field theoretical modelling of the qcd phase diagram · 2010-01-28 · dependency is one focus in...

Field theoretical modelling of the

QCD phase diagram

Diploma Thesisby

Simon Roßner

May, 2006

Technische Universitat Munchen

Physik-Departement

T39 (Prof. Dr. Wolfram Weise)

Abstract

We investigated the confinement-deconfinement phase transition of quark matter. This is accom-plished by an NJL-type model: the Polyakov loop extended NJL-model known as PNJL-model.This model enables us to explore the region of low temperatures up to approximately two timesthe deconfinement temperature, that are not accessible to perturbative QCD. This model com-bines features of spontaneous chiral symmetry breaking in two flavors and confinement. Theconfinement is built into the model on the basis of an effective Polyakov loop potential. Forthis effective potential a Ginzburg-Landau type ansatz is used. Pure gluonic lattice calculationsprovide the input for this ansatz. To extend the applicability of the PNJL-model to moderatequark chemical potentials color superconducting diquark degrees of freedom (2SC condensates)are explicitly included into this PNJL-model. We mainly consider models with scalar couplingterms as the vector channels do not contribute to pressure and quark density to an extend, thatexceeds the current accuracy of the model. This enables us to explore the QCD phase structureincluding confinement in regions not accessible to prevalent lattice calculations.

In a first approach the model is treated in a self consistent way in mean field theory. Meanfield theory however puts additional constraints on the field configurations. The general problemarising in mean field approximations is related to complex actions and the fermion sign problemencountered in lattice calculations at finite chemical potentials. To release these constraintsthe model has to be treated in an less crude approximation. This improved treatment allowsfor investigations on the difference of the thermal expectation values of Polyakov loop and itscomplex conjugate 〈Φ∗ − Φ〉.

The result of both mean field calculations and calculations in the improved approximationscheme are confronted with lattice calculations. The comparison suggests that the effectivegluonic part of the model is not yet treated in an appropriate way. To reach better agreement anew ansatz for the loop effective potential is chosen. This ansatz is based on group theoreticalconsiderations of the gauge group of QCD SU(3)C. First calculations using this new ansatzshow better agreement with lattice data. This suggests that the group structure of the gaugegroup is of major importance for gluon dynamics. The tests with the PNJL model includingquarks could lead to a better understanding of the gluon behavior.

Contents

1 Introduction 5

2 Basics of Quantum Chromo Dynamics 7

2.1 The QCD Lagrangian and its symmetries . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Perturbative regime of QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Non-perturbative aspects of QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Lattice calculations of QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 The QCD phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Nambu and Jona-Lasinio models 15

3.1 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 From quarks to mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.1 Self energies in Hartree-Fock approximation . . . . . . . . . . . . . . . . . 16

3.2.2 Interaction terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.3 Random Phase Approximation . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.4 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.5 Parameter fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 Color superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.1 The two flavor case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.2 The three flavor case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 Path integral approach and thermodynamics of NJL-models . . . . . . . . . . . . 29

3.4.1 Consistency of the condensates . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4.2 Bosonization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4.3 Nambu-Gor’kov space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4.4 The Wick rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4.5 The Matsubara formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4.6 Evaluation of Matsubara sums . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4.7 Self consistency equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.8 Sets of condensates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 The PNJL-model in mean field approximation 43

4.1 Gluon dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1.1 Polyakov loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1.2 The Polyakov loop model . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Incorporating gluon dynamics into a PNJL-model . . . . . . . . . . . . . . . . . . 50

4.2.1 Coupling quarks and Polyakov loop . . . . . . . . . . . . . . . . . . . . . 51

4.2.2 Effect of the loop on symmetries and its interplay with condensates . . . 51

4.2.3 PNJL-models without diquarks . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.4 PNJL-models including scalar diquarks . . . . . . . . . . . . . . . . . . . 55

4.2.5 PNJL-models including scalar and vector diquarks . . . . . . . . . . . . . 56

2

CONTENTS 3

4.3 Complex Euclidean actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.3.1 Origin of complex actions . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.3.2 Impact of a complex action on mean field approximation . . . . . . . . . 59

4.4 Numerical results in mean field approximation . . . . . . . . . . . . . . . . . . . 604.4.1 Fields and Polyakov loop . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.4.2 PNJL model predicitons on the QCD phase diagram . . . . . . . . . . . 644.4.3 Comparing PNJL results with lattice QCD results . . . . . . . . . . . . . 70

5 The PNJL-model beyond mean field approximation 75

5.1 Shortcomings of mean field theory and their improvement . . . . . . . . . . . . . 755.2 Second order corrections to the mean field potential . . . . . . . . . . . . . . . . 76

5.2.1 Thermodynamic expectation values . . . . . . . . . . . . . . . . . . . . . . 765.2.2 Thermal expectation values as corrections to mean field values . . . . . . 775.2.3 Correction to the mean field thermodynamic potential . . . . . . . . . . . 795.2.4 Corrections to the self consistency equations . . . . . . . . . . . . . . . . 805.2.5 Aspects of calculating corrections numerically . . . . . . . . . . . . . . . . 81

5.3 Numerical results approximating the action to second order . . . . . . . . . . . . 825.3.1 Fields and Polyakov loop . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.3.2 The phase diagram beyond mean field approximation . . . . . . . . . . . 845.3.3 Comparison with results from lattice QCD . . . . . . . . . . . . . . . . . 84

6 Summary 89

6.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

A Conventions 92

B Fierz Transformations 94

B.1 Fierz identities for Dirac matrices and elements of SU(N). . . . . . . . . . . . . . 94B.1.1 Dirac structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94B.1.2 SU(N) structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

B.2 Fierz transformation of the color current interaction . . . . . . . . . . . . . . . . 96

C Evaluation of Feynman graphs 98

C.1 Derivation of Feynman rules for NJL-models . . . . . . . . . . . . . . . . . . . . 98C.2 Derivation of the gap equation via the Dyson equation . . . . . . . . . . . . . . . 98

Chapter 1

Introduction

The high momentum region of the strong interaction has been studied in both experiment andtheory. Experiments at SLAC, CERN, DESY and RHIC have provided a large amount ofexperimental data. The prevailing theory of the strong interaction Quantum-Chromo-Dynamics(QCD) was able to explain these experimental results to remarkable accuracy, confirming thediscovery of asymptotic freedom [GW73, Pol73] and the systematics of Q2 evolution in QCD.

The physics in the low-energy region of QCD is qualitatively different and subject of ongoingintensive research activities. Due to the running of the coupling in QCD the theory cannot beexplored in the low energy regions with the perturbative methods that have been enormouslysuccessful at high momenta. At very low energies QCD shows a large variety of bound states.All these states are color singlets. That is, the quantity dominating the high energy regiondoes not enter the stage in the low energy region at all. The mechanism that hides color iscalled confinement [Sus79, MS81a, Pol78]. The other important feature of low energy QCD isthe observation that there is an energy gap between the pseudoscalar mesons and the scalarmesons. This is surprising as the QCD Lagrangian is chirally symmetric in the limit of mass-less quarks. The experimental findings can however be described by a spontaneous symmetrybreaking mechanism. I. e. the pseudoscalar mesons are identified with Goldstone modes.

This work is trying to contribute to the exploration of the intermediate momentum regimewhere the low energy structures dissolve and their constituents reappear in the better knownregime at high energies. The focus here is put on the thermodynamics of QCD. Low energiesare identified with low temperatures, where the system is in the hadronic phase we are livingin. At high energies corresponding to high temperatures the system is expected to be in thequark-gluon-plasma (QGP) state. In addition, to the temperature the quark chemical potentialdependency is one focus in this report, i. e. the phase diagram in wide ranges of the temperatureand chemical potential plane is explored here by means of a field theoretical model. There areseveral other predictions on the phase diagram of QCD motivated by results of different sources[Raj99, Alf03, Sch03]. One important source of information are numeric lattice calculations.Lattice calculations can produce correct solution to the QCD field equations, they do howevernot provide deeper understanding of the machinery at work in QCD. The lattice data still needto be interpreted to find the mechanisms and principles that lead to these numeric results.

In this setting we want to incorporate spontaneous chiral symmetry breaking and confinementat low temperatures into a unified field theoretical model. At high temperatures the model shalldescribe the QGP. As input for the model one has to rely on the numeric lattice calculationas there is no direct comparison with experimental data available yet. But from the latticeside only the pieces that are well under control are taken as input. Namely these are puregluonic calculations, that are numerically much cheaper than calculations including fermions.In addition, these calculations do not suffer from artifacts like fermion doubling that have to befixed by Wilson action or staggered fermion action or other methods. At non-zero quark chemical

5

6 Chapter 1. Introduction

potential strongly reduced numeric convergence still complicates systematic calculations on thelattice.1 Pure gluonic calculations do not only minimize statistical but also systematic errors.

The model constructed here is based on the Nambu and Jona-Lasinio model (NJL model)[NJL61a, NJL61b]. This model was built such, that it implements spontaneous symmetry break-ing. Even thought in the original version by Nambu and Jona-Lasinio the degrees of freedomwere not identified with quarks but with nucleons, this model is still used today. Nowadays thedegrees of freedom are interpreted in terms of quarks. The spontaneous symmetry breaking pro-vided by this model in this interpretation is the chiral symmetry breaking [VW91, Kle92, HK94].The NJL model does however not confine.

The effective potential arising from confinement in the pure gluonic sector can be describedby Ginzburg-Landau type approaches [KSS82, SY82, GK84]. The order parameter needed forthe Ginzburg-Landau potentials is the Polyakov loop which is a timelike Wilson line in Euclideanspace-time. This effective theory can be coupled to the NJL model [MMO02, MST04, MRAS04].In this work we follow the method discussed in [Fuk04].

Even though this model combines confinement and spontaneous chiral symmetry breakingin qualitative correct way, the quantitative results still show discrepancies with results found inlattice QCD [KZ05, A+05]. This comparison is also part of this report. To improve on thesediscrepancies a new form for the effective loop potentential is chosen.

This thesis is organized as follows: In Chapt. 2 the main features of QCD relevant for thefollowing considerations are outlined. Chapt. 3 gives a brief review of the methods applied inNJL-models.

In Chapt. 4 Polyakov loop models are presented and connected to the quark dynamics. Theso-called PNJL-model is presented. In addition, to previous calculations with this PNJL-model,diquark degrees of freedom are explicitly introduced. The mean field equations for this modelis solved numerically and the solutions are presented. Properies of the QCD phase diagramare predicted. With the inclusion of the diquarks this model can predict on the high quarkchemical potential region of the phase diagram. The results for pressure difference and quarknumber density are viewed in comparison with lattice results. Shortcomings are discussed andimprovements to the Polyakov loop part of the model are outlined. Finally some problemsarising in mean field approximation are discussed.

In Chapt. 5 the discussion on the shortcoming in mean field approximation discovered inChapt. 4 is resumed and a solution is proposed. This new approximation for the thermodynamicpotential is worked out. In the sections thereafter the changes in comparison with the meanfield approach are discussed.

This thesis ends with a short summary in Chapt. 6. In this chapter a conclusions of the workdone is drawn and a brief outlook on future work is given.

1The problem is founded in complex weights and known as the fermion sign problem.

Chapter 2

Basics of

Quantum Chromo Dynamics

In this chapter a brief introduction to Quantum-Chromo-Dynamics (QCD) is given [TW]. QCDis the theory that is most likely to describe the strong interaction in a correct way. QCD is aquantized gauge field theory based on simple principles. Even though emanating from simpleand clear principles the solution of the field equations causes major problems. This is due to thenon-Abelian structure involved in QCD. This property is most likely at the origin of confinement.However there is yet no strict prove that QCD confines. I. e. that QCD is a proper descriptionof nature that only seems to implement color singlets. There are several approaches to enlightenthe mechanisms of QCD.

This work focuses on a model. This so-called PNJL-model adopts important symmetries ofQCD and incorporates other effective theories. The PNJL-model therefore is no theory on itsown but a contribution to the understanding of the dynamics of QCD.

2.1 The QCD Lagrangian and its symmetries

The central starting point of quantized gauge field theories is the Lagrangian. For QCD theLagrangian is

LQCD = Lquark0 + LI + Lglue

0 , (2.1)

where the different contributions are defined by

Lquark0 = ψ [iγµ∂µ − m]ψ LI = ψγµAµψ

Lglue0 = − 1

4g2Gµνa Gaµν = − 1

4g2trcG

µνGµν .(2.2)

In the second and third term the gauge fields (gluon fields) where redefined by Aµ → g Aµ. Hereg is the universal coupling constant of the theory. The mass operator m indicates that there aredifferent masses for the different quark flavors. In many cases it is handy to use the covariantderivative defined as

Dµ = ∂µ − iAµ = ∂µ − iAµata = Dµa ta. (2.3)

With this definition the gluonic field strength tensor Gµν can be written as

Gµν = Gµνa ta =i

g[Dµ, Dν ] =

i

g[∂µ, ∂ν ]︸︷︷︸

=0

+1

g

(∂µA

aν − ∂νAaµ

)ta +

1

gfabcAbµA

aν tc. (2.4)

7

8 Chapter 2. Basics of Quantum Chromo Dynamics

Above the normalized generators of SU(3)C were used. They are defined as ta = 12λa, where λa

are the Gell-Mann matrices. The ta are normalized such that trc [ta, tb] = 12δab. The numbers

fabc are the structure constants of the gauge group. In QCD this gauge group is SU(3)C. Thestructure constants are defined by

[ta, tb] = ifabctc. (2.5)

This non-Abelian structure is the origin of three and four-gluon vertices in perturbation theory,that are not present in Abelian gauge theories like quantum electro dynamics (QED). This isconnected to the observation that gluons carry color charge unlike photons, that are electricallyneutral. This self interaction of the gauge part of the theory is the source of asymptotic freedomand confinement discussed in the following sections.

As symmetries play a central role in quantum field theories the symmetries of the QCDLagrangian are discussed. The most important symmetry is of course the gauge symmetry. InQCD the gauge group is a local SU(3). This Lie group is generated by the color degrees offreedom.

In contrast to the gauge symmetry, all other symmetries of the QCD Lagrangian are globalsymmetries. This discussion starts in the case of vanishing current quark masses. In this settingthe right and the left handed quark fields completely decouple. There exists a symmetry forthe right handed and the left handed quark fields. The right and left handed quark fields of themassless QCD Lagrangian are representations of U(N)R and U(N)L (see first line of Fig. 2.1).Only the three lightest flavors up (u), down (d) and strange (s) are considered here, as they arethe only flavor degrees of freedom accessable at the temperatures of interest here.

The group U(N) can be decomposed into SU(N)×U(1). The decomposition into right andleft handed fields is not unique. By taking linearly independent combinations of the generators,the symmetries of left and right handed particles can be transformed. By adding (subtracting)the generators of the symmetry groups of left and right handed quarks one obtains new genera-tors. These belong to the vector (axial vector) symmetry group. The axial U(1)A is anomalouslybroken. Anomalous effects are effects appearing on the quantum level only. I. e. the solutionsof the theory on the quantum level do no longer respect this symmetry due to quantum effects.

The symmetry SU(3)L×SU(3)R still is a good symmetry at the level of the QCD Lagrangianat vanishing quark mass. On the basis of the QCD Lagrangian these groups are on equalfooting. This equivalence of right and left handed quarks is however not observed in nature.We observe an octet1 of light pseudoscalar mesons. The corresponding scalar mesons havemuch higher masses. The difference in the masses is usually called mass gap. Nambu andJona-Lasinio explained this mass gap with the mechanism of spontaneous symmetry breaking[NJL61a]. In the model by Nambu and Jona-Lasinio, the symmetry is not broken on the levelof the Lagrangian but only on the level of the solutions to the field equations of the stronginteraction2. The symmetry broken is the axial part of SU(3)L×SU(3)R, while the vector partof the symmetry is respected by the solutions to the field equations. The chiral condensategenerated by spontaneous chiral symmetry breaking acts as a quark mass term. This lead tothe interpretation of the pseudoscalar mesons as massless Goldstone bosons, while the scalarmesons are regular massive particles.

Non-vanishing, degenerate current quark masses do not break the vector part of the chiralsymmetry.3 The vector symmetry can be identified with the symmetry of the three light flavors.

1The eight pseudoscalar mesons correspond to the N2 − 1 generators of SU(N) for N = 3.2QCD was not known in 1961.3The axial part of SU(3)L×SU(3)R is broken by non-vanishing, degenerate current quark masses (mu = md =

ms 6= 0). On the basis of the QCD-Lagrangian including degenerate current quark masses the axial symmetry isbroken in the Lagrangian already. From this perspective the spontaneous chiral symmetry breaking happens onthe background of a Lagrangian with reduced symmetry. This is most likely the reason why the axial symmetryis broken spontaneously and not the vector symmetry.

2.2. Perturbative regime of QCD 9

mu = md = ms = 0 SU(3)C × U(3)L

decomposition in subgroupsEE

EEE

""EE

EEE

× U(3)R

!!CC

CCCC

CCCC

CC

SU(3)C × SU(3)L × U(1)L ×

anomalous breaking of U(1)AEE

EEE

""EE

EEE

SU(3)R × U(1)R

SU(3)C × SU(3)L

spontaneous chiral symmetry breaking of SU(3)A

× SU(3)R

yyyy

y

||yyyy

y

× U(1)V

SU(3)C × SU(3)V

introduction of current quark masses mu = md = ms

× U(1)V

mu = md = ms 6= 0 SU(3)C × SU(3)f

non-degenerate current quark masses mu 6= md 6= ms

× U(1)V ⇒ deg. meson octet

mu 6= md 6= ms 6= 0 SU(3)C ×

SU(3)f × U(1)V ⇒ non-deg. meson octet

Figure 2.1: The symmetries of the massless QCD-Lagrangian (top) and the symmetry breakingpattern generated by the axial anomaly, spontaneous chiral symmetry breaking and explicitsymmetry breaking by non-vanishing current quark masses.

The vectorial group SU(3)V is therefore often called flavor symmetry and denoted as SU(3)f .When the current quark masses of the different flavors are non-vanishing and different from eachother, this flavor symmetry is explicitly broken. This explicit breaking of the flavor symmetryis responsable for the non-degenerate masses in the pseudoscalar meson octet. The explicitlysymmetry breaking mass terms in the QCD Lagrangian can be treated in perturbation theory.The pseudoscalar meson octet is in the case of vanishing current quark masses an octet ofGoldstone Bosons. As the current quark masses are small, the Goldstone Boson propertiesare only perturbed slightly. The pseudoscalar mesons are therefore often referred to as pseudoGoldstone bosons.

The only true symmetries that remain are the gauge symmetry SU(3)C and the vectorsymmetry U(1)V. According to Noether’s theorem the conserved charge generated by thisU(1)V is the Baryon number.

2.2 Perturbative regime of QCD

Even though perturbative calculations are not in the center of this report a short overview ofthis way of treating QCD is given. These approaches have been very successful in the regionwell above ΛQCD.

Perturbative methods expand around some free theory with known solutions. A handyapproach to this expansion is the path integral formalism using generating functionals [IZ, PS,TW]. The generating functional of the full theory is approximated by the generating functionalof the free theory corrected by a finite amount of corrections, calculated from the part of thefull generating functional not included in the free generating functional. The full generating


functional of QCD is

Z =

∫DA

∫Dψ

∫Dψ exp

i

∫d4xLQCD + η(x)ψ(x) + ¯ψ(x)η(x) +Aaµ(x)J

aµ(x)

. (2.6)

The generating functional is a functional of the external sources η, η and Jaµ. For explicitperturbative calculations one has to explicitly choose a specific gauge. One way to ensure this,is to break the gauge symmetry in the generating functional explicitly by a so-called gauge fixingterm.4 A wide spread way to implement the gauge fixing is the use of Faddeev-Popov ghosts[FP67]. The Lagrangian is then extended by a Faddeev-Popov contribution responsable for thepropagation of the ghost fields and a gauge fixing term that determines which gauge is usedthroughout the calculation.

The free parts of the generating functional that is expanded about are

Zquark0 (η, η) = exp

−i∫

d4xd4y η SF(x, y) η

, (2.7)

Zgluon0 (Jaµ) = exp

− i

2

∫d4xd4y JaµDab

µν(x, y)Jbν

, (2.8)

Zghost0 (ξ, ξ⋆) = exp

−i∫

d4xd4y ξa⋆Dabµν(x, y) ξ

b

, (2.9)

where SF(x, y) is the Feynman propagator andDabµν(x, y) is the gluon propagator. The generating

functional can now be rewritten as

Z = exp

i

∫d4xLint

(−i δ

δJaµ,−i δ

δξa⋆,−i δ

δξa,−i δ

δη,−i δ

δη

)Zquark

0 Zgluon0 Zghost

0 , (2.10)

where ξa⋆ and ξa are the external sources for the ghost fields. The first term can now beexpanded in its arguments yielding the Feynman rules, that are not discussed here.

When going through the renormalization procedure one finds the renormalization groupequation. The solution to this equation in QCD in one loop order is

αS(Q2) =

αS(µ2)

1 + αS(µ2)4π β0 log Q2

µ2

with β0 = 11− 3

4Nf , (2.11)

where Nf is the number of active flavors, i. e. those flavors for which the current quark massm0 / Q2. This dependence of the coupling constant (and the coupling constant) on the mo-mentum transfer Q2 is called running. It determines the change in the model parameters whenchanging the momentum scale that is the basis for the definition of the free generating func-tional. The running is the origin of the asymptotic freedom of QCD [GW73, Pol73]. I. e. the fullgenerating functional at high momentum scales behaves asymptotically like the free generatingfunctional. In the infinite momentum limit only a free theory is left over. Asymptotic freedomimplies that the coupling gets weaker for larger momentum scales. For smaller momenta thecoupling will get ever so larger. This phenomenon is directly coupled to the negative β-functionoriginating in gauge symmetry SU(3)C. The perturbative expansion breaks down once the ex-

pansion coefficient αS = g2

4π becomes large. In Eqn. 2.11 this will definitely be the case once thedenominator vanishes. This happens for

√Q2 = ΛQCD ⇒ ΛQCD = µ exp

− 2π

β0αS(µ2)

. (2.12)

Depending on the calculation and the regularization scheme one finds ΛQCD ≈ 0.3 GeV (inMS-scheme).

4As all gauges are equivalent the generating functional (2.6) contains the same content in infinitely many”copies” the so-called Gribov copies [Gri78].

2.3. Non-perturbative aspects of QCD 11

2.3 Non-perturbative aspects of QCD

Running the coupling constant from the perturbative high momentum region down to low mo-menta implies rising αS. Near the momentum scale ΛQCD ≈ 0.3 GeV the renormalization groupflow of QCD predicts a coupling constant αS larger than one. Perturbative techniques with thebasic fields of QCD (quarks and gluons, ghosts) can only produce reliable results well above thisscale. The regime below Q2 / 1 GeV2 is therefore called the non-perturbative regime.

There are two mechanisms that govern the non-perturbative regime, spontaneous chiral sym-metry breaking and confinement.5 Spontaneous chiral symmetry breaking is the phenomenonthat the solutions to the field equations do no longer respect all the of the original QCD La-grangian.

Spontaneous symmetry breaking is realized if there exists a non-vanishing vacuum expecta-tion value

〈0| [Q(t), φ(0)] |0〉 = v 6= 0, (2.13)

where φ(x) is some elementary or composite field and Q(t) is the charge operator belongingto some conserved current originating from a continuous symmetry. Goldstone’s theorem thenstates that there exists a massless Goldstone boson. In the case of spontaneous chiral symmetrybreaking with three massless quark flavors the charge operators in (2.13) belong to the conservedcurrents generated by the axial part of the flavor symmetry. As the Lagrangian of QCD in themassless case is parity invariant it is a priori not clear why it is the axial part of SU(3)L×SU(3)Rthat is broken and not the vector part of this symmetry. It is the experiment that tells us thatnature prefers to break the axial symmetry. The explicit form related to (2.13) in the case ofspontaneous chiral symmetry breaking is

〈0|jµ5a (x)πb(p)〉 = −ipµfπδabeip·x, (2.14)

where fπ is the pion decay constant and jµ5a = ψγµγ5ta is the axial vector current, and ta are

the generators of the SU(3)f flavor group.It can be shown that the eight non-vanishing vacuum expectation values 2.14 are consistent

with a non-vanishing scalar quark condensate

〈ψψ〉 = −iTr limz→x+

[SF(x, z)− SF(x, z)

]≈ − (0.25 GeV)3 , (2.15)

where SF is the Feynman propagator in the non-perturbative vacuum while SF is the Feynmanpropagator in the perturbative vacuum.6

The real QCD Lagrangian has a non-vanishing mass term i. e. chiral symmetry is not onlybroken spontaneously but in addition explicitly. This small explicit breaking of the axial sym-metry is most likely responsable for the fact that the spontaneous symmetry breaking appearsin the axial symmetry and not in the unbroken vector part of flavor symmetry. Explicit sym-metry breaking dilutes the strong statement of Goldstone’s theorem that there are masslessdegrees of freedom. Explicit symmetry breaking is at the origin for the finite masses of thepseudoscalar mesons. Even though pseudo Goldstone degrees of freedom in presence of explicitsymmetry breaking term are no longer exact Goldstone degrees of freedom, we still refer to themas Goldstone modes.

There is much less known about the mechanism of the other non-perturbative effect, namelyconfinement. Confinement names the phenomenon that all color charges are completely screened

5The anomalous breaking of U(1)A does play a role in the non-perturbative regime. It is however usuallycalculated in perturbation theory. Anomalous breaking is not limited to the non-perturbative regime and thereforenot a downright non-perturbative effect.

6From the existence of a non-vanishing chiral condensate 〈ψψ〉 it cannot be inferred, that it is the SU(3)A-symmetry, that is broken spontaneously.


in the far away region of the color source. In an non-physical system with infinitely heavy quarksconfinement means that a system containing a static color source and a static color sink at infiniteseparation has an infinite free energy. More precisely, the free energy of a system with a staticcolor source and a static color sink is proportional to the separation of source and sink. Thequotient of free energy change and distance is usually called string tension. If one thinks abouta flux tube connection source and sink the proportionality of energy and distance implies thatin the large volume limit the field strength across the tube is independent of the distance in thedirection connecting source and sink. The proportionality of energy and distance does not allowthe flux to spread over infinite space. The existence of a flux tube of finite diameter limits theinfluence of color source and sink to a small space-time region.

2.4 Lattice calculations of QCD

Starting from the path integral of QCD one could ask, if it is possible to perform these inte-grations. Analytically there is no way known to the present day. The effort using numericalmethods is often referred to as lattice QCD. As the path integral is an infinite-dimensional in-tegral, for numeric calculations phase space has to be limited to a finite subspace. This is doneby discretizing space-time and by periodically (or antiperiodically) continuing space-time. Afour-dimensional hypercube replaces the infinite space-time of QCD.

The infinite-dimensional path integral is replaced by a Nx×Ny×Nz×Nt×Ndof -dimensionalintegral, where Ndof is the number of degrees of freedom at a single point in space-time. Alreadysmall spatial and temporal extensions lead to an enormously high-dimensional integral. For suchhigh-dimensional numeric integrals Monte-Carlo methods are suited best. The convergence ofthese methods of integration crucially depends on the integrand. Convergence gets very slow,if the integrand is a complex quantity. To obtain a real measure these integrations are carriedout in Euclidean space-time.

The first calculations were done for pure gluonic systems ignoring fermions. Besides savingadditional degrees of freedom for these calculations the fermionic part of the path integral neednot be evaluated. The fermionic part, often called the fermion determinant7, is numericallyvery expensive. The next generation of calculations were the so-called quenched approximationswhere the fermion determinant is approximated by a constant, that does not depend on thegluonic fields. In contrast to these quenched calculations dynamic calculations do respect thedependence of the fermion determinant on the gluon fields. These calculations do however need alot more computing power. Within the dynamic calculations one can still distinguish connectedand disconnected calculations. Connected calculations do not respect fermion loops withoutconnection to real particles. Disconnected calculations however do take vacuum fluctuationsinto account.

The inclusion of quarks into the calculations causes even more problems. The simplestdiscretized implementations of fermions suffer from fermion doubling. That is, the dispersionrelation for discretized fermions has several degenerate minima. One is in the center of theBrillouin zone and several others are at the borders of the zone. To remedy this, additionalterms can be added to the lattice action, as done in the so-called Wilson action. Other proce-dures introduce an additional quantum number to distinguish the different minima (staggeredfermions).

The enormous numerical costs limit the number of points in space-time dramatically. As aconsequence the lattice spacings (the distance of two neighboring points) cannot be chosen too

7The Path integral is the functional determinant of the inverse fermion propagator. In discretized space-time the functional determinant is transformed into an ordinary determinant. When the fermionic degrees offreedom are integrated out, the resulting fermion determinant is still inside the integration of the gluonic degreesof freedom. It cannot be separated, as the fermion determinant depends on the gluonic degrees of freedom.

2.5. The QCD phase diagram 13

small, and the overall size of the lattice cannot be chosen too big. This leads to sizeable effects,namely discretization errors and finite volume effects. As the grid cannot be chosen to be veryfine, the current quark masses are typically increased above their physical values. This reducesthe relative energy gap between nucleons and pions. These unphysical quark masses howevercause systematic errors of unknown size. In addition, the discretized version of QCD is heavilyreduced in symmetry in comparison to full QCD.

In thermodynamic applications lattice calculations face big problems once the quark chemicalpotential is non-vanishing. Calculations at non-zero chemical potential are numerically verycostly as for µ 6= 0 the Euclidean action assumes in general complex values. This leads to anon-positive definite fermion determinant, which in turn spoils the convergence of the availableMonte-Carlo integration methods.8 One way to obtain information about the non-zero chemicalpotential region is to expand the thermodynamic potential obtained at µ = 0 [A+05]. The rangeof such expansions is limited to µ

T / 1. A second method uses reweighting techniques. Here in afirst step an ensemble at µ = 0 on the transition is created according to their Boltzmann weights.In a second step this ensemble is carried along a ”line of constant physics” [FK02, FK04] to apoint at non-vanishing chemical potential µ and possibly of different temperature. The authorsof [FK02, FK04] claim better convergence through this procedure. A third approach analyticallycontinues the action in the imaginary chemical potential direction [dFP02, dFP03, DL03], toretain a positive real measure once µ 6= 0.

2.5 The QCD phase diagram

Typically not all dimensions in parameter space are discussed. For example quite often thequark chemical potential is chosen equal for all flavors. This need in general not be the case.These non-symmetric cases could be of interest for astrophysical applications. In this report werefer to the phase diagram in the temperature and chemical potential plane.

