Shear Viscosities of Interacting Bose Gases - TUM · 2010. 11. 10. · manifold Rd 3 7!g( ) 2Gis...

Shear Viscosities ofInteracting Bose Gases

Diploma Thesisby

Robert Lang

October, 2010

Technische Universitat Munchen

Physik-Department

T39 (Prof. Dr. Wolfram Weise)

Contents

1 Introduction 5

2 Quantum Chromodynamics 72.1 QCD in the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Yang-Mills Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Symmetries of QCD . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Chiral Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 QCD Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Hydrodynamics of Relativistic Heavy-Ion Collisions 233.1 First Principles of Relativistic Hydrodynamics . . . . . . . . . . . . . . . . 233.2 Relativistic Hydrodynamics of Perfect and Dissipative Fluids . . . . . . . . 263.3 1 + 1 Dimensional Example . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4 Connections to Experimental Results from RHIC . . . . . . . . . . . . . . 33

4 Non-Equilibrium Thermodynamics and Transport Phenomena 354.1 Statistical Operator of Non-Equilibrium Systems . . . . . . . . . . . . . . . 354.2 Kubo-Type Formulas for Transport Coefficients . . . . . . . . . . . . . . . 37

5 Bosonic Thermal Field Theory and Imaginary Time Formalism 435.1 Path Integral Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2 Statistical Physics and Matsubara Formalism . . . . . . . . . . . . . . . . 465.3 Neutral Scalar Field at Finite Temperature . . . . . . . . . . . . . . . . . . 48

5.3.1 Free Neutral Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . 485.3.2 Neutral Fields with φ4 Interaction . . . . . . . . . . . . . . . . . . . 515.3.3 Corrections to lnZ in Thermal φ4 Theory . . . . . . . . . . . . . . 52

5.4 Spectral Representation of Green’s Functions . . . . . . . . . . . . . . . . . 545.5 Discussion of Thermal Quantum Field Theories . . . . . . . . . . . . . . . 55

6 Shear Viscosity in φ4 Theory 576.1 Skeleton Expansion and Matsubara Propagator . . . . . . . . . . . . . . . 586.2 First-Order Correction to the Propagator . . . . . . . . . . . . . . . . . . . 626.3 Second-Order Corrections to the Propagator . . . . . . . . . . . . . . . . . 636.4 Third-Order Corrections to the Propagator . . . . . . . . . . . . . . . . . . 686.5 Numerical Evaluation of the Shear Viscosity . . . . . . . . . . . . . . . . . 69

7 Shear Viscosity of a Pion Gas 757.1 Skeleton Expansion in Chiral Perturbation Theory . . . . . . . . . . . . . . 757.2 First-Order Correction to the Propagator . . . . . . . . . . . . . . . . . . . 777.3 Second-Order Correction to the Propagator . . . . . . . . . . . . . . . . . 787.4 Numerical Evaluation of the Shear Viscosity . . . . . . . . . . . . . . . . . 80

3

Contents

8 Results in Comparison 878.1 Short Digression: AdS/CFT Correspondence . . . . . . . . . . . . . . . . . 878.2 η/s in Chiral Perturbation Theory and φ4 Theory . . . . . . . . . . . . . . 89

9 Summary and Outlook 939.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 939.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

A Appendix 95A.1 Conventions and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . 95A.2 Master Formulas for Bosonic Matsubara Sums . . . . . . . . . . . . . . . . 96A.3 Matsubara Propagator in Spectral Representation . . . . . . . . . . . . . . 97A.4 Curie’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98A.5 Feynman Rules for L2 in χPT . . . . . . . . . . . . . . . . . . . . . . . . . 99A.6 Dimensional Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4

1 Introduction

On 23rd November 2009 the first measurement with the ALICE detector at the CERNLarge Hadron Collider (LHC) took place. At a center-of-mass energy

√s = 900 GeV the

pseudorapidity density dN/dηPS of charged particles after a proton-proton collision wasdetermined and already five days later published [Col10]. It is planned to explore suchcollisions at

√s = 14 TeV. In addition, LHC will provide up to 28 times more power-

ful heavy-ion collisions compared to the preceding experiments at the BNL RelativisticHeavy-Ion Collider (RHIC) during the last ten years. One of the main experimental re-sults of RHIC is the finding that in such collisions a quark-gluon phase is created in theform of an ideal fluid [Hei05]. The heavy-ion-collision program at the LHC is expected toconfirm and extend the results of RHIC to even higher temperatures and energy densities.

In this thesis we first start with a phenomenological approach to the hydrodynamicsof relativistic heavy-ion collisions and discuss how the dissipative parameters, namely theshear and bulk viscosity, determine energy and entropy production. Using the Bjorkenspacetime picture we relate the free parameters to experimental results from RHIC. Inorder to get a deeper theoretical insight how these macroscopic parameters emerge fromquantum field theory we use non-equilibrium statistical mechanics and derive the Kubo-type formula for the shear viscosity η in detail. Explicit expressions will be calculated forthe shear viscosity of particles with φ4 interaction and for an interacting pion gas withinthe framework of chiral perturbation theory (χPT). The first mentioned calculation servesas a model introducing and explaining all relevant techniques needed to determine theshear viscosity for any given quantum field theory.

The results from RHIC also indicate that the η/s ratio of the system, where s denotesits entropy density, is minimal at the phase transition between confined (hadronic) anddeconfined (quark-gluon) matter. In 1998, Maldacena proved a version of the AdS/CFTcorrespondence between string theory and quantum field theory [Mal99]. Using this cor-respondence, it is possible to derive under certain conditions a lower limit, η/s ≥ 1/4π,which is compatible with experimental results so far. It is not clear if this limit does alsohold for non-superconformal theories such as QCD, but in [KSS05], it is conjectured thatit remains valid for all relativistic quantum field theories. In this thesis we compare theratio obtained in the two considered theories here to the AdS limit.

The structure of this thesis is as follows: in chapter 2 we give a brief introduction toQCD starting with the general Yang-Mills theory and the quantization of non-Abeliangauge theories. The symmetries of QCD in different limits are considered and the group-theoretical aspects of chiral perturbation theory (χPT) are discussed in detail. Further-more, we present the QCD phase diagram, the different phases and the transitions betweenthem. In chapter 3 we discuss a phenomenological approach to relativistic heavy-ion col-lisions and to the hydrodynamics of perfect and dissipative fluids. There, we introducethe dissipative tensor, parameterized by the shear and bulk viscosity. These are relatedto the energy and entropy production in a heavy-ion collision. Chapter 4 covers thederivation of the Kubo-type formula for the shear viscosity. The key ingredients for thisissue are non-equilibrium statistical physics, linear-response theory and Curie’s theorem.

5

1 Introduction

The formalism of thermal field theory for bosons is prepared in chapter 5. We usethe Matsubara formalism which relates temperature to imaginary time and apply thismethod to a neutral scalar field at finite temperature. In addition, we discuss the conceptof temperature within a relativistic quantum field theory. In chapter 6 scalar fields withφ4 interaction are introduced as a toy model in order to explore the techniques neededto calculate the shear viscosity of a more realistic pion gas in the subsequent chapter.We introduce the skeleton expansion of correlators and derive a criterion for whether adiagram in the perturbative expansion of the Matsubara propagator contributes to theshear viscosity or not. In the end, we give both numerical and, under some approxima-tions, analytical results for the temperature dependence of the shear viscosity. For thecalculation of the shear viscosity of a pion gas in chapter 7 we use again the skeletonexpansion and the criterion derived in the previous chapter. Finally, the numerical resultsfor the temperature and mass dependence of the shear viscosity are shown. Chapter 8briefly summarizes the most important terms concerning the AdS/CFT correspondence.We determine the η/s ratio using the results of the two preceding chapters and comparewith the AdS limit that is conjectured to be valid for all relativistic quantum field theories.In chapter 9 we summarize our most important results and discuss the approximationsand restrictions we have made in this thesis. The appendix displays conventions andnotations and some derivations of technical details.

6

2 Quantum Chromodynamics

In this chapter we give a brief survey of quantum chromodynamics (QCD) regarding itsgeneral geometric and group-theoretical background as one special Yang-Mills theory. Inaddition we investigate its classical and quantum mechanical symmetries which are keyingredients for the construction of chiral perturbation theory (χPT), an effective fieldtheory approach that represents QCD in its low-energy region. Furthermore we describethe most important features of the QCD phase diagram and define our region of interest.In this chapter we refer mostly to [CL06, Ryd05, PS95] and [Sch02, FH91].

2.1 QCD in the Standard Model

2.1.1 Yang-Mills Theory

Consider a symmetry group G which is assumed to be a d-dimensional matrix Lie groupSU(N), SO(N) or Sp(N). We find d(SU(N)) = N2 − 1, d(SO(N)) = 1

2N(N − 1) and

d(Sp(N)) = 12N(N + 1). Our theory, for instance QCD, is descibed by a Lagrangian L[ψ]

depending on a fermionic field ψ : M → S, which is defined on Minkowski space M andmaps to spinor space S. The action, S =

∫d4x L, remains unchanged under the global

transformation g ∈ G:ψ(x) 7→ gψ(x) = eiα

aTaψ(x) . (2.1)

We have introduced the generators Ta of the Lie group G:

Ta ≡ −i∂g(α1, . . . , αd)

∂αa

∣∣∣∣α=0

. (2.2)

These generators can be interpreted as a basis of the corresponding Lie algebra G: theTa span the tangent space of id∈G, because the parameterization of the d-dimensionalmanifold Rd 3 α 7→ g(α) ∈ G is assumed to be chosen in such a way that g(0) = id. Aslong as the Lagrangian is independent of derivatives ∂µψ(x) it remains unchanged alsounder local transformations

ψ(x) 7→ g(x)ψ(x) = eiαa(x)Taψ(x) , (2.3)

with spacetime-dependent functions αa : M → R. The derivative ∂µψ(x) involves space-time points which are infinitesimally separated and which transform differently underthe local transformation. Therefore, the derivative term has a highly non-trivial behav-ior under local transformations of the field ψ(x). Considering a unitary matrix fieldU(x, y) : M2 → CN×N , with U(x, x) = 1 and the transformation property

U(x, y) 7→ g(y)U(y, x)g(x) , (2.4)

we can expand U(x, y) for x ≈ y = x−εn, where ε > 0 is small and n ∈M is a normalizedspacetime vector with n · n = −1:

U(x+ εn, x) = 1 + igεnµAaµ(x)Ta +O(ε2) . (2.5)

7


Here we have introduced d vector fields Aaµ(x) with some convenient prefactor g in orderto recover the Abelian case QED within this formalism. Now we are able to define thecovariant derivative

Dµ ≡ ∂µ − igAaµTa , (2.6)

which has, in contrast to ∂µ itself, the same local transformation behavior as ψ(x):

Dµψ(x) 7→ g(x)Dµψ(x) . (2.7)

Having defined the vector fields Aaµ, we are able to construct the field strength tensor F aµν ,

which does not depend on the fermionic fields:

F aµνTa ≡

i

g[Dµ, Dν ] = ∂µA

aνTa − ∂νAaµTa − ig

[AbµTb, A

cνTc]. (2.8)

In the case of matrix Lie groups, the multiplication in the corresponding Lie algebra G,the so-called Lie bracket [ · , · ] : G × G → G, is just the usual commutator. With this, wecan define the antisymmetric structure constants fabc by

[Ta, Tb] ≡ ifabcTc , (2.9)

which lead to a more explicit form of the field strength tensor:

F aµν = ∂µA

aν − ∂νAaµ + gfabcA

bµA

cν . (2.10)

In the case of an Abelian Lie group, for instance G = U(1) with d = 1 in QED, thestructure constants vanish, fabc = 0, and we recover the well-known Maxwell field strengthtensor Fµν . It is important to note that a single F a

µν is not gauge-invariant, because itdepends on the direction a in gauge space G. However, performing the trace we find aninvariant quantity:

TrG(F aµνTa · F µν

b Tb)

= TrG(F aµνF

µνb TaTb

)= C(r)F a

µνFµνa . (2.11)

Here we have used the fact that Tr (Ta(r)Tb(r)) = C(r)δab with some positive constantC(r) depending on the representation r of the gauge fields Aaµ. In the most important case,G = SU(N), we find in the fundamental representation r = N the prefactor C(N) = 1

2

directly from definition (2.2). Switching to another representation of Ta(r) means findingother generators Ta or, in the case of matrix Lie groups, other matrices which obeycondition (2.9). Note, that for non-vanishing fabc, hence for non-Abelian groups, we havecubic and quartic terms of the vector fields Aaµ in the product F a

µνFµνa , hence interaction

terms in the gauge sector.Finally, we are able to write down the Yang-Mills Lagrangian LYM, [YM54], which is

locally invariant under transformations g(x) ∈ G of a matrix Lie group G:

LYM = ψ(i /D −m

)ψ − 1

4F aµνF

µνa . (2.12)

The infinitesimal transformations of its components are as follows:

ψ(x) 7→ (1 + iαa(x)Ta)ψ(x) ,

Dµψ(x) 7→ (1 + iαa(x)Ta)Dµψ(x) ,

Aaµ(x) 7→ Aaµ(x) +1

g∂µα

a(x) + fabcAbµ(x)αc(x) = Aaµ(x) +1

gDµα

a(x) ,

F aµν(x)Ta 7→ F a

µν(x)Ta +[iαb(x)Tb, F

cµν(x)Tc

]= F a

µν(x)Ta + fabcFbµν(x)αc(x)T a .

(2.13)

8


The general formalism developed in the Yang-Mills theory is essential for the quantumfield theoretical approach to the strong force, QCD. Coleman and Gross actually showedthat a theory must necessarily be non-Abelian in order to feature asymptotic freedom[CG73, Gro05].

So far we have discussed the Yang-Mills theory only on a classical level. The quantiza-tion of the theory requires fixing the gauge freedom, the key ingredient when constructingthe Lagrangian (2.12). We consider now the partition function of the gauge theory:

Z =d∏a=1

4∏µ=0

∫DAaµ(x) eiS[Aaµ] ≡

∫DA(x) eiS[A] . (2.14)

Here, we suppress the Minkowski and isospin indices of the gauge field in the path integralmeasure DA(x). Two field configurations A(x) and A′(x) are called equivalent, if thereexists a gauge transformation α(x), (2.13), with A(x) 7→ A′(x) ≡ Aα(x). There areinfinitely many configurations Aα(x) which are in the same equivalence class A(x) = 0,hence (2.14) is divergent. We can get rid of this unphysical counting by introducing agauge-fixing condition following Faddeev and Popov [FP67]: for arbitrary scalar functionsωa : M→ C we define

Ga[A] ≡ ∂µAaµ(x)− ωa(x) = 0 . (2.15)

This is a Lorentz-invariant gauge-fixing condition, since ωa is a scalar function. For ωa ≡ 0we have the well-known Lorentz gauge. Inserting the formal decomposition of the identityin functional space into the partition function,

1 =

∫Dα(x) det

(δG[Aα(x)]

δα(x)

)δ(G[Aα(x)]) , (2.16)

we ensure that only one field configuration of each equivalence class contributes to thepartition function. Since this argument holds for all scalar functions ωa we can integratethem out and, neglecting an unphysical constant prefactor, we arrive at:

Z =

∫Dω exp

[−i∫d4x

(ωa(x))2

2ξ

] ∫Dα

∫DA det

(δG[Aα(x)]

δα(x)

)δ(G[Aα(x)]) eiS[A] =

=

∫Dα

∫DA det

(δG[Aα(x)]

δα(x)

)eiS[A] exp

[−i∫d4x

1

2ξ

(∂µAaµ(x)

)2],

(2.17)where ξ > 0 is the so-called gauge parameter. Physical obervables must be independentof ξ, hence we can choose it suitably. Note, that due to the fact that α(x) is a gaugetransformation, we have S[Aα] = S[A]. The choice of a Gaussian weight ensures theconvergence of Z. The functional determinant reads

det

(δGa[Aα]

δαa(x)

)(2.13)= det

(1

g∂µDµ

), (2.18)

for all a = 1, . . . , d. In the Abelian case, fabc = 0, we obtain Dµ 7→ ∂µ, hence thisdeterminant is independent of the gauge fields A(x) and can be absorbed in the normal-ization of the partition function. However, in the non-Abelian case we have to take careabout the functional determinant explicitly: we need to perform the Gaussian integrationin Grassmann space with the so-called Faddeev-Popov ghost fields c(x) and c(x). Using

9


these anti-commuting scalar fields instead of usual numbers ensures that the Gaussianintegration results in a determinant in the numerator and not in the denominator:

det(O) ∼∫Dc(x)Dc(x) exp

[−i∫d4x cOc

]. (2.19)

On the one hand the ghost fields are Lorentz scalars, but on the other hand they are anti-commuting Grassmann numbers, hence they violate the Spin-Statistic Theorem [Pau40].Indeed these fields are unphysical degrees of freedom and cannot appear in external states,but they ensure the S matrix to be unitary and the Optical Theorem, which is just acorollary of the unitarity of S, to hold in the non-Abelian case, too.

The identity (2.19) is valid for arbitrary operators O. From equation (2.18) we find inour case the operator

O =1

g∂µDµ =

1

g

(− i∂µAaµTa

). (2.20)

Putting everything together we can state the partition function of the pure gauge sector:

Z =

∫DA(x)

∫Dc(x)Dc(x) ei

∫d4x L , (2.21)

where the Lagrangian L includes both the gauge-fixing and Faddeev-Popov terms:

L = −1

4F aµνF

µνa −

1

2ξ

(∂µAaµ

)2 − ca(δac + gfabc∂µAbµ

)cc . (2.22)

In order to get the usual kinetic structure in the ghost sector, as well, we have redefinedthe ghost fields: ca 7→ ca

√g. Again, as already done in (2.13), we used the adjoint

representation A of the generators (Ta)bc = ifabc. In summary, the quantization of a non-Abelian Yang-Mills theory forces us to introduce both the gauge-fixing parameter ξ andthe anti-commuting scalar ghost fields ca and ca. Furthermore, we can state that there areghosts also in the Abelian case, for instance in QED, but inspecting the Lagrangian (2.22),the coupling between gauge fields Abµ and ghosts ca, cc is proportional to the structureconstants fabc, hence there is no ghost interaction in the Abelian case at all. We skip theexplicit non-Abelian Feynman rules and refer for instance to [PS95].

2.1.2 Symmetries of QCD

QCD is a special Yang-Mills theory with color gauge symmetry G = SU(N = 3) describedby the Lagrangian

LQCD = ψ(i /D −m)ψ − 1

4GaµνG

µνa , (2.23)

where ψ = (ψ1, ψ2, . . . , ψNf )T collects all Nf quark flavors and ψi = (ψri , ψ

gi , ψ

bi )T collects

all three colors, hence ψ ∈ S3Nf . The diagonal mass matrix m contains Nf blocks ofthe type mf · id3×3, hence the quark masses are degenerate in color space. The covariantderivative (2.6) is degenerate in both flavor and color space and the field strength tensor(2.8) enters the Lagrangian with the gauge bosons Aµa in fundamental representation N.For the symmetry group G = SU(N) the dimension of the Lie algebra G = su(N) is justd = N2 − 1. This leads to a su(3) basis with eight generators

Ta =λa2, a = 1, . . . , 8 , (2.24)

10


where λa denote the Gell-Mann matrices. The unphysical two-color case, N = 2, wouldbe described by the Pauli matricies σa, a = 1, 2, 3 .

Furthermore, QCD possesses additional global symmetries in flavor space: withoutany constraints we find the global U(1)V symmetry ψf 7→ exp (−iα)ψf for all flavors f .

The corresponding conserved Noether charge B =∑

f

∫d3x ψ†fψf is the baryon number.

This is consistent with all experimental results, so far. In contrast, theories beyond theStandard Model include the possibility that proton decay might take place [Pat00].

In the massless limit m → 0, the so-called chiral limit, we introduce left- and right-handed fields defined as

ψL ≡1

2(1− γ5)ψ , ψR ≡

1

2(1 + γ5)ψ , (2.25)

where we use γ5 in the chiral representation. The remaining Lagrangian separates intoleft- and right-handed terms and no mixed terms appear:

L0QCD ≡ LQCD|m=0 = iψL /DψL + iψR /DψR −

1

4GaµνG

µνa . (2.26)

Therefore, in the limit of exactly massless quarks, we have an additional chiral symmetry,SU(Nf )L × SU(Nf )R, which is assumed to be an approximative symmetry of QCD forNf ≤ 3, even with non-zero quark masses. This seems resonable because the currentquark masses of the three lightest quarks, u, d, s, are lower than ΛQCD ≈ 0.2 GeV asshown explicitly in table 2.2.

On the classical level we have in the chiral limit additionally the global U(1)A symmetry:ψf 7→ exp (−iαγ5)ψf for all flavors f . In general, the divergence of the axial vectorcurrent, jµ5 ≡ ψγµγ5ψ, is given by two terms: the mass-dependent term which breaksthe axial symmetry explicitely and the anomalous Adler-Bell-Jackiw (ABJ) term [Adl69,BJ69, CL06]:

∂µjµ5 = 2i

Nf∑f=1

mf ψfγ5ψf +g2Nf

32π2εµνρσGa

µνGaρσ . (2.27)

Even in the chiral limit the axial vector current is not conserved due to quantum fluctu-ations. The actual problem of the ABJ anomaly is given by the fact that it is impossibleto have both vector and axialvector symmetry in flavor space, but in principle one couldsplit the anomaly to both the vector and axialvector current. Due to the observed baryon-number conservation we avoid to break the vector current anomalously. In QED, U(1)Vdenotes just the gauge symmetry and it would be disastrous to have violations of a gaugesymmetry due to quantum fluctuations. Indeed, one can show that there is no anomalousbreaking of gauge symmetries within the Standard Model at all [PS95]. Another way tointerpret this anomalous term in (2.27) is given by the path integral approach to a quan-tized QCD: only in the vector case U(1)V the path integral measure

∫DψDψ is invariant

under the (classical) symmetry transformation. This does not hold in the axialvector caseU(1)A and leads to the anomalous term [Fuj79]. The corresponding anomalous breakingof U(1)A in QED gives rise to the pion decay π0 → γγ, which is observed in exerperiment.

