er of contributions to the Hdibaryon op erator Since

two quarks elds of each avor are contained in each part the corresp onding correlation

function G x hH x H i involves terms of the structure

U U U U D D D D S S S S

11 22 12 21 11 22 12 21 11 22 12 21

Extracting Masses from Correlation Functions

Each quark can b e combined with the rst or the second one of the same avor in the

conjugate op erator The second term gets a minus sign due to the anticommutation

relations of the quark elds as already shown in the simple case of the diquarks Taking

the symmetry prop erties of the tensor and the C matrix under the interchange of

two indices into account and considering only degenerate u and d quark avors then yields

the Hdibaryon correlation function in the compactied representation

G x h C C C C C C

H 5 5 5 5 5 5

abc def g hi j k l

f i f l dg ag

cl bh ek bk eh ci dj aj

S S U U U U S S U U U f U

f h f i

ci bk el bl ek ch

S S U U U U S S

f k f h

eh ek bi cl bl ci

U U U U S S S S

f l f k

ek el bh ci bi ch

U U U U S S S S

f g f k eg

bh ci bi ch ek aj dl al dj

S S S S g i U U U U U U U U

Note that the actual calculation of this correlator in a lattice simulation involves lo ops

over all the color and spinor indices which is naturally very time consuming Therefore it

is absolutely necessary to consider that three out of four combinations of the spinor indices

give no contribution to the correlation function due to the structure of the C matrix

see app endix A Moreover only of p ossible combinations of the color indices in the

tensor yield a nonzero result Nevertheless the evaluation of the Hdibaryon correlation

function involves the calculation of O indep endent terms and requires a factor

more CPUtime than the calculation of a lamb da baryon correlator

Extracting Masses from Correlation Functions

As mentioned b efore particle masses can b e obtained from the long range b ehavior of the

correlation functions In the case of the temp oral correlator this yields the p ole mass while

the exp onential decay of the spatial correlator usually in the z direction determines the

screening mass

G dx dy dz Gx y z

Z Z

d dx dy Gx y z Gz

The following considerations ab out the temp oral correlation function and the p ole mass

can also b e translated corresp ondingly for the spatial correlator and the screening mass

Once the correlation function is calculated a rst estimate of the particle mass can b e

obtained from its eective mass at Euclidean time which is dened as

m log

ef f

Chapter Strange Hadron Spectrum

For small time separations the eective mass usually shows contributions of excited states

while it turns into a plateau of a constant ground state mass at larger times For this

reason one usually denes correlations functions which pro ject onto states with denite

momentum Performing a Fourier transformation one obtains

i p x

Gx e Gp

E (p) E (p)(N )

n n

A pe B pe

n n

At large time separations this correlator at xed momentum is dominated by the ground

m p The pro jection onto zero momentum then yields simply state energy E p

the reduction to the ground state mass E m The amplitudes ob ey the relation

0 0

B p A p if the correlator Gp is even o dd in the temp oral direction Hence

n n

the correlation function for p is antisymmetric around N which is expressed in

the dierent functional forms

G A e cosh m

sy m 0 0

sinh m G A e

0 asy m 0

By using the appropriate ansatz as t function a Jackknife analysis see section

of the particle mass can b e p erformed Ideally this pro cedure is completed by a twomass

t according to

N N

m m

1 0

2 2

A e G A e cosh m cosh m

1 sy m 0 0 1

in order to eliminate eciently the contributions of excited states at the rst few time

slices Furthermore a stable plateau of the ground state mass should b e obtained by

leaving out successively data p oints at small time separations

Extrap olation to the Physical Point

The resulting particle masses at dierent values of the bare quark mass have to b e ex

trap olated to the chiral limit or to the physical quark mass value resp ectively As already

intro duced in section the quark mass is related to the squared mass of the

pseudoscalar meson in leading order chiral p erturbation theory Therefore a linear de

p endence of m on the quark mass can b e assumed

b m

where is the critical hopping parameter at which the pseudoscalar meson mass vanishes

For all other particle masses a similar relation is applicable

m a b

Improvement through the Fuzzing Technique

In the case of particles consisting of dierent quark avors like the strange hadrons inves

tigated in the present study the linear functions are understo o d in terms of the average

quark mass

This leads to a mo died version of the relation for the mass of the strange pseudo

scalar K meson

c s c

and similarly to a mo dication of for the other particles Obviously an extrap olation

to the physical values of the degenerate u and d quark mass as well as to the s quark mass

is necessary By keeping xed the second term in b ecomes simply a constant

which allows to p erform a linear t in and an extrap olation to the physical bare

quark mass The latter can b e determined from the ratio of two nonstrange hadrons like

the pion rho or nucleon or alternatively from the ratio of one of these with the string

tension

Having xed the physical the rst term of the relation is only a constant

Therefore a linear extrap olation in leads to the physical p oint The physical can

s s

b e determined from the ratio of a strangeness carrying particle to a nonstrange one In

the quenched approximation the dierent choices of the input particle masses pro duce

deviations in a range compared to the exp erimental values whereas this eect

can b e reduced in simulations with dynamical fermions

Improvement through the Fuzzing Technique

It was already mentioned that the excited states interfere with the clean exp onential

decay of the ground state in the correlation function This observation inspired Gupta

et al to develop an improvement pro cedure which was further explored by the

UKQCD collab oration The socalled fuzzing technique provides a b etter overlap

with the ground state since it takes into account to some extent the physical size of the

particle The separation of a quark antiquark pair by a suitable distance R thus will

maximize the ground state contribution relative to the ones of the excited states already

at small Euclidean times The fuzzed quark eld is constructed to b e symmetric in all

space directions see gure

y y R

U x U x R x R x

=13

U x U x R x R

Such a fuzzed quark eld is only used at the sink while the one at the source remains lo cal

For a general meson the fuzzed op erator and the corresp onding correlator can b e written as

Chapter Strange Hadron Spectrum

R y

Gx hM x M i with

R R

x M x x

This combination is referred to as lo calfuzzed LF corre

lator It would not b e appropriate to use a fuzzedfuzzed

FF correlator since some of the fuzzed links cancel each

other which would resemble the purely lo cal correlator

Figure Fuzzed fermion

For the baryons two p ossible combinations arise the LLF

op erator with radius R

singlefuzzed and the LFF doublefuzzed correlator They

are visualized in gure where each of the extended links

is understo o d as a sum over the six spatial orientations Doubling the quark content of a

baryon leads to a dibaryon op erator in the LLLLFF and the LLFFFF representation

Meson Baryon Baryon Dibaryon Dibaryon (LF) (LLF) (LFF)

(LLLLFF) (LLFFFF)

Figure Fuzzing for mesons baryons and dibaryons

In addition to the quark elds also the distribution of the gluon elds can b e improved

The desired approach should imitate the gluon cloud which surrounds the quark elds

This is commonly achieved by the iterative APE smearing pro cedure

U x P c U x U x

new

ol d stapl es

SU (3)

with three links p erp endicular to the gauge eld and the time where the staples

direction are added to the simple link This is followed by the back pro jection to the

SU symmetry group Some trials with dierent parameters c and varying numb er of

iteration steps have shown that the choice of c and at least eight iteration steps

leads to an ecient smearing of the gauge elds

The eect of the fuzzing and smearing pro cedures is now further explored for the H

dibaryon correlation function It is obvious from the correlator as well as from the eec

tive mass plot in gure that already the fuzzing of two quarks yields a considerable

reduction of contributions from excited states The b est results were obtained by applying

the fuzzing technique for four of the six quarks where an almost constant value for up to

ve time separations could b e achieved The remaining contributions of excited states at

small time separations were eciently absorb ed in the ts with two exp onentials

Hadron Spectrum

10 no quarks fuzzed meff no quarks fuzzed G( τ ) 2 quarks fuzzed 2 quarks fuzzed 1 4 quarks fuzzed 4 quarks fuzzed 8

3 1e-05 lattice size 16 x 30 lattice size 163 x 30 6 κ = 0.147 κ u u = 0.147 1e-10 κ = 0.145 κ s 4 s = 0.145

1e-15 2 τ τ 1e-20 0

0 5 10 15 20 25 30 0 1 2 3 4 5 6 7 8 9

Figure Inuence of the fuzzing technique on the Hdibaryon correlation function and

the eective masses

Hadron Sp ectrum

For the detailed analysis of the stability of the Hdibaryon a sp ectrum calculation was

p erformed in quenched QCD with improved gauge and fermion actions In the gauge sector

the Symanzik improved action was used with a gauge coupling The

lattice spacing was determined from the string tension to afm or equivalently

a GeV The Clover fermion action was used with treelevel clover co ecient

since nonp erturbatively improved co ecients are not yet available for the employed gauge

action see also section In order to study the nite size eects simulations of the

strange hadrons as well as the Hdibaryon were p erformed on four lattice sizes

The numb er of congurations ranges from for the smallest to for the largest lattice

They are separated by sweeps of four overrelaxation and one heatbath step each The

correlators were calculated for three dierent and up to ve values On these fairly

coarse lattices a fuzzing radius R has b een chosen in order to obtain a broad plateau

for eective masses as an approximation for the Hdibaryon ground state mass

The physical value was determined from a previous simulation of the light hadron

sp ectrum with the same parameters as ab ove but on a lattice and on the

and basis of in total quark propagators Adjusting the particle ratios m m m

to the exp erimental values leads to very similar results for the hopping parameter m

which corresp onds to the physical value of the light quark masses

Strange Particle Masses

The strange particle masses were determined with one and twostate ts during a Jack

knife analysis see sec as illustrated the section The masses obtained in this way

are summarized in the tables BB of the app endix The strange particle masses for all

Chapter Strange Hadron Spectrum

1.4 2 physical 1/κ 3 mK, mK*,Σ,Λ s Lambda 24 x30 1.2 163x30 123x30 83x30 1.0 Sigma 243x30 163x30 3 0.8 12 x30 83x30 K* Meson 243x30 0.6 163x30 123x30 83x30 0.4 K Meson 243x30 163x30 3 0.2 12 x30 3 κ 8 x30 1/ s 0.0 6.7 6.8 6.9 7.0 7.1 7.2 7.3 7.4 7.5 7.6

4.0

mH,2Λ 3.5 κ physical 1/ s 3.0

2.5

H-Dibaryon 243x30 2.0 163x30 3 1.5 12 x30 83x30 3 1.0 2*Lambda 24 x30 163x30 3 0.5 12 x30 κ 83x30 1/ s 0.0

6.7 6.8 6.9 7.0 7.1 7.2 7.3 7.4 7.5 7.6

Figure Strange hadron top and Hdibaryon masses b ottom at physical mass ud

Hadron Spectrum

lattice sizes are displayed in gure where the linear extrap olation to the physical

value was already p erformed Hence the particle masses dep end only on the strange quark

mass The lines should guide the eye and indicate the results of linear ts for the

lattice data

In general one observes a go o d agreement of the strange hadron masses on the dierent

lattice sizes Only the results of the smallest lattice are slightly overestimated This

eect is more pronounced for the baryons in comparison to the mesons since the nite

size eects increase with the size of the simulated particle That is even more obvious

from gure b ottom where strong deviations can b e observed for the Hdibaryon on

the smallest lattice which has an spatial extension of only L fm

Before the dierence b etween the Hdibaryon and two masses is investigated in detail

the plotted region for the physical s quark mass should b e explained In order to absorb

as many quenching eects as p ossible the physical should b e set by the ratio m m

s N

Here the contact to physical units was made by the nucleon mass m m a MeV

N N

as an input which yields a GeV This pro cedure leads to only small deviations

in the range compared to the exp erimental values of the baryon and the K meson

on the largest lattices while the K meson was found to b e ab out heavier This eect

could also b e observed in a similar study

Inspired by the go o d agreement of the and K meson masses with the exp erimental

values the scale was set alternatively by the ratios m m and m m The obtained

N K N

particle masses for dierent input choices are collected in table B Since it is desirable to

dene a common physical value for a comparison of the results on all lattice sizes the

average over the values on the two larger lattices was taken which yields deviations of

no more than for the nonadjusted particle masses The mean value is

indicated as physical region in the plots and The particle masses for this mean

value can also b e found in table B

HDibaryon Stability

Now all prerequisites are available to discuss the investigation of the Hdibaryon stability