The phase diagram of QCD is still quite unknown. But some statements about QCD phasestructure can be formulated from universality principles. E. g. the universality of the phasetransition leads to the result that the order of the chiral phase transition depends on the numberof flavors and on their current quark masses [PW84]. The quantitative discussion of the order ofthe phase transition however cannot be decided by general arguments. As part of this thesis, themodel presented here, the PNJL model, predicts on the order and the position of the differentphase transitions. Other calculations are discussed in [Raj99, Alf03, Sch03].

Lattice QCD calculations show a first order phase transition in a pure gluonic system atTc ≈ 0.27 GeV [KLP01]. Such a system is of course unphysical. When quarks are added to thelattice QCD calculation this transition temperature is decreased to Tc ≈ 0.17 GeV in two flavorcalculations [KLP01, dFP02, dFP03] and to Tc ≈ 0.154 GeV in a 2+1 flavor calculation [FK04].The transition including quarks is predicted to be a cross-over transition. On the experimentalside heavy ion collision measurements at the high energies are expected to shed more light on thistransition, as temperatures of about 168 MeV have already been reached in Pb-Pb collistions atSPS [BMHS99], and slightly higher temperatures are reached at RHIC with about T = 174 MeV[BMMRS01].

On the chemical potential axis effective models predict a first order phase transition at acritical chemical potential value [AY89, BR99, CD99]. From the existenc of the cross-over onthe T -axis we conclude that there has to be some point in the phase diagram, where the twotransition lines come together. At this point the order of the phase transition changes, i. e. thereexists a critical point in the QCD phase diagram. There are lattice QCD predictions on this

8The Monte-Carlo integration methods use importance sampling. This method breaks down for non-positivevalues of the integrand. The complex phase causes ”interference” phenomena leading to extinctions of differentfield configurations. Such extinctions increase statistical errors.


critical point available [FK02, FK04]. For µ 6= 0 lattice QCD calculations suffer however fromconvergence problems.

There are two other points accessable to the experiment already: the vacuum at T = 0 andµ = 0 and the point where nuclear matter is realized. This second point is at a quark chemicalpotential µ = 1

3 (MN − Ebind.) ≈ 0.308 GeV. For these points it is known, that they are in thehadronic confined phase. Confinement implies chiral symmetry breaking [Sus79, Pol78].

There are many theoretical indications that a color superconducting phase must exist in theQCD phase diagram at µ > 0.308 GeV [ARW99, BR99, CD99]. Depending on the model thissuperconducting phase is in direct contact with the hadronic phase, being separated by the firstorder transition mentioned above. There could however be a quark gluon plasma (QGP) phasein between. It is not likely that there will be experiments that can reach these extreme regionsin the phase diagram. In high-energy collisions of Si with several other nuclei, the freeze-outtemperature was determined in [BMSWX95] to be T ≈ 0.12−0.14 GeV, which is still rather highcompared to the temperatures at which diquark condensation is expected. The baryon chemicalpotential on the other hand has not been high enough, it was estimated to be µB ≈ 0.54 GeV,corresponding to µq ≈ 0.18 GeV.

The QGP phase is the high temperature phase of QCD. Due to asymptotic freedom quarksshould deconfine at high momenta, that can be realized at temperatures above the transitiontemperature (Tc ≈ 0.17 GeV). How ”free” the quarks are at temperatures just above the tran-sition is still an open question. There are indications that the QGP near the transition behavesrather like a fluid than like gas or a well known electromagnetic (chemical) plasma [Shu05].

Chapter 3

Nambu and Jona-Lasinio models

The Nambu and Jona-Lasinio model was originally written down to describe the dynamics of thestrong interaction [NJL61a, NJL61b]. In 1961 when Yoichiro Nambu and Giovanni Jona-Lasinioinvented this model there was no established theory of the strong interaction, and the degreesof freedom in the model were identified with mesons and nucleons. Today the fermionic degreesof freedom are identified with quarks [VW91, Kle92, HK94]. From this historical perspective itis quite obvious that the lack of confinement in the NJL model was not considered a problem.

The NJL model was constructed in analogy to the BCS theory [BCS57] of superconductivitypublished a few years earlier. This similarity to the BCS theory makes it particularly simpleto include diquark degrees of freedom. The diquark condensate is the direct analog to Cooperpairing and the superconducting phase in BCS theory [Bub05]. While in BCS theory the phaseis electrically superconducting, the superconductivity in the NJL model is with respect to color.

3.1 Symmetries

Nambu and Jona-Lasinio models can be used, within certain limits, to mimic QCD. The symme-try structure is what mainly determines the behavior of a field theory. The gauge symmetry ofQCD is based on the group SU (3)C . It is this symmetry that is most likely responsible for theconfining properties of QCD. As the model we want to construct is a model dealing with bulkproperties of strong interacting matter, we integrate out effects on small length scales / 0.2 fm.Doing this we replace the local SU (3)C gauge symmetry by a global symmetry. The effects ofgluons are limited to the pressure the gluon gas generates and to the strength of the quark-quarkcoupling.

NJL models are designed to describe the dynamics below ΛQCD, so only the three lightflavors may significantly influence the behavior of the system. As a further simplification thenumber of flavors will be restricted here to Nf = 2. In addition, in explicit calculations isospinsymmetry will be preserved, i. e. the up-quark mass equals the down-quark mass (mu = md).The NJL Lagrangian is of the form

LNJL = ψ(/p− m

)ψ + δLNJL, (3.1)

where the mass term m is m0 1 and δLNJL is an interaction term. It is usually implied that NJLmodels have only point like interaction terms. In most cases this is even restricted to 4-fermionpoint interactions. So the generic interaction Lagrangian is of the form

δLNJL = g(ψΓaµψ

) (ψΓµaψ

). (3.2)

The standard interaction term used by Nambu and Jona-Lasinio [NJL61a, NJL61b] is

δLNJL =G

2

([ψψ]2

+[ψ iγ5 ~τ ψ

]2), (3.3)

15

16 Chapter 3. Nambu and Jona-Lasinio models

where ~τ = (τ1, τ2, τ3)T with the Pauli spin matrices τ1, τ2 and τ3.

One of the important features of QCD below ΛQCD is spontaneous chiral symmetry breaking.This symmetry breakdown is realized in NJL-models. The massless Lagrangian of NJL modelsis chirally symmetric. It may however happen — dependent on the effective coupling strengthof the model — that a quark mass is generated dynamically. This dynamically generated quarkmass breaks the chiral symmetry: SU (2)R×SU (2)L −→ SU (2)V , just as it happens in QCD.

3.2 From quarks to mesons

In this subsection we rewiev the way how the Nambu and Jona-Lasinio model with its quarkquasiparticle degrees of freedom can reproduce the meson spectrum. This works well with threeflavors [Kle92, VW91]. Here only the two flavor case is considered for simplicity. I. e. only thepion has to be reproduced in its mass. In fact pion properties will be used to fix key parametersof the model.

3.2.1 Self energies in Hartree-Fock approximation

One of the most important features about the Nambu and Jona-Lasinio model, is the sponta-neous symmetry breaking. This mechanism was already used by Bardeen, Cooper and Schrieffer[BCS57], whose work motivated the study of Nambu and Jona-Lasinio. The spontaneous sym-metry breaking mechanism is implemented into the model via a so-called gap-equation. Oneway to derive the gap-equation is to calculate the vacuum energy of the quarks with a Dysonequation (Fig. 3.1). Using the Feynman rules given in App. C.1. the graphs in this equationmay be rewritten as

M = m+ iG

∫d4q

(2π)4TrS(q) with S(p) =

(/p−M = iǫ

)−1. (3.4)

The mass M is the dressed quark mass while m is the unrenormalized or bare quark mass. Afterusing the explicit form of the propagator S(q) the gap equation reads

M = m+ 4iGNcNf

∫d4q

(2π)4M

q2 −M2 + iǫ. (3.5)

The Dyson equation (Fig. 3.1) is evaluated in App. C.2 in Hartree-Fock approximation. As weare dealing in this model with a local 4-point interaction it is possible to cast Fock or exchangeterms in the form of Hartree or direct terms. This is illustrated in Fig. 3.2. In this sense Hartreeapproximation can incorporate all types of graphs included in Hartree-Fock approximation. Theexchange terms can be added to the Lagrangian explicitly. Then the Hartree approximation willyield the same result as the Hartree-Fock calculation after a redefinition of the constants. If theparameters are determined to reproduce physical observables, this redefinition will automaticallybe included.

In the Bogoliubov-Valatin approach to the gap equation, the broken vacuum is explicitlyconstructed. The new ground state is

|Ω〉 =∏

~p

(cos θ(p) + sin θ(p)b†~pd

†−~p

)|0〉 (3.6)

in analogy to BCS theory [BCS57]. In contrast to BCS theory, where fermions are paired, herequark antiquark pairs of zero momentum and spin are added to the perturbative vacuum |0〉.The new ground state energy is evaluated and minized with respect to θ(p), where θ(p) describesthe extent to which the vacuum is polarized. The necessary condition for an extremum

∂ 〈Ω|HNJL|Ω〉∂θ(p)

= 0. (3.7)

3.2. From quarks to mesons 17

= +

Figure 3.1: Dyson equation for the calculation of the self energy in Hartree approximation.The thin arrowed lines symbolize a bare quark propagating, the thick arrowed lines symbolize adressed quark propagating. The vertex coupling strength is G (see App. C.1).

Fock

contraction−−−−−−−→to a point

contraction←−−−−−−−to a point

Hartree

Figure 3.2: If the boson mediating the interaction is contracted to a point (middle), it is nolonger necessary to distinguish the direct (Hartree) term (right) and the exchange term (left).

is equivalent to the gap equation 3.4 without exchange terms. Here HNJL is the Hamiltonian ofthe system.

Yet another way to recover the gab equation is to evaluate the effective action. This approachis presented in Sec. 3.4.

3.2.2 Interaction terms

A common starting point for the interaction part of the Lagrangian is a color current inter-action. The color current interaction can be motivated by the gluon exchange in perturbativeQCD. If the gluon propagator is approximated by a localized function, the resulting interactionterm is exactly of the NJL color current type (see Fig. 3.3). In the previous chapter it wasstated that Hartree approximation is as good as Hartree-Fock approximation. Note howeverthat this statement is meant in the sense that for 4-fermion point interactions the interactionLagrangian in Hartree-Fock approximation may be replaced by another 4-fermion point inter-action Lagrangian treated only in Hartree approximation such that the dynamics produced isequivalent. This Hartree interaction Lagrangian is in general not equal to the interaction La-grangian in Hartree-Fock approximation. So the task is to construct the interaction Lagrangianthat produces the same results in Hartree approximation as for example the color current in-teraction in Hartree-Fock approximation. This is done by explicitly calculating the direct andexchange terms of the interaction Lagrangian and adding them.

Now this recasting of the exchange terms into the form of a direct term has to be performed.One way to obtain the exchange term to a given direct interaction Lagrangian is the Fierztransformation. Let the original interaction Lagrangian be of the form in 3.2.

Lint = gΓijΓklqiqj qkql =

−gΓijΓklqiqlqkqj ≡ Lex.

gΓijΓklqiqkqlqj ≡ Lqq(3.8)

I. e. a relation is needed to interchange the indices of the gamma structure. This is provided bythe Fierz transformations.

ΓijΓkl =∑

m∈M

c(ex.)m Γmil Γmkj =

∑

n∈N

c(qq)n ΓnilΓ

nkj (3.9)


ψ

ψ

= ig0 γµta

×

= δab

[−gµν + (1− λ)

qµqνq2+iǫ

]i

q2+iǫ

approx.−−−−→in NJL

− δab ρ(q2) gµν

×

ig0 γ

νtb =

ψ

ψ

= −ρ(q2)[ψ (ig0 γ

µta)ψ] [ψ (ig0 γµt

a)ψ]

= 4π αQCD Λ−2 Θ(~q2 − Λ2)[ψγµtaψ

] [ψγµt

aψ]

Figure 3.3: One motivation for taking a color current current interaction as starting point for NJLmodels mimicking QCD dynamics. The function ρ(q2) contains the regularization procedure.

This identity can always be established, as the 16 bilinear covariants from a basis of Dirac space.

The coefficients c(ex.)m and c

(qq)n are the coordinates according to the new basis. The next step is

to add Ldir., Lex. and Lqq. Using this Lagrangian in Hartree approximation obviously includesexchange terms. More details are shown in App. B.2.

3.2.3 Random Phase Approximation

The NJL model reproduced the mass spectrum of the light mesons which were pure Goldstonebosons in the case of vanishing current quark mass. As the NJL model should model the chiralsymmetry break down at low temperatures, it also should incorporate information about theGoldstone bosons, or what is left of them after explicitly breaking the chiral symmetry. We builda meson from a quark and an antiquark demanding that this pair may propagate together. ABethe-Salpeter equation can be used to evaluate the mass of such a compound state (Fig. 3.4).First the lefthand side in Fig. 3.4 is equated to the first term in the second line. This is done inHartree approximation1. The vertex connecting a pion to quark and antiquark Γπqq has to beproportional to γ5 in Dirac and τa in flavor space as the pion is a pseudoscalar particle.

Γaπqq = iG γ5τa

∫d4q

(2π)4Tr[iγ5τbSF(p− q)ΓcπqqSF(q)

]

+ iG γ5τa

∫d4q

(2π)4Tr[SF(p− q)ΓcπqqSF(q)

]︸︷︷︸

=0

(3.10)

This ends up being

1 = 4NcNfiG

∫d4q

(2π)4M

p2 −M2 + iǫ, (3.11)

which is the gap equation in Hartree approximation for vanishing current quark mass m.Equating the lefthand side and the very right in Fig. 3.4 evaluates as follows

M(p2)ab = ΓaπqqSπ(p2)Γbπqq = iγ5τa

[iG

1−GΠ(p2)

]iγ5τb (3.12)

Obviously the pion mass is given by the poles of the pion propagator. The pion propagator isproportional to the expression in the square brackets in (3.12). I. e. the roots of the denominatordetermine the pion mass:

1−GΠ(p2) = 0 ⇐⇒ p2 = m2π (3.13)

1We do not consider the exchange term in the vertex function evaluated in App. C.1.


Obviously, for vanishing current quark mass one findes mπ = 0 in accordance with Goldstone’stheorem. The residue of the term on the righthand side of (3.12) fixes the coupling strength gπqqof pion to a quark-antiquark pair, which is encoded in Γaπqq = igπqqγ5τa. This coupling strengthis determined by the equation

gπqqi

p2 +m2π + iǫ

gπqq =iG

1−GΠ(p2), (3.14)

from which is derived that

g−2πqq =

∂

∂p2

1−GΠ(p2)

G

∣∣∣∣p2=m2

π

=∂Π(p2)

∂p2

∣∣∣∣p2=m2

π

. (3.15)

This method of determining the spectrum of a mode (here as an example the pseudoscalarmode) is equivalent to the so-called random phase approximation (RPA) familiar from manybody theory.

Up to now only pseudoscalar modes were considered, which are identified by the pions. Butit is of course possible to proceed analogously with other modes. The self-energy Π and thecoupling g are therefore labeled with an index M distinguishing different channels.

ΠM (p2) = i

∫d4q

(2π)4Tr[OMSF(q +

p

2)OMSF(q − p

2)]

and g−2Mqq =

∂ΠM (p2)

∂p2

∣∣∣∣p2=m2

π

(3.16)

Now the integrals determining mπ and gπqq are computed more explicitly, as they are neededin the latter.

Ππa(p2) = 4iNcNf

∫d4q

(2π)4q2 − p2

4 +M2

[(q + p

2

)2 −M2 + iǫ] [(

q − p2

)2 −M2 + iǫ] (3.17)

= 2iNcNf

∫d4q

(2π)4

[1

(q + p

2

)2 −M2 + iǫ+

1(q − p

2

)2 −M2 + iǫ

]

− 2iNcNf p2

∫d4q

(2π)41[(

q + p2

)2 −M2 + iǫ] [(

q − p2

)2 −M2 + iǫ]

(3.18)

The integral in the last term is often abbreviated by

I(p2) =

∫d4q

(2π)41[(

q + p2

)2 −M2 + iǫ] [(

q − p2

)2 −M2 + iǫ] . (3.19)

The first two terms in (3.18) look as if q could be shifted by ±p2 such that all contributions of q

are eliminated. But this is not really true as this integral is is highly divergent as it stands. Aregularization will in general break translational invariance. We will do this simplification any-way arguing that the symmetry breaking by the regularization will take place at high momentaq where the value of the integrand is suppressed.

Ππa = 4iNcNf

∫d4q

(2π)41

q2 −M2 + iǫ− 2iNcNf p

2 I(p2) (3.20)

With the gap equation in the form of (3.5) this may be written rather short

Ππa(p2) =1

G

(1− m

M

)− 2iNcNf p

2 I(p2). (3.21)


= + + + . . .

= + =

1 −

Figure 3.4: Bethe-Salpeter equation in Hartree approximation

〈0 |Aµa(x) |πb(p)〉 = ipµfπδabe−ip·x = µ, a

iγµγ5τa2

Γbπqqp, b

Figure 3.5: Matrix element connecting the vacuum to the pion, via the axial current. The vertexfunction is defined as Γbπqq = iγ5gπqq

τb2 . The dotted line is the axial vector current, the dashed

double line represents the pion.

(3.13) now provides an explicit constraint for m2π

m2π = −m

M

1

2iGNcNf I(m2π). (3.22)

This equation nicely retrieves the chiral limit in which m2π −→ 0. As for the two light quarks

up and down the explicit symmetry breaking is small, (3.22) will be simplified by evaluating theIntegral I at m2

π = 0.The coupling constant gπqq has not been used, yet. It is involved in the pion decay constant.

This connection gets visible more clearly in Fig. 3.5, which can be evaluated to

ipµfπδab = gπqq

∫d4q

(2π)4Tr[iγµγ5

τa2SF(q +

p

2)iγ5τbSF(q − p

2)]

(3.23)

= gπqqpµ 4NcMδab

∫d4q

(2π)41[(

q + p2

)2 −M2 + iǫ] [(

q − p2

)2 −M2 + iǫ] (3.24)

= gπqqpµ 4NcMδabI(p

2) (3.25)

The coupling gπqq is calculated with (3.22) and (3.16) at p2 = 0:

g−2πqq = −2iGNcNfI(0). (3.26)

Now this is combined with the squared (3.25) and evaluated at p2 = 0:

f2π = −8i

Nc

NfM2 I(0). (3.27)

3.2.4 Regularization

Basically all 4-momentum integrals introduced in the previous section are divergent. There areseveral ways to remedy this. All methods are based on the idea to evaluate the 4-momentumintegrals over a finite momentum region characteristic of the range of applicablility of the model.


This can be motivated by the counterterm argument, stating that counter terms absorbing theinfinities are introduced. A more physical interpretation would be such that at the cutoff theinteraction is switched off. Of course there are divergent terms from the free particles, butthose contributions may be subtracted without changing the theory. In QCD-based models thisswitching off of the interaction at high momenta does indeed make sense. This is motivatedby the QCD interaction, which gets weaker at higher momenta, and finally vanishes at infinitemomentum transfer.

The first method is the non-covariant 3-momentum cutoff scheme. Its advantage is firstof all its simplicity. Secondly it tends to reproduce Fermi-Dirac statistics. This happens as acertain configuration of the system is only discriminated against by the Boltzmann factor andthe Pauli principle, and not additionally by the regularization scheme. Other schemes thatdo not reproduce Fermi-Dirac statistics typically interfere with this thermodynamic weighingprocedure by explicitly restricting the energy of the system. The non-covariance is not anobstacle since our aim is to work in medium i. e. with µ 6= 0. This breaks Lorentz invarianceexplicitly. The 3-momentum cutoff scheme also breaks gauge symmetry. In the NJL model thelocal color symmetry of QCD has been replaced by a global symmetry, so there is no additionalshortcoming caused by the 3-momentum cutoff scheme. An important point is that chiralsymmetry and Goldstone’s theorem are strictly maintained after regularizing.

There are other regularization schemes that could be used. Some are shortly discussed here.First there is the 4-momentum cutoff scheme. This scheme simply stops the integration over4-momentum once a certain q2 = q20 − ~q 2 is reached. This prescription is obviously covariantbut it cannot result in a system obeying Fermi-Dirac statistics, for reasons discussed in theprevious paragraph. The proper time regularization scheme is another covariant regularizationscheme working with a similar mechanism, not in momentum but in coordinate space. Finallythe Pauli-Villars scheme is mentioned here. It is a covariant scheme respecting gauge invariance.

3.2.5 Parameter fixing

The model so far has three parameters, the 3-momentum cutoff Λ, the effective quark-quarkcoupling G and the current quark mass m0. The first quantity this model reproduces is the massof the Goldstone boson mode. In two flavors they correspond to the pions. In Sec. 3.2.3 the pionmass was determined by equation (3.22). We eliminate the integral I(m2

π) by approximation itwith I(0) and substituting it using equation (3.27). Additionally M = m0 −G 〈qq〉 was used toobtain

f2πm

2π =

4

N2f

(m0 〈qq〉+

m20

G

), (3.28)

which is the Gell-Mann-Oakes-Renner relation [GMOR68] for the isospin symmetric case (m0 =mu = md) — at least to the first non-vanishing order in m0. In agreement with the Gell-MannOakes Renner relation the higher order term in m0 is dropped in the determination of theparameters.

The second quantity to reproduce is the pion decay constant fπ. Here equation (3.27) isused:

f2π =

2Nc

NfM2

∫ Λ d3q

(2π)31

E3q

=2Nc

NfM2

∫ Λ

0

dq q2

2π2E3q

, (3.29)

where the upper bound Λ of the integral indicates that these integrals need to be regularized.

The third quantity we use for the determination of the model parameters is the chiral con-densate 〈qq〉. With M = m0 −G 〈qq〉 and equation (3.5) the chiral condensate is:

〈ψψ〉 = −2NcNfM

∫ Λ d3q

(2π)31

Eq. = −2NcNfM

∫ Λ

0

dq q2

2π2Eq. (3.30)


Λ [GeV] G[GeV−2

]m0 [MeV]

0.651 10.08 5.5

|〈uu〉| 13 [GeV] fπ [MeV] mπ [MeV]

0.251 94.0 140.5

Table 3.1: The chosen Parameters for the physical point (left table) reproduce the physicalquantities (right table). The constituent quark mass ends up being M = .325 GeV. The numberof flavors and colors are Nf = 2 and Nc = 3

The Gell-Mann-Oakes-Renner relation (3.28), (3.29) and (3.30) are now solved simultane-ously to determine the three parameters Λ, G and m0. An additional (weak) constraint is thatM , the constituent quark mass, is approximately 1

3 MN. Taking this as a strict requirement thesystem would be overdetermined. Dependent on how much emphasis is put on this additionalrelationship it is the coupling constant G that is mainly constrained to a certain range:

1

3MN ≈M = m0 −G 〈qq〉 = m0 −G

(〈uu〉+ 〈dd〉

)≈ −2G 〈uu〉 (3.31)

The last step implied isospin symmetry and the approximation m0 ≈ 0. The numeric outcomeis G ≈ 10 GeV−2. The parameter set used here is given in Tab. 3.1. These parameters are usedin [RTW06].

In Fig. 3.6 the dependence of cutoff Λ, Coupling G and current quark mass m0 is plotted as afunction of the pion mass mπ and the pion decay constant fπ, using the Gell-Mann Oakes Rennerrelation and equations (3.29) and (3.30). This illustrates that the pion mass mπ mainly fixesthe current quark mass m0 (through the Gell-Mann-Oakes-Renner relation) but leaves cutoff Λand quark-quark coupling G practically unchanged.

On the other hand it is interesting to see, what happens to the physical quantities when

changing the model parameters. The dependence of the chiral condensate |〈uu〉| 13 , the pionmass mπ and the pion decay constant fπ on quark-quark coupling G and cutoff Λ are illustratedin Fig. 3.7. The dependence of the chiral condensate, the pion mass and the pion decay constanton the current quark mass is shown in Fig. 3.8. That is, this model makes a prediction of thechange of the ”nucleon mass” and the pion decay constant with varying squared pion mass (seeFig. 3.10). This relation is similar to what is obtained in lattice calculations.

To get some insight into the changes happening from the 2 flavor case to the 3 flavor case the

number of flavors Nf was treated as a variable in Fig. 3.9. In the chiral condensate |〈uu〉| 13 =

0.251 GeV was kept constant, i. e.∣∣〈ψψ〉

∣∣ 13 = N13f |〈uu〉|

13 . The additional flavor was assumed to

have the current quark mass as the already existing two.

3.3 Color superconductivity

3.3.1 The two flavor case

In analogy to the Cooper pairs in BCS theory [BCS57] diquark condensates are introduced intothe NJL model. The diquark condensate is defined as the expectation value

⟨qTMq

⟩. (3.32)

q denotes a quark spinor field and M is an operator in Dirac-, in flavor- and in color-space.Without loss of generality we demand for a single diquark condensate that M factorizes2 in

2The requirement, that the operator M factorizes into its subcomponents in Dirac-, flavor- and color spacedoes not constrain our problem. This is due to the fact that for a NV = NU ×NW -dimensional vector space V ,

3.3. Color superconductivity 23

0.1

0.12

0.14

0.16

0.180.2

mΠGeV

0.08

0.085

0.090.095

fΠGeV

2.5

5

7.5

10m0

MeV

0.1

0.12

0.14

0.16

0.180.2

mGeV

2.5

5

7.5

10

0.1

0.12

0.14

0.160.18

0.2

mΠGeV

0.08

0.085

0.090.095

fΠGeV

6

8

10

12

GGeV-2

0.1

0.12

0.14

0.160.18

0.2

mGeV

6

8

10

12

0.1

0.12

0.14

0.16

0.180.2

mΠGeV

0.08

0.085

0.090.095

fΠGeV

0.65

0.7

0.75L

GeV

0.1

0.12

0.14

0.16

0.180.2

mGeV

0.65

0.7

0.75

Figure 3.6: Illustration of the dependence of the current quark mass m0 (top left), the quark-quark coupling G (right) and the cutoff Λ (bottom) on the pion mass mπ and the pion decay

constant fπ at fixed chiral condensate |〈uu〉| 13 = 0.251 GeV and Nf = 2, Nc = 3.


57.5

1012.5

15

GGeV-2

0.4

0.50.6

0.70.8

0.9

LGeV

00.10.20.3

0.4

< u u >13

GeV

57.5

1012.5

15

GGeV-2

57.5

10

12.5

15

GGeV-2

0.4

0.6

0.8

LGeV

0.15

0.2

0.25

0.3

mΠGeV

57.5

10

12.5

15

GGeV-2

57.5

10

12.5

15

GGeV-2

0.4

0.5

0.6

0.70.8

0.9

LGeV

0

0.05

0.1

0.15

fΠGeV

57.5

10

12.5

15

GGeV-2

Figure 3.7: Illustration of the prediction of the NJL model for varying model parameters forNf = 2 and Nc = 3. Here the coupling G and the cutoff Λ are changed. The prediction for the

chiral condensate |〈uu〉| 13 (top left), for the pion mass mπ (right) and the pion decay constantfπ (bottom). Note that in the graph at the top left the absolute value of the chiral condensateis plotted. The chiral condensate is negative however |〈uu〉| < 0.


0 0.05 0.1 0.15m0 GeV

0.25

0.255

0.26

0.265

0.27

0.275

È<q

q>È1

3G

eV

0 0.05 0.1 0.15m0 GeV

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

mΠG

eV

0 0.05 0.1 0.15m0 GeV

0.094

0.096

0.098

0.1

0.102

0.104

0.106

f ΠG

eV

Figure 3.8: Illustration of the prediction of the NJL model for varying model parameters forNf = 2 and Nc = 3 for varying current quark mass m0. The prediction for the chiral condensate

|〈uu〉| 13 (top left), for the pion mass mπ (top right) and the pion decay constant fπ (bottom).Note that in the leftmost graph the absolute value of the chiral condensate is plotted. The chiralcondensate is negative however |〈uu〉| < 0.


1 1.2 1.4 1.6 1.8 2.Nf

3

3.5

4

4.5

5

5.5

m0M

eV

1 1.2 1.4 1.6 1.8 2.Nf

8.5

9

9.5

10

10.5

GG

eV-

2

1 1.2 1.4 1.6 1.8 2.Nf

0.65

0.7

0.75

0.8

0.85

0.9

LG

eV

Figure 3.9: Illustration of the changes of the model parameters with changing number of flavorsNf . The current quark mass m0 (top left), the quark-quark coupling G (top right) and the cutoffΛ (bottom).

0. 0.1 0.2 0.3 0.4 0.5

mΠ2 GeV2

92.5

95.

97.5

100.

102.5

105.

f ΠM

eV

0. 0.1 0.2 0.3 0.4 0.5

mΠ2 GeV2

1.

1.1

1.2

1.3

1.4

MN

»3

MG

eV

Figure 3.10: Illustration of the changes of the pion decay constant fπ and the ”nucleon mass”MN = 3M with changing pion mass to the square.


these three subspacesM =MDirac ⊗Mflavor ⊗Mcolor. (3.33)

As quarks are fermions they obey anticommutation relations. This means that the operatorMhas to be completely antisymmetric:

qTMq = qiMijqj = −qjMijqi = −qTMT q ⇒ Mij = −Mji. (3.34)

In Dirac space we have potentially 16 different operators at hand. They can be classified bytheir behavior under Lorentz transformation, i. e. scalar, pseudoscalar, vector, axial vector andtensor.

In two-dimensional flavor space we have N2f = 4 different operators, 1 ≡ τ0, τ1, τ2 and τ3,

where τi is equal to the Pauli matrix σi. In color space we have N2c = 9 different contributions,

1 ≡ λ0 and λi with i ∈1, 2, . . . 8 = N2

c − 1. There are several ways to compose an antisym-

metric operator M. Either all three subspace operators in (3.34) have to be antisymmetric ortwo have to be symmetric while one suboperator is antisymmetric (see Tab. 3.2).

The one-gluon exchange interaction suggests that the color antitriplet is attractive, while thesextet is repulsive. Therefore, quark-quark pairing will most likely be strong in the antitripletchannel and most likely not be present in the sextet channel. The spontaneous symmetry break-down in NJL models analogously to the spontaneous symmetry breakdown in QCD, suggeststhat quark-quark pairing will be stronger in scalar, vector and tensor channels and weaker in thepseudoscalar and axialvector channels. This is due to the explicit breaking of chiral symmetry.At zero quark mass and unbroken chiral symmetry, parity even and odd condensates can betransformed into each other via a U(1) symmetry [PR99]. This would allow the condensate totake any direction in this U(1) group. In the presence of a symmetry breaking term the con-densates will automatically choose a direction in this U(1) favored by this symmetry breakingterm. I. e. condensates with even parity will be favored. The three condensates that promise tobe strong are listed in Tab. 3.3.

