2+1 flavor qcd at imaginary chemical potential1 flavor qcd at imaginary chemical potential...

2+1 Flavor QCD at Imaginary ChemicalPotential

Master’s Thesis

submitted byFlorian Meyer

Fakultat fur PhysikUniversitat Bielefeld

July 2012

Supervisor and 1st corrector: Prof. Dr. Edwin Laermann2nd corrector: Dr. Christian Schmidt

Contents

1. QCD in the continuum 31.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2. QCD - fundamental relations . . . . . . . . . . . . . . . . . . . . . . . . 41.3. Gauge fixing and renormalization . . . . . . . . . . . . . . . . . . . . . . 6

2. QCD on the lattice 92.1. Naive discretization and gauge principle . . . . . . . . . . . . . . . . . . 92.2. Discretized gauge action . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3. Wilson fermion action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4. Staggered action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5. Action used in the simulation . . . . . . . . . . . . . . . . . . . . . . . . 14

3. Finite temperature on the lattice 173.1. Introducing temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2. Phase diagram and phase structure . . . . . . . . . . . . . . . . . . . . . 183.3. Polyakov loop and Z3 symmetry . . . . . . . . . . . . . . . . . . . . . . . 193.4. Chiral symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5. Critical phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4. Numerical Simulation of Lattice QCD 274.1. Importance sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2. Metropolis algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3. The fermion determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.4. Hybrid Monte Carlo methods . . . . . . . . . . . . . . . . . . . . . . . . 304.5. HMC applied to QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5. Lattice QCD at finite density 355.1. Introduction of chemical potential . . . . . . . . . . . . . . . . . . . . . . 355.2. Sign problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.3. Taylor expansion and imaginary µ . . . . . . . . . . . . . . . . . . . . . . 36

6. Methods and observables 396.1. Method of noisy estimators . . . . . . . . . . . . . . . . . . . . . . . . . . 396.2. Jacknife method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.3. Computation of observables . . . . . . . . . . . . . . . . . . . . . . . . . 41

1

Contents

7. Results at imaginary chemical potential 457.1. Outline / simulation details . . . . . . . . . . . . . . . . . . . . . . . . . 457.2. Chiral condensate and susceptibility . . . . . . . . . . . . . . . . . . . . . 467.3. Quark density and susceptibility . . . . . . . . . . . . . . . . . . . . . . . 527.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

A. Computation of coefficients 59

B. Parameterization of scaling functions 63

C. Tables 67

Acknowledgements 73

Disclaimer 75

2

1. QCD in the continuum

1.1. Introduction

In the second half of the 19th century physics faced a severe crisis. While prior to that itseemed possible for many physicists that most of the field had been discovered, over timea few problems emerged. The black body spectrum, the photo effect, the existence ofthermodynamical equilibrium etc. withstood all attempts to solve them using theoreticaltools at hand. By the end of the century Planck, Rutherford and many others opened thedoor to what today is called quantum mechanics. Since was sufficiently finished in 1927,it has proved to be the language of the microscopic world. Moreover, fused with specialrelativity and the mathematical concept of symmetry and group theory, it provides thefield of modern particle physics with the concept of four independent forces. Three ofthem were successfully formulated as quantum field theories, namely the electromagneticforce, the strong force and the weak force. For the gravitational force, in contrast, noclosed framework using quantum mechanics exists. The Standard Model is a collectionof theories describing the first three forces, i.e.

the electromagnetic force is described by the theory of quantumelectrodynamics (QED).Mediating between electrically charged particles, it is transmitted by photons asforce carriers.

The strong force is described by the theory of quantumchromodynamics (QCD). Me-diating between colorcharged particles, it is transmitted by gluons as force carriers.

The weak force is described by the Glashow Weinberg Salem model (GWS). Mediatingbetween particles via the exchange of Z/W+/− bosons.

In addition, a set of elementary particles is known, which are not composed of otherparticles and are listed in Table 1.1. Among the fermions, the six quarks up, down,

ParticlesGeneration Fermions Bosons Theory

Quarks Leptons Mediators1st u d e νe γ QED2nd s c µ νµ W+/−, Z GWS3rd b t τ ντ g QCD

Table 1.1.: Elementary particles in the Standard Model

3


strange, charm, bottom and top, are the only ones to be exposed to the strong force.In addition to this color charge, they also have an electric charge and hence interactelectromagnetically. Of the six leptons, only the electron e, the muon µ and the tauon τhave an electric charge, while their corresponding neutrinos, νe,νµ and ντ , have neithercolor nor electric charge. Their creation and annihilation hence is a characteristic ofweak processes. Among the bosons, the photon is the only one not interacting with itsown kind, while the eight existent gluons can interact with each other, as well as there isinteraction between a W+, a W− and a Z. This feature renders the theory of QCD muchmore difficult to compute than QED, but also provides a large spectrum of observablecomposite particles. With three identified color degrees of freedom and the requirementof observable particles to be colorless, one can describe a variety of quark/antiquarkstates qq and three quark states qqq, called mesons and baryons, respectively. The selfinteraction is also related to the fact that colored particles, at least below a certainenergy threshold, are not observable, i.e. only colorless particles can be detected.

Incorporating the feature of finite temperature into QCD is an important step towardsan accurate description of nature. While e.g. collider experiments with rather lightparticles are sufficiently well described at zero temperature, collisions of heavy ions (e.g.lead or gold) urge theorists to also include thermodynamical considerations into thedescription of occurring phenomena. The outcome of a heavy ion collosion experimentis a hot ’fireball’ of matter, whose temperature depends on the former energy of the twoions heading towards each other. At a certain energy a phase transition is expected,leading to a phase which is called the deconfined phase, where colored particles are notconfined to form color neutral particles anymore. Hence, it differs a lot in the physicsand phenomena from the confined phase mentioned above, and is under investigationexperimentally as well as theoretically. In order to come even closer to a physical pictureof heavy ion collisions, not only the notion of temperature, but in addition also thenotion of finite density is needed. This is technically very difficult to persue, particularlywithin the framework of Lattice Gauge Theory, which allows for a theoretical descriptionstarting from first principles. This topic is addressed in the present work with respectto two techniques developed which at least give rise to results for small densities.

1.2. QCD - fundamental relations

QCD is the theory of strong interactions incorporated into the Standard Model. It isconstructed starting with a non-interacting Euclidean Lagrangian given by

L =∑f

ψα,af (x)(γµα,β∂µδ

a,b +mfδα,βδa,b

)ψβ,bf (x), (1.2.1)

where multiple occurring indices are summed over. Here the ’lower’ greek letters (α, β...)refer to the Dirac structure of the fields, the ’higher’ ones (µ in this example) refer to theLorentz structure, while the ’lower’ latin letters refer to the color structure. The indexf is summed over explicitly and shows that the Lagrangian does not couple different

4

1.2. QCD - fundamental relations

flavours. Keeping in mind the mathematical structure instead of writing it down andsetting the sum over flavours aside (the Lagrangian splits anyway), one obtains

L = ψ(x)(γµ∂µ +m

)ψ(x). (1.2.2)

It is invariant under a global rotation in color space, i.e. an SU(3) transformation actingon the color indices of the fields,

ψ → V ψ, V ∈ SU(3) (1.2.3)

The deep and fundamental idea of gauge theory is now to promote the global trans-formation to a local one, to make the parameters of SU(3) transformations space timedependent. Because due to the derivative term the Lagrangian is not invariant any-more, one adds a spin-1 vector field Aµ to the derivative term and calls this constructthe covariant derivative Dµ. Its transformation properties are chosen to be

Dµ → V DµV†, (1.2.4)

such that the Lagrangian is now invariant under these local SU(3) transformations. Asa consequence the final Lagrangian, reading

L = ψ(x) (γµDµ +m)ψ(x) = ψ(x) (γµ [∂µ + igAµ(x)] +m)ψ(x), (1.2.5)

now inhibits a coupling of the gauge field Aµ to the quark variables. Mathematicallythey couple via color indices, i.e.

Aabµ (x) = Aiµ(x)T iab, (1.2.6)

with T i being the eight generators of SU(3). Thus, as a linear combination of those, thenew vector field is an element of the algebra su(3). The generators are hermitean andtraceless, fulfilling the ’angular momentum’ commutation relations[

T i, T j]

= if ijkT k. (1.2.7)

Using the color field strength tensor F µν ≡ 1ig

[Dµ, Dν ] ≡ F µνi Ti, the kinetic term for the

gauge fields can be written as

Lkin.,gauge =1

2tr [F µνFµν ] =

1

4F µνi F i

µν , (1.2.8)

which is obviously Lorentz invariant and, due to the trace and the definition of F µν

as a commutator of the covariant derivative, also gauge invariant. The last equalitystems from the normalization condition of the generators, tr [TiTj] =

δij2

. The completeLagrangian of QCD thus reads

LQCD = ψ(x) (γµDµ +m)ψ(x)− 1

4F iµνF

µνi

≡ ψ(x)Mψ(x)− 1

4F iµνF

µνi .

(1.2.9)

5


Using the Lagrangian, expectation values of operators can be defined as

〈O〉 =1

Z

∫DψDψDA O e−SE , (1.2.10)

and hence look similar to the corresponding expectation value when considering spinsystems. The mathematical difference is of course that in the case of field theory onedeals with a functional integral, whereas in the theory of spin systems one has finite (butnevertheless large!) sums over degrees of freedoms. Feynman gave an interpretation tothe notion of a functional integral, and indeed many known methods from functionanalysis can be generalized to be used in a functional manner. This anology allowsutilizing methods from statistical mechanics in field theory. A representation, which willturn out to be useful in later chapters, is given by integrating out the fermionic degreesof freedom, ending up with

〈O〉 =1

Z

∫DAO detMe−SG , (1.2.11)

where detM is called quark or fermion determinant.

1.3. Gauge fixing and renormalization

When computing expectation values via (1.2.10) naively, the results obtained will beinfinite. This has two fundamental reasons.

First, the above integration over the gauge fields is not well defined. The constructAµ is a massless spin-1 vector field, hence it has two independent degrees of freedom.But in (1.2.10) all four components are integrated over, making the integral divergent,see for example [1] or [2]. This problem’s cause can also be addressed in terms of gaugesymmetry. The gauge principle was used to generate interactions between the quarks,but now the freedom of performing transformations blows up the integral. In order to getrid of this problem one has to remove this freedom again, i.e. fix the gauge accordinglyby imposing conditions on the vector fields. A standard procedure that can be found inmost basic textbooks on quantum field theory, see for example [3], is the Fadeev Popovmethod. By this method unphysical ghost fields are introduced into the theory, whichnevertheless may occur in feynman diagrams as virtual particles.

Second, there is another source of infinities when computing observables. Naivelydoing computations within the framework of quantum field theory does not yield anyphysical results. In perturbative expansions, which were the main tool in QED, internal(momentum) loop degrees of freedom in the Feynman diagrams have to be integratedout, unfortunately being divergent. In order to obtain valid results the theory has to befirst regularized and then renormalized. Regularisation means rendering the theory finite

6

1.3. Gauge fixing and renormalization

by some (unphysical) measure. There are various ways to perform regularization, forexample introducing momentum cutoffs, dimensional regularization, Pauli-Villars massesetc. Important in the context of this work is the notion of lattice regularization. This willbe addressed in the next chapter. After regularising the theory, certain quantities likemasses, coupling constants and field operators, have to be reexpressed by their physical,or renormalized, counterparts. Doing this perturbatively in the continuum results inexpressions which do not depend anymore on whatever has been introduced to renderthem finite.

7

2. QCD on the lattice

Lattice gauge theory means expressing QCD not in a continuous space time, but ona discrete space time lattice. Discretizing the theory thereby means to put equation(1.2.9) on a lattice. There is large freedom in doing this as long as when removing thelattice, i.e. taking the limit a→ 0, the correct continuum expression can be reobtained.In general terms this means that the fermions are put onto the sites of a space timelattice as

ψ(x)→ ψ(an) ≡ ψ(n) (2.0.1)∫d4x→ a4

∫d4n ≡ a4

∑n

, (2.0.2)

where a is the lattice spacing, dropped in the spinor argument for convenience, and thesum extends over all sites on the lattice. For the introduction of interaction terms viathe gauge principle a naive discretization is helpful. Later other discretizations of thefermion action will be presented. The following two subchapters are standard derivationsto be found in e.g. [4] or [5].

2.1. Naive discretization and gauge principle

The derivative ∂µ acting on the field ψ(n) can be discretized in many ways, for exampleby a ’central’ or ’symmetric’ difference quotient

∂µψ(n)→ ψ(n+ µ)− ψ(n− µ)

2a. (2.1.1)

Here µ is a canonical basis vector of the lattice. As a result, the discretized free fermionaction reads

S = a4∑n

ψ(n)

(∑µ

γµψ(n+ µ)− ψ(n− µ)

2a+mψ(n)

). (2.1.2)

In order to couple the fermion fields to gauge bosons, again the action is demanded tobe invariant under a local SU(3) transformation of the fields,

ψ(n)→ V (n)ψ(n), ψ(n)→ ψ(n)V †(n). (2.1.3)

The mass term again is invariant, but the derivative terms are not, e.g.

ψ(n)ψ(n+ µ)→ ψ(n)V †(n)V (n+ µ)ψ(n+ µ). (2.1.4)

9


To render this combination invariant a new field Uµ(n) is introduced between the twospinors, which is set to transform under the local SU(3) transformation like

Uµ(n)→ V (n)Uµ(n)V †(n+ µ)

U †µ(n)→ V (n+ µ)U †µ(n)V †(n).(2.1.5)

These new variables are link variables, SU(3) elements

Uµ(n) = eiagAµ(n) = I + iagAµ(n) +O(a2), (2.1.6)

and lattice counterparts of the corresponding continuum SU(3) algebra fields Aµ. Withthis definition of the transformation behaviour, one finds for the other term

ψ(n)ψ(n− µ)→ ψ(n)U †µ(n− µ)ψ(n− µ). (2.1.7)

One defines for this adjoint link variable

U−µ(n) ≡ U †µ(n− µ), (2.1.8)

and hence the gauge invariant action reads

S = a4∑n

ψ(n)

(∑µ

γµUµ(n)ψ(n+ µ)− U−µ(n)ψ(n− µ)

2a+mψ(n)

). (2.1.9)

Using (2.1.6) and ψ(n± µ) = ψ(n) +a∂µψ(n) +O(a2) one obtains the continuum actionincluding Lagrangian (1.2.5), i.e. expression (2.1.9) properly implements the covariantderivative on the lattice. The discretization errors are of order O(a2).

2.2. Discretized gauge action

Since the SU(3) matrices Uµ are the gauge fields, an expression for their kinetic term isneeded. A common Ansatz is the Wilson gauge action,

SG =2

g2

∑n

∑µ<ν

< tr (I3x3 − Uµν(n)) , (2.2.1)

where Uµν = Uµ(n)Uν(n + µ)U †µ(n + ν)U †ν(n) is called plaquette. Since the plaquetteconstitutes a closed path the trace operation renders the action gauge invariant. Sendingthe lattice spacing to zero one should retrieve the expression for the continuum gaugeaction. The link variables are group elements of SU(3), hence utilizing the Baker-Campbell-Hausdorff formula,

eaAeaB = exp

(aA+ aB +

a2

2[A,B] +O(a3)

), (2.2.2)

10

2.3. Wilson fermion action

and Taylor expanding the fields A around n, one obtains up to O(a2):

Uµν(n) = exp(iga2 ∂µAν(n)− ∂νAµ(n) + ig [Aµ(n), Aν(n)]+O(a3)

)= exp

(iga2Fµν(n) +O(a3)

).