Note, that in principle we could add both a P and CP, and therefore also T violatingterm with θ ∈ C to the QCD Lagrangian:

Lθ = − θ

64π2εµνρσGµνGρσ . (2.28)

11


QCD Lagrangian QCD vacuum

global flavor symmetry full QCD isospin limit chiral limit 〈ψψ〉 6= 0

SU(Nf )L × SU(Nf )R × × X ×SU(Nf )V × X X X

U(1)V X X X X

U(1)A × × cl. ×

Table 2.1: Summary of symmetries in flavor space, Nf ∈ 2, 3, of the QCD Lagrangian(2.23) and the vacuum state. The × denotes an absent symmetry and X denotes a presentsymmetry. On the Lagrangian level, symmetries are broken explicitely, whereas the vacuumstate breaks symmetries spontaneously. The U(1)A symmetry in the chiral limit is only presentin the non-quantized Lagrangian due to the ABJ anomaly.

Although being a total derivative, (2.28) affects physical observables: according to thecurrent-algebra analysis within the framework of an effective field theory approach theθ-term contributes to the electric dipole moment of the neutron [W+79]:

dn ≈ gπNN0.038 θ

4π2mN

lnmN

mπ

= 5.2 · 10−16θ e cm , (2.29)

where mN and mπ denote the neutron (nucleon) and pion mass, respectively, and gπNNdenotes the effective pion-nucleon-nucleon coupling: π · Nτ (iγ5gπNN)N . So far, no elec-tric dipole moment of the neutron was found [B+06]: |dn| < 2.9 · 10−26 e cm. There-fore, we find the upper limit |θ| < 10−10. Within the Standard Model there is al-ready the Glashow-Weinberg-Salem (GWS) theory which gives rise to a non-zero electricdipol moment of the neutron, but nowadays the experimental bound is too high [KZ82]:dGWS

n ≈ 2.0 · 10−32 e cm. However, we set θ = 0 in this thesis.

In table 2.1 we summarize the flavor symmetries of the QCD Lagrangian for two or threeflavors Nf ∈ 2, 3: in full QCD, Nf different quark masses enter the Lagrangian (2.23).We assume mf = m, for all flavors f , in the isospin limit. In the chiral limit no mass termsenter at all. The chiral condensate 〈ψψ〉 in the last column relates to the spontaneousbreaking of chiral symmetry which will be discussed in the next chapter in detail. Theresults in the table hold in the one-flavor case, Nf = 1, except for SU(1)V = id, whichbecomes a symmetry of full QCD, too. In the table there are all possible patterns forsymmetry breaking present: explicit, spontaneous and anomalous breaking.

2.2 Chiral Perturbation Theory

In this section we give a brief introduction to the fundamental ideas of chiral perturba-tion theory (χPT), a systematic approximation scheme of the chiral affective field theorythat describes the low-energy region of QCD. Due to (color) confinement which we willdiscuss in section 2.3, at low energies the actual degrees of freedom are no longer quarksand gluons, but baryons and mesons. At zero chemical potential this confined region islocated at temperatures T . ΛQCD ≈ 0.2 GeV. The construction of χPT is based on twoexperimental facts:

12


1. In the hadronic spectrum there are eight pseudoscalar particles with small massescompared to the hadronic scale 4πfπ/

√Nf ≈ 1 GeV.

2. There is no parity doubling in the low-energy mesonic and baryonic spectrum.

In table 2.2 we show the values of the current quark masses from the Particle Data Group[N+10]. Only the u, d, s quarks have masses below ΛQCD ≈ 0.2 GeV, hence we expect thechiral limit to be a feasible approximation of QCD only for Nf ≤ 3. Furthermore, all ninemeson masses are below the hadronic scale, but the mass of η′ is about a factor two largerthan the η-meson mass. This is known as the η′ puzzle that can be explained by the ABJanomaly and instantons [tH76]. The two experimental facts lead to the following workinghypothesis:

i. QCD features in the low-energy region a spontaneous breakdown of the chiral sym-metry G ≡ SU(Nf )L × SU(Nf )R to H ≡ SU(Nf )V .

ii. This spontaneous breakdown is characterized by a non-vanishing quark-antiquarkcondensate 〈ψψ〉 6= 0 .

We introduce the left- and right-handed Noether currents, jµa,L and jµa,R, by

jµa,L = ψLγµTaψL, jµa,R = ψRγ

µTaψR , (2.30)

where the ψL/R have been defined in (2.25). For the further discussion we need to considerthe vector, jµa , and axialvector current j5µ

a :

jµa ≡ jµa,R + jµa,L = ψγµTaψ ,

j5µa ≡ jµa,R − jµa,L = ψγµγ5Taψ .

(2.31)

Due to Noether’s Theorem the corresponding vector and axialvector charge are conserved:

0 = [QaV ,H0

QCD] =

∫d3x [j0

a(x, t),H0QCD] ,

0 = [QaA,H0

QCD] =

∫d3x [j5,0

a (x, t),H0QCD] .

(2.32)

Furthermore, the parities of QaV and Qa

A differ,

P QaV P

−1 = +QaV , P Q

aA P

−1 = −QaA . (2.33)

mu md ms

1.7 – 3.3 MeV 4.1 – 5.8 MeV 80 – 130 MeV

mc mb mt

1.2 – 1.3 GeV 4.1 – 4.4 GeV 170 – 174 GeV

mπ0 mπ± mK± mK0 ,mK

0 mη mη′

135.0 MeV 139.6 MeV 493.7 MeV 497.6 MeV 547.9 MeV 957.8 MeV

Table 2.2: Masses of the current quarks in MS at 2 GeV and the nine pseudoscalar mesons

13


In order to avoid a parity doubling at low energies, it is necessary to have QaA|0〉 6= 0 .

This claim can be proven as follows: let |i,−〉 denote a meson i with negative parity as itis observed in the hadronic spectrum, for instance i = π0. We know the parity and energyeigenvalues of this meson:

P |i,−〉 = −|i,−〉 , H0QCD|i,−〉 = εi|i,−〉 . (2.34)

Consider now the state |Φai 〉 ≡ Qa

A|i,−〉. Because the axial charge QaA is a conserved

quantity, [QaA,H0

QCD] = 0, the state |Φai 〉 has the same energy as |i,−〉,

H0QCD|Φa

i 〉 = H0QCDQ

aA|i,−〉 = Qa

AH0QCD|i,−〉 = εi|Φa

i 〉 , (2.35)

but opposite, positive parity:

P |Φai 〉 = P Q

aA P

−1︸︷︷︸−QaA

P |i,−〉︸︷︷︸−|i,−〉

= +|Φai 〉 . (2.36)

Indeed, we have shown that for any given pseudoscalar meson i = 1, . . . , 8 with energyeigenvalue εi, there exists a degenerate scalar state |Φa

i 〉 with opposite parity. But we havenot shown so far that |Φa

i 〉 can be expanded in such physical states. Assuming our claimis wrong, Qa

A|0〉 = 0, this state can be related to the meson states |j,+〉, which belong tothe irreducible representation of the vector part of SU(Nf )L × SU(Nf )R. From current-algebra analysis it is known that the creation operators of scalar (b†) and pseudoscalar(a†) states are related by

[QaA, a

†i ] = −taijb†j , (2.37)

with some group-theoretical constants taij [Sch02]. Now, we can state

|Φai 〉 = Qa

A|i,−〉 = QaAa†i |0〉 = [Qa

A, a†i ]|0〉+ a†i Q

aA|0〉︸︷︷︸=0

= −taij|j,+〉 , (2.38)

hence we would expect an octet of scalar mesons (JP = 0+) which is degenerate with thepseudoscalar octet (JP = 0−). From the second experimental statement, such an octet isnot observed in the hadronic spectrum at low energies. Therefore, our assumption waswrong and Qa

A|0〉 6= 0 holds.The fact that the vacuum |0〉 is not invariant under the axialvector charge Qa

A inducesthe definition of the so-called pion decay constant fπ: due to (2.32) we define the non-vanishing correlator

0 6= 〈0|j5µa (0)|φb(p)〉 ≡ ipµfπδab , (2.39)

where we have used the axialvector current at the origin, j5µa (0), and a state |φb(p)〉 of

the pseudoscalar meson octet. The definition of the meson fields is chosen in such a waythat the flavor factor δab arises naturally: 〈φa|φb〉 = δab. However, the right-hand side isjust a parametrization due to the Lorentz structure of the correlator with the prefactorfπ. Taking the divergence of relation (2.39) yields

0 6= 〈0|∂µj5µa (0)|φb(p)〉 = m2

πfπδab , (2.40)

where we have used p ·p = m2π in the isospin limit. Promoting this relation to the operator

level we arrive at the so-called partially conserved axialvector current (PCAC) hypothesis :

∂µj5µa (p) = m2

πfπφa(p) . (2.41)

14


The pion decay constant can be determined experimentally from the decay π+ → l+ + νl[N+10]. Later in chapter 7 we will use fπ = 93 MeV for numerical calculations.

Now we want to establish the relationship between our second working hypothesis〈ψψ〉 6= 0 and the PCAC discussion. Again, just from current analysis, it is possibleto relate the quark-antiquark condensate 〈ψψ〉 to the axialvector current Qa

A and thepseudoscalar quark density Pa [Sch02]:

〈0|[P a, QaA]|0〉 =

4

3〈ψψ〉 , (2.42)

where the pseudoscalar density P a for a = 1, . . . , N2f − 1 is defined by

P a ≡ ψγ5Taψ . (2.43)

Relation (2.42) implies〈ψψ〉 6= 0 ⇒ Qa

A|0〉 6= 0 , (2.44)

hence our second working hypothesis of a non-vanishing quark-antiquark condensate〈ψψ〉 6= 0 is a sufficient, but not necessary condition for spontaneous symmetry breakingin QCD. Indeed, it is not necessary, since also a diquark condensate 〈ψψ〉 6= 0 wouldviolate chiral symmetry of the vacuum state. Such a diquark condensate is assumed tobe created at a high chemical potential.

Let us now discuss the group-theoretical background of the spontaneous symmetrybreaking of the chiral symmetry group G, induced by 〈ψψ〉 6= 0:

G = SU(Nf )L × SU(Nf )R = SU(Nf )V × SU(Nf )A〈ψψ〉6=0

−−−−−→ H = SU(Nf )V . (2.45)

The vacuum remains invariant under the subgroup H of G. It is important to note thatH is no normal subgroup, hence we cannot expect the quotient G/H ≡ gH | g ∈ G =SU(Nf )A to obey group structure anymore. Indeed, we find that SU(Nf )A itself is not agroup, because the corresponding “Lie algebra” is not closed:

[Ta, Tb] = ifabcTc ,

[Ta, γ5Tb] = ifabcγ5Tc ,

[γ5Ta, γ5Tb] = ifabcTc .

(2.46)

We know from the third line that the left and right coset of H differ: gH 6= Hg. Thisshows that H is not a normal subgroup.

Compare the relations (2.46) to the well-known case of the Lorentz group SO(1, 3):so(1, 3) = C ⊗ su(2): the rotations form a group, SU(2)R, whereas the boosts do not,SU(2)B. Physical consequences of this fact can be observed, for instance, in the so-calledThomas precession: there is a relativistic factor of two in the precession frequency of thedoublet separation in the hydrogen fine structure.

In χPT we identify the basis of the left coset gH = G/H with N2f − 1 Goldstone

bosons, φa = ψγ5Taψ. Goldstone’s Theorem ensures that the resulting bosons carrythe same quantum numbers as the broken generators, hence there are N2

f − 1 masslesspseudoscalars which are identified in the three-flavor case with the light meson octet andin the two-flavor case with the three pions. Because the chiral limit is not realized innature because of explicit mass terms in the QCD Lagrangian, we cannot expect to findmassless, but light bosons in the spectrum. Since the isospin limit is not realized in natureeither, the masses of π, K and η differ.

15


Note, that the subgroup H of G = SU(Nf )L × SU(Nf )R can be identified with thediagonal of G:

H = (V, V ) ∈ G | V ∈ SU(Nf ) = SU(Nf )V . (2.47)

Furthermore, the left coset g0H for any g0 = (L0, R0) ∈ G can be identified with R0L†0,

as the following calculation shows:

g0H 3 (L0V,R0V ) = (L0V,R0L†0L0V ) = (id, R0L

†0) (L0V, L0V )︸︷︷︸

∈H

∈ (id, R0L†0)H . (2.48)

From this, U ≡ R0L†0 can be used to define the left coset g0H, hence it is also related to

the Goldstone boson fields φ = (φ1, . . . , φd), d = N2f − 1. The transformation of left coset

U under the whole symmetry group G reads:

UG7→RUL† , (2.49)

because we find for any g = (L,R) ∈ G:

gg0H = (L,RR0L†0)H = (id, RR0L

†0L†) (L,L)H︸︷︷︸

=H

= (id, R(L0R†0)†L†)H . (2.50)

In order to find the relation between the left coset U and the Goldstone boson fields φ interms of a group operation ϕ, we have to introduce two sets:

F ≡ φ : M→ Rd | φ = (φ1, . . . , φd), φi : M→ R , (2.51)

collecting all Goldstone boson fields in a complex vector space. Furthermore, we introducethe set of functions

M ≡ X : M→ SU(Nf ) | X continuous in φ(x) for all φ ∈ F , (2.52)

which is not a vector space, since SU(Nf ) is not closed under adding. However, the mapϕ : G×M →M , defined by

ϕ[(L,R), X](x) ≡ RX(x)L† ∈ SU(Nf ) , (2.53)

is a group operation on M because it fulfilles the defining properties

ϕ[ id, X] = X for all X ∈M,

ϕ[ g1, ϕ( g2, X)] = ϕ[ g1 g2, X] for all g1, g2 ∈ G, X ∈M .(2.54)

Because M is not a vector space, the group operation is not linear in its second argu-ment, hence we call ϕ a realization instead of representation. Promoting (2.49) to a localtransformation law, definition (2.53) of the group operation ϕ shows that we are able toidentify the left coset U(x) and the function X(x):

M 3 x 7→ φ(x)X7→X(x) ≡ U(x) = R0L

†0 ∈ SU(Nf ) . (2.55)

The explicit choice how the function X(x) ∈ F depends on φ(x) is free in parametrization.For instance, there is the exponential realization

U(x) = X(x) ≡ exp

(2i

f0

φa(x)Ta

), (2.56)

16


where the constant f0 is related to the pion decay constant by f0 = fπ + O(m2π, p

2). Inthis thesis we restrict ourself to the two-flavor case, Nf = 2, hence it is possible to usethe equivalent square-root realization with Ta = σa/2 and φa = πa:

U(x) = X(x) ≡ 1

f0

(√f 2

0 − φ2(x) + 2i φa(x)Ta

). (2.57)

Both realizations of the Goldstone boson fields are equivalent because their Feynmanrules coincide in the physical on-shell case. We skip the explicit construction of effectiveLagrangians and refer to literature [Sch02]. In appendix A.5 we consider the second-orderLagrangian L2 and derive the corresponding Feynman rules.

Considering the vacuum, φ = 0, hence U0(x) = id ∈ SU(Nf ), we see that this state isnot invariant under the whole symmetry group G, but only under the subgroup H. Wefind for all x ∈M:

ϕ[(L,R), U0](x) = RU0(x)L† = RL†!

=U0 = id ⇒ R = L . (2.58)

This condition selects just the diagonal of G, defined as subgroup H, which remainsa symmetry of the QCD vacuum state. The identification of the left coset U = gHwith the Goldstone boson fields φ is thus representative of the spontaneous breaking ofchiral symmetry G in the low-energy region. The key ingredient is the non-vanishingquark-antiquark condensate 〈ψψ〉 6= 0 which is only sufficient, but not necessary, forspontaneous symmetry breaking as discussed before.

The question why it is not possible to choose a representation of the Goldstone bosonscan be answered using group theory: considering the two- and three-flavor case, we showthat if one deals with a representation, one factor of the symmetry group G becomestrivial. First we investigate G = SU(2)L × SU(2)R. A representation of SU(2) can becharacterized by one parameter, j, related to the dimension of the representation byd(j) = 2j + 1, as known, for instance, from angular-momentum analysis [Gre05]. TheGoldstone bosons transform under G in adjoint representation A, hence the product-grouprepresentation must fulfill

3!

= d(jL)d(jR) = (2jL + 1)(2jR + 1) ⇒ ji = 0 for some i ∈ L,R . (2.59)

Since d(0) = 1, one factor of G must be in the trivial representation of SU(2), in contra-diction to the transformation laws of the pion fields.

Also in the three-flavor case, G = SU(3)L × SU(3)R, this argument holds: SU(3) hastwo Casimir operators, hence its representations can be characterized by two parameters,p and q. The dimension of a representation (p, q) reads: d(p, q) = 1

2(p+1)(q+1)(p+q+2).

Again, the Goldstone bosons transform under G in adjoint representation A, hence

8!

= d(pL, qL)d(pR, qR) ⇒ (pi, qi) = 0 for some i ∈ L,R , (2.60)

therefore, one factor of G is trivial.

Our discussion shows that assuming a linear realization ϕ of the Goldstone bosons Uleads to wrong transformation properties of the mesons under the chiral symmetry groupG, hence ϕ is necessarily non-linear.

17


2.3 QCD Phase Diagram

Strongly interacting matter seems to occur in nature in manifold phases governed bythree fundamental properties of QCD: asymptotic freedom, confinement and symmetries.The questions which phases really exist in nature and what kinds of phase transitionstake place are investigated intensely by both experimental and theoretical physicists. Asdiscussed in the previous section in (2.29) and the following discussion, even the questionwhether QCD conserves or violates parity symmetry remains open until an electric dipolmoment of the neutron, dn, with 2.0 · 10−32 e cm < dn < 2.9 · 10−26e cm is discovered.

In 2004 Gross, Politzer and Wilczek reveived the Nobel Price “for the discovery ofasymptotic freedom in the theory of the strong interaction” more than 30 years ago[WG73, Pol73]. The β function of QCD is defined by

β(g) ≡ µ∂g

∂µ, (2.61)

where g is the coupling introduced in (2.5) and µ is some renormalization scale. Thereare two contributions to the β function of QCD with opposite sign: the quark degrees offreedom give rise to screening effects, whereas the gluons obey anti-screening. In contrastto the photons in Abelian QED the gluons are self interacting, hence there are additionalloop terms. In summary the well-known β function of QCD with gauge symmetry SU(N)and Nf quarks reads in the chiral limit at one-loop level:

β(g) = − g3

(4π)2

(11

3N − 2

3Nf

). (2.62)

From this, the scale dependent strong coupling is given by:

g2(k) =g2(µ)

1 + g2(µ)(4π)2

(113N − 2

3Nf

)ln (k2/µ2)

. (2.63)

For sufficiently small numbers of fermionic degrees of freedom, Nf <336N |N=3 ≤ 16, we

find a negative β function, hence limk→∞ g(k) = 0. Such theories, as QCD in the physicalcase N = 3 and Nf = 6, are called asymptotically free and can be treated perturbativelyat high energies or, equivalently, at small distances. In terms of renormalization grouptheory, asymptotically free theories have a trivial (Gaussian) ultraviolet fixed point, sincefor vanishing coupling the theory becomes free. As stated in section 2.1.1, asymptoticfreedom is a characteristic property of non-Abelian Yang-Mills theories [CG73, Gro05].

The third main property of QCD matter is the fact that quarks and gluons cannot beobserved individually. This phenomenon is called (color) confinement and so far it is notfully understood [JW00]. Using the β function at low energies where the coupling is largedoes not provide an explanation for confinement, since low energies k < ΛQCD ≈ 0.2 GeVare not in the domain of a perturbatively calculated coupling g(k). From Lattice QCD itis confirmed that there is a linear quark-antiquark potential at large distances, but it isnot understood how it comes about. Several theories for confinement are summarized in[AG07]: magnetic monopoles could create an electric flux tube in which the color-chargedquarks are confined. This picture describes a dual superconducter, since in a commonsuperconducter electric charged Cooper pairs create a magnetic flux in which magneticcharges are confined [Man79]. A further approach to confinement is given by the infraredbehavior of the Schwinger-Dyson equations and renormalization group theory [Fis06].

18


〈ψψ〉 = 0

〈ψψ〉 = 0

〈Φ〉 = 0

〈ψψ〉 = 0

〈Φ〉 = 0

hadron gas

QGP

CFL

μB [GeV]

T [GeV]

1

0.15

0

nuclearmatter

Figure 2.1: Sketch of the QCD phase diagram

It turns out that at small energies a non-Abelian theory in Landau gauge is governedby the Faddeev-Popov determinant, hence by ghost dynamics. Because this functionalapproach does not implement the Lorentz gauge, additional constraints for the gaugefixing must be introduced in order to avoid an unphysical counting of Gribov copies. Usingstochastic quantization as key ingredient, this analysis is descibed in [Zwa04]. A quite newapproach to confinement is given by the AdS/CFT correspondence. There, long-range,non-perturbative quantum field theories can be related to weak-coupling gravity duals.However, QCD is neither a supersymmetric nor a conformal theory. In chapter 8 we givea brief survey of the most important notions concerning AdS/CFT correspondence.