The dierence b etween the Hdibaryon and twice the lamb da mass is illustrated in detail

in gure The slop e of a linear t in would b e quite dierent for varying lattice

sizes For the largest lattice almost a constant b ehavior can b e observed whereas the

values of the lattice are rising with increasing strange quark mass On the

contrary the data p oints of the two smaller lattices exhibit a negative slop e tted linear

in Nevertheless the results on all lattice sizes have in common that a value around

zero is exp ected in the physical region This would mean that the examined Hdibaryon

state is simply the unb ound comp osition of two lamb da baryons

Finally the dep endence of this result on the dierent choices of the input particle mass

should b e investigated Figure shows the dierence in mass m m m at the

Chapter Strange Hadron Spectrum

1.2 3 (mH-2mΛ)/2mΛ 24 x30 1.0 163x30 123x30 0.8 83x30 0.6 κ physical 1/ s 0.4 0.2 0.0 -0.2 -0.4 κ 1/ s -0.6

6.7 6.8 6.9 7.0 7.1 7.2 7.3

Figure Dierence of Hdibaryon and *lamb da masses for all lattice sizes

Data p oints are slightly displaced in for b etter visibility s

0.6 (m -2m )/2m Negele Λ input H Λ Λ Λ 0.5 input Σ input K* input 0.4 mean input 0.3

0.2

0.1

0.0

-0.1 L [fm] -0.2

0 1 2 3 4 5

Figure Dierence of Hdibaryon and *lamb da masses for dierent input

Data p oints are slightly displaced in L for b etter visibility

Hadron Spectrum

physical ud and s quark values for the exp erimental input of the and K masses

as well as for the formerly describ ed mean value of The nite size eects are obvious

for the smallest lattice with L fm esp ecially for the input On the intermediate

lattices the values scatter around zero dep ending on the dierent input masses while the

p oints for the largest lattice with L fm lie slightly ab ove Here the limited statistics

of only congurations on the lattice should b e taken into account

Taking the results on all lattice sizes into account leads to the conclusion that an unb ound

Hdibaryon state consisting of two lamb da baryons has b een observed Otherwise a lighter

Hdibaryon should have b een found as in the case of the smaller lattice L fm of

the study of Negele et al which is also indicated in gure The authors could not

rule out considerable nite size eects on this lattice Therefore a further investigation

on a larger lattice with L fm had b een p erformed which is in much b etter

agreement with the results of the present thesis Hence a common conclusion arises from

the recent and present studies The Hdibaryon do es not exist as stable particle in the

vacuum It seems to b e unb ound at least within quenched QCD

Two previous attempts to calculate the Hdibaryon mass on the lattice were rep orted in the

literature more than a decade ago see also table The rst calculation of Mackenzie

and Thacker on a fairly small lattice resulted also in an unb ound Hdibaryon while

Iwasaki et al observed a strongly b ound Hdibaryon in a further simulation with the

renormalization group improved gauge action This calculation suered from the limited

statistics of only congurations and quite large bare quark masses In a subsequent

pap er the authors realized that the dierence in mass m m was considerably

decreased in a simulation with the standard plaquette action which sheds a dierent light

on the observation of a strongly b ound Hdibaryon

1.0 ∆G(t)/G(t) 0.8 H-Dibaryon Lambda κ u = 0.147 0.6 κ s = 0.145

0.4 lattice size 163x 30

0.2

t 0.0

0 2 4 6 8 10 12 14 16

Figure Relative error of Hdibaryon and lamb da correlation functions

Chapter Strange Hadron Spectrum

In order to complete this section ab out strange hadronic matter a short outlo ok on further

lattice simulations in this eld should b e added In gure the relative error of the H

dibaryon and lamb da correlators are illustrated It can b e observed that the error rises only

linearly with the baryon numb er of the investigated state Therefore it seems p ossible in

the future to examine larger strangelets or even multiquark clusters in lattice simulations

once the eciency of the available parallel machines has increased suciently

Chapter

Hadron Sp ectral Functions

So far only hadron sp ectra at zero temp erature have b een discussed Now the fo cus is on

the mo dications of sp ectra which may arise from the presence of a thermal medium In

this last chapter the p ossibilities of the sp ectral analysis of hadronic correlation functions

with the Maximum Entropy Metho d MEM are describ ed The rst section gives

an intro duction into the relevance of sp ectral functions in QCD After a presentation of

the main principles of MEM in section the numerical algorithm will b e explained in

greater detail in

Section shows the indep endence of the obtained sp ectral function under changes of

the input parameters The following two parts illustrate the resulting sp ectral functions for

meson sec and diquark sec correlation functions at zero temp erature whereas

the last section demonstrates the applicability of this metho d to mesonic correlation

functions at nite temp erature

Sp ectral Functions in QCD

Recent lattice calculations of static observables like hadron masses and decay constants

at zero temp erature have reached a quite satisfactory precision At nite temp er

ature it is desirable to get access to dynamic quantities in particular sp ectral functions

and realtime correlation functions starting from the temp oral correlators in Euclidean

time calculated on the lattice Furthermore mo dications of hadronic prop erties at nite

temp erature and density could b e studied in terms of changes of the sp ectral shap e The

analysis of temp oral as well as spatial correlation functions in the vicinity of the critical

temp erature gives evidence of substantial changes in the prop erties of hadronic states b e

tween the conned phase and the hot plasma phase of QCD In order to

judge if the observed mo dications of hadron correlators are indeed related to the disap

p earance of hadron b ound states a detailed study of the structure of sp ectral functions is

required

Chapter Hadron Spectral Functions

Moreover sp ectral functions are directly related to the exp erimental annihilation cross

sections For example a temp erature dep endence of the vector meson mass and width

is linked to changes of the dilepton sp ectrum whereas thermal eects on the pseudo

scalar correlator inuence the chiral condensate These phenomena can b e examined in

relativistic heavy ion collisions

At very high temp eratures b eyond the phase transition thermo dynamic observables can

b e calculated p erturbatively in the framework of the hard thermal lo op HTL resummed

theory In a recent study the HTLresummed scalar sp ectral function was cal

culated for vanishing momentum In comparison with the free thermal sp ectral function

an enhancement at low energies as well as a suppression in the high energy regime can b e

observed The former is due to the interaction b etween quarks and gluons in the heatbath

whereas the latter results from the inuence of the thermal quark mass Debye screening

The obtained mesonic correlation functions can b e compared to lattice results for tem

p eratures larger than T T The HTLresummed vector sp ectral function diverges for

small energies hence the corresp onding correlator is divergent Therefore a direct com

parison of the vector correlation function calculated on the lattice with HTLresummed

p erturbation theory is not p ossible This makes it even more imp ortant to p erform lattice

calculations on large spatial lattices in order to explore the nonp erturbative part of the

sp ectral functions at low momenta

In general the thermal twop oint function for a sp ecic hadronic observable O at Euclidean

time T in co ordinate space is given by the innite sum

G x hO x O i

d p

i( px)

e G p T

The Fourier transformed correlation function G p dep ends on the discrete Matsubara

mo des i e n T for b osons The sp ectral function A p can b e determined from

the imaginary part of the momentum space correlator

A p

d A p Im G p G p

n n

Using b oth the equations and together with the identity

e e

T with

i T

for the Fourier transform of the free b oson propagator the sp ectral representation of the

thermal correlation function in the co ordinate space at xed momentum can b e written

e G p G p T

T A p d

n n

Principles of the Maximum Entropy Method

A p d

( 1T )

e e

A p d

cosh T

A p d

sinh T

K A p d

The function K is the integral kernel in the continuum representation which is

essential for the MEM analysis It can easily b e veried that this kernel reduces to

K e at zero temp erature T This version will b e referred to as exp onen

tial kernel In the following only correlation and sp ectral functions at xed momentum

p will b e considered therefore the lab el p is omitted from now on

The last line of equation shows that a given data set D G for the correlator

and the corresp onding sp ectral function A are related by an inverse Laplace transform

Unfortunately lattice calculations can only provide the correlation function for a discrete

set of Euclidean times The numb er of data p oints is much smaller than the desired

numb er of sampling p oints needed to reconstruct the sp ectral function This is a typical

example for an illp osed problem where the standard tting pro cedure is inapplicable

Furthermore the data are noisy due to the Monte Carlo sampling

So far the analytic continuation from imaginary to realtime correlation functions was

p erformed by strict assumptions on the sp ectral shap e for example a function repre

senting the p ole mass plus a continuumlike structure Such an approach

inhibits to prob e the ne structure of the sp ectral function Further diculties arise at

nite temp erature where very little is known ab out the sp ectral shap e The Maximum

Entropy Metho d is a new approach to tackle this problem by use of Bayesian metho ds of

interference It provides a pro cedure to estimate suitable sp ectral functions from given

data in Euclidean time which requires no a priori assumptions on the sp ectral shap e

Principles of the Maximum Entropy Metho d

The Maximum Entropy Metho d is a well known technique in condensed matter physics

image reconstruction and astronomy In lattice QCD it has recently b een

applied to analyze meson correlation functions at zero temp erature It could b e

demonstrated that this metho d correctly detects the lo cation of p oles in the correlation

function and moreover is sensitive to the contribution of higher excited states in the

correlators Furthermore the sp ectral functions allow the determination of decay

constants from the area b elow sharp p eaks

The framework of Bayesian interference in probability theory provides a p owerful to ol to

nd the most probable sp ectral function The basic formula is given by the Bayes Theorem

Chapter Hadron Spectral Functions

of conditional probability P X jY for the event X given Y

P Y jX P X

P X jY

P Y

The p osterior probability for the sp ectral function A given the data p oints D for

a correlation function in imaginary time is now accordingly determined as

P D jAH P AjH

P D jAH expL

with P AjDH

P AjH expS

P D jH

where P D jH is only a normalization factor which is indep endent of the sp ectral function

P D jAH is called the likeliho o d function and P AjH the prior probability

H includes all prior knowledge ab out the sp ectral function such as the p ositivity A

for

The central limit theorem provides the functional form of the likeliho o d function P D jAH

in for the case of a large numb er of measurements It can b e expressed in terms of

the usual distribution

L F D C F D

i i j j

where D is the average over all measurements C denotes the symmetric covariance

i ij

matrix which has already b een dened in equation Employing equation the

t function

K A d F K A

i i ij j

is obtained in the discretized version by means of a predened kernel K K and

ij i j

the sp ectral function A A

j j

Maximizing the likeliho o d function P D jAH corresp onds to minimizing which is the

commonly used tting pro cedure where only a few parameters are adjusted But for the

addressed illp osed problem the inuence of the prior probability plays an imp ortant role

At variance with the maximum likeliho o d metho d in addition one has to maximize the

prior probability P AjH expS which dep ends on the factor and the entropy S

The ShannonJaynes entropy can b e constructed from axiomatic requirements such as

subset and system indep endence co ordinate invariance and scaling This leads

to the following expression for the entropy

d S A m A log

A m A log

j j j

j j

Principles of the Maximum Entropy Method

1.0 -10 P[α|Dm] * 108 log(P[α|Dm]) 0.8 -20

0.6 -30 0.4

-40 0.2 log(α) log(α) 0.0 -50

-15 -10 -5 0 5 10 -15 -10 -5 0 5 10

Figure Sharply p eaked p osterior distribution P jD m of the weight factor given

the data left and its logarithm log P jD m right for the pion at T at

where the default mo del m incorp orates any a priori knowledge ab out the sp ectral

function A In the absence of any information ab out the data the entropy is maximized

with A m For the studies of meson correlation functions m m is used as

the initial ansatz for A This default mo del is motivated by the asymptotic

b ehavior of the sp ectral function at high energies which can b e studied in p erturbative