(2.2.3)

When the exponential is Taylor expanded and inserted in equation (2.2.1), one findsthat the unit matrices cancel each other, and the terms of order O(a2), O(a3) and O(a5)are purely imaginary due to (1.2.6) and (1.2.7). Only the real part of the O(a4) termdoes not vanish and represents the discretized form of the continuum kinetic action:

SG =2a4

g2

g2

2

∑n

tr F 2µν(n) +O(a6) =

a4

2

∑n

F bµν(n)F b

µν(n) +O(a6), (2.2.4)

where the sum is taken over all µ and ν. In the limit a → 0 the continuum expression(1.2.8) is retrieved and the discretization errors are of order O(a2).

2.3. Wilson fermion action

As already mentioned in the last chapter, discretizing a theory is a method of regular-isation. To be more precise, it introduces a momentum cutoff in terms of restrictingmomenta to the Brillouin Zone. This is a very general phenomenon, because for anarbitrary function f(x) holds

f(p) =

∫dxe−ipxf(x) ≈ a

∑n

e−ipnaf(na) (2.3.1)

⇒ f(p+2π

a) = a

∑n

e−ipnae−2iπnf(na) = a∑n

e−ipnaf(na) = f(p), (2.3.2)

due to the periodicity of the complex exponential. Applied to lattice gauge theory, thismeans that pµ ∈

[−πa, πa

]∀ µ. This has strong consequences for the discretization of a

fermionic theory, as can be seen by considering the fermion propagator of the naivelydiscretised theory (2.1.2), as derived in [4],

D−1(p) =a∑

µ γµ sin(apµ)

i∑

µ sin2(apµ). (2.3.3)

Here for p = (0, 0, 0, 0) the propagator has a pole. By taking the limit a → 0 first, onesees that this is the correct expression for the continuum propagator and this pole refersto its physical pole. But since the Brillouin Zone extends up to π

a, being also roots of

sin(apµ), there are also poles if one or more of the components of pµ are πa, so after all

one is left with∑4

n=0

(4n

)= 16 poles, 15 of them being relics of the (unphysical) Brillouin

Zone. These additional particles are called doublers and are true particles of the theoryif discretised in the naive way. In order to obtain real continuum QCD, the occurrence

11


of these particles should be avoided, for in a dynamical picture they will influence thebehaviour of the continuum theory for example by doubler-antidoubler production (see[6, 7] ). Wilson found a solution to this problem just by adding another gauge invariantterm to the Lagrangian, which in spatial basis reads

δLWilson = −a∑µ

ψ(n)Uµ(n)ψ(n+ µ)− 2ψ(n) + U−µ(n)ψ(n− µ)

2a2, (2.3.4)

and is just a discretization of the continuum Laplace operator. The Lagrangian inmomentum space is modified by

δLWilson =1

a

∑µ

(1− cos(pµa)) , (2.3.5)

hence for a → 0 the contribution leaves the physical pole pµ = 0 ∀ µ untouched butmodifies the mass term for pµ = π

afor any µ with a contributon proportional to 1

a,

thus decoupling these modes from the theory. Unfortunately these Wilson fermionsbreak chiral symmetry even in the limit of massless quarks due to the additional ’mass’term. As a consequence other discretizations have been developed, one of them beingthe staggered action used in the simulations involved in this work.

2.4. Staggered action

The naive fermion action (2.1.2) suffers from unphysical particles placed at the cornersof the Brillouin zone. The idea of the staggered action is to reduce the size of theBrillouin zone so the doublers are simply removed from the momentum domain of thetheory. This is possible by starting out with a lattice and placing different componentsof spinors on different lattice sites [5]. Because this mixes space time and Dirac indices,the space time dimension fixes the number of quarks described by such an action toNf = 2

d2 , where d is the space time dimension.

The staggered action can be obtained from the naive discretisation by a transformation

ψ(n)→

(4∏i=1

γnii

)ψ(n) ≡ Γ(n)ψ(n)

ψ(n)→ ψ(n)

(1∏i=4

γnii

)≡ ψΓ†(n).

(2.4.1)

It is important to respect the order of the Dirac matrices in the transformation of theDirac adjoint spinor, hence the reverse numbering in the product. The mass term isinvariant, the derivative term requires some γ algebra, and in the end one is left with

S = a4∑n

ψ(n)

(4∑

µ=1

ηµ(n)ψ(n+ µ)− ψ(n− µ)

2a+mψ(n)

)(2.4.2)

where ηµ(n) =(−1)∑µ−1j=0 nj , n0 ≡ 0. (2.4.3)

12

2.4. Staggered action

The γ matrices now became phases η, and the whole Lagrangian is diagonal in Diracspace. Thus different Dirac components do not mix and the staggered action is obtainedby taking only one of the Dirac components, hence not summing over the Dirac indicesanymore. By the principle of gauge invariance the interacting staggered action thenreads

S = a4∑n

χ(n)

(4∑

µ=1

ηµ(n)Uµ(n)χ(n+ µ)− U−µ(n)χ(n− µ)

2a+mχ(n)

), (2.4.4)

where χ and χ are one-component of ψ and ψ, respectivley.

Since the degrees of freedom described by this action are distributed over the sitesof a subset (called a hypercube) of the original lattice, quark degrees of freedom can beconstructed as linear combinations of χ and χ. Assuming an even number of sites ineach direction of the lattice, different sites can be addressed via 2hµ + sµ, the first fourvector labeling the hypercube, the latter one labeling the corners of the correspondinghypercube, hence sµ = 0 or sµ = 1. Due to γ2

µ = 1 ∀ µ, the phases and especiallytransformation (2.4.1) are independent of the position of the hypercube,

Γ(2h+ s) = Γ(s), η(2h+ s) = η(s). (2.4.5)

One then defines new quark fields (for the non-interacting case) via

qα,t(h) ≡ 1

8

∑s

Γα,t(s)χ(2h+ s)

qα,t(h) ≡ 1

8

∑s

χ(2h+ s)(Γα,t)†

(s),(2.4.6)

where α is a Dirac index and t labels four different quark degrees of freedom, calledtastes in order to distinguish them from usual flavour. Hence one has four degeneratefermion tastes on each hypercube, being separated from neighboring hypercubes by twolattice spacings. Thus the Brillouin zone is effectively cut in half, and the four tastes donot have doublers.

In order to reduce these taste degrees of freedom the method of rooting is commonlyused, but nevertheless controversial. The idea is that the tastes couple on the lattice,i.e. at finite lattice spacing, but they decouple from each other in the limit a→ 0. Thequark matrix M then aquires block diagonal form, hence one takes the fourth root of thequark determinant to reduce the number of tastes to one. This way one would describee.g. two flavours by

Z =

∫DU (detM1)1/4 (detM2)1/4 e−SG . (2.4.7)

Since simulations take place at a > 0, the controversy is whether extrapolating lattice re-sults to the continuum is allowed or not, because e.g. the universality class of the theory

13


described might change. Though mathematically not fully understood, lattice simula-tions using rooting techniques confirm experimental data. See e.g. [7] for further details.

However, one reason to use staggered quarks is that chiral symmetry is, to a partialextent, preserved on the lattice. One can show that the (massless) staggered actionwritten in ’taste basis’ (2.4.6) is invariant under a global rotation

qα,t → eiξγα,β5 γt,s5 qβ,s, qα,t → qβ,seiξγ

β,α5 γs,t5 (2.4.8)

where the second γ5 matrix acts in taste space and ξ is the transformation angle. Thisis a U(1) remnant of the chiral symmetry in the continuum, and hence at least someaspects of chiral symmetry breaking may be explored on the lattice.

2.5. Action used in the simulation

As mentioned at the beginning of this chapter, one has large freedom in discretizing atheory. In the naive continuum limit, actions reproduce the continuum counterfeit up toa certain order in lattice spacing, e.g. the discretizaton errors of the Wilson gauge actionvanish like O(a2), the staggered formulation has artifacts like the naive formulation, i.e.O(a2). Hence one is in principle allowed to add terms to a discretized action in orderto improve their cutoff behaviour. The action used in this work is a tree level improvedSymanzik action for the gauge part and a p4fat3 staggered action for the fermionic part.The gauge part of the action is obtained by adding all possible of 2× 1 and 1× 2 planarWilson loops to the original Wilson gauge action, leaving lattice artifacts up to orderO(a4) at tree level in the continuum limit. Hence the expression for the improved gaugeaction reads

SG =∑n

∑µ<ν

(βpl(1− Pµν) + βrt(1−Rµν)

), (2.5.1)

with Pµν being the plaquette term and Rµν containing the 1× 2 planar Wilson loops:

Pµν =1

3< tr Uµ(n)Uν(n+ µ)U †µ(n+ ν)U †ν(n)

Rµν =1

6< tr

(Ux,µUx+µ,µUx+2µ,νU

†x+µ+ν,µU

†x+ν,µU

†x,ν + (µ↔ ν)

).

(2.5.2)

The fermionic part is constructed by fattening all link variables of (2.4.4), i.e. addingall possible staple-shaped gauge paths consisting of three links to each link variable,

U fatµ (n) =

1

1 + 6ω

(Uµ(n) + ω

∑ν 6=µ

[Uν(n)Uµ(n+ ν)U †ν(n+ µ)

+U †ν(n− ν)Uµ(n− ν)Uν(n+ µ− ν)])

,

(2.5.3)

14

2.5. Action used in the simulation

with a choice of ω = 0.2. Then another term B (without fattened links) is added to theaction, which consists of all possible 2 × 1 and 1 × 2 ’L-shaped’ paths, such that theDirac matrix is given by

Mij = m δij +(c1Aij + c12Bij

). (2.5.4)

For the p4 -action, c1 = 3/8 and c12 = 1/96. The Matrices A and B are given by

Aij =∑µ

ηµ(i)(U fati,µδij−µ − U

fat†i−µ,µδij+µ

)Bij =

∑µ

ηµ(i)∑ν 6=µ

[(Ui,µUi+µ,νUi+µ+ν,νδij−µ−2ν − U †i−ν,νU

†i−2ν,νU

†i−µ−2ν,µδij+µ+2ν

)+(Ui,νUi+ν,νUi+2ν,µδij−µ−2ν − U †i−µ,µU

†i−µ−ν,νU

†i−µ−2ν,νδij+µ+2ν

)+(U †i−ν,νU

†i−2ν,νUi−2ν,µδij−µ+2ν − U †i−µ,µUi−µ,νUi−µ+ν,νδij+µ−2ν

)+(Ui,µU

†i+µ−ν,νU

†i+µ−2ν,µδij−µ+2ν − Ui,νUi+ν,νU †i−µ+2ν,µδij+µ−2ν

)].

(2.5.5)

15

3. Finite temperature on the lattice

In the preceeding chapters everything discussed was cold, no temperature has been con-sidered yet. The formalism of lattice gauge theory is nevertheless well suited to describephysics at finite temperature, opening a window of opportunity for the investigationof hot QCD matter. Moreover, the perturbative treatment even in the large temper-ature regime faces difficulties due to infrared divergencies appearing in higher orders,and inherently non perturbative phenomena at moderate temperatures cannot be ana-lysed at all [6]. Hence Lattice QCD and especially its numerical simulation are of greatimportance for finite temperature studies.

3.1. Introducing temperature

The partition function introduced so far is very similar in its structure to a partitionfunction describing a system in the framework of quantum statistical mechanics. Ex-pressing this as a path integral, evolving it over all possible field configurations andrequiring it to eventually be in the same state as before, one pictorially resembles therole of the trace in that analogy. The idea is that one can consider a single spin evolvingin an amount of time according to a time evolution operator U = eiHt, with H being aHamiltonian. Following [8] loosely, one imagines a series of snapshots of the spin’s statesduring this evolution and puts them next to each other, forgetting about the notion oftime. Thus one ends up with a spin chain in a certain state. Identifying the time extenti · t ≡ −τ with an inverse temperature β = 1/T , the time evolution operator becomes

P (T ) = e−βH . (3.1.1)

This expression is the Boltzmann distribution and resembles the distribution of states ofthe spin chain. Hence, with the above definitions, one has found an analogy to statisticalmechanics. Because of the relation i · t ≡ 1

T, the time extend is kept finite, i.e. the time

integration int the action has an upper bound β. This is sensible because, unlike incomputations of scattering processes, the notion of initial and final eigenstates at aninfinite time has no meaning in this analogy to statistical mechanics. The resultingaction is given by

SE =

∫ β

0

dτ

∫d3xLE. (3.1.2)

Since β = Nτa in lattice units, large time extents correspond to zero temperature com-putations, and in this sense the effect of temperature is nothing but a desired finitesize effect. Taking the true continuum limit this way means to send the physical spatialvolume to infinity and keeping T = 1

Nτafixed.

17


3.2. Phase diagram and phase structure

The possibility to investigate hot QCD invites for investigations of the thermodynami-cal phase structure in form of a phase diagram. In Figure 3.1 one sees what the phasediagram may look like, according to phenomenological model calculations. Its left part,which is the region of interest in this thesis, is divided into a phase of hadronic matterand a quark gluon plasma phase. These two phases were the first ones assumed to bepresent in the thermodynamics of QCD. The red line bending leftwards in between thoseis first order for higher densities, finally running into a second order endpoint when de-creasing the density. Subsequently the transition is supposed to be a crossover. Thephases at large chemical potential and low temperature are accessible via e.g. large Nc

models. A very important part of the phase diagram is the top left corner, i.e. strongly

Quark Gluon Plasma

Hadronic Region

T

mu

Quarkyonic Phase

Color Superconductor

Figure 3.1.: A rough scheme of the phase diagram of QCD

interacting matter at high temperatures and rather small chemical potentials. It is im-portant because this type of matter is produced in heavy ion collision experiments. AtLHC/CERN scientists aim at very hot matter having around zero baryon density inthe center of collision. According to [9], in a hydrodynamical view two nuclei collidingleave behind matter with zero baryon density. The investigation of more dense matteris the aim of the FAIR project, which reaches lower temperatures in turn. Becauseinteractions in this regime of baryon density µ and temperature T are expected to bedominated by the strong force, lattice QCD calculations can help to clarify the natureof the deconfinement and chiral transitions. As a consequence, many simulations havebeen done since the early eighties, investigating the phase structure of quenched theories,theories with only light quarks and theories with two light plus one heavier (strange)quark. Especially the latter case is useful in the context of the deconfinement and chiraltransitions, since heavier quarks, starting from the charm quark, are unlikely to play a

18

3.3. Polyakov loop and Z3 symmetry

role in this energy regime. The invention of techniques which make it possible to simu-late dynamical quarks at finite chemical potential allows to come closer to the situationsobserved in heavy ion collisions.

On the lattice it is easiest to start considering pure gauge theory, because it alreadycontains the feature of a deconfinement transition. Adding dynamical quarks then mod-ifies this picture, but is much more difficult from the technical point of view.