In figure 2.1 we show a schematic QCD phase diagram combining theoretical calcula-tions, experimental results and expectations. We have chosen the baryon-chemical poten-tial µB instead of the baryon-number density ρB, since the first one is continuous at thephase transition lines. The latter one would lead to transition regions instead of singletransition lines.

Due to confinement, the actual degrees of freedom at low energies are not quarks andgluons but hadrons. In the physical case of three colors, N = 3, there are color-neutralmesons and baryons. Using the constituent-quark model, one quark and one antiquarkform a meson, whereas three quarks form a baryon. However, there might be exotic color-neutral objects like pentaquarks consisting of four quarks and one antiquark, or glueballshaving no constituent quark at all. So far there is no assured experimental evidence forsuch particles.

The theoretical framework of low-energy QCD is given by chiral effective field theoryintroduced in the section 2.2. Chiral perturbation theory (χPT will be used intensely inchapter 7 for calculating the shear viscosity of a two-flavor pion gas. Furthermore, atlow temperatures and baryon-chemical potential µB = mN − Ebinding ≈ (938− 16) MeVthere is the phase of nuclear matter. It is separated from the hadron gas by a first-ordertransition line with critical end point at T nucl

c ≈ 17 MeV. So far, this is the only first-orderphase transition in hadronic matter which has been confirmed experimentally [N+02].

For very high temperatures quarks and gluons are asymptotically free and form a

19


quark-gluon plasma. In-between two phase transitions take place: the confinement-deconfinement transtition and the chiral restoration. The latter one is due to a melt-ing quark-antiquark condensate 〈ψψ〉 6= 0 for T < T chi

c to 〈ψψ〉 = 0 for T > T chic . The

quark-antiquark condensate 〈ψψ〉 serves as order parameter for the chiral transition in thechiral limit. However, the order parameter of the confinement-deconfinement transtitionis given by the expectation value of the Polyakov loop:

〈Φ(x)〉 ≡ 1

N

⟨Tr

[P exp

i

∫ β

0

dτ A4(τ,x)

]⟩. (2.64)

Here, P denotes the path-ordering symbol, β is the inverse temperature and Aµ the gaugefield. The trace must be evaluated in color space SU(N). The Polyakov loop is related tothe free energy F of a single quark by [MS81, Hel10]

〈Φ(0)〉 = e−12βF . (2.65)

In the confined phase for T < T confc where no free quarks exist, the free energy diverges:

F → ∞, hence 〈Φ〉 → 0. The deconfined phase for T > T confc is related to 〈Φ〉 → 1.

We want to emphasize that both 〈ψψ〉 and 〈Φ〉 are only approximative order parameters :only in the chiral limit QCD obeys chiral symmetry which is, in this case, restored forT > T chi

c . Furthermore, relation (2.65) is only valid in a pure gauge case without quarks,Nf = 0. However, the Polyakov loop can be used in the Polyakov-loop-extended Nambu–Jona-Lasinio (PNJL) model [Fuk04]. It describes a confinement scenario not includedin the preceding Nambu–Jona-Lasinio (NJL) model in which gluonic degrees of freedomare not considered. Historically, [NJL61a, NJL61b], the NJL model was introduced as amodel for nucleons with non-linear four-fermion interaction in order to derive the nucleonmass. The historical NJL Lagrangian,

LNJL = −ψ /∂ψ − g0

2

[(ψγµψ

)2 −(ψγµγ5ψ

)2], (2.66)

obeys the flavor symmetry SU(2)L × SU(2)R × U(1)V , which is present in the two-flavorQCD in the chiral limit, too. However, the (anomalous) breaking of U(1)A is already im-plemented at the classical level. Nowadays the NJL model is used for two- and three-flavorQCD and its fermionic degrees of freedom are interpreted as quarks. The synthesis ofNJL and Polyakov-loop thermodynamics in terms of the PNJL model applied successfullyin modelling QCD phases [VW91, RTW06, HRCW10].

The critical temperatures T chic and T conf

c for the chiral and the confinement-deconfine-ment phase transition, respectively, can in principle be different. The standard symmetrybreaking pattern would suggest that chiral restoration happens at higher temperaturesthan the deconfining transition, T conf

c < T chic , since chiral symmetry is always sponta-

neously broken in a confined phase. Computations from Lattice QCD at zero baryon-chemical potential [C+10, A+09] indicate that the chiral and deconfinement transitiontemperatures seem to coincide within an overlapping window along the correspondingcrossover transitions: Tc ≈ T conf

c ≈ T chic ≈ (160± 10) MeV. In general, Lattice QCD is

only applicable for µB/T 1 due to the so-called sign problem.Let us now investigate the question what kind of (chiral) phase transition takes place

at zero chemical potential: first order, second order or a crossover region? Consideringa linear σ model which has the same universality class as QCD, in [PW84] the chiralrestoration is identified as a first-order phase transition for three quark flavors in the chiral

20


?

?

phys.point

00

N = 2

N = 3

N = 1

f

f

f

m s

sm

Gauge

m , mu

1st

2nd orderO(4) ?

2nd orderZ(2)

2nd orderZ(2)

crossover

1st

d

tric

∞

∞

Pure

Figure 2.2: Columbia Plot at zero baryon-chemical potential (adapted from [LP03])

limit. In the so-called Columbia Plot [BBC+90], shown in figure 2.2, this case is locatedat the origin. In the two-flavor case this transition is supposed to be of second-order inthe chiral limit and a crossover in the isospin limit. In the vicinity of the pure-gaugecase where no fermionic degrees of freedom contribute at all, there is again a first-orderphase transition. The large crossover region in figure 2.2 is separated from the first-order-transition regions by a line of second-order transitions with universality class Z(2). Here,Z(N) ≡ z ∈ SU(N) | zg = gz for all g ∈ SU(N) denotes the center subgroup of SU(N),which is the largest Abelian subgroup of SU(N). In contrast, for vanishing u- and d-quarkmasses and strange-quark masses larger than the so-called tricritical mass, ms > mtric

s ,the second-order transitions belongs to the universality class O(4) = SU(2) × SU(2).However, so far it is not clear where the physical point is located. In chapter 7 we aregoing to calculate the shear viscosity of a two-flavor pion gas with physical pion massmπ = 140 MeV. Therefore, according to the Columbia Plot, there should be a crossoverregion and we will use for the crossover temperature the same number as for the criticaltemperature Tc ≡ 155 MeV at zero baryon-chemical potential.

Going to large chemical potentials, µB 1 GeV, but not too high temperatures, T .100 MeV, there is a phase predicted where a non-vanishing diquark condensate violateschiral symmetry: 〈ψψ〉 6= 0. It is called color-flavor-locked (CFL) phase, since the Cooperpairs of this superconducting phase connect color and flavor: SU(N) ⊕ SU(Nf ). Thisphase of QCD matter is supposed to be realized in the center of dense neutron stars. Fordetails to the CFL phase and the QCD phase diagram in general we refer, for instance,to [Ste06] and the references therein.

In this thesis we are interested in the temperature axis of the phase diagram for µB = 0.The φ4 theory in chapter 6 is considered in the high-temperature limit, T > Tc, whereaschiral perturbation theory in chapter 7 is applicable for T < Tc.

21

3 Hydrodynamics of RelativisticHeavy-Ion Collisions

In this chapter we give an introduction to the relativistic hydrodynamics of perfect anddissipative fluids, a useful framework in order to describe relativistic heavy-ion collisions.The experimental results of the Relativistic Heavy-Ion Collider (RHIC) at BrookhavenNational Laboratory show that it is possible to describe such collisions in terms of aperfect fluid, taking small dissipative effects into account [Hei05]. We introduce the bulk-and shear-viscosity parameters and discuss entropy production on a phenomenologicallevel without explicit reference to a microscopic theory of the collision.

3.1 First Principles of Relativistic Hydrodynamics

Relativistic heavy-ion collisions are run with nuclear masses of the ions large compared toa single proton, for instance gold (A = 195) or lead (A = 207). The center of mass energyper nucleon for the gold experiments at RHIC in Brookhaven was

√sAA = 200AGeV, the

new LHC in Geneva is expected to provide√sAA = 5600AGeV for lead ions [YHM08,

Table 15.1]. A sketch of a central relativistic heavy-ion collision is shown in figure 3.1.Due to Lorentz contraction, the colliding ions are flattened, hence the diameter at rest,D0, reads in the lab (collider) frame: D = D0/γ.

ions about to collide collision plasma formation expansion towardshadronization

Figure 3.1: Sketch of a central ultra-relativistic heavy-ion collision (adapted from [RHI])

The initial state of a relativistic heavy-ion collision consists of two well-separated ionspropagating along the z-axis. We assume that only central collisions with impact pa-rameter b = 0 occur. For b > 0 the ions can be split into interacting and spectatingcomponents. The latter, without deflection, can be reconstructed due to a clear signaturein the detector. Such a collision can be described as a central one with a reduced effectivenuclear mass A′ of the interacting components.

In the Bjorken spacetime picture [Bjo83], the time after the collision is divided intothree parts:

pre-equilibrium: The very first time after the collision is assumed to be governed byhuge entropy production because the matter is in a very hot and dense state. Due to

23

3 Hydrodynamics of Relativistic Heavy-Ion Collisions

Figure 3.2: Bjorken spacetime picture of a central relativistic heavy-ion collision (adaptedfrom [YHM08])

the high energies, the quarks and gluons are almost free, so that the spatial collisionregion is expanding rapidly. This state is sketched in the second picture of figure3.1. In order to investigate that, we would need to describe a thermal non-Abeliangauge theory in a non-equilibrium state. This is beyond the scope of this thesis. Inprinciple the kinetic theory using the Boltzmann equation is one possible approachto investigate this demanding state of matter [CDOW10].

local equilibrium: After the thermalization has taken place, a local equilibrium isreached in a characteristic time τ0. The quarks and gluons are no longer free butstill deconfined; they form a hot plasma, sketched in the third picture of figure3.1. One possible approach to describe this state of matter is given by relativistichydrodynamics. The calculations within this formalism agree quite well with ex-perimental results from RHIC. In this chapter, relativistic hydrodynamics is usedto describe the quark-gluon plasma as a dissipative fluid. Later in chapters 6 and7, thermal field theory is employed to calculate the dissipative parameters of thehydrodynamical equations.

hadronization: The temperature decreases rapidly as the plasma extends with time.Therefore, we expect a transition to the confined hadronic phase of QCD matter.After a characteristic time τf , the chemical freeze-out takes place, and the annihila-tion and production of particles is balanced. Finally only hadrons and leptons cancreate a signal in the detector. This state can be described by some effective theory,for instance by chiral perturbation theory.

In figure 3.2 the Bjorken spacetime picture is sketched for a 1 + 1 dimensional collision.The hyperbolas denote points in spacetime with constant proper time τ =

√t2 − z2. We

have stated that the quark-gluon plasma is formed when the pre-equilibrium state hasreached local equilibrium. We need to define this term:

24

3.1 First Principles of Relativistic Hydrodynamics

A system in local thermodynamic equilibrium is a macroscopic system Ω inspacetime which can be divided into mesoscopic zones Ωi in such a way thateach zone by itself is in thermodynamic equilibrium. (compare figure 3.3.)

The definition of local equilibrium implies that all thermodynamical quantities becomefields on Minkowski space M. For instance, we find for the four-velocity of the system:

u : Ω ⊆M→ S , uµ(x) ≡ dxµ

dτ= γ(x)(1,v(x)) . (3.1)

Several conceptual problems arise at this point: the first problem is that a local transfor-mation γ(x) is no longer an element of the Poincare group. This means that in generalthere exists no Lorentz transformation which transforms all Ωi simultaneously to the sameframe. Consequently, we have to do the Lorentz analysis carefully and more explicitlythan usual. In particular expressions with spacetime derivatives are dangerous becauseneighboring zones Ωi are involved that need not be in the same frame. Also the ideato evaluate a Lorentz scalar at rest must be rechecked carefully, for instance for (3.20).Indeed, Lorentz invariance is broken by the fact that the thermal medium (heat bath) “atrest” necessarily defines a distinguished frame of reference.

The second problem is encoded in the expression “mesoscopic“ : on one hand, due tothe fact that we want to do calculus on Ω, the zones Ωi should be as small as possible;in the best case a single point. On the other hand, we assume that each zone is inthermodynamic equilibrium, which cannot be true for a single point. Hence, given thestatistical foundation of thermodynamics, the zones should be as large as possible. Onepossible escape from this contradiction is to introduce a Minkowski space made of non-standard numbers R∗ [LR94]. There, each zone is indeed just a real number (vector)and the thermodynamics takes place in its monade, which is the set of infinitesimallyneighboring numbers (vectors). In this work we will assume that it is possible to do realcalculus on zones in thermodynamic equilibrium.

The first principles of relativistic hydrodynamics are expressed in one vector and onescalar conservation law. Every theory governed by a translation invariant Lagrangian

Ω ⊆ M

Ω1

Ω2Ω3

Ω4

Ω5

Ω6

Figure 3.3: Visualization of a system in local thermodynamic equilibrium

25


with additional global U(1)V flavor symmetry provides five Noether currents:

∂µTµν(x) = 0 , ∂µj

µB(x) = 0 , (3.2)

where T µν(x) denotes the energy-momentum tensor field and jµB(x) denotes the baryon-number current, defined by

jµB(x) ≡ nB(x)uµ(x) . (3.3)

The baryon density nB(x) is defined in the local rest frame uµ(x) = (1, 0, 0, 0). It isnot a Lorentz scalar. QCD, the fundamental theory we are interested in to describe thequark-gluon plasma, provides all needed symmetries for these basic conservation laws.Altogether we have five equations for the six characterizing quantities of a relativisticliquid: baryon density nB(x), energy density ε(x), pressure P (x) and velocity v(x). Thelatter one comes from the four-velocity uµ(x). With the normalization uµuµ = 1 thatfollows directly from its definition (3.1), the temporal component is determined by thespatial ones. The missing equation to describe the system is completely given by theequation of state (EOS) that connects pressure P (x) and energy density ε(x). Derivingthe EOS from first principles of a theory is demanding and indeed a research area byitself. Especially the EOS for nuclear matter is an involved problem due to its locationin the confined, low energy region of QCD.

3.2 Relativistic Hydrodynamics of Perfect andDissipative Fluids

First we take a look at perfect fluids. We give a definition of this term:

A perfect fluid is a fluid whose pressure fulfills Pascal’s law: its pressure exertsno transverse forces and disperses isotropically.

Therefore, the energy-momentum tensor of a perfect fluid at local rest frame is directlyobtained by

T µν(x) =

ε(x) 0 0 0

0 P (x) 0 00 0 P (x) 00 0 0 P (x)

. (3.4)

Pascal’s law forces the pressure tensor to be proportional to unity. Because T µν is asecond rank tensor, we also know it in any frame:

T µν(x) = ΛµρΛν

σTρσ(x) = [ε(x) + P (x)]uµ(x)uν(x)− gµνP (x) . (3.5)

From the conservation of the energy-momentum tensor we can derive a longitudinal(scalar) and a transverse (vector) part. Contracting ∂µT

µν = 0 with the four-velocityuν leads to the scalar part:

uµ∂µε+ (ε+ P )∂µuµ = 0 . (3.6)

Inserting the first law of thermodynamics in the case of vanishing chemical potentials,

Ts = ε+ P, (3.7)

where we use intensive quantities only, we arrive at the conservation of the entropy-densitycurrent :

∂µ(suµ) ≡ ∂µ(sµ) = 0 . (3.8)

26

3.2 Relativistic Hydrodynamics of Perfect and Dissipative Fluids

This result is quite natural for a perfect fluid and it is just a consequence of spacetimesymmetries and the first law of thermodynamics. Contracting ∂µT

µν = 0 in (3.2) withgµν − uµuν , which is orthogonal to the four-velocity uν , one gets the transverse part ofthe energy-momentum conservation. Combining its spatial and temporal part, we derivethe relativistic Euler equation:

1

c

dv

dt= −1− v2/c2

ε+ P

(c∇P +

v

c

∂P

∂t

). (3.9)

In order to easily calculate its non-relativistic limit, we have explicitly inserted the speed oflight c into the equation. In the limit v/c 1, we find ε+P = ρc2+1

2ρv2+O(v4)+P → ρc2.

Additionally the ∂P/∂t contribution in (3.9) vanishes, hence we arrive at

dv

dt= −∇P

ρ. (3.10)

Indeed, this is the well-known Euler equation of continuum mechanics, a corollary ofNewton’s second law.

If we want to describe a dissipative fluid, the starting point is to extend the energy-momentum tensor of a perfect fluid by the dissipative tensor τµν :

T µν = uµuν(ε+ P )− Pgµν + τµν . (3.11)

The more the system Ω with local equilibrium deviates from global equilibrium, the morezones Ωi we have to introduce and the more boundaries appear. The thermodynamicquantities change only on these boundaries. Hence, the derivative ∂µuν measures how farthe system is away from global equilibrium. Global equilibrium is reached if and only if∂µuν(x) = 0 for all x ∈ Ω.

In order to construct the dissipative tensor at first order we have to meet the followingthree conditions:

uµτµν = 0, a direct consequence of (3.11), evaluated at rest,

∂µsµ ≥ 0, second law of thermodynamics,

only first order derivatives ∂µuν .

These conditions imply that the dissipative tensor has the following form [YHM08, Wei72]:

τµν = η

[∂µ⊥u

ν + ∂ν⊥uµ − 2

3∆µν(∂⊥ · u)

]+ ξ∆µν(∂⊥ · u) , (3.12)

with the abbreviations

∆µν ≡ gµν − uµuν , ∂µ⊥ ≡ ∂µ − uµ(u · ∂) = ∆µν∂ν . (3.13)

In (3.12) we introduced two dissipative parameters: the shear viscosity η, describing thetraceless part of τµν , and the bulk viscosity ξ. At this level, both coefficients are just someunknown quantities, but later on, we will calculate them within microscopic theories. Thesecond law of thermodynamics ensures that both dissipative parameters are non-negativenumbers: η, ξ ≥ 0.

27


The second rank tensor ∆µν is a projector, transverse to the four velocity, with thefollowing properties:

∆µν = ∆νµ , uµ∆µν = 0 , ∆µν∆ρµ = ∆νρ , ∆µ

µ = 3 . (3.14)

Apart from the shear and bulk viscosity, there is in general a third dissipative parameter,the heat conductivity κ. In addition to the energy-momentum tensor, the baryon-numbercurrent (3.3) and the entropy-density current (3.8) must be modified, [YHM08, (11.32)-(11.34)]. In this thesis we restrict ourself to ultra-relativistic systems, assuming thattemperature provides the dominant energy scale. In terms of the QCD phase diagram, wediscuss the quark-gluon plasma for zero chemical potential. In [DG85] it was derived thatheat conduction is suppressed compared to other dissipative phenomena if µB/T 1, sothat we can neglect the heat conductivity.

Similar to the derivation of the relativistic Euler equation (3.9), we could derive therelativistic version of the Navier-Stokes equation which takes the dissipative parametersinto account. Within the setup of general relativity, this is shown by [Kub01]. We are notgoing to work with these equations explicitly, since they are just corollaries of the morefundamental Noether currents T µν(x) and jµB(x). This means that extending the Euler orNavier-Stokes equation to their relativistic generalizations is not sufficient to describe arelativistic fluid. Indeed, we will apply the conservation laws (3.2) explicitly and therebyuse the relativistic Euler and Navier-Stokes equation implicitly.

3.3 1 + 1 Dimensional Example

We now consider a relativistic heavy-ion collision in the z, t-plane in the time windowbetween quark-gluon-plasma formation and the onset of hadronization. Instead of thenon-scalar quantities z and t, we use the proper time τ and the rapidity Y , defined by

τ ≡√t2 − z2 , Y ≡ artanh βz =

1

2ln

t + z

t− z, (3.15)

with their inverse maps

z = τ sinhY , t = γτ = τ coshY . (3.16)

In these coordinates, the four-velocity reads

uµ = (t/τ, 0, 0, z/τ) = (coshY, 0, 0, sinhY ) . (3.17)

For further calculations we need to know how to transform derivatives in terms of z, tto derivatives in terms of τ, Y :

(∂t∂z

)=

(coshY − sinhY− sinhY coshY

)(∂τ

1τ∂Y

). (3.18)

28


To verify this, consider a function f = f(t = τ coshY, z = τ sinhY ):[coshY ∂τ −

1

τsinhY ∂Y

]f(t, z) =

= coshY

(∂f

∂t

∂t

∂τ+∂f

∂z

∂z

∂τ

)− sinhY

τ

(∂f

∂t

∂t

∂Y+∂f

∂z

∂z

∂Y

)=

= cosh2 Y∂f

∂t+ coshY sinhY

∂f

∂z− sinh2 Y

∂f

∂t− sinhY coshY

∂f

∂z=∂f

∂tX[

− sinhY ∂τ +1

τcoshY ∂Y

]f(t, z) =

= − sinhY

(∂f

∂t

∂t

∂τ+∂f

∂z

∂z

∂τ

)+

1

τcoshY

(∂f

∂t

∂t

∂Y+∂f

∂z

∂z

∂Y

)=

= − sinhY coshY∂f

∂t− sinh2 Y

∂f

∂z+ coshY sinhY

∂f

∂t+ cosh2 Y

∂f

∂z=∂f

∂zX

(3.19)In order to derive first order differential equations for the energy and entropy density, weneed the following relations:

uµ∂µ =

∂

∂τ, ∂µuµ =

1

τ. (3.20)

We derive them by a direct calculation, using (3.17) and (3.18):

uµ∂µ = coshY ∂t + sinhY ∂z =

= coshY

(coshY ∂τ −

1

τsinhY ∂Y

)+ sinhY

(− sinhY ∂τ +

1

τcoshY ∂Y

)=

= cosh2 Y ∂τ −1

τcoshY sinhY ∂Y − sinh2 Y ∂τ +

1

τsinhY coshY ∂Y = ∂τ X

∂µuµ =

(coshY ∂τ −

1

τsinhY ∂Y

)coshY +

(− sinhY ∂τ +

1

τcoshY ∂Y

)sinhY =

= −1

τsinh2 Y +

1

τcosh2 Y =

1

τX

(3.21)Note, that both results are indeed Lorentz scalars. The first result can be confirmed easilyby evaluating uµ∂

µ at rest, because uµ = (1, 0, 0, 0) takes only the time component of ∂µ,namely ∂t = ∂τ . As discussed around (3.1), it does not work to evaluate the second resultat rest, because we are only locally at rest. Taking this Poincare violating aspect not intoaccount by a careful analysis as done in (3.21), ∂µu

µ would simply vanish.We now consider again a perfect fluid. Inserting the identities (3.20) in (3.6) and (3.8),

we arrive at two ordinary differential equations (ODEs) of first order:

∂ε

∂τ= −ε+ P

τ,

∂s

∂τ= −s

τ. (3.22)

In order to solve these equations, we need the EOS of an ultra-relativistic system: ε = 3P .Due to the dominant temperature scale T m, we deal with massless and non-interactingparticles which is consistent with the weak QCD coupling at high energies. Therefore wefind for the first law of thermodynamics:

Ts = ε+ P =4

3ε . (3.23)

29


0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

ΤΤ0

Figure 3.4: Solution of the differential equations (3.22) for a perfect fluid in the ultra-relativistic limit: energy density ε/ε0 (red), entropy density s/s0 (green) and temperatureT/T0 (blue).