QCD The inuence of the choice of the mo del on the nal sp ectral function is analyzed

in section

The real and p ositive factor in the prior probability P AjH P Ajm e controls

the relative weight b etween the entropy and likeliho o d function For large the t is

mostly inuenced by the default mo del whereas for small the sp ectral function tends to

t the lattice data The nal result is indep endent of b ecause an integration over this

parameter is p erformed

The most probable sp ectral function A for given is then obtained by maximizing

P AjDH P AjD m It can easily b e seen from that this is equivalent to nding a

maximum of Q S L The nal sp ectral function A is determined from a weighted

average over

Z Z Z

A D A d A P AjD mP jD m d A P jD m

The ab ove approximation is valid for a sharply p eaked distribution P AjD m The weight

factor P jD m can b e obtained in applying Bayes Theorem

P jD m D A P D jAm P Ajm P jm

P jm D A expS L

It usually turns out that the weight factor P jD m is sharply p eaked around a unique

value for data with a small error see gure Therefore a Gaussian approximation

of P jD m can b e used which will b e explained in the next section

Chapter Hadron Spectral Functions

The most probable sp ectral function can b e found from the condition for an extremum

maximum making use of the Bryan algorithm see also sec

r Q r S r L

A A A

The uncertainty of the solution is then contained in the second derivative of Q with

resp ect to the sp ectral function A namely rrQ Since one is interested in error bars

for certain regions p eaks of A the average sp ectral function in the frequency range

I must b e calculated for a given

k l

A d

j I

hA i

d l k

The error in this region is then obtained from the variance

rrQ d d

I I

h A i

d d

I I

where rrQ is given by equation Having calculated the mean and variance

for the sp ectral function in the desired ranges at every given the nal result can b e

calculated via the integration over similar to equation

hA i d hA i P jD m

I I

2 2

h A i d h A i P jD m

I I

Details of the Algorithm

This section should serve as a guideline through the algorithmic details describ ed in

by R K Bryan The most complicated part of MEM is maximizing Q S L or

equivalently solving rQ rS rL in order to obtain a global maximum First of

all the derivatives with resp ect to the sp ectral function A can b e calculated

log A m with A m A m rS

j j j j j j

L L F

K rL

A F F

For the kernel K a singular value decomp osition SVD is p erformed The N N kernel

matrix can b e written as pro duct K V U where U and V are orthogonal matrices

and is a diagonal matrix with the ordered singular values Since some of the singular

values are very small or even zero for an illconditioned matrix the space can b e reduced to

the singular space with the dimension N N Using a condition like

s min max

to determine the size of this space one obtains the N singular values the N N matrix

s s

(s) (s)

U and N N matrix V in the reduced space

Details of the Algorithm

Since the sp ectral function is p ositive semidenite it now can b e parametrized using the

(s)

orthogonal matrix U U and a vector u in the singular space

A m exp U u

j j

j k k

k =1

This yields the enormous advantage that the N O dimensional discretized sp ectral

function can b e expressed in terms of an only N dimensional vector u

Inserting the formulas and into equation the condition for an ex

tremum in the singular space can b e rewritten as

g u V

Now a standard Newton search can b e p erformed for the function f u u g starting

with any arbitrary uvector

f u g

u f u l u u g

u u

The derivative of g with resp ect to u can b e calculated by

2 2

g L L

T T 1

V V U diag fAg U with C

2 2

u F F

In the context of MEM it is more suitable to set u in the b eginning which corresp onds

to a start with the default mo del see equation In order to guarantee the lowest

order approximation of the Newton search the step size u must b e restricted Therefore

an auxiliary parameter is intro duced in formula

u u g l

T T

A is fullled The iteration starts with such as u U diag fAg U u O

and pro ceeds eg in multiplying by b eginning with If the norm

condition is satised the step size is small enough for a reasonable up date of the solution

vector u u u This pro cedure is then iterated until the general stopping criterion for

convergence is reached which checks for equal length of the gradients

L S

U diag fAg U u

u u

with

L S

U diag fAg U g

u u

Once the convergence is reached the solution vector u can b e substituted back in equation

in order to obtain the most probable sp ectral function A for given

The numerical eort can b e further reduced if the matrix g u in the Newton search

is diagonalized The addressed pro cedure involves quite a numb er of new matrices

and a subspace division b ecause some of the required eigenvalues can b ecome very small

or even negative Since this section should only b e an outline of the Bryan algorithm the

interested reader is referred to the detailed description in

Chapter Hadron Spectral Functions

After the solution A was obtained the probability P jD m and the covariance matrix

rrQ are still to b e calculated Applying a Gaussian approximation to the

A A j probability can b e rewritten in terms of the eigenvectors of

i j i ij

A=A A A

i j

see for details

P jD m P jm exp S L log

The rst factor can b e chosen as P jm which is absorb ed in the integration mea

sure d d log The probability must b e correctly normalized to P jD m d

alternatively the nal sp ectral function in equation can b e divided by P jD m d

which yields in the discretized version

o n

P P

min

S L log log A exp

2 + max

n o

P P

min

exp log S L log

max

2 +

Numerically it is convenient to start at with P jD m P jD m Then

max max

the contributions A P jD m for each are summed up in steps of log until a

similar probability as the initial one is reached for at the other side of the sharply

min

p eaked distribution see gure

For the calculation of the error on the sp ectral function in a certain range the covariance

matrix in the Gaussian approximation is needed It can b e calculated in terms of the

singular space quantities

T T 1

Y U diag fAg rrQ diag fAg U Y diag

where the matrix Y can b e derived from the diagonalized matrix g u in the Newton

search Inserting equation in yields nally the desired sp ectral function with

error bars in sp ecic regions

Dep endence on Input Parameters

After having implemented the MEM co de describ ed in the last section it is imp ortant to

examine the inuence of the input parameters as well as the statistics on the resulting

sp ectral function Such a test was p erformed with a data set for the pion correlator

which is describ ed in the b eginning of section First of all a large numb er of almost

indep endent gauge eld congurations is needed to obtain a reliable data set for the

observable The inuence of the numb er of congurations is displayed in gure a for

the ground state p eak of the pion at zero temp erature It can easily b e seen that an

insucient numb er of congurations results in a attening and broadening of the p eak

Furthermore the p osition of the p eak is shifted slightly towards smaller masses This

can b e b est observed in comparison with the black line which indicates the pion mass

obtained with a conventional twoexp onential t The masses extracted from the sp ectral

Dependence on Input Parameters

30 30 ρ(ω) ρ(ω) NC = 292 Nτ = 32 25 146 25 24 73 22 20 36 20 20

15 15

10 10

5 5 (a) ω [GeV] (b) ω [GeV] 0 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

30 30 ρ ω ρ ω ( ) ∆ω = 0.005 ( ) continuum 25 0.01 25 lattice 0.02 exp 20 0.05 20 smooth exp

15 15

10 10

5 5 (c) ω [GeV] (d) ω [GeV] 0 0

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure Inuence of the input parameters on the sp ectral function a numb er of

congurations N b numb er of time slices N c discretization interval d kernel

denition displayed for the enlarged ground state region for the zero temp erature pion

with the black line indicates the p osition obtained from the exp onential t

functions with and congurations clearly agree with the simple t result whereas

b oth sp ectral functions with less statistic would lead to a somewhat smaller value

A second parameter which is restricted by the limited available computer time is the

lattice volume It is certainly desirable to p erform lattice calculations on large lattices

but in practice most of the calculations were carried out on relatively small lattices The

volume dep endence on the correlation and sp ectral functions will b e further explored in

section Here I only want to direct the attention to the inuence of the temp oral

extent of the lattice Since no data sets with various numb er of time slices were available

the situation could b e simulated in leaving out data p oints for larger time separations in

the MEM analysis This would corresp ond to data with the same lattice spacing but a

smaller numb er of time slices The resulting ground state p eaks for the pion can b e found

in gure b The p osition of the p eak is only shifted for the smallest N In general

the p eak broadens and is reduced in its height for decreasing numb er of time slices This

eect is esp ecially visible for N

Chapter Hadron Spectral Functions

The MEM analysis works with a sp ectral function determined at discrete frequencies

Since the computer time rises exp onentially with the numb er of p oints N in the range

explored it is necessary to compromise b etween a fast computation and a go o d resolution

This is of particular imp ortance for the exact analysis of p eak p ositions Keeping the

maximal frequency on the lattice xed at ab out a the frequency range was divided into

discrete p oints The ground state p eak of the pion can easily b e resolved for the two

smallest frequency separations in gure c For larger the correct determination of

the p eak p osition b ecomes increasingly complicated and would result in an overestimated

error on the particle mass Therefore it is advisable to use several hundred p oints for the

discretization of the frequency range

Another imp ortant asp ect of the MEM analysis is the choice of the integral kernel in

equation At zero temp erature resp ectively N the p erio dic continuum kernel

reduces to the simple exp onential kernel exp Note that this kernel do es not take

into account the p erio dicity on a lattice with nite N The p eak of the sp ectral function

obtained with the continuum kernel is well p ositioned on the black line of the conventional

t in gure d whereas the one from the exp onential kernel is clearly shifted to the

right The correct p eak p osition for this kernel could only b e obtained with the smo othing

technique for the covariance matrix see section for details A further p ossibility to

cho ose the integral kernel arises from the sp ecial lattice kernel which should comp ensate

the eects of the nite lattice spacing The discrete frequencies for a b oson propagator on

the lattice sinn N can b e inserted in formula

nl at

e A d T D

nl at

Then the lattice kernel is determined through

N 1

expi n N

l at

K N

sin n N

n=0

Using the lattice kernel in the MEM pro cedure results in a sp ectral function very similar

to the one obtained with the continuum kernel which can b e observed in gure d

The eect of the lattice kernel is more imp ortant for the nite temp erature sp ectral func

tions where the continuum part gets more dominant compared to the ground state p eak

Short distance distortions in the correlation function due to the lattice discretization thus

are enhanced Since the numb er of time slices is usually smaller at high temp erature

T N a the height of the p eaks is reduced additionally in analogy to gure b

At nite temp erature it is convenient to express the sp ectral functions in units of T

Accordingly the correlation functions should b e divided by a factor T The inuence of

the dierent integral kernels is shown in gure ab The sp ectral function of the pion

at T is a bit smo other for the lattice kernel in particular the broad continuum bump

has attened out whereas the ground state p eak nearly remains the same The eect of

the lattice kernel is even more pronounced for the sp ectral function of a free meson In this

case all unphysical bumps and hills disapp ear and the wellknown tanhlike sp ectral shap e

is obtained Further details ab out the free sp ectral functions are describ ed in section

Dependence on Input Parameters

1.4 0.1 ρ ω ( ) continuum ρ(ω) continuum 1.2 lattice 0.08 lattice 1.0 0.8 0.06

0.6 0.04 0.4 (a) 0.02 (b) 0.2 ω/T ω/T 0.0 0 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

1.8 2.0 ρ(ω) mf ρ ω ω /T = 256 1.6 ( ) max 5*mf 128 1/5*m 1.4 f 1.5 64 48 1.2 1.0 1.0 0.8 0.6 0.5 0.4 (c) (d) 0.2 ω/T ω/T 0.0 0.0

0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

Figure Inuence of the input parameters on the sp ectral function at nite temp erature

kernel dep endence for the pion at Tc a and for the free scalar meson b dep endence

on the default mo del parameter m f m c and N d

0 max

The last two parameters which inuence the nite temp erature sp ectral functions are the

factor m of the default mo del and the maximal frequency Since the default mo del

0 max

is assumed to have the functional form m m which is motivated by p erturbative

calculations the only freedom is contained in the variation of m Therefore the

inuence of a factor f on m f m with f is explored in gure c This

factor alters only the height of the p eaks but not their p osition The correct normalization

2 2

constant is m for the pseudoscalar and m for the axialvector

f f

meson which yields the free correlation function for a massless meson with

D N Multiplying the data and the default mo del with the same factor only results

in the enhancement of the amplitude of the sp ectral function by this factor Furthermore

the relevant range is shifted towards smaller values for a factor f which could

sometimes b e advantageous in the numerical calculation

In gure d it is shown that the sp ectral functions for smaller exhibit a b ehav

max

ior similar to that obtained with a small m Moreover the p eaks are shifted towards

lower frequencies For values T or larger the sp ectral functions display only

max

neglectable dierences Such a value of T roughly corresp onds to the maximal avail max

Chapter Hadron Spectral Functions

able momentum on the lattice In general a reasonably large range should b e chosen to

obtain reliable sp ectral functions

After studying the inuence of the various input parameters on the nal sp ectral function

one can pro ceed to the systematic analysis of correlation functions with the Maximum