3.3. Polyakov loop and Z3 symmetry

Phase transitions are in many cases accompanied by a spontaneous breakdown of asymmetry. In the case of pure gauge theory the expectation value of the Polyakov Loop,defined by

P (~n) =∏n4

U4(~n, n4), (3.3.1)

is an exact order parameter. In this context an (exact) order parameter is supposed to bezero in one phase and non zero in the other, thus providing a clear distinction betweenthe two phases. The pure gauge QCD Lagrangian is invariant under multiplicationof all temporal link variables, i.e. those in 4-direction, located at the same time slicex4 = const with one of the three elements of the group Z3. This can be seen easily sinceall elementary quantities in the Lagrangian are plaquettes, so

U34(n)→ U3(~n, n4)zU4(~n+ 3, n4)U †3(~n, n4 + 1)U †4(~n, n4)z† (3.3.2)

= U3(~n, n4)U4(~n+ 3, n4)U †3(~n, n4 + 1)U †4(~n, n4) (3.3.3)

= U34(n), (3.3.4)

since Z3 is the center of SU(3) and hence

[U, z] = 0 ∀ U ∈ SU(3), z ∈ Z3. (3.3.5)

In contrast, the Polyakov Loop closes via boundary conditions, so there is no secondelement of Z3 for cancelation and it transforms like P → zP . As a result,

〈P 〉 =1

3〈P + zP + z2P 〉 =

1

3

(1 + z + z2

)〈P 〉 = 0, (3.3.6)

the Polyakov Loop expectation value is exactly zero. Considering the fact that it isrelated to the confining potential of a heavy quark-antiquark pair like

〈P (~n)P †(~m)〉 ∼ e−βVqq(|~n−~m|), (3.3.7)

and that the Polyakov Loop factorizes for large distances,

〈P (~n)P (~m)〉 −→r→∞

|〈P 〉|2, r = |~n− ~m|, (3.3.8)

19


i.e. one can use the spatial average over all Polyakov Loops, it turns out that its expec-tation value is zero as soon as quarks are confined. In turn interpreting the expectationvalue of a single Polyakov Loop as the probability of seeing a single color charge, as donein [4],

|〈P 〉| ∼ e−Fq/T , (3.3.9)

one can argue that if the free energy of a single quark is finite, then the system is ina state of deconfinement and hence |〈P 〉| aquires a value different from zero. The La-grangian still has the Z3 symmetry, but relation (3.3.6) does not hold anymore, so theZ3 symmetry is spontaneously broken. Thus the Polyakov Loop indeed may serve as anorder parameter in pure gauge theory.

When switching on dynamical fermions this is in general not true anymore. Pictoriallyspoken, the energy in the linear confining potential might at some point become largeenough to produce another quark anti-quark pair and thus never reaches infinity. Hence,the Polyakov Loop in a theory with fermions is never exactly zero in a confined phase,which leaves it useless as an order parameter in the strict theoretical sense. Neverthe-less, simulations show that it still develops a noticeable rise when approaching a phasetransition [5].

3.4. Chiral symmetry

The continuum Lagrangian of zero temperature QCD posseses another very importantfundamental symmetry. Depending on the masses of quarks, a set of transformationsacting on the fermion fields can be identified which leave the Lagrangian invariant. Thelargest symmetry group is found in case of a classical Lagrangian with massless quarks,where it reads

SU(Nf )A × SU(Nf )V × U(1)V × U(1)A. (3.4.1)

The transformation matrices are vector transformations, reading

ψ → eiαTaψ, ψ → ψe−iαTa , (3.4.2)

in addition to axial transformations, reading

ψ → eiαγ5Taψ, ψ → ψeiαγ5Ta , (3.4.3)

with Ta being the generators of SU(Nf ). The two phase transformations U(1)V andU(1)A are obtained by setting Ta = I in the above formulas. Defining projection opera-tors

PL =1

2(I− γ5) , PR =

1

2(I + γ5) (3.4.4)

⇒ ψR = PRψ, ψL = PLψ, ψR = ψPL, ψL = ψPR, (3.4.5)

20

3.4. Chiral symmetry

classifiying the fields to be either righthanded or lefthanded, the Lagrangian splits intotwo parts, L = LL + LR. As mentioned above, equation (3.4.1) is the symmetry groupfor the classical, massless Lagrangian. Upon quantization the U(1)A is broken explicitlydue to the transformation behaviour of the fermion determinant, called axial anomaly.Furthermore, introducing a massterm breaks the SU(Nf )A explicitly because

γ5, γµ = 0, but [γ5, I] = 0, (3.4.6)

so the massterm is not invariant under the transformation. The SU(Nf )V remains validif the masses of the fields are chosen to be degenerate. In this case the remainingsymmetry group is

SU(Nf )V × U(1)V (3.4.7)

If the mass term loses its form of a unit matrix, i.e. the masses are not degenerateanymore, SU(Nf )V is also broken and the only symmetry left is U(1)V . This is themost general case and the one that applies to full Nf = 6 QCD. Its conserved current isnothing but the baryon number.

Nevertheless it is instructive to have a look at the case of Nf = 2 massless quarks,being a good approximation for QCD at low energies where the effect of heavier quarksis suppressed. As can be seen in [10] the ρ and the a1 particles can be transformedinto each other by an SU(2)A transformation. Thus one expects their masses to bedegenerate. However,

mρ ≈ 770MeV, ma1 ≈ 1260MeV, (3.4.8)

are measured in nature. Of course the assumption of massless quarks is not entirely sat-isfied, but such a big difference in the mass cannot be understood by the comparativelysmall u and d quark masses. A similar conclusion can be made by considering the nucleonN and its parity partner N∗, see [4]. The mechanism of spontaneous symmetry breakingcan explain these phenomena. The SU(2)A symmetry, although still being a symmetryof the theory on the level of the Lagrangian, is broken spontaneously and hence themasses of the particles are not degenerate anymore. Furthermore, Goldstone modes, i.e.massless particles in the spectrum, are expected to appear in the case of a continuoussymmetry being broken spontaneously. Since at small, but finite, u and d quark massesthe symmetry is broken slightly, but explicitly, the corresponding Goldstone modes areexpected to aquire a small mass. This is an explanation for the comparatively smallpion masses observed.

At finite temperature there are predictions for the chiral condensate to vanish, i.e.chiral symmetry to be restored at a temperature Tch. Evidence comes from modelcalculations, e.g. the NJL model, see [11]. An order parameter for a chiral transition isthe chiral condensate, which is defined as

Σ = 〈ψψ〉. (3.4.9)

21


In the limit of vanishing quark masses it is an exact order parameter, signaling a restora-tion of the spontaneously broken chiral symmetry above Tch. Since at non vanishingquark masses this symmetry is explicitly broken, it is not an exact order parameter, butstill exhibits conspicuous behaviour at a transition point. Because pure gauge theory canformally be thought of as a theory whose quark masses are infinite, the cases where 〈L〉and Σ, respectively, are exact order parameters are extreme cases of a ’mixed’ situation,when considering finite quark masses. In this case these two transitions, deconfinementand chiral restoration, are suggested to possibly coincide, see [5].

3.5. Critical phenomena

The theory of critical phenomena developed very powerful statements concerning mi-crosopically different systems in the context of phase transitions. Physically relevantphase transitions are split into two classes [12].

first order phase transitions may be classified by a discontinuity of the order parameteras a function of a parameter (e.g. temperature) at the transition point.

second order phase transitions may be classified as the order parameter being a con-tinuos function of a parameter at the transition point.

There is a striking feature of the latter ones which may also serve as a definition todistinguish both types. When approaching a second order phase transition the corre-lation length ξ becomes very large and diverges at the point of transition. This meansthat at such a critical point the microscopic features of a theory become unimportant.Investigation of critical features of a theory can thus be done by integrating out smalllength scales, arriving at an effective Hamiltonian which governs the behaviour of thesystem at a transition point. For a discrete system with Hamiltonian H, this can bedone by the method of real space renormalization. H in general is a function of severalparameters or ’couplings’ ~x. Grouping together several sites on the lattice to form anew lattice can lead to a system with exactly the same functional form, but differentparameters ~x′. The lattice spacing therefore changes like

a→ ba, b > 1. (3.5.1)

There are several schemes how to do the blocking, see e.g. [12]. In general, invariance ofthe functional form will hold at least approximately, so that upon this operation (calledR)

H(~x)R→ H(~x′), (3.5.2)

~x′ = R(~x). (3.5.3)

Thus under repeated transformations the system is taken through parameter space,following a trajectory. This process generally has fixed points,

~xfix = R(~xfix), (3.5.4)

22


which are either repelling, attracting or a mixture of these under renormalization. Ofthese the mixed fixed point is of special interest in the context of a phase transition,therefore such points are also called critical fixed points.

In the immediate vicinity of such a critical point it is possible to linearize the trans-formation R, as

R(~x) = ~xfix +Mδ~x+O(δ~x2), Mij =∂Ri(~x)

∂xj, (3.5.5)

with a diagonalizable Jacobian M . Thus near a mixed point one will generally haveeigenvalues |λ| < 1 and |λ| > 1, the first ones being called irrelevant and attracting ~xtowards ~xfix, and the latter ones being called relevant and repulsing ~x from ~xfix. Theeigenvectors in this basis will be called ~zi and chosen orthonormally, so near ~xfix

~x ≈ ~xfix +∑i

δzi~zi (3.5.6)

⇒ R(~x) ≈ ~xfix +∑i

λiδzi~zi, (3.5.7)

where δzi, the coefficients in this basis, are scaling variables. Since two subsequenttransformations with scale factors b1 and b2 fulfil the group property

λ(b1)λ(b2) = λ(b1b2)⇒ λi = byi , (3.5.8)

with yi < 0 for attractive and yi > 0 for repulsive eigenvalues, the eigenvalues of atransformation near a critical point are related to the scale change b. One can also showthat the free energy per lattice site f of a system, which is here written as a function ofthe scaling variables, transforms like

− ln(e−H( ~δz)) ∼ f( ~δz′) = bd

(f( ~δz)− g( ~δz)

), (3.5.9)

independent of the blocking scheme. The function g is a remnant of the blocking proce-dure and especially is an intensive quantity for large systems, hence yields only regularcontributions, and thus critical behaviour comes from f , more precisely

f( ~δz) = fr( ~δz) + fs( ~δz), (3.5.10)

where the subscripts s and r mean singular and regular, respectively.

With the ’global’ scaling behaviour of the free energy density (3.5.9) and the scalingof the eigenvalues of the linearized transformation (3.5.8) one now can construct, by

specific choices of b = δz−1/yii , where δzi is supposed to be one of the system’s relevant

scaling variables, several forms of scaling equations. Starting from

fs( ~δz′) = fs(b

y1δz1, by2δz2, ...) = bdfs(δz1, δz2, ...) (3.5.11)

23


and choosing b = δz−1/y1

1 , the equation becomes

fs(δz1, δz2, ...) = δzd/y1

1 fs(1, δz2/δzy2/y1

1 , ...). (3.5.12)

Any variable zi belonging to an irrelevant eigenvalue will vanish, since for yi < 0 andy1 > 0

δzi

δzyi/y1

1

= δziδz|yi/y1|1 −→

δz1→00, (3.5.13)

when approaching the critical point. Hence, having two relevant degrees of freedom,

fs(δz1, δz2) = δzd/y1

1 Ψ1

(δz2

δzy2/y1

1

). (3.5.14)

This is a formal variant of the Widom homogeneity hypothesis, which was derived in 1965only empirically, see [12]. One can now consider a spin system, e.g. the two dimensionalIsing model, characterized by

H = J∑<i,j>

sisj +H∑i

si. (3.5.15)

Many models with this structure can be built, e.g. the n-state Potts models, the Heisen-berg model etc; the Ising model is just taken as a representative. The quantities J andH couple different (neighboring) spins and act as an external field, respectively. Theseare the two relevant degrees of freedom, and thus the scaling variables are just the coeffi-cients when expressing the difference vector δ~x in this (orthogonal) basis. Furthermore,in such systems the external field H breaks the symmetry of the spin coupling part (Z(2)in case of the two dimensional Ising model) explicitly. Hence, Phase transitions in cou-pling constant space occur only at a critical Hc = 0, which is, in particular, independentof the temperature. In contrast, the critical pair coupling Jc depends on the temperatureeven at the critical point, because of the factor 1/T in the partition function,

Z =∑si

e−H/T , (3.5.16)

where si means a whole spin configuration. Knowing this, the critical coupling Jc canbe expressed via the critical temperature Tc. Hence, one can choose parameterizations

δz1 = cHH +O(Ht), δz2 = ctt+O(t2, H2), (3.5.17)

with t = (T − Tc)/Tc, for the scaling variables, and the above considerations ensuringthat at least to first order δz1 depends only on H, hence being called the magnetic scal-ing variable, and δz2 depends only on t, being called the temperature like scaling variable.

In general, at a phase transition several quantities of a system diverge (in the thermo-dynamic limit) and, furthermore, they do so in a controlled way. Namely, there are crit-ical exponents defined which govern these divergencies, see table 3.1 for their definition.

24


exponent definition conditions

α c ∼ (|T−Tc|/Tc)−α−1α

T → Tc, H = 0

β M ∼ (Tc − T )β T Tc, H = 0γ χT ∼ |T − Tc|−γ T → Tc, H = 0δ M ∼ B1/δ H → 0, T = Tcη G

(2)c (r) ∼ 1

rd−2+η H = 0, T = Tcν ξ ∼ |T − Tc|−ν T → Tc, H = 0

Table 3.1.: Six critical exponents and their definition

O(2) O(4)α -0.01722 -0.2131β 0.349 0.380γ 1.319 1.453δ 4.780 4.824

Table 3.2.: Critical exponentsfor the O(2) andO(4) universalityclasses

Examples for critical exponents of the O(2) and O(4)universality classes, taken from [13], are shown in Table3.2. In general, critical exponents are universal quanti-ties, i.e. systems with totally different microscopic be-haviour might have the same critical exponents, sincenear criticality the notion of microscopic descriptionloses its meaning. If so, the two systems are said tobe in the same universality class. This is indeed a verypowerful statement, since inside one universality classthe function Ψ1(x) in (3.5.14) is the same function ofa rescaled parameter, a universal scaling function. So,if the scaling function is known from one system, it provides the possibility to makestatements about other systems in the same universality class.

The quantities whose divergence behavior defines the critical exponents are now ex-plained in more detail.

c =(∂ε∂T

)in general labels the specific heat of a system..

M = − ∂f∂H

often called magnetization and thought of in connection with spin models,but it has analoga in totally different systems, e.g. the chiral condensate ψψ inQCD. It can be approached varying any of the two relevant degrees of freedom.

χ = ∂M∂H

is the (magnetic) susceptibility.

G(2)c (r) = 〈φ(0)φ(r)〉 − |〈φ〉|2 , with φ being an order parameter of the system, is a

connected correlation function. It defines the anomalous dimension η.

ξ , the correlation length, is related to the connected correlation function Gc(r) ∼ e−r/ξ

rd−2+η .

Not all of the exponents introduced in Table 3.1 are independent. In fact, from thedivergence behaviour of the quantities above, four equations relating them can be de-duced. These are called scaling laws and were discovered empirically way before the

25


methods of renormalization were at hand to proof them.

2β + γ = 2− α2βδ − γ = 2− α

γ = ν(2− η)

νd = 2− α,

(3.5.18)

where d is again the dimensionality of the system. By comparison of equation (3.5.14)with the relations from Table 3.1 and using the scaling variables (3.5.17), one sees thatall critical exponents are given by the values of y1 and y2.