Now both ODEs in (3.22), with initial conditions ε0, s0 and T0, can be solved analytically:

ε(τ) = ε0

(τ0

τ

)4/3

, s(τ) = s0τ0

τ, T (τ) = T0

(τ0

τ

)1/3

. (3.24)

Their dependence on τ is shown in figure 3.4.Taking dissipative effects into account, the energy-momentum tensor depends on the

dissipative tensor τµν (3.12). In order to calculate all components, we derive in a waysimilar to (3.20):

uµ∂µ⊥ = 0, ∂µ⊥uµ =

1

τ. (3.25)

Here, ∂µ⊥ is defined in (3.13) and the proof of (3.25) is analogous to (3.21) and thereforeomitted. Because we are restricted to only one spatial dimension, we have τµν = 0 forµ, ν ∈ 1, 2. Since τµν is a symmetric tensor, only four components of the dissipativetensor contribute to T µν :

τ 00 = −(

43η + ξ

)1τ

sinh2 Y , τ 00 = 0 ,

τ 33 = −(

43η + ξ

)1τ

cosh2 Y , τ 33 6= 0 ,

τ 03 = −(

43η + ξ

)1τ

sinhY coshY , τ 03 = 0 .

(3.26)

Therefore, the energy-momentum tensor is no longer diagonal. Even in the local restframe, Y = 0, the spatial part describing the pressure is not isotropic, because τ 33 6= 0.Of course, this anisotropy at rest is not caused by the beam pipe along the z-axes butby the dissipative character of the fluid, parametrized by η and ξ. Here we can seeclearly that the definition of a perfect fluid in terms of Pascal’s law is not fulfilled in thecase of dissipative fluids. Furthermore, the time reversal acting on τ 03 produces a minussign, so that the energy-momentum tensor is not invariant under time reversal. Froma thermodynamical point of view, we can therefore state that dissipative fluids lead toirreversible processes. Note, that in the large proper-time limit, τ → ∞, all componentsof τµν vanish and the fluid becomes a perfect one.

30


We verify (3.26) by a straightforward calculation using (3.25) and

∆00 = − sinh2 Y , ∆33 = − cosh2 Y , ∆03 = − sinhY coshY ,

∂γ

∂t= −1

τsinh2 Y , z

∂γ

∂z= γ sinh2 Y ,

∂γ

∂τ= 0 .

(3.27)

The relations (3.27) lead to

∂0⊥u

0 =[∂0 − u0(u · ∂)

]γ(x) = [∂t − γ∂τ ] γ(x) = ∂tγ(x) = −1

τsinh2 Y

⇒ τ 00 = η

[−2

τsinh2 Y +

2

3τsinh2 Y

]− ξ

τsinh2 Y = −

(4

3η + ξ

)sinh2 Y

τX

(3.28)

For τ 33, the calculation is very similar to (3.28), but τ 03 has a different structure:

∂3⊥u

0 =[−∂z − γ

z

t∂τ

]γ = −∂zγ = −1

zγ sinh2 Y = −sinhY coshY

τ,

∂0⊥u

3 = [∂t − γ∂τ ] γz

t= ∂t

z

t2 − z2= − zt

τ 3= −γz

τ 2= −sinhY coshY

τ

⇒ τ 03 = −(

4

3η + ξ

)sinhY coshY

τX

(3.29)

Generalizing equation (3.22) we find from the longitudinal part of the energy-momentumtensor of a dissipative fluid the following differential equations for the energy and entropydensity:

∂ε

∂τ= −ε+ P

τ+

1

τ 2

(4

3η + ξ

),

∂s

∂τ= −s

τ+

43η + ξ

τ 2 T.

(3.30)

The differential equations (3.22) and (3.30) coincide with (42)–(44) in [Mur01]. For thesake of simplicity, we will drop the bulk viscosity ξ in the following numerical calculationand use again the EOS for an ultra-relativistic system with massless, non-interactingparticles as done in (3.23). As we will see in (6.84), the shear viscosity in the φ4 theoryis given by

η(τ) = η0

(T (τ)

T0

)3

, (3.31)

with temperature T and initial viscosity and temperature η0, T0. Inserting this model-dependent result into (3.30), we find using the first law of thermodynamics, Ts = ε+ P ,two coupled differential equations:

∂ε

∂τ= − 4ε

3τ+

44η0

34T 30

ε3

τ 2s3,

∂s

∂τ= −s

τ+

43η0

33T 30

ε2

τ 2s2.

(3.32)

In figure 3.5, these equations are solved numerically for different initial shear viscositiesη0. Note, that the diagram for η0 = 0 is the well-known case of a perfect fluid in figure 3.4.Already in the simplest case of a scalar theory with φ4 interaction and the EOS of an ultra-relativistic system we can observe some interesting aspects of relativistic hydrodynamics:

31


0 5 10 15 200.0

0.5

1.0

1.5

2.0

ΤΤ0

Η0 = 0

0 5 10 15 200.0

0.5

1.0

1.5

2.0

ΤΤ0

Η0 = 0.3

0 5 10 15 200.0

0.5

1.0

1.5

2.0

ΤΤ0

Η0 = 0.8

0 5 10 15 200.0

0.5

1.0

1.5

2.0

ΤΤ0

Η0 = 1

Figure 3.5: Numerical solutions of the coupled differential equation system (3.32) for a(dissipative) fluid in the ultra-relativistic limit: energy density ε/ε0 (red), entropy density s/s0

(green) and temperature T/T0 (blue). The case η0 = 0 describes the perfect fluid case. It isnot sufficient to have η0 > 0 in order to have entropy production, but the initial shear viscositymust exceed a threshold. The temperature evolution depends only moderately on η0, but theenergy and entropy density are strongly affected by the initial shear viscosity.

if the initial shear viscosity η0 is large enough, there is a huge entropy production asmentioned at the beginning of this chapter. As the diagram for η0 = 0.3 shows, it iswrong to state that it is typical for a dissipative system to produce entropy. In fact, thereis a threshold for the inital shear viscosity which determines if the system produces entropyor not. In our case the threshold is located at η0 ≈ 0.45. Furthermore, for the perfectfluid and the dissipative fluid at first order in ∂µuν , the results agree with figures 1–4 in[Mur01]. There, the second order result is shown additionally. This extended theory isdiscussed quite frequently in literature, for instance by [Mur03, Mur01, RG96, IS79, Isr76].Neglecting the second order in the construction of the dissipative tensor τµν leads to aparabolic equation of motion, which violates causality. For instance, the Navier-Stokesequation is also a parabolic differential equation with the same behavior. We are notgoing to discuss the second-order effects because we are not interested in the equation ofmotion but in the explicit calculation of the dissipative parameters η and ξ. The second-order theory would lead to a hyperbolic equation of motion, which preserves causality asdiscussed in literature [HL87, HL85].

32

3.4 Connections to Experimental Results from RHIC

3.4 Connections to Experimental Results from RHIC

Now we are going to discuss some results of the four detectors of RHIC at BrookhavenNational Laboratory [RHI]: PHOBOS & BRAHMS, STAR and, the largest one, PHENIX.In order to match the theoretical description of heavy-ion collisions using relativistichydrodynamics to experimental results, we need to introduce the pseudo rapidity, ηPS,which is determined by the geometry of the detector:

ηPS ≡ − ln tanθ

2, (3.33)

where θ is just the polar angle between the collision line along the z-axes and the producedhadrons. In the ultra-relativistic limit, the pseudo rapidity ηPS and the usual rapidity Y(3.15) coincide:

Y(3.15)=

1

2lnt+ z

t− z =1

2ln

1 + β

1− β =1

2lnE + pzE − pz

→

−−−−→m→0

1

2lnp+ pzp− pz

= − ln

√1− cos θ

1 + cos θ= − ln tan

θ

2= ηPS .

(3.34)

In figure 3.6, the inverse relation of (3.33) is shown: the smaller the angle θ, the higher therapidity in +z-direction. A baryon emitted orthogonal to the beam pipe has, of course,zero rapidity. In the Bjorken picture [YHM08, Chapter 10.3], one expects the baryonmultiplicity per rapidity to form a plateau with dN/dηPS is approximately a constant.

In [BW09] it is shown that measurements at RHIC for Au-Au collisions at center ofmass energy

√sAA = 200AGeV are well-reproduced by hydrodynamical calculations as

done in the previous section for 1 + 1 dimensions. Taking also transverse directions intoaccount leads to a very successful theory to describe heavy-ion collisions. The central-ity of the collision can be described by the centrality class. The lower the centralityclass is, the closer we are to a central collision. As shown in figure 3.7, for a decreas-ing centrality class more baryons of a certain rapidity are created as we would expectintuitively. The flanks remain unchanged under switching the centrality class, while theplateau altitude changes. The maximal plateau height depends on the collision centerof mass energy

√sAA: in [BB+04], again for Au-Au collisions, plateaus are shown for√

sAA = 19.6AGeV and√sAA = 200AGeV, hence decreasing

√s also leads to a smaller

-4 -2 0 2 40

Π

2

Π

Pseudo rapidity ΗPS

Θ=

2A

rcta

ne-

ΗP

S

Figure 3.6: Relation between pseudo rapidity ηPS and polar angle θ

33


PS

η -6 -4 -2 0 2 4 6

0

200

400

600

PHOBOS c=0-6%,...,45-50%BRAHMS c=0-5% =150MeVfT

=165MeVfTPS

ηdN/d

Figure 3.7: Pseudorapidity distribution of charged particles for centrality classes 0-6% (up-per), 6-15%, 15-25%, 25-35%, 35-45% and 45-55% (lower) calculated for the freeze-outtemperatures Tf = 165 MeV and Tf = 150 MeV (solid and dashed lines, respectively) com-pared to PHOBOS Collaboration data (dots) [BB+04]. The squares represent the BRAHMSCollaboration data for centrality 0-5% [AB+03]. (adapted from [BW09])

plateau width. Therefore the collision energy changes the flanks of the plateau. In 2009new experimental results from RHIC (PHOBOS) were published for Cu-Cu collision at√sAA = 22.4AGeV, 62.4AGeV, 200AGeV [AB+09]. By combining older results from

Au-Au collisions, the experiments using copper ions allow to get very precise ratios ofrapidity distributions because the same detector setup was used and systematic errorscancel.

An estimate for the initial energy density ε0 was derived by [Bjo83]:

εBj0 =

1

πτ0r2

dETdηPS

. (3.35)

Here r and ET denote the radius of the ion and the transverse energy, respectively. Themeasured plateaus provide numbers for dN/dηPS ≈ 1000 and dET/dηPS = f · dN/dηPS

with f ≈ 0.5 GeV. A more involved approach by [YHM08, Chapter 11.4] estimates theinitial energy density to

ε0 =3

4 3√a

(4

πτ0r2

dN

dηPS

)4/3

. (3.36)

We use the radius r ≈ r03√A, r0 = 1.5 fm, for the flattened ions, the mass A ≈ 100 and

a typical time scale τ0 ≈ 1 fm. In [YHM08, Exercise 11.5], a = 4·37π2

90is calculated. Now

we arrive atεBj

0 = 6.5 GeV/fm3, ε0 = 4.6 GeV/fm3 . (3.37)

The theoretical estimates (3.35) and (3.36) for the initial energy density produce, using theexperimental input dN/dηPS, reasonable numbers between 1 GeV/fm3 and 10 GeV/fm3.This is in agreement with the numerical estimates by [Bjo83].

34

4 Non-Equilibrium Thermodynamicsand Transport Phenomena

So far we have used the shear and bulk viscosity η and ξ as parameters of the dissipativetensor. In fact, it is possible to learn something about their values from experiments, butin this thesis we are interested in a microscopic description of the transport phenomena.In order to develop such a formalism, we consider statistical physics of non-equilibriumsystems and derive the Kubo-type formula for the shear viscosity.

4.1 Statistical Operator of Non-Equilibrium Systems

The key ingredient for calculating observables in statistical physics is the statistical opera-tor ρ, which determines expectation values via 〈 · 〉 = Tr (ρ · ). It can be constructed usingZubarev’s approch [Zub74] which describes non-equilibrium systems in the Schrodingerpicture

dρ

dt= 0 . (4.1)

Therefore we introduce the operator

B(x) ≡ F µ(x, t)T0µ(x, t)−∫ t

−∞dt′ [F µ(x, t′)∂t′T0µ(x, t′) + ∂t′F

µ(x, t′)T0µ(x, t′)] (4.2)

with the four vectorF µ(x, t) ≡ βs(x, t)u

µ(x, t) . (4.3)

Using γ = (1− v2)−1/2

, we defined the inverse proper temperature βs by

βs ≡1

Ts≡ γ

∂S

∂E=∂S

∂m. (4.4)

Since E = mγ = p0 is the temporal component of the four momentum pµ, βs is a Lorentzscalar. At rest it coincides with the usual inverse temperature β. Using this generalizedtemperature term ensures that F µ is indeed a four vector. Furthermore, we are able toconstruct a Lorentz covariant theory of non-equilibrium thermodynamics, knowing at thesame time that the fundamental concepts of temperature and heat bath in grand canonicalensembles break the Lorentz symmetry.

Using the Fundamental Theorem of Calculus shows directly that B(x) is indeed time-independent. Now, the statistical operator is defined by

ρ ≡ 1

Qexp

[−∫d3x B(x)

], (4.5)

with Q ≡ Tr exp[−∫d3x B(x)

]. Equation (4.5) defines an operator in Schrodinger pic-

ture as claimed in (4.1). Using energy-momentum conservation (3.2), ∂tT0µ = −∂iTiµ, we

35

4 Non-Equilibrium Thermodynamics and Transport Phenomena

can rewrite the integrated second term of (4.2). After integration by parts and neglectingthe surface term we arrive at:∫

d3x

∫ t

−∞dt′ [F µ∂t′T0µ − ∂t′F µT0µ] = −

∫d3x

∫ t

−∞dt′ [F µ∂iTiµ + ∂t′F

µT0µ] =

= −∫d3x

∫ t

−∞dt′ [−∂iF µTiµ + ∂t′F

µT0µ] = −∫d3x

∫ t

−∞dt′ Tµν∂

µF ν .

(4.6)

Therefore, the statistical operator (4.5) reads:

ρ(4.5)=

1

Qexp

[−∫d3x B(x)

](4.6)=

=1

Qexp

[−∫d3x FµT

0µ +

∫d3x

∫ t

−∞dt′ Tµν∂

µF ν

].

(4.7)

Note, that the operator B itself is not a Lorentz scalar, but with the additional spatialintegration, the statistical operator ρ is a Lorentz scalar. The discussion preceding figure3.3 showed that the more temperature and four velocity vary in different zones, the furtherthe system is away from global equilibrium and the more important dissipative effects are.

The constructed statistical operator (4.7) is one possible approach to non-equilibriumthermodynamics. There are other approaches to derive the shear viscosity: for instance,in [NC07] the linearized Boltzmann equation is used to describe the shear viscosity of thedeconfined phase. In fact, it is not necessary to introduce an operator B as done in (4.2),but our result (4.7) seems to be reasonable: first because it depends on the dissipativeforces ∂µF ν , a measure for the dissipative level of the system. Secondly it reproduces thewell-known statistical operator ρ0 for ideal systems at rest:

Fµ = βsuµ = βδ0µ , T 00 = ε .

⇒ ρ0 ≡ ρ|∂µF ν=0 =1

Qexp

[−β∫d3x ε

]=

1

Qe−βH .

(4.8)

Note, that ρ0 is a Lorentz scalar, since the product βH is a Lorentz scalar, too.

However, in order to decompose a given energy-momentum tensor T µν we now define:

ε ≡ uµuνTµν , p ≡ −1

3∆µνT

µν ,

Pµ ≡ ∆µρuσTρσ , πµν ≡

(∆µν∆ρσ −

1

3∆µρ∆νσ

)T ρσ .

(4.9)

Using exclusively the definitions (4.9), the energy-momentum tensor can be written inthe following way:

T µν = εuµuν − p∆µν + P µuν + P νuµ + πµν . (4.10)

Evaluating at rest identifies ε = T 00 = ε as the energy density, hence its nomenclature isvalid. With the abbreviation

T ≡ 1

3

(T 11 + T 22 + T 33

)(4.11)

36

4.2 Kubo-Type Formulas for Transport Coefficients

we arrive at the dissipative tensor

πµν =

0 0 0 0

0 T 11 − T 0 0

0 0 T 22 − T 0

0 0 0 T 33 − T

6= 0 . (4.12)

Note, that in the general dissipative case we find p = T 6= P and therefore πµν 6= 0. Only

for the perfect fluid case (3.4) with T ii = P , (i = 1, 2, 3), we find p = P , hence πµν = 0.

The tensor πµν is traceless, because Tr πµν = 0, or we can evaluate in a general frame

πµµ =

(∆µρ∆

µσ −

1

3∆µµ∆ρσ

)T ρσ

(3.14)= 0 , (4.13)

with ∆µν = gµν − uµuν defined in (3.13). From this we observe that the tensor πµν is ameasure for the dissipative character of a system. Remembering the traceless part of thedissipative tensor τµν (3.12), we can now proceed to find a relationship between η, so farjust a parameter, and πµν , governed by the energy-momentum tensor of a microscopicquantum field theory.


In this section we will derive the Kubo-type formula for the shear viscosity η(x, t) usinglinear-response theory within the framework of statistical physics of non-equilibrium sys-tems. The Kubo formulas are general fluctuation-dissipation theorems and were derivedfirst in [Kub57]. We start with the non-equilibrium statistical operator (4.7):

ρ =1

Qexp

[−∫d3x B(x)

]≡ 1

Qe−A+B , (4.14)

with

A ≡∫d3x′ F µ(x′, t)T0µ(x′, t)

(4.8)= βH ,

B ≡∫d3x′

∫ t

−∞dt′ Tµν(x

′, t′)∂µF ν(x′, t′) .

(4.15)

Using a version of the Baker-Campbell-Hausdorff formula for non-abelian operators Aand B,

e−A+B = e−A(

1 +

∫ 1

0

dy eAyBe−Ay +O(B2)

), (4.16)

the statistical operator (4.14) reads at leading order in the dissipative forces ∂µF ν :

ρ =e−A+B

Tr (e−A+B)≈ e−A

Tr (e−A) + Tr (e−AB)

(1 +

∫ 1

0

dy eAyBe−Ay)

(4.8)=

=ρ0

1 + 〈B〉0

(1 +

∫ 1

0

dy eAyBe−Ay)≈ ρ0

(1 +

∫ 1

0

dy eAyBe−Ay − 〈B〉0).

(4.17)

37


Therefore, the linear response of the energy-momentum tensor Tµν to the dissipative forces∂σF ρ is given by:

〈Tµν(x, t)〉 = Tr (ρ Tµν(x, t)) = 〈Tµν(x, t)〉0 +

+

∫d3x′

∫ t

−∞dt′∫ 1

0

dy 〈Tµν(x, t)[eβHyTρσ(x′, t′)e−βHy − 〈Tρσ(x′, t′)〉0

]∂ρF σ(x′, t′)〉0 ≡

≡ 〈Tµν(x, t)〉0 +

∫d3x′

∫ t

−∞dt′ (Tµν(x, t), Tρσ(x′, t′)) ∂ρF σ(x′, t′) .