Entropy Metho d The next three sections deal with a detailed analysis of meson and

diquark sp ectral functions at zero temp erature as well as meson sp ectral functions at

nite temp erature

Meson Sp ectral Functions

Previously pro duced data samples in quenched QCD with Wilson fermions could b e used

to analyze meson correlation functions at zero temp erature The gauge eld congurations

have b een generated on a lattice of size with a treelevel Symanzik improved action

2 1

at a gauge coupling g The lattice cuto a GeV was determined

by means of the string tension In total congurations were xed to Landau gauge

see sec which op ens the p ossibility to calculate correlation functions for the color

carrying diquark op erators For the fermion sector the SheikholeslamiWohlert action with

a treelevel clover co ecient was used The fermion matrix has b een inverted for eight

dierent quark mass values and with source vectors at four dierent lattice sites Therefore

the data sample is based on quark propagators which provides sucient statistics for

the Maximum Entropy analysis The meson masses obtained from conventional two

exp onential ts were already published in

The meson correlation functions at zero temp erature see gure a show a clean

exp onential decay therefore a sharp like p eak in the sp ectral function is exp ected This

is indeed the case for the pseudoscalar meson sp ectral function displayed in gure b

A much stronger inuence of excited states can b e observed for the vector meson which

is already visible in the curvature of the correlation function at small time separations

1e+01 30 D(τ) pion ρ(ω) pion rho 20*rho 25 1e-01 20

1e-03 15

10 1e-05 5 (a) τ ω [GeV] 1e-07 0

0 5 10 15 20 25 30 0 0.5 1 1.5 2 2.5 3 3.5

Figure Correlation functions left and corresp onding sp ectral functions right for

the zero temp erature pseudoscalar and vector meson at

Meson Spectral Functions

In the sp ectral function this is reected by the continuumlike structure at high energies

Recently it has b een prop osed that this broad p eak at high energies ab out GeV

GeV for the pion rho can b e interpreted as b ound state of two fermion doublers of

the Wilson quark action Therefore this state may b e a lattice artifact with a divergent

mass in the continuum limit Since the data set at zero temp erature was only generated for

one lattice spacing this assumption could not b e veried in the framework of the present

analysis The error bars shown in gure need a little explanation Each horizontal bar

marks the range within which the average of the sp ectral function was taken and its

height reects this average value The vertical error bar then indicates the variance of the

average sp ectral function in this range Mean and variance were calculated by applying

equation

Another imp ortant p oint is to understand the p eak width and height of the reconstructed

sp ectral functions since the area under the p eaks is related to the decay constants of the

resp ective particle The simulations were carried out in the quenched approximation

which only admits interactions of the fermions with the gluonic background elds In

this case the exp ected width of the ground state p eak would b e zero in the pseudoscalar

channel and very small for the vector meson but this assumption holds true only for an

innite numb er of gauge eld congurations and an innite temp oral extent of the lattice

In gure ab it was already shown that an insucient numb er of congurations and

a limitation on the numb er of time slices results in a attening and broadening of the

p eaks Nevertheless the width of the pion ground state p eak is much smaller than the

one of the rho meson see g The statistical noise has more inuence for the vector

meson op erator therefore a larger numb er of congurations is needed to obtain a similar

accuracy in the rho meson mass This eect is also visible in the larger error bars for the

rho masses obtained in twoexp onential ts

Figure displays a whole set of pseudoscalar and vector meson sp ectral functions at

several quark mass values They were obtained with the continuum kernel in a frequency

range up to a with separations of a The pseudoscalar sp ectral

max

functions show very sharp pion ground state p eaks and only small contributions of excited

70 1.2 ρ(ω) pion κ = 0.148 ρ(ω) κ = 0.148 0.147 0.147 60 0.146 1 0.146 0.145 0.145 50 0.144 0.144 0.142 0.8 0.142 40 0.6 30 0.4 20

10 0.2 ω [GeV] ω [GeV] 0 0

0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5

Figure Sp ectral functions for the pseudoscalar left and the vector meson right at

dierent values

Chapter Hadron Spectral Functions

1.2 2 mπ mρ 1.0

κ = 0.14922(1) 0.8 c

0.6

0.4

0.2 1/κ 0.0

6.7 6.8 6.9 7 7.1 7.2

Figure Comparison of the MEM results diamonds with conventional twoexp onential

t results squares

states The p eaks for smaller quark masses are more pronounced in this representation

of the sp ectral function but they are of approximately equal height in the full sp ectral

function A

On the other hand the broadening and drop of the low mass contributions seen for the

lightest quark masses in the vector sp ectral function can b e addressed to insucient statis

tics for this correlator This is also apparent from conventional exp onential ts which in

these cases lead to large errors on the lightest vector meson masses in gure In general

the pion and rho meson masses obtained with MEM are in very go o d agreement with the

results from twoexp onential ts For the lightest rho meson mass even an improvement

could b e achieved Instead of a drop of the mass with a large error bar a much more

reasonable p oint with a smaller error could b e obtained with MEM

An error on the p eak p osition of the sp ectral functions can b e obtained in dierent ways

One p ossibility is to employ a t with a Gaussian form to extract the p osition and the vari

ance of the p eak but this pro cedure usually overestimates the error on the particle masses

For the results quoted in this thesis a Jackknife analysis with six blo cks was utilized This

metho d includes the MEM analysis for the whole data set as well as for six reduced data

sets After the determination of the p eak p osition on each of the reconstructed sp ectral

functions the Jackknife error could easily b e obtained as describ ed in section A lin

ear extrap olation in leads to a vanishing pion mass at which p erfectly

coincides with the value obtained from a conventional exp onential t Extrap

olating the rho meson mass for the ve lowest quark masses to the chiral limit one obtains

m MEM in lattice units which is compatible with m expt The

values for each quark mass can b e found in table B of app endix B

Diquark Spectral Functions

30 ρ(ω) pion 5*rho 25 fuzzed pion fuzzed 5*rho

5 ω [GeV] 0

0 0.5 1 1.5 2 2.5 3 3.5

Figure Inuence of the fuzzing technique on the sp ectral functions

Concluding this section ab out meson sp ectral functions at zero temp erature the inuence

of the fuzzing technique see section on the contributions of excited states to the

sp ectral functions is investigated In this case a smaller data set of congurations on a

lattice was available All other parameters remain the same as describ ed at the

b eginning of this section

The fuzzing technique was applied for spatially extended op erators with a radius R

fm In gure the pseudoscalar and vector meson sp ectral functions with and with

out fuzzing can b e examined After the fuzzing pro cedure the height of the ground state

p eaks remains almost the same whereas their width is considerably reduced Moreover

the contributions of excited states at higher energies vanish As a consequence only the

sharp ground state p eaks are left in the sp ectral functions They thus uncover unambigu

ously that the application of the fuzzing technique eliminates the excited states almost

completely

Diquark Sp ectral Functions

This section deals with diquark correlation and sp ectral functions at zero temp erature

obtained from the quark propagators mentioned at the b eginning of the previous

section The correlation functions were calculated for four dierent diquark op erators

namely the lighter color antitriplet states and as well as the color sextet states

and These abbreviations corresp ond to the avor spin colorrepresentation as

already intro duced in section

Chapter Hadron Spectral Functions

1e+01 − − D(τ) 3 03 diquark − 613 diquark 1e-01 − 3 16 diquark 606 diquark 1e-03

1e-05

1e-07

(a) τ 1e-09 0 5 10 15 20 25 30

0.4 ρ ω − − ( ) 1/10* 3 03 diquark − 613 diquark − 0.3 3 16 diquark 606 diquark

0.2

0.1

(b) ω [GeV] 0

0 0.5 1 1.5 2 2.5 3 3.5

Figure Correlation functions top and corresp onding sp ectral functions b ottom

for the zero temp erature diquarks at

Diquark Spectral Functions

The diquark correlation functions obtained from this data set are shown in gure a It

can clearly b e observed that the diquark correlation function exhibits the characteristic

exp onential decay known from mesons To a certain extent this is also the case for the

diquark The correlation functions of the color sextet diquarks are strongly inuenced

by higher excited states Furthermore larger error bars are attached to the p oints which

already indicates the insucient statistics for these noisy correlators

The sp ectral functions for the color antitriplet diquarks are similar to the ones for mesons

see gure b In particular the diquark sp ectral function shows a pronounced

ground state p eak and is therefore reduced by a factor in the plot However already

in the case of the diquark the ground state p eak broadens and b ecomes comparable

in magnitude to the continuum structure found at higher energies This broad continuum

b ecomes even more dominant in the color sextet channels Although it is unclear to a

certain degree whether a b ound state exists in these quantum numb er channels this series

of diquark states is an excellent example to study the p ossible variations from b ound to

probably unb ound states in terms of the sp ectral functions

Nevertheless the ground state p eak p osition for all diquark states could b e determined

by a Jackknife error analysis at all quark masses These results are summarized in

table B of app endix B Note that the smo othing of the covariance matrix b ecomes

increasingly imp ortant for the analysis of

color sextet sp ectral functions This

the 0.2

was describ ed in detail in sec metho d ρ(ω) full covariance matrix

smoothed C, NR=6

tion Figure visualizes the in

0.15 smoothed C, NR=5

uence of the pro cedure on the sp ec

tral function of the diquark Since

0.1

the numb er of congurations is insucient

for this noisy correlator the covariance

has several to o small eigenvalues

matrix 0.05

Using the full covariance matrix for the

ω [GeV]

analysis results in an unphysical MEM 0

0 0.5 1 1.5 2 2.5 3 3.5

state b elow the ground state p eak More

over the ground state p eak is shifted to

higher frequencies By retaining only the

Figure Inuence of the covariance matrix

largest eigenvalues the unphysical p eak

smo othing pro cedure N denotes the num

can b e eliminated or at least reduced

b er of retained eigenvalues see sec

Having obtained the diquark masses from a Jackknife analysis of the sp ectral functions

they can directly b e compared with the ones obtained from conventional twoexp onential

ts For the color antitriplet diquarks this is displayed at the top of gure The MEM

values lie systematically a bit lower than the exp onential t results but they clearly agree

within their error bars This leads to slightly lower masses in the chiral limit extrap olated

from the values at the ve lightest quark masses Table can b e consulted for the exact

values

Chapter Hadron Spectral Functions

1.6 m 1.4

1.2 κ c= 0.14922(1) 1.0

0.8

0.6 − 613 diquark − − 0.4 303 diquark

0.2 1/κ 0.0 6.7 6.8 6.9 7 7.1 7.2

1.6 m 1.4 κ 1.2 c = 0.14922(1)

1.0

0.8

0.6 606 diquark − 0.4 316 diquark

0.2 1/κ 0.0

6.7 6.8 6.9 7 7.1 7.2

Figure Comparison of the MEM results diamonds with twoexp onential ts b oxes

for the color antitriplet diquarks top and sextet diquarks b ottom

Thermal Spectral Functions

Diquark state

ma MEM result

(FSC )

ma exp t

(FSC )