26

4. Numerical Simulation of LatticeQCD

Lattice regularization is not only a useful analytical tool, it also enables a numericalapproach to theory, be it the determination of critical exponents in spin systems or thecomputation of mass spectra in field theories. Anything that may not be accessibleanalytically, nevertheless, may be approachable in a numerical way. In modern QCDnumerical simulations have become an important part ever since Mike Creutz publishedhis first results on the string tension in 1980. Since then, the field has emerged verymuch with the result of Lattice QCD being one pillar of the investigation of stronginteractions.

4.1. Importance sampling

The expectation value of an operator O was defined as

〈O〉 =

∫DUDψDψO(U)e−SE∫DUDψDψe−SE

. (4.1.1)

The integrals to be evaluated numerically are extremely high dimensional, making themunfeasable to treat with methods like gaussian quadratures; the computational costwould be extremely high for even relatively small lattices, and the errors of those meth-ods generally depend on the dimension. In contrast, the method of Monte-Carlo inte-gration only exhibits a statistical error independent of the dimensionality of the problemand is also much easier to perform in this case. As good as this sounds, fermion fieldswill nevertheless be covered later, because with methods introduced in this chapter sim-ulating full dynamical fermions still takes way too much time even on modern computers.

The main idea of Monte-Carlo integration is to evaluate the (multidimensional) in-tegrand on randomly chosen sets of values xn from the domain of definition. Via themean value theorem an average over the obtained numbers can be taken, e.g. in onedimension ∫ b

a

dxf(x) ≈ (b− a)

N

N∑n=1

f(xn). (4.1.2)

This is the result of the integration up to a statistical error of ∼ 1√N

, see [4], with Nbeing the number of random values used in the integration. A more refined version

27

4. Numerical Simulation of Lattice QCD

would not take homogeneous random values to evaluate the integrand, but values whoseprobability distribution resembles the integrand in a way, e.g.∫ b

a

dxexf(x) =

∫dyf(y) ≈ 1

N

N∑n=1

f(xn), (4.1.3)

where dy = 1eb−ea e

xdx is a properly normalized probability distribution, i.e. the xn aredistributed accordingly, while y is homogeneous.

This idea is the core of simulations in spin systems and lattice QCD. Using the meanvalue theorem, one writes

〈O〉 =1

N

∑U

O(U), (4.1.4)

where U denotes a set of link configurations on the lattice and N is the number ofthese configurations. Comparing this expression to (4.1.1) one sees that the distributionfunction of the link variables is

dP =e−SEDU∫DUe−SE

. (4.1.5)

In order to arrive at this distribution one can start with an initial configuration of linkmatrices (e.g. all the same, or all totally random) and evolve them step by step usingone of the many known algorithms, e.g. Metropolis or Heat Bath. The algorithm usedhas to constitute a Markov process, i.e. newly built configurations may only depend onthe last configuration. If, in addition, the thus obtained Markov chain

1. can reach any possible state in a finite amount of steps (irreducibility of the Markovchain)

2. has a non zero probability of returning to its current state for any number offurther steps (aperiodicity of the Markov chain)

3. the mean number of steps to return to the current state for the first time is finite(positivity of the Markov chain),

then there is a theorem that ensures the existence and uniqueness of a limiting distri-bution. It can be shown that the requirement of detailed balance,

e−S(C)P (C → C ′) = e−S(C′)P (C ′ → C), (4.1.6)

with e−S(C) being a Boltzmann factor dependent on link configuration C and P (C → C ′)being the probability to arrive at configuration C ′ when starting with configuration C,suffices to end up with the desired Boltzmann distribution (4.1.5).

28

4.2. Metropolis algorithm

4.2. Metropolis algorithm

In the last chapter Markov chains and the process of converging to an equilibrium dis-tribution were introduced. The machinery that is applied in order to arrive at theBoltzmann distribution is a set of algorithms, like Metropolis and Heat Bath, but alsoones that were not yet mentioned like Overrelaxation and Hybrid Monte-Carlo. Thesimplest one, but also the most general and versatile, is the Metropolis algorithm, soit will be covered here. The Hybrid Monte-Carlo will be discussed when dynamicalfermions are treated.

Having a configuration of links C the Metropolis algorithm consists of choosing a newcandidate configuration C ′ with a probability P0. Usually this probability is taken tobe symmetric, i.e. P0(C → C ′) = P0(C ′ → C). Then an acceptance test is performed,deciding wether to accept the candidate as the new configuration or keep the old one asa valid new configuration. In case of a symmetric probability P0 the test reduces to theacceptance probability being

PA = min

(1,e−S(C′)

e−S(C)

). (4.2.1)

In words, if the action is lowered by the new configuration, it is accepted. If the ac-tion grows, it is accepted with probability e−S(C′)/e−S(C). Physically spoken, this wayquantum fluctuations are incorporated in the evolution of a systems gauge configuration.

Within this framework one is free to do the updating procedure locally or globally.The first method is just choosing one new link variable per step and thus apply e.g.the Metropolis algorithm with low computational cost per step, because only the localnext neighbor part of the action has to be recomputed. The latter method amountsto choosing a completely new configuration and thus involves much more expensive perstep computations, but in contrast being done with a whole lattice update in one step (ifthe new configuration is accepted!). In general, it will be better to perform local updates(or updates in clusters which are small compared to the lattice) in purely gluonic theory,because doing global updates the change in the action is proportional to the volume V ofthe lattice [4], and hence the acceptance rate, using the Metropolis algorithm, will dropdrastically. The situation is different if one includes dynamical fermions into the theory,where the per-step-computational-cost is extremely high and global updates (using morerefined algorithms than Metropolis) are the faster way.

As already mentioned, starting from an initial configuration of links distribution(4.1.5) is to be reached first by thermalizing the system. This is done by consider-ing certain observables, e.g. plaquette values, while generating new link configurations.Once their distribution is close to (4.1.5), one expects a characteristic behaviour of theseobservables, e.g. the plaquette values saturate and do not vary much anymore. Then,with equilibrated configurations, the measurements of observables of interest can be per-

29


formed, i.e. the actual simulation of QCD starts at this point.

4.3. The fermion determinant

It has already been mentioned that including dynamical fermions into a simulation isa nontrivial task. The underlying reasons are to be discussed now. In principle onecould perform a simulation with dynamical fermions just as as described above, butintegrating (anticommuting) Grassman fields into a simulation needs a lot of memory,and also local updates take a lot of computation time. A strategy to avoid this is tointegrate out the fermion fields, yielding the fermion determinant in the numerator,∫

DUDψDψe−SG(U)−ψM(U)ψ =

∫DU det(M)e−SG(U). (4.3.1)

The fermion matrix M in general has as many rows/columns as there are lattice sites,multiplied by the corresponding ’interaction indices’ of space time and color. For largelattices this number grows very large and the evaluation of the fermion determinantbecomes extremely expensive. Hence doing local updates means also evaluating the de-terminant frequently, and thus the actual reason for local updates, i.e. the fact thatmost of the action could be ingnored in a single step, does not apply here anymore.

The idea of treating it as an observable, i.e. computing

〈O〉 =

∫DU det(M)Oe−SG(U)∫DU det(M)e−SG(U)

=〈O det(M)〉G〈det(M)〉G

, (4.3.2)

where 〈〉G denotes a pure gauge link expectation value, would reduce this cost becausethe whole updating procedure does not involve a single computation of det(M) andusually one even waits several updates before measuring the observable to take care ofunwanted correlation. Unfortunately this leads to large fluctuations of the expectationvalue and hence is only feasible in special cases like two dimensions or when very muchstatistics is available, see [4]. As a consequence, the determinant has to be taken intothe measure and thus to play a role in the updating process.

4.4. Hybrid Monte Carlo methods

A way out is to use global updating procedures instead and try to increase the acceptancerate up to a level where this method becomes usable. In 1987 Duane, Kennedy, Pendle-ton and Roweth proposed a molecular dynamics method which lacks any discretizationerrors, i.e. it is exact up to numerical accuracy [14]. This method is called Hybrid MonteCarlo and is a refinement of earlier methods. The general idea is to influence the changeof link variables when updating globally in a very physical way. This will be sketched

30

4.5. HMC applied to QCD

for a scalar field theory for simplicity and generalised later.

In anology to classical mechanics, one introduces canonical momenta π additive tothe action and treats the action SE as a potential term, i.e. one obtains a kind ofHamiltonian

H(φ, π) =1

2πiπi + S(φ), (4.4.1)

with the momenta being integrated over in the path integral. The partition function doesnot change up to a normalization factor, because the gaussian momentum integrationcan be performed. Since expectation values are to be computed, the normalization factoris irrelevant. Via Hamilton’s equations of motion,

φi =∂H

∂πi= πi and πi = −∂H

∂φi= − ∂S

∂φi, (4.4.2)

one can obtain the dynamics of the scalar fields, which can here be thought of as theanalogue to the link variables in a full QCD formulation, and thus evolve the system incomputertime (i.e. time in the sense of a Markov process). This constitutes a trajec-tory in phase space, finally ending up with a new or, better, alternative configuration.While the scalar fields are chosen only initially and then are just evolved trajectoryafter trajectory according to (4.4.2), the momenta are chosen newly at random aftereach trajectory according to their gaussian distribution. This puts randomness into the,otherwise strictly deterministic, evolution through phase space, and thus yields ergodic-ity (which corresponds to the markov chain being irreducible). This evolution throughconfiguration phase space ends (when the equations of motion are solved exactly) in newconfigurations with exactly the same ’energy’ H. Furthermore, in the thermodynamiclimit the canonical ensemble average of an observable is equal to the microcanonicalensemble average at a fixed energy [5]. So by staying close to this so called moleculardynamics trajectory one obtains new configurations resembling a Boltzmann distribu-tion. Since solving these equations analytically is impossible in practice, one has to doit numerically. Numerical integration of such equations of course leads to numericalerrors, which are not under control. The idea of the HMC is simple, but essential: theprocedure may finally be finished by a Metropolis acceptance test, making the algorithmexact since one can take the computed configuration just as a candidate configuration,and then either accept or reject it.

In order to fulfil the condition of detailed balance the algorithm used for solving (4.4.2)has to preserve the integration measure and has to be reversible. Both conditions aresatisfied with the so called leapfrog integrator.


The reason for using the HMC algorithm is to increase the performance of simulationsincluding dynamical fermions. To incorporate e.g. two degenerate flavors of fermions

31


into the process of Markov chain evolution one can use the fact that a Grassmann integralcan be rewritten as a c-number integral as follows,∫

DψDψe−ψ1Mψ1−ψ2Mψ2 = (detM)2 = detM detM † = detMM †

∼∫DφRDφIe

−φ†(MM†)−1φ.

(4.5.1)

A factor on the right hand side has been dropped since it will cancel in an expectationvalue anyway. At this stage the method becomes limited to simulating an even numberof fermions, because one sees from the result above, that the expression in the exponentof the c-number integral should be positive definite (achieved by MM †), otherwise theintegral will not converge. In this work a method is used to circumvent this, as willbe explained at the end of this chapter. The fields φ, also called pseudofermions, arebosonic variables and can be treated in the molecular dynamics step as an external fieldcontributing to the action. The equations of motion presented above can be rewrittenfor SU(3) gauge fields and conjugate momentum fields. The link variables can be pa-rameterized by 8 real parameters ωaµ, and conjugate to each of them is a real momentumP aµ , thus

Uµ(n) = eiωaµ(n)Ta , Pµ(n) = P a

µ (n)T a. (4.5.2)

The T a are the generators of the gauge group SU(3). In terms of the su(3) algebra fieldPµ the momentum contribution to the Hamiltonian is

1

2π2 =

∑n,µ

tr(P 2µ(n)

). (4.5.3)

When computing a trajectory for the gauge fields, one first refreshes the momentum andpseudofermion variables, i.e. chooses

P aµ ← e−

12π2

and χ← e−χ†χ ⇒ φ = Mχ.

(4.5.4)

With the help of an integration scheme, like e.g. leapfrog, one can then solve theequations

Uµ(n) = iPµ(n)Uµ(n)

Pµ(n) = − ∂SG∂Uµ(n)

− φ†∂(MM †)−1

∂Uµ(n)φ

(4.5.5)

with φ as a constant background field. The second term on the right hand side of thesecond equation is called fermion force and contains the whole fermionic part of the in-teraction. It is the computationally most expensive part in the calculation. For furtherdetails of how to generalize the leapfrog algorithm for use in Lattice QCD, and how todefine the derivative with respect to the gauge links, see [7, 4, 5].

32


In order to be capable of simulating 2 + 1 (staggered) fermions, the rational hybridmonte carlo (RHMC) method has been used in this work. This method proves to bevery effective in case the matrix expression in the exponent of (4.5.1) is of non-integer

power (see [15]). The expression(M †M

)−α, with |α| < 1 being a rational number, is

approximated by

(M †M

)−α ≈ a0 +m∑n=1

anM †M + bn

≡ r2(M †M). (4.5.6)

The heatbath is evaluated according to

χ← e−χ†χ ⇒ φ = r(M †M)−1χ, (4.5.7)

and from here the recipe for the HMC algorithm can be used.

33

5. Lattice QCD at finite density

Chapter 3 dealt with the introduction of finite temperature into QCD, enabling oneto do thermodynamics and investigation of the phase structure of the theory. A nextstep to come even closer to real QCD, as seen for example in collider experiments, is tointroduce a non-zero density of fermions or, in this context, quarks. QCD is a relativistictheory, and hence the number of particles is not constant. Nevertheless, without theintroduction of a finite chemical potential, the net quark density n ≡ nq − nq is alwayszero.

5.1. Introduction of chemical potential

In order to obtain a non-zero net quark density n, one utilizes thermodynamics in agrand canonical ensemble and hence sets up the grand canonical partition function

Z = Z(T, µ) = tr e−(H−µN)/T , (5.1.1)

where N is the quark number operator and µ is the chemical potential. This way theproduction of quarks over antiquarks is favoured and the quark number density, whichis the net quark number normalized to the Volume, can be obtained by

n =1

V〈N〉 =

T

V

∂ lnZ(T, µ)

∂µ= − 1

V

∂Ω (T, µ)

∂µ, (5.1.2)

with Ω(T, µ) = −T lnZ(T, µ) being the grand canonical potential. To put the chemicalpotential explicitly into the lattice action, one transforms all temporal links like

U4(n)→ eaµU4(n),

U−4(n)→ e−aµU−4(n),(5.1.3)

see [16]. Plaquette operators are not affected by this since the additional factors cancelexactly, but loops closed via the boundary conditions in time direction pick up as manyfactors as the number of times they wind around the lattice. Those terms are not presentin the gauge action, but in the fermion determinant, and its modification has seriousimplications for numerical simulations.

5.2. Sign problem

At zero density the Dirac matrix is γ5-hermitean,

M = γ5M†γ5, (5.2.1)

35


so for the determinant one obtains

detM = det(γ5M

†γ5

)= det

(M †) = (detM)∗

⇒ detM ∈ R.(5.2.2)

This is vital to the numerical approach of importance sampling described in chapter 4,since the determinant together with the Boltzmann term is used as a probability weight.The situation changes at finite chemical potential. Modified by equation (5.1.3), thesign of the µ-dependence switches, because for real µ the exponentials are not affectedby hermitean conjugation and hence they break γ5-hermiticity,

M(µ) = γ5M†(−µ)γ5. (5.2.3)

ThusdetM (µ) = (detM (−µ))∗

⇒ detM (µ) ∈ C,(5.2.4)

and in general, importance sampling becomes impossible. Several ways around thisproblem have been figured out, for example evolving the configurations via a Langevinalgorithm or taking the absolute of the determinant (phase quenching). Other methodsevolve the configurations at zero density and multiply the quantity to be measured witha correction factor, the so called reweighting method. In this work two methods arecombined, both valid for rather small values of the chemical potential.