(4.18)In the last line we defined the correlator (Tµν(x, t), Tρσ(x′, t′)), which reads in general

(X, Y ) ≡∫ 1

0

dy 〈X[eβHyY e−βHy − 〈Y 〉0

]〉0 . (4.19)

Using the decomposition of the energy-momentum tensor (4.10), we can decompose thecontracted dissipative forces Tρσ∂

ρF σ in (4.18), too:

Tρσ∂ρF σ = (εuρuσ − p∆ρσ + Pρuσ + pρuσ + πρσ) ∂ρF σ =

= ε(u · ∂)βs + εuρuσβs∂ρuσ︸︷︷︸

=0

−pβs(∂ · u)−p(u · ∂)βs + p(u · ∂)βs︸︷︷︸=0

+Pρuσβs∂ρuσ︸︷︷︸

=0

+

+ Pρ∂ρβs + Pσuρβs∂

ρuσ + Pσuρuσ∂ρβs︸︷︷︸

=0

+πρσβs∂ρuσ + πρσu

σ∂ρβs︸︷︷︸=0

=

= ε(u · ∂)βs − pβs(∂ · u) + βs(P · ∂) + βsPσ(u · ∂)uσ + βsπρσ∂ρuσ .

(4.20)In this derivation we have used the orthogonality relations uµ∂

νuµ = 0, Pµuµ = 0 and

πµνuµ = 0, resulting directly from their definition. The contractions u · ∂ = ∂

∂τand

∂ ·u = 1τ

have been derived in (3.20). Due to energy-momentum conservation, ∂µTµν = 0,

and the linearity of the trace operator, we find with 〈Pµ〉0 = 0 and 〈πµν〉0 = 0:

0 = uν∂µ〈T µν〉0(4.10)= (u · ∂)〈ε〉0 + h(∂ · u) = (u · ∂)βs ·

∂〈ε〉0∂βs

+ h (∂ · u) , (4.21)

where the enthalpy density h = ε+ p has been used. The last step is allowed because theenergy density depends only on temperature, hence the chain rule is applicable. Usingthe thermodynamic relation

h = T∂〈p〉0∂T

= −βs∂〈p〉0∂βs

, (4.22)

we can express the temperature derivative (u · ∂)βs in terms of a four-vector derivative,

(u · ∂)βs(4.21)= − ∂βs

∂〈ε〉0h (∂ · u)

(4.22)= βs

∂〈p〉0∂〈ε〉0

(∂ · u) . (4.23)

Using the contractions (3.20), this relation reads:

τ∂βs∂τ

= βs∂〈p〉0∂〈ε〉0

. (4.24)

Now we can use (4.23) for the decomposition of Tρσ∂ρF σ in (4.20) and arrive at

Tρσ∂ρF σ = βsp

′(∂ · u) + βs(P · ∂) + βsPρ(u · ∂)uρ + βsπρσ∂ρuσ =

= βsp′(∂ · u) + βsPρ

[β−1s ∂ρβs + (u · ∂)uρ

]+ βsπρσ∂

ρuσ ,(4.25)

38


where we have introduced the abbreviation

p′ ≡ ε∂〈p〉0∂〈ε〉0

− p . (4.26)

Since the correlator (X, Y ), defined in (4.19), must vanish if X and Y are tensors of dif-ferent ranks, only (p(x), p(x′)), (P µ(x), P ν(x′)) and (πµν(x), πρσ(x′)) must be considered.This is known as Curie’s Theorem, discussed briefly in appendix A.4.

For our purpose it turns out that the ππ-correlator exclusively determines the shearviscosity η, hence we discuss this correlator in detail: because πµν is traceless and sym-metric, Πµνρσ(x, x′) ≡ (πµν(x), πρσ(x′)) must be a tensor of fourth rank with the followingproperties:

Πµνρσ = Πνµρσ = Πµνσρ = Πνµσρ , Πµµρσ = Πµνρρ = 0 . (4.27)

The building block for constructing this tensor is just ∆µν , which is a combination ofthe metric gµν and the four velocity uµ (3.13). The following ansatz with some unknownscalar function f(πµν , πρσ) meets the conditions (4.27):

Πµνρσ(x, x′) = f(πµν(x), πρσ(x′))

(∆µρ∆νσ + ∆µσ∆νρ − 2

3∆µν∆ρσ

). (4.28)

In general, if a scalar function f depends on a second rank tensorXµν evaluated at differentspacetime points, f(Xµν(x), Xρσ(x′)), there are only 3 = 1

2

(42

)possible contractions:

Tr X(x)Tr X(x′), Xµν(x)Xµν(x′) and Xµν(x)Xνµ(x′). (4.29)

In our case we have Xµν = πµν , a traceless and symmetric tensor, hence f(x, x′) =f(πµν(x)πµν(x

′)). In order to determine f explicitly, we evaluate Πµνµν(x, x

′):

(πµν(x), πµν(x′)) = f(πµν(x)πµν(x

′))

(∆µµ∆ν

ν + ∆µν∆ν

µ −2

3∆µν∆µν

)(3.14)=

= f(πµν(x)πµν(x′)) · 10 .

(4.30)

This yields the complete form of the ππ-correlator:

Πµνρσ(x, x′) =1

10

(παβ(x), παβ(x′)

)(∆µρ∆νσ + ∆µσ∆νρ − 2

3∆µν∆ρσ

). (4.31)

Now, using the explicit form (4.31), we are able to determine the linear response of theπ-tensor to the dissipative forces:

〈πµν(x, t)〉 (4.18)=

∫d3x′

∫ t

−∞dt′ (πµν(x, t), Tρσ(x′, t′)) ∂ρF σ(x′, t′)

(4.25)=

= βs

∫d3x′

∫ t

−∞dt′ (πµν(x, t), πρσ(x′, t′)) ∂ρuσ(x′, t′)

(4.31)=

=βs10

∫d3x′

∫ t

−∞dt′(παβ(x, t), παβ(x′, t′)

)·

·(

∆µρ∆ν

σ + ∆µσ∆ν

ρ −2

3∆µν∆ρσ

)∂ρuσ(x′, t′)

(3.13)=

= η(x, t)

(∂µ⊥u

ν + ∂ν⊥uµ − 2

3∆µν(∂⊥ · u)

).

(4.32)

39


Substituting Tρσ∂ρF σ in the first line, we used the fact, that correlators between tensors of

different rank vanish, hence only the last term in (4.25) contributes to the linear response.Furthermore, in the last line, we identified the same structure as in the parametrizationof the dissipative tensor τµν (3.12). There, η was defined as the coefficient of the tracelesspart. Now we are able to determine the shear viscosity within a microscopic theory,encoded by the energy-momentum tensor T µν , thus by the π-tensor:

η(x, t) =βs10

∫d3x′

∫ t

−∞dt′ (πµν(x, t), πµν(x

′, t′)) . (4.33)

In order to find a calculable form of the shear viscosity, we transform the correlator(πµν(x, t), πµν(x

′, t′)) into a retarded one. The following calculation is valid in the largetime limit t t′. This is physically motivated by the fact that linear-response theory,used the first time in (4.18), is a reasonable approximation only if the local equilibriumsystem is close to the global equilibrium state and dissipative forces, measured by ∂µF ν ,are small. In fact, (4.33) ensures that t′ ∈ (−∞, t), hence t t′ is just an approximation.In this large time limit the expectation value of a product factorizes:

(πµν(x, t), πµν(x′, t′))

(4.18)=

∫ 1

0

dy 〈πµν(x, t)(eβHyπµν(x

′, t′)e−βHy − 〈πµν(x′, t′)〉0)〉0 τ=βy

=

=1

β

∫ β

0

dτ (〈πµν(x, t)πµν(x′, t′ − iτ)〉0 − 〈πµν(x, t)〉0〈πµν(x′, t′)〉0) ≈

≈ 1

β

∫ β

0

dτ 〈[πµν(x′, t′), πµν(x, t)]〉0 θ(t− t′) .(4.34)

In the last step we used besides the large-time limit the peridocity of the trace,

〈X(t)Y (t′ + iτ)〉0 = Tr[ρ0 X(t)e−HτY (t′)eHτ

]= 〈Y (t′)X(t)〉0 , (4.35)

for arbitrary operators X and Y . Neglecting the surface πµν(x′,−∞), we arrive at:

(πµν(x, t), πµν(x′, t′))

(4.34)≈ − 1

β

∫ β

0

dτ

∫ t′

−∞dt 〈πµν(x, t) d

dtπµν(x

′, t)〉0 θ(t− t′) =

= − 1

iβ

∫ β

0

dτ

∫ t′

−∞dt 〈 d

dτπµν(x

′, t+ iτ)πµν(x, t) 〉0 θ(t− t′) =

= − 1

iβ

∫ t′

−∞dt 〈πµν(x′, t+ iβ)πµν(x, t)− πµν(x′, t)πµν(x, t)〉0 θ(t− t′) =

= − 1

β

∫ t′

−∞dt 〈[πµν(x, t), πµν(x′, t)]〉0(−i)θ(t− t′) (5.62)

=

= − 1

β

∫ t′

−∞dt 〈πµν(x, t), πµν(x′, t)〉ret .

(4.36)

Plugging this into (4.33) leads to the shear viscosity η(x, t) in terms of a retarded corre-lator. The prefactor βs/β = γ → 1 reduces to one when transforming to local restframe.In the following we consider the shear viscosity only at rest. There, the shear viscosity isgiven by the Kubo-type formula

η(x, t) = − 1

10

∫d3x′

∫ t

−∞dt′∫ t′

−∞dt 〈πµν(x, t), πµν(x′, t)〉ret . (4.37)

40


This result is a factor two smaller compared to [HST84], due to our definition of the shearviscosity coefficient (3.13) and the matching (4.32). With the here used definition wecoincide with [YHM08] and [HDL06].

For completeness we state also the expression for the bulk viscosity ξ, the second param-eter of the dissipative tensor describing its non-traceless part and for the heat conductivityκ which can be neglected for small chemical potentials:

ξ(x, t) = −∫d3x′

∫ t

−∞dt′∫ t′

−∞dt 〈p′(x, t), p′(x′, t)〉ret , (4.38)

κ(x, t) =1

3

∫d3x′

∫ t

−∞dt′∫ t′

−∞dt 〈P µ(x, t), Pµ(x′, t)〉ret , (4.39)

where p′ and P µ were defined in (4.26) and (4.9), respectively. Note, that all quantities,the shear viscosity (4.37), the bulk viscosity (4.38) and the heat conductivity (4.39) aregoverned by a retarded correlator. In order to evaluate the retarded correlator for the shearviscosity we will use the power of (quantum) field theories, in particular a perturbative(φ4 theory in the next chapter), and an effective approach (chiral perturbation theory inchapter 7).

41

5 Bosonic Thermal Field Theory andImaginary Time Formalism

In this chapter we give an introduction to the imaginary time formalism in order todescribe field theories at finite temperatures T > 0. In this approach we deal with fieldtheories on a d + 1-dimensional Euclidean cylinder Rd × [0, β], β ≡ 1/T , where the timeaxis is compactified. There is another, equivalent approach using a real time formalismwhich leads to a doubling of all field degrees of freedom [KG06], but will not be consideredin this thesis. We discuss scalar fields within φ4 theory and pseudoscalar pions within theframework of chiral perturbation theory, hence we restrict the main discussion to bosons.

5.1 Path Integral Approach

Let |Ψ(t)〉S denote a quantum state in the Schrodinger picture and H be a time-independentHamiltonian. This state is related to a state |Ψ〉H in the Heisenberg picture by

|Ψ(t)〉S = exp(−iHt

)|Ψ〉H = U(t, 0)|Ψ〉H , (5.1)

where |Ψ〉H is time-independent and

U(tf , ti) ≡ exp(−iH(tf − ti)

)(5.2)

is the time-evolution operator, defined for tf ≥ ti. Since the position space states form acomplete set |q〉 : q ∈ Rd, the vectors defined by

|q, t〉 ≡ U †(t, 0)|q〉 (5.3)

are also complete for all t ∈ R. Then, the wavefunction (in position space) is given by

Ψ(q, t) ≡ 〈q|Ψ(t)〉S = 〈q, t|U †(t, 0)|Ψ(t)〉S = 〈q, t|Ψ〉H . (5.4)

Let us define the propagator

K(qf , tf ;qi, ti) ≡ 〈qf , tf |qi, ti〉 = 〈qf |U(tf , ti)|qi〉 , (5.5)

where the equal sign arises from the group property U(t′′, t) U(t, t′) = U(t′′, t′) of time-evolution operators. To call K a propagator is also justified by

Ψ(qf , tf ) = 〈qf , tf |Ψ〉H =

∫Rdddqi 〈qf , tf |qi, ti〉〈qi, ti|Ψ〉H =

=

∫Rdddqi K(qf , tf ;qi, ti)Ψ(qi, ti) .

(5.6)

43

5 Bosonic Thermal Field Theory and Imaginary Time Formalism

Here we used the completeness of |q, t〉 : q ∈ Rd for any t ∈ R. Assuming that theHamiltonian of the system is given by

H =p2

2m+ V (q) , (5.7)

we first calculate the propagator for a free particle H0 = H(V = 0), using Gaussianintegration in d dimensions:

K0(qf , tf ;qi, ti) = 〈qf | exp

(−i p

2

2m(tf − ti)

)|qi〉 =

=

(m

2πi(tf − ti)

)d/2exp

(im

2

(qf − qi)2

tf − ti

).

(5.8)

In order to construct the path integral, we introduce a time segmentation: dividing theinterval [ti, tf ] into n+ 1 equidistant segments of length ε = (tf − ti)/(n+ 1), (n denotesthe number of intersection points), the propagator reads

K(qf , tf ;qi, ti)(5.5)= 〈qf | exp

(−iH(tf − ti)

)|qi〉 =

= 〈qf | exp

(−iH

n∑j=0

tf − tin+ 1

)|qi〉 =

=n∏j=0

∫Rdddqj 〈qj+1| exp

(−iHε

)|qj〉 ,

(5.9)

where we used the identification q0 ≡ qi, qn+1 ≡ qf and∫Rdddq0 ≡ 1. Factorizing the

exponential of every single factor yields

〈qj+1| exp(−iεH

)|qj〉 = 〈qj+1| exp

(−iεH0

)exp

(−iεV (q)

)|qj〉+O(ε2) =

= 〈qj+1| exp(−iεH0

)|qj〉 exp

(−iεV (qj)

)+O(ε2)

(5.8)=

=( m

2πiε

)d/2exp

iε

[m

2

(qj+1 − qj

ε

)2

− V (qj)

]+O(ε2) .

(5.10)In the limit n → ∞ (ε → 0) this factorization becomes exact and the propagator iswritten as a path integral:

K(qf , tf ;qi, ti) = limn→∞

n∏j=0

∫Rdddqj

( m

2πiε

)d/2exp

iε

[m

2

(qj+1 − qj

ε

)2

− V (qj)

]=

= limn→∞

(( m

2πiε

)d(n+1)/2n∏j=0

∫Rdddqj

)exp

iε

n∑j=0

[m

2

(qj+1 − qj

ε

)2

− V (qj)

].

(5.11)The prefactor nn is super-exponentially divergent for n→∞, but it is assumed that it ispossible for this prefactor to be interpreted as the integral measure of the path integral:∫

Dq ≡ limn→∞

(( m

2πiε

)d(n+1)/2n∏j=0

∫Rdddqj

). (5.12)

44

5.1 Path Integral Approach

Note, that the path integral measure has no mass dimension, since [ε] = −1, hence[∫Dq]

=[(m/ε)d(n+1)/2

]− d(n+ 1) = 0 . (5.13)

In contrast to the path integral measure, for the exponential function in (5.11) the con-tinuum limit n→∞ is well-defined, since it is just the definition of the Riemann integral:

limn→∞

exp

iε

n∑j=0

[m

2

(qj+1 − qj

ε

)2

− V (qj)

]= exp

i

∫ tf

ti

dt L[q(t)]

. (5.14)

Altogether we can express the propagator K as a path integral, an operator-free repre-sentation of the transition amplitude:

K(qf , tf ;qi, ti) =

∫Dq exp

i

∫ tf

ti

dt L[q(t)]

=

∫Dq eiS[q] . (5.15)

The precise mathematical definition of the path integral is problematic, because the inte-gral measure is actually divergent, but the path integral provides a quite intuitive way tounderstand a quantum-mechanical transition amplitude: it is just the weighted coherentsum of all paths with q(ti) = qi and q(tf ) = qf , which reflects the quantum characterof the system as it can be observed, for instance, in the double slit experiment. In thispicture, the path integral can be interpreted as a generalization to infinitely many slitsin infinitely many plates. In the classical limit the path with stationary action gives themost important contribution and the path integral collapses to a delta sum.

To give a more rigorous mathematical meaning to the path integral, let us briefly look atthe theory of stochastic processes [Kle08]: every single path q : [ti, tf ]→ Rd contributingto K is supposed to be a d-dimensional Brownian motion. This is reasonable for instancebecause the path t 7→ q(t) ∈ Rd is continuous. With this we identify

∫Dq eiS[q] as Ito

integral ∫ tf

ti

Hτ dqτ , (5.16)

with Hτ : Rd × [ti, tf ] → R. Here it is assumed that the Lagrangian has no explicittime dependence. Looking at the Lagrangian L =

∫Rdddx L(x), we note that L contains

derivatives of q describing the kinetic part of the theory. This is in contradiction toour assumption that q is a Brownian motion, since the Paley-Wiener-Zygmund Theoremstates, that every realization of q is nowhere differentiable [Kle08, theorem 21.17]. Thismeans that the path integral sums over all typical continuous functions with the boundaryconditions q(ti) = qi and q(tf ) = qf , since a random continuous function is almost surelynowhere differentiable. To avoid this contradiction, we may interpret the field derivativesas weak derivatives, which are defined for any distribution. However, in this thesis wewill use the path integral just as a formal generalization of the Lebesgue integral. In allcalculable observables the awkward integral measure drops out, resulting in well-definedphysical quantities.

45


5.2 Statistical Physics and Matsubara Formalism

The central object in statistical physics is the partition function Z from which all oberv-ables can be derived. Introducing β = 1/T , the partition function of the grand canonicalensemble with conserved particle numbers Ni reads

Z = Tr exp−β(H − µiNi)

. (5.17)

From this, the statistical operator is defined by

ρ ≡ 1

Zexp

−β(H − µiNi)

. (5.18)

The trace in (5.17) can be evaluated using position space states |q〉 : q ∈ Rd:

Z =

∫Rdddq 〈q|e−β(H−µiNi)|q〉 ≡

∫Rdddq k(q, β) . (5.19)

Comparing the definition of K (5.5) and k (5.19) we obtain the relation

k(q, β) = K(q,−iβ;q, 0) . (5.20)

This means that the fields are forced to be periodic in imaginary time with ti = 0 andtf = −iβ ∈ iR. Using the path integral representation of K (5.15) and doing a Wickrotation τ ≡ it ∈ R, we can express the partition function (5.17) as

Z =

∫Rdddq K(q,−iβ;q, 0) =

=

∫Rdddq

∫q(0)=q(−iβ)

Dq exp

i

∫ −iβ0

dt L[q(t)]

≡

≡∫q(0)=q(−iβ)

Dq exp

∫ β

0

dτ L[q(−iτ)]

≡∮Dq eSE [q] .

(5.21)

We suppress the overall prefactor of Z, because it does not change the thermodynamics.The measure Dq denotes the additional integration

∫Rdddq, and the circle integral denotes

the β-periodicity of the paths: q(t= 0) = q(t=−iβ) ⇔ q(τ = 0) = q(τ = β). In thefollowing we will suppress the tilde on the integral measure. The partition function ofstatistical physics converges in the limit T → 0 to the partition function of a Euclideanfield theory in one real time and d real space dimensions.

In order to express the partition function in terms of a path integral (5.21), we haveto compactify the Wick-rotated time axis to [0, β]τ . Hence, the Fourier expansion of thetemporal part of a general bosonic or fermionic field Ψ ∈ φ, ψ reads

Ψ(τ) =1

β

∑n∈Z

Ψne−iωnτ . (5.22)

Only countably many frequencies ωn instead of countinuous modes k in the spatial partappear. These so-called Matsubara frequencies need to fulfill the periodic boundary con-dition which states that at τ = 0 and τ = β the same physical state is realized. Theboundary conditions for bosons and fermions read:

ωBn = 2nπT , ωFn = (2n+ 1)πT , (5.23)

46

5.2 Statistical Physics and Matsubara Formalism

with n ∈ Z. In order to see this, we define in a common way the time ordering symbolfor bosons and fermions by

T Bτ[φ(q1, τ1)φ†(q2, τ2)

]≡ φ1φ

†2θ(τ1 − τ2) + φ†2φ1θ(τ2 − τ1) ,

T Fτ[ψ(q1, τ1)ψ(q2, τ2)

]≡ ψ1ψ2θ(τ1 − τ2)− ψ2ψ1θ(τ2 − τ1) .

(5.24)

For fields Ψ ∈ φ, ψ we also define the thermal Green’s function in the same way as inquantum field theories.

Gβ(q1,q2; τ1, τ2) ≡⟨Tτ[Ψ(q1, τ1)Ψ(q2, τ2)

]⟩=

= Tr(ρ Tτ

[Ψ(q1, τ1)Ψ(q2, τ2)

]).

(5.25)

Let us now consider the Green’s function for a field Ψ for 0 = τ2 < τ1 = τ < β. Using thecyclic property of the trace and the abbreviation K ≡ H − µiNi, we can derive:

Gβ(q1,q2; τ, 0) =1

ZTr(e−βKΨ(q1, τ)Ψ(q2, 0)

)=

=1

ZTr(

Ψ(q2, 0)e−βKΨ(q1, τ))

=

=1

ZTr(e−βKeβKΨ(q2, 0)e−βKΨ(q1, τ)

)=

=1

ZTr(e−βKΨ(q2, β)Ψ(q1, τ)

)(5.24)=

=1

ZTr(±e−βK Tτ

[Ψ(q1, τ)Ψ(q2, β)

])=

= ±Gβ(q1,q2; τ, β) .