Table Comparison of diquark masses in the chiral limit

The color antitriplet as well as the sextet states have in common that their masses are

close to each other at larger bare quark mass and split up at smaller quark masses due

to the avorspin interaction The qq interaction is attractive repulsive for the color

antitriplet sextet diquark states see table or in detail This b ehavior was

covered in statistical noise of the correlators for the heavier sextet states and could not

b e resolved by exp onential ts The masses of the diquark state were only slightly

underestimated but had large errors whereas the exp onential t results for the diquark

were misled through the unphysical state visible in g For the color sextet diquarks

an enormous improvement could b e achieved with the Maximum Entropy Metho d

Extrap olating the masses to the chiral limit the same value as in the exp onential t

is obtained for the diquark but with an error half as large as b efore Note that

this value has changed in comparison to the publication where the inuence of the

covariance matrix smo othing had not yet b een explored The mass values of the

diquark now extrap olate to a considerably higher value in the chiral limit which is in

much b etter agreement with the strong repulsion b etween the two quarks exp ected from

the avorspin qq interaction in this channel In general one may question for all but

the diquark state whether the leading p eak in the sp ectral function at all corresp onds

to a b ound state The sp ectral function favors an interpretation in terms of a threshold

followed by a broad continuum This question could b e resolved in a more detailed sp ectral

analysis with higher statistics

Thermal Sp ectral Functions

In the last two sections the sp ectral functions at zero temp erature were analyzed and

results on masses were compared to conventional twoexp onential ts This gave some

insight into the parameter dep endence of the maximum entropy analysis and has shown

how the sp ectral functions vary with dierent content in the correlation function It could

b e shown that the MEM analysis correctly detects b ound states like the pion as well as

the more unb ound and continuumlike states in the color sextet diquark channel Since

this exp erience is very encouraging an application of the metho d at nite temp erature

could b e ventured In this case it is absolutely mandatory that MEM needs no previous

assumptions ab out the sp ectral shap e b ecause very little is known ab out thermal sp ectral

functions so far

Chapter Hadron Spectral Functions

Free Meson Sp ectral Functions in the Continuum

In the high temp erature limit the meson correlation functions are exp ected to reect the

dynamics of freely propagating quark antiquark pairs This can b e describ ed to leading

order by the free sp ectral function in the degenerate pseudoscalar channel

A tanh T

As a rst step it thus is imp ortant to test whether such a sp ectral shap e can b e repro duced

on lattices with nite temp oral extent Using the continuum expression see equation

cosh T

D A d

sinh T

the free meson correlation function can b e calculated for an arbitrary numb er of time

slices N The prefactor N of the sp ectral function ensures that the obtained

correlation function is normalized to one at T This is shown in gure a for

several numb er of time slices Note that the correlation functions at nite temp erature are

displayed in units of T the sp ectral functions are determined corresp ondingly in terms

of T

For the purp ose of constructing a suitable mo ck data set for the MEM analysis of free

meson correlators a Gaussian noise with the variance

b D for N

b D N for N N

was added to the exact data values obtained from equation and The prefactor

b controls the noise level In order to guarantee a compatible noise level for data sets with

dierent temp oral extent the pro duct b N should b e kept xed

The reconstructed sp ectral functions for dierent N and noise level are combined in gure

b All curves are in go o d agreement with the free sp ectral function except the one

with the largest noise level b They were obtained with the continuum kernel and

quite a large frequency range T N The resulting sp ectral functions with

max

smaller T would have some bumps in addition to the free curve For the sp ectral

max

functions with N it can clearly b e observed that the decrease of the noise level

from to results in a reduction of the dierence b etween the reconstructed and

the original free sp ectral function Even with only N p oints and comparable low

noise level the sp ectral shap e can b e reconstructed almost p erfectly Sometimes it may

b e useful to combine data sets on smaller lattices in order to get additional temp oral grid

p oints by varying N This is illustrated in gure b for a combination of N

and which also yields a convincing reconstruction of the free sp ectral function

Thermal Spectral Functions

τ 3 D( T)/T Nτ = 32 16 1000 12 10

100

(a) τT 1 0 0.2 0.4 0.6 0.8 1

0.05 ρ(ω) 0.04

0.03 free spectral function N = 32 b = 0.1 0.02 τ Nτ = 32 b = 0.01 Nτ = 32 b = 0.005 N = 16 b = 0.01 0.01 τ Nτ = 10,12,16 b = 0.01 (b) ω/T 0

0 5 10 15 20

Figure Discretized free thermal meson correlation function top and reconstructed

sp ectral functions b ottom with dierent noise level b

Chapter Hadron Spectral Functions

Free Meson Sp ectral Functions on the Lattice

A step further towards the examination of thermal sp ectral functions is the calculation of

free meson correlation functions on the lattice which are restricted to the Euclidean time

interval T In the high temp erature limit the coupling g vanishes and the gauge

elds approach unit matrices in color space This allows the analytic calculation of the

quark propagators see eq and thus the free meson correlators for any given lattice

size Alternatively lattice calculations may b e p erformed with a unit conguration

These free correlation functions on the lattice can b e used to explore the nite volume and

cuto eects which are exp ected to mo dify the results obtained on small lattices Figure

a displays the free pseudoscalar correlation functions for various spatial extents of

the lattice At equal spatial and temp oral extension the curve is enhanced and attened

in the middle This leads to mo dications at low frequencies in the corresp onding sp ectral

function It can b e observed that a ratio of N N is needed to eliminate the

nite volume eects almost completely

The inuence of the nite lattice cuto is illustrated in gure b for several lattice

sizes The largest deviations from the continuum free curve are visible for small Euclidean

times in particular for the rst two p oints N Unlike in the continuum limit

the lattice correlation function is not divergent at for N The free meson

correlation function on the N lattice was utilized at rst in a MEM analysis with

the continuum kernel However this attempt to reconstruct the sp ectral shap e failed

unexp ectedly Even leaving out the p oints at and which have the strongest cuto

eects yields the curve with two bumps visible in gure In this case the lattice kernel

dened in equation is a resource to repro duce the free sp ectral function correctly

Instead of the fairly large range which is used for the continuum kernel the maximal

frequency is restricted to T N for the lattice kernel which corresp onds to the

max

maximal available momentum on the lattice The sp ectral shap e could b e repro duced

D(τT)/T3 D(τT)/T3 3 1000 Nσ=16 1000 32 x8 24 483x12 32 643x16 100 64 100

10 10 (a) (b) τ T τ T 1 1

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Figure Finite volume eects with xed N left and cuto eects with xed

ratio N N right for the free meson correlation function on the lattice the black line

indicates the continuum free correlator

Thermal Spectral Functions

0.1

ρ(ω) lattice kernel Nτ= 8 12 0.08 16 continuum kernel Nτ=16 0.06

0.04

0.02 ω/T 0

0 10 20 30 40 50

Figure Reconstructed free lattice sp ectral functions with xed noise level b N

well already with the correlators obtained on lattices with small temp oral extent N

and Apparently the cuto eects are comp ensated eciently in applying the lattice

kernel

At rst sight an accurate reconstruction should also b e p ossible for a combined data set

eg for N and This attempt fails unfortunately since only the cuto eects for

a sp ecial N value can b e absorb ed in the choice of the lattice kernel and an appropriate

choice of T It is obvious from gure b that correlators with evidently diering

max

p oints at the same time separation cannot b e represented by an unique sp ectral function

For the combined data set analyzed with the continuum kernel g this was no

hindrance since all correlators with dierent N lay on a single smo oth curve

Considering the success of the fuzzing technique for an optimized pro jection onto the

ground state at zero temp erature see gure one might b e tempted to use this metho d

also at nite temp erature where the p eaks are less pronounced compared to the continuum

contribution However the entire concept of fuzzing has to fail when a single state is not

well separated from higher excited states The situation will naturally b e even worse when

only a continuum exists In the case T T one can no longer b e sure to pro ject on an

actual ground state since even the fuzzing of the free meson correlation function leads to

sharp p eaks instead of the broad continuum This is illustrated in gure for dierent

fuzzing radii R The fuzzing causes the reduced curvature of the correlator at small time

separations which is corresp ondingly translated into the denitely unphysical p eaks in the

sp ectral function Therefore it is mandatory for T T to use the unmo died p ointp oint

correlator in the MEM analysis to preserve the full information ab out ground and excited

states as well as the continuum contribution

Chapter Hadron Spectral Functions

1.6 ρ ω D(τT)/T3 ( ) R=0 1.4 R=1 1000 R = 0 1.2 R=2 R = 1 R=3 R = 2 1.0 100 R = 3 0.8 0.6 10 0.4 0.2 τ T ω/T 1 0.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30

Figure Fuzzing of the free sp ectral function

Meson Sp ectral Functions at Finite Temp erature

Having tested the Maximum Entropy Metho d with continuum and lattice free meson

correlation functions we can pro ceed to the analysis of data sets from lattice simulations

at nite temp erature Instead of using anisotropic lattices which involves the precise

calibration of the anisotropy parameters we decided to p erform simulations on fairly

large isotropic lattices in quenched QCD As shown b efore for the free meson correlation

function on the lattice already data sets with temp oral extents N seem

sucient to obtain reliable sp ectral functions Table summarizes the parameters used

for the simulations b elow the critical temp erature p erformed on the APE machines in

Bielefeld and those employed ab ove T within our pro ject on the Cray TE in Julich

1 3

N T T a GeV N N values value

c C c

Table Summary of simulation parameters T T and a are determined from the

string tension on lattices with N values interp olated from

Thermal Spectral Functions

The gauge eld congurations are separated by sweeps of overrelaxation and one

heatbath step each They were generated with the standard plaquette action which allows

us to apply the improved Clover fermion action with nonp erturbatively determined clover

co ecients see sec Temp oral and spatial meson correlators were calculated for up

to four quark mass values b elow T and at almost zero quark mass at estimated T

c c

ab ove the critical temp erature The Coulomb gauge was xed on the congurations ab ove

T in applying the Los Alamos algorithm combined with the overrelaxation metho d see

sec for details This op ened the p ossibility to calculate gaugevariant quark and gluon

correlators

A temp erature of T is reachable in heavy ion collisions whereas the larger temp erature

of T was chosen to allow a comparison with previous results from HTLresummed

p erturbation theory Furthermore the results in this high temp erature region can

b e compared to the free meson correlation and sp ectral functions discussed ab ove The

simulations b elow the critical temp erature were p erformed in order to analyze the changes

in meson sp ectral functions in the whole range b etween the well known zero temp erature

case and the deconning phase transition as well as to study nite volume and cuto

eects

Mesons at T in comparison with T

Having explored the zero temp erature correlators and sp ectral functions in the sections

and the results can b e compared to the ones at T well b elow the phase

transition The sp ectral functions obtained for the pseudoscalar scalar and vector meson

are displayed in gure for several values of the bare quark mass The b ehavior of the

pseudoscalar and vector meson sp ectral functions is qualitatively the same as observed for

zero temp erature g In general the height of the ground state p eaks is reduced their

width is increased and the errors on the p eaks are enlarged due to the limited statistics

and the cuto eects since the numb er of congurations is only a fth and the numb er

of time slices N is halved in comparison to the T simulation These eects were

investigated in section and are summarized in the gures a and b Nevertheless

a slight temp erature eect could play a role in the changes of the sp ectral shap e

4 0.5 0.5 ρ ω ρ ω ρ(ω) κ = 0.1348 ( ) κ = 0.1348 ( ) κ = 0.1348 0.1342 0.4 0.1342 0.4 0.1342 3 0.1332 0.1332 0.1332 0.1324 0.1324 0.1324 0.3 0.3 2

pseudo-scalar 0.4 Tc 0.2 scalar 0.4 Tc 0.2 vector 0.4 Tc

1 0.1 0.1 ω/T ω/T c c ω/T 0 0.0 0.0 c

0 5 10 15 20 0 5 10 15 20 0 5 10 15 20

Figure Meson sp ectral functions at T for dierent values c

Chapter Hadron Spectral Functions

The sp ectral functions of the scalar meson in the middle of gure show a much less

systematic b ehavior The continuum structure dominates for all bare quark masses similar

to the color sextet diquark states which were investigated in section see eg gure

The sp ectral shap e resembles the scalar sp ectral function at low temp erature in a

simulation of the O linear mo del Since only the connected avor nonsinglet

parts of the scalar correlator were calculated in the lattice QCD simulation the scalar

channel corresp onds to the meson intro duced in section The present MEM results

suggest that no b ound state exists in this channel at low temp eratures which needs to b e

conrmed by additional simulations with larger statistics

0.8 ≈ T 0.4Tc 0.7 κ 1/ c 0.6 mVa 0.5 0.4 0.3 2 (mPSa) 0.2 T=0 MEM 0.1 temp spatial 1/κ 0.0

7.30 7.35 7.40 7.45 7.50 7.55 7.60

Figure Meson masses at T obtained with MEM and twoexp onential ts the

band indicates the range obtained for the dierent data sets

The ground state p eak p ositions of the pseudoscalar and vector meson can b e estimated

in a Jackknife analysis of the sp ectral function The MEM results for the meson masses at

all temp eratures b elow T are combined in table B of the app endix The extrap olation

to the chiral limit is visualized in gure The p oints indicated as T are adopted

ockeler et al on a lattice at and similar from a simulation of G

values In addition to the MEM analysis conventional ts with one and two exp onen

tials were p erformed for the temp oral and the spatial correlator see table B For the

pseudoscalar meson mass a go o d agreement is obtained b etween the MEM and T

results The masses from the exp onential ts of the spatial and temp oral correlator lie

slightly ab ove the others but extrap olate to similar values of the critical hopping param

eter The deviations are more pronounced for the vector meson The exp onential t of

the temp oral correlator tends obviously to overestimated values due to the inuence of

continuum contributions Since the MEM analysis allows a distinction b etween the p ole