5.3. Taylor expansion and imaginary µ

The idea of the Taylor expansion approach is straightforward: the grand canonical po-tential can be considered at small µ and thus Taylor expanded around µ = 0. Therefore,it is redefined to become a dimensionless grand canonical potential density, as in [17].Using the same symbol the potential now reads

Ω(T, µ) ≡ 1

V T 3lnZ. (5.3.1)

For three flavours u, d and s the Taylor expansion in µi/T is given by

Ω(T, ~µ) =∑i,j,k

cudsijk (T )(µuT

)i (µdT

)j (µsT

)k, (5.3.2)

with the factorials contained in the definition of the coefficients

cudsijk (T ) =1

i!j!k!

∂i

∂(µu/T )i∂j

∂(µd/T )j∂k

∂(µs/T )kΩ(T, ~µ)

∣∣∣∣~µ=~0

. (5.3.3)

In case of two flavors of degenerate u and d quarks, called q from now on, the expansionis simplified to

Ω(T, ~µ) =∑i

cqsij (T )(µqT

)i (µsT

)j, (5.3.4)

36

5.3. Taylor expansion and imaginary µ

with coefficients

cqsij (T ) =1

i!j!

∂i

∂(µq/T )i∂j

∂(µs/T )jΩ(T, ~µ)

∣∣∣∣~µ=~0

. (5.3.5)

Due to CP symmetry the partition function is even in µ/T , so the odd coefficients of(5.3.4), i.e. those whose indices i and j add up to an odd number, vanish identicallywhen evaluated at zero chemical potential, see [17]. Two important quantities can easilybe constructed from the Taylor expansion and thus approximated at finite density. Thequark number density nq is obtained by

nq(T, ~µ)

T 3=∂Ω(T, ~µ)

∂µq/T=∑i,j

icqsij (T )(µqT

)i−1 (µsT

)j, (5.3.6)

and the (diagonal) quark number susceptibility χq by

χq(T, ~µ)

T 2=∂2Ω(T, ~µ)

∂(µq/T )2=∑i,j

i(i− 1)cqsij (T )(µqT

)i−2 (µsT

)j. (5.3.7)

The coefficients cqsij (T ) are at hand for the action used and were e.g. employed in theconstruction of baryonic number and electric charge fluctuations, see [18, 19].

The chiral critical line in the T − µ plane presumably ends in a 2nd order point, asdiscussed in Chapter 3. This will be visible as a singularity in the grand canonical poten-tial, which therefore leads to a breakdown of the Taylor expansion. Since the presence ofsingularities in general limits the radius of convergence of such series, its determinationyields a method for locating the critical point in the T − µ plane, see [20].

The imaginary µ approach is based on the observation that equation (5.2.4) for µ ∈ Creads

detM (µ) = (detM (−µ∗))∗ , (5.3.8)

and thus, for purely imaginary chemical potential, the sign of µ = iµI again switchesand the determinant is real. This enables one to carry out simulations at finite chemicalpotential, though the range of validity of this method is limited, because the partitionfunction is periodic in the chemical potential variable with a period of ∆µI = 2πT/3, asshown by Roberge and Weiss in [21]. Furthermore, a nontrivial phase structure emergesat finite imaginary µ, as the free energy has a discontinuity at µI = 2πT/6 for high T .This so called Roberge-Weiss transition is hence of first order and marks the tunnelinginto another Z3 sector. For low temperatures a smooth crossover connects the sectors.Along the line, these two regions are seperated by a 2nd order endpoint. Furthermore,it is generally expected that the high and low temperature regions are seperated by acritical line in the chiral limit. This situation is illustrated in Figure 5.1. Lattice simula-tions with four staggered quarks as in [22] have provided evidence that the chiral criticaland confinement/deconfinement transition lines conincide and connect to the endpointof the Roberge-Weiss line. These results have also been analytically continued to real µ

37


T

mu_I

Figure 5.1.: A rough sketch of the phase diagram at imagninary chemical potential. Redlines mark Roberge Weiss transitions, blue points mark their endpoints. Thebent lines represent chiral transitions.

[23, 24], which amounts to real baryochemical potentials of µB ≈ 500MeV , and hencethis method reaches, for four staggered quarks, the region relevant in heavy ion collisions.

Since simulations are equally well to perform at finite imaginary µ as at zero chemicalpotential, one can combine the two approaches presented above, and Taylor expandaround finite values of µI . In this case expansion (5.3.4) becomes

Ω(T, ~µ) =∑k,l

cqskl(T )

(µq − µq,0

T

)k (µs − µs,0

T

)l−→

∑k,l

ik+lcqskl(T )

(µI,q − µI,q,0

T

)k (µI,s − µI,s,0

T

)l,

(5.3.9)

with the coefficients being evaluated at ~µI = ~µI,0. This in turn implies, since the coeffi-cients are not evaluated at zero imaginary chemical potential, that the odd coefficientsdo not vanish anymore and have to be taken into account in the expansion. Further-more, odd coefficients are purely imaginary, while even coefficients are purely real whenevaluated at imaginary chemical potential.

38

6. Methods and observables

6.1. Method of noisy estimators

The computation of the pressure coefficients involves taking traces over combinations ofinverse Dirac matrices and derivatives of Dirac matrices with respect to µ. This is donenumerically by the method of noisy estimators, which in this case is performed usingZ(2) noise. A d-dimensional Z(2) random vector has components of either 1 or −1, i.e.

ri = ±1 ∀ i = 1, ..., d.

Defining the random vector product by multiplying two components of the same randomvector and taking the sum over several different random vectors,

< rirj >r≡1

N

N∑n=1

r(n)i r

(n)j , (6.1.1)

one finds that this product has the property

< rirj >r −→N→∞

δij. (6.1.2)

This relation holds, because the product of two equal components yields

r(n)i r

(n)i = (±1)2 = 1 ∀ i, n, (6.1.3)

while a product of different components in general is

r(n)i r

(n)j =

+1 with a probability of 50%

−1 with a probability of 50%. (6.1.4)

Hence, a sum over a large number of different random vectors will be equal to N if thesame components are multiplied and will average to zero if different components aremultiplied. This can be used to compute traces, as is shown in the following. In orderto compute the trace of M−1, with M ∈ Cd × Cd, one can solve the linear system

Mijxj = ri, (6.1.5)

where x ∈ CN and r is a random vector, multiply the same random vector from the leftand use relation (6.1.2), yielding

xi = M−1ij rj =⇒ rixi = riM

−1ij rj

=⇒ < rixi >r=1

N

N∑n=1

r(n)i M−1

ij r(n)j ≈Mii = tr M−1.

(6.1.6)

39


This can be in principle done for an arbitrarily large product of matrices. To computethe product of two traces one must be careful not to include terms contributing to thetrace of the product of matrices, so

tr A tr B =1

N(N − 1)

N∑n=1m 6=n

r(n)i Aijr

(n)j r

(m)k Bklr

(m)l . (6.1.7)

Products of three or more traces can be obtained accordingly.

6.2. Jacknife method

Since the number of configurations available to compute expectation values is finite, amethod for error estimation is needed. In this work the jacknife method is used. Havinga set of N measurements of a quantity x, one can subdivide these into M groups of sizen = N/M . If n is not an integer number, then it is brought down to a round figure, thusleaving out a number of measurements. Defining the average of each group

γi ≡1

n

n−1∑j=in

xj ∀ i = 0, ...,M − 1, (6.2.1)

and further defining the average of all M subsets but one,

Γi ≡1

M − 1

M−1∑j=0j 6=i

γj, (6.2.2)

an expectation value with error estimation is then given by

〈x〉 → 1

N

N−1∑i=0

xi ± δx

with δx =

√√√√M − 1

M

M−1∑i=0

(Γi −

M−1∑j=0

γj

)2

.

(6.2.3)

This method expects uncorrelated data, i.e. the blocking size n chosen such thatthe resulting blocked values are uncorrelated. There are two methods typically used todetermine n, namely computing autocorrelation times for the data at hand, and the datablocking method. An autocorrelation time is obtained by computing the autocorrelation,defined as

Cx(t) = 〈xixi+t〉 − 〈xi〉〈xi+t〉, (6.2.4)

40

6.3. Computation of observables

for measured data points xi seperated by the (computer) time t, see [4]. Assuming anexponential decay of the normalized autocorrelation

Cx(t)

Cx(0)∼ e−t/τ , (6.2.5)

where τ is the autocorrelation time, one can obtain τ by a linear fit at a logarithmicscale. The method of data blocking amounts to computing the error of a quantity x forincreasing block sizes. The error increases as a function of the block size, but reaches aplateau when the data becomes independent. Both methods were used in this work.


The observables treated in this work are exclusively the Polyakov loop, the chiral conden-sate, its susceptibility and the coefficients of the Taylor expansion discussed in Chapter5.3. The coefficients cqs10 and cuds100, evaluated at finite (imaginary) µ, are equal to thequark number densities for two degenerate flavors and one flavor, respectively. The co-efficients cqs20 and cuds200 are equal to half the value of the corresponding quark numbersusceptibilities. While the simulation itself takes all three flavors into account, the mea-sured quantities are one and two flavor observables, respectively.

The chiral condensate can be obtained from the partition function by differentiatingwith respect to the mass, e.g.

〈ψψ〉u ≡1

N3σNt

∂ lnZ

∂mu

, (6.3.1)

for the u quark, with Nσ and Nt being the spatial and temporal lattice extent, respec-tively. The partition function is given by

Z =

∫DU (detMu)

1/4 (detMd)1/4 (detMs)

1/4 e−SG . (6.3.2)

The only mass dependence of Z is inside the quark determinant. Furthermore, the massterm of the action is diagonal (i.e. ∂M

∂m= I), and hence the chiral condensate can be

measured via

〈ψψ〉u =1

N3σNt

1

Z

∫dU

∂ (detMu)1/4

∂mu

(detMd)1/4 (detMs)

1/4 e−SG

=1

4N3σNt

1

Z

∫dU (detMu)

1/4 (detMd)1/4 (detMs)

1/4 ∂ (tr lnMu)

∂me−SG

=1

4N3σNt

〈tr M−1u 〉.

(6.3.3)

In the last step, it is used that

∂ detM = ∂∏i

λi =∑j

∏i 6=j

λi∂ (λj) =∑j

∏i

λi∂ (λj)

λj=∏i

λi∑j

∂ (λj)

λj

= detM∂ (tr lnM) .

(6.3.4)

41


The corresponding u quark susceptibility is obtained by again differentiating with respectto the u quark mass, i.e.

χψψ,u ≡ N3σNt

∂〈ψψ〉u∂mu

=1

16N3σNt

(〈[tr M−1

u

]2〉 − 〈tr M−1u 〉2

), (6.3.5)

with the quark line connected part ∼ 〈tr M−2u 〉 being left out and unconsidered in the

analysis.

The coefficients of the Taylor expansion are obtained similarly by differentiating lnZwith respect to the chemical potential. For degenerate u and d quarks, the one flavorcoefficients are given by equation (5.3.3),

cuds100 =∂Ω(T, ~µ)

∂µu/T, cuds200 =

1

2

∂2Ω(T, ~µ)

∂(µu/T )2. (6.3.6)

The two flavor counterparts can be computed in the same fashion via

cqs10 =

(∂

∂µu/T+

∂

∂µd/T

)Ω(T, ~µ)

cqs20 =1

2

(∂

∂µu/T+

∂

∂µd/T

)2

Ω(T, ~µ),

(6.3.7)

or, which is computationally easier, by utilizing equation (5.3.5) and taking the partitionfunction to be

Z =

∫DU (detMq)

2/4 (detMs)1/4 e−SG . (6.3.8)

Since the derivatives affect only Mu and Mq, respectively, one can effectively write

Z =

∫DU (detM)Nf/4 e−SG (6.3.9)

for the computation. Equations (5.3.3) and (5.3.5) then yield

cn(T ) ≡ 1

n!

N3−nt

N3σ

∂n lnZ

∂µn

∣∣∣∣µ=µ0

, (6.3.10)

with aµq = µq being the chemical potential in lattice units. Unlike for mass derivatives,∂µM ≡ M (1) 6= I, thus the terms produced involve traces of products over inverse anddifferentiated Dirac matrices, i.e.

∂Z

∂µ≡ ∂µZ =

Nf

4

∫DU (detM)Nf/4 tr

(M−1M (1)

)e−SG

≡ Nf

4〈tr M−1M (1)〉0.

(6.3.11)

42


Higher derivatives can be generated similarly, using ∂µM−1 = M−1M (1)M−1 and 1

Z〈O〉0 =

〈O〉, one finds for the first two coefficients

c1 =N2t

N3σ

1

Z∂µZ =

N2t

N3σ

Nf

4〈tr M−1M (1)〉

c2 =Nt

N3σ

((Nf

4

)2 〈[trM−1M (1)

]2〉 − 〈trM−1M (1)〉2

+Nf

4

〈trM−1M (2)〉 − 〈tr

[(M−1M (1)

)2]〉)

.

(6.3.12)

Measuring one flavor and two flavor coefficients then amounts to choosing Nf = 1 andNf = 2, respectively:

cuds100 = c1|Nf=1 cqs10 = c1|Nf=2

cuds200 = c2|Nf=1 cqs20 = c2|Nf=2

(6.3.13)

The expressions for the coefficients c1 to c4 can be found in appendix A, since the com-putation of the higher derivatives is straightforward, but involves many more terms.

For each gauge link configuration evaluated the simulation program outputs expres-sions of the form r

(n)i Pijr

(n)j , where the r(n) is a random vector enumerated by n =

1, 2, ..., N . Here P is either the inverse quark matrix (for the chiral condensate/susceptibility),or a product of inverse matrices and derivatives with respect to µ (for the coefficients).The traces can then be evaluated by taking the random vector product over all N ran-dom vectors r(n) as in equation (6.1.6). In this thesis, N = 10 random vectors are usedfor the chiral observables and N = 50 for the computation of the coefficients.

43

7. Results at imaginary chemicalpotential

7.1. Outline / simulation details

In this work simulations are performed with 2 + 1 dynamical quarks at finite imaginarychemical potential µ = iµI . In [23] one flavor chiral condensate and Polyakov loop sus-ceptibilities were considered as functions of aµI ≡ µI and used for a rough estimate onthe curvature of a parametrization of the chiral critical line, which is of second order inµ/T . This is extended by using chiral susceptibilities as an indicator of a chiral transi-tion. MEoS scaling ansatze for both the O(2) and O(4) universality classes are fitted tothe measured chiral condensate data, using a parameterization of the critical line. Fur-thermore, the one and two flavor quark number densities nu and nq, respectively, andquark susceptibilities χu and χq, respectively, are computed at finite values of imaginarychemical potential. The results are compared to Taylor expansions around µ/T = 0,which are analytically continued to purely imaginary chemical potential. Similar to thescaling fits to the chiral condensate, also for the one flavor quark density and quarksusceptibility proper fit ansatze are computed and used to recompute the curvature pa-rameter of the chiral critical line.