The scalar and the vectorial condensates in Table 3.3 depend on τA and λA′ , where τAand λA′ are representations of the groups SU(2)f and SU(3)C. The SU(2)f and the SU(3)csymmetry allow to rotate such that τA and λA′ end up being equal to any one of the generatorsor a normalized linear combination. It is therefore only necessary to consider one of thosecondensates (see also Sec. 3.4.1).3 We choose A = 2 and A′ = 2. Additionally only the zerocomponents of the vectorial and the zero component of the first index of the tensorial4 condensateare considered here. At non-zero quark chemical potential Lorentz invariance is broken, whichpermits such Lorentz non-invariant condensates. The most prominent example for a Lorentznon-invariant expectation value at non-vanishing quark chemical potential is the quark densitynq = 〈q†q〉. There could in principle be expectation values of the spatial part of a vectorialquantity. Such spatial expectation values break rotational invariance. They do however notchange the outcome of the model, as they compete with the time component. I. e. φ2

0 − ~φ2 willalways assume the same value. This changes, if we allow different coupling constants for thetime and the space component of vectorial and tensorial condensates, which would be legitimate

that is the tensor product of the NU and NW -dimensional vectorspaces U and W (V = U ⊗W ) there exists abasis BV =

˘eV1 , e

V2 , . . . e

VNU×NW

¯, that is constructed from the bases of U and W , BU =

˘eU1 , e

U2 , . . . e

UNU

¯and

BW =˘eW1 , eW

2 , . . . eWNW

¯in the following way: eV

i = eUi div NW

⊗ eWi mod NW

. If we are interested in a condensatewhich is not factorizable we can always write it as a sum of condensates with operators M, that can be factorized.

3Even though a specific direction in color space is chosen, color neutrality is guaranteed as diquark condensatesand anti-diquark condensates appear symmetrically.

4The tensorial condensate represents a condensate where the spins of the two quarks pair to a spin-1 triplet.The second component indexed with j in the tensorial consdensate does not index an spatial direction but thecorrelation of the direction of the spin ~S ant the total angular momentum ~J . So the index j is analogous to themagnetic quantum number of e. g. hydrogen in an 2P state. This tensorial condensate is quite similar to thesuperfluid phase in He3.


Dirac SU(2)f SU(3)c

antisymmetricCiγ5, scalarC, pseudoscalar

Cγµγ5, vectorτ2 singlet

λ2

λ5

λ7

antitriplet

symmetricCγµ, axialvectorCσµν , tensor

1

τ1τ3

triplet

1, λ1

λ3, λ4

λ6, λ8

sextet

Table 3.2: Symmetry properties of diquark Dirac structures, SU(2)f and SU(3)c generatorsunder transposition (particle exchange). τi = σi, i ∈ 1, 2, 3 with σi the Pauli spin matrices,and λi, i ∈ 1, 2, . . . , 8 the Gell-Mann matrices. C denotes the charge conjugation matrixdefined here as C = i γ0γ2. For the derivation of the Dirac structures see also App. B.1.

Condensate JP Dirac Color Flavor

δAA′=〈qTCγ5τAλ′Aq〉 , δ = δ22 0+ scalar antitriplet 3 singlet

δµAA′=〈qTCγ0γ5τAλ′Aq〉, δ0 = δ022 1+ vector antitriplet 3 singlet

ζj =〈qTCσ0jτAλSq〉 1+ tensor antitriplet 3 singlet

Table 3.3: The condensates (channels) with the strongest attraction. The condensate δ isconsidered in all calculations presented in this work. The condensate δ0 is considered only infew cases below, the tensorial condensate ζi is not subject to the work presented here. C denotesthe charge conjugation matrix defined here as C = i γ0γ2.

as the effective Lagrangian is Lorentz non-invariant. In this case it would be inappropriate tosimply treat the time component of a vectorial expectation value.

3.3.2 The three flavor case

The three flavor case is only shortly discussed here, as it is not subject of the calculations in thiswork. The three flavor case opens up many possibilities for diquark condensates. Classifyingthese condensates is a matter of finding subgroups to SU(3)f ⊗ SU(3)c ⊗ U(1). The 2SC (two-color-superconducting) phase that was considered in Sec. 3.3.1 only breaks SU(3)c down toSU(2)rg

5. But it is of course also possible to find subgroups combining the generators of thecolor and the flavor group differently. This is done in the case of the CFL (color-flavor-locked)phase first suggested in [ARW99].

If we want to consider diquark condensates, in the scalar (or vectorial) color antitripletchannel, the flavor structure has to be antisymmetric. A basis in flavor and color space for thesecondensates is τA ⊗ λA′ , A and A′ numbering the generators of the antisymmetric subgroups ofSU(Nf) and SU(Nc). Suppose that τi and λj are of the form of the Gell-Mann matrices. In thisbasis the scalar condensates are

sAA′ = 〈qTCγ5τAλA′q〉 , (3.35)

where τA and λA′ are the antisymmetric generators of SU(Nf) and SU(Nc). That is, sAA′ is amatrix of the dimension of the antisymmetric subgroups of SU(Nf) and SU(Nc). In the case

5SU(2)rg is a subgroup of SU(3)c. If the rows and columns in the 3-dimensional representation of SU(3)c arenamed ”red”, ”green” and ”blue”, SU(2)rg is the subgroup obtained by disregarding the ”blue” part.

3.4. Path integral approach and thermodynamics of NJL-models 29

whereNf = 3 andNc = 3 this is a complex 3×3-matrix [Bub05]. The eighteen degrees of freedommay be reduced by a rotation in color space. This can be realized even if there are explicitsymmetry breaking mass terms, as these terms are proportional to the identity in color space.In flavor space there are only the two diagonal rotations that leave explicit symmetry breakingmass terms invariant. Finally there is an overall phase, i. e. a transformation proportional tothe identity in both color and flavor space. I. e. this matrix of condensates actually has onlyseven real degrees of freedom [ARW99].

The 2SC-phase described in Sec. 3.3.1 in the terms of this matrix is characterized by s22 6= 0and sAA′ = 0 for A 6= 2 or A′ 6= 2. For the CFL phase it is sA=A′ 6= 0 and sA 6=A′ = 0. I want toleave it at this point at these two most prominent diquark condensates. It should be mentionedthat there is a whole class of condensates left, which is not described here: condensates in whichthe spin of the quarks couple to a spin-1-triplet [Sch05].

3.4 Path integral approach and thermodynamics of NJL-models

In this section another approach to the solution of the field equations (the so-called self con-sistency equations) is presented. While the calculations in Sec. 3.2 were performed using aDyson-Schwinger or a Bethe-Salpeter equation, in this section an approach using the thermody-namic potential is shown. The thermodynamic potential is proportional to the effective action.The idea is in principle quite simple: In thermal equilibrium of a system the thermodynamicpotential is minimal. This approach is quite handy, but of course not always applicable, as thethermodynamic potential cannot always be evaluated easily. The field equations are then justthe gradients of the thermodynamic potential with respect to the fields, set to zero. In meanfield approach the effective action is approximated by the classic action. I. e. the fields aretreated as classical fields. This is equivalent to omitting the path integral. Corrections to thismean field approximation will be calculated in Chap. 5.

3.4.1 Consistency of the condensates

As it was outlined in Sec. 2.1 for the special case of QCD, the solutions to a quantized fieldtheory do not need to possess all the symmetries of the underlying Lagrangian. In the case ofQCD there are two mechanisms involved in this reduction of symmetry, anomalous symmetrybreaking and spontaneous symmetry breaking. Here we only focus on the latter, spontaneoussymmetry breaking. In the presence of spontaneous symmetry breaking the vacuum expectationvalue of at least one field is non-vanishing. Expanding the Lagrangian about this ground stateyields a Lagrangian with reduced symmetry. The non-vanishing expectation value of the fieldhas spontaneously broken parts of the symmetries of the original Lagrangian.

The situation becomes more complicated, if several fields at the same time have non-vanishingvacuum expectation values. Each one of the fields is now acting according to a Lagrangian ofalready reduced symmetry — reduced by spontaneous symmetry breaking of the other field witha non-vanishing vacuum expectation value. Fig. 3.11 illustrates how two symmetries brokenspontaneously may be located in the total symmetry group of the original Lagrangian. Theblank rectangle in Fig. 3.11 a) represents the symmetry of the original Lagrangian in absenceof spontaneous symmetry breaking. In Fig. 3.11 b) and c) only one of the two fields assumes anon-vanishing vacuum expectation value. The hatched areas represent the part of the symmetrythat is spontaneously broken. If the two fields break the symmetry at the same time, there arethree different possibilities how the broken symmetries may be located within the total symmetrygroup of the original Lagrangian. The three settings are illustrated in Fig. 3.11 d), e) and f).

A Lagrangian always generates a effective potential that has the same symmetries as theLagrangian itself. In the case of spontaneous symmetry breaking the absolute minimum of such


a potential is no longer at vanishing fields, but at a finite vacuum expectation value of the fields.In addition, the original symmetries require several degenerate minima. The new ground statewill exist in one of these minima. The neighborhood of this new ground state is in general quitedifferent from the original one.

If there is just one field that is broken spontaneously the new ground state is shifted alonga line in the field configuration space that is orthogonal to the unbroken symmetries.6 If twofields at a time are spontaneously broken, there exist two such shifts, and two sets of unbrokensymmetries. These two shifts combined are in general no longer orthogonal to both sets ofunbroken symmetries. A violation of this orthogonality condition would be in contradictionto the existence of the symmetries of the Lagrangian in the first place. This orthogonalitycondition is certainly fulfilled, if the two condensates break symmetries in distinct subspaces.This corresponds to case d) in Fig. 3.11. If the two subspaces are not distinct, it is certainlypossible to choose the condensates such that their combination again fulfills the orthogonalitycondition. However the two condensates were introduced because this choice is not known in thefirst place and may change once model parameters like temperature and chemical potential arevaried. For this reason the two condensates are chosen such that they can approximately fulfillthe orthogonality condition. Such choices are represented by the cases illustrated in Fig. 3.11 e)and f). Hence, it is necessary to introduce a third field such that the combination of all threefields can satisfy the orthogonality condition. Firstly, this requires the three fields to be linearlyindependent. Secondly, the additional field must not break any of the unbroken symmetries.

For this reason cases e) and f) in Fig. 3.11 in general require the presence of an additionalfield. Choices of fields that can in general satisfy the orthogonality condition are usually calledfully self consistent. If the fields are chosen such that the violation of the orthogonality is onlysmall it may be justified to abandon the concept of a fully self consistent set of condensates.

This discussion matters for NJL models once diquark condensates and chiral condensatesare introduced. Both condensates break chiral symmetry, while only the chiral condensate is acolor singlet. The cases of interest here are discussed in more detail in Sec. 3.4.8 for NJL modelsand in Sec. 4.2.2 for PNJL models.

3.4.2 Bosonization

Bosonization refers to the transformation which eliminates fermionic degrees of freedom andreplaces them by bosonic degrees freedom. In this subsection a method on the level of thegenerating functional is presented.

The starting point in this subsection is a generic NJL-Lagrangian

L = ψS−1F ψ +

G

2

(ψΓαψ

)2, (3.36)

where S−1F is the inverse propagator and Γα is some Dirac7 structure indexed8 with α. The

generating functional is defined as

Z [η, η] =1

N

∫Dψ

∫Dψ exp

i

∫d4x

[L(ψ, ψ

)+ ψη + ηψ

], (3.37)

where η and η are the fermion and antifermion sources. In order to replace the four-fermionterms we use the path integral of a gaussian of a new (bosonic) field φr. The index r does not

6Orthogonality of fields and symmetries here refers to the orthogonality of the Dirac-, color-, flavor-, . . . ,structure of the fields and the generators of the symmetries.

7The formalism presented can easily extended on larger spaces, i. e. on Dirac, color and flavor space.8If the formalism is extended to larger spaces, α stands for a set of necessary indexes to cover all space. In

the latter upper and lower indices are used. This only is of importance meaning in the case of space-time indices,where it has the usual meaning, i. e. φµ = gµνφ

ν . In all other cases upper and lower indexed variables are equal.


a) b) c)

d) e) f)

symmetries unbroken sym. broken by condensate A

sym. broken by condensate B sym. broken by condensate A and B

Figure 3.11: Illustration of possible symmetry breaking patterns of two different condensates.The rectangle in a) symbolizes the full symmetry of the Lagrangian. In b) and c) two differentcondensates breaking different parts of the symmetry are introduced. There are three cases howthese symmetries may be located within the total symmetry group: d) the parts of the symmetrybroken by the two condensates are completely distinct, e) the parts of the symmetry broken bythe two condensates overlap partially, f) the symmetry broken by the second condensate liescompletely inside the region broken by the first symmetry.

need to be specified so far.

∫Dφr exp

i

∫d4x

∑

rs

φrMrsφs

= const. (3.38)

Shifting φr by some vector ±12

∑sM−1

rs as gives

N ′ =

∫Dφr exp

i

∫d4x

(∑

rs

φrMrsφs ±

∑

r

φrar +1

4

∑

rs

arM−1rs a

s

). (3.39)

By doing some simple algebra this is transformed to

⇒∫Dφr exp

i

∫d4x

(∑

rs

φrMrsφs ±

∑

r

φrar

)

= N ′ exp

−i∫

d4x1

4

∑

rs

arM−1rs a

s

. (3.40)

In the case of the Lagrangian (3.36), this reads

N ′ exp

i∫

d4x∑

αβ

(ψΓαψ

) Gδαβ2

(ψΓβψ

)

=

∫Dφα exp

i∫

d4x

−

∑

αβ

φαδαβ2G

φβ ±∑

α

φα(ψΓαψ

) (3.41)

Note thatM in almost all NJL cases of interest is proportional to the identity matrix. This can


be substituted into the formula for the generating functional (3.37).

Z [η, η] =1

N N ′

∫Dψ

∫Dψ

∫Dφ

exp

i

∫d4x ψ

(S−1

F ±∑

α

φαΓα

)ψ − 1

2G

∑

α

φαφα + ψη + ηψ

(3.42)

So far only an auxiliary field φ was introduced. This allowed to eliminate the four-fermion terms,and to absorb them into the bosonic fields. In exchange additional interaction terms appeared.One couples the fermion field to the auxiliary field φ. The other couples the auxiliary bosonfields. Now (3.42) can be evaluated further by integrating out the fermions. Before this canbe done the fermion sources η and the anti-fermion sources η have to be uncoupled from thefermion and anti-fermion fields ψ and ψ. This is achieved by the substitution ψ −→ ψ−S η andψ −→ ψ − S η. Additionally a shifted inverse propagator is defined by

S−1 = S−1F ±

∑

α

φαΓα. (3.43)

The generating functional becomes

Z [η, η] =1

N N ′

∫Dψ

∫Dψ

∫Dφ exp

i

∫d4x ψ S−1 ψ −

∑

α

φαφα2G

exp

−i∫

d4x η S η

=1

N N ′

∫Dφ

[exp−i Tr log S−1(y, x)

]∣∣∣y→x+

exp

− i∫

d4x η S η +1

2G

∑

α

φαφα

.

(3.44)

Here Tr is the functional trace plus the trace over Dirac, color and flavor space. From (3.44)the bosonized Lagrangian is read off as

L(x) = − Tr log S−1(y, x)∣∣y→x+ −

1

2G

∑

α

φα(x)φα(x). (3.45)

At this point it is useful to use the identity tr log = log det. The bosonized Lagrangian (3.45) isno longer a function of the fermion fields, but of the bosonic field φ, that was just introduced.In a strict sense this bosonic Lagrangian is an effective Lagrangian with respect to the fermionfields that are integrated out.

3.4.3 Nambu-Gor’kov space

In the previous section (Sec. 3.4.2) the bosonization for quark-antiquark pairs was outlined. Inthis work not only quark-antiquark condensates, but in particular diquark condensates are con-sidered. Diquark condensates can be bosonized analogously to the method sketched in Sec. 3.4.2.There is a difficulty however. After bosonizing the propagator is modified by a term propor-tional to the condensate. It can be seen in (3.43) that this additional term is proportional toboth the quark and the antiquark field. When bosonizing a diquark field this term will becomeproportional to two quark or two antiquark fields. As such it cannot be absorbed into thepropagator without changes. In this section a method will be outlined how the propagator canbe generalized such that it will be possible to absorb terms appearing in the bosonization ofdiquarks.

The idea is to write all appearing terms (quark-antiquark and quark-quark terms) in a singleunified structure. In analogy to a quadratic form that can be written as a polynomial as wellas in matrix notation in two-dimensional space the quark and antiquark fields are written in


the form of a two-dimensional vector notation. The natural connection of quark and antiquarkfields is the charge conjugation transformation.

The first step is therefore to split the Lagrangian in two equal parts. One of the parts isthen written in terms of the charge conjugate fields [PS, IZ],

ψC = C ψT , ψC = ψT C, (3.46)

where C = iγ0γ2 is the charge conjugation matrix.9 A bilinear covariant can be rewritten asfollows

ψ Γψ =(−C ψC

)TΓ(−ψCC

)T=(ψC)TC ΓC

(ψC)T

=[(ψC)TC ΓC

(ψC)T ]T

= −ψCC ΓT CψC = ψCΓCψC . (3.47)

In the last step of the first line it was used that the transposed of a number is the numberitself. It is important to note that transposing interchanges two fermionic fields giving theminus sign in the second line. Additionally operators contained in Γ acting to the right (left)after transposition act to the left (right).10 From (3.47) it can be read off that

←−Γ C = −C −→Γ T C. (3.48)

For example a Lagrangian

L = ψS−1ψ +G

2

(ψΓψ

)2(3.49)

=1

2ψS−1ψ +

1

2ψC[S−1

]CψC +

G

8

(ψΓψ + ψCΓCψC

)2(3.50)

can be transformed to a quadratic matrix structure by introducing

Ψ =

(ψψC

)Ψ =

(ψ, ψC

)Γ =

(Γ 00 ΓC

): (3.51)

L =1

2Ψ

( [S−1

]0

0[S−1

]C)

Ψ +G

8

(ΨΓΨ

)2. (3.52)

Coupling terms in the diquark channel are now easily incorporated. A generic coupling term inthe diquark channel is of the form

δLH =H

2

(ψ ΓH C ψ

T) (ψT C ΓH ψ

). (3.53)

A transformation analogous to the transform applied in the step from (3.49) to (3.50) gives

δLH =H

8

(ψ ΓH ψ

C) (ψC ΓH ψ

)− H

8

(ψ ΓH ψ

C) (ψT C ΓH C

(ψC)T)

− H

8

((ψC)TC ΓH C ψ

T) (ψC ΓH ψ

T)

+H

8

((ψC)TC ΓH C ψ

T)(

ψT C ΓH C(ψC)T)

(3.54)

=H

8

(ψ ΓH ψ

C) (ψC ΓH ψ

)+H

8

(ψ ΓH ψ

C) (ψC ΓCHψ

)

+H

8

(ψΓCH ψ

C) (ψC ΓH ψ

T)

+H

8

(ψΓCH ψ

C) (ψC ΓCHψ

C)

(3.55)

=H

8

[Ψ

(0 ΓH

ΓCH 0

)Ψ

]2

. (3.56)

9The charge conjugation matrix has the following properties: C†C = 1, C2 = −1, CT = −C and C∗ = C.10The direction in which an operator acts is indicated by an arrows above the operator.


What actually happend is that a new two-dimensional space has been introduced. Thecomplete space is now the tensor product of this two-dimensional space with the spaces usedbefore. This two-dimensional space is called the Nambu-Gor’kov space [Gor58, Nam60].

With this construction the bosonization can be done as shown in Sec. 3.4.2. The onlydifference is that the space in which the structures are defined is no longer

[Dirac]⊗ [color]⊗ [flavor] but [Nambu-Gor’kov]⊗ [Dirac]⊗ [color]⊗ [flavor]

instead. The inverse propagator (now called Nambu-Gor’kov propagator) with both quark-antiquark condensates and diquark condensates has the form11

S−1 =1

2

( [S−1

][ΓH ]

[ΓH ]C[S−1

]C). (3.57)

3.4.4 The Wick rotation

The Wick rotation usually appears in two different contexts. The Wick rotation was first usedby Dyson [Dys49] to evaluate loop integrals. This method has to be seen in connection withcomplex analysis [PS, IZ]. The integral along the real time axis can be completed to a closedcontour. Knowing that the total integral is determined by the poles inside the closed contour,it is often possible to relate the integral along the real time axis with the integral along theimaginary time axis. Interestingly the connection is even deeper. The analytic properties ofthe n-point functions may be used to continue them to the whole complex time plane. If it ispossible generate n-point functions from a Lagrangian, then there exists a generating functionalfor producing the Wick rotated n-point functions. The Lagrangian corresponding to this newgenerating functional is the Wick rotated Lagrangian originally defined for real times. Thisone-to-one correspondence exists for a whole class of field theories [Wig56, Wig57]. However forgauge theories this has not been rigorously proven yet. Nevertheless it is possible to constructthe generating functional for the Wick rotated correlators. In [Sch59] this has been done for thecase of quantum electro dynamics (QED).

For this work the more important application of the Wick rotation concerns quantum fieldtheories at non-zero temperatures [LB96, Kap89]. In quantum field theories the partition func-tion is defined as

Z = Tr eβH =

∫∑

n

〈n|e−βH |n〉 , (3.58)

where | n〉 n∈N is an orthonormal basis of the Hilbertspace.12 The Hamilton operator H is

defined as∫

d3x H, where H is the Hamiltonian density of the theory. The trace Tr is definedhere by the second equality in (3.58).13 Calculating thermal expectation values it is useful tointroduce the density

ρ =1

Ze−βH . (3.59)

Thermal expectation values are calculated by the formula

〈A〉β = Tr ρA. (3.60)

In real time it may be useful to define the unitary operator U(tb, ta) that propagates operators

11Note that the inverse Nambu-Gor’kov propagator is defined here with a factor 12

in front.12It may be useful to take the normalized eigenvectors of H.13Note that this definition of cTr is to be distinguished from the meaning of Tr thoughout the rest of this work.


and states from time ta to time tb.

|φ(tb)〉 = U(tb, ta) |φ(ta)〉 (3.61)

〈φ(tb)| = 〈φ(ta)|U(ta, tb) = 〈φ(ta)|U †(tb, ta) (3.62)

A(tb) = U(tb, ta)A(ta)U(ta, tb) = U(tb, ta)A(ta)U†(tb, ta) (3.63)

U(tb, ta) can be written as

U(tb, ta) = T exp

−i∫ tb

ta

dt H(t)

, (3.64)

where T is the time ordering operator. The similarity of U to Z ρ is quite obvious. TransformingU into Z ρ is to change the integrational limits such that −iβ = tb − ta. When doing this it ishowever necessary to generalize the definition of the time ordering. The time ordering has tobe defined such that the Hamiltonian at times at the lower integrational bound acts first and atthe upper bound last.14

The operator U(tb, ta) can be written in terms of a path integral

U(tb, ta) =1

N

∫Dπ∫Dφ∫Dψ

∫Dψ exp

i

∫ tb

ta

dt

∫d3xL(φ, ∂φ;ψ, ψ)

, (3.65)

where φ is a wildcard for all bosonic fields, π is the conjugate momentum to φ, ψ is a wildcard forall fermionic fields. It is important to note that the path integral over the conjugate momentmumπ of φ quite often is not written down explicitly. This is due to the fact that the Lagrangianby nature is a function of φ and φ. When a theory is built on a Lagrangian there is no πdependence. Therefore the path integral over π will only lead to an unimportant constant. Inthis work the path integral over π will be omitted as well. Additionally it is important that thecorrespondence of a Hamilton density operator H used as generator for the time propagationand a Hamilton density function H as Legendre transform of the Lagrangian is not unique.The Hamilton density operator H usually is build up by field operators. In general these fieldoperators do not commute, and therefore have a distinct order, that cannot be changed withoutgetting additional terms. The Hamilton density function H is a function of real numbers whichdo commute. Depending on how the path integral is derived different ordering prescriptions arefound.15 The corresponding generating functional16 is

Z[J, η, η] =

∫Dφ∫Dψ

∫Dψ exp

i

∫d4xL+ J(x)φ(x) + ηψ + ψη

. (3.66)

As already stated in most cases the generating functional can be Wick rotated to obtain thegenerating functional of the Euclidean quantum field theory. Setting the sources in Z[J, η, η] tozero and restricting the integral in time direction gives U(tb, ta), which therefore can also be Wickrotated to give an operator propagating in imaginary time direction. One finds U(0,−iβ) = Z ρ.All that needs to be done to evaluate thermal expectation values is to take the trace. As thetrace sums matrix elements of equal final and initial states the path integral representation isconstrained at the upper and the lower integration limit tb and ta. Rigorous calculations lead todifferent boundary conditions for bosons and fermions. While bosons require periodic boundaryconditions fermions require antiperiodic boundaries, i. e. φ(0) = φ(−iβ) and ψ(0) = ψ(−iβ).The conjugate momenta π and ψ† = ψγ0 are not subject to additional constraints.

14Usually the integral is performed form t = 0 to t = −iβ. Therefore the time ordering in imaginary time τ = it

is defined such that τ decreases from left to right.15Prominent orderings are the normal ordering and the Weyl ordering. The first puts all annihilators to the

right and all creators to the left, while the Weyl ordering is completely symmetric.16Here the path integral over π has already been omitted.


Thermal expectation values therefore can be written in the Lagrange formalism as follows

〈A〉β =1

Z

∫Dφ∫Dψ

∫DψA(φ, ψ, ψ) exp

−∫ β

0dτ

∫d3xLE

, (3.67)

where Z is the partition function in the path integral formalism

Z =

∫Dφ∫Dψ

∫Dψ exp

−∫ β

0dτ

∫d3xLE

. (3.68)

This partition function does not equal the partition function defined in (3.58). The differencehowever is just an unimportant constant, i. e. the physical content is equivalent. Again both in(3.67) and in (3.68) the path integrals are subject to the constraint that bosonic field need tobe β periodic, while fermionic fields are β antiperiodic.

The periodicity conditions for fermions and bosons are related to the Kubo-Martin-Schwingerrelation (KMS) [Kub57, MS59] for the thermal Green’s function. We define

SF(ta, tb)β = 〈T[ψ(ta)ψ(tb)

]〉β

= 〈ψ(ta)ψ(tb)〉β Θ(ta − tb)− 〈ψ(tb)ψ(ta)〉β Θ(tb − ta), (3.69)

= S>(ta, tb)βΘ(ta − tb) + S<(ta, tb)βΘ(tb − ta), (3.70)

D(ta, tb)β = 〈T [φ(ta)φ(tb)]〉β = 〈φ(ta)φ(tb)〉β Θ(ta − tb) + 〈φ(tb)φ(ta)〉β Θ(tb − ta), (3.71)

= D>(ta, tb)βΘ(ta − tb) +D<(ta, ta)βΘ(tb − ta). (3.72)

Also of interest is a relationship of the fields at times ta = 0 and tb = −iβ, with only S>β and

D>β considered at this point.

S>(ta, tb)β = Tr[ρψ(ta)ψ(tb)

]=

1

ZTr[e−βH e+βHψ(ta + iβ) e−βH ψ(tb)

]

=1

ZTr[e−βH ψ(tb)ψ(ta + iβ)

]= Tr

[ρ ψ(tb)ψ(ta + iβ)

]

= −S<(ta + iβ, tb)β (3.73)

D>(ta, tb)β = Tr [ρφ(ta)φ(tb)] =1

ZTr[e−βH e+βHφ(ta + iβ) e−βH φ(tb)

]

=1

ZTr[e−βH φ(tb)φ(ta + iβ)

]= Tr [ρφ(tb)φ(ta + iβ)]

= D<(ta + iβ, tb)β (3.74)

From (3.73) and (3.74) one finds

ψ(−iβ) =

∫d3xS>(−iβ, 0)β ψ(0) = −

∫d3xS<(0, 0)β ψ(0) = −ψ(0) (3.75)

φ(−iβ) =

∫d3xD>(−iβ, 0)β φ(0) = +

∫d3xD<(0, 0)β φ(0) = +φ(0), (3.76)

which are the constraints set for the path integral.

3.4.5 The Matsubara formalism

When evaluating the partition function Z in (3.68) an integral∫ −iβ0 needs to be calculated.

In this section a method for the evaluation of this integral is presented. Instead of evaluatingthis integral in direct imaginary time space the action can be transformed to momentum space.To do this we write the integrand as a Fourier series. This is possible as the fields obey (anti-)periodic boundary conditions at t = t0 and t = t0−iβ. (This is the KMS relation, see Sec. 3.4.4).


This is fulfilled by the following definition of the Fourier series. The fields in momentum spaceare for convenience normalized such that they are dimensionless.

ψ(τ = 0) = −ψ(τ = β) ⇒ ωn =π (2n+ 1)

T

ψ(τ, ~x) =√V∑

n

∫d3p

(2π)3ψn(~p) e

i(ωnτ+~p·~x)

with ψn(~p) =T√V

∫ β

0dτ

∫d3xψ(τ, ~x) e−i(ωnτ+~p·~x)) (3.77)

φ(τ = 0) = −φ(τ = β) ⇒ ωn =2πn

T

φ(τ, ~x) =

√V

T

∑

n

∫d3p

(2π)3φn(~p) e

i(ωnτ+~p·~x)

with φn(~p) =

√T 3

V

∫ β

0dτ

∫d3xφ(τ, ~x) e−i(ωnτ+~p·~x)) (3.78)

This normalization is particularly useful when calculating the partition function and theEuclidean action. The Euclidean action for fermions and bosons can be written in the followingform

SfermionE = Trfunctional

[ψ(SβF

)−1ψ

]= V

∑

n

∫d3p

(2π)3ψ(SβF(~p)

)−1ψ (3.79)

SbosonE =

1

2Trfunctional

[φ∗(Dβ)−1

φ

]=V

2

∑

n

∫d3p

(2π)3φ†(Dβ(~p)

)−1φ (3.80)

This can be seen as a generalized quadratic form (vTAv with A = AT ). The partition functioncan then be evaluated in analogy to a Gaussian integral over all fields.