Thermal Spectral Functions

and continuum contributions much more reasonable values can b e extracted from the

same correlation function In the chiral limit the extrap olated screening mass obtained

from exp onential ts in the more extended spatial direction as well as the T data

p oint coincide with the MEM values within the error bars This leads to the implication

that the observed b ehavior at T is not dierent from the one at T

Finite size eects b elow T

At the somewhat higher temp erature of T 4 c

ρ ω 3

sets on three dierent lattice sizes op en ( ) NC = 30 32 x 16 data 3

NC = 60 24 x 16

p ossibility to study the nite size eects on 3 3 the

NC = 240 16 x 16

the lattices investigated b elow T Figure

ws the pseudoscalar sp ectral functions at

2 sho

smallest quark mass value for pseudo-scalar 0.6 Tc the

κ=0.1354 3

lattice size with the indi

1 a

cated numb er of congurations The ground ω

/Tc

p eak gets narrower and is increased by

0 state

hanging the spatial extent of the lattice from

0 5 10 15 20 25 30 c

N to Only marginal dierences are

observed enlarging N to which could

Figure Finite size eects on the

b e aected by the limited statistics of only

pseudoscalar sp ectral function at T

congurations on the largest lattice However

a shift of the ground state p eaks towards lighter pion masses can b e observed continuously

by enlarging the lattice volumes The indicated error bars b elong to the sp ectral function

of the lattice data set The larger one illustrates mean and variance calculated

with relation in the given range whereas the smaller one represents solely the

statistical error which is obtained in a Jackknife analysis of the average sp ectral function

in this range The divergence at very small frequencies can b e traced back to insucient

statistics In such a case the sp ectral function approaches not strictly prop or

tional to which leads to the unphysical b ehavior for the displayed sp ectral function

In the following the sp ectral functions b elow T refer to the lattice In general

go o d agreement of the meson masses on lattices with N and could b e observed

but due to the limited statistics at N the errors are still at least twice as large as

with N For example the resulting masses extracted in a Jackknife error analysis of

24 32

the ab ove mentioned sp ectral functions yield m a and m a

PS PS

Pseudoscalar and vector mesons at T T T

c c

The inuence of increasing temp erature on the sp ectral functions in the pseudoscalar

and vector channel can b e illustrated more clearly in a combined plot such as gure

Below the critical temp erature the sp ectral functions are shown at similar quark masses

m MeV while the ones ab ove T have b een obtained at m which was

c q

checked with the relation The sp ectral functions at and T remain very c

Chapter Hadron Spectral Functions

1.2 0.3 ρ(ω) pseudo-scalar 0.4 Tc ρ(ω) vector 0.4 Tc 0.6 Tc 0.6 Tc 1.0 0.9 Tc 0.9 Tc 1.5 Tc 1.5 Tc 0.8 3.0 Tc 0.2 3.0 Tc

0.6

0.4 0.1

ω 0.2 /Tc ω/T 0.0 0.0 c

0 5 10 15 20 25 30 0 5 10 15 20 25 30

Figure Sp ectral functions of the pseudoscalar left and vector meson right for

several temp eratures

similar in b oth channels whereas a doubling of the width and a reduction in height of the

ground state p eak is visible at T Furthermore the p eak of the vector meson is shifted

towards larger frequency values At present it is not clear whether this eect is an actual

temp erature induced mo dication of the sp ectral shap e or only an artifact due to limited

statistics since the exp onential t of the spatial correlator yields an almost unchanged

screening mass from T up to T see table B

c c

Ab ove the critical temp erature the broad p eak in the vector channel vanishes almost

completely in favor of a continuum structure However b etter statistics is clearly desirable

for the sp ectral functions at T In the pseudoscalar channel the sharp ground state

p eak broadens considerably ab ove T and is shifted towards larger energies Nevertheless

deviations from the shap e of the free sp ectral functions can b e still observed which will

now b e illustrated in more detail

At high temp eratures a scaling of the sp ectral function with the temp erature is exp ected

Therefore it is more appropriate to investigate the changes of the sp ectral shap e in units

of the temp erature rather than in physical units like T The correlators and corresp ond

ing sp ectral functions in the pseudoscalar and vector channel are illustrated in gure

Note that in the vector channel the sum of the contributions with all matrices

k is shown which yields twice the pseudoscalar correlator in the free case At

and T the comp onent of the correlation functions is simply a constant which

c 4

eliminates the divergence at T in the vector meson sp ectral function Furthermore

the nonp erturbative renormalization factors for each value in the pseudoscalar vec

tor channel were taken into account which are summarized in table B The

correlation functions of the vector meson at and T lie almost on top of each other

and agree very well with the free meson correlator indicated as solid black line Slight

deviations can b e observed at intermediate time separations which are also evident from

the corresp onding sp ectral functions at smaller frequencies Such a mo dication of the

sp ectral shap e is exp ected in HTLresummed p erturbation theory In this case the

sp ectral function in the vector channel is even linear divergent at low frequencies which

renders the temp oral correlator infrared divergent

Thermal Spectral Functions

0.3 3 ρ(ω) G(τT)/T free spectral function (pseudo-)scalar 1.5 Tc 1000 pseudo-scalar 1.5 T c 3.0 Tc 3.0 T c 0.2 0.5 * vector 1.5 Tc 0.5 * vector 1.5 T c 3.0 Tc 100 3.0 Tc T = ∞ 0.1 10

τ T ω/T 1 0 10 20 30 40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure Meson correlation left and sp ectral functions right ab ove T

For the pseudoscalar correlator a gradual approach towards the corresp onding free me

son correlation function can b e observed by increasing the temp erature from to T

Nevertheless quite large deviations from the free quark b ehavior still p ersist at the highest

temp erature investigated which conrms the previous ndings of Boyd et al This

observation is even more emphasized considering the pseudoscalar sp ectral functions in

gure In addition to a continuumlike structure a broad p eak at lower frequencies

is still evident The p osition of this bump is quite sensitive to the numb er of retained

eigenvalues in the covariance matrix smo othing pro cedure see sec Higher statistics

thus is needed to clarify this asp ect of the sp ectral function in more detail The large

deviations from the free sp ectral functions found in the scalar channel cannot b e su

ciently explained by HTL medium eects like thermal quark masses and Landau damping

In the pseudoscalar channel this leads to the conclusion that the b ehavior in the

deconned plasma phase is characterized by strongly correlated quarks and gluons in the

temp erature range up to T The QCDTARO collab oration rep orted similar nd

ings in a recent lattice simulation with Wilson fermions on anisotropic lattices Since they

have used smeared meson propagators the results however might b e strongly inuenced

by the continuum contributions and might pro ject on unphysical ground states as shown

in the end of section with the application of the fuzzing technique for the free meson

sp ectral function

The temp erature dep endence of the obtained correlation functions can b e investigated

more directly after normalizing them with the corresp onding free meson correlator on

the lattice Such ratios are displayed in gure for the pseudoscalar scalar and

vector channel and the same quark masses as describ ed b efore At low temp eratures

they can b e compared to previous investigations with Wilson fermions and over

lap fermions as well as to predictions of the interacting instanton liquid mo del

for T T In general a go o d qualitative agreement with the rep orted results

can b e observed in all channels The deviations from one at time could b e

reduced by taking the nonp erturbative renormalization factors into account Since the

Zfactors are only available in the chiral limit they cannot b e obtained for the correlators

b elow the critical temp erature In order to allow a comparison of the correlator ratios for

Chapter Hadron Spectral Functions

100 100 pseudo-scalar 0.4 Tc pseudo-scalar 0.4 Tc 0.6 Tc 0.6 Tc 0.9 Tc 0.9 Tc 1.5 Tc 1.5 Tc 3.0 Tc 3.0 Tc 10 10

r [fm] r T 1 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5

6 6 scalar 0.4 Tc scalar 0.4 Tc 0.6 T 0.6 T 5 c 5 c 0.9 Tc 0.9 Tc 1.5 T 1.5 T 4 c 4 c 3.0 Tc 3.0 Tc 3 3

2 2

1 1 r [fm] r T 0 0 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5

3 3 vector 0.4 Tc vector 0.4 Tc 0.6 Tc 0.6 Tc 0.9 Tc 0.9 Tc 1.5 T 1.5 T 2 c 2 c 3.0 Tc 3.0 Tc

1 1

r [fm] r T 0 0

0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5

Figure Ratio of meson and free thermal meson correlator on the lattice

Thermal Spectral Functions

dierent temp eratures no renormalization factors were used in gure In the following

the correlators will b e referred to as attractive if the ratio is larger than one while a ratio

smaller than one corresp onds to a repulsive interaction Such an interpretation is inspired

by the op erator pro duct expansion of these ratios presented in

In the pseudoscalar channel the strong attractive interaction b etween the quarks is

visible in the steep rise of the ratio note the logarithmic scale which reects the prop erty

of the pion as Goldstone b oson b elow T The attening of the ratio at larger distances

could also b e observed in at a similar value of the quark mass m m

Ab ove the critical temp erature the ratio is considerably decreased but still well ab ove

one which conrms the exp ectations of the instanton liquid mo del Such a b ehavior

indicates the p ersistent attractive interaction in this channel even ab ove T as already

observed in the analysis of the corresp onding sp ectral function Note that at such high

temp eratures mainly the short distance b ehavior of the correlators can b e investigated

since the extent of the temp oral direction shrinks with increasing temp erature which is

obvious from the left panels of gure

The situation is clearly dierent in the scalar channel Apart from a weak attractive

interaction similar to the pseudoscalar channel at small distances the general b ehavior

b elow T can b e characterized by a strong repulsion in the range r f m which

was also found in The strength of the repulsive interaction is increasingly reduced

when approaching the critical temp erature Ab ove T the ratio is very similar to the one in

the pseudoscalar channel indicating the restoration of the chiral symmetry see sec

This phenomenon is further explored b elow in terms of the sp ectral functions

Although the ratio in the vector channel lo oks similar to the scalar ratio at low tem

p eratures it remains much closer to one note the dierent scale The ratio attens even

more with increasing temp erature and reaches almost a constant at T This b ehavior

is consistent with the picture of a rapidly melting resonance contribution indicating

the free propagation of the quarks at T T describ ed solely by the continuum contribu

tion which is in accordance with the analysis of the vector meson sp ectral functions at

and T in gure and the resp ective annotations

Chiral symmetry restoration in the pseudoscalar channel

Finally a somewhat closer lo ok is addressed to the development of the pseudoscalar

correlation and sp ectral function for increasing temp erature For a simulation with dy

namical quarks it is exp ected that the pseudoscalar and scalar correlators b ecome

degenerate at the critical temp erature if the axial U symmetry is eectively restored

see sec The correlators and sp ectral functions are shown in gure for nearly

equal quark masses at and T as well as for almost zero quark mass at T The

c c

b ehavior of the correlation functions at T resembles the b ehavior at zero temp erature

A clear pion ground state p eak can b e observed in the corresp onding sp ectral function of

the pseudoscalar meson while the one of the scalar meson is characterized by a much

broader continuumlike structure

Chapter Hadron Spectral Functions

1.2 D(τT)/T3 pseudo-scalar ρ(ω) pseudo-scalar scalar 1.0 1000 scalar 0.8

100 0.6 Tc 0.6 0.6 Tc

0.4 10 0.2 ω/T τ T c 1 0.0 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30

1.2 D(τT)/T3 pseudo-scalar ρ(ω) pseudo-scalar scalar 1.0 1000 scalar 0.8

100 0.9 Tc 0.6 0.9 Tc

0.4 10 ω 0.2 /Tc τ T 1 0.0 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30

1.2 ρ ω D(τT)/T3 pseudo-scalar ( ) pseudo-scalar scalar 1.0 scalar 1000 0.8

100 1.5 Tc 0.6 1.5 Tc

0.4 10 0.2 τ ω/T T 0.0 c 1 0 5 10 15 20 25 30

0 0.2 0.4 0.6 0.8 1

Figure Restauration of the chiral symmetry b etween the scalar and pseudoscalar

meson error bars indicate statistical errors obtained in a Jackknife analysis

Thermal Spectral Functions

At T rst changes of the scalar correlator can b e recognized since the values around