β 3.320 3.335 3.351T [MeV ] 204.8 209.6 218.2

Table 7.1.: Translation betweencoupling and tempera-ture

Dynamical quarks used in this work have massesmu = md ≡ ml, chosen such that the Goldstonepion mass is ∼ 220MeV , while the kaon takes onits physical mass. The strange quark mass is set tobe ms = 10ml. Gauge configurations are producedusing the RHMC algorithm with a p4fat3 staggeredfermion action and a tree level improved Symanzikgauge action, see chapter 2.5. Subsequent configu-

rations are separated by 10 molecular dynamics trajectories of length 0.5, each carriedout with a stepsize of ε = 8.33333 · 10−2. The lattice used has 163× 4 sites. The criticaltemperature at µ = 0 is determined to be Tc = 202MeV , see [25] for further detailsand the method used for setting the scale. Simulations are performed at three differ-ent values of the coupling β = 6

g2 , see Table 7.1 for the corresponding temperatures.For these couplings Taylor coefficients, computed around zero chemical potential, are athand, which are compared to the coefficients computed directly at imaginary chemicalpotential below. Configurations are computed for several values of imaginary chemicalpotential µI at each β, Tables C.1-C.3 provide also an overview of the numbers of con-

45

7. Results at imaginary chemical potential

figurations used for evaluation. Most of these have been computed earlier and used in[23]. In order to achive better accuracy, three values of µI for each β were added in thevicinity of the corresponding Polyakov loop susceptibility peak shown in that paper. Forthese, statistics of ∼ 1000 to ∼ 2000 configurations could be taken.

7.2. Chiral condensate and susceptibility

The Polyakov loop and the one flavor chiral condensate were computed at three temper-atures for imaginary chemical potentials as shown in Tables C.1 and C.2, respectively.The results are shown in Figures 7.1, the data for the chiral condensate is given in

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.05 0.1 0.15 0.2 0.25

aµI

T = 205 MeV

L

ψ-

ψ

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.05 0.1 0.15 0.2 0.25 0.3

aµI

T = 210 MeV

L

ψ-

ψ

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.05 0.1 0.15 0.2 0.25 0.3

aµI

T = 218 MeV

L

ψ-

ψ

Figure 7.1.: Polyakov loop and chiral condensate as functions of imaginary chemicalpotential.

Table C.4. For all three temperatures considered, Polyakov loop and chiral condensatebehave very similar. While for the lowest temperature the rise of the chiral condensateat increasing µI is almost linear, for β = 3.351 it remains near its initial value for a widerange of µI , before rapidly rising.

The chiral susceptibility is plotted in Figures 7.2, for all three couplings and valuesof the chemical potential shown in table C.2. The position of the peak indicates the

46


pseudocritical value of µI . For β = 3.335, a peak is visible around µI = 0.15, besidesthe data point at µI = 0.125, which deviates from the otherwise very pronounced peakstructure. At β = 3.351 the peaking data point is located at µI = 0.22. For the lowest

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.05 0.1 0.15 0.2 0.25

aµI

T = 205 MeV

χl

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.05 0.1 0.15 0.2 0.25 0.3

aµI

T = 210 MeV

χl

0

1

2

3

4

5

0 0.05 0.1 0.15 0.2 0.25 0.3

aµI

T = 218 MeV

χl

Figure 7.2.: One flavor chiral susceptibility, peaks indicate a critical value of µI .

coupling, however, no clear peak structure is visible. In fact, the plateau for low valuesof the chemical potential reflects no sign of a transition. The discussion of the scaling fitsfor the quark number observables will provide further evidence for this. Nevertheless,the peak positions for the other two couplings suffice to provide a first estimate on thecurvature of the chiral critical line parameterized by t = const., with

t =T − TcTc

+∑f

κf

(µfT

)2

=T − TcTc

+ 3 · 16κµ2 =T − TcTc

− 3 · 16κµ2I . (7.2.1)

Here, Tc is the critical temperature at µ = 0 and in the first step the curvature κis taken to be degenerate for all three flavors of quarks. A fit in µ2

I leaves κ as theonly parameter, the result is κ = 0.0346(2). The error, yielded by the fit routine, isunderestimated. This has to be compared with the result obtained in [26]. Becausethere a two flavor curvature κq = 0.059(2)(4) is measured, one should compare witha value of 2κ = 2 · 0.0346(2) = 0.0692(4). There are, however, technical differencesbetween these two determinations of the curvature.

47


• First, Taylor expansions around zero chemical potential are done, as opposed tosimulations at finite values of imaginary chemical potential, which are done in thisthesis. In particular, the Taylor expansion is performed in two degenerate lightquark flavors, as opposed to a finite strange quark chemical potential, degeneratewith the light quark chemical potential, being incorporated in the simulation. Thusthere is no curvature for the strange quark at all in that publication, while in thisthesis, all three curvatures are taken to be degenerate.

• Second, the results in this thesis are not extrapolated to the chiral limit, as opposedto the two flavor result from [26].

The errors of all data in this work are obtained by blocking the data to an extentwhere it can be considered to be uncorrelated, and then computed via the jacknifealgorithm, as explained in Chapter 6.2. In general, proper block sizes are chosen by eithercomputing autocorrelation times, or by using the method of data blocking. For the chiralcondensate data, an example of an autocorrelation function is given in Figure 7.3. Theautocorrelation time is computed from the slope of the logarithmic autocorrelation atsmall times. At larger times, the autocorrelation function in general becomes unreliabledue to its own errors. For the data used, autocorrelation times of ∼ 10− 50 trajectoriesare typically found, and at least a value of 20 is used as the proper block size to assureuncorrelated data.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

2 4 6 8 10 12 14 16 18

computer time

autocorrelation

0.001

0.01

0.1

1

2 4 6 8 10 12 14 16 18

computer time

autocorrelation

Figure 7.3.: Left: The autocorrelation function for β = 3.351 and µI = 0.125. Right:The same with logarithmic y axis.

The chiral condensate in QCD can be treated analogously to the magnetization of aspin system. There the average magnetization can be expressed as

M = − ∂f∂H

= h1/δfG(z), fG(z) = −(

1 +1

δ

)ff (z) +

z

δβf ′f (z), (7.2.2)

with H = h0h being the external field and ff being a universal function related to thesingular part of the free energy as

fs = h0h1+1/δff (z), (7.2.3)

48


see [27]. This is the magnetic form of the Widom hypothesis (3.5.14). Thus, knowing theuniversal function ff (z) (or equivalently fG(z)) of the system’s universality class, onecan describe the magnetization using the scaling variable z = t/h1/(βδ). In the contextof QCD, the chiral condensate can be considered as a magnetization, with the lightquark mass in lattice units ml acting as the external field. H is for all temperatureschosen to be H = ml/ms = 1/10 to get rid of multiplicative renormalizations. Hence,the magnetic scaling variable h is a constant. The temperature-like scaling variable t isthe parameterization of the critical line given by equation (7.2.1). When defining thenon-subtracted order parameter, as done in [13], as

Mb ≡M = N4t ms〈ψψ〉u, (7.2.4)

the chiral condensate can be parameterized as

〈ψψ〉u =h1/δ

N4t ms

fG(z). (7.2.5)

O(2) (O4)Tc [MeV ] 194.6 194.3

t0 0.0041 0.0049h0 0.0028 0.0020

Table 7.2.: Scaling parametersfor both universal-ity classes.

Since for the system under consideration the systemspecific scale factors t0, h0 and the critical tempera-tures in the chiral limit Tc have been determined forthe action used and the used quark mass ratio, the onlyunspecified parameter left is the curvature of the crit-ical line, κ. (see Table 7.2). Furthermore, the massratio of H = 0.1 seems to be in the scaling regime [13].Using parameterizations of the scaling functions (seeAppendix B) for both the O(2) and O(4) universalityclasses, above ansatze are fitted to the chiral condensatedata. Results are given in table 7.3 and the fits are plotted in Figures 7.4. The purpledata points are not included in the fitting process, so the fits extrapolate nicely back toµI = 0. The values of κ for O(2) seem to be slightly larger than the ones for O(4), atleast for the lowest temperature. Comparing 2 · κ to the value κq = 0.059(2)(4) from[26], one sees that the obtained results are all smaller by amounts of ∆κ ≈ 0.010−0.015,

β = 3.320 β = 3.335 β = 3.351O(2) κ 0.0255(9) 0.0307(6) 0.0246(29)

χ2

d.o.f.19.00 11.08 130.35

O(4) κ 0.0219(9) 0.0282(5) 0.0247(20)χ2

d.o.f.19.38 11.70 78.14

Table 7.3.: Results of the fits to the chiral condensate for both universality classes.

except for β = 3.335. The χ2/d.o.f. are rather large at all three temperatures, in partowing to the small errors of the data points. However, for β = 3.351, one sees bothfrom the plot and the χ2 that the scaling approach becomes unreliable there. Trying toimprove the difference between data points and fit function, adding an ansatz for the

49


0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0 0.05 0.1 0.15 0.2 0.25

aµI

T = 205 MeV

ψ-

ψO(4) fitO(2) fit

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.05 0.1 0.15 0.2 0.25

aµI

T = 210 MeV

ψ-

ψO(4) fitO(2) fit

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.05 0.1 0.15 0.2 0.25 0.3

aµI

T = 218 MeV

ψ-

ψO(4) fitO(2) fit

Figure 7.4.: Scaling fits to the chiral condensate data.

regular part of the free energy density might suffice. First attempts to do this did notsucceed in improving the fit, so this is work in progress. There are qualitative features,regarding the scaling, to be discussed. The range of z values taken for the obtained κvalues is plotted in Figure 7.5. The horizontal lines mark the pseudocritical value zp,which is just the peak position of the scaling function susceptibility

χM(t, h) =∂M

∂H=h−1+1/δ

h0

fχ(z)

⇒ fχ(z) =1

δ

(fG(z)− z

β

∂fG(z)

∂z

),

(7.2.6)

and z = 0, respectively. The peak position of the chiral susceptibility should correspondto a pseudocritical value zchiral peak ≈ zp. For the lowest temperature, one sees from theplot that the pseudocritical value is not reached at all, i.e. z < zp ∀ µI . This explains theplateau structure observed for the susceptibility at the lowest temperature, as opposedto a clear peak. For the other two temperatures this provides a check, and approximatelythe peak positions coincide with the pseudocritical values zp, although these show to beat slightly smaller µI for β = 3.335, and at slightly larger µI for β = 3.351, than thepeak positions in the data. The data for the medium temperature shows an inflectionpoint around µI ≈ 0.17, which seems not to be reflected by the magnetization ansatz.

50


-2

-1

0

1

2

3

4

5

0 0.05 0.1 0.15 0.2 0.25 0.3

z

aµI

O(2)

β = 3.320β = 3.335β = 3.351

zp = 1.56(10)z = 0

-1

0

1

2

3

4

0 0.05 0.1 0.15 0.2 0.25 0.3

z

aµI

O(4)

β = 3.320β = 3.335β = 3.351

zp = 1.33(5)z = 0

Figure 7.5.: Values taken by the variable z as a function of µI , for values of κ as obtainedfrom the fits to the chiral condensate.

Considering the form of fG(z), shown in Figure B.1, the inflection point is located atz = 0, which is reached at µI ≈ 0.23− 0.24 in Figure 7.5. First, this inflection point isnot very distinct, i.e. above z = −1 it will be hardly visible in the scaling ansatz, andsince for µI ≈ 0.3, which is already beyond the Roberge-Weiss transition line, it takesa value z ≈ −1, one cannot expect the ansatz to reflect this inflection point. Second,the inflection point in the data might be due to the proximity of the Roberge-Weisstransition line, i.e. an effect due to the periodicity of the grand canonical potential inthe T − µI plane beyond this line. This is speculative and needs further investigation.

For the two peaks in Figures 7.2, one can do it the other way around and put theirlocation, together with the values of zp for O(2) and O(4), respectively, into expression(7.2.1) and solve for κ,

κ =1

48µ2I,peak

(T

Tc− t0h1/(βδ)zp − 1

). (7.2.7)

κβ O(2) O(4)

3.335 0.0203(30) 0.0214(32)3.351 0.0267(23) 0.0272(27)

Table 7.4.: Curvaturescomputed viaz directly.

The result, for both the O(2) and O(4) universality classesgiven in Table 7.4, shows that from this viewpoint the crit-ical line, when making an ansatz as in (7.2.1), dependsslightly on temperature, at least for O(2). For O(4) the val-ues match within errors. These are obtained by assuming∆µI = 0.01 for the peak postitions. This could, however,mean that one cannot naively neglect higher order contri-butions to the parameterization of the critical line, sincea finite chemical potential does not break chiral symmetryexplicitly, but indirectly via its appearance in the variable t.

51


7.3. Quark density and susceptibility

The Quark number density and susceptibility (and also the third and fourth diagonalcoefficient) are computed for all three temperatures and chemical potentials listed inTable C.3. The results are given in Tables C.6-C.8. It should be mentioned that thereare contributions to the third and fourth coefficient which fluctuate severely when eval-uated with only 50 random vectors. Hence, these have very large errors. See e.g. [28] forthe influence of the number of random vectors on the stability of the coefficients. Theirerrors are obtained using the method of data blocking. As opposed to the computationof errors for chiral condensate data, measuring autocorrelation times turns out to be

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 20 40 60 80 100 120

blocksize

error

Figure 7.6.: The error of cqs20 for β = 3.335 and µI = 0.175 as a function of the block size.

unreliable in this case. Good results are obtained by computing the error using suc-cessively increasing block sizes, and plotting the obtained error against the block size.An expample is shown in Figure 7.6. It is clearly visible that the error stops increasingat some value of the block size, which in this particular example would be taken to be∼ 40 − 50. At this point one expects the blocked data to be uncorrelated and thus itcan be used as input for a jacknife algorithm.

With the help of equation (6.3.12) the two flavor density and susceptibility are ex-panded around zero chemical potential. Because n/T 3 = c1 and χ/T 2 = 2c2 also forimaginary chemical potential, the density is purely imaginary and the susceptibility ispurely real. Since all coefficients are available up to sixth order, i.e. including i+ j = 6,the complete expansion is, for imaginary µ, given by

nq(µ)

T 3=4

(2cqs20 + cqs11

)µI − 64

(4cqs40 + 3cqs31 + 2cqs22 + cqs13

)µ3I

+ 1024

(6cqs60 + 5cqs51 + 4cqs40 + 3cqs33 + 2cqs24 + cqs15

)µ5I .

(7.3.1)

52


Similarly, the quark susceptibility expansion reads

χq(µ)

T 2=2cqs20 − 16

(12cqs40 + 6cqs31 + 2cqs22

)µ2I

+ 256

(30cqs60 + 20cqs51 + 12cqs42 + 6cqs33 + 2cqs24

)µ4I .