Zfermion =

∫ψ

∫ψ e−SE = Det

[(SβF

)−1]

(3.81)

Zboson =

∫φ e−SE =

(Det

[1

2

(Dβ)−1

])− 12

(3.82)

Where Det is the determinant over all spaces including the functional space. Using the identityDet = exp Tr log the determinant can be replaced by the trace. Most often it is convenient to

evaluate the functional trace in momentum space. Here one finds Tr = V∑

n

∫ d3p(2π)3

Tr, where

Tr is the trace over all remaining subspaces of the fields. The thermodynamic potentials17 thenevaluate to

Ωfermion = −T Tr log

[(SβF

)−1]

= −T∑

n

∫d3p

(2π)3Tr log

[(SβF(~p)

)−1], (3.83)

Ωboson =T

2Tr log

[(Dβ)−1

]=

T

2

∑

n

∫d3p

(2π)3Tr log

[(Dβ(~p)

)−1]. (3.84)

The propagators in momentum and frequency space (for fermions and bosons) are

(SβF

)−1= β (iωnγ0 + ~γ · ~p+m) ,

(Dβ)−1

= β2(ω2n + E2

~k

). (3.85)

17Indeed this is the thermodynamic potential density, i. e. the thermodynamic potential per volume.


Note that here the fields in momentum space as well as the functional trace were normalizedsuch that they are dimensionless. This normalization leads to the factors of β and β2 in theexplicit form of the propagators SβF and Dβ. These propagators are here referred to as thermalpropagators. It is easy to see that the thermal propagators are dimensionless, as they shouldbe. For distinction of these thermal propagators a superscript β was added. The thermalpropagators and the conventional propagators are connected via the relation

β SβF = SF, β2Dβ = D. (3.86)

Explicitly using the free propagators (3.85) the Matsubara sums can be evaluated. Additionaldegrees of freedom (color and flavor) were neglected:

Ωfermion = −4T

∫d3p

(2π)3

[1

2βE~k + log

(1 + e−βE~k

)](3.87)

Ωboson = −T∫

d3p

(2π)3

[1

2βE~k + log

(1− e−βE~k

)](3.88)

When the whole calculation is carried out at non-zero chemical potentials (3.87) and (3.88)is modified to:

Ωfermion = −2T

∫d3p

(2π)3

[βE~k + log

(1 + e−β(E~k

−µ))

+ log(1 + e−β(E~k

+µ))]

(3.89)

Ωboson = −1

2T

∫d3p

(2π)3

[βE~k + log

(1− e−β(E~k

−µ))

+ log(1− e−β(E~k

+µ))]

(3.90)

3.4.6 Evaluation of Matsubara sums

In the previous section the derivation of the thermodynamic potential for free fermions andbosons was sketched. The models of interest are of course not models with only free particles.The NJL-models belong to a class of models with 4-quark point interactions. In all modelstreated here it is possible to bosonize the quark fields (see Sec. 3.4.2). This procedure absorbsall interaction terms. What remains is a quasi-free theory. In Sec. 3.4.2 it was shown thatfirstly an additional potential appears in the Lagrangian (3.45). Secondly the free propagator ismodified to the form given in (3.43). This propagator actually contains an interaction term ofthe fermionic field with the condensates. Nevertheless the procedure for the calculation of thethermodynamic potential in the form in (3.83) outlined in Sec. 3.4.5 can be followed with theinverse propagator in (3.43) as well, as no assumptions on the form of the propagator had to bemade. The calculations then leading to (3.89) can be repeated for an inverse propagator of theform in (3.43). In the fermion case this calculation shall be sketched in brief here.

∑

n

∫d3p

(2π)3Tr log

[SβF

]−1=∑

n

∫d3p

(2π)3log Det

[SβF

]−1(3.91)

The evaluation of Det[SβF]−1 can be performed, as the inverse thermal propagator is a matrix inNambu-Gor’kov, in Dirac, in color and in flavor space. I. e. its dimension is N = 2×4×Nc×Nf .The determinant is then a polynomial of degree N .

First the calculation of free fermions is reviewed. An explicit calculation quickly shows that

det[/p−m

]=(m2 − p2

)2=(−p2

0 + λ2)2

with λ2 = E2~k. (3.92)

The naive way of Wick rotating this expression and performing the Matsubara sum fails as theinfinite Matsubara sum diverges. The standard trick used is to differentiate with respect to λ


first, perform the summation and to finally integrate the summed expression with respect to λ.

∑

n

log[ω2n + λ2

]=

∫dλ∑

n

∂λ log[ω2n + λ2

]=

∫dλ∑

n

2λ

ω2n + λ2

=

∫dλβ tanh

[βλ

2

]= 2 log cosh

[βλ

2

]= βλ+ log

[1 + e−βλ

](3.93)

Due to spin symmetry it is possible to choose a unitary transform of S−1 such that S−1 isat least in block diagonal form with two equal blocks. The same argument holds for isospin,if isospin symmetry is unbroken. Throughout this work all problems are spin and isospin sym-metric. As a consequence detS−1 = 0 can be unitarily transformed such that S−1 is at least inblock diagonal form with four equal blocks. In simple cases of spin and isospin symmetry thisunitary transformation is particularly simple: It is just build up from permutations of rows andcolumns of the matrix S−1(~p).

There is even more symmetry within S−1 due to the Nambu-Gor’kov structure. The con-struction of Nambu-Gor’kov space doubled the dimensionality of the Hilbert space by explicitlyadding the charge conjugate fields. Therefore all energy values will always appear with bothsigns. The determinant of the inverse Nambu-Gor’kov propagator detS−1 is a polynomial ofdegree N in p0. detS−1 therefore can be factorized to

detS−1 =∏

a

(p0 − λa

), (3.94)

where λa are the roots of the equation detS−1 = 0. The fact that all energy values appear withboth signs can be stated more formally:

∀ λi ∈W =λ|detS−1(p0 = λ) = 0

∃ λi 6= λj ∈W , such that λi = −λj . (3.95)

In the latter the pairs λi,j are indexed only once: λk = λi with Re[λi] > 0. The statement(3.95) is equivalent to the statement that detS−1 is a polynomial in p2

0. This is of great helpas this allows to write detS−1 as a product of terms of the form (−p2

0 + λ2i ) = (ω2

n + λ2i ). Then

log[detS−1] can be written in the form∑

i log[ω2n + λ2

i ]. This of course allows to proceed inanalogy to (3.93).

3.4.7 Self consistency equations

By analogy to the derivation of the thermodynamic potential for free fermions in Sec. 3.4.5,the thermodynamic potential for the bosonized Lagrangian (3.45) can be calculated. For thefollowing considerations it is of no importance, if the propagator in this bosonized Lagrangian isthe Nambu-Gor’kov propagator or the original one.18 Through the bosonization procedure thecalculation of the partition function from Lagrangian (3.45) involves additional path integrals.

Z =

∫ ∏

α

Dφα∫Dψ

∫Dψ e−SE (3.96)

The fermionic integrals can be evaluated giving the so-called fermion determinant

Zf [φα] = exp Tr log S−1[φα](y, x)∣∣∣y→x+

. (3.97)

18Note however that the Nambu-Gor’kov propagator is defined here with an additional factor 12

in front (seeSec. 3.4.3).


The partition function then can be expressed as

Z =

∫ ∏

α

Dφα Zf [φα] exp

−∫ β

0dτ

∫d3x

∑α φ

α(x)φα(x)

2G

. (3.98)

Unfortunately it is very expensive to solve this problem exactly. A common simplification isthe mean field approximation. This approximation contains two main features: First of allfields are treated as constants in space and imaginary time. The average fields are defined byφα = T

V

∫ β0 dτ

∫d3xφα(x). This already simplifies the partition function dramatically

Z =

∫ ∏

α

dφα Zf [φα] exp

−β V

∑α φ

αφα

2G

. (3.99)

The second simplification is based on the argument that for large volume the remaining integralsare basically determined by those configurations with minimal Euclidean action. Non-extremalconfigurations are neglected.

∫ ∏

α

dφα e−SE[φα] ≈ e−SE[φα]∣∣∣φα=φ

min.α

(3.100)

The conditions φα = φmin.α are the so-called self consistency equations. In this approximation

the thermodynamic potential is of the form

Ω = −TV

logZ ≈ T

VSE

[φα]∣∣φα=φ

min.α

= ΩMF (3.101)

From this equation the constraint in the Euclidean action SE

[φα]

= minimal can be trans-

formed in a constraint on the mean field thermodynamic potential ΩMF

[φα]

= minimal. Thenecessary condition for this minimization is

∂ΩMF

∂φγ= 0 , for all fields φγ , (3.102)

which is the most common formulation of the self consistency equations.

3.4.8 Sets of condensates

In this section commonly used condensates will be presented. Off course these sets of condensatesshall satisfy what is usually called ”fully self consistent” (see Sec. 3.4.1).

The original interaction term applied by Nambu and Jona-Lasinio G2

[ (ψψ)2

+(ψγ5~τψ

)2 ]

leads to the so-called chiral condensate

σ = G 〈ψψ〉 . (3.103)

The chiral condensate is an isoscalar field, i. e. JP = 0+. The ”dressed” propagator afterintroducing σ, as outlined in Sec. 3.4.2, has an effective mass M = m0 − σ. This effectivemass is usually called the constituent quark mass. In all calculations in this work are isospinsymmetric. This is why there is no isovector condensate, even though there potentially is acoupling term.

To this original coupling a vector coupling can be added. It is of the form GV2

[ (ψγµψ

)2+(

ψγµγ5~τψ)2 ]

. The condensate resulting from this additional coupling term only contains thetime like component. Off course one could in principle also include the spatial components.This however would not change the outcome of the model as time like and spatial components


are competing degrees of freedom. The quantity 〈Φµ〉〈Φµ〉 is independent of the space-timedirection, the condensate 〈Φ〉 is chosen to point to. The most convenient choice is 〈Φµ〉 = δµ0 〈Φ0〉.The condensate resulting from this is

v = GV 〈ψγ0ψ〉 . (3.104)

According to Sec. 3.4.1 the chiral condensate and the vector condensate do not require anyadditional condensate for the model to remain fully self consistent, as they break differentsymmetries. This corresponds to case d) in Fig. 3.11. The vector condensate v does not breakchiral symmetry, as terms of the form ψR/L γ

0v ψL/R = 0.19 On the other hand the chiralcondensate σ does not break Lorentz invariance, while the vector condensate v does.

Another coupling that can be added to the original Nambu and Jona-Lasinio model is ascalar diquark coupling

H

2

(ψ ΓH C ψ

T) (ψT C ΓH ψ

), (3.105)

with ΓH = γ5τAλA′ . A and A′ are the indices for the antisymmetric subgroups of color andfavor symmetry groups SU(Nc)c and SU(Nf)f . When a condensate arises from this coupling thedirection in color and flavor space may be chose to A = 2 = A′ (see Sec. 3.3.1). This couplingtherefore leads to the condensate

∆ = H 〈ψTCγ5τ2λ2ψ〉 = H 〈ψCγ5τ2λ2ψT 〉∗ . (3.106)

Like σ this condensate breaks the chiral symmetry as

ψTCγ5τ2λ2ψ −→ ψT eiγ5θCγ5τ2λ2eiγ5θψ = ψTCγ5τ2λ2e

2iγ5θψ 6= ψTCγ5τ2λ2ψ. (3.107)

So there is an overlap in the broken symmetries. The chiral SU(2)L×SU(2)R symmetry is brokendown to SU(2)V by both σ and ∆. In addition, ∆ breaks SU(3)C down to a SU(2) subgroupof SU(3)C. In full QCD where the color symmetry is the gauge symmetry the spontaneousbreaking of SU(3)C leads to massive gluons. In NJL-models however the gluons are integratedout. This scenario corresponds to case e) in Fig. 3.11. Therefore an additional condensate isneeded to allow for all possible symmetry breaking patterns. A condensate is needed that breaksan independent set of symmetries but respects all those symmetries, that are still intact.20 Acondensate meeting all requirements is

σ8 = G8 〈ψλ8ψ〉 , (3.108)

originating from the coupling term

δL =G8

2

(ψλ8ψ

)2. (3.109)

The fact that ∆ breaks the global U(1) symmetry, while σ and σ8 do not, does not result in anynew condensate rather than in non-zero quark densities, whenever the diquark condensate ∆ isnon-vanishing.

Finally these two condensates, the vector condensate and the diquark condensate, are com-bined with the chiral condensate. When introducing a scalar diquark coupling and a vectorialcoupling a consequent way to proceed is not to exclude a vectorial diquark channel. The couplingterm in the vectorial diquark channel (3.105) is now realized with the structure ΓH = γ0γ5τAλA′ .Again A and A′ are the indices for the antisymmetric subgroups of color and favor symmetry

19This is due to PL/Rγ0PL/R = γ0PR/LPL/R = γ0 0 = 0 where PL/R are the projectors on the left and right

handed subspaces.20If the new condensate cannot meet the second requirement even more condensates would have to be added.


Lorentz

U(1)

U(1) A

SU(2) f,V

SU(2) f,A

SU(3) C

SU(2) C,rg

vector

diqu

ark

vector

&

diqu

ark

v

v8

σ

σ8

∆

∆0

Table 3.4: Overview over the symmetries of the discussed fields and their compatibility.

groups SU(NC) and SU(Nf). This condensate introduces breaking of Lorentz invariance incombination with color symmetry breaking. For the same reasons given the paragraph aboveand in Sec. 3.4.1 it is necessary to include yet another condensate

v8 = G8V 〈ψγ0λ8ψ〉 . (3.110)

This is the”color density”, the color non-invariant analog of v.In full generality all these condensates should be included. Actually all condensates one

could possibly think of, should be included. This however is obviously not possible and themodifications due to all those condensates would be way smaller than the precision and reliabilityof the model as an approximation to QCD. This section is therefore concluded with an overviewover the discussed fields. Tab. 3.4 lists which symmetries that are broken or left unbroken bythe fields discussed above.

Chapter 4

PNJL: the Polyakov-loop extended

NJL-model in mean field

approximation

The NJL-model described in Chapt. 3 is widely used to study and model chiral symmetrybreaking in QCD. There is however another important feature involved in QCD dynamics, theconfinement (see Sec. 2.3). The NJL model by its nature cannot account for that, as the gluonicdegrees of freedom have been integrated out. This leads to non-negligible quark densities attemperatures well below the transition temperature.

In this chapter an approach is presented that incorporates the effects of confinement intothe NJL model. The aim is to unify the deconfinement phase transition and the chiral phasetransition. This is of interest as there are indications, that the two phase transitions appear tobe basically coincident. This suggests that the coinciding transitions are due to some mechanismintertwining chiral restoration and deconfinement phenomena.

4.1 Gluon dynamics

Gluon dynamics is a key to the understanding of QCD, as it is responsible for confinement.At temperatures of interest for a NJL model (T / Λ ≈ 0.6–1.0 GeV) gluons are in the non-perturbative regime. For this reason the knowledge of gluon dynamics is mainly limited to dataobtained in lattice QCD calculations. Fortunately the calculations in the pure gluonic sector,i. e. without fermions are numerically much cheaper than calculations including fermions. Forthis reason there is a quite good knowledge on quantitative grounds of the behavior of a puregluonic system. One has to mention however that this knowledge unfortunately is limited tocertain characteristic quantities and does not include a deeper understanding of the mechanismsgenerating these figures. In the following we want to exploit lattice calculations as a source ofaccurate numbers. It is made the effort here to give these numbers a meaning in the context ofa PNJL-model, which is simple enough to understand the functioning of this machinery.

4.1.1 Polyakov loops

The most important results of lattice QCD calculations in the pure gluonic sector are ther-modynamic quantities. One of the obtrusive questions, when considering these quantities, iswhich quantity works as an order parameter of this system. Even if there are no fermions in thesystem there exists confinement, as quarks themselves are color charged objects. Confinementcertainly is realized, if two static color sources are bound infinitely deep. Infinitely deep means,that the free energy of the system diverges, when going to larger separations of the two color

43

44 Chapter 4. The PNJL-model in mean field approximation

sources. Confinement in this definition is an idealization. In a real system with quarks of finitemass confinement cannot be strictly proven according to this definition. There will always bescreening effects generated by pariticle antiparticle production spoiling this infinity in the freeenergy.

The Wilson loop is a transformation operator that connects the configuration on the gaugegroup manifold of one point in space-time to the configuration of another point.

W = P exp

i

∫ point B

point AdxµAµ(x)

(4.1)

After analytically continuing the formalism to the complex time plane, the Wilson loop can aswell connect two points distinct in imaginary space-time. A timelike Wilson line connecting twopoints at t = 0 and t = −iβ is called Polyakov loop1

L = P exp

i

∫ β

0dτ A4(τ, ~x)

. (4.2)

The gauge fields2 are part of the covariant derivative Dµ = ∂µ − iAµ, that is acting on quarkfield. When integrating the covariant derivative acting on a quark field a general relation ofthese quark fields at the two endpoints of the path is established. This connection may be usedto connect quark fields at t = 0 and t = −iβ. The quark fields at these endpoints are connectedwith the formula for the thermal expectation values (3.67). This connection is exploited in[MS81a, MS81b] to establish

e−βFqq(~x−~y,T ) = 〈Φ(~x)Φ†(~y)〉β with Φ(~x) =1

NCtrC [L(~x)] , (4.3)

where Fqq(~x − ~y, T ) is the free energy at temperature T of two color sources q and q with thespatial separation ~x− ~y. For infinite separation ~x− ~y the correlation of Φ(~x) and Φ†(~y) die out.Which leads to the free energy for a quark antiquark pair at infinite distance

F∞ = limr→∞

Fqq(r = |~x− ~y| , T ) = −T log |〈Φ〉|2 . (4.4)

Even though the free energy of a single static quark does not have a physical meaning, the freeenergy is defined by

〈Φ〉 = e−βFq . (4.5)

〈Φ〉 is also called renormalized Polyakov loop. The renormalized Polyakov loop is of the form

〈Φ〉 = e−12βF∞(T ) (4.6)

It is quite clear, that in the idealized confined phase the free energy of a single quark hasto diverge. The Polyakov loop in this case vanishes. Whenever deconfinement sets in therenormalized Polyakov loop will have a non-zero value. In a pure gluonic system the renormalizedPolyakov loop therefore is an order parameter. This however is only true in the idealized caseof one single quark in whole space. As soon as quarks enter the system on thermodynamic basisrather than on the basis of a static probe the idealization of infinitely separated color sourcescan no longer hold. As these color sources would have infinite energy, a thermodynamic systemwould always prefer configurations of lower energy, i. e. configurations with finite separation ofthe color sources. Hence in a thermodynamic system with quarks the free energy of a quark

1In contrast to many publications in this work the Polyakov loop denoted L is a quantity, that still has aSU(Nc)c group structure, while Φ is defined as the normalized trace of L: Φ(~x) = 1

NctrC [L(~x)].

2The coupling constant g was absorbed by the gauge field redefinition (See Sec. 2.1).

4.1. Gluon dynamics 45

does not diverge meaning that strictly 〈Φ〉 > 0. In this case the renormalized Polyakov loop isonly approximately an order parameter. The informational contents of an order parameter inthe Polyakov loop decreases even further at non-zero chemical potentials.

It is important to note, that in the case of non-zero quark chemical potential the free energiesof a quark and an antiquark are of course different. Charge conjugating the quarks means tointerchange color sources and color sinks. The charge conjugation interchanges Φ and Φ† in(4.3). Following again the derivation of the renormalized Polyakov loop 〈Φ〉 one finds that thePolyakov loop is a measure for the free energy of the quarks, while the hermitian conjugatedetermines the free energy of the antiquarks.

〈Φ〉 = e−12βF q

∞(T ) 〈Φ†〉 = e−12βF q

∞(T ) (4.7)

Consequently, 〈Φ〉 is a measure for the confinement of the quarks, while 〈Φ†〉 indicates thestrength of the confinement of antiquarks. And indeed there is no reason why at non-zerochemical potential (µ 6= 0) the confinement-deconfinement transition of quarks and antiquarksshould happen simultaneously.

The confinement-deconfinement transition in a pure gluonic system is connected to an evenmore fundamental process, namely spontaneous symmetry breaking of the Z(NC)center sym-metry of SU(NC) [Wei81]. The gauge field nature of the gluon fields allows to perform gaugetransformations U satisfying the periodicity conditions at t = 0 and t = −iβ:

U(t = −iβ) = U(t = 0). (4.8)

There are however gauge transformations that maintain the periodic boundary conditions onthe gauge fields Aµa(t = 0) = Aµa(t = −iβ) but do not fulfill (4.8). Such transformations satisfyless strict boundary conditions

U(−iβ) = e2πik

Nc U(0) with k = 0, 1, . . . , Nc − 1. (4.9)

Of course this does not change the nature of U(t = −iβ) and U(t = 0) being elements of

SU(NC). In fact e2πk

Nc with k = 0, 1, . . . , NC − 1 is an element of Z(NC), which is the centersymmetry of SU(NC). The Polyakov loop is not invariant under such transformations either:

〈Φ〉 gauge trafo (4.9)−−−−−−−−−−→ e−2πi kNc 〈Φ〉 . (4.10)

The statement that these transformations are allowed due to symmetry is equivalent with thestatement 〈Φ〉 = 0. As explained above 〈Φ〉 = 0 corresponds to confinement (Fq → ∞) while〈Φ〉 6= 0 automatically allows for deconfinement (Fq < ∞). From this it can be concludedthat confinement is present whenever the Z(NC) center symmetry is unbroken. Spontaneoussymmetry breaking of the Z(NC) center is equivalent to deconfinement.

With this knowledge the statement above, that confinement is an idealized situation at zeroquark density, can be put on more solid grounds [Wei82]. In presence of quark the transfor-mations (4.9) are connected to the thermodynamic boundary conditions of the quarks by themechnism used to derive (4.3). If a gauge transformation is applied to the quark fields withantiperiodic boundary conditions ψ(−iβ) = ψ(0) these boundary conditions are spoilt:

ψ(0)gauge trafo (4.9)−−−−−−−−−−→ ψ′(0) = U(0)ψ(0)

ψ(−iβ)gauge trafo (4.9)−−−−−−−−−−→ U(−iβ)ψ(−iβ) = e2πi

kNc U(0)ψ(−iβ) = −e2πi

kNc U(0)ψ(0)

⇒ − e2πik

Nc ψ′(0) 6= −ψ′(0). (4.11)

I. e. there is no Z(NC) center symmetry in the presence of quarks and 〈Φ〉 6= 0. The author of[Wei82] concludes that static external sources are not confined, but are screened by dynamic colorsources generated through pair production. The presence of explicit and spontaneous symmetrybreaking implies, that the deconfinement transition can only be a cross-over transition.


4.1.2 The Polyakov loop model

As the Polyakov loop is an order parameter of the pure gluonic theory, it can be used to describethe dynamics. A straight forward way to do that is a Ginzburg-Landau type ansatz. At thisstage there are no quarks resulting in

〈Φ〉 = 〈Φ†〉 (4.12)

Additionally the ansatz for the effective potential should incorporate the spontaneous symmetrybreaking of the center Z(NC) symmetry of SU(NC), whenever the system is in the deconfinedphase. The spontaneous symmetry breaking implies, that the underlying potential has to respectthe symmetry, that is to be broken spontaneously.3 On the other hand, the potential must notcontain a higher symmetry. This constrains the terms allowed in the potential. Functions of|〈Φ〉| are allowed as they respect Z(NC = 3), but they are not sufficient, as they are in fact U(1)-symmetric. Therefore there needs to be a term that breaks this U(1) respecting Z(NC = 3).Such terms are functions of Re

[Φ3].4

The ansatz for the potential has to allow for a first order phase transition. I. e. for T = T0 thepotential U(Φ, T ) needs to have two distinct minima at Φmin.,1 and Φmin.,2 with U(Φmin.,1, T0) =U(Φmin.,2, T0). This requires the potential to contain at least a second and a fourth order termin Φ. If the simplest U(1)-breaking term is added one ends up with the minimal ansatz for thepotential

U(Φ)

T 4= −1

2b2(T )Φ∗Φ− 1

3b3(T ) Re

[Φ3]+

1

4b4(T ) (Φ∗Φ)2 . (4.13)

The signs are chosen such that the coefficient are positive for T = T0, where the potential U hastwo minima at Φmin.,1 = 0 and Φmin.,2 > 0. The coefficients b2, b3 and b4 also need to includethe temperature dependence of the potential. The temperature dependence needs to be suchthat there is a phase transition at T = T0. This means that the global minimum at T < T0 isat Φ = Φmin. = 0. While for T > T0 the global minimum is at Φ = Φmin. with 0 < Φmin. < 1.One way this can be implemented is by choosing b3(T ) = b3 = const., b4(T ) = b4 = const. and

b2(T ) = a0 + a1

(T0

T

)+ a2

(T0

T

)2

+ a3

(T0

T

)3

. (4.14)

The parameters of this Ginzburg-Landau ansatz are now to be fixed to reproduce the thermo-dynamic potentials. As there are no pure gluonic systems in nature, one has to rely on latticesimulations of gluon dynamics.

The ansatz (4.13) and (4.14) is fitted to the energy density, the entropy density and thepressure simultaneously, which were calculated from the data given in [B+96]. This mean squarefit and the data are plotted in Fig. 4.1. As a cross check the prediction of this fit for the Polyakovloop can be compared to lattice predictions [KKPZ02] (see Fig. 4.1). The resulting values forthe Polyakov loop overestimate the lattice results. The parameter fit given in [RTW06] is shownin Fig. 4.2. This fit was optimized to describe the behavior of the Polyakov loop. This requiredto take a worse fit for the pressure, the entropy density and the energy density. For the sake ofcompatibility these parameters are used in the mean field calculations presented in this chapter.

One main difference of the two parametes sets is the parameter b3 determining the strengthof the breaking of U(1) down to Z(NC). This stronger breaking seems to be crucial for thecalculations that go beyond mean field approximation (see Chapt. 5).

There are however other possibilities to chose an ansatz for the effective loop potential.Following [Fuk04] the fourth order term is replaced by log [J(Φ)], where J(Φ) is a Jacobi de-terminant, in which the volume of the excess degrees of freedom was integrated out.5 If all

3If the potential does not respect the broken symmetry, this is called explicit symmetry breaking.4Potentially one could additionally have terms that are funcitons of Re

ˆΦ3k

˜, k ∈ N.

5The group volume can be integrated out via the Haar measure.


a0 a1 a2 a3 b3 b4

Data taken from [RTW06] 6.75 −1.95 2.625 −7.44 0.75 7.5

Fit to data from [B+96] 4.07 −2.42 4.31 −7.31 8.84 12.9

Table 4.1: The parameterization of the gluon potential. The parameterization given in [RTW06]is used for the mean field calculations, the second parameterization is used for the calculationsbeyond mean field approximation.

0 1 2 3 4 5TTc

0

1

2

3

4

5

3 pT4

3 s4 T3

¶

T4

0 0.5 1 1.5 2 2.5 3TTc

0

0.2

0.4

0.6

0.8

1

Fm

inL

renHTL

Lattice data

Ueff HFmin L=min.

Fmin ,

Figure 4.1: Left: Mean square fit and data given in [B+96]. Right: Comparison of the Polyakovloop dynamics with the data in [KKPZ02].

0 1 2 3 4 5TTc

0

1

2

3

4

5

3 pT4

3 s4 T3

¶

T4

0 0.5 1 1.5 2 2.5 3TTc

0

0.2

0.4

0.6

0.8

1

Fm

inL

renHTL

Lattice data

Ueff HFmin L=min.

Fmin ,

Figure 4.2: Left: Fit given in [RTW06] and data given in [B+96]. Right: Comparison of thePolyakov loop dynamics determined by the parameters of [RTW06] with the data in [KKPZ02].


non-diagonal group elements of SU(3) are integrated out one finds6

1

VSU(3)

∫· · ·∫ ∏

i= 1,2,4,5,6,7

dφi = J(φ3, φ8) (4.15)

=1

6π2

(sin(2φ3)− sin(φ3 +

√3φ8) + sin(φ3 −

√3φ8)

)2(4.16)

=2

3π2

(cos(φ3)− cos(

√3φ8)

)2sin2(φ3). (4.17)

This can however be written in terms of the normalized trace of the exponential of the repre-sentation of the group elements λ3 and λ8

Φ =1

3tr [exp (i (φ3λ3 + φ8λ8))] (4.18)

and the complex conjugate thereof. The name Φ was chosen, because this is exactly the definitionof the Polyakov loop in mean field approximation, where φ3,8 = βA3,8.

J(Φ,Φ∗) =9

8π2

(1− 6Φ∗Φ + 4

(Φ∗3 + Φ3

)− 3 (Φ∗Φ)2

)(4.19)

Note however that now J is no longer the Jacobian with respect to the variables Φ and Φ∗.Changing variables would introduce additional Jacobian determinants. The actual integrationhowever becomes cumbersome as the limits of integration are difficult to handle.7

The form of the potential log [J(Φ∗,Φ)] is motivated by the path integral∫DA

∫Dψ

∫Dψ e−SE

=∏

ω,~p

∫· · ·∫ [ ∏

j= 1,2,4,5,6,7

dAω,~p,j

]∫dAω,~p,3

∫dAω,~p,8

∫dψω,~p

∫dψω,~p e

−SE

=

∫DA3

∫DA8

∫Dψ

∫Dψ[∏

ω,~p

J(Aω,~p,3

T,Aω,~p,8T

)]e−SE

=

∫DA3

∫DA8

∫Dψ

∫Dψ exp

− SE +

∑

ω,~p

log

[J(Aω,~p,3

T,Aω,~p,8T

)],

(4.20)

where the gloun fields A(t, ~x) were expressed in terms of the basis λi ei(ωt−~p·~x):

A(t, ~x) =∑

ω,~p

N2C−1∑

j=1

Aω,~p,jλj ei(ωt−~p·~x),

where λi are the Gell-Mann matrices acting in color space. This procedure left us with the term∑ω,~p log

[J(Aω,~p,3

T ,Aω,~p,8

T

)]in the exponential that acts like an effective potential. Integration

out all spatial and temporal fluctuations as it is done in mean field approximation leaves us with

Veff = −k log

[J(A3

T,A8

T

)], (4.21)

6The result of the partially integrated Haar measure shown here is based on calculations by Norbert Kaiser,who made this formula accessable to me.

7For the φ3, φ8 parameterization the region of integration is periodic of course. One periodic patch is givenby the inequalities 0 ≤ φ3 + φ8√

3< 2π and 0 ≤ −φ3 + φ8√

3< 2π or any other patch, that can be mapped via the

periodicity onto this patch. It can be helpful to introduce new variables θ1 = φ3 + φ8√3

and θ2 = −φ3 + φ8√3.


0 1 2 3 4 5TTc

0

1

2

3

4

5

3 pT4

3 s4 T3

¶

T4

0 0.5 1 1.5 2 2.5 3TTc

0

0.2

0.4

0.6

0.8

1

Fm

inL

renHTL

Lattice data

Ueff HFmin L=min.