T are increased and approach the pseudoscalar correlation function The scalar

sp ectral function remains essentially the same while the ground state p eak of the pseudo

scalar sp ectral function broadens and is reduced in height Ab ove the critical temp erature

at T the pseudoscalar and scalar correlator are very similar One therefore can observe

the restoration of the chiral symmetry already in the quenched sp ectrum calculations The

degeneracy of the pseudoscalar mesons is also reected in the corresp onding sp ectral

functions which show a go o d agreement Since only the correlators for the avor non

singlets mesons and were calculated in the present lattice simulations the observed

chiral symmetry restoration at T actually refers to the axial U symmetry in the

c A

pseudoscalar channel Predictions ab out the SU N SU N chiral symmetry

L R

f f

restoration could b e obtained in simulating the avor singlet mesons and which

would involve the calculation of disconnected propagators Such investigations are still at

an exploratory stage since the signal is hidden in a high level of statistical noise

and thus requires the application of sto chastic variance reduction techniques

Outlo ok on future applications

The quark and gluon propagators ab ove the critical temp erature calculated with the gauge

eld congurations in Coulomb gauge see table are currently under investigation

Preliminary results obtained for temp oral correlators at zero and nite momenta will b e

presented in the near future Although the propagators are quite noisy MEM is

capable to extract the p osition of the p oles In addition it would b e desirable to explore

their prop erties in other gauges eg the Landau gauge where the propagators are less

inuenced by the statistical noise Such a study would require the adaptation of MEM

for nonp ositive sp ectral functions First suggestions for mo died versions of the entropy

were rep orted recently

Another interesting eld of application for MEM would b e lattice simulations in full QCD

In this case the prop erties of the p eaks in the reconstructed sp ectral functions are directly

related to the resonance widths and decay constants in exp erimental sp ectra such as

For this purp ose it is mandatory to understand the nonzero width of the ground

state p eaks obtained in the MEM analysis of quenched lattice data It is evident from the

present investigation that insucient statistics and a limited temp oral extent of the lattice

inuence the height and width of the p eaks considerably A more quantitative analysis of

these dep endencies is certainly still needed in quenched lattice QCD calculations b efore

one can pro ceed to QCD simulations with light dynamical quarks

Chapter Hadron Spectral Functions

Conclusions

Two asp ects of QCD were considered in the present thesis In the rst part the strange

hadron sp ectrum was investigated which was calculated in quenched QCD on four dierent

lattice sizes covering a large physical volume of fm fm Strong nite size

eects could b e observed for the smallest lattice which was manifest in overestimated

hadron masses esp ecially for the strange baryons and the Hdibaryon

A go o d agreement of the strange hadron masses could b e found on the three larger lattices

Compared to exp erimental values the obtained K and masses show a maximal

deviation b elow dep endent on the particle used to set the physical scale Such a

b ehavior is common for simulations in the quenched approximation

The mass of the Hdibaryon is compatible with twice the mass of the baryon on all

investigated lattice sizes Moreover a quantitative agreement with previous studies on a

similarly large lattice could b e reached No evidence for a b ound Hdibaryon is apparent

from the current calculations in quenched QCD therefore such six quark state may b e

considered as unb ound assembly of two baryons

The exp erience gained with the sp ectrum calculations at T also form the basis for

studies of the hadron sp ectrum at nite temp erature which has b een p erformed in the

second part of this thesis An essential new feature of this study is the application of

the Maximum Entropy Metho d which made an investigation of hadron sp ectral functions

obtained from correlators on the lattice p ossible for the rst time This approach included

the implementation test and application of the metho d for the sp ectral analysis of hadron

correlation functions in the vacuum as well as in a thermal medium

For the meson correlators at zero temp erature it could b e demonstrated that the p osition

of the p ole mass is repro duced correctly in the corresp onding sp ectral function Moreover

the application of MEM yields useful additional information in comparison to conventional

exp onential ts as is evident from the analysis of the diquark correlators The color anti

triplet diquark states exhibit similar clean ground state p eaks as the mesons but for the

color sextet diquarks the continuum contribution dominates the sp ectral function which

explains the severe problems observed in ts with two exp onentials

Conclusions

The applicability of MEM for nite temp erature correlation functions was examined for

the rst time It could b e shown for the free thermal meson correlator that the continuum

structure is indeed repro ducible For the analysis of the thermal correlation function

calculated on the lattice a sp ecially adapted lattice kernel is required to absorb the cuto

eects

The investigation of p ole and screening masses obtained from temp oral and spatial corre

lators revealed that MEM is capable to extract the correct p ole mass value even when the

conventional exp onential ts tend to overestimate the mass due to the short extent of the

time direction in simulations at nite temp erature The sp ectral representation facilitates

the distinction b etween the p ole and continuum contributions which makes the MEM

analysis sup erior to conventional t metho ds

At the lowest investigated temp eratures and T the prop erties of the meson sp ectral

functions remain qualitatively the same as at zero temp erature Deviations of the sp ectral

shap e can b e explained by the short extent of the time direction and limited statistics First

changes in the sp ectral functions like a broadening of the p eaks and a shift towards higher

masses could b e observed at T in the vector channel Whether this eect p ersists

with larger statistics is still unclear and should b e further claried on larger lattices with

reduced nite volume and cuto eects

The sp ectral function of the scalar meson is dominated by the continuum contribution at

low temp eratures which might indicate that no b ound state exists in this channel at least

at low temp eratures At T the scalar correlator approaches already the one of the

pseudoscalar meson giving rst evidence of chiral symmetry restoration Finally at

T the sp ectral functions in the pseudoscalar channel are almost p erfectly in agreement

This observation leads to the prediction that the U symmetry b etween the and

meson gets approximately restored in this temp erature range

At the highest temp erature of T the b ehavior of the correlation and sp ectral functions

in the vector channel are comparable with the propagation of almost free quarks Only

a slight enhancement is visible at large time separations and corresp ondingly small fre

quencies In contrast to that considerable deviations from the free sp ectral function can

b e observed in the pseudoscalar channel A broad bump in the low energy regime still

p ersists at and even T indicating only a gradual approach towards the free quark

b ehavior The sp ectral functions obtained with MEM conrm the earlier ndings in lat

tice simulations and are in qualitative agreement with the predictions of the interacting

instanton liquid mo del On a more quantitative level their structure cannot b e suciently

explained by medium eects taken into account in HTLresummed p erturbation theory

In this exploratory application of the Maximum Entropy Metho d at nite temp erature it

could clearly b e shown that such an analysis of lattice correlators leads to new insights into

the thermal mo dication of hadronic prop erties MEM provides a sensitive to ol to study

thermal eects in hadron sp ectra without making a priori assumptions on the sp ectral

shap es Nevertheless more statistics and larger lattices are needed to gain more exp eri

ence with this new approach Furthermore it would b e desirable to study the interacting

physical medium eects more quantitatively

App endix A

Conventions

Chapter A Conventions

Dirac Matrices

The euclidian matrices in the nonrelativistic representation are selfadjoint

and ob ey the anticommutation relation f g

i i

B B B C C C

i i

B B B C C C

1 2 3

A A A

i i

B B C C

1 2 3 4 5 4

A A

Charge Conjugation Matrix

For the diquark baryon and dibaryon op erators the charge conjugation matrix C is

needed in combination with to ensure the appropriate combination of the spinor indices

see section

B C

4 2 1 3 5

B C

App endix B

Numerical Results

Chapter B Numerical Results

KMeson

u s

KMeson

u s

SigmaBaryon

u s

Lamb daBaryon

u s

HDibaryon

u s

extrap olated values

Table B Particle masses on the lattice congurations

KMeson

u s

KMeson

u s

SigmaBaryon

u s

Lamb daBaryon

u s

HDibaryon

u s

Table B Particle masses on the lattice congurations

Chapter B Numerical Results

KMeson

u s

KMeson

u s

SigmaBaryon

u s

Lamb daBaryon

u s

HDibaryon

u s

Table B Particle masses on the lattice congurations

KMeson

u s

KMeson

u s

SigmaBaryon

u s

Lamb daBaryon

u s

HDibaryon

u s

Table B Particle masses on the lattice congurations

Chapter B Numerical Results

lattice congurations

input input K input mean

s s s s

lattice congurations

input input K input mean

s s s s

lattice congurations

z z z

input input K input mean

s s s s

lattice congurations

z z z

input input K input mean

s s s s

values averaged to mean value

s s

Table B Physical particle masses on all lattice sizes in M eV

MEM exp t

m m m m

Table B Comparison of meson masses obtained from MEM and twoexp onential ts

at MEM and exp vanishing pion mass m

c c

MEM exp t

m m m m m m m m

606 606

613 613 3 03 316 303 316

Table B Comparison of diquark masses obtained from MEM and twoexp onential ts

Chapter B Numerical Results

T a fm a GeV L fm N

c C

m a m GeV m a m GeV m a m GeV

T a fm a GeV L fm N

c C

m a m GeV m a m GeV m a m GeV

T a fm a GeV L fm N

c C

m a m GeV m a m GeV m a m GeV

Table B MEM results b elow T on the lattice c

T a fm a GeV L fm N

c C

t z t z t z

m a m a m a m a m a m a

T a fm a GeV L fm N

c C

t z t z t z

a a a a m m m a m a m m

T a fm a GeV L fm N

c C

t z t z t z

a a a a m m m a m a m m

Table B Twoexp onential t results b elow T on the lattice c

Chapter B Numerical Results

Temp erature value Z Z

PS V

Table B Renormalization factors calculated nonp erturbatively for the vector channel