(7.3.2)

As mentioned in Chapter 5, all odd coefficients vanish when evaluated at zero chemicalpotential. These two expansions are compared in Figures 7.7 and 7.8 to the density andsusceptibility computed at imaginary chemical potential, given by

nq(µ)

T 3= cqs10(µ),

χq(µ)

T 2= 2cqs20(µ). (7.3.3)

The errorbands are obtained by taking the errors of the expansion coefficients into ac-count. Since the expansions are linear in the coefficients, their errors were added and

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.05 0.1 0.15 0.2

aµI

T = 205 MeV

nq(µ)

6th

order4

th order

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3

aµI

T = 210 MeV

nq(µ)

6th

order4

th order

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.05 0.1 0.15 0.2 0.25 0.3

aµI

T = 218 MeV

nq(µ)

6th

order4

th order

Figure 7.7.: Taylor expansion of the (imaginary part of the) quark density compared tothe density measured at finite µ.

subtracted in order to obtain the maximum and minimum boundary curve, respectively.These errors are thus overestimated. For all temperatures the expansion works reason-ably well up to almost µI ≈ 0.2. For the two lowest temperatures can be clearly seen

53


-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.05 0.1 0.15 0.2

aµI

T = 205 MeV

χq(µ)

6th

order4

th order

-6

-4

-2

0

2

0 0.05 0.1 0.15 0.2 0.25 0.3

aµI

T = 210 MeV

χq(µ)

6th

order4

th order

-5

-4

-3

-2

-1

0

1

2

3

0 0.05 0.1 0.15 0.2 0.25 0.3

aµI

T = 218 MeV

χq(µ)

6th

order4

th order

Figure 7.8.: Taylor expansion of the (real part of the) quark susceptibility compared tothe susceptibility measured at finite µ.

that the fourth order expansion is too high in its value and properly corrected by thesixth order part. This is not the case for the result obtained at the highest temperature,because the abrupt decrease of the measured density and susceptibility is not at allcaught by the expansion. Since it is the highest temperature, one might argue that, forincreasing chemical potential, a region in the proximity of the Roberge-Weiss endpointis reached.

Similar to the scaling of the condensate, a proper scaling ansatz can be found todescribe the coefficient data measured at imaginary chemical potential. Assuming thesingular part of the free energy to yield the dominant contributions, one obtains for theone flavor coefficients by chain rule

c1 ≡ cuds100 =∂fs

∂µu/T=∂fs∂t

dt

d (µu/T ). (7.3.4)

Utilizing the scaling equation equation (7.2.3) and using the chain rule again to switchto the scaling variable z yields

∂fs∂t

= h0h1+1/δ dff (z)

dz

dz

dt. (7.3.5)

54


With z = t/h1/(δβ) and considering the most general form of the parameterization of thecritical line,

t =1

t0

(T − TcTc

+ κu

(µuT

)2

+ κd

(µdT

)2

+ κs

(µsT

)2), (7.3.6)

the result is

c1 = 2Hh1/δ−1/(βδ)κuµut0T

dff (z)

dz. (7.3.7)

Switching to imaginary chemical potential and, now that the derivatives are taken,setting µ ≡ µu = µd = µs and κ ≡ κu = κd = κs, one finds

=(c1) =8κµIt0T

Hh1/δ−1/(δβ)dff (z)

dz(7.3.8)

as a fitting ansatz for c1. Differentiating (7.3.7) again with respect to µu/T and multi-plying a factor of 1/2!, see (5.3.3), yields an ansatz for the coefficient c2 ≡ cuds200,

< (c2) = <(

∂c1

∂µu/T

)= h0h

1+1/δ−2/(βδ)κut0

[h1/(βδ)dff (z)

dz+

2κut0

(µuT

)2 d2ff (z)

dz2

]≡ h0h

1+1/δ−2/(βδ) κ

t0

[h1/(βδ)dff (z)

dz− 32κ

t0µ2I

d2ff (z)

dz2

].

(7.3.9)

Fits to the data for the coefficients c1 and c2 suggest that the regular part of the freeenergy density might not be negligible. Hence, a polynomial ansatz, as in [13], is usedto account for the regular part, i.e.

f(t, h) = fs(t, h) + At+B

⇒ =(c1)→ =(c1)− 2AκµI

⇒ < (c2)→ < (c2)− 2Aκ,

(7.3.10)

thus including A as a second parameter into the fit. The results for the constant Aand the curvature κ are shown in Table 7.5, while plots of the fits are shown in Figure7.9. For the two lowest temperatures, the fits work quite well. Again, the data pointsat µI = 0 are not taken into account in the fit. The errorbars for c2 become large atlarge chemical potential for β = 3.335, but nevertheless for both universality classesthe results of c1 and c2 are consistent. At the highest temperature, for small values ofchemical potential (µI ≤ 0.2), the fits still work well. However, for larger µI , as for theTaylor expansions, also the O(N) fit functions are not able to reproduce the course of thedata points, yielding χ2/d.o.f. ≈ 50. This might also be an indication for the presenceof a Roberge-Weiss endpoint, since it is governed not by an O(N) symmetry, but Z(3),thus being a possible reason for the scaling to fail in this regime. Excluding data pointswith µI > 0.2 from the fit, as opposed to fitting with all points included, does notchange the result for κ much. Due to the problems with the scaling mentioned above,

55


O(2) β = 3.320 β = 3.335 β = 3.351c1 κ 0.0213(7) 0.0208(6) 0.0126(10)

A -7.28(149) 18.12(146) 6.63(1076)χ2

d.o.f.3.35 4.11 7.27

c2 κ 0.0226(10) 0.0215(8) 0.0143(14)A -1.00(26) 2.47(26) 2.54(118)χ2

d.o.f.2.55 4.07 9.3

O(4) β = 3.320 β = 3.335 β = 3.351c1 κ 0.0387(13) 0.0355(14) 0.0181(16)

A -11.82(84) -10.02(112) -38.83(831)χ2

d.o.f.3.37 6.04 8.30

c2 κ 0.0418(21) 0.0355(13) 0.0208(24)A -1.52(15) -1.09(18) -3.57(95)χ2

d.o.f.2.85 4.89 11.10

Table 7.5.: Fit results for the curvature parameter and the constant of the regular partfor both universality classes.

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.05 0.1 0.15 0.2 0.25

aµI

T = 205 MeV

c1O(4) fitO(2) fit

c2O(4) fitO(2) fit

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 0.05 0.1 0.15 0.2 0.25 0.3

aµI

T = 210 MeV

c1O(4) fitO(2) fit

c2O(4) fitO(2) fit

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

0 0.05 0.1 0.15 0.2 0.25 0.3

aµI

T = 218 MeV

c1O(4) fitO(2) fit

c2O(4) fitO(2) fit

Figure 7.9.: Scaling fits to the coefficients evaluated at finite µ. For the last plot, pointswith µI > 0.2 are discarded in the fit.

56

7.4. Conclusion

the values of the fit parameters in Table 7.5 and the plots in Figure 7.9 are performedleaving out all µI > 0.2 for β = 3.351, thereby decreasing the value of χ2/d.o.f. to acomparable to the results of the two lower temperatures. The results for κ are consistentwithin each universality class, but are for O(4) roughly a factor of two larger than thecorresponding O(2) values. This is in contrast to the results of the fits to the chiralcondensate, where the O(2) values are slightly larger, although by much less than afactor of two. The obtained values for the second fit parameter, A, are not consistent,for they should coincide at fixed temperature and universality class. This might be fixedby using a different ansatz for the expansion of the regular part. One has to comparetwice the result for κ with κq = 0.059(2)(4) from [26]. The results for O(2) are too lowby roughly ∆κ ≈ 0.015− 0.02 , while the results for the O(4) universality class are toohigh by roughly ∆κ ≈ 0.01− 0.02.

7.4. Conclusion

The Polyakov loop, chiral condensate and chiral susceptibility are measured for a widerange of finite imaginary chemical potentials, almost up to the location of the Roberge-Weiss transition. An approximate attempt to determine the curvature of the chiralcritical line in the T − µI plane, using the peak positions of the chiral susceptibilities,results in a value which is higher than the value from [26], the study used for comparison.Fits to the chiral condensate data, done with ansatze using both the O(2) and O(4)universality class scaling functions, result in lower values. The occurrence of an inflectionpoint in the data is not properly visible in the scaling ansatz, which might be related tothe periodicity of the grand canonical potential in the T−µI plane. Further investigationis needed at that point. Furthermore, the quark number density and quark susceptibilityare measured for a comparable range of µI . The coefficients of a Taylor expansion aroundµ/T = 0 are at hand and thus utilized to compare the expansion, which is computed upto both fourth and sixth order, to the measured values of the two flavor quark densityand quark susceptibility. The expansion is conclusive at the two lowest temperatures.In contrast to this, the expansion at the highest temperature taken into account doesnot work at all. By measuring not only the diagonal coefficients c1 and c2, but alsomixed coefficients, the Taylor expansion could be performed around imaginary chemicalpotentials (see end of Chapter 5.3). This would yield a cross check on the expansionaround µ = 0 and might also help to clarify the breakdown of the expansion at the highesttemperature for large µI . Analogously to the scaling fits to the chiral condensate, fitansatze for the coefficient data are obtained and used to extract the curvature of thechiral critical line as a fit parameter. For these, regular parts are required to improvethe quality of the fit. The fits work well apart from the ones at highest temperature.Comparing all fit results for κ (i.e. fits to the chiral condensate and to the coefficients)with the value κq = 0.059(2)(4) from [26], one finds deviations of up to ∆κ ≈ 0.02 inboth directions. However, since in this work the obtained values are not extrapolated tothe chiral limit and the simulations are being done at finite values of the strange quarkchemical potential and with threefold degenerate curvatures, in contrast to the study

57


compared with, this work has to be understood as being exploratory and the results areclose enough to the above κq to leave room for further investigation. Another interestingaspect is that both the breakdown of the fits to the coefficients and the breakdown ofthe Taylor expansion at the highest temperature might indicate the proximity of theRoberge-Weiss endpoint. Hence, scaling behavior different from O(2) and (O4) wouldbe observable there.

58

A. Computation of coefficients

At finite imaginary chemical potential the coefficients cudsi00 and cqsi0 were measured fori = 1, ..., 4. For brevity, the partition function is taken to be

Z =

∫DU (detM)n e−SG , (A.0.1)

with n ≡ Nf/4, since the derivates only act on one determinant.Defining cn ≡ (n!N3

σ/N3−nt )cn, the derivatives of lnZ with respect to µq are

c1 =∂ lnZ

∂µq≡ ∂µ lnZ =

1

Z∂µZ

c2 = ∂2µ lnZ =

1

Z∂2µZ −

1

Z2(∂µZ)2

c3 =1

Z∂3µZ − 3

1

Z2∂2µZ∂µZ + 2

1

Z3(∂µZ)3

c4 =1

Z∂4µZ − 4

1

Z2∂3µZ∂µZ + 12

1

Z3∂2µZ (∂µZ)2

− 31

Z2

(∂2µZ)2 − 6

1

Z4(∂µZ)4 .

(A.0.2)

Using equation (6.3.4) for the derivative of the determinant, one arrives at

∂2µZ = n2〈

[tr M−1M (1)

]2〉0 + n〈tr M−1M (2)〉0

− n〈tr[(M−1M (1)

)2]〉0

∂3µZ = n3〈

[tr M−1M (1)

]3〉0+ 3n2〈tr M−1M (1)

tr M−1M (2) − tr

[(M−1M (1)

)2]〉0

+ n〈tr M−1M (3) − 3tr M−1M (2)M−1M (1) + 2tr[(M−1M (1)

)3]〉0

59

A. Computation of coefficients

∂4µZ = n4〈

[tr M−1M (1)

]4〉0+ 6n3〈

[tr M−1M (1)

]2 tr M−1M (2) − tr

[(M−1M (1)

)2]〉0

+ 4n2〈tr M−1M (1)

tr M−1M (3) − 3tr M−1M (2)M−1M (1)〉0

+ 8n2〈tr M−1M (1)tr[(M−1M (1)

)3]〉0

+ 3n2〈tr M−1M (2)

tr M−1M (2) − 2tr[(M−1M (1)

)2]〉0

+ 3n2〈[tr[(M−1M (1)

)2]]2

〉0 + n〈tr M−1M (4)〉0

− 4n〈tr M−1M (3)M−1M (1)〉0 + 12n〈tr M−1M (2)(M−1M (1)

)2〉0− 3n〈tr

[(M−1M (2)

)2]〉0.

With these derivatives of the partition function, the coefficients can be computed via(A.0.2). Using 1

Z〈O〉0 = 〈O〉, one obtains

c1 = n〈tr M−1M (1)〉

c2 = n2〈[tr M−1M (1)

]2〉 − 〈tr M−1M (1)〉2

+n〈tr M−1M (2)〉 − 〈tr

[(M−1M (1)

)2]〉

c3 = n3〈[tr M−1M (1)

]3〉+ 2〈tr M−1M (1)〉3 − 3〈tr M−1M (1)〉〈[tr M−1M (1)

]2〉+n2

3〈tr M−1M (1)tr M−1M (2)〉 − 3〈tr M−1M (1)tr

[(M−1M (1)

)2]〉

−3〈tr M−1M (2)〉〈tr M−1M (1)〉+ 3〈tr M−1M (1)〉〈tr[(M−1M (1)

)2]〉

+n〈tr M−1M (3)〉 − 3〈tr M−1M (2)M−1M (1)〉+ 2〈tr

[(M−1M (1)

)3]〉

60

c4 = n4〈[tr M−1M (1)

]4〉 − 6〈tr M−1M (1)〉4 − 4〈tr M−1M (1)〉〈[tr M−1M (1)

]3〉+12〈tr M−1M (1)〉2〈

[tr M−1M (1)

]2〉 − 3〈[tr M−1M (1)

]2〉2+n3

6〈[tr M−1M (1)

]2tr M−1M (2)〉 − 6〈

[tr M−1M (1)

]2tr[(M−1M (1)

)2]〉

+12〈tr M−1M (1)〉〈tr M−1M (1)tr[(M−1M (1)

)2]〉

−12〈tr M−1M (1)〉〈tr M−1M (1)tr M−1M (2)〉

+12〈tr M−1M (1)〉2(〈tr M−1M (2)〉 − 〈tr

[(M−1M (1)

)2]〉)

+6〈[tr M−1M (1)

]2〉(〈tr [(M−1M (1))2]〉 − 〈tr M−1M (2)〉

)+n2

4〈tr M−1M (1)tr M−1M (3)〉 − 12〈tr M−1M (1)tr M−1M (2)M−1M (1)〉

+8〈tr M−1M (1)tr[(M−1M (1)

)3]〉+ 3〈

[tr M−1M (2)

]2〉+3〈

[tr[(M−1M (1)

)2]]2

〉 − 6〈tr M−1M (2)tr[(M−1M (1)

)2]〉

−4〈tr M−1M (1)〉〈tr M−1M (3)〉+ 12〈tr M−1M (1)〉〈tr M−1M (2)M−1M (1)〉

−8〈tr M−1M (1)〉〈tr[(M−1M (1)

)3]〉 − 3〈tr M−1M (1)〉2

+6〈tr M−1M (2)〉〈tr[(M−1M (1)

)2]〉 − 3〈tr

[(M−1M (1)

)2]〉2

+n〈tr M−1M (4)〉 − 4〈tr M−1M (3)M−1M (1)〉 − 3〈tr

[(M−1M (2)

)2]〉

+12〈tr M−1M (2)(M−1M (1)

)2〉 − 6〈tr[(M−1M (1)

)4]〉.