Fmin ,

Figure 4.3: Simultaneous fit to data given in [B+96] (left) and [KKPZ02] (right). The parametersfound for the potential (4.22) are listed in Tab. 4.2.

where the constant k describes all contributions from the functional trace∑

ω,~p . We do notwant to do explicit calculations of that trace here, as we aim at fitting b4 to lattice data, thatof course have all this information about SU(3)C incorporated already.

A nice feature about this effective potential Veff is that U(1) is already broken by the Jacobian(4.19). As the U(1) breaking is already implemented we leave out the explicit Re

[Φ3]-term in

our ansatz for the loop potential

U(Φ)

T 4= −1

2b2 (T ) Φ∗Φ + b4 (T ) log

[1− 6Φ∗Φ + 4

(Φ∗3 + Φ3

)− 3 (Φ∗Φ)2

]. (4.22)

Here b2 (T ) is chosen as in (4.14) where b4 (T ) is defined by

b4 (T ) = b4

(T0

T

)3

. (4.23)

This choice of temperature dependence is motivated by the calculation (4.20). The prefactor k

in (4.21) has emerged from a functional trace which can be written as Trfunc. = V∑

ωn

∫ d3p(2π)3

.

Therefore k should scale as VT , which in turn means that b4 (T ) scales with T−3.8

The parameters were fixed, such that the model describes both the lattice data for pressure,entropy density and energy density [B+96] and additionally the behavior of Polyakov loop pre-dicted by lattice calculations [KKPZ02]. The fit for the thermodynamic quantities and for thePolyakov loop are shown in Fig. 4.3. It can be seen that this fit nicely fits to both pressure,entropy density and energy density on the one side and the Polyakov loop on the other side. Itfits to the high temperature loop quite well, and produces a strong transition at T = T0. The fitwas constrained such that the first order phase transition takes place at T = T0 and such thatthe Stefan-Boltzmann limit is satisfied. The parameters obtained are listed in Tab. 4.2. Thispotential now diverges once the Polyakov loop Φ approaches 1. This can be seen quite nicely inFig. 4.4.

Having a second look at (4.20) we find that k should approximate the number of accessablestates. k is connected to b4 via

k = −T 4 b4(T )Trfunc.constant fields−−−−−−−−−−−→

in space and timek = −β V

(T T 3

0 b4)

= −V T 30 b4. (4.24)

8Note that the normalization in (4.22) simply results in an additive constant. The normalization was chosensuch that b4 is a dimensionless constant just as a0,...,3.


a0 a1 a2 a3 b4

Fit to data taken from [B+96, KZ05] 3.51 −2.56 15.2 −0.617 −1.68

Table 4.2: The parameterization of the gluon potential (4.22) explicitly including SU(3)C-constraints.

0 0.2 0.4 0.6 0.8 1Φ

-1

-0.5

0

0.5

1

1.5

UT4

0.5 T0

0.75 T0

1.0 T0

1.25 T0

2.0 T0

Figure 4.4: The potential (4.22) plotted as a function of the Polyakov loop. A first order phasetransition is located at T = T0 as the minimum Φmin. jumps from zero to a finite value. Thepotential has SU(3)C-constraints build in: The potential diverges for Φ→ 1.

If we now set k to the number of states N we find

n =N

V= −T 3

0 b4 ≈ 0.033 GeV3 ≈ 4.3 fm−3 (4.25)

for the density of states. If we assume a three momentum cutoff Λ this density can be translatedinto a momentum scale:

n =1

(2π)34π

3Λ3 ⇒ Λ =

3√

6π2n ≈ 1.25 GeV (4.26)

Interestingly Λ is of the order of the cutoffs used in PNJL-models.

4.2 Incorporating gluon dynamics into a PNJL-model

In the previous the NJL-model and the Polyakov loop model have been presented. The NJL-model describes chiral symmetry breaking, while the Polyakov loop model incorporates thedynamics of the quarks. It has already been discussed, that the presence of quarks in a systemwith gluons changes the situation dramatically. On the other hand, the effect of the gluons andthe phenomenon of confinement is most likely having big impact on the quarks in a system. Inthis section a way to combine these two models in such a way that it is indeed a combinationand not just the sum of two individual models. The method shown here that combines the NJL-model and a Polyakov loop model aimes towards a simultaneous transition of chiral symmetryrestoration and deconfinement [Fuk04].

4.2. Incorporating gluon dynamics into a PNJL-model 51

4.2.1 Coupling quarks and Polyakov loop

The Polyakov loop is a functional of the gluon fields (4.2). The interaction of the quarks andthe loop is therefore most easily implemented by adding the standard QCD interaction term tothe model

δL =(ψγµtaψ

)Aµa . (4.27)

Spatial fluctuations are neglected here, just as it is usually done in NJL-models. Therefore it isnot necessary to consider the spatial components of the gluon fields. Only the time componentis treated

Aµa = δµ0 A0a. (4.28)

Still there are N2c − 1 = 8 degrees of freedom in these fields. The Polyakov loop Φ and its

complex conjugate Φ∗ however are only two real degrees of freedom. If fluctuations are neglectedconsistently, six out of eight gluon fields do not contain any essential information. These sixdegrees of freedom are contained in the SU(3)C group structure. In models in which the fieldsare treated as spatially constant a local gauge symmetry degenerates to a global symmetry.Therefore the elements of SU(3)C can be rotated by an arbitrary transformation. It is thereforepossible to rotate such that only group elements with diagonal representation contribute. In theGell-Mann representation the diagonal matrices are λ3 and λ8. The six redundant degrees offreedom are absorbed in the rotation to the element of diagonal representation. The remainingtwo degrees of freedom are the ones contained in Φ and Φ∗.

There is however a slight flaw in the argument above. When rotating in color space notonly the interaction term (4.27) under the limitation (4.28) is transformed. The rest of theLagrangian is transformed as well. Here the rotation is however only done for the loop couplingterm and not for the whole Lagrangian. The only constraint this model has to meet is a goodapproximation of QCD. Terms of prime importance are taken into account, while unimportantones shall be dropped for simplicity. The interaction term here is:

δL = ψ (γ0 ta=3,8)ψA0a=3,8. (4.29)

In Euclidean time this reads

δLE = i ψ (γ0 ta=3,8)ψA4a=3,8, with A4 = A4 = iA0. (4.30)

When the gauge fields in Euclidean time are limited as in (4.30) the Polyakov loop is pa-rameterized by

Φ =1

3

(ei

„

A43

T+

A48√

3 T

«

+ ei

„

−A4

3T

+A4

8√3 T

«

+ ei

„

−2A4

8√3 T

«). (4.31)

Here the gauge fields were treated as constants in time. On the one hand, the value for Φ inthis parameterization determines the value of the effective gluon potential. On the other hand,the interaction term (4.30) changes the structure of the quark propagator. The quark behaviorinfluences the loop transition and vice versa.

4.2.2 Effect of the loop on symmetries and its interplay with condensates

The way the Polyakov loop was coupled to the NJL model by (4.27) affects the symmetryproperties of the Lagrangian. The gluon mean fields are formally added to the propagator atthe same level as other condensates. Therefore the discussion in Sec. 3.4.8 is applicable to thegluonic mean fields parameterizing the Polyakov loop as well.

The crucial step in Sec. 3.4.8 was the classification of the symmetry properties of the conden-sates. This classification is shown in Tab. 4.3. The condensate

(ψ γ0t3 ψ

)A4

3 breaks the subgroupSU (2)C,rg of SU (3)C. Following the procedure in Sec. 3.4.8 this would require many additional


Lorentz

U(1)

U(1) A

SU(2) f,V

SU(2) f,A

SU(3) C

SU(2) C,rg

(ψ γ0t3 ψ

)A4

3 (ψ γ0t8 ψ

)A4

8

Table 4.3: Classification of the symmetries broken or unbroken by the presence of the loopcoupling term. This table extends Tab. 3.4.

symmetry breaking terms and condensates. The explicit solutions to the mean field equations dohowever not break the subgroup SU (2)C,rg. This can be shown by gereral arguments presentedin the following paragraphs.

When charge conjugating the gluon fields they simply change sign. In the second step thefirst two indices9 in the 3×3-dimensional matrix representation10 of color space are interchanged.This is done on both rows and columns of the inverse propagator. In cases treated in this workthis procedure leads to the rule

A43 −→ +A4

3 and A48 −→ −A4

8. (4.32)

All other field present in the explicit forms of the Lagrangian used in Sec. 4.2.3, Sec. 4.2.4 andSec. 4.2.5 remain unchanged. Therefore the thermodynamic potential fulfills

Ω(A48) = Ω(−A4

8) =1

2

(Ω(A4

8) + Ω(−A48)). (4.33)

From this it can be concluded that the thermodynamic potential has an extremum at A48 = 0.

This is one of the self consistency equations, that is present in all PNJL models in this work.This of course does not mean that there cannot be other solutions. However I have not foundany that allow to fulfill the other self consistency equations simultaneously. Substituting backA4

8 = 0 in (4.31) immediately gives

Φ ∈ R ⇒ Φ∗ = Φ. (4.34)

This is true whenever the mean field self consistency equations are used to calculate the fieldexpectation values.

The reasoning in the paragraph above can be applied to A43 as well. Here it is however

possible to find solutions other than A43 = 0 that allow to fulfill the remaining self consistency

equations at the same time. If a solution at A43 6= 0 exists it is in all cases we encountered

energetically favored over A43 = 0. I. e. if both A4

3 6= 0 and A43 = 0 are solutions to the self

consistency equations A43 6= 0 minimizes the thermodynamic potential in all cases we found.

Then at A43 = 0 there exists a local maximum of the thermodynamic potential.

Nevertheless the symmetry of A43 is very useful in the discussion of the necessity of additional

condensates. When A48 = 0 there exists a symmetry under sign change of A4

3

Ω(A43) = Ω(−A4

3). (4.35)

9The first two indices are named ”red” and ”green” here.10This of course relies on the representation chosen here. Namely it relies on the fact that the Gell-Mann λ3 is

only non-zero in the upper left 2 × 2 block while λ8 is the identity in the upper left 2 × 2 block. In the presenceof diquarks it also relies on the choice of direction in the color anti-triplet 3 subspace. It is essential that thediquark fields are represented by the Gell-Mann λ2 as this representation again shows the 2 × 2 block structurein the upper left. When the first two colors (say ”red” and ”green”) are interchanged in rows and colomns for arepresentation of the diquark fields by ∆ ∝ λ2 the diquark fields change sign. This sign change however can beabsorbed into the complex phase of the diquark condensate, which does not change the thermodynamic potential.


scalar

vector

diqu

ark

vector

&

diqu

ark

v

v8

σ

σ8

∆

∆0 (ψ γ0t3 ψ

)A4

3 (ψ γ0t8 ψ

)A4

8

Table 4.4: Fully self consistent sets of condensates including the gluon fields parameterizing thepolyakov loop (compare Tab. 3.4).

Additional fields could potentially have two behaviors under this transformation: Either theychange sign as well, or they do not change sign. In the first case the same reasoning as for A4

8

can be applied to show that the self consistency equation is satisfied if the field vanishes. In thesecond case the field will couple to the first two directions in color space (say ”red” and ”green”)equally. It is a singlet in the SU (2)C,rg subspace of SU (3)C. Such an additional field hasalready been introduced, namely σ8. In the vector case, v8 is needed in addition. Even thoughthe interaction term

(ψ γ0t3 ψ

)A4

3 breaks the color subgroup SU (2)C,rg in general, the explicit

solution of the mean field equations (A48 = 0) allows to absorb the symmetry breaking effects

into other symmetries. This is necessary for diquark fields ∆ and ∆0. These fields change signunder the discussed symmetry operation taking A4

3 → −A43. The complex phase of the diquark

fields is free however, which allows for solutions of the mean field equations with non-vanishingdiquark fields. We summarize that no fields other than σ8 and v8 are needed when includinga Polyakov loop interaction terms of the form (4.30). This is even true in the presence of thediquark fields ∆ and ∆0 (see Tab 4.4).

4.2.3 PNJL-models without diquarks

In this section the most simple PNJL model for Nf = 2 is presented. It consists of an NJL partcontaining only the chiral field. The connection to the Polyakov loop model is accomplished withthe method presented in Sec. 4.2.1. The condenates chosen here are not in compliance with theprocedure discribed in Sec. 4.2.2. Although the gluon fields parametrizing the Polyakov loopbreak color symmetry, in this section the condensate σ8 = 〈ψλ8ψ〉, that breaks color symmetry,is not considered here. This simplifies the computations significantly and can be justified by thesmall changes in the thermodynamic potential caused by this field.

The Lagrangian for this most simple model is

LPNJL = ψ (iγµDµ − m0)ψ +

G

2

[(ψψ)2

+(ψiγ5~τψ

)2]− U(Φ,Φ∗, T ), (4.36)

where the mass matrix m0 is isospin symmetrically chosen, m0 = 1m0 with the current quarkmass m0. Φ again is the normalized trace of the Polyakov loop. The covariant derivative isdefined as

Dµ = ∂µ − iAµ with Aµ = δµ0 A0, A0 = g A0

a

λa2

and A0a = δa3A

03 + δa8A

08. (4.37)


Proceeding along the lines sketched in Sec. 3.4.5 the thermodynamic potential can be eval-uated to

Ω = U(Φ,Φ∗, T )− T∑

n

∫d3p

(2π)3Tr log

[βS−1(iωn, ~p)

]+σ2

2G, (4.38)

where ωn = (2n + 1)π T are the Matsubara frequencies for fermions and the inverse Nambu-Gor’kov propagator11 is

S−1(ωn, ~p) =1

2

(iγ0ωn − ~γ · ~p−M − γ0(µ+ iA4) 0

0 iγ0ωn − ~γ · ~p−M + γ0(µ+ iA4)

).

(4.39)The mass M here is the dynamically generated quark mass, defined as M = m0 − σ.

This can be evaluated explicitly to

Ω = −Nf T

∫ Λ′d3p

(2π)3

(Tr log

[1 + L e−β(E~p−µ)

]+ Tr log

[1 + L† e−β(E~p+µ)

])

− 6Nf

∫ Λ d3p

(2π)3E~p +

σ2

2G+ U(Φ,Φ∗, T ). (4.40)

Here the divergent integral is regularized by the three momentum cutoff Λ. The first, nondivergent integral labeled with the upper integrational boundary Λ′ is integrated from 0 to Λusing the dynamically generated quarkmass and from Λ to infinity using the current quarkmass.12 The reason why this ”soft” cutoff is implemented is the asymptotic freedom of QCD atlarge spatial momenta. In this cutoff scheme quarks are freed at the spatial momenta of order Λ.The energy E~p is defined as usual E~p =

√~p2 +M2 where M = m0 − σ is the constituent quark

mass. The first term in (4.40) can be evaluated further by using the identity13 tr logA = log detA[RTW06]

log det[1 + L e−β(E~p−µ)

]+ log det

[1 + L† e−β(E~p+µ)

]=

log[1 + 3

(Φ + Φ∗ e−β(E~p−µ)

)e−β(E~p−µ) + e−3β(E~p−µ)

]

+ log[1 + 3

(Φ∗ + Φ e−β(E~p+µ)

)e−β(E~p+µ) + e−3β(E~p+µ)

]

(4.41)

This simplification is possible as long as the field σ8 = 〈ψλ8ψ〉 is neglected.The expression in (4.40) is valid, if the mean field configuration is chosen such that the

thermodynamic potential is minimal. That is, the self consistency equations need to be fulfilledat any time (see Sec. 3.4.7). In this example the self consistency equations are

∂Ω

∂σ=

∂Ω

∂A43

=∂Ω

∂A48

= 0 (4.42)

When considering the parameterization of the Polyakov loop Φ (4.31) one can establish theinfinitesimal connection between loop Φ and its parameterization A4

3 and A48.

(dΦdΦ∗

)=

[∂Φ∂A4

3

∂Φ∂A4

8∂Φ∗

∂A43

∂Φ∗

∂A48

](dA4

3

dA48

)=

[∂ (Φ,Φ∗)

∂(A4

3, A48

)](

dA43

dA48

)(4.43)

11Note that the Nambu-Gor’kov propagator is defined in Eqn. (3.57) with an additional factor 12. In other

calculations this factor 12

is part of the thermodynamic potential.12In many other calculations this ”soft” cutoff is not implemented.13The logarithm on the left can be evaluated, when the matrix A is rewritten as A = T−1DT , where D is a

diagonal matrix. The lefthand side then is tr(T−1 log[D]T ) = tr logD =P

i log λi, where λi are the eigenvaluesof A. Now the sum can be pulled to the right of the logarithm turning into a product

Pi log λi = log

Qi λi. But

the product of all eigenvalues of A is just the determinant of A giving the desired result.


As long as the Jacobian matrix connecting the two sets of variables is non-singular the followingis true:

∂Ω

∂A43

=∂Ω

∂A48

= 0 ⇐⇒ ∂Ω

∂Φ=

∂Ω

∂Φ∗= 0 (4.44)

This means that the self consistency condition for the loop Φ and those of their parameterizationwith the gluon fields are equivalent.

The symmetries in the Nambu-Gor’kov fields can be exploited to solve the rightmost selfconsistency equation in (4.42) analytically. This has been done in Sec. 4.2.2. The self consistencyequation is solved by A4

8 = 0.

4.2.4 PNJL-models including scalar diquarks

In this section the simplest PNJL model for Nf = 2 is generalized for the first time, to includediquarks. The NJL-part of this model is treated in greater detail in [Bub05]. The condensatesneeded for a consistent treatment of the problem are listed in the third column on the right ofTab. 4.4.

The Lagrangian for this most simple model with explicit inclusion of diquark degrees offreedom is

LdiquarkPNJL = LPNJL+

H

2

[(ψCγ5τ2λ2ψ

T) (ψTγ5τ2λ2Cψ

)]+G8

2

[(ψλ8ψ

)2+(ψiγ5~τλ8ψ

)2], (4.45)

where LPNJL was defined in Sec. 4.2.3. The ratios of the three coupling constants are fixed by theconstruction of the NJL Lagrangian. This construction starting at a color current interaction ispresented in detail in App. B.2. The direction of the diquark condensate in the anitsymmetriccolor anti-triplet channel was chosen to point in λA′=2 direction. The thermodynamic potentialevaluates to

Ω = U(Φ,Φ∗, T )− T∑

n

∫d3p

(2π)3Tr log

[βS−1(iωn, ~p)

]+σ2

2G+

σ28

2G8+

∆∗∆

2H. (4.46)

The inverse Nambu-Gor’kov propagator including diquarks is now of a non-block-diagonal form

S−1(ωn, ~p) =1

2

(iγ0ωn − ~γ · ~p− M − γ0(µ+ iA4) ∆γ5τ2λ2

−∆∗γ5τ2λ2 iγ0ωn − ~γ · ~p− M + γ0(µ+ iA4)

).

(4.47)The mass M is no longer a color singlet: M = m0 1C−σ 1C−σ8λ8. The dynamically generatedmasses for different colors no longer have to be equal. The expression Tr log

[βS−1(iωn, ~p)

]is

now evaluated as sketched in Sec. 3.4.6. To do this the roots of det S−1(p0) = 0 as a polynomialin p2

0 need to be determined. Then we made use of (3.93). The energies in (3.93) equal the rootsof p2

0 = 0 with positive real part. These six roots are

Eb± =√p2 +M2

b ± µb Er±(a),±(b)= Er±(a)

±(b)iA4

3

with Er±(a)=√p2 +M2

r + µ2r + |∆|2 ±

(a)2µr√p2 +M2

r ,(4.48)

where the generated masses and the generalized chemical potentials are now color dependentand given by

Mb = m0 − σ +2√3σ8 Mr = m0 − σ −

1√3σ8

µb = µ− 2√3iA4

8 µr = µ+1√3iA4

8 A4 = A43λ3 +A4

8λ8.

(4.49)


Applying the Matsubara formalism (see Sec. 3.4.5) the thermodynamic potential becomes

Ω = −2NfT

∫ Λ′d3p

(2π)3

∑

j∈J

log[1 + e−βEj

]− 6Nf

∫ Λ d3p

(2π)3

∑

j∈J

Ej

+σ2

2G+

σ28

2G8+

∆∗∆

2H+ U (Φ,Φ∗, T ) . (4.50)

Where J is an index set numbering all 6 energies in equations (4.48). The applied cutoffprocedures indicted by Λ and Λ′ are implemented as in Sec. 4.2.3.14 Note that in the caseincluding off-diagonal diquark terms it is no longer possible to factor out the classical fieldvalues of the Polyakov loop Φ and its complex conjugate Φ∗ as it was done in (4.41). Finallythe sum over j ∈ J in (4.50) is evaluated explicitly

Ω = −2NfT

∫ Λ′d3p

(2π)3log[1 + e−βE

b+

]+ log

[1 + e−βE

b−]

+ log[1 + 2 cos

(β A4

3

)e−βE

r+ + e−2βEr

+

]+ log

[1 + 2 cos

(β A4

3

)e−βE

r+ + e−2βEr

+

]

− 6Nf

∫ Λ d3p

(2π)3

[E

(b)+ + E

(b)− + 2E

(r)+ + 2E

(b)−

]+σ2

2G+

σ28

2G8+

∆∗∆

2H+ U (Φ,Φ∗, T ) . (4.51)

The self consistency equations that can be derived from this thermodynamic potential are

∂Ω

∂σ=

∂Ω

∂σ8=

∂Ω

∂∆=

∂Ω

∂A43

=∂Ω

∂A48

= 0 (4.52)

4.2.5 PNJL-models including scalar and vector diquarks

In this section the PNJL model of Sec. 4.2.4 is extended. The NJL-part of this model is treated ingreater detail in [Bub05]. In addition the model in this section includes vector degrees of freedom.The condensates needed for a consistency are listed in the rightmost column of Tab. 4.4.

The Lagrangian LdiquarkPNJL of Sec. 4.2.4 is supplemented by additional vector coupling terms

Lvector & diquarkPNJL = Ldiquark

PNJL +GV

2

[(ψγ0ψ

)2+(ψγ0γ5~τψ

)2]

+GV8

2

[(ψγ0λ8ψ

)2+(ψγ0γ5~τλ8ψ

)2]

+H0

2

[(ψCγ0γ5τ2λ2ψ

T) (ψTγ0γ5τ2λ2Cψ

)]. (4.53)

Again the direction of the diquark condensate in the anitsymmetric color anti-triplet channelwas chosen to point in λA′=2 direction. The isovector coupling terms were added in accordancewith the original Nambu and Jona-Lasinio coupling [NJL61b]. In the isoscalar case (Nf = 2,mu = mu) these coupling terms do not affect the model in mean field approximation. Theadditional coupling constants are fixed by the construction of the NJL Lagrangian starting at acolor current interaction (see App. B.2). The thermodynamic potential is of the form

Ω = U(Φ,Φ∗, T )− T∑

n

∫d3p

(2π)3Tr log

[βS−1(iωn, ~p)

]

+σ2

2G+

σ28

2G8+

v2

2GV+

v28

2GV8+

∆∗∆

2H+

∆∗0∆0

2H0. (4.54)

14The upper integrational bound Λ indicates a common 3-momentum cutoff. I. e. it is only integrated over ~pwith |~p| < Λ. The integrals labeled with an upper integrational bound Λ′ are integrated up to |~p| < Λ in theusual way. From |~p| = Λ onwards to higher 3-momenta the integrand is evaluated with the mean fields set tozero. This procedure mimiks the asymptotic freedom of QCD albeit in a very primitive way.


The inverse Nambu-Gor’kov propagator including diquarks is now of the non-block-diagonalform

S−1(ωn, ~p) =1

2

(iγ0ωn − ~γ · ~p− M − γ0(µ+ iA4) ∆γ5τ2λ2 + ∆0γ0γ5τ2λ2

−∆∗γ5τ2λ2 + ∆∗0γ0γ5τ2λ2 iγ0ωn − ~γ · ~p− M + γ0(µ+ iA4)

).

(4.55)The mass M is defined as in Sec. 4.2.4. The generalized chemical potential µ is defined byµ = µ 1C − v 1C − v8λ8. The thermodynamic potential is is evaluated in analogy to Sec. 4.2.4.The six roots of det S−1(p2

0) = 0 in this case are

Eb± =√p2 +M2

b ± µb Er±(a),±(b)= Er±(a)

±(b)iA4

3

with Er±(a)=√p2 +M2

r + µ2r + |∆|2 + |∆0|2 ±

(a)2s

and s =

√(µ2r + |∆0|2

)~p2 + (Mrµr − Re [∆∆∗

0])2 ,

(4.56)

where the generated masses and the generalized chemical potentials are again color dependentand given by

Mb = m0 − σ +2√3σ8 Mr = m0 − σ −

1√3σ8

µb = µ− v +2√3

(v8 − iA4

8

)µr = µ− 1√

3

(v8 − iA4

8

)A4 = A4

3λ3 +A48λ8.

(4.57)

Applying the Matsubara formalism (see Sec. 3.4.5) we find for the thermodynamic potential

Ω = −2NfT

∫ Λ′d3p

(2π)3

∑

j∈J

log[1 + e−βEj

]− 6Nf

∫ Λ d3p

(2π)3

∑

j∈J

Ej

+σ2

2G+

σ28

2G8+

v2

2GV+

v28

2GV8+

∆∗∆

2H+ U (Φ,Φ∗, T ) , (4.58)

where J is an index set numbering all 6 energies in equations (4.56). The applied cutoff pro-cedures indicted by Λ and Λ′ are implemented as in Sec. 4.2.3.14 Finally the sum in (4.58) isevaluated explicitly

Ω = −2NfT

∫ Λ′d3p

(2π)3log[1 + e−βE

b+

]+ log

[1 + e−βE

b−]

+ log[1 + 2 cos

(β A4

3

)e−βE

r+ + e−2βEr

+

]+ log

[1 + 2 cos

(β A4

3

)e−βE

r+ + e−2βEr

+

]

− 6Nf

∫ Λ d3p

(2π)3

[E

(b)+ + E

(b)− + 2E

(r)+ + 2E

(b)−

]

+σ2

2G+

σ28

2G8+

v2

2GV+

v28

2GV8+

∆∗∆

2H+ U (Φ,Φ∗, T ) . (4.59)

The self consistency equations that can be derived from this thermodynamic potential are

∂Ω

∂σ=

∂Ω

∂σ8=

∂Ω

∂v=

∂Ω

∂v8=

∂Ω

∂∆=

∂Ω

∂A43

=∂Ω

∂A48

= 0 (4.60)


4.3 Complex Euclidean actions

In this section it is outlined why complex actions appear in PNJL models and what impact theyhave on mean field calculations.

One of the main advantages when working in Euclidean spacetime is that the Euclideanaction is real in most cases. A real action is interesting because it allows to set up a hierarchy ofimportance. If SE ∈ R, the value of e−SE indicates the importance of a certain field configuration.This hierarchy is used for example in the Monte-Carlo methods used in lattice QCD calculations.The Euclidean action however does not have to be real by principal construction. This causesproblems in numeric calculations and is known as the fermion sign problem. The name is areminder that the fermion determinant obtained by integrating out fermions introduces complexvalues into the path integral. This suggests, that it is the anticommutation relations of fermions,that generate the complex action.

4.3.1 Origin of complex actions

The origin of the complex action is enlightened in this section on a technical basis. In PNJLmodels the contribution of the fermion determinant to the Euclidean action is

δLE = V Tr[log(βS−1

)]= V

∑

n

∫d3p

(2π)2Tr[log(βS−1

)]

= V∑

n

∫d3p

(2π)2log[Det

(βS−1

)]. (4.61)

The determinant denoted Det is the determinant in Dirac, color and flavor space. The deter-minant can be calculated by evaluating the product of all eigenvalues. If the eigenvalues are allreal the determinant is real.

S−1vn = λn vn, with λn ∈ R ⇐⇒ S−1 =[S−1

]†, hermitian (4.62)

To check where the non hermitian parts, responsable for non-real eigenvalues, come in, one needsto look at the contributions in the inverse propagator explicitly.

The mass term is diagonal and real and therefore hermitian. The spatial momentum term~γ · ~p is antihermitian. The sign can however be absorbed by a redefinition of spatial momentum.One is free to do this, as the theory is rotationally symmetric. The term γ0iωn is antihermition.Here a similar argument holds, if the Matsubara frequencies are redefined by ωn −→ −ωn = ω−n.This is allowed, as the Matsubara sum is symmetric under a sign change of n. The coupling termsto scalar fields are hermitian, as the coupling terms are diagonal and real. The coupling terms tovector fields are hermitian as they are proportional to γ0. For the diquark fields the propagatorhas to be considered in Nambu-Gor’kov space. The diquark terms (4.55) are antihermitian.The sign can however be absorbed into the diquark fields, that have a free complex phase.Redefining the sign simultaneously to conjugating will keep the diquark contribution hermitian.The chemical potential comes with a γ0 in front and is hermition therefore, too. The only non-hermitian terms are the terms coupling the fermions to the (temporal) gauge fields. The Diracstructure in Euclidean space here is iγ0.

In the case of vanishing quark chemical potential µ = 0 the Euclidean action is real even inthe presence of temporal gauge fields A4 6= 0. This is due to the fact that at µ = 0 there is aquark charge conjugation symmetry. The symmetry transformation will change the sign of thecolor charge of the quarks and the quark chemical potential. If the quark chemical potential iszero, the action and its change conjugate can be added to give a real action [DPZ05].

In the case of non-zero quark chemical potential the reasoning in the previous paragraph isno longer valid. The term coupling quark and gluon fields introduces an antihermitian term,

4.3. Complex Euclidean actions 59

that cannot be made quasi-hermitian by absorbing the sign in another symmetry. Thus theEuclidean action is in general complex.

4.3.2 Impact of a complex action on mean field approximation

The mean field approximation typically contains two approximations. Firstly all fields aretreated as spatially and temporally constant. This reduces the degrees of freedom of the pathintegral dramatically. In fact there is no path integral left after this approximation.

The second approximation is usually implied in the mean field approximation is the infinitevolume limit. As the action is an extensive quantity, this implies that only few importantfield configurations contribute considerably to the path integral, while all other configurationscontribute negligibly.

∫DX e−SE =

∫dX

∫D[δX] e−SE

MF approx. No. 1−−−−−−−−−−−→∫

dX e−SE (4.63)

MF approx. No. 2−−−−−−−−−−−→ e−SE∣∣φmax.

(4.64)

Here φmax. means the field configuration with the most important contribution to the pathintegral. Of course in both steps the normalization, which is of no physical importance needs tobe adjusted.