(5.26)

The bosonic time ordering operator induces the plus sign, whereas the fermionic caseresults in the minus-sign. Since q1 and τ are arbitrary, we end with

φ(q2, 0) = φ(q2, β) , ψ(q2, 0) = −ψ(q2, β) , (5.27)

which implies our claim (5.23) for the Matsubara frequencies. In the following we will usethis result to calculate partition functions and we want to emphasize that in (5.22)−(5.27)the bosonic fields φ are c-numbers and the fermionic fields ψ are Grassmann-(a)-numbers,but no operators.

Later on we need to evaluate sums over the bosonic Matsubara frequencies, hence it isuseful to state here the master formulas for this issue. For x /∈ iZ we have

S(1)(x) ≡∑n∈Z

1

in+ x= 2π

(1

2+

1

e2πx − 1

), (5.28)

and for x, y /∈ Z and y 6= x we define

S(2)(x, y) ≡∑n∈Z

1

(n+ x)(n+ y)=π(cotπx− cotπy)

y − x . (5.29)

From this, the physically motivated Matsubara sums are just corollaries:

T∑n∈Z

1

iωn + ω=

1

2+ n(ω) , (5.30)

47


T∑n∈Z

1

ω2n + ω2

=1

ω

(1

2+ n(ω)

), (5.31)

where n(ω) is the Bose distribution:

n(ω) =1

eβω − 1. (5.32)

The identities (5.28)-(5.31) are proven in appendix A.2.

5.3 Neutral Scalar Field at Finite Temperature

The Lagrangian of a neutral scalar field φ(q, t) in Minkowski space, d = 3, reads

L =1

2(∂µφ)(∂µφ)− 1

2m2φ2 − U(φ) . (5.33)

Considering the simplest renormalizable interacting theory, the interaction term is givenby U(φ) = λφ4 for some λ ∈ (0, 1). We want to emphasize that φ is just a c-numberand not an operator. This holds also in the quantized theory, since the path integralapproach provides an operator-free representation of the transition amplitude. Due to itsinteraction term, this theory is called φ4 theory.

5.3.1 Free Neutral Scalar Fields

In order to calculate the partition function of a free neutral scalar field at finite temper-ature, (5.21) tells us to determine the Euclidean action SE:

SE[φ] =

∫ β

0

dτ

∫d3q LE[τ ] =

∫ β

0

dτ

∫d3q L[−iτ ] =

=1

2

∫ β

0

dτ

∫d3q

[(∂φ

∂(−iτ)

)2

− (∇φ)2 −m2φ2

]=

= −1

2

∫ β

0

dτ

∫d3q

[(∂φ

∂τ

)2

+ (∇φ)2 +m2φ2

]=

= −1

2

∫ β

0

dτ

∫d3q φ

[− ∂2

∂τ 2−∆ +m2

]φ .

(5.34)

In the integration by parts, the bosonic boundary condition (5.27) was used to eliminatethe surface contribution. In order to calculate the partition function Z (5.21), we use thegeneralization of the Gaussian integral for the real path integral:∫

Dφ exp

−1

2

∫d4q φ(q) O φ(q)

=(

det O)−1/2

. (5.35)

Here we refer to rescaled dimensionless quantities φ = βφ, O = β2O and q = q T , sincethe partition function Z has no mass dimension and the temperature T , equivalentlyβ = 1/T , is the only intrinsic mass scale in thermal field theory. Hence, in (5.34) we dealwith the dimensionless operator

O = β2O = β2

[− ∂2

∂τ 2−∆ +m2

]FT⇒ β2

[ω2n + p2 +m2

]. (5.36)

48


The last step here indicates that the Fourier transform along the compactified imaginarytime interval replaces −∂2/∂τ 2 by ω2

n, the squared Matsubara frequencies. In the followingwe will use the identity

ln[(2πn)2 + β2ω2

]= ln

(1 + (2πn)2

)+

∫ β2ω2

1

dθ2

θ2 + (2πn)2, (5.37)

but the β-independent term gives only a non-physical contribution to the partitition func-tion and will be dropped. The determinant and the trace of an operator are independentof their basis, so we are allowed to evaluate them in momentum space and obtain withthe definition ω2 ≡ p2 +m2:

lnZ(5.21)= ln

∮Dφ eSE [φ] (5.35)

= ln(det β2

[ω2n + p2 +m2

])−1/2=

= −1

2Tr ln

[β2ω2

n + β2ω2] (5.23)

= −V2

∫d3p

(2π)3

∑n∈Z

ln((2πn)2 + β2ω2

) (5.37)=

= −V2

∫d3p

(2π)3

∫ β2ω2

1

dθ2∑n∈Z

1

θ2 + (2πn)2

(5.31)=

= −V2

∫d3p

(2π)3

∫ β2ω2

1

dθ2 1

θ

(1

2+ n(θT )

)=

= −V2

∫d3p

(2π)3

(βω + 2 ln(1− e−βω)

).

(5.38)

In the calculation we dropped all β-independent additive terms and neglected an overallprefactor coming from the rescaling to dimensionless quantities. For the last line we useddθ2 = 2θdθ. Knowing the partition function for the free φ4 theory, we can derive allthermodynamic quantities by the well-known relations

P =∂(T lnZ)

∂V, Ni =

∂(T lnZ)

∂µi, S =

∂(T lnZ)

∂T. (5.39)

For instance, in the ultra-relativistic limitm→ 0, we derive the well-known T 4-dependenceof the pressure of a non-interacting spin-0-bosons:

P =∂(T lnZ)

∂V

(5.38)=

T

VlnZ =

= T

∫d3p

(2π)3

(−1

2βp− ln(1− e−βp

)=

= −4πT

8π3

∫ ∞0

dp p2 ln(1− eβp) =

(− T

2π2

)(− π4

45β3

)=π2

90T 4 .

(5.40)

For a non-interacting photon gas, an additional factor of two arises due to the multiplicityof massless spin-1-bosons. In the calculation (5.40) we have dropped the divergent term,which describes the pressure of the quantum vacuum. This contribution is observable onlyin embedded systems as seen for instance in the Casimir effect or in General Relativity,where the vacuum energy contributes to the energy-momentum tensor.

Now we define the free Matsubara propagator ∆(0)M (q − q′) for q, q′ ∈ [0, β]τ × R3 in

thermal φ4 theory by(− ∂2

∂τ 2−∆ +m2

)∆

(0)M (q − q′) ≡ δ(4)(q − q′) . (5.41)

49


Performing a Fourier transformation of this definition we arrive at:

∆(0)M (q − q′) =

∑n∈Z

∫d3p

(2π)3∆

(0)M (ωn,p) exp −i [p(q− q′) + ωn(τ − τ ′)] , (5.42)

with the free Matsubara propagator in momentum space:

∆(0)M (ωn,p) =

1

ω2n + p2 +m2

. (5.43)

The generating functional of the thermal neutral scalar field is defined in a similar wayas in quantum field theory:

Z[J ] ≡∮Dφ exp

SE[φ] +

∫ β

0

dτ

∫d3q J(q)φ(q)

. (5.44)

If the external source J vanishes, the generating functional is again the partition function(5.21) of a thermal theory, hence Z[0] = Z. Using (5.34) for the Euclidean action, thegeneralized Gaussian integral∫

dξ exp

(−1

2(ξ,Oξ)− (b, ξ)

)= (det O)−1 exp

(1

2(b,O−1b)

), (5.45)

and the identity

J(q) =

∫ β

0

dτ ′∫d3q′δ(q − q′)J(q′) ,

we can rewrite the generating functional (5.44) in the following form:

Z0[J ] =

∮Dφ exp

−1

2

∫ β

0

dτ

∫d3q

(φ(q)

[− ∂2

∂τ 2−∆ +m2

]φ(q)− J(q)φ(q)

)=

= Z0[0] exp

1

2

∫ β

0

dτ

∫d3q J(q)

[− ∂2

∂τ 2−∆ +m2

]−1

J(q)

=

= Z0[0] exp

1

2

∫ β

0

dτ

∫d3q

∫ β

0

dτ ′∫d3q′ J(q)∆

(0)M (q − q′)J(q′)

.

(5.46)In the last line we just used the definition of the Matsubara propagator (5.41). Re-membering the connection between the n-point function and the generating functional inquantum field theory, we find similary for any thermal scalar theory

Gβ(q1, . . . , qn)(5.25)= Tr (ρ Tτ [φ(x1) . . . φ(xn)]) =

=1

Z

∮Dφ (φ(x1) . . . φ(xn)) eSE [φ] =

=1

Z

(n∏k=1

δ

δJ(xk)

)Z[J ]

∣∣∣∣∣J=0

.

(5.47)

In the case of a free neutral scalar field we get for the two-point function (5.25)

Gβ(q1, q2)(5.47)=

1

Z0

δ

δJ(q1)

δ

δJ(q2)Z0[J ]

∣∣∣∣J=0

(5.46)= ∆

(0)M (q1 − q2) . (5.48)

Also in the case of interacting fields, the Matsubara propagator is just the thermal two-point Green’s function.

50


5.3.2 Neutral Fields with φ4 Interaction

Taking interactions into account, we can write the generating functional (5.44) in a formwith exponentiated derivative operators, inspired by (5.47). This is possible as long asthe Lagrangian Lint is an analytic function of φ, usually a polynomial:

Z[J ] =

∮Dφ exp

∫ β

0

dτ

∫d3q (L0(φ) + Lint(φ) + J(q)φ(q))

=

= exp

∫ β

0

dτ

∫d3q Lint

(δ

δJ(q)

)Z0[J ] .

(5.49)

From the four-point function we can extract the Feynman rule for the vertex in thermalφ4 theory, hence we expand the exponential in (5.49) for Lint(φ) = −λφ4 up to first orderin λ:

exp

∫ β

0

dτ

∫d3q Lint

(δ

δJ(q)

)= 1− λ

∫ β

0

dτ

∫d3q

(δ

δJ(q)

)4

+O(λ2) . (5.50)

The result for the four-point vertex reads:

= −λ · βδ(

4∑i=1

ωni

)· (2π)3δ(3)

(4∑i=1

pi

). (5.51)

The temporal part has a Kronecker delta, whereas the three dimensional spatial parthas a delta function. Both arise from Fourier transforming the identity and they ensureenergy-momentum conservation at the vertex.

For an internal line the free Matsubara propagator is given by (5.43). Furthermore, wehave to integrate over all internal lines by

loop =

∫d4p

(2π)4= T

∑n∈Z

∫d3p

(2π)3. (5.52)

The discrete sum arises from the compact time dimension τ ∈ [0, β]τ .Now we are prepared to go back to the partition function of φ4 theory. The interac-

tion term can generally be separated from the non-interacting part in the following way,suppressing the additional index E for the Euclidean action:

Z(5.21)=

∮Dφ exp S0 + Sint =

∮Dφ eS0

∞∑n=0

1

n!Snint =

=

∮Dφ eS0

[1 +

∞∑n=1

∮Dφ eS0 1

n!Snint∮

Dφ eS0

].

(5.53)

The logarithm of the partition function is then decomposed as

lnZint = ln

[1 +

∞∑n=1

∮Dφ eS0 1

n!Snint∮

Dφ eS0

]= ln

[1 +

∞∑n=1

1

n!〈Snint〉0

]=

=∞∑k=1

(−1)k+1

k

(∞∑n=1

1

n!〈Snint〉0

)k

≡ lnZ1 + lnZ2 + . . .

(5.54)

51


The 〈·〉0 denote the fact that the thermal expectation value has to be calculated withrespect to the non-interacting theory. The lnZi collect all contributions to lnZint of orderO(λi). We list the first four terms explicitly:

lnZ1 = 〈Sint〉0 ,

lnZ2 =1

2

⟨S2

int

⟩0− 1

2〈Sint〉20 ,

lnZ3 =1

6

⟨S3

int

⟩0− 1

2

⟨S2

int

⟩0〈Sint〉0 +

1

3〈Sint〉30 ,

lnZ4 =1

24

⟨S4

int

⟩0− 1

6

⟨S3

int

⟩0〈Sint〉0 −

1

8

⟨S2

int

⟩2

0+

1

2〈Sint〉20

⟨S2

int

⟩0− 1

4〈Sint〉40 .

(5.55)

To lnZi contribute 2i−1 terms of the form∏m

j=1〈Snjint〉

lj0 with

∑mj=1 njlj = i.

5.3.3 Corrections of lnZ in Thermal φ4 Theory

Because the corrections lnZi, i ≥ 1, to lnZ0 depend on the vacuum expectation value〈Snint〉0, only fully contracted diagrams contribute. This can be seen by calculating 〈Snint〉0order by order, which is done, at least for lnZ1, in [KG06]. To avoid such calculations,we switch to the equivalent operator representation of the expectation value. Wick’sTheorem tells us that all fully contracted terms contribute to the expectation values.These expectation values are taken with respect to the vacuum of a non-interacting theory.Therefore, no contractions of fields with the vacuum are possible and no diagrams withexternal legs contribute. One finds:

〈Snint〉0 =

⟨0

∣∣∣∣∣(∫ β

0

dτ

∫d3q (−λ)φ4(q)

)n∣∣∣∣∣ 0⟩

=

= (−λ)n

(n∏k=1

∫ β

0

dτk

∫d3qk

)⟨0

∣∣∣∣∣n∏k=1

φ4(qk)

∣∣∣∣∣ 0⟩.

(5.56)

The first- and second-order thermal corrections to lnZ become:

lnZ1 =⟨S1

int

⟩0

=∑

contractions

= 3 =

= −3λβV

(T∑n∈Z

∫d3p

(2π)3∆

(0)M (ωn,p)

)2

.

(5.57)

The symmetry factor 3 = 12

(42

)of lnZ1 is due to all possible indistinguishable contractions

of two lines. The factor βV appears due to the vertex, because the energy-momentumconservation (5.51) is trivially fulfilled in a vacuum bubble and we have βδ (0) = β and(2π)3δ(3) (0) = V .

For lnZ2 there are two different types of diagrams with symmetry factors 72 = 2(

42

)(42

)and 24 = 4!, respectively:

52


lnZ2 =1

2

⟨S2

int

⟩0− 1

2

⟨S1

int

⟩2

0=

=1

2

∑contr.

⊕

− 1

2

(3 ⊗ 3

)=

=1

2

(3 ⊗ 3 + 72 +

+24

− 1

2

(3 ⊗ 3

)=

= 36 + 12 .

(5.58)Using the Feynman rules for thermal φ4 theory, this diagrammatic expression reads:

lnZ2 = 36βV (−λ)2βδ(ωn1 + ωn2)(2π)3δ(3)(p1 + p2)

(T∑n

∫d3p

(2π)3∆

(0)M (ωn,p)

)2

·

·(T∑n1

∫d3p1

(2π)3∆

(0)M (ωn1 ,p1)

)(T∑n2

∫d3p2

(2π)3∆

(0)M (ωn2 ,p2)

)+

+ 12βV (−λ)2βδ(ωn1 + ωn2 + ωn3 + ωn4)(2π)3δ(3)(p1 + p2 + p3 + p4) ·

·4∏i=1

(T∑ni

∫d3pi(2π)3

∆(0)M (ωni ,pi)

).

(5.59)Corrections to all thermodynamic quantities can be obtained from lnZ using (5.39). Forinstance, the pressure up to next-to-leading order in φ4 theory is given by [KG06, (3.51)]

P = T 4

(π2

90− λ

48

)+O(λ3/2) . (5.60)

Later on we will use lnZ to determine the self energy corrections Σβ in (6.30) using themodified result [KG06, (3.35)]:

Σβ(ωn,p) = − 2

βV

δ lnZ

δ∆(0)M (ωn,p)

∣∣∣∣∣OPI

. (5.61)

This identity encodes the following recipe:

First draw all diagrams which contribute to lnZint up to a given order, thendifferentiate with respect to ∆

(0)M , and, lastly, throw away the one-particle

reducible diagrams. This yields the diagrammatic expansion of Σβ.

We will use this recipe to derive the first- and second-order corrections to the self energyof scalar fields with φ4 interaction. In chapter 6.4 we will discuss briefly the structure oflnZ3 and the third-order corrections to the self energy.

53


5.4 Spectral Representation of Green’s Functions

Besides the thermal Green’s function (5.25), there are the retarded and advanced Green’sfunctions for qi ∈ Rd, ti ∈ R , introduced in quantum field theory by

GR(q1,q2; t1, t2) ≡ −iθ(t2 − t1)⟨[φ(q2, t2), φ†(q1, t1)

]⟩,

GA(q1,q2; t1, t2) ≡ −iθ(t1 − t2)⟨[φ(q2, t2), φ†(q1, t1)

]⟩.

(5.62)

The −i in the definition is introduced in order to have (5.68) below without additionalcomplex constants. Both the retarded and the thermal Green’s function are related to asingle spectral function ρ(p), defined by

Gβ(ωn,p) ≡∫ ∞−∞

dω′ρ(ω′,p)

ω′ − iωn. (5.63)

From this definition, the spectral function for a free particle with energy ω =√

p2 +m2

reads:

ρF (p) = sgn(p0)δ((p0)2 − ω2) . (5.64)

This is verified by examining

Gβ(p)|ρF =

∫ ∞−∞

dω′ sgn(p0)δ((p0)2 − ω2)

ω′ − iωn=∑ξ=±ω

∫ ∞−∞

dω′sgn(ω′)

|2ξ|δ(ω′ − ξ)ω′ − iωn

=

=1

2ω(ω − iωn)+

1

2ω(ω + iωn)=

1

ω2 − (iωn)2=

1

ω2n + p2 +m2

,

(5.65)

which is just the free Matsubara propagator in momentum space. So far, Gβ is a functionwith domain Z×R3, hence we define a continuation G : M→ C by

G(p) ≡∫ ∞−∞

dω′ρ(ω′,p)

ω′ − p0. (5.66)

With a very similar calculation as done in (5.65), the spectral representation of G(p) inthe free case reads

G(p)|ρF = − 1

(p0)2 − p2 −m2= − 1

p2 −m2. (5.67)

From this we find the important relation between the Matsubara propagator, the advancedand retarded propagator, and the spectral function:

Gβ(ωn,p)−−−−−−−−−−→iωn 7→p0+iε

− 1

p2 −m2 + iε= G(p0 + iε,p) = −GR(p) ,

Gβ(ωn,p)−−−−−−−−−−→iωn 7→p0−iε

− 1

p2 −m2 − iε = G(p0 − iε,p) = −GA(p) .(5.68)

This leads finally to the familiar spectral representation of the retarded Green’s function:

GR(p) =

∫ ∞−∞

dω′ρ(ω′,p)

p0 + iε− ω′ . (5.69)

54

5.5 Discussion of Thermal Quantum Field Theories

For the following statement (5.72) we use the principal value integral for arbitrary numbersξ0, κ, ε ∈ R and ε 6= 0:

PV

∫ ∞−∞

dξκ

ξ − (ξ0 + iε)= sgn(ε)πiκ . (5.70)

To prove this, first for ε > 0, we use Cauchy’s Residue Theorem with the semi-circleξ = ξ0 + δeiϕ, δ > 0, ϕ ∈ [π, 2π], in the lower half-plane of ξ:∮

dξκ

ξ − ξ0

= limδ→0

[(∫ ξ0−δ

−∞dξ +

∫ ∞ξ0+δ

dξ

)κ

ξ − ξ0

+

∫ 2π

π

dϕiκδeiϕ

δeiϕ

]=

= PV

∫ ∞−∞

dξκ

ξ − ξ0

+ πiκ!

= 2πiκ .

(5.71)

In the case ε > 0, the contour integral is closed in the upper half-plane, hence the integra-tion is counter-clockwise and, due to its residue, (5.71) reads +2πiκ. In the other case,ε < 0, the semi-circle can be parametrized by ϕ ∈ [π, 0] and the contour integral is closedin the lower half-plane, which means a clockwise integration. Hence, the residue resultsin −2πiκ and the claim is shown. However, equation (5.70) holds also in the case if κ isan analytic function, therefore (5.69) yields

ImGR(p) = −πρ(p) . (5.72)

We will use this key identity between the spectral function and the retarded propagatorto calculate the shear viscosity η in the following chapters.

5.5 Discussion of Thermal Quantum Field Theories

In the previous sections we have introduced the imaginary time formalism, primarily forbosonic fields, in order to describe these fields at finite temperature. We might ask thequestion if we have constructed indeed a relativistic quantum field theory at finite tem-perature? The answer to this question is negative, since compactifying only the temporalpart of the Minkowski space M is not a Lorentz-invariant concept. Doing such a com-pactification, we try to establish a quantum field theory on a four-dimensional cylinder(x, y, z, τ) with x, y, z ∈ R and τ ∈ [0, β]. In fact, it is possible to discuss quantum fieldtheory on compactified manifolds, for example on a four dimensional torus, [Nak03], butcylindric symmetry is too weak to ensure Lorentz invariance. The heat bath introducedin the (grand-)canonical statistical description (5.17) is not a Lorentz invariant concepteither. The imaginary time formalism is thus not a relativistic, Lorentz invariant quantumfield theory.

The reason for loosing Lorentz invariance is given by the ordinary concept of temper-ature which already violates Galilean invariance: temperature is related to the secondmoment of the velocity of a statistical system: T ∼ 〈v2〉. This statement holds in adistinguished frame of reference, namely the rest frame of the heat bath, and only there.

The definition of temperature in statistical physics, T ≡ dE/dS, provides another wayof looking at this. According to this definition it is clear that temperature is not a Lorentzscalar: it is the time component of a four-vector, E = x0, divided by a pure number. Theentropy S of a system does not change under Galilean or Lorentz transformations; thetime component does. In chapter 4 we have constructed a Lorentz covariant formalism

55


in order to describe statistical systems in a non-equilibrium state. There, in (4.4) wehave introduced a “proper” temperature, Ts ≡ T/γ, which is indeed a Lorentz scalar andcoincides at rest with the usual statistical definition of temperature.