61

B. Parameterization of scalingfunctions

The scaling functions fG(z) and ff (z) for O(2) and O(4), respectively, are taken fromdifferent sources. A parameterization for the O(4) scaling function is given in [27]. TherefG and ff are obtained as sixth order Taylor polynomials for −2 < z < 1.95, while theregions for large negative and large positive z are found by utilizing proper asymptoticexpansion ansatze. Hence, fG(z) is given by

fG(z) =

(−z)β∑n=0

d−n (−z)−n∆/2 z < −2∑n=0

b−n zn −2 ≤ z ≤ 0∑

n=0

b+n z

n 0 < z ≤ 1.95

z−γ∑n=0

d+n z−2n∆ z > 1.95.

(B.0.1)

The related function ff (z) is given by

ff (z) =

(−z)2−α∑n=0

c−n (−z)−n∆/2 z < −2∑n=0

a−n zn −2 ≤ z ≤ 0∑

n=0

a+n z

n 0 < z ≤ 1.95

z2−α∑n=0

c+n z−2n∆ z > 1.95,

(B.0.2)

where the coefficients are related by

a±n =∆b±n

α + n− 2, c+

n+1 =−d+

n

2(n− 1), c−n+2 = − 2d−n

n+ 2, (B.0.3)

with ∆ ≡ βδ. The values for all coefficients are given in Table B.1 and Table B.2,respectively. These forms are directly incorporated into the fit functions (7.2.5), (7.3.8)and (7.3.9).

Such a parameterization is unfortunately not at hand for the scaling function of theO(2) universality class. Instead, an implicit parameterization is taken from [13], which

63

B. Parameterization of scaling functions

cannot be directly used inside a fit ansatz. It is given by

y = f−δG , x = zf1/βG , (B.0.4)

with x and y being related by the function

x(y) = xs(y)yp0

yp0 + yp+ xl(y)

yp

yp0 + yp. (B.0.5)

This function interpolates between the two asymptotic functions

xs(y) = −1 + (c1 + d3)y + c2y1/2 + d2y

3/2, (B.0.6)

for small y, andxl(y) = au1/γ + by(1−2β)/γ, (B.0.7)

for large y. In order to obtain an analytic expression for fG(z), which can easily beincorporated into the fit functions like in the O(4) case, first fG(z) was obtained from

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

-3 -2 -1 0 1 2 3

z

fG

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

-0.5 0 0.5 1 1.5 2 2.5

z

reconstructed fGfG

Figure B.1.: Left: The function fG(z). Right: Comparison of the different parameteri-zations of fG. The one labeled ’reconstructed’ is obtained from the implicitdefinition (B.0.4).

the above relations numerically. Then, following the method shown in [27], the functions(B.0.1) were fitted to this data, with the coefficients b±n and d±n as fit parameters. Thus,the respective O(2) coefficients are shown in Table B.1. When computing the coefficientsfor the parameterization of ff , c

±0 are neither determined by (B.0.3), nor fixed by the

normalization conditions of the scaling function. They can be obtained via

c+0 =

∆

2− α

∫ ∞0

dssα−2

(∂fG∂z

∣∣∣∣s

− ∂fG∂z

∣∣∣∣0

− s ∂2fG

∂(z)2

∣∣∣∣0

), (B.0.8)

and

c+0 =

−∆

2− α

∫ 0

−∞ds (−s)α−2

(∂fG∂z

∣∣∣∣s

− ∂fG∂z

∣∣∣∣0

− s ∂2fG

∂(z)2

∣∣∣∣0

), (B.0.9)

64

see [27].

In fact parameterization (B.0.4) is given for both the O(2) and O(4) universalityclasses, see [13] for the values of the parameters p, y0, ci etc. Thus the above procedurecan be done for O(4) as well. The result can be compared to the parameterizationdirectly obtained from [27]. This is done and both thus obtained scaling functions fG(z)for O(4) are compared in Figure B.1. As expected, the asymptotical expressions for theimplicit parameterization (B.0.4) and the ones from (B.0.1) cannot be distinguished byeye, but in an interval from z ≈ 0.5 to z ≈ 1.5 the reconstructed function has a slightlylarger value. This means, since the O(4) curve from [27] is obtained by fits with hugestatistics, this region is probably less reliably parameterized in the O(2) case, comparedto the O(4) case.

fG(z) O(2) O(4)d−0 0.996921(42) 1(0)d−1 0.235929(249) 0.273651(2933)d−2 0.023149(286) 0.0036058(4875)b−0 1(0) 1(0)b−1 -0.27924(20) -0.3166125(534)b−2 -0.009617075(357113) -0.04112553(129000)b−3 0.0143736(2669) 0.00384019(66700)b−4 -0.0083097(4098) 0.007100450(160000)b−5 -0.00951141(25520) 0.0023729(950)b−6 -0.00200099(5385) 0.000272312(21000)b+

0 1(0) 1(0)b+

1 -0.27924(20) -0.3166125(534)b+

2 -0.009617075(357113) -0.04112553(129000)b+

3 0.00282602(30640) 0.00384019(66700)b+

4 0.0125069(3919) 0.006705475(1704000)b+

5 0.012628(435) 0.0047342(14290)b+

6 -0.00271348(14300) -0.001931267d+

0 1.34804(33) 1.10599(555)d+

1 -2.59298(778) 1.31829(10870)d+

2 4.65776(3313) 1.5884(4646)

Table B.1.: Coefficients of the parameterizations of fG(z).

65

B. Parameterization of scaling functions

ff (z) O(2) O(4)c−0 0.785594(34245) 0.229176(10669)c−1 0(0) 0(0)c−2 -0.996921(42) -1(0)c−3 -0.157286(166) -0.182434(1955)c−4 -0.011575(143) -0.001803(2438)a−0 -0.82699(0) -0.828304(0)a−1 0.457948(328) 0.478434(807)a−2 0.931672(34596) 0.353768(11097)a−3 0.0243985(4530) 0.0089459(15538)a−4 -0.00699137(34479) 0.00728411(16414)a−5 -0.00531958(14273) 0.00156081(6249)a−6 -0.000838131(22556) 0.000131818(10165)a+

0 -0.82699(0) -0.828304(0)a+

1 0.457948(328) 0.478434(807)a+

2 0.931672(34596) 0.353768(11097)a+

3 0.00479703(52010) 0.0089459(15538)a+

4 -0.0105227(3297) 0.00687892(174808)a+

5 0.00706263(24301) 0.00311398(93994)a+

6 -0.00113656(5989) -0.000934866(151029)c+

0 1.00947(3429) 0.42206(1595)c+

1 -0.67402(16) -0.552995(2775)c+

2 0.648245(1945) 0.329572(27175)c+

3 -0.776293(5522) -0.264733(77433)

Table B.2.: Coefficients of the parameterizations of ff (z).

66

C. Tables

β = 3.320 β = 3.335 β = 3.351µI # µI # µI #

0.03 9990 0.05 5010 0.05 193900.05 9970 0.10 4920 0.10 189900.06 11310 0.115 25860 0.125 131800.07 9970 0.125 26170 0.15 124600.08 10160 0.1375 18180 0.175 125000.09 8780 0.145 22660 0.20 131200.10 9980 0.15 26800 0.21 200300.15 0050 0.155 12620 0.22 137100.16 9960 0.1625 17870 0.225 30100.17 9950 0.175 22940 0.235 200800.20 9950 0.20 23900 0.25 29700.22 996 0.22 10430

0.24 101700.26 6530

Table C.1.: Simulation couplings and chemical potentials (with statistics) used forPolyakov loop

67

C. Tables

β = 3.320 β = 3.335 β = 3.351µI # µI # µI #

0.03 999 0.05 501 0.05 19390.05 997 0.10 492 0.10 18990.06 2257 0.115 2529 0.125 13180.07 997 0.125 2617 0.15 12460.08 1947 0.1375 1818 0.175 12500.09 1334 0.145 2209 0.20 13120.10 998 0.15 2680 0.21 9880.15 995 0.155 1260 0.22 12710.16 996 0.1625 1787 0.225 3010.17 995 0.175 2294 0.235 19140.20 995 0.20 23900.22 996 0.22 1043

0.24 10170.26 653

Table C.2.: Simulation couplings and chemical potentials (with statistics) used for chiralcondensate and susceptibility

β = 3.320 β = 3.335 β = 3.351µI # µI # µI #

0.06 2181 0.05 224 0.05 2740.08 1860 0.10 88 0.125 7990.09 1234 0.115 2436 0.15 7590.10 991 0.125 1575 0.175 2350.15 989 0.1375 1080 0.20 6570.16 983 0.145 2129 0.21 13250.17 988 0.15 1592 0.22 1274

0.155 1162 0.225 21810.1625 421 0.235 19280.175 1309 0.25 2100.20 13470.22 1350.24 1030.26 131

Table C.3.: Simulation couplings and chemical potentials (with statistics) used for mea-surement of coefficients

68

β = 3.320 β = 3.335 β = 3.351µI 〈ψψ〉u µI 〈ψψ〉u µI 〈ψψ〉u0.0 0.0896(11) 0.0 0.0519(12) 0.0 0.0361(4)0.03 0.084847(1348) 0.05 0.0549532(12301) 0.05 0.0346108(2764)0.05 0.0868673(11786) 0.10 0.0615773(13244) 0.10 0.0391008(4414)0.06 0.0942234(9707) 0.115 0.0690693(7245) 0.125 0.0372087(4604)0.07 0.090734(1276) 0.125 0.0745976(8610) 0.15 0.0424532(5735)0.08 0.0959998(8735) 0.1375 0.0771576(8151) 0.175 0.0494272(7675)0.09 0.0999309(13014) 0.145 0.0843078(9668) 0.20 0.060372(1013)0.10 0.103114(1431) 0.15 0.0835636(9488) 0.21 0.0695977(18867)0.15 0.125925(1375) 0.155 0.0867398(14147) 0.22 0.0827451(21733)0.16 0.130054(1012) 0.1625 0.0902291(13355) 0.225 0.0808124(28189)0.17 0.127418(838) 0.175 0.103869(801) 0.235 0.100757(1101)0.20 0.141274(752) 0.20 0.116498(689)0.22 0.144981(664) 0.22 0.125413(651)

0.24 0.132357(640)0.26 0.13335(77)

Table C.4.: Results for the chiral condensate

β = 3.320 β = 3.335 β = 3.351µI χl µI χl µI χl

0.03 3.10268(41098) 0.05 1.32013(18544) 0.05 0.565987(39181)0.05 2.37469(20418) 0.10 1.86403(32105) 0.10 0.965612(64539)0.06 2.97136(24723) 0.115 2.19111(18809) 0.125 0.73863(5577)0.07 2.8775(2761) 0.125 3.02533(24208) 0.15 1.1126(776)0.08 2.40616(18009) 0.1375 2.58925(18965) 0.175 1.46798(14140)0.09 2.84145(310431) 0.145 3.00756(18705) 0.20 2.24153(21116)0.10 2.91075(27934) 0.15 3.42169(29362) 0.21 2.83574(34446)0.15 2.27884(37508) 0.155 2.98545(30166) 0.22 3.87653(59821)0.16 1.82844(15444) 0.1625 2.71215(32394) 0.225 3.0218(4668)0.17 1.58171(21184) 0.175 2.44542(17684) 0.235 2.32843(26605)0.20 1.4744(1557) 0.20 2.01757(13288)0.22 1.21834(19058) 0.22 1.20074(12621)

0.24 1.03759(10550)0.26 0.877179(76338)

Table C.5.: Results for the chiral susceptibility

69

C. Tables

β = 3.320µI =(c1) < (c2) =(c3) < (c4)

0.06 0.1637(31) 0.3412(78) 0.1165(166) 0.1321(393)0.08 0.2129(27) 0.3320(61) 0.1094(139) 0.0528(405)0.09 0.2336(49) 0.2805(141) 0.1949(352) 0.3051(1052)0.10 0.2435(78) 0.2528(201) 0.1528(389) 0.0539(616)0.15 0.2467(97) 0.0861(199) 0.1109(378) -0.0603(1113)0.16 0.2202(113) 0.0320(424) 0.1490(850) -0.0802(2284)0.17 0.2524(84) 0.0451(257) 0.1514(439) 0.0481(1330)

Table C.6.: Results for the coefficients at β = 3.320.

β = 3.335µI =(c1) < (c2) =(c3) < (c4)

0.05 0.1781(29) 0.4531(38) 0.05161(149) 0.0702(121)0.10 0.3486(14) 0.4201(158) 0.01347(81) -0.0036(8)0.115 0.3670(37) 0.3495(105) 0.1651(248) 0.1483(438)0.125 0.38168(58) 0.3024(171) 0.2506(536) 0.3057(1559)0.1375 0.4103(69) 0.2879(133) 0.1751(319) 0.1249(564)0.145 0.3929(069) 0.2049(120) 0.2731(317) 0.0190(879)0.15 0.3977(85) 0.1895(266) 0.3186(780) 0.3292(2015)0.155 0.3879(147) 0.1089(464) 0.4605(1221) 0.1487(2845)0.1625 0.3751(170) 0.1344(494) 0.3882(1074) 0.1764(1546)0.175 0.3453(105) 0.0361(275) 0.1409(674) -0.0959(1917)0.20 0.3061(136) -0.0765(444) 0.3519(1054) -0.3633(2863)0.22 0.3353(506) -0.2068(2003) -0.6521(7477) 1.5533(20307)0.24 0.1998(640) -0.5586(4749) -3.3995(28130) 9.5456(100777)0.26 0.1055(404) -0.1892(385) 0.2179(3425) -1.2792(8385)


β = 3.351µI =(c1) < (c2) =(c3) < (c4)

0.05 0.1968(15) 0.4998(77) 0.0361(64) 0.0400(301)0.125 0.4705(21) 0.4471(45) 0.0826(100) 0.0657(216)0.15 0.5405(34) 0.4068(62) 0.1182(134) 0.0542(207)0.175 0.5931(68) 0.3470(102) 0.1594(188) 0.0402(458)0.20 0.6060(114) 0.1975(233) 0.2930(492) 0.0575(1218)0.21 0.5591(149) 0.0132(583) 0.6171(2044) 0.5060(4717)0.22 0.5101(194) -0.2066(820) 0.9529(3921) 0.7227(16297)0.225 0.5222(471) -0.2556(1865) 0.4775(5595) -2.7123(21261)0.235 0.3511(176) -0.3667(756) 0.5900(2366) -0.8480(12852)0.25 0.2772(1053) -1.2134(9147) -4.1816(406985) -11.0130(247680)


70

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72

Acknowledgements

I would like to kindly say ’Thank you, ...’

• ... Edwin Laermann, for giving me the opportunity to write this thesis and a lotof illuminative discussions, besides all the help I got when I needed it.

• ... Maria-Paola Lombardo, for the great lecture on finite density and her ideasabout our T − µ plane.

• ... Christian Schmidt, for providing the coefficients at zero chemical potential andenduring my questions.

• ... Nirupam, for telling me about the violins.

• ... Sama and Ioan, for long office hours.

Furthermore I would like to thank my family, who always encourage me and never letme down. Last, but not least, I want to thank Gudrun for her patience. Te iubesc,luceafarul meu. Look at the page number...

73

Disclaimer

I hereby declare that the work done in this thesis is that of the author alone with thehelp of no more than the mentioned literature and auxiliary means.

Bielefeld, 19.07.2012

Florian Meyer

75

2+1 flavor qcd at imaginary chemical potential1 flavor qcd at imaginary chemical potential...

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