and in tadp oleimproved p erturbation theory c for the pseudoscalar channel

Bibliography

H MeyerOrtmanns Phase transitions in quantum chromo dynamics Rev Mod

Phys

F Karsch and M Lutgemeier Deconnement and chiral symmetry restoration in

an SU gauge theory with adjoint fermions Nucl Phys B

heplat

M Fukugita M Okawa and A Ukawa Finite size scaling study of the

deconning phase transition in pure SU lattice gauge theory Nucl Phys

B Beinlich F Karsch E Laermann and A Peikert String tension and

thermo dynamics with tree level and tadp ole improved actions Eur Phys J C

heplat

F Karsch E Laermann A Peikert C Schmidt and S Stickan Flavor and

quark mass dep endence of QCD thermo dynamics Nucl Phys Proc Suppl

heplat

M Stephanov K Ra jagopal and E Shuryak Eventbyevent uctuations in

heavy ion collisions and the QCD critical p oint Phys Rev D

E Laermann Chiral transition in avor staggered QCD Nucl Phys Proc

Suppl A

R D Pisarski and F Wilczek Remarks on the chiral phase transition in

chromo dynamics Phys Rev D

G Boyd F Karsch E Laermann and M Oevers Two avour QCD phase

transition heplat

J B Kogut J F Lagae and D K Sinclair Top ology fermionic zero mo des and

avor singlet correlators in nite temp erature QCD Phys Rev D

heplat

R V Gavai S Gupta and R Lacaze Quenched QCD at nite temp erature with

chiral fermions heplat

Bibliography

P Vranas Dynamical lattice QCD thermo dynamics and the UA symmetry

with domain wall fermions heplat

G Boyd et al Hadron prop erties just b efore deconnement Phys Lett B

heplat

E Laermann and P Schmidt Meson screening masses at high temp erature in

quenched QCD with improved Wilson quarks heplat

C DeTar and J B Kogut Measuring the hadronic sp ectrum of the quark

plasma Phys Rev D

QCDTARO Collab oration P de Forcrand et al Meson correlators in nite

temp erature lattice QCD Phys Rev D heplat

M Jarrel and J E Gub ernatis Bayesian inference and the analytic continuation

of imaginarytime quantum Monte Carlo data Phys Rep

M Asakawa T Hatsuda and Y Nakahara Maximum entropy analysis of the

sp ectral functions in lattice QCD Prog Part Nucl Phys

heplat

M A Halasz A D Jackson R E Shro ck M A Stephanov and J J M

Verbaarschot On the phase diagram of QCD Phys Rev D

J Po cho dzalla et al Probing the nuclear liquid gas phase transition Phys

Rev Lett

J Bardeen L N Co op er and J R Schrieer Theory of sup erconductivity

Phys Rev

A D Rujula H Georgi and S L Glashow Hadron masses in a gauge theory

Phys Rev D

afer E V Shuryak and J J M Verbaarschot Baryonic correlators in the T Sch

random instanton vacuum Nucl Phys B hepph

afer E V Shuryak and M Velkovsky Diquark Bose condensates R Rapp T Sch

in high density matter and instantons Phys Rev Lett

hepph

D Bailin and A Love Sup eruidity and sup erconductivity in relativistic fermion

systems Phys Rept

M Alford K Ra jagopal and F Wilczek Coloravor lo cking and chiral

symmetry breaking in high density QCD Nucl Phys B

afer and F Wilczek Continuity of quark and hadron matter Phys Rev T Sch

Lett hepph

E Farhi and R L Jae Strange matter Phys Rev D

Bibliography

C Greiner and J SchanerBielich Physics of Strange Matter

nuclth

C Alco ck E Farhi and A Olinto Strange stars Nucl Phys Proc Suppl B

E Witten Cosmic separation of phases Phys Rev D

W Busza R L Jae J Sandweiss and F Wilczek Review of sp eculative

disaster scenarios at RHIC Rev Mod Phys

hepph

R Jae Perhaps a stable dihyp eron Phys Rev Lett

T Sakai K Shimizu and K Yazaki H dibaryon Prog Theor Phys Suppl

nuclth

S V Bashinsky and R L Jae Quark states near a threshold and the unstable

Hdibaryon Nucl Phys A hepph

P B Mackenzie and H B Thacker Evidence against a stable dibaryon from

lattice QCD Phys Rev Lett

Y Iwasaki T Yoshie and Y Tsub oi The H dibaryon in lattice QCD Phys

Rev Lett

T Yoshie Y Iwasaki and S Sakai Hadron sp ectrum on a x lattice

Nucl Phys Proc Suppl

A Po chinsky J W Negele and B Scarlet Lattice study of the H dibaryon

Nucl Phys Proc Suppl B heplat

I Wetzorke F Karsch and E Laermann Further evidence for an unstable

Hdibaryon Nucl Phys Proc Suppl heplat

H J Rothe Lattice gauge theories An intro duction Singap ore Singap ore

World Scientic p

I Montvay and G Munster Quantum elds on a lattice Cambridge UK Univ

Pr p Cambridge monographs on mathematical physics

K Symanzik Mathematical Problems in Theoretical Physics Lecture notes in

Physics SpringerVerlag Berlin

CPPACS Collab oration S Aoki et al Quenched light hadron sp ectrum

Phys Rev Lett heplat

H B Nielsen and M Ninomiya No go theorem for regularizing chiral fermions

Phys Lett B

B Sheikholeslami and R Wohlert Improved continuum limit lattice action for

QCD with Wilson fermions Nucl Phys B

Bibliography

M Luscher S Sint R Sommer P Weisz and U Wol Nonp erturbative Oa

improvement of lattice QCD Nucl Phys B

heplat

K Jansen and R Sommer The nonp erturbative Oalpha improved action for

dynamical Wilson fermions Nucl Phys Proc Suppl

heplat

J Kogut and L Susskind Hamiltonian formulation of Wilsons lattice gauge

theories Phys Rev D

P H Ginsparg and K G Wilson A remnant of chiral symmetry on the lattice

Phys Rev D

M Luscher Ab elian chiral gauge theories on the lattice with exact gauge

invariance Nucl Phys B heplat

D B Kaplan A metho d for simulating chiral fermions on the lattice Phys Lett

B heplat

R Narayanan and H Neub erger A construction of lattice chiral gauge theories

Nucl Phys B hepth

H Neub erger Chiral fermions on the lattice Nucl Phys Proc Suppl

heplat

R Gupta Intro duction to lattice QCD heplat

N Metrop olis A W Rosenbluth M N Rosenbluth A H Teller and E Teller

Equation of state calculations by fast computing machines J Chem Phys

N Cabibb o and E Marinari A new metho d for up dating SUN matrices in

computer simulations of gauge theories Phys Lett B

A D Kennedy and B J Pendleton Improved heat bath metho d for Monte Carlo

calculations in lattice gauge theories Phys Lett B

S L Adler An overrelaxation metho d for the Monte Carlo evaluation of the

partition function for multiquadratic actions Phys Rev D

M Creutz Overrelaxation and Monte Carlo Simulation Phys Rev D

S L Eisenstat et al J Num Anal

Y Oyanagi An incomplete LDU decomp osition of lattice fermions and its

application to conjugate residual metho ds Comp Phys Comm

H van der Vorst SIAM J Sc Stat Comp

M R Hestenes and E Stiefel J Res Nat Bur Standards

Bibliography

A Frommer V Hannemann B No ckel T Lipp ert and K Schilling Accelerating

Wilson fermion matrix inversions by means of the stabilized biconjugate gradient

algorithm Int J Mod Phys C heplat

T A DeGrand and P Rossi Conditioning techniques for dynamical fermions

Comput Phys Commun

L Giusti M L Paciello C Parrinello S Petrarca and B Taglienti Problems in

Lattice Gauge Fixing heplat

C Bernard D Murphy A Soni and K Yee Lattice quark propagator in xed

gauges Nucl Phys Proc Suppl B

A Cucchieri and T Mendes Gauge xing and gluon propagator in

lamb dagauges heplat

C T H Davies et al Fourier acceleration in lattice gauge theories Landau

gauge xing Phys Rev D

P de Forcrand and R Gupta Multigrid Techniques for Quark Propagator Nucl

Phys Proc Suppl B

R Gupta et al The hadron sp ectrum on a x lattice Phys Rev D

J E Mandula and M Ogilvie Ecient gauge xing via overrelaxation Phys

Lett B

I Wetzorke Quarks Diquarks und Hadronen in der Gitter QCD Diploma

Thesis Bielefeld

M Hess F Karsch E Laermann and I Wetzorke Diquark Masses from Lattice

QCD Phys Rev D heplat

C Michael and A McKerrell Fitting correlated hadron mass sp ectrum data

Phys Rev D heplat

J F Donoghue and K S Sateesh Diquark clusters in the quark gluon plasma

Phys Rev D

L Y Glozman and D O Riska The sp ectrum of the nucleons and the strange

hyp erons and chiral dynamics Phys Rept hepph

F E Close An Intro duction to Quarks and Partons Academic Press London

R L Jae Color spin and avordep endent forces in quantum

chromo dynamics hepph

E Golowich and T Sotirelis Oalphas mass contributions to the H dibaryon

in a truncated bag mo del Phys Rev D

Bibliography

CPPACS Collab oration A A Khan et al Light Hadron Sp ectroscopy with

Two Flavors of Dynamical Quarks on the Lattice heplat

R Gupta D Daniel and J Grandy Bethesalp eter amplitudes and density

correlations for mesons with Wilson fermions Phys Rev D

heplat

UKQCD Collab oration P Laco ck A McKerrell C Michael I M Stopher and

P W Stephenson Ecient hadronic op erators in lattice gauge theory Phys

Rev D heplat

APE Collab oration M Albanese et al Glueball masses and string tension in

lattice QCD Phys Lett B

S Aoki Unquenched QCD simulation results Nucl Phys Proc Suppl

heplat

G Boyd S Gupta F Karsch and E Laermann Spatial and temp oral hadron

correlators b elow and ab ove the chiral phase transition Z Phys C

heplat

R Rapp and J Wambach Chiral symmetry restoration and dileptons in

relativistic heavyion collisions Adv Nucl Phys hepph

H Satz Colour deconnement in nuclear collisions Rept Prog Phys

E Braaten and R D Pisarski Soft amplitudes in hot gauge theories A general

analysis Nucl Phys B

F Karsch M G Mustafa and M H Thoma Finite temp erature meson

correlation functions in HTL approximation Phys Lett B

hepph

M C Chu J M Grandy S Huang and J W Negele Correlation functions of

hadron currents in the QCD vacuum calculated in lattice QCD Phys Rev D

heplat

D B Leinweb er Testing QCD sum rule techniques on the lattice Phys Rev

D nuclth

C Allton and S Capitani Study of lattice correlation functions at small times

using the QCD sum rules continuum mo del Nucl Phys B

heplat

N Wu The Maximum Entropy Metho d SpringerVerlag Berlin p

S F Gull Developments in Maximum Entropy Data Analysis in Maximum

Entropy and Bayesian Methods Kluwer Academic Publishers London

Bibliography

M Oevers C Davies and J Shigemitsu Towards the application of the

maximum entropy metho d to nite temp erature Upsilon sp ectroscopy Nucl

Phys Proc Suppl heplat

CPPACS Collab oration T Yamazaki et al Sp ectral function and excited

states in lattice QCD with maximum entropy metho d heplat

Y Nakahara M Asakawa and T Hatsuda Hadronic sp ectral functions in lattice

QCD Phys Rev D heplat

J Skilling Classic Maximum Entropy in Maximum Entropy and Bayesian

Methods Kluwer Academic Publishers London

R K Bryan Maximum entropy analysis of oversampled data problems Eur

Biophys J

L J Reinders H Rubinstein and S Yazaki Hadron prop erties from QCD sum

rules Phys Rept

I Wetzorke and F Karsch Testing MEM with diquark and thermal meson

correlation functions Proceedings of the International Workshop on Strong and

Electroweak Matter Edt CP KorthalsAltes World Scientic

p heplat

W Florkowski and B L Friman Spatial dep endence of meson correlation

functions at high temp erature Z Phys A

F Karsch E Laermann P Petreczky S Stickan and I Wetzorke Finite

Temp erature Meson Correlation Functions in preparation

T Hatsuda Sp ectral change of hadrons and chiral symmetry hepph

ockeler et al Scaling of nonp erturbatively Oa improved Wilson fermions M G

Hadron sp ectrum quark masses and decay constants Phys Rev D

heplat

ockeler et al Perturbative renormalisation of bilinear quark and gluon M G

op erators Nucl Phys Proc Suppl heplat

ockeler et al Nonp erturbative renormalisation of comp osite op erators in M G

lattice QCD Nucl Phys B heplat

T DeGrand Short distance current correlators Comparing lattice simulations to

the instanton liquid heplat

afer and E V Shuryak Hadronic correlation functions in the interacting T Sch

instanton liquid Phys Rev D hepph

E V Shuryak Correlation functions in the QCD vacuum Rev Mod Phys

Bibliography

SESAM Collab oration T Struckmann et al Flavor singlet pseudoscalar masses

in Nf QCD Phys Rev D heplat

UKQCD Collab oration C McNeile and C Michael The eta and eta mesons in

QCD Phys Lett B heplat

P Petreczky et al Temp oral quark and gluon propagators measuring the

quasiparticle masses to appear in the proceedings of Lattice

K Langfeld H Reinhardt and J Gattnar Gluon propagators and quark

connement hepph

Acknowledgements

Meinem Doktorvater Prof Dr Frithjof Karsch danke ich fur die hervorragende Betreuung

ahrend der vier Jahre wissenschaftlicher Forschung die zu meiner Diplomarb eit und w

ochte ich schlielich zur vorliegenden Dissertation gefuhrt hab en Besonders hervorheb en m

andige Diskussionsb ereitschaft und die Anregungen und Ideen die zum Gelin dab ei seine st

ochte ich Prof Dr Edwin Laermann fur gen dieser Arb eit b eigetragen hab en Auerdem m

seine Bereitwilligkeit danken sich jederzeit mit oenen Fragen auseinanderzusetzen und

auftretenden Problemen auf den Grund zu gehen

Weiterhin danke ich Peter Petreczky und insb esondere Sven Stickan fur die optimale

Zusammenarb eit im Rahmen unseres Pro jektes Finite Temp erature Meson Correlation

Functions Darub er hinaus sei den vielen fruheren und heutigen Diplomanden und Dok

toranden der Arb eitsgrupp e gedankt die die Mittags und Kaeepausen mit physikalischen

on an und nichtphysikalischen Diskussionen b ereichert hab en Nicht zuletzt ein Dankesch

arinnen Gundrun Eickmeyer und Susi von Reder die eb enfalls zur angenehmen die Sektret

are b eigetragen hab en Arb eitsatmosph

Abschlieend geht ein b esonderer Dank an meine Eltern und meinen Bruder fur die

ahrend meines gesamten Studiums Unterstutzung und Motivation w