In the case of a real Euclidean action the most important contribution is easily determined.It is the minimum of the action. The necessary condition for this minimum is

∂SE(φmin.)

∂φ= 0 . (4.65)

For a complex Euclidean action there are two mechanisms, that could determine, which fieldconfiguration contributes most to the simplified path integral (4.63). Firstly there is a minimumin the real part of the action, secondly there is the mechanism known from path integrals inMinkowski space as the so-called points of stationary phase. To resolve this question the infinitevolume limit is of great help here.

Z ∝∫

dX e−SE =

∫dX e−V sE =

∫dX e−V Re[sE] e−iV Im[sE] (4.66)

The imaginary part of the action causes the integrand to rotate its phase in the complexplane. When scaling the system up in volume, these oscillations happen at higher frequencies,f ∝ V . When integrating over these oscillations complete periods approximately cancel. So theremainder of the cancellation δI scales with the length of one period of the oscillations δI ∝ V −1.The real part of the action determines the absolute value of the integrand. When scaling thesystem up in volume the integrand will scale exponentially δI ∝ eV/V0 . This exponential scalingin the large volume limit is much stronger than the cancellations caused by rotating complexphases. This is why it is the real part of the action that determines which point contributesmost to the path integral.

The field configuration that contributes most importantly to the path integral, is the con-figuration with minimal real part of the Euclidean action. The necessary condition for thisminimum is

∂ Re [SE(φmin.)]

∂φ= 0 . (4.67)

In the Sec. 4.2.3, Sec. 4.2.4 and Sec. 4.2.5 the Euclidean action was in fact complex. Thereforethe mean field equations (4.42), (4.52) and (4.60) have to be understood in the context of (4.67),i. e. only the real part of the derivative of the Euclidean action or the thermodynamic potentialhas to vanish.


In mean field approximation there are even more restrictions to the Euclidean action. Dueto the simplifications (4.63) and (4.64) the partition function is proportional to the exponentialof minus the Euclidean action at the field configuration minimizing the action:

Ω = −TV

log [Z] = −TV

log[e−SE

∣∣φ=φmin.

]=T

VSE |φ=φmin.

(4.68)

Because of the role the thermodynamic potential plays in macroscopic physics, it has to be a realfunction. This of course implies, that the Euclidean action at the minimal field configurationφmin. has to be real as well. In all cases we encountered this constraint was fulfilled. In Sec. 4.2.2it was shown that A4

8 = 0 is part of the minimizing field configuration, which lead to Φ = Φ∗ ∈R. Substituting this into the explicit forms immediately shows that for the thermodynamicpotentials (4.40), (4.51) and (4.59) the constraint (4.68) is not violated.

4.4 Numerical results in mean field approximation

In this section the numerical results mainly for the model including scalar diquarks (see Sec. 4.2.4)are presented. The model including scalar and vector diqarks shown in Sec. 4.2.5 is presentedwhenever there are important differences. In Sec. 4.4.1 the results for the field configurationsminimizing the thermodynamic potential are presented. In Sec. 4.4.2 the phase diagrams ofthe PNJL models are discussed. In this section changes due to changing the deconfinementtemperature and the current quark mass are presented. Finally in Sec. 4.4.3 the predictions ofthe PNJL model are confronted with lattice QCD calculations.

4.4.1 Fields and Polyakov loop

This subsection mainly presents the field configurations, that minimize the thermodynamicpotential derived in Sec. 4.2.4. The additional fields included in Sec. 4.2.5 are presented at theend of this subsection. Throughout this section the parameters given in [RTW06] are used forthe loop effective potential. The current quark mass was set to m0 = 5.5 MeV and the first ordertransition temperature of the loop effective potential was set to T0 = 0.27 GeV (see Tab. 3.1).

First the mean field values that satisfy the self consistency equations for the PNJL modelpresented in Sec. 4.2.4 are illustrated. The constituent quark mass M = m0− σ = m0−G 〈ψψ〉is plotted in Fig. 4.5 as a function of chemical potential and temperature. The scalar diquarkcondensate ∆ = H 〈ψTCγ5τ2λ2ψ〉 is plotted in Fig. 4.6 as a function of chemical potential andtemperature. The chemical potential is increased up to the order of the 3-momentum cutoff. Forchemical potentials and temperatures close to the cutoff the model starts to lose its predictivepower.

The diquark condensate and the chiral condensate are separated by a first order phasetransition at finite µ. This transition gives rise to a discontinuity in both chiral and diquarkfield (see Fig. 4.13 and Fig. 4.14)15. The diquark condensate and the high temperature phaseare separated by a second order phase transition. The second order phase transition manifestsitself in a kink in the condensate as a function of temperature. I. e. the gradient of the field∆ is discontinuous. The phase of spontaneously broken chiral symmetry at low temperaturesand low chemical potentials is separated from the high temperature phase of restored chiralsymmetry by a cross-over transition. A cross-over transition is no phase transition in the strictsense, as there is no symmetry, that is broken or restored at a certain critical temperature orchemical potential. A cross-over transition connects two regions in which a symmetry is broken

15In Fig. 4.5 and Fig. 4.6 the rendering procedure chosen cannot account for sharp edges. The transition issmoother than it should be. That there indeed is a first order phasetransition can be seen in Fig. 4.13 andFig. 4.14.

4.4. Numerical results in mean field approximation 61

00.10.20.30.4 ΜGeV

0

0.1

0.2

TGeV

0.1

0.2

0.3

MGeV

00.10.20.30.4

Figure 4.5: The constitutent quark massM = m0 − σ = m0 −G 〈ψψ〉 plotted as afunction of temperature and quark chemicalpotential.

0.30.40.50.6

ΜGeV

0

0.05

0.1

0.15

TGeV

0

0.05

0.1D

GeV

0.30.40.50.6

0

0.05

Figure 4.6: The scalar diquark field∆ = H 〈ψTCγ5τ2λ2ψ〉 plotted as a func-tion of temperature and quark chemicalpotential.

to a different extend. The cross-over transition in the chiral symmetry appears, because bothspontaneous and explicit symmetry breaking are present. The explicit symmetry breaking iscaused by the current quark mass, while the spontaneous symmetry breaking happens due todynamic effects. The best way to quantify the position of a cross-over transition is to determinethe position of maximal susceptibility in temperature or chemical potential direction.

Confinement is realized in PNJL-models via the Polyakov loop. The dependence of thePolyakov loop on chemical potential and temperature is plotted in Fig. 4.716. When having acloser look one can see the footprint of the second order phase transition separating the diquarkphase from the high temperature phase. This transition manifests itself in a slight kink in Φ, asit did in the diquark condensate ∆. In regions where the Polyakov loop Φ is close to zero, theconfinement is strong. For rising Polyakov loop the confinement gets weaker and finally vanishesat Φ = 1.17

As discussed in Sec. 4.2.2 the introduction of the Polyakov loop parameterized by the gluonmean fields breaks color symmetry. This is manifest in the appearance of the additional fieldσ8 = G8 〈ψλ8ψ〉. This field results in a difference of the constituent quark masses for ”red”(”green”) quarks and ”blue” quarks. This mass difference is plotted in Fig. 4.8 as a functionof chemical potential and temperature. Especially in the region of the cross-over transition ofchiral symmetry restoration this field is non-vanishing18. Although the diquark field breaks colorsymmetry, the field σ8 does not show as large contribution in the diquark phase. I. e. there seems

16Note that this diagram is plotted in a different orientation than Fig. 4.5 and Fig. 4.6. This was done for thesake of a better visibility of the temperature and chemical potential dependence.

17When the Polyakov loop reaches one the model predicts a second order transition as the loop has a kink.This is an artefact of the model. There is no symmetry that is broken or restored. This fake transition appearsbecause the model does not know about the volume of SU(3)C. Near Φ = 1 the volume element of SU(3)C,that belongs to a Polyakov loop close to one, vanishes and becomes zero at Φ = 1. The group SU(3)C does onlyallow a Polyakov loop value of exactly one at one single point on the N2

C − 1-dimensional group manifold. Thissingularity will dampen the approach to Φ = 1. It is the loop potential that was fitted to lattice data, that doknow about SU(3)C. It is the deficit of our ansatz, that it cannot capture these SU(3)C features encoded in thelattice data. This is due to the fact that the potential is truncated at some finite order in Φ and T .

18Note however that the condensate σ8 is given in MeV while all other condensates are plotted in GeV. I. e.the overall contribution of σ8 to the thermodynamic potential (the pressure) is not of major importance.


0

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ΜGeV0

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0.2

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0.4

TGeV

0

0.25

0.5

0.75

1

F

0.2

0.40.6

ΜGeV

Figure 4.7: The Polyakov loop configurationsthat solve the self consistency equations plot-ted as a function of chemical potential andtemperature.

0

0.2

0.40.6

ΜGeV0

0.1

0.2

0.3

0.4

TGeV

0

2

4nqT3

0

0.2

0.40.6

ΜGeV

Figure 4.8: The difference of the constituentquark mass of ”red” (”green”) quarks and”blue” quarks plotted as a function of chemi-cal potential and temperature.

to be some mechanism at work that compensates the color symmetry breaking contributions.

When the mean field configurations are determined, these can be used to find the quark den-sity and the pressure of the system. The quark density normalized by T−3 is plotted in Fig. 4.9as a function of temperature and chemical potential. In the confined phase the quark densityis basically zero, as it should be. In the diquark phase the quark density is finite such that thenormalized quark density diverges with T−3. The pressure of QCD thermodynamics predictedby the PNJL-model normalized by T−4 is plotted in Fig. 4.10 as a function of temperature andchemical potential. The pressure is zero in the confined phase. When deconfinement sets inthe pressure starts to rise as the quark density did. At large chemical potentials the pressure isfinite for vanishing temperatures, so that the normalized pressure divergers with T−4.

Of special interest for comparisons with lattice calculations is the pressure difference,

∆p(µ) = p(µ 6= 0)− p(µ = 0). (4.69)

The pressure difference normalized by T−4 is plotted in Fig. 4.11. These results are confrontedwith lattice calculations in Sec. 4.4.3.

The phases of broken chiral symmetry, at low chemical potential, where σ ≫ m0, and athigh chemical potential, where ∆ 6= 0, are separated by a phase transition of first order. Thefirst order phase transition is characterized by a discontinuity in the chiral field σ and the scalardiquark condensate ∆. These discontinuities are illustrated in Fig. 4.13 for an almost vanishingtemperature, T = 1 MeV. The first order phase transition also has an impact on the pressure ofthe system19. The pressure is non-differentiable at the phase transition. This can quite clearlybe identified in Fig. 4.14.

Finally some results for the PNJL-model additionally including the vector diquark conden-sate (see Sec. 4.2.5) are discussed. The vector coupling constants are coupled to the scalarcoupling constants via (B.16). The vector condensate and vector diquark condensate are plot-ted in Fig. 4.15 and Fig. 4.16. The presence of a non-vanishing diquark condensate discriminatesone color against the other two. This leads to differences of the constituent quark mass and the

19Here we use the thermodynamic potential synonymous to the thermodynamic potential per Volume which

equals the negative pressure: Ω = ΩV

= −p.


00.1

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TGeV

0

0.2

0.4

ΜGeV

0

1

2

3

4

nqT3

0.10.2

0.30.4

T

Figure 4.9: The quark density predicted by thePNJL model normalized by T−3 plotted as afunction of temperature and quark chemicalpotential.

00.1

0.20.3

0.4

TGeV

0

0.2

0.4

ΜGeV

0

1

2

3

4

5

DpT4

0.10.2

0.30.4

T

Figure 4.10: The pressure predicted by thePNJL model normalized by T−4 plotted as afunction of temperature and quark chemicalpotential.

00.1

0.20.3

0.4

TGeV

0

0.1

0.2Μ

GeV

0

0.25

0.5

0.75

1DpT4

0.10.2

0.30.4

T

Figure 4.11: The pressure difference definedin (4.69) and normalized by T−4 plotted as afunction of temperature and quark chemicalpotential.

0 0.1 0.2 0.3 0.4 0.5ΜGeV

0

2

4

6

8

vrg - vb

MeV

Mrg - Mb

MeV

D0MeV

Figure 4.12: The difference of the constituentquark mass of ”red” (”green”) and ”blue”quarks, the corresponding difference in thevector condensate and the vectorial diquarkcondensate ∆0 plotted as a function of chem-ical potential.


0 0.1 0.2 0.3 0.4 0.5ΜGeV

0.05

0.1

0.15

0.2

0.25

0.3

DGeV

MGeV

Figure 4.13: The constituent quark massM = m0 − σ = m0 −G 〈ψψ〉 and the scalardiquark condensate ∆ as a function of chemi-cal potential plotted at almost vanishing tem-perature T = 1 MeV. (This plot was producedwithout vector coupling)

0 0.1 0.2 0.3 0.4 0.5ΜGeV

0

0.001

0.002

0.003

0.004

p in GeV4

Figure 4.14: The pressure plotted as a func-tion of chemical potential plotted at almostvanishing temperature T = 1 MeV. (This plotwas produced without vector coupling)

vector condensate for different colors. The vector diquark condensate and these differences ofdifferently colored condensates are plotted in Fig. 4.12.

The phase transitions predicted by our PNJL model are of three different orders. Thetransition lines necessarily meet at some point. This point is a so-called triple point. If the triplepoint coincides with the critical point, i. e. a point where the order of a transition changes, thispoint is called tri-critical point. The tri-critical point and its position is discussed in Sec. 4.4.2.

4.4.2 PNJL model predicitons on the QCD phase diagram

Having seen the self consistent solutions in Sec. 4.4.1 we now focus on the QCD-Phase diagrampredictions of the PNJL-model. In the course of the discussion several parameters of the modelwill be changed to explore the dependence of the phase structure on these parameters. Firstof all we will show, that the coincidence of the chiral symmetry restoration transition and thedeconfinement transition are not in general coincident in the PNJL-model. For that reasonthe dependence on the regularization and the effective loop transition temperature is explored.In preparation of the comparison of the PNJL-model with the lattice results the current quarkmass is changed to higher values where prevalent lattice calculations are performed at the presentstage.

We start our discussion with the phase diagram of the PNJL-model with the parametersfor the loop effective potential and the parameters for the NJL-model used in [RTW06]. Thecurrent quark mass is set m0 = 5.5 MeV and the critical loop temperature to T0 = 0.27 GeV.With this parameter set we find phase transition lines of three different orders20 in the phasediagram. There is a first order phase transition separating the phase of dynamically broken chiralsymmetry at low T and µ and the diquark phase at low T and high µ21. The diquark phase is

20For the definition of the criteria we chose to determine the order of a phase transition see App. A.21If one would trust the model up to chemical potentials close to the 3-momentum cutoff and beyond one would

find that at high chemical potential the diquark condensate will decay again. The chemical potential range wherethe diquark condensate is non-zero is therefore sometimes called intermediate chemical potential range. We referto this region by calling it the high chemical potential region.


0.20.4

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ΜGeV

0.1

0.2

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0.4

TGeV

0

0.05

0.1

0.15

vGeV

0.20.4

0.6

0

0.05

Figure 4.15: The vector condensatev = −GV 〈ψγ0ψ〉 plotted as a function ofchemical potential and temperature.

0.30.40.50.6

ΜGeV

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0.05

0.1

0.15

TGeV

0

2

4

6

8

-D0

MeV

0.30.40.50.6

0

2

4

Figure 4.16: The vector diquark condensate∆0 = −GV 〈ψγ0ψ〉 plotted as a function ofchemical potential and temperature.

separated from the high temperature phase by a second order phase transition. Finally there isthe chiral symmetry restoration transition that appears when going to higher temperatures inthe low chemical potential regime. As the phases are characterized by a specific set of symmetriesthese transition lines cannot simply end. The endpoints of the transition lines have to cometogether such that there are no loose ends. The point where the transition lines come togetheris called triple point. Points where a phase transition changes order is called critical point. Atriple point emerging from three transition lines of different order is called tri-critical point.

The chiral phase transition and the deconfinement phase transition nearly coincide. Thishowever is a sensitive balance, that is not saved by any mechanism in the PNJL-model. Onecan for example change the regularization prescription used. For all calculations we used a3-momentum cutoff to regularize all divergent integrals. The pieces, that remain finite whenintegrating over 3-momentum, are integrated up to the cutoff, taking into account all dynamicallygenerated condensates. From the cutoff onwards all condensates are set to zero. This procedureshall account for the asymptotic freedom of QCD as implemented in the model. The standardregularization procedure is however to integrate the convergent pieces up to infinity withoutrestricting the existence of the fields to low momentum scales. The two schemes produce similarphase diagrams (see Fig. 4.17). The main difference produced by the two schemes is the relativeposition of the cross-over transition of the loop and of the chiral condensate.22 This indicatesthat in PNJL models the coincidence of the chiral symmetry breakdown and of confinement isinfluenced by the model parameters and the cutoff schemes.23

The deconfinement phase transition in Fig. 4.17 still happens at rather high temperatureswhen comparing to lattice data [KLP01]24. The sensitive reaction of the transitions temperatureon slight changes to the regularization raises the question, what influence the loop criticaltemperature has on the deconfinement transition temperature in the presence of quarks and

22As cross-over transitions do only involve continuous changes of all quantities, the position of a cross-overtransition depends on the definition of the transition criteria. Here the point of maximal susceptibility waschosen.

23It can in fact not be expected that changing the regularization scheme does not have effects on the outcome ofthe model as the regularization scheme is part of the model. I. e. in Fig. 4.17 two different models are comparedmaking different predictions on the coincidence of chiral and deconfinement phase transition.

24For 2 + 1 flavors data on the transition temperature can be found in [FK02, FK04].


0 0.1 0.2 0.3 0.4 0.5ΜGeV

0

0.05

0.1

0.15

0.2

TG

eV

critical point2nd order

1st order

loopchiral

hadronicphase

quark-gluonphase

diquarkphase

0 0.1 0.2 0.3 0.4 0.5ΜGeV

0

0.05

0.1

0.15

0.2

TG

eV


1st order

loopchiral

hadronicphase

quark-gluonphase

diquarkphase

Figure 4.17: Comparison of the phase diagram for different regularization schemes. The param-eters used are m0 = 5.5 MeV and T0 = 0.27 GeV. Left: regularization used in this work. Right:regularization scheme mainly used in the literature. The comparison shows the sensitivity ofthe coincidence of the chiral and deconfinement phase transition.

the coincidence of chiral and deconfinement phase transition. To investigate these questionsthe critical temperature of the loop effective potential was modified such that the cross-overtransition of deconfinement is in agreement with lattice calculations [KLP01]. For comparison ofthe phase diagrams for an unchanged current quark mass of m0 = 5.5 MeV and a loop transitiontemperature T0 = 0.27 GeV and T0 = 0.19 GeV respectively see Fig. 4.20 and Fig. 4.21. Startingat these two parameter sets the parameter space of the phase diagram in the current quark massdirection is explored. At higher current quark mass the critical point, which coincides with thetriple point is shifted to higher quark chemical potentials (see Fig. 4.22 and Fig. 4.23).

The critical point coincides with the triple point throughout the range from 5.5−50.MeV. Inwide ranges of the region below 5.5 MeV this is the case, too. The critical point coincides with thetriple point already at quark masses lower than the physical quark mass. At vanishing currentquark mass this coincidence is no longer existent (see Fig. 4.18 and Fig. 4.19). The critical pointat vanishing quark mass is shifted along the transition line to higher temperatures and lowerchemical potentials. In addition, a second triple point appears. In the phase diagrams withm0 = 5.5 MeV and m0 = 50.MeV the chiral cross-over transition and the deconfinement cross-over converged at the only triple point. In the case of vanishing quark mass the chiral transitionis no longer a cross-over but a second order transition. This qualitative change also changedthe relative position of the chiral and the deconfinement transition lines. The deconfinementcross-over now meets the second order transition line of chiral symmetry restoration in thesecond triple point at higher temperatures and lower chemical potential than the first one. Inthe calculation with the loop transition temperature T0 = 0.19 GeV this second triple pointcoincides with the critical point. The case of vanishing current quark mass shown here is specialin the sense that the chiral transition only is of second order when m0 strictly vanishes. Theexistence of the additional critical point is limited to this case where the chiral phase transitionis of second order. Therefore this second triple point is of no importance for this discussion ofthe current quark mass dependence of the phase structure.25

When the current quark mass reaches finite values below the physical value, the criticalpoint shifts along the transition line to lower temperatures and higher chemical potential till it

25The additional triple point appearing in the case of vanishing current quark mass is not listed in Tab. 4.5 andTab. 4.6.


0 0.1 0.2 0.3 0.4 0.5ΜGeV

0

0.05

0.1

0.15

0.2

TG

eV


1st order

loopchiral

hadronicphase

quark-gluonphase

diquarkphase

Figure 4.18: The phase diagram in the tem-perature chemical potential plane for vanish-ing current quark mass m0 = 0.0 MeV and aloop transition temperature T0 = 0.27 GeV.In this case chiral symmetry restoration is as-sociated with a second order phase transition,while the transition of the loop is a crossoverin the low µ region.

0 0.1 0.2 0.3 0.4 0.5ΜGeV

0

0.05

0.1

0.15

0.2

TG

eV


1st order

loopchiral

hadronicphase

quark-gluonphase

diquarkphase

Figure 4.19: The phase diagram in the tem-perature chemical potential plane for a cur-rent quark mass m0 = 0.0 MeV and a looptransition temperature T0 = 0.19 GeV. In thiscase chiral symmetry restoration is associatedwith a second order phase transition, while thetransition of the loop is a crossover in the lowµ region.

0 0.1 0.2 0.3 0.4 0.5ΜGeV

0

0.05

0.1

0.15

0.2

TG

eV


1st order

loopchiral

hadronicphase

quark-gluonphase

diquarkphase

Figure 4.20: The phase diagram in the tem-perature chemical potential plane for a currentquark mass m0 = 5.5 MeV and a loop transi-tion temperature T0 = 0.27 GeV.

0 0.1 0.2 0.3 0.4 0.5ΜGeV

0

0.05

0.1

0.15

0.2

TG

eV


1st order

loopchiral

hadronicphase

quark-gluonphase

diquarkphase

Figure 4.21: The phase diagram in the tem-perature chemical potential plane for a currentquark mass m0 = 5.5 MeV and a loop transi-tion temperature T0 = 0.19 GeV.


0 0.1 0.2 0.3 0.4 0.5ΜGeV

0

0.05

0.1

0.15

0.2

0.25

TG

eV


1st order

loopchiral

hadronicphase

quark-gluonphase

diquarkphase

Figure 4.22: The phase diagram in the tem-perature chemical potential plane for a currentquark mass m0 = 50.MeV and a loop transi-tion temperature T0 = 0.27 GeV.

0 0.1 0.2 0.3 0.4 0.5ΜGeV

0

0.05

0.1

0.15

0.2

TG

eV


1st order

loopchiral

hadronic phase

quark-gluonphase

diquarkphase

Figure 4.23: The phase diagram in the tem-perature chemical potential plane for a currentquark mass m0 = 50.MeV and a loop transi-tion temperature T0 = 0.19 GeV.

0 10 20 30 40 50 60m0 MeV

0.3

0.32

0.34

0.36

0.38

0.4

Μcr

itG

eV

PNJL Nf =2

0 20 40 60 80m0 MeV

0.1

0.2

0.3

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0.5

Μcr

itG

eV

prev

alen

tla

ttice

calc

ulat

ions

Fodor & Katz H2002LNf =2+1

Fodor & Katz H2004LNf =2+1

PNJL Nf =2

Figure 4.24: Left: The dependence of the critical chemical potential on the current quark mass.Right: Comparison with lattice calculations taken from [FK02, FK04]. The lattice data hasbeen calculated in 2 + 1 flavors while the PNJL-model is a 2 flavor calculation.


current quark mass Critical Point Triple Pointm0 [MeV] µ [GeV] T [GeV] µ [GeV] T [GeV]

0.0 .269 .156 .294 .120

5.5 .314 .115 → critical point

50. .395 .122 → critical point

Table 4.5: The position of the triple points and the critical points26 corresponding the plots inFig. 4.18, Fig. 4.17 and Fig. 4.22 showing the phasediagrams for T0 = 0.27 GeV. The criticalpoint is defined as the point where the order of the transition changes from 1st to 2nd order(m0 = 0.0 MeV) or to a crossover (m0 > 0.0 MeV). The triple point is the point where thediquark transition line and the deconfinement transition line meet.


0.0 .262 .132 .298 .092



Table 4.6: The position of the triple points and of the critical points26 corresponding the plotsin Fig. 4.19, Fig. 4.21 and Fig. 4.23 showing the phasediagrams for T0 = 0.19 GeV. The criticalpoint is defined as the point where the order of the transition changes from 1st to 2nd order(m0 = 0.0 MeV) or to a crossover (m0 > 0.0 MeV). The triple point is the point where thediquark transition line and the deconfinement transition line meet.

reaches the triple point of the diquark condensate. Once the critical point as reached this triplepoint it is stabilized in temperature by the interplay of the diquark condensate and the chiralcondensate. This interplay also fixes the position of the triple point. Raising the current quarkmasses to values well above the physical quark mass, therefore does not lead to a dependenceof the critical temperature on the current quark mass (see Fig. 4.22 and Fig. 4.23). The criticalchemical potential however moves together with the triple point to higher chemical potentials.

The critical points were extracted from the calculations. Critical points are labeled in allgraphs by a black triangle. If there is a coincidence with the triple point the critical point isusually called tri-critical point. The numeric values for the triple points and the critical pointsfor the loop transition temperature of T0 = 0.27 GeV are given in Tab. 4.526. For the modifiedloop transition temperature T0 = 0.19 GeV the numeric values are given in Tab. 4.626.

The current quark mass dependence of the critical chemical potential is plotted in Fig. 4.24.The graph shows a quite significant discrepancy between lattice predictions and our PNJL-calculation. This may be founded in the different number of flavors treated in these calculations.The shown lattice results [FK02, FK04] are calculated for 2 + 1 flavors while the PNJL-modelpresented here is a 2 flavor calculation. It is quite unlikely that the values of the lattice calcu-lations, that are quite low, show the same coincidence of triple point and critical point. Thecritical chemical potential of these lattice calculations are way below the chemical potentialinside nuclei. I. e. the presence of a diquark condensate at such low chemical potential would in-dicate color superconductivity in nuclei, which is not observed. This suggests that the inclusionof strange degrees of freedom leads to a decoupling of critical point and triple point. There could

26The precision of the values given in Tab. 4.5 and Tab. 4.6 is estimated to ≈ ±3 MeV. The finite numberof points calculated on the transition lines is responsable for this uncertainty. This is due to the fact that for afinite amount of points, i. e. a discontinuous sample of a continuous curve, it is difficult to test the underlyingcontinuous curve on steadiness or differentiability.


0 0.5 1 1.5 2TTc

0.1

0.2

0.3

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DpT4

Μ=.55 Tc

lattice

SB-limit

PNJL

NJL

0 0.5 1 1.5 2TTc

0.2

0.4

0.6

0.8

1

nqT3

Μ=.55 Tc

lattice

SB-limit

PNJL

NJL

Figure 4.25: Comparison of the NJL-model and the PNJL-model with lattice results [A+05].Left: The pressure difference defined in (4.69) plotted as a function of temperature. Right: Thequark number density plotted as a function of temperature. (This graph has been producedusing the second parameterization in Tab. 4.1.)

however also be other effects caused by parts of the PNJL-model, that are not yet implementedin an optimal way namely the effective loop dynamics. This decoupling of critical point andtriple point is however already observed at vanishing current quark masses (see Fig. 4.18 andFig. 4.19). This decoupling however vanishes already at very small current quark masses.

4.4.3 Comparing PNJL results with lattice QCD results

In this section the PNJL-model is confronted with predictions made by lattice calculations.The first step in this direction is the comparison of the NJL-model and the PNJL-model withlattice results [A+05]. The pressure difference defined in (4.69) and the quark number densityare plotted in Fig. 4.25 and are compared to lattice calculations. It can be seen quite clearly,that the PNJL-model shows much better agreement with lattice calculation. The deficits ofthe pure NJL calculation are obviously, that pressure and density rise already in a temperatureregion where no free quarks are allowed due to confinement. It can be seen, that the confiningmechanism in the PNJL-model works quite well.

There are however discrepancies between the PNJL-model and the lattice calculation abovethe critical temperature. The PNJL-model approaches the Stefan-Boltzmann limit immediatelyabove the transition, while the lattice calculations predict pressure differences and densities,that are approximately 20 % below the Stefan-Boltzmann limit. When comparing the meanfield value of the Polyakov loop with lattice calculations [KZ05] one finds that the mean fieldvalue of the Polyakov loop approaches the high temperature limit Φ→ 1 too fast (see Fig. 4.26).This indicates that there is something mistuned in the effective loop potential following theansatz (4.13). In addition, the loop shows a kink, once it reaches one. This is due to the factthat the loop potential used here does not know about the structure of SU(3)C. The volumeof SU(3)C near Φ = 1 is singular. Only a single point on the group manifold corresponds toΦ = 1. Therefore the loop potential should always keep Φ below one.

To sort out the source of discrepancies caused by the not yet optimally chosen ansatz forthe effective loop potential the two different parameterizations in Tab. 4.1 are compared (seeFig. 4.27 in comparison with Fig. 4.28). This suggests that the first parameterization worksbetter in the temperature region around the transition temperature and the second one worksbetter in the temperature regime below the transition temperature. The same comparison can


0 0.5 1 1.5 2TTc

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F

PNJL

lattice

0 0.5 1 1.5 2TTc

0

0.2

0.4

0.6

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1

F

PNJL

lattice

Figure 4.26: Comparison of the mean field expectation values of the Polyakov loop and latticecalculations [KZ05]. Left: First parameterization for the effective loop potential in Tab. 4.1.Right: Second parameterization for the effective loop potential in Tab. 4.1.

be done in the pure gluonic sector (see Fig. 4.1 and Fig. 4.2). The fit that was adjusted such thatit reproduces the Polyakov loop much better [RTW06] (see Fig. 4.2 and Fig. 4.27) works betterat temperatures above the transition. The pure mean square fit to pressure, entropy density andenergy density predicts a stronger transition (see Fig. 4.1 and Fig. 4.28) and therefore producesa more accurate transition behavior.

Up to now the PNJL-calculation at finite chemical potential has been compared to latticeresults. These lattice data are however not produced at finite chemical potential. Due to numericreasons the finite chemical potential data available is just an expansion in the parameter µ

T aboutthe zero chemical potential axis. Such an expansion can of course be established using the PNJLmodel as well. In Fig. 4.29 the second and fourth moment are compared to lattice results [A+05].The coefficients are defined by

∆( p

T 4(µ))

=p

T 4

∣∣∣T,µ− p

T 4

∣∣∣T,0

=∞∑

p=1

cp(T )(µT

)p,

⇒ c2 =1

2!