Concerning the renormalizability of thermal field theories, temperature does not playany role since it only affects infrared physics and is not relevant at short distances. We canstate that a thermal field theory is renormalizable if and only if the corresponding quantumfield theory at zero temperature is renormalizable. A formal proof of this statement canbe found, for instance, in [Bel00]. Note, that the limit T → 0 of a thermal field theoryis not sufficient to create a quantum field theory since this limit obeys Euclidean metric.Additionally, one has to do an inverse Wick rotation in order to recover the Minkowskimetric.

Indeed, in the infrared region, finite temperature affects the structure of perturbationtheory within such field theories. For instance, scalar fields with φ4 interaction generatea thermal mass, m = λ1/2 T (6.38), which leads finally to a breakdown of the usualperturbative series: it is no longer possible to make a power expansion in the couplingparameter λ, since thermodynamic quantities are non-analytical functions of λ [KG06].This has already been outlined in (5.60), where the pressure P up to next-to-next-to-leading-order corrections is given by

P = T 4

(π2

90− λ

48+λ3/2

12π

)+O(λ2) . (5.73)

This behavior of thermal field theories in the infrared region is well-known in gaugetheories like QED and QCD as the so-called Linde problem [Lin80].

56

6 Shear Viscosity in φ4 Theory

In this chapter we calculate the shear viscosity for a real scalar field with φ4 interaction.The calculations within this chapter help to understand the basic ingredients for the moreinvolved calculation for a pion gas which will be considered in the next chapter withinthe framework of chiral perturbation theory. Consider the bare Lagrangian of real scalarfields with φ4 interaction:

L =1

2(∂µφ)(∂µφ)− 1

2m2

0φ2 − λφ4 . (6.1)

Only the kinetic part of the energy-momentum tensor,

Tµν =∂L

∂(∂µφ)∂νφ− gµνL = ∂µφ∂νφ− gµνL , (6.2)

contributes to the viscous-stress tensor

πµν = (∆ρµ∆σ

ν −1

3∆µν∆

ρσ)Tρσ = (∆ρµ∆σ

ν −1

3∆µν∆

ρσ)∂ρφ∂σφ . (6.3)

In general, the shear viscosity (4.37) is a field η : M → R, but in the case that localequilibrium is close to global thermodynamic equilibrium, the shear viscosity η reducesto a single number, because global homogeneity is restored approximately:

η = − 1

10

∫ 0

−∞dt

∫ t

−∞dt′∫d3x′ 〈πµν(0), πµν(x

′, t′)〉ret =

= − 1

10

∫ 0

−∞dt

∫ t

−∞dt′∫d3x′ (−i)θ(−t′) 〈[πµν(0), πµν(x

′, t′)]〉 ≡

≡ − 1

10

∫ 0

−∞dt

∫ t

−∞dt′ ΠR(t′) .

(6.4)

Here, ΠR(t′) is the spatially-integrated retarded Green’s function (5.62) for the viscous-stress tensor. Using the real-time continuation (5.68), we need to calculate the spatially-integrated thermal Green’s function (5.25):

Πβ(τ) ≡∫d3x 〈Tτ [πµν(0)πµν(x, τ)]〉 . (6.5)

For this, we determine the fully contracted viscous-stress tensor product:

πµν(0)πµν(x, τ) = (∆µρ∆νσ − 1

3∆µν∆ρσ)(∂ρφ)(∂σφ)(∆α

µ∆βν −

1

3∆µν∆

αβ)(∂αφ)(∂βφ) =

= (∆µρ∆νσ∆αµ∆β

ν −1

3∆µν∆ρσ∆α

µ∆βν −

1

3∆µρ∆νσ∆µν∆

αβ+

+1

9∆µν∆ρσ∆µν∆

αβ)(∂ρφ)(∂σφ)(∂αφ)(∂βφ)(3.14)=

= (∆ρα∆σβ − 1

3∆ρσ∆αβ)(∂ρφ(0))(∂σφ(0))(∂αφ(x, τ))(∂βφ(x, τ)) .

(6.6)

57


To get rid of the derivatives, (6.5) is Fourier transformed using ∂µ 7→ ipµ. Then, we arriveat the momentum-integrated thermal Green’s function

Πβ(ωn) =

∫ β

0

dτ eiωnτ∫

d3p

(2π)3(∆µρ∆νσ − 1

3∆µν∆ρσ) pµpνpρpσ ·

· 1

V 2〈Tτ [φ(0)φ(0)φ(p, τ)φ(p, τ)]〉 .

(6.7)

Note the interval τ ∈ [0, β] of the Fourier transform, which is due to the periodic boundarycondition (5.27) of the bosonic fields in thermal field theory.

6.1 Skeleton Expansion and Matsubara Propagator

First, we discuss the four-point correlator 〈Tτ [φ(0)φ(0)φ(p, τ)φ(p, τ)]〉. In order to dothat, we recall the covariance of two random variables X and Y :

Cov(X, Y ) ≡ 〈(X − 〈X〉)(Y − 〈Y 〉)〉 = 〈XY 〉 − 〈X〉〈Y 〉 . (6.8)

This means that the expectation value of a product factorizes if and only if the randomvariables are uncorrelated. We introduce the skeleton expansion, which is an expansionin full propagators subject to interactions between them treated perturbatively [Hun92].Applying this to the four-point correlator, we get in leading-order skeleton expansion:

1

V 2〈Tτ [φ(0)φ(0)φ(p, τ)φ(p, τ)]〉 ≈ 2

V 2〈Tτ [φ(0)φ(p, τ)]〉〈Tτ [φ(0)φ(p, τ)]〉+

+1

V 2〈Tτ [φ(0)φ(0)]〉〈Tτ [φ(p, τ)φ(p, τ)]〉 ≈

≈ 2

V 2(〈Tτ [φ(0)φ(p, τ)]〉)2 ≡ 2G2

β(p, τ) .

(6.9)

The first approximation comes from counting all possible contractions that lead to fullpropagators and factorizing them using equation (6.8). The second one is due to theexpansion of the vacuum state of the full theory, |Ω〉 = |0〉 + O(λ), the vacuum bubbledoes, hence, not contribute at leading order. Diagrammatically, the skeleton expansionin φ4 theory can be written up to next-to-leading order O(λ2):

1

V 2

⟨Tτ[φ2(0) φ2(p, τ)

]⟩= + + . . . (6.10)

In the diagrams double lines denote full propagators. The next-to-next-to-leading-orderdiagram would be of order O(λ4), but we take only the leading-order contribution ofthe skeleton expansion into account. Consequently, the momentum-integrated thermalGreen’s function (6.7) reads:

Πβ(ωn) = 2

∫ β

0

dτ eiωnτ∫

d3p

(2π)3(∆µρ∆νσ − 1

3∆µν∆ρσ) pµpνpρpσ G

2β(p, τ) . (6.11)

The spatially Fourier transformed thermal Green’s function, Gβ(p, τ), is related to thethermal Green’s function by a temporal Fourier transformation

Gβ(p, τ) =1

β

∑n∈Z

e−iωnτGβ(p, ωn) . (6.12)

58


This one-dimensional Fourier transformation in addition to the replacement τ → ωn in(6.11) is necessary for the evaluation of Gβ(p, τ) in the Matsubara formalism. Insertingthe spectral representation of the thermal Green’s function, Gβ(p) = Gβ(p, ωn), we obtainfor τ ∈ [0, β]:

Gβ(p, τ)|τ∈[0,β]

(5.63)= − 1

β

∑n∈Z

e−iωnτ∫ ∞−∞

dωρ(ω,p)

iωn − ω=

= −∫ ∞−∞

dω ρ(ω,p)1

β

∑n∈Z

e−iωnτ

iωn − ωA.3=τ>0

= −∫ ∞−∞

dω ρ(ω,p)n(−ω) e−ωτω 7→−ω

=

= −∫ ∞−∞

dω ρ(−ω,p)n(ω) eωτ =

=

∫ ∞−∞

dω ρ(ω,p) n(ω) eωτ .

(6.13)

In the last line we have used the identity ρ(−ω,p) = −ρ(ω,p). In order to check thecorrect periodicity, we calculate Gβ(p, τ) similarly for τ ∈ [−β, 0]:

Gβ(p, τ)|τ∈[−β,0] = −∫ ∞−∞

dω ρ(ω,p)1

β

∑n∈Z

e−iωnτ

iωn − ωA.3=τ<0

=

∫ ∞−∞

dω ρ(ω,p)n(ω) e−ωτ .

(6.14)

Hence, on the interval τ ∈ [−β, β] the thermal Green’s function Gβ(p, τ) reads:

Gβ(p, τ) =

∫ ∞−∞

dω ρ(ω,p)n(ω) eω|τ | . (6.15)

Note the expected periodicity of the Matsubara propagator in imaginary time:

Gβ(p, τ − β) = Gβ(p, τ) for τ ∈ [0, β] . (6.16)

Next, return to the skeleton-expanded momentum-integrated thermal Green’s function(6.11) and insert the spectral representation (6.15):

Πβ(ωn) = 2

∫ β

0

dτ eiωnτ∫

d3p

(2π)3(∆µρ∆νσ − 1


·∫ ∞−∞

dω1

∫ ∞−∞

dω2 ρ(ω1,p)n(ω1) ρ(ω2,p)n(ω2) e|τ |(ω1+ω2) =

= 2

∫d3p

(2π)3(∆µρ∆νσ − 1

3∆µν∆ρσ) pµpνpρpσ

∫ ∞−∞

dω1

∫ ∞−∞

dω2 ·

· ρ(ω1,p)n(ω1) ρ(ω2,p)n(ω2)

[1

iωn − ω1 − ω2

− 1

iωn + ω1 + ω2

].

(6.17)

In the last line we have used the periodicity (6.16), in particular∫ ∞−∞

dωi ρ(ωi,p)n(ωi) eβωi =

∫ ∞−∞

dωi ρ(ωi,p)n(ωi) for i = 1, 2 . (6.18)

59


Now, using (5.68) we continue (6.17) to its retarded version:

ΠR(p0) = − Πβ(ωn)|iωn 7→p0+iε =

= −2

∫d3p

(2π)3(∆µρ∆νσ − 1

3∆µν∆ρσ) pµpνpρpσ

∫ ∞−∞

dω1

∫ ∞−∞

dω2 ρ(ω1,p)n(ω1) ·

· ρ(ω2,p)n(ω2)

[1

p0 + iε− ω1 − ω2

− 1

p0 + iε+ ω1 + ω2

].

(6.19)The temporal Fourier-transformed version of (6.19) is needed in (6.4), hence η reads

η(6.4)= − 1

10

∫ 0

−∞dt

∫ t

−∞dt′ ΠR(t′) =

= − 1

10

∫ 0

−∞dt

∫ t

−∞dt′∫dp0

2πe−ip

0t′ ΠR(p0) =

= − 1

10

∫ 0

−∞dt′∫ 0

t′dt︸︷︷︸

=−t′

∫dp0

2πe−ip

0t′ ΠR(p0) .

(6.20)

After chancing the order of the two-dimensional (t, t′) integral in the last line the integra-tion over t′ can be performed. With

−t′e−ip0t′ = −i ddp0

e−ip0t′ (6.21)

we arrive at

η =i

10

d

dp0

∫dp0

2πΠR(p0)

∫ 0

−∞dt′ e−ip

0t′︸︷︷︸≈2πδ(p0)

≈ i

10

d

dp0ΠR(p0)

∣∣∣∣p0=0

. (6.22)

Note, that the δ(p0) approximation becomes exact for large t because the system con-verges, coming from non-equilibrium along local equilibrium, to global equilibrium. Theassumption (6.4) that only one constant shear viscosity describes the whole system is onlyvalid near to the global equilibrium. Therefore, calculating η using the δ(p0) approxima-tion is feasible.

We conclude that we can easily calculate the shear viscosity using (6.22). But thisrequires the momentum-integrated retarded Green’s function ΠR(p0) in (6.19), whichdepends on the spectral function. Here, the connection (5.72), πρ(p) = −ImGR(p),between the spectral function and the full retarded propagator is needed to calculate(6.22): we have to calculate the full propagator

GR(p) =1

(p0)2 − p2 −m20 − ΣR(p) + iε

=1

1−G(0)R (p)ΣR(p)

G(0)R (p) =

=∞∑n=0

(G

(0)R (p)ΣR(p)

)nG

(0)R (p) ,

(6.23)

with the one-particle-irreducible (OPI) self-energy ΣR(p), corresponding to the amputateddiagrams, which are diagrams without external legs. We have diagrammatically:

= + +

+ + . . .(6.24)

60


It is instructive to decompose the self-energy into its real and imaginary part. Assumingthat a Feynman expansion of the OPI self-energy is possible, both Re ΣR(p) and Im ΣR(p)are small and we can neglect higher order terms. We find for the propagator

GR(p)−1 = (p0)2 − p2 −m20 − Re ΣR(p)− iIm ΣR(p) + iε ≈

≈(p0 − i

2p0Im ΣR(p)

)2

−(p2 +m2

0 + Re ΣR(p))

+ iε ≡

≡(p0 + iΓ(p)

)2 − E2p + iε .

(6.25)

Here, we have defined the spectral width by

Γ(p) ≡ − 1

2p0Im ΣR(p) , (6.26)

which governs the shear viscosity through the spectral function

ρ(p)(5.72)= − 1

πImGR(p)

∣∣∣∣ε=0

(6.25)=

1

π

2p0Γ(p)((p0)2 − Γ(p)2 − E2

p

)2+ 4(p0)2Γ(p)2

=

=1

2πi

(1

(p0 − iΓ(p))2 − E2p

− 1

(p0 + iΓ(p))2 − E2p

).

(6.27)

We will justify the choice of the Γ(p) prefactor a posteriori in (6.45). For the time beingit should be considered as a definition.

From the above we can state the following important line of argument:

Im ΣR(p) = 0(6.26)⇒ Γ(p) = 0

(6.27)⇒ ρ(p) = 0(6.19)⇒ ΠR(p) = 0

(6.22)⇒ η = 0 .(6.28)

In order to calculate the spectral function (6.27) and the retarded correlator (6.19), wenow determine the self-energy ΣR(p) up to sufficiently high order. Transforming (6.23)to finite temperatures using (5.68), we find

Gβ(ωn,p) =1

ω2n + p2 +m2

0 + Σβ(ωn,p), (6.29)

where we have introduced

Σβ(ωn,p) ≡ ΣR(iωn,p) . (6.30)

We emphasize the important criterion following from (6.28) and (6.30):

In order for a diagram in the Feynman expansion of Σβ(ω,p) to contributeto the shear viscosity η, it is necessary that this diagram is momentum, i. e.(ωn,p) dependent.

This is true because the Matsubara propagator is real in momentum space, hence, in orderto give ΣR(iωn,p) an imaginary part, Σβ must be momentum dependent. For instance,we will see that the one-loop result is independent of the incoming momentum, hence itdoes not contribute to the shear viscosity.

61


6.2 First-Order Correction to the Propagator

In order to determine the full retarded propagator (6.23) we need to calculate the thermalself-energy Σβ via (6.30) and perform the continuation via (5.61). At first order wecalculate in both an analytical and in a diagrammatic way; higher orders will be calculatedwith the latter technique. We find for the self-energy at first order:

Σ(1)β (ωn,p)

(5.61)=−2

βV

δ lnZ1

δ∆(0)M (ωn,p)

∣∣∣∣∣OPI

(5.57)=

= 6λδ

δ∆(0)M

(T∑m∈Z

∫d3p′

(2π)3∆

(0)M (ωm,p

′)

)2

=

= 12λ

(T∑m∈Z

∫d3p′

(2π)3∆

(0)M (ωm,p

′)

)· T∑m∈Z

∫d3p′

(2π)3

δ∆(0)M (ωm,p

′)

δ∆(0)M (ωn,p)

.

(6.31)

In general, the dimension of a functional derivative is[δf(y)δg(x)

]= [f ]− [g]− [x]. This also

holds in our case with [f ] = [g] = [∆(0)M ] = −2 and [x] = [(ωn,p)] = 4:

δ∆(0)M (ωm,p

′)

δ∆(0)M (ωn,p)

= βδm,n(2π)3δ(3)(p′ − p) . (6.32)

Then, the first-order self-energy (6.31) is just

Σ(1)β (ωn,p) = 12λ T

∑m∈Z

∫d3p′

(2π)3∆

(0)M (ωm,p

′) . (6.33)

Another way to calculate the self energy is given by the diagrammatic technique. Here,the derivative with respect to the Matsubara propagator means cutting one loop, since, aswe have seen, the functional derivative transforms one loop integral to 1/(βV ). Therefore,we get

Σ(1)β (ωn,p) = − 2

βV

δ lnZ1

δ∆(0)M (ωn,p)

∣∣∣∣∣OPI

(5.57)= − 6

βV

δ

δ∆(0)M (ωn,p)

=

= −12 .

(6.34)

Using our Feynman rules for thermal φ4 theory (6.33) and (6.34) are identical. In orderto calculate a number for the first-order self-energy, we first perform the Matsubara sum,then divide the integral into a convergent and a divergent vacuum term. We use theabbreviation θ2 = p2 +m2

0 for the bosonic Matsubara sum (5.31):

Σ(1)β = −12 = 12λ T

∑n∈Z

∫d3p

(2π)3

1

ω2n + p2 +m2

0

(5.31)=

= 12λ

∫d3p

(2π)3

1√p2 +m2

0

[1

2+

1

eβ√

p2+m20 − 1

]=

=3λ

π2

∫ ∞0

dpp2√

p2 +m20

+

∫ ∞0

dp2p2(

eβ√p2+m2

0 − 1)√

p2 +m20

.(6.35)

62

6.3 Second-Order Corrections to the Propagator

The first term is, at leading order, quadratically divergent

Σ(1,div)β = lim

Λ→∞

3λ

π2

∫ Λ

0

dpp2√

p2 +m20

=

= limΛ→∞

3λ

π2

[Λ

2

√Λ2 +m2

0 −m2

0

2ln

(2Λ + 2

√Λ2 +m2

0

)]→

−−−−→m0→0

limΛ→∞

3λ

π2Λ2 ,

(6.36)

hence we have to regularize the theory, e. g. by introducting a counter term δm2. It ischosen in such a way that there is no vacuum contribution to the self energy:

Σ(1,ren)β = Σ

(1,div)β + δm2 ≡ 0 . (6.37)

The remaining convergent part of the self energy reads

Σ(1,conv)β =

6λ

π2

∫ ∞0

dpp2(

eβ√p2+m2

0 − 1)√

p2 +m20

−−−−→m0→0

λT 2 . (6.38)

Consequently, the first-order self-energy is independent of the particle momentum. There-fore, as discussed along with (6.30), it does not contribute to the shear viscosity. Now wecan write the Matsubara propagator (6.29) at order λ,

Gβ(ωn,p) =1

ω2n + p2 +m2

0 + δm2 + λT 2, (6.39)

and identify the physical mass:

m2 = m20 + δm2 + λT 2 . (6.40)

The thermal mass at one-loop level, m = Σ(1,conv)β = λ1/2 T , has no imaginary part and

thus does not contribute to the shear viscosity at this order.


We derive the second-order contribution to the self-energy of the Matsubara propagatorusing the diagrammatic functional-derivative technique:

Σ(2)β (ωn,p) = − 2

βV

δ lnZ2

δ∆(0)M (ωn,p)

∣∣∣∣∣OPI

(5.58)=

= − 2

βV

δ lnZ2

δ∆(0)M (ωn,p)

36 + 12

∣∣∣∣∣∣OPI

=

= −72 · 2 − 24 · 4 .

(6.41)

63


From the criterion (6.30) only momentum-dependent diagrams contribute to the shearviscosity. Hence only the second diagram in (6.41) will be important in the presentcontext. The first diagram gives a second-order contribution to ΣR(p), and, thus, toE2p = p2 +m2

0 + δm2 + λT 2 +O(λ2) in (6.40). We now evaluate the second diagram:

Σ(2)β (ωn,p)

∣∣∣η

= −96 =

= −96(−λ)2

3∏i=1

(T∑ni∈Z

∫d3pi(2π)3

∆(0)M (ωni ,pi)

)·

· βδ (ωn1 + ωn2 + ωn3 − ωn) (2π)3δ(3) (p1 + p2 + p3 − p) =

= −96λ2 T∑n1∈Z

∫d3p1

(2π)3T∑n2∈Z

∫d3p2

(2π)3·

·∆(0)M (ωn − ωn1 ,p− p1)∆

(0)M (ωn2 ,p2)∆

(0)M (ωn1 − ωn2 ,p1 − p2) .