∂2(p/T 4

)

∂ (µ/T )2

∣∣∣∣∣T

=1

2!

1

T 2

∂2 p

∂µ2, c4 =

1

4!

∂4(p/T 4

)

∂ (µ/T )4

∣∣∣∣∣T

=1

4!

1

T 4

∂4 p

∂µ4, (4.70)

where the second equality for the expansion coefficient holds as we perform this differentiationat fixed temperature. The uneven expansion coefficients have to vanish as the setting is chargeconjugation invariant, i. e. symmetric to the T -axis. The expansion coefficients are a sensitivemeasure. Here the discrepancies of PNJL-model and lattice calculations get most obvious.

The trend of the pressure and density predicted by the PNJL model shows a pretty goodagreement with lattice calculations. Motivated by the apparently correct quark dynamics and thedeficiencies in describing the gluon dynamics, the third ansatz for an effective loop potential inSec. 4.1.2 is employed. The pressure and the density are plotted in Fig. 4.30. The Polyakov loopin the presence of quark now is in good agreement with lattice data (see Fig. 4.31). Finally themoments are calculated and compared with lattice data (see Fig. 4.32). Even here the agreementis now astonishingly good. Note that the PNJL calculation reaches the Stefan-Boltzmann limitin all calculation. For the second moment the Stefan-Boltzmann limit is c2 → nf

2 = 1 and for thefourth moment it is c4 → nf

4π2 ≈ 0.0507. Once the effective loop potential is modeled such thatit can account for the structure of SU(3)C the unphysical kink is removed from the Polyakovloop.


0 0.5 1 1.5 2TTc

0

0.05

0.1

0.15

0.2

Dp

T4

Μ=0.61 Tc

Μ=0.46 Tc

Μ=0.30 Tc

Μ=0.15 Tc

SB-limit

PNJL

lattice

0 0.5 1 1.5 2TTc

0.2

0.4

0.6

0.8

n qT

3

Μ=0.61 Tc

Μ=0.46 Tc

Μ=0.30 Tc

Μ=0.15 Tc

SB-limit

PNJL

lattice

Figure 4.27: Comparison of the PNJL-model with lattice results [A+05]. Left: The pressuredifference defined in (4.69) plotted as a function of temperature. Right: The quark numberdensity plotted as a function of temperature. This graph has been produced using the firstparameterization in Tab. 4.1.

0 0.5 1 1.5 2TTc

0

0.1

0.2

0.3

0.4

Dp

T4

Μ=0.71 Tc

Μ=0.54 Tc

Μ=0.36 Tc

Μ=0.18 Tc

SB-limit

PNJL

lattice

0 0.5 1 1.5 2TTc

0.2

0.4

0.6

0.8

1

n qT

3

Μ=0.71 Tc

Μ=0.54 Tc

Μ=0.36 Tc

Μ=0.18 Tc

SB-limit

PNJL

lattice

Figure 4.28: Comparison of the PNJL-model with lattice results [A+05]. Left: The pressuredifference defined in (4.69) plotted as a function of temperature. Right: The quark numberdensity plotted as a function of temperature. This graph has been produced using the secondparameterization in Tab. 4.1.


0 0.5 1 1.5 2TTc

0

0.2

0.4

0.6

0.8

1

c2

SB-limit

PNJL

lattice

0 0.5 1 1.5 2TTc

0

0.05

0.1

0.15

0.2

c4

SB-limit

PNJL

lattice

Figure 4.29: The second and fourth moment of the pressure difference defined in (4.70) plottedas a function of temperature. Compared are the PNJL-predictions with lattice results [A+05].

0 0.5 1 1.5 2TTc

0

0.05

0.1

0.15

0.2

0.25

0.3

Dp

T4 Μ=0.74 Tc

Μ=0.56 Tc

Μ=0.37 Tc

Μ=0.19 Tc

SB-limit

PNJL

lattice

0 0.5 1 1.5 2TTc

0.2

0.4

0.6

0.8

1

n qT

3

Μ=0.74 Tc

Μ=0.56 Tc

Μ=0.37 Tc

Μ=0.19 Tc

SB-limit

PNJL

lattice

Figure 4.30: Comparison of the PNJL-model with lattice results [A+05]. Left: The pressuredifference defined in (4.69) plotted as a function of temperature. Right: The quark numberdensity plotted as a function of temperature. This graph has been produced using a new ansatzfor the effective loop potential (4.22). The parameters in this ansatz were fitted to the puregluonic sector. The fit is given in Tab. 4.2.


0 0.5 1 1.5 2TTc

0

0.2

0.4

0.6

0.8

F

PNJL

lattice

Figure 4.31: Comparison of the mean field expectation values of the Polyakov loop and latticecalculations [KZ05]. This graph has been produced using a new ansatz for the effective looppotential (4.22). The parameters in this ansatz were fitted to the pure gluonic sector. The fit isgiven in Tab. 4.2.

0 0.5 1 1.5 2TTc

0

0.2

0.4

0.6

0.8

1

c2

SB-limit

PNJL

lattice

0 0.5 1 1.5 2TTc

0

0.05

0.1

0.15

0.2

c4

SB-limit

PNJL

lattice

Figure 4.32: The second and fourth moment of the pressure difference with respect to chemicalpotential plotted defined by (4.70) as a function of temperature. Compared are the PNJL-predictions with lattice results [A+05] using a new ansatz for the effective loop potential (4.22).

Chapter 5

The PNJL-model beyond

mean field approximation

In this chapter the mean field approximation used in the previous chapter is abandoned. Inexchange a new way to determine thermal expectation values is worked out, with the aim torelease constraints imposed by the mean field procedure. This allows to treat the thermalexpectation values of the Polyakov loop and its complex conjugate as independant degrees offreedom.

5.1 Shortcomings of mean field theory and their improvement

In general it can be shown, that despite the complex Euclidean action, the partition functionand the thermal expectation values of real fields have to be real. In mean field theory this is notensured. In Sec. 4.3.1 it was shown, that the gauge fields are coupled to the quarks such, thatthe fermion determinant is no longer real.

To show that the partition function is real, even if the Euclidean action is complex this sectionfollows the reasoning in [DPZ05]. Demanding that the PNJL model is symmetric under chargeconjugation, enables us to calculate the partition function by adding the partition function ofthe charge conjugated theory and to the original one:

Z =1

2

(Z + ZC

)=

1

2

(∫DA

∫Dφ e−SE +

∫DAC

∫DφC e−SC

E

). (5.1)

Here φ is a wildcard variable for all fields other than the gauge fields. It is assumed that thesefields are all real here. Since AC = −A and the path integral containing all field configurationsone can write

Z =1

2

(∫DA

∫Dφ e−SE +

∫D(−A)

∫DφC e−SC

E

)

=1

2

∫DA

(∫Dφ e−SE +

∫DφC e−SC

E

). (5.2)

Similar symmetries in the other fields allow to go even further1

Z =1

2

∫DA

∫Dφ(e−SE + e−S

CE

). (5.3)

1For a charged fermion field the charge conjugation has the effectRDψ

RDψ −→

RDψ

RDψ, for boson fields

the path integral obtains a sign according to the charge of the boson. In path integrals sign-changes in theintegration variables do not change the path integral, as the path integral counts the number of configurations.This number is independant of the sign of the integration variable.

75

76 Chapter 5. The PNJL-model beyond mean field approximation

This calculation does not hold for the mean field approximation as soon as φmin. 6= φCmin. orAmin. 6= −ACmin.. The symmetries of the models treated in Chapt. 4 and the fact that the meanfield configurations satisfy equalities such, that the mean field configuration φmin. and Amin.

are equivalent to the charge conjugated field configurations φCmin. and ACmin. is not given by anyprinciple.

Following the argument that led to (5.3) thermal expectation values can be evaluated. Thethermal expectation values of purely real fields will of course remain real. Interestingly thethermal expectation value of the complex Polyakov loop has to be real [DHL+04]. The gaugefields change sign under charge conjugation. Therefore the Polyakov loop will change into itscharge conjugate. In combination with an action satisfying SCE = S∗

E, one finds for the Polyakovloop

〈Φ〉 =1

2Z

∫DA

∫Dφ(Φ e−SE + Φ∗ e−S

∗E

)=

1

Z

∫DA

∫Dφ Re

(Φ e−SE

)∈ R. (5.4)

The same can of course be written down for 〈Φ∗〉. The expectation values 〈Φ〉 and 〈Φ∗〉 howeverdo not have to be equal. In general they will differ. As the thermal expectation values of thePolyakov loop and its complex conjugate play an important role in the PNJL models we define

l = 〈Φ〉 and l∗ = 〈Φ∗〉 . (5.5)

In the following a scheme is derived that keeps the partition function real by a mechanismworking analogous to the one in the derivation of (5.3). It is obvious that this can only work,if any approimation of the path integral is carried out very carefully. It has to be done suchthat the remaining integral contains both the fields and their charge conjugate symmetrically.The first mean field approximation (4.63) that integrates out all spatial fluctuations can stillbe performed as before. This does not alter the one-to-one correspondence of fields and chargeconjugate fields in the integral. The second approximation (4.64) however does not preservethis correspondence. The integral over the time and space averaged fields has to be worked out.With a general form of the Euclidean action this integral may be quite complicated. This is whywe choose the simplification of expanding the Euclidean action in a Taylor series. In Sec. 5.2the approximated path integrals are performed for a Taylor expanded action truncated after thequadratic term.

5.2 Second order corrections to the mean field potential

In this section an approximate path integral is evaluated. This is done for the simplified case ofa Taylor expanded Euclidean action truncated after the second order term. The approximationof the path integral is based on integrating out all temporal and spatial fluctuations. Thefluctuations still treated explicitly are variations around the time and space averages. I. e. eachpath integral is approximated by an ordinary integral,

∫Dφ(x) e−SE −→

∫dφ e−SE . (5.6)

This procedure guarantees, that thermal expectation values and thermodynamic potentials re-main real (see Sec. 5.1).

5.2.1 Thermodynamic expectation values

In the approximation scheme (5.6) thermal expectation values are calculated as follows

〈A〉β =1

Z

∫· · ·∫ ∞

−∞

n∏

k=1

dφk A(φ1 . . . φn)e−SE . (5.7)

5.2. Second order corrections to the mean field potential 77

Here the fields φk, k ∈ 1, 2, . . . n stand for all fields the action depends on. These fieldsare constants in space and time, i.e. they are the temporal and spatial averages over the fieldconfigurations in the path integral. The temporal and spatial fluctuations are neglected. Thequantity A is some function of the fields.2 The thermal expectation value is normalized by thepartition function

Z =

∫· · ·∫ ∞

−∞

n∏

k=1

dφk e−SE . (5.8)

Depending on the functional dependence of the Euclidean action the integrals in (5.7) and (5.8)are rather hard to evaluate. For this reason we will expand the Euclidean action in a Taylorseries and truncate at some order. The field configuration the action is expanded about has yetto be determined. Small deviations of the expansion from the true value of the action will beenlarged by the exponential of the action:

e−SE − e−Strunc.E = e−SE

(1− e−Strunc.

E +SE

)≈ ∆SE e

−SE (5.9)

The expansion should have small errors were the factor e−SE is large. But this is exactly thepoint where φk = φk,min. ∀ k ∈ 1, . . . n (see Sec. 4.3.2). Therefore we expand around the meanfield configuration determined by the self consistency equations (4.67). (4.67) is written here forthe case of the fields φk, k ∈ 1, 2, . . . n .

∂ Re [SE (φ1, . . . , φn)]

∂φi

∣∣∣∣φk=φmin

k

= 0, ∀i, k ∈ 1, 2, . . . n (5.10)

The Euclidean action is Taylor expanded up to second order:

Strunc.E = c0 +

∑

i

ci1ϕi +∑

ij

ϕi cij2 ϕj with ϕi = φi − φmin

i

c0 = SE|φk=φmink

ci1 =∂SE

∂φi


k

cij2 =1

2

∂2SE

∂φi∂φj


k

(5.11)

The evaluation of partition function and thermal expectation value now involves Gaussian inte-grals. These integrals can be evaluated analytically.

5.2.2 Thermal expectation values as corrections to mean field values

In this section the Gaussian integrals that appeared in the previous subsection are evaluatedanalytically. With the explicit form for these Gaussian integrals the partition function and thethermal expectation values can be evaluated. It turns out that the thermal expectation valuescan be cast into the form of the mean field results slightly shifted by small corrections.

The real part of the Euclidean action has a vanishing field gradient (5.10). Therefore c1 ispurely imaginary. Defining k by c1 = k

i = Im [c1] the approximated path integral in the partitionfunction takes the form of a Fourier integral:

Z =

∫· · ·∫ ∏

i

Dφie−SE =

∫· · ·∫ ∞

−∞

∏

i

dφie−SE

=

∫· · ·∫ ∞

−∞

∏

i

dϕie−c0−

P

ij ϕi cij2 ϕje−i

P

i kiϕi .

(5.12)

2In the path integral formalism all operators are transformed into functions respecting the ordering that isgiven by the Hamiltonian (see Sec. 3.4.4).


As A in the path integral formalism is no longer an operator but a function in the fields, it canbe rewritten in terms of a basis of the functional space the fields φ live in. An appropriate basisis the monomial basis em1,...,mn mk∈N0 ∀ k∈ 1,...n = ϕm1 · · ·ϕmn .

A (ϕ1, . . . , ϕn) =∞∑

m1,...,mn=0

am1,...,mn em1,...,mn =∞∑

m1,...,mn=0

am1,...,mn ϕm1 · · ·ϕmn (5.13)

Using the linearity of the integral it is sufficient to evaluate the thermal expectation values ofthe basis functions em1,...,mn = ϕm1 · · ·ϕmn . Within the framework of Fourier transformationsthis is particularly simple, as each ϕi simply has to be replaced by i∂ki . This derivative canthen be placed in front of the integral. It is therefore only necessary to evaluate the partitionfunction Z, which is the Fourier transformed of a Gaussian. The partition function is

Z =

∫· · ·∫ ∞

−∞

∏

i

dϕie−c0−

P

ij ϕi cij2 ϕje−i

P

i kiϕi =e−c0−

14

P

ij ki[c−12 ]

ijkj

2n√π |det [c2]|

. (5.14)

As we are interested in the thermodynamic potential, we have to consider the large volume limit.For its calculation it is helpful to make the substitution

SE −→ V SE =V

TΩMF

c0 −→ V c0, c1 −→ V c1, c2 −→ V c2 , k −→ V k.(5.15)

Applying this rule the thermal expectation values of the basis functions are

〈em1...mn〉β = 〈ϕm1 · · ·ϕmn〉β

=

e

−V c0−V4

P

ij ki[c2−1]ijkj

2n√πV n |det [c2]|

−1

k=Im[c1]

(

n∏

i=1

(i

V∂ki

)mi)e−V c0−

V4

P

ij ki[c2−1]ijkj


k=Im[c1]

.

(5.16)

In the large volume limit V −→∞ all inverse powers of V vanish. When expanding the thermalexpectation values in terms of negative powers in V only the 0th-order term will survive thelarge volume limit. As each derivative comes with a factor V −1 each derivative has to producea factor V , if the result shall contribute to the 0th-order term. When applying the productrule of differentiation, only those terms survive for which each partial derivative acts directly on

the exponential factor exp−V

4

∑ij ki

[c2

−1]ijkj

. This reasoning allows to write (5.16) in the

limit V −→∞ as

limV→∞

〈em1...mn〉β = limV→∞

〈ϕm1 · · ·ϕmn〉β =n∏

i=1

i

V∂ki

−V

4

∑

ij

ki[c2

−1]ijkj

k=Im[c1]

mi

=n∏

i=1

−1

2

∑

j

[c2

−1]ijc1j

mi

=n∏

i=1

−1

2

∑

j

[c−12

]ijcj1

mi

. (5.17)

Effectively this is equivalent to replacing each power of ϕi by the factor −12

∑j

[c−12

]ijcj1.

〈A (ϕ1, . . . , ϕn)〉β = A(− 1

2

∑

j

[c−12

]1jcj1, . . . ,−

1

2

∑

j

[c−12

]njcj1

)(5.18)


Or if we go back to the original variables φi we have

〈A (φ1, . . . , φn)〉β = A

φmin

1 − 1

2

∑

j

[c−12

]1jcj1, . . . , φ

minn − 1

2

∑

j

[c−12

]njcj1

(5.19)

This procedure can be expressed as a correction to the mean field approximation. This approxi-mation by construction produces real expectation values. This has happend rather coincidentalfor the mean field approximation. It is important to note however, that this will only be trueup to second order in the fields, as the Taylor expansion of the action was truncated after thesecond order.3

The calculation so far was based on a non-singular Hessian of the action. Once the Hessianbecomes singular the Fourier transform is no longer a Gaussian. The singular parts of theHessian can however be separated from the non-singular part of the Hessian.4 The non-singularpart the calculation can be performed as it was done above. For the singular part a generalizedFourier transform yields a Dirac-δ at φ = 0. In these directions the mean field values remainuncorrected. The correction term in (5.19) does not seem to be continuous once c2 is singularin some direction. The considerations above concering the Fourier integral however ensure thecontinuity of the correction term. This implies that the coefficient c1 has to vanish in thosedirections where the Hessian 1

2 c2 is singular. In an explicit numeric calculation numeric errorswill spoil this exact cancellation of zeros in c1 with singularities in c2.

From (5.15) and (5.16) it can be seen, that contributions of the action higher than secondorder cannot contribute to the dynamics of a system of infinite size. All contributions of orderα will die out with V −α+2. Quantities that vanish in the infinite volume limit cannot be treatedin the framework of this approximation. E. g. thermal expectation values of fluctuations cannotbe treated. Quantities of the form 〈φ2〉 − 〈φ〉2 are predicted to vanish5 in this approximationas 〈φ2〉 − 〈φ〉2 ∝ V −1. When computing thermal expectation values of such fields all termsproducing a V −1 dependence have to be taken into account. This has to be done both whenTaylor expanding the action and when computing the derivatives that lead from (5.16) to (5.17).

5.2.3 Correction to the mean field thermodynamic potential

For the calculation of the thermodynamic potential the standard formula

Ω = −TV

logZ = −TV

log

∫· · ·∫ ∏

i

Dφi e−SE (5.20)

is used. As in the previous the fields φi, i ∈ 1 . . . n represent all degrees of freedom of thesystem. With equation (5.8) and the rules (5.15) we find the thermodynamic potential in the

3In all cases treated here all thermal expectation values are strictly real. This is founded in the real solutionsto the mean field equations, which give the configuration expanded about here. If this was not the case thecorrections would bring the imaginary part to zero to the same order the action was expanded to.

4The space spanned by the eigenvectors to zero eigenvalue can be separated from the subspace spanned ofthose eigenvectors to non-zero eigenvalues.

5On a more technical level the correction can be expressed as a correction term in the elemenary fields ofthe approximated path integral: 〈f(φ)〉 = f(φMF + δφ). From this we conclude that 〈f2(φ)〉 = f2(φMF + δφ) =[f(φMF + δφ)]2 = 〈f(φ)〉2.


large volume limit

Ω = limV→∞

−TV

logZ = limV→∞

−TV

log

e

−V c0+V4

P

ij c1i[c2−1]

ijc1

j


= limV→∞

−TV

−V c0 +

V

4

∑

ij

c1i[c2

−1]ijc1j − log

(2n√πV n |det [c2]|

)

= T c0 −T

4

∑

ij

c1i[c2

−1]ijc1j + T lim

V→∞

1

2Vlog(22nV nπ |det [c2]|

)

= T c0 −T

4

∑

ij

c1i[c2

−1]ijc1j (5.21)

=T

V

c0 −

1

4

∑

ij

ci1[c−12

]ijcj1

(5.22)

In the last step rule (5.15) was applied in reverse direction. Substituting the expansion coeffi-cients by their explicit forms in terms of the Euclidean action of the mean field thermodynamicpotential, the improved thermodynamic potential can be expressed as:

Ω =T

V

SE|φ=φmin − 1

2

∑

ij

(∂SE

∂φi

)[∂2SE

∂φi∂φj

]−1(∂SE

∂φj

) (5.23)

= ΩMF|φ=φmin − 1

2

∑

ij

(∂ΩMF

∂φi

)[∂2ΩMF

∂φi∂φj

]−1(∂ΩMF

∂φj

), (5.24)

in terms of the mean field thermodynamic potential and a correction term arising from theintegration over the field averages.

5.2.4 Corrections to the self consistency equations

In pure mean field theory we use the real self consistency equations of (3.102) which are equivalentto (5.10). It is now shown that the new thermodynamic potential Ω fulfills a new set of selfconsistency equations, at least to low orders. This new set is the analog of (3.102) in thisapproximation beyond mean field theory.

∂Ω

∂φi= 0 +O (α) . (5.25)

We use the short notation ∂i = ∂∂φi

and imply summation over repeated indices. As a firststep of the derivation of a new set of self consistency equations the thermodynamic potential isexpanded about the mean field configuration φmin.

∂bΩ|φmina +φa

= ∂bΩMF|φmina +φa

− 1

2∂b

(∂iΩMF|φmin

a

[∂i∂jΩMF|φmin

a

]−1∂jΩMF|φmin

a

)∣∣∣∣φmin

a +φa

= ∂bΩMF|φmina− ∂α∂bΩMF|φmin

a

([∂α∂jΩMF]−1 ∂jΩMF

)∣∣∣φmin

a

− 1

2∂b

(∂iΩMF|φmin

a

[∂i∂jΩMF|φmin

a

]−1∂jΩMF|φmin

a

)∣∣∣∣φmin

a

+1

2∂β ∂b

(∂iΩMF|φmin

a

[∂i∂jΩMF|φmin

a

]−1∂jΩMF|φmin

a

)∣∣∣∣φmin

a

([∂β∂jΩMF]−1 ∂jΩMF

)∣∣∣φmin

a

(5.26)


To perform the derivatives of the second Taylor expansion coefficient the identity

∂A−1

∂x= −A−1∂A

∂xA−1 , with A : C −→ C

n×n, (5.27)

is of great help. The final result is

∂bΩ|φmina +φa

= −1

2∂iΩMF|φmin

a

(∂b

[∂i∂jΩMF|φmin

a

]−1∣∣∣∣φmin

a

)∂jΩMF|φmin

a

+[∂b∂iΩMF|φmin

a

](∂β

[∂i∂jΩMF|φmin

a

]−1)∂jΩMF|φmin

a

[∂β∂kΩMF|φmin

a

]−1∂kΩMF|φmin

a

+1

2∂iΩMF|φmin

a

(∂b∂β

[∂i∂jΩMF|φmin

a

]−1)∂jΩMF|φmin

a

[∂β∂kΩMF|φmin

a

]−1∂kΩMF|φmin

a.

(5.28)

This formula only contains higher derivatives of the mean field thermodynamic potential (orrespectively the Euclidean action). When applying mean field theory to systems with complexEuclidean action, the self consistency equations (3.102) have to be restricted to the real part ofthe action. There may be a remaining imaginary gradient. The formalism we have developedeliminates this imaginary gradient as can be seen from (5.28). The first terms that appear areproportional to third and forth derivatives of the action. Therefore the function Ω is indeed thecorrect thermodynamic potential up to second order.

5.2.5 Aspects of calculating corrections numerically

The PNJL Lagrangian’s dependence on l and l∗ forbids to first solve the mean field self con-sistency equation (5.10) and to calculate all corrections in a second independent step. Insteadwe have to calculate the thermal average of Φ and Φ∗ at the same time as we solve the selfconsistency equations for the mean field problem.

One first way to address this problem is straight forward, simply inserting the formula forthe thermal averages into the mean field self consistency equations. The other way to do thiswould be to introduce two constraint equations demanding that the thermal expectation valuesare correctly determined:

l − 〈Φ〉 = 0, l∗ − 〈Φ∗〉 = 0. (5.29)

We add those two equations to the mean field self consistency equations written with theirexplicit l and l∗ dependence and their explicit A4

3 and A48 dependence. Then we treat l and l∗

as additional variables when solving this multi-dimensional problem numerically. Note that forthe determination of the coefficients c0, c1 and c2 (or c0, c1 and c2) we must resolve the A4

3 andA4

8 dependence encoded in U (l, l∗, T ). Only when solving the self consistency equations (5.10)

plus the constraint equations (5.29) we do treat l, l∗ and A(3)4 , A

(8)4 as independent variables.

It turned out to be unfavorable to insert the thermal averages directly into the self consistencyequations (5.10) before solving them. Instead it is faster to choose the second path and addfurther variables. Equivalently to equations (5.29), we introduced a linear combination of twoadditional variables

lIm = 〈Im [Φ]〉 =l − l∗

2lRe = 〈Re [Φ]〉 =

l + l∗2

. (5.30)

The additional variables are then determined by these two constraint equations, carrying nomore information than equations (5.29).


0

0.2

0.40.6

ΜGeV0

0.1

0.2

TGeV

0

0.2

0.4

0.6

0.8

1

l* + l

2

0.2

0.40.6

ΜGeV

Figure 5.1: The thermal expectation value12 (l∗ + l) = 1

2 〈Φ∗ + Φ〉 plotted in the temper-ature chemical potential plane.

0

0.20.4

0.6

ΜGeV0

0.1

0.2

TGeV

-0.05

0

0.05

0.1

l* - l

2

0.20.4

0.6

ΜGeV

Figure 5.2: The thermal expectation value12 (l∗ − l) = 1

2 〈Φ∗ − Φ〉 plotted in the temper-ature chemical potential plane.

5.3 Numerical results approximating the action to second order

This section presents the results obtained from the improved PNJL-model going beyond meanfield approximation.6 The most important difference is now the appearance of an additionaldegree of freedom 〈Φ∗ − Φ〉. This degree of freedom could not be treated in the mean fieldapproach in Chapt. 4. Besides presenting the temperature and chemical potential dependence of〈Φ∗ − Φ〉, the changes to the phase diagram are discussed. The improved approximation schemepresented in this chapter is confronted with lattice QCD results. The discrepancies observed arereduced by employing a different form of the effective loop potential. The problems with theform for the effective loop potential (4.1.2) have already appeared in Sec. 4.4.3 and are againeliminated by the new ansatz for the effective loop potential.

5.3.1 Fields and Polyakov loop

In comparison to the PNJL-models described in Sec. 4.4.1 the models in this chapter can treatthe thermal expectation value of the Polyakov loop l = 〈Φ〉 and the thermal expectation value ofits complex conjugate l∗ = 〈Φ∗〉 separately. I. e. the two expectation values can assume differentvalues. We are working with the fields 1

2 (l∗ + l) and 12 (l∗ − l). These fields are plotted in the

temperature chemical potential plane in Fig. 5.1 and Fig. 5.2

It can be seen in Fig. 5.3 and Fig. 5.4 that the ”difference” field 12 (l∗ − l) reaches its greatest

values in the region near the phase transition. In Fig. 5.3 the temperature dependence of thePolyakov loop at low chemical potential is shown. In Fig. 5.4 we present the temperaturedependence of the Polyakov loop at high chemical potential. The dependence of l and l∗ on thechemical potential in the low temperature region is shown in Fig. 5.5. The high temperatureregime is plotted in Fig. 5.6.

6For the calculations in this chapter going beyond the mean field approach the second parameter set in Tab. 4.1is used. For the mean field calculations presented in Chapt. 4 the first parameter set in Tab. 4.1 was employed.The crutial difference to the first set taken from [RTW06] is the stronger transition. This stronger transition wasnecessary to obtain realistic pressure differences and quark densities. With a new ansatz for the effective looppotential (see Sec. 4.1.2) this problem does no longer exist.

5.3. Numerical results approximating the action to second order 83

0 0.25 0.5 0.75 1 1.25 1.5 1.75TTc

0

0.2

0.4

0.6

0.8

1

Μ = 0.6 Μc

<F* >

<F>

Figure 5.3: The behavior of the Polyakov loopas a function of temperature in the chiral con-densate regime at low chemical potential.

0 0.25 0.5 0.75 1 1.25 1.5 1.75TTc

0

0.2

0.4

0.6

0.8

1

Μ = 1.2 Μc

<F* >

<F>

Figure 5.4: The behavior of the Polyakov loopas a function of temperature in the diquarkcondensate regime at high chemical potential.

0 0.5 1 1.5 2ΜΜcrit

0

0.1

0.2

0.3

0.4

0.5

T = .43 Tcrit

<F* >

<F>

Figure 5.5: The behavior of the Polyakov loopas a function of chemical potential at low tem-peratures.

0 0.5 1 1.5 2ΜΜcrit

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

T = 1.4 Tcrit

<F* >

<F>

Figure 5.6: The behavior of the Polyakov loopas a function of chemical potential at hightemperatures.


5.3.2 The phase diagram beyond mean field approximation

The phase diagram of the improved approximation presented in this chapter is very similar tothe mean field phase diagram. A comparison of the phase diagrams is shown in Fig. 5.7. Theimproved approximation introduces only slight changes to the phase diagram obtained in meanfield approximation. The dependence of triple points and the critical points on current quarkmasses and effective loop transition temperatures are not very much distinct from mean fieldcalculations. The positions of critical points and triple points for T0 = 0.27 GeV are listed inTab. 5.17, and for the modified loop transition temperature T0 = 0.19 GeV in Tab. 5.27. Thecase of vanishing quark mass now has the particularity that there is no critical point any more.This is due to the fact that the chiral symmetry restauration appears in a first order phasetransition. This however is due to the approximation procedure chosen. The mean field resultis corrected by a small term. This small correction term is enough to change a second ordertransition into a first order transition. A cross-over transition, i. e. a smooth change in the fieldswill however always remain smooth when expanding up to finite order, as the corrections aresmooth functions.

The determination of triple points is complicated in this new approximation scheme. Atthe triple point the effective action necessarily has a singular Hessian in the field expectationvalues.8 As the action was expanded about the mean field results up to second order, the Hessianof the action near the mean field triple point is close to singular. The finite numeric precisionof all real calculations spoils a cancellation of these singularities with vanishing gradients (seeSec. 5.2.2). Physically the singularity of the Hessian of the action is synonymous with divergingfluctuations. These problems in explicit calculations will make it difficult to determine triplepoints. The solution near the triple point is however the solution of the mean field equations,at least in those eigenvector directions of the Hessian with vanishing eigenvalue (see Sec. 5.2.2).

Near critical points connecting a first and a second order transition the setting is similar tothose at triple points. For those field directions that correspond to a vanishing eigenvalue ofthe Hessian of the action, the solution is equivalent to the mean field solution (see Sec. 5.2.2).In the other directions corresponding to non-zero eigenvalues there still are corrections to themean field result. Near the critical point it is legitimate to neglegt the correction terms for thesubspace of field configurations subject to vanishing eigenvalues. It is not possible to neglegtthe corrections for those field directions with non-zero eigenvalues. The numeric results clearlyshow that this certainly is the case for the field 1

2 〈Φ∗ − Φ〉, as it assumes its largest values inthe vincinity of the critical point. The separation of the two subspaces is however quite difficult.In addition the precision without taking these considerations into account7 is sufficient for thepurposes of this model.

5.3.3 Comparison with results from lattice QCD

The pressure difference defined in (4.69) and the quark number density calculated in the im-proved approximation are plotted in Fig. 4.25. They are compared to lattice calculations [A+05].The discrepancies above the critical temperature between version of the PNJL-model, that usesthe Polyakov loop effective potential of the form (4.13) and the lattice calculation, are againvisible (see Fig. 5.8). The PNJL-model approaches the Stefan-Boltzmann limit immediately

7The precision of the values given in Tab. 5.1 and Tab. 5.2 is estimated to ≈ ±3 MeV. The finite numberof points calculated on the transition lines is responsable for this uncertainty. This is due to the fact that for afinite amount of points, i. e. a discontinuous sample of a continuous curve, it is difficult to test the underlyingcontinuous curve on steadiness or differentiability.

8This can easily be seen when imagining the effective action as a landscape in the fields. At the triple pointthree valleys meet. As the effective action is differentiable everywhere. The tangential plane at the triple point istangential to the minima coming along the three valleys. This setting requires a singular Hessian of the effectiveaction near the triple point.


0 0.1 0.2 0.3 0.4 0.5ΜGeV

0

0.05

0.1

0.15

0.2

TG

eV


1st order

loopchiral

hadronicphase

quark-gluonphase

diquarkphase

0 0.1 0.2 0.3 0.4 0.5ΜGeV

0

0.05

0.1

0.15

0.2

TG

eV


1st order

loopchiral

hadronicphase

quark-gluonphase

diquarkphase

Figure 5.7: Comparison of the phase diagrams in mean field theory and the improved approx-imation of the path integral. The diagrams were calculated with the second parameter set inTab. 4.1 for the effective loop potential. The current quark mass was chosen to equal the physicalcurrent quark mass m0 = 5.5 MeV and T0 = 0.27 GeV.


0.0 no critical endpoint .120 .293



Table 5.1: The position of the triple points and critical points7. The critical point is definedas the point where the order of the transition changes from 1st order to a cross-over transition.The triple point is the point where the diquark transition line and the deconfinement transitionline meet.


0.0 no critical endpoint .091 .296



Table 5.2: The position of the triple points and critical points7. The critical point is defined asthe point where the order of the transition changes from 1st order or to a cross-over transition.The triple point is the point where the diquark transition line and the deconfinement transitionline meet.


0 0.5 1 1.5 2TTc

0

0.05

0.1

0.15

0.2

0.25

0.3

Dp

T4 Μ=0.71 Tc

Μ=0.54 Tc

Μ=0.36 Tc

Μ=0.18 Tc

SB-limit

PNJL

lattice

0 0.5 1 1.5 2TTc

0.2

0.4

0.6

0.8

1

n qT

3

Μ=0.71 Tc

Μ=0.54 Tc

Μ=0.36 Tc

Μ=0.18 Tc

SB-limit

PNJL

lattice

Figure 5.8: Pressure difference and quark number density calculated with the PNJL-model usingthe improved approximation of the path integral in comparison with lattice results [A+05]. Left:The pressure difference defined in (4.69) plotted as a function of Temperature. Right: The quarknumber density plotted as a function of temperature.

above the transition while the lattice calculations predict pressure differences and densities thatare approximately 20 % under the Stefan-Boltzmann limit. The source of discrepancies is causedby the not yet optimally chosen ansatz for the effective loop potential. The quark dynamicshowever seems to agree pretty well with the lattice data [A+05] that is shown in comparison tothe PNJL calculations in Fig. 5.8.

In Sec. 4.4.3 the mean field results of the PNJL-model were compared with the secondand fourth moment of the pressure difference defined by (4.70). The corresponing comparisonof the PNJL-calculations using the Polyakov loop effective potential (4.13) in the improvedapproximation scheme with data taken from [A+05] is plotted in Fig. 5.9.

The deficits of overestimating the pressure and the quark density can be fixed in the improvedapproximation scheme by introducing a different loop potential. The pressure difference and thequark density calculated with the ansatz (4.22) are shown in Fig. 5.11. This goes togetherwith the improved agreement of the temperature dependence of Polyakov loop calculated inthe framework of the PNJL-model and on the lattice [KZ05] (see Fig. 5.10). The improvedapproximation as it possesses an additional degree of freedom contains more information aboutthermodynamic fluctuations. Therefore the describtion of the fourth moment given in (4.70)is even better (compare the mean field calculation Fig. 4.32 with Fig. 5.12 calculated in theimporved PNJL-model). The PNJL result for the second moment overestimates the latticepoints at high temperatures, which is due to the fact that the PNJL model has to reproduce theStefan-Boltzmann limit at high temperatures. Yet the agreement with lattice data is satisfying.


0 0.5 1 1.5 2TTc

0

0.2

0.4

0.6

0.8

1

c2

SB-limit

0 0.5 1 1.5 2TTc

0

0.05

0.1

0.15

0.2

0.25

c4

SB-limit

Figure 5.9: The second and fourth moment of the pressure difference with respect to the chemicalpotential plotted as a function of temperature. Compared are the PNJL-predictions in theimproved approximation scheme with lattice results [A+05].

0 0.5 1 1.5 2TTc

0

0.2

0.4

0.6

0.8

1

F

PNJL

lattice

0 0.5 1 1.5 2TTc

0

0.2

0.4

0.6

0.8

F

PNJL

lattice

Figure 5.10: Comparison of the mean field expectation values of the Polyakov loop and latticecalculations [KZ05]. The graph on the left has been produced using the effective loop potential(4.13), the graph to the right has been produced using a new ansatz for the effective Polyakovloop potential (4.22).


0 0.5 1 1.5 2TTc

0

0.05

0.1

0.15

0.2

0.25

0.3

Dp

T4 Μ=0.74 Tc

Μ=0.56 Tc

Μ=0.37 Tc

Μ=0.19 Tc

SB-limit

PNJL

lattice

0 0.5 1 1.5 2TTc

0.2

0.4

0.6

0.8

1

n qT

3

Μ=0.74 Tc

Μ=0.56 Tc

Μ=0.37 Tc

Μ=0.19 Tc

SB-limit

PNJL

lattice

Figure 5.11: Comparison of pressure difference and quark number density calculated in improvedPNJL-model with lattice results [A+05]. Left: The pressure difference defined in (4.69) plottedas a function of Temperature. Right: The quark number density plotted as a function oftemperature. This graph has been produced using a new ansatz for the effective loop potential(4.22). The parameters in this ansatz were fitted to the pure gluonic sector. The fit parametersare given in Tab. 4.2.

0 0.5 1 1.5 2TTc

0

0.2

0.4

0.6

0.8

1

c2

SB-limit

PNJL

lattice

0 0.5 1 1.5 2TTc

0

0.05

0.1

0.15

0.2

c4

SB-limit

PNJL

lattice

Figure 5.12: The second and fourth moment of the pressure difference defined in (4.70) plotted asa function of temperature. Compared are the PNJL-predictions in the improved approximationscheme with lattice results [A+05]. This graph has been produced using a new ansatz for theeffective loop potential (4.22). The parameters in this ansatz were fitted to the pure gluonicsector. The fit parameters are given in Tab. 4.2.

Chapter 6

Summary

6.1 Discussion

In this report a field theoretical model for QCD thermodynamics with Nf = 2 quark flavors hasbeen discussed. This model aims at the region ranging from zero temperature up to tempera-tures of about twice the deconfinement temperature. The model was constructed such that itincludes both features of spontaneous chiral symmetry breaking and confinement. This allowsto study the nature of the confinement-deconfinement phase transition in a system includingquarks and its interplay with chiral symmetry restauration. A possible coincidence of chiral andconfinement-deconfinement phase transition can be examined with this model as well.

In a first step the model has been treated in mean field approximation. Possible problems,that can arise from this rather crude approximation, have been discussed and a new way toimprove mean field approximations has been proposed. This new method is better suited todescribe thermal fluctuations of the fields involved. It has been shown that the field 〈φ∗ − φ〉βcannot be treated in mean field approximation at all. The improved approximation however wasable to treat this degree of freedom in an appropriate way.

In this work first calculations of the PNJL model with explicit inclusion of diquark degreesof freedom have been performed. The inclusion of diquarks allows for predictions at quarkchemical potentials of about 0.5–0.6 GeV. In the region of large quark chemical potential andlow temperature this model predicts a color superconducting phase. In addition, this modelproduces a first order phase transition separating the diquark phase from the hadronic phase ofspontaneously broken chiral symmetry.

We have made predictions concerning the critical point, where the cross-over1 phase tran-sition of chiral symmetry restoration and the first order phase transition line coming from lowtemperatures and high chemical potentials meet. Triple point and critical point are predictedto coincide for physical quark masses. As in this point three phase transitions of different or-der meet the critical point is in fact a so-called ”tri-critical” point. The dependence of this”tri-critical” point on the current quark mass has been examined. Up to the precision of thecalculation this dependence has been only a dependence of the critical chemical potential andnot of the critical temperature. The critical temperature is most likely stabilized by the presenceof the diquark condensate.

Apart from the analysis of the phase diagram the predictions of the models developed herehave been compared to QCD calculations on the lattice, showing good agreement at the level of10−20%. Comparing the mean field calculation with the newly developed approximation schemewe have shown that especially in the region near the critical point where thermal fluctuations

1For vanishing quark mass the model predicts a second order phase transition as chiral symmetry is completelyunbroken in the high temperature phase. At non-zero quark mass there is a cross-over phase transition due tofact that chiral symmetry cannot be restored completely due to the explicitly symmetry breaking terms.

89

90 Chapter 6. Summary

are large, the improved version of the model predicts a faster cross-over phase transition. Thatis, susceptibilities show more articulate structures than in mean field approximation.

To improve on the discrepancies of PNJL calculations and lattice QCD results, a new ansatzfor the Polyakov loop model has been proposed. The corresponding new effective potential hasbeen motivated by group theoretical considerations. First calculations have shown promisinglygood agreement with lattice calculations, at a level of comparison considerably better than10%. This supports the expectation that the loop dynamics is strongly influenced by the groupstructure of SU(3)C.

6.2 Conclusion

The new scheme of approximating the path integrals is able to describe thermal expectation val-ues in a much better way than pure mean field calculations. At the origin of the short comingsof mean field calculations is the complexity of the Euclidean action. The complex structure ofthe potential landscape may lead to large contributions to the path integral used to calculatethermal expectations values that are not located at the mean field solution. These contributionsto the thermal expectation values arise in the neighborhood of the mean field solutions. As thevicinity of the mean field solution becomes important here this way of approximating thermalexpectation values partially takes into account thermal fluctuations. Thermal expectation val-ues of fluctuations are quantities that die out in the large volume limit: 〈φ2〉 − 〈φ〉2 ∝ V −1.The approximation presented here however neglects these terms. For the large volume limitthe expansion of the action truncated at second order is sufficient. For the proper calcula-tion of expectation values of thermal fluctuations this approximation needs to be generalizedsystematically taking into account all terms ∝ V −1.

At critical points and triple points the second derivative of the thermodynamic potential issingular. Close to these points the second derivative of the thermodynamic potential is badlyconditioned. This leads to instable numeric solutions. Therefore orders of phase transitions inthe vicinity of triple points and critical points as well as the positions of triple and critical pointsare rather hard to determine in this scheme.

The merit of this new approximation ansatz is the calculation of the thermal expectationvalues of Polyakov loop and its complex conjugate in the presence of diquarks. The two expec-tation values are both real and in general different at non-vanishing quark chemical potential asit is stated in another approach [DPZ05].

The PNJL-model treated here makes predictions on the critical point and its quark massdependence. This dependence is of interest when comparing with lattice results as many latticecalculations still run with unphysically large quark masses. The available lattice predictionson the critical point are however not in agreement with the model described here. This couldbe due to the fact that those lattice results were obtained for 2 + 1 flavors while our modelhas been working with only two flavors. Another source of the discrepancies could of coursebe systematic errors of the lattice calculations, that are not yet completely under control asthese extrapolations to high quark chemical potential are extremely difficult and numericallyexpensive.

The discrepancies of the model predictions and lattice results suggest that the built-inPolyakov loop model does not capture important features of gluon dynamics. Therefore we haveintroduced an improved ansatz for an effective loop potential allowing to encode constraintsoriginating in the group structure of SU(3)C. First calculations have shown better agreementwith lattice calculations. From this promising agreement it can be inferred that this ansatzcomes closer to the effective dynamics of pure gluonic QCD. From the results including quarksone possibly could learn more about the dynamics of pure gluonic systems.

6.3. Outlook 91

6.3 Outlook

By generalizing the presented approximation of thermal expectations values it may becomepossible to describe expectation values of thermal fluctuations in a correct way. The fluctuationsof the Polyakov loop across the deconfinement transition is of special interest here.

The extension of the PNJL model to 2+1 quark flavors in the presence of diquarks is one ofthe next steps to be made. 2 + 1 flavor calculations could possibly resolve current discrepanciesof model and lattice QCD predictions on the critical point. This is not unlikely as the orderof phase transitions reacts sensitively to small changes in the parameters or the regularizationscheme.

The approximation of all fields by the temporal and spatial averages could have great impacton the order of a phase transition. This is indicated by the sensitive dependence on changesto regularization scheme observed. If a physically motivated momentum dependence of theNJL coupling strengths is built into the model the influence of possible regularizations shouldbe reduced. The running coupling of QCD should be implemented systematically in furtherdevelopments of models.

Appendix A

Conventions

In this little section I would like to give some of the conventions used in this work.

Traces

• The separate trace over Dirac, color or flavor indices is denoted explicitly as trDirac = trd,trc or trf .

• The trace over Dirac, color and flavor indices is denoted as Tr = trdtrctrf

• The trace denoted Tr denotes the trace defined by

Tr [· · · ] =

∫∑

n

〈n| · · · |n〉

• In analogy the determinant over Dirac, color or flavor indices is denoted Det (in contrastto det)

• A functional trace with an additional trace over Dirac, color and flavor indices is denotedas

TrO =

∫d4p

(2π)4Tr 〈p|O|p〉 =

∫d4xTr 〈x|O|x〉

Path integrals

• The generating functional is denoted as Z, while the grand canonical partition function isdenoted Z.

• Whenever a path integral with integration limits τ = 0 and τ = β (or t = 0 and t = −iβ)is written down, periodic boundary conditions are implied for bosons and antiperiodicboundary conditions are implied for fermions, i. e. for bosons we use φ(τ = 0) = φ(τ = β),and for fermions we use φ(t = 0) = −φ(t = −iβ).

• The fields φ1...n in path integrals and approximations thereof are wildcards for any kindof field. φ1...n represent whatever degrees of freedom there are in the system. Note that incontrast to small φ, capital Φ is the Polyakov loop defined as Φ = 1

3trc L

• The angles 〈. . .〉 are used for vacuum expectation values at T = 0, while thermal expecta-tion values are labeled with a subscript β: 〈. . .〉β.

92

93

Polyakov loops

• The Polyakov loop L is defined as in (4.2). L is an operator in color space.

• The normalized trace of the Polyakov loop L is denoted as Φ = 1NC

trc [Φ].

• The thermal expectation value of the traced Polyakov loop Φ is denoted as l = 〈Φ〉

• The thermal expectation value of the complex conjugate of the traced Polyakov loop Φ isdenoted as l∗ = 〈Φ∗〉. Note that that the complex conjugate of l does not equal l∗.

Notations of SU(N) structures

• The generators of SU(N) are denoted τa with a = 1, . . . , N2 − 1. In the special caseSU(3)C the generators are sometimes denoted λa.

• The generators are normalized such that tr [τaτb] = 2δab.

• The N ×N unity matrix is denoted 1.

• The generators are supplemented by τ0 =√

2/N 1.

• The symmetric generators including τ0 are denoted τS with τTS = τS (or λS respectively).

• The antisymmetric generators are denoted τA with τTA = −τA (or λA respectively).

Classification of the order of phase transitions

• First order phase transitions are transitions where the first derivative of the thermodynamicpotential with respect to the fields is discontinuous.

• Second order phase transitions are transitions where the second derivative of the thermo-dynamic potential with respect to the fields is discontinuous.

• A cross-over transition is not a phase transition in the strict sense as the correspondingsymmetry is broken on both sides of the transition line. Such situations arise for examplewhen both spontaneous and explicit symmetry breaking are present. In this work thetransition line is plotted at maximal susceptibility. It is important to realize however thatthe position of a cross-over as we plot it is just one criteria we chose to work with. Forexample field redefinitions may change the point of maximal susceptibility. This indicatesthat the cross over transition is no phase transition in the strict sense. Its position is fuzzyand subject to our choice of fields and transition criteria.

Appendix B

Fierz Transformations

In this appendix the Fierz transformations and their use in NJL models is enlightend. Fierztransformations are transformations of Dirac structures or matrice representations of SU(N)group elements. Fierz tranformations in general are of the form

Γ(X)ij Γ

(X)kl =

∑

n∈N

c(Y )n Γ

(Y )il Γ

(Y )kj , (B.1)

where Γ represents a matrix structure either in Dirac space or an element of a SU(N). TheFierz transformation in NJL models is used to construct an effective Lagrangian that producesthe Hartree-Fock result when calculating in the simpler Hartree formalism. This approach ispossible here, as direct and exchange terms for point interactions are of the same general form(see Sec. 3.2.2). The effective Lagrangian is constructed by adding direct and exchange terms

Leff. = Ldir. + Lex. (B.2)

B.1 Fierz identities for Dirac matrices and elements of SU(N).

B.1.1 Dirac structures

The coefficients defined in (B.1) are given in many standard textbooks [IZ, PS]. The coefficientsare given here in matrix form.

(1)ij (1)kl

(iγ5)ij (iγ5)kl

(γµ)ij (γµ)kl

(γµγ5)ij (γµγ5)kl

(σµν)ij (σµν)kl

=

14 −1

414 −1

418

−14

14

14 −1

4 −18

1 1 −12 −1

2 0

−1 −1 −12 −1

2 0

3 −3 0 0 −12

(1)il (1)kj

(iγ5)il (iγ5)kj

(γµ)il (γµ)kj

(γµγ5)il (γµγ5)kj

(σµν)il (σµν)kj

(B.3)

These relations show the transformation to the qq-channel. The qq-channel is obtained whencharge conjugating two of the fields connected by one of the Γ-structures.

(qΓ(X)q

)(qΓ(X)q

)=(qΓ(X)q

)(qCCΓ(X)CCq

)=(qΓ(X)q

)((qC)TCΓ(X)C

(qC)T)

(B.4)

= −(qΓ(X)q

)(qCC

[Γ(X)

]TCqC

)=(qΓ(X)q

)(qC[Γ(X)

]CqC)

(B.5)

94

B.1. Fierz identities for Dirac matrices and elements of SU(N). 95

Then the Fierz transformation (B.3) is applied. Finally the charge conjugated fields are trans-

formed back. Doing this it was used that[Γ(X)

]C=[±(S,P,A)

(V,T)

](X) [Γ(X)

]. Where the minus sign

appears in the vector, the axialvector and the tensor term.

(qiΓ

(X)ij qj

)(qkΓ

(X)kl ql

)=[±(S,P,A)

(V,T)

](X) (qiΓ

(X)ij qj

)(qCl Γ

(X)lk qCk

)

Fierz trafo−−−−−−→∑

n∈N

[±(S,P,A)

(V,T)

](X)c(Y )n Γ

(Y )ik Γ

(Y )lj qiqj q

Cl q

Ck (B.6)

= −∑

n∈N

[±(S,P,A)

(V,T)

](X)c(Y )n Γ

(Y )ik Γ

(Y )lj qiq

Ck q

Cl qj

= −∑

n∈N

[±(S,P,A)

(V,T)

](X)c(Y )n

(Γ(Y )C

)ik

(C Γ(Y )

)ljqiq

Tk q

Tl qj

= −∑

n∈N

[±(S,P,A)

(V,T)

](X)c(Y )n

(q Γ(Y )C qT

)(qT C Γ(Y ) q

)(B.7)

This procedure gives a new matrix of coefficients

(1)ij (1)kl

(iγ5)ij (iγ5)kl

(γµ)ij (γµ)kl

(γµγ5)ij (γµγ5)kl

(σµν)ij (σµν)kl

=

14 −1

414 −1

4 −18

−14

14

14 −1

418

1 1 −12 −1

2 0

1 1 12

12 0

−3 3 0 0 −12

(iγ5C)ik (C iγ5)lj

(C)ik (C)lj

(γµγ5C)ik (C γµγ5)lj

(γµC)ik (C γµ)lj

(σµν C)ik (C σµν)lj

. (B.8)

Note that the vector on the right is again of the order (S, P, V, A, T). This however madeinterchanges of S ↔ P and V ↔ A necessary.

B.1.2 SU(N) structures

The following notations are used in this subsection:

• The generators of SU(N) are denoted τa with a = 1, . . . , N2 − 1.

• The generators are normalized such that tr [τaτb] = 2δab.

• The N ×N unity matrix is denoted 1.

• The generators are supplemented by τ0 =√

2/N 1.

• The symmetric generators including τ0 are denoted τS. They satisfy τTS = τS.

• The antisymmetric generators are denoted τA. They satisfy τTA = −τA.

For the transform to the quark-antiquark channel the identity(

(1)ij (1)kl

(τa)ij (τa)kl

)=

(1N

12

2 N2−1N2 − 1

N

) ((1)il (1)kj

(τa)il (τa)kj

)(B.9)

is used.For the transform to the diquark channel the identity

((1)ij (1)kl

(τa)ij (τa)kl

)=

(12

12

N−1N −N+1

N

) ((τS)ik (τS)lj

(τA)ik (τA)lj

)(B.10)

is of help.

96 Chapter B. Fierz Transformations

B.2 Fierz transformation of the color current interaction

The color current interaction that was used as ansatz is

δLint = −gN2

C−1∑

a=1

(ψ γµ λaψ

) (ψ γµ λaψ

). (B.11)

Here the generators for color space SU(NC)C fulfilling the normalization condition in App. B.1.2are the Gell-Mann λ matrices. The generators of flavor space SU(Nf)f are denoted τa.

When calculating the exchange term in the quark-antiquark channel one finds

Lex.,qq =(N2

C − 1)

N2C

g

N2f −1∑

a=0

[(qτaq)

2 + (qiγ5τaq)2 − 1

2(qγµτaq)

2 − 1

2(qγµγ5τaq)

2]

− 1

2NCg

N2f −1∑

a=0

[(qτaλa′q)

2 + (qiγ5τaλa′q)2 − 1

2(qγµτaλa′q)

2 − 1

2(qγµγ5τaλa′q)

2]

(B.12)

=1

4g

N2C−1∑

a′=0

N2f −1∑

a=0

[(qτaλa′q)

2 + (qiγ5τaλa′q)2 − 1

2(qγµτaλa′q)

2 − 1

2(qγµγ5τaλa′q)

2],

(B.13)

where in the last line the normalization of the SU(NC)C generators was redefined. Here thenormalization is

tr[λaλb

]= NCδab λ0 =

√4(N2

C − 1)

N2C

1 ⇒ tr[λ0λ0

]=

4(N2

C − 1)

NC. (B.14)

The exchange term in the diquark channel is

Lex.,qq =Nc + 1

2Ncg

N2f −1∑

b=0

∑

A′

[(qiγ5CτbλA′ qT )(qTCiγ5τbλA′q) + (qCτbλA′ qT )(qTCτbλA′q)

− 1

2(qγµγ5CτbλA′ qT )(qTCγµγ5τbλA′q) − 1

2(qγµCτbλA′ qT )(qTCγµτbλA′q)

]

−Nc − 1

2Ncg

N2f −1∑

b=0

∑

S′

[(qiγ5CτbλS′ qT )(qTCiγ5τbλS′q) + (qCτbλS′ qT )(qTCτbλS′q)

− 1

2(qγµγ5CτbλS′ qT )(qTCγµγ5τbλS′q) − 1

2(qγµCτbλS′ qT )(qTCγµτbλS′q)

].

(B.15)

This fixes the ratio of the coupling constants for a Lorentz invariant interaction to

G : G8 : GV : GV8 : H : H0 =

2(N2C − 1)

NfN2C

: − 1

NfNC: − (N2

C − 1)

NfN2C

:1

2NfNC:

Nc + 1

2Nc: − Nc + 1

4Nc. (B.16)

If one chooses to use different coupling strengths for scalar and vector part of the interaction,the scalar and the vector couplings have again fixed ratios. The ratio of the scalar couplingconstants is

G : G8 : H =2(N2

C − 1)

NfN2C

: − 1

NfNC:Nc + 1

2Nc. (B.17)

B.2. Fierz transformation of the color current interaction 97

The ratio of the vector coupling constants is

GV : GV8 : H0 = − (N2C − 1)

NfN2C

:1

2NfNC: − Nc + 1

4Nc. (B.18)

When NC = 3 and Nf = 2 one finds

G : G8 : H = GV : GV8 : H0 = 1 : − 3

16:

3

4(B.19)

Appendix C

Evaluation of Feynman graphs

C.1 Derivation of Feynman rules for NJL-models

We add a brief derivation of the propagators and vertex functions. This derivation will beperformed only for a simple Lagrangian which is basically the Lagrangian used in the originalpapers by Nambu and Jona-Lasinio [NJL61a], [NJL61b].

L = ψ(i/∂ − m

)ψ +

G

2

(ψψ)2

+(ψiγ5~τψ

)2(C.1)

Here the mass term m is a matrix in the two dimensionsal flavor space. It will be necessary atsome point to generalize some of the results.

First the vertex function for the bare fields is derived. In momentum space this gives theFeynman rule for the vertex. All momenta are incoming.

= Γ(x1, . . . , x4) =i δ4S

[ψ, ψ

]

δψα(x1)δψβ(x2)δψγ(x3)δψδ(x4)

∣∣∣∣∣ψ=ψ=0

(C.2)

⇒ Γ(p1, . . . , p4) = iG(δαγδβδ + δαδδβγ + [iγ5τa]αγ [iγ5τa]βδ

+ [iγ5τb]αδ [iγ5τb]βγ)δ(4)(p1 + · · ·+ p4) (C.3)

Now the quark propagator is calculated. Again all momenta are incoming.

S−1F (x1, x2) = Γ

(2)Born(x1, x2) =

δ2S[ψ, ψ

]

δψα(x1)δψβ(x2)

∣∣∣∣∣ψ=ψ=0

(C.4)

=

∫d4x δ(4)(x− x1)

(i/∂ − m

)αβδ(4)(x− x2) (C.5)

⇒ S−1F (p1, p2) = δ(4)(p1 + p2)

(/p− m

)(C.6)

= SF(p) =(/p− m+ iǫ

)−1. (C.7)

C.2 Derivation of the gap equation via the Dyson equation

In this part of the appendix a more explicit derivation of the Dyson equation will be presented.Applying the Feynman rules to the graphs in Fig. 3.1 we find

Sdressed = Sbare + Sbare K Sdressed (C.8)

98

C.2. Derivation of the gap equation via the Dyson equation 99

Sdressed is the dressed propagator, Sbare is the bare propagator and K is the Kernel. This canbe multiplied by the inverse dressed propagator from the right and the inverse bare propagatorfrom the left giving

S−1dressed = S−1

bare −K. (C.9)

When applying the Feynman rules the Kernel K which is just a tadpole graph can be evaluatedto

K =

∫d4q

(2π)4Γ

/q −M + iǫ(C.10)

= iG

∫d4q

(2π)4

(δαγδδβ + δαβδγδ + [iγ5τa]αγ [iγ5τa]δβ + [iγ5τb]αβ [iγ5τb]γδ

)( 1

/q −M + iǫ

)

γδ

= iG

∫d4q

(2π)4

((1

/q −M + iǫ

)

αβ

+ δαβTr

[1

/q −M + iǫ

]+

(1

/q −M + iǫ

)

αβ

+ [iγ5τa]αβ Tr

[1

/q −M + iǫiγ5τa

])(C.11)

= iGδαβ

∫d4q

(2π)4Tr

1

/q −M + iǫ+ 2iG

∫d4q

(2π)42

(1

/q −M + iǫ

)

αβ

(C.12)

= iGδαβ

∫d4q

(2π)4Tr

1

/q −M + iǫ+ iGδαβ

∫d4q

(2π)4Tr

(1

/q −M + iǫ

)(C.13)

=⇒M = m+ 4iG

(NcNf +

1

2

)∫d4q

(2π)4M

q2 −M2 + iǫ(C.14)

The factor 12 originates from the exchange term. The exchange term appears because this

calculation is performed in Hartee-Fock approximation. It is however sufficient to calculate inHartree approximation, if the Lagrangian was modified such that all direct terms already includethe exchange terms of the original Lagrangian. This is explained in more detail in Sec. 3.2.1and Sec. 3.2.2.

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Acknowledgements

This thesis would not have been possible without the help and support of many people. So Iwould like to drop a few lines fraught with thanks to these fantastic people, knowing too wellthat these crummy lines are not in the slightest adequate to express my gratitude. This includespeople I worked with in the resent year and of course those people that supported me all alongthe way. As it is virtually impossible to make this complete, I would like to apologize to allthose that I have not listed by name but contributed to this thesis in one way or another.

I would like to thank Prof. Dr. Wolfram Weise for guiding me to the interesting questions,for motivation and for his constant interest. He has been pointing to the details to be understoodand to the problems to be solved.

The enlightening discussions with Claudia Ratti and Michael Thaler helped me understandespecially the technical issues involved in this thesis. I thank Norbert Kaiser for supplying mewith important answers to group theoretical questions. Thanks to the computer expertise ofMichael Thaler and Bernhard Musch computer problems remained unknown to me.

It was a great pleasure to work together with my office mates who created a fantastic workingatmosphere — always relaxed but even more so productive. I am obligated to all members ofT39 for the great work climate and for all the inspiring discussions.

But all this would not have been possible without the support both before and during thetime I dedicated to this diploma thesis. Here I thank the Studienstiftung des deutschen Volkesfor supporting me not only financially thoughout the course of my studies.

I am most indebted to my parents who have always supported me in so many ways.

104

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