(6.42)

In the last line we have performed the (n3,p3) sum and integral which eliminate theKronecker delta and the delta function, respectively. We have shifted n1 7→ n − n1 andp1 7→ p − p1, hence only the first Matsubara propagator depends on ωn. In order tocalculate the retarded Green’s function we continue Σβ(ωn,p) via (6.30):

Σ(2)R (p0,p)

∣∣∣η

= Σ(2)β (−ip0,p)

∣∣∣η

=

= −96λ2 T∑n1∈Z

∫d3p1

(2π)3T∑n2∈Z

∫d3p2

(2π)3∆

(0)M (ωn2 ,p2) ·

· 1

(−ip0 − ωn1)2 + (p− p1)2 +m2

∆(0)M (ωn1 − ωn2 ,p1 − p2) =

= −96λ2 T∑n1∈Z

∫d3p1

(2π)3T∑n2∈Z

∫d3p2

(2π)3∆

(0)M (ωn2 ,p2) ·

·[A∆

(0)M (ωn1 − ωn2 ,p1 − p2)

A2 + 4(p0)2ω2n1

− 2ip0ωn1 ∆(0)M (ωn1 − ωn2 ,p1 − p2)

A2 + 4(p0)2ω2n1

],

(6.43)

where we have introduced

A ≡ Re (∆(0)M (−ip0 − ωm1 ,p− p1))−1 = −(p0)2 + (p− p1)2 + ω2

n1+m2 . (6.44)

From (6.43) we obtain directly the imaginary part of the retarded Green’s function,

Im Σ(2)R (p0,p) = −2p0Γ(p) , (6.45)

where we have used Γ(p) to denote the remaining sums and integrals. Note, that theprefactor −2p0 arises naturally from (6.43) and we meet the definition (6.26). However,

the real part of Σ(2)R (p) contributes indirectly to the shear viscosity, because it contributes

to the dynamical mass (6.40) at order O(λ)2, the same as the first diagram in (6.41).Leaving the spectral width Γ(p) > 0 unevaluated at this point, we insert the spectral

function (6.27) into (6.19) and get:

ΠR(p0) = −2

∫d3p

(2π)3

⟨∆∆ p4

⟩ ∫ ∞−∞

dω1

∫ ∞−∞

dω2 ρ(ω1,p)ρ(ω2,p) n(ω1)n(ω2) WR(ω12, p0) ,

(6.46)

64


where we have defined ⟨∆∆ p4

⟩≡ (∆µρ∆νσ − 1

3∆µν∆ρσ) pµpνpρpσ,

WR(ω12, p0) ≡

[1

p0 + iε− ω12

− 1

p0 + iε+ ω12

],

(6.47)

with ω12 ≡ ω1 + ω2. The subscript R denotes the retarded structure p0 + iε of W . Notethe important properties

WR(ω12,−p0) = WA(ω12, p0) , WR(−ω12,−p0) = −WA(ω12, p

0) . (6.48)

Inserting ΠR(p0) into (6.22), we get for the shear viscosity

η =i

10

d

dp0ΠR(p0)

∣∣∣∣p0=0

= −2

∫d3p

(2π)3

⟨∆∆ p4

⟩ ∫ ∞−∞

dω1

∫ ∞−∞

dω2 ·

· U(ω1, ω2)V (ω1, ω2)i

10

d

dp0WR(ω12, p

0)

∣∣∣∣p0=0

.

(6.49)

If WR(ω12, p0) were an even function in p0, its derivative at p0 = 0 would vanish. Indeed,

we are very close to this case, but as (6.48) shows, the even symmetry is weakly broken,R→ A, by the pole structure ε. This structure gives rise to the non-zero contribution:

i

10

d

dp0WR(ω12, p

0)

∣∣∣∣p0=0

= − i

10

4iεω12

(ε2 + ω212)2

=i

10

d

dω12

2πi · επ(ε2 + ω2

12)→

Lorentz−δ−−−−−−→

ε→0

i

102πi

d

dω12

δ(ω12) =π

5(−δ′(ω12)) .

(6.50)

Transforming the two-dimensional integral, ω12 = ω1 + ω2 and ω12 = ω1 − ω2,∫R2

dω1dω2 =1

2

∫R2

dω12dω12 , (6.51)

we arrive with the additional substitution ω12 ≡ 12ω at

η =i

10

d

dp0ΠR(p0)

∣∣∣∣p0=0

= −2

∫d3p

(2π)3

⟨∆∆ p4

⟩ 1

2

∫ ∞−∞

dω12 ·

· π5

d

dω12

U

(1

2(ω12 + ω12),

1

2(ω12 − ω12)

)V

(1

2(ω12 + ω12),

1

2(ω12 − ω12)

)∣∣∣∣ω12=0

=

= −π5

∫d3p

(2π)3

⟨∆∆ p4

⟩ (−4

π2

)∫ ∞−∞

dω ·

· 2ω2eβω

(eβω − 1)2

βΓ2(E2p − (ω − iΓ)2

)2 (E2p − (ω + iΓ)2

)2 .

(6.52)The integrand in (6.52) reads

F (ω; β,Γ, Ep) ≡2ω2eβω

(eβω − 1)2

βΓ2(E2p − (ω − iΓ)2

)2 (E2p − (ω + iΓ)2

)2 (6.53)

and possesses countably many poles in the complex plane: for the exponential we findpoles at ω = iωn = 2πinT , for n ∈ Z \ 0, where ωn are the bosonic Matsubara

65


ω

2πiT

4πiT

6πiT

−2πiT

−4πiT

−6πiT

−Ep − iΓ Ep − iΓ

Ep + iΓ−Ep + iΓ

Figure 6.1: Pole structure of the integrand F (ω) in (6.52). Note, that for small Γ > 0 theEp-dependent poles are close to the real axis. The Matsubara poles at the imaginary axis haveno contribution to the shear viscosity at leading order.

frequencies. Indeed, at ω = 0, the integrand has no singularity. Furthermore, there areeight poles, i. e. four poles of second order: ωi = ±Ep ∓ iΓ, for i = 1, 2, 3, 4. None ofthese poles is located on the real axis. We use Cauchy’s Residue Theorem to evaluate theintegral, closing the contour in the upper half plane:∫ ∞

−∞dω F (ω; β,Γ, Ep) = 2πi

[∞∑n=1

ResF (iωn) +2∑i=1

ResF (ωi)

]. (6.54)

The two residua ω1,2 = ±Ep + iΓ are calculated analytically (with some lengtly result)and its contribution to the integral (6.52) can be expanded in a Laurent series:

2πi2∑i=1

ResF (ωi) =eβEp

(eβEp − 1)2

πβ

16E2pΓ

+O(Γ1) . (6.55)

Note, that there is no constant term in the expansion. The spectral width Γ is smallbecause it originates from a perturbative expansion of the thermal propagator. The Γ−1

contribution is much more important than the linear one. Calculating the contributionof the n-th Matsubara frequence, one finds:

ResF (iωn) =

=16nβ6Γ2

(48n2iπ4 + 8n2iE2

pπ2β2 − iE4

pβ4 − 8n2Iπ2β2Γ2 − 2iE2

pβ4Γ2 − iβ4Γ4

)π(4n2π2 + E2

pβ2 − 4nπβΓ + β2Γ2

)3 (4n2π2 + E2

pβ2 + 4nπβΓ + β2Γ2

)3 ∼

∼ β6Γ2

n9, for small Γ, β and large n .

(6.56)The sum of ResF (iωn) over n ∈ N converges and gives some function of order O(Γ2) thatcan be neglected for small Γ compared to the Γ−1 contribution in (6.55). After evaluating

66


the integral to leading order, we arrive at

η =β

20

∫d3p

(2π)3

(∆µν∆ρσ − 1

3∆µρ∆νσ

)pµpνpρpσ

eβEp

(eβEp − 1)2

1

E2pΓ(Ep)

. (6.57)

This is our final result for the shear viscosity η. Note, that for Γ(Ep) > 0 also η > 0follows, as expected from the second law of thermodynamics (3.12).

Finally, we determine a more suitable form for the spectral width Γ(p0) than the onegiven implicitly in (6.43). Therefore, we go back to (6.42) and perform a partial fractiondecomposition for the free Matsubara propagator:

1

ω2 + E2=∑s=±1

s

2E

1

iω + sE. (6.58)

We have E2i = m2 + p2

i and E23 = m2 + (p − p1 − p2)2, owing to the delta functions.

Carrying out the Matsubara sums yields:

Σ(2)β (ωn,p)

∣∣∣η

(6.42)= −96λ2 T

∑n1∈Z

∫d3p1

(2π)3T∑n2∈Z

∫d3p2

(2π)3·

· 1

ω2n1

+ E21

1

ω2n2

+ E22

1

(ωn − ωn1 − ωn2)2 + E2

3

(6.58)=

= −96λ2T 2∑n1,2∈Z

∫1

∫2

∑si=±1

s1s2s3

8E1E2E3

1

iωn1 + s1E1

1

iωn2 + s2E2

1

i(ωn − ωn1 − ωn2) + s3E3

=

= −96λ2T∑n1∈Z

∫1

∫2

∑si=±1

s1s2s3

8E1E2E3

1

iωn1 + s1E1

[n(s2E2)− n(−s3E3)]

s2E2 + s3E3 + i(ωn − ωn1)=

= −96λ2

∫1

∫2

∑si=±1

s1s2s3

8E1E2E3

[n(s2E2)− n(−s3E3)] [n(s1E1)− n(−s2E2 − s3E3)]

s1E1 + s2E2 + s3E3 + iωn.

(6.59)In the calculation we could not apply the master formula (5.30) directly, because thefrequencies ωn1 , ωn2 occur in two Matsubara propagators and couple with one another.Therefore, we have used again a partial-fraction decomposition:

T∑n∈Z

1

iωn + A

1

i(−ωn) +B= T

∑n∈Z

1

A+B

(1

iωn + A− 1

iωn −B

)(5.30)=

=1

A+B[n(A)− n(−B)] .

(6.60)

Note, that because of ωn= 2πnT we have n(A+iωn) = n(A) for A 6= 0 and all n ∈ Z. Now,

after performing the Matsubara sums, we switch to Σ(2)R (p0,p), the analytical continuation

of (6.59), iωn 7→ p0 + iε, and calculate its imaginary part which determines the spectralwidth (6.26). Taking the limit ε→ 0, we arrive at a similar argument as in (6.50):

limε→0

Ima+ ibε

x+ iε= lim

ε→0

−ε (a− bx)

x2 + ε2= − (a− bx) πδ(x) = −aπδ(x), for all b ∈ R . (6.61)

The sum∑

si=±1 for i = 1, 2, 3 in (6.59) has eight terms with four different signatures:(+ + +), (− + +), (− − +) and (− − −), where the signs are given with respect to p0

as shown in figure 6.2. The first case will lead to δ(p0 + E1 + E2 + E3) = 0 because all

67


(+ + +) (−+ +) (−−+) (−−−)

Figure 6.2: Signatures of the sum∑

si=±1 for i = 1, 2, 3 in (6.59). Only the case (−−+)gives an on-shell contribution (6.62) for the spectral width.

energies are positive numbers, hence this term will not contribute at all. The (− + +)and (−−−) structures are not realized because they describe the decay of a particle intothree copies of it. Indeed, on-shell this is not possible for massive particles. Only threeterms with (−−+) signatures remain:

[n(−E2)− n(−E3)]︸︷︷︸=n(E3)−n(E2)

[n(−E1)− n(E2 − E3)︸︷︷︸=n(E3−E2)−n(E1)

=

= −n(E1)n(E3) + n(E1)n(E2) + n(E3 − E2)[n(E3)− n(E2)]︸︷︷︸=−n(E3)n(−E2)

=

= −n(E1)n(E3) + n(E1)n(E2)− n(E3)[1 + n(E2)] =

= −n(E3)− n(E1)n(E3)− n(E2)n(E3) + n(E1)n(E2) .

(6.62)

From the signature (− − +) the delta function δ(p0 − E1 − E2 + E3) arises and we cancalculate further:

(6.62) = −n(E1)n(E2)n(E3)[eβ(E1+E2) − eβE3

]=

= −n(E1)n(E2)[1 + n(E3)]n−1(p0) .(6.63)

Equation (6.26) gives us the final form of the spectral width Γ(p). Indeed, Γ is a positivenumber because four minus signs contribute to the prefactor, 72 = (−)(−96)(−1

2)(−3

2),

coming from (6.63), (6.59), (6.26) and (6.61). We arrive at:

EpΓ(Ep) = −1

2Im ΣR(p0,p) =

= 72λ2 n−1(Ep)3∏i=1

∫d3pi

(2π)3 2Ein(E1)n(E2)[1 + n(E3)] (2π)4δ(4)(p− p1 − p2 + p3) .

(6.64)Note, that for a given energy Ep the spectral width Γ(Ep) is just a number and can becalculated numerically.

6.4 Third-Order Corrections to the Propagator

Let us briefly look at the third-order corrections to the Matsubara propagator. From(5.55) we find all connected diagrams:

lnZ3 =1

6

S1 + S2 + S3

,

(6.65)

68

6.5 Numerical Evaluation of the Shear Viscosity

with the symmetry factors S1 =(

42

)3 · 4 · 2 = 1728, S2 = 3 ·(

42

)2 · 4! = 2592 and

S3 = 3 ·(

42

)· 8 · 4 · 3! = 3456. From this, using (5.61), we get for the third-order

contribution to the self-energy of the Matsubara propagator, dropping all one-particlereducible diagrams:

Σ(3)β (ωn,p) = −2S1

6· 3 − 2S2

6· 2 −

− 2S3

6

+ 3

.

(6.66)

Due to the continuation (6.30) only the last diagram contributes to the shear viscositybecause all other diagrams are independent of the incoming momentum. Therefore, alsoat third-order perturbation theory in λ, our results in the last chapter hold qualitativelybecause the only difference between contributions at both orders to the shear viscosity istaken into account by the replacement of one free Matsubara propagator with a first-ordercorrected Matsubara propagator:

∆(0)M (ωn2 ,p2) 7→ ∆

(1)M (ωn2 ,p2) in (6.42) . (6.67)

Without loss of generality, we can identify the (ωn2 ,p2) propagator as the corrected one.


In the previous sections we derived the shear viscosity within the φ4 theory using a skeletonexpansion in full propagators. In second-order perturbation theory, (6.57), we obtained

η =β

20

∫d3p

(2π)3

(∆µν∆ρσ − 1

3∆µρ∆νσ

)pµpνpρpσ n(Ep) [1 + n(Ep)]

1

E2pΓ(Ep)

, (6.68)

where the spectral width is given in (6.64) by a nine-dimensional phase-space integral

EpΓ(Ep) = 72λ2 n−1(Ep)3∏i=1

∫d3pi

(2π)3 2Ein(E1)n(E2)[1+n(E3)] (2π)4δ(4)(p−p1−p2+p3) .

(6.69)In order to calculate both quantities numerically we need to get rid of the delta functionsand reduce the numbers of integrals to arrive at a better conditioned problem. First notethat

∫d3p1 δ

(4)(p− p1− p2 + p3) = δ(Ep−E1−E2 +E3), hence E1 = Ep +E3−E2. Thedelta function for the three-momentum gives

E21 = m2 + (p + p3 − p2)2 = E2

p + |p3 − p2|2 + 2p|p3 − p2| cos θp , (6.70)

so that the integrand depends on the direction cos θp of the incoming particle. Of course,by construction, the shear viscosity Γ(Ep) is an isotropic quantity. We use

1 =1

4π

∫dΩ =

1

2

∫ 1

−1

d cos θp (6.71)

69


to integrate out this formal angular dependence and arrive at:

1

4π

∫dΩ

3∏i=1

∫d3piEi

δ(4)(p− p1 − p2 + p3) =

=

∫ ∞0

dp2

∫ ∞0

dp3

∫dΩ2

∫dΩ3

p22

E2

p23

E3

∫ 1

−1

d cos θp1

2E1

δ(Ep − E1 − E2 + E3) .

(6.72)

With E1 = E1(cos θp) ≥ 0, we can get rid of the remaining delta function:∫ 1

−1

d cos θp1

2E1

δ(Ep − E1 − E2 + E3) =

∫ 1

−1

d cos θp δ((Ep − E2 + E3)2 − E21) =

=

(∂E2

1

∂ cos θp

)−1(6.70)=

1

2p|p3 − p2|.

(6.73)

With the abbreviation Q ≡ |p3−p2| ≥ 0 and again (6.70), the condition −1 ≤ cos θp ≤ 1reads

− 1 ≤ (E3 − E2)(E3 − E2 + 2Ep)−Q2

2pQ≤ 1

⇔ F (Q) ≡ 4p2Q2 −[(E3 − E2)(E3 − E2 + 2Ep)−Q2

]2 ≥ 0 .

(6.74)

Note, that F (Q2) is just a concave-down parabola with F (Q2) ≥ 0 for Q21 ≤ Q2 ≤ Q2

2

with roots Q21 = (p − p1)2 and Q2

2 = (p + p1)2 of F (Q2). Because Q ≥ 0, we arriveat Q1 = |p − p1| and Q2 = p + p1, the only two non-negative roots of F (Q). UsingEi dEi = pi dpi, we find:

(6.72) =1

2p

∫ ∞m

dE3

∫ Ep+E3−m

m

dE2

∫dΩ2

∫dΩ3

p2p3

|p3 − p2|Θ(F (Q)) . (6.75)

The upper boundary of the E2 integral just reflects energy conservation:

E2 = Ep + E3 − E1 ≤ Ep + E3 −m. (6.76)

Furthermore, by definition, Q2 = p22 +p2

3−2p2p3 cos θ23, hence the integrand depends onlyon one angle θ23 = arccos (p2 · p3). The angular integrals of (6.75) can be carried outusing QdQ = −p2p3 d cos θ23:∫

dΩ2

∫dΩ3

p2p3

|p3 − p2|Θ(F (Q)) = 4π · 2π

∫ 1

−1

d cos θ23p2p3

QΘ(F (Q)) =

= 8π2

∫ p2+p3

|p2−p3|dQ Θ(F (Q)) = 8π2(Q+ −Q−) Θ(Q+ −Q−) ,

(6.77)

withQ− ≡ max |p2 − p3|, |p− p1| ,Q+ ≡ min p2 + p3, p+ p1 .

(6.78)

Note, that the boundaries p2+p3 and |p2−p3| come from the extremal values of Q withouttaking Θ(F (Q)) into account. Of course, we have Q1 ≤ Q2, hence, the Q-integral has nocontribution if Q > Q2 = p + p1 or if Q < Q1 = |p − p1|. This leads to the max/minstructure of Q±. In addition, we have to ensure that Q+ ≥ Q−, which implies the factor

70


0 1 2 3 4 5

1

10

100

1000

p @GeVD

L@f

mD

Figure 6.3: Mean free path Λ(p) in φ4 theory for λ = 1. Different colors (bands) denotedifferent temperatures: 75 MeV (red), 150 MeV (green), 300 MeV (blue). Within a givenband the upper curve denotes the highest mass: 1 MeV, 10 MeV, 20 MeV, 100 MeV, in thisupward order.

Θ(Q+ − Q−). Indeed, for the crossed terms in Q± this is not clear from energy andmomentum conservation. Since n(Ei) is independent of the angles, we arrive at

EpΓ(Ep) =9λ2

(2π)3

n−1(Ep)

p

∫ ∞m

dE3

∫ Ep+E3−m

m

dE2 (Q+ −Q−) ·

· n(E1)n(E2)[1 + n(E3)] Θ(Q+ −Q−) ,

(6.79)

of course with E1 = Ep−E2 +E3. Instead of (6.69), this is a suitable form of the spectralwidth Γ(Ep), that can be used to obtain numerical results. We introduce the mean freepath Λ(p) by

Λ(Ep) ≡p

Γ(Ep)Ep−−−−→m→0

1

Γ(p), (6.80)

which is mainly the inverse spectral width. Its numerical results for different temperaturesand masses are shown in figure 6.3. We see that the temperature dependence is muchmore important than the mass dependence of Λ. For higher temperatures the mean freepath decreases while it becomes larger with increasing masses. For different colors (bands)the temperature is changing only by a factor of two. Varying the mass by a factor tenleads to much smaller changes of the mean free path. For high momenta p T , themean free path is approximately proportional to the momentum: Λ(p) ∼ p. This is clearfrom dimensional analysis, since p is the only remaining energy scale in this limit. Laterin (6.83) we will calculate the prefactor analytically in some approximation.

71


0.5 1.0 1.5 2.00

1

2

3

4

5

T T0

ΗT

3

Figure 6.4: Temperature dependence of the shear viscosity η/T 3 in φ4 theory for λ = 1and T0 = 150 MeV. Different lines denote different masses: 0.1 MeV (dotted line), 10 MeV(dashed line), 20 MeV (solid line). For finite masses the temperature dependence of η becomesweaker.

Using the results for Λ(Ep;T,m), respectively Γ(Ep;T,m), we can derive numericallythe shear viscosity η, equation (6.68), as function of the temperature T with mass m asa parameter. These results are shown in figure 6.4. We see that in the (almost) masslesscase the η ∼ T 3 dependence is valid, as expected from dimensional analysis. A finitemass m > 0 weakens the temperature dependence in such a way that the cubic exponentis decreasing: η ∼ T 3 · f(m/T ) with f(x) → 1 for x → 0 ⇔ (m → 0, T → ∞). Thedependence of f on m/T is also clear from dimensional analysis, since we already knowthe massless limit. Numerically we find η/T 3 ≈ const. ≈ 1.14, hence with T0 = 150 MeV:

η ≈ 109

λ2

MeV

fm2

(T

T0

)3

. (6.81)

Now we compare the numerical result (6.81) with an approximative one which can beobtained by switching to the massless limit m = 0 and to the large-momentum casep T . First of all, we find in the massless limit, due to p1 = p− p2 + p3, the simplifiedQ±, which characterize the two-body phase space:

Q− = |p2 − p3| ,

Q+ =1

2((p2 + p3) + (p+ p1)− |(p2 + p3)− (p+ p1)|) =

= p+ p3 − |p− p2| .(6.82)

72


In a second step we can simplify the integrand (6.69) by neglecting the Bose distributionsin the initial and final states: 1 + n(p) = eβpn(p) ≈ 1. This means: we switch fromthe Bose to the Boltzmann statistics which is justified in the limit T p consideredhere. Neglecting two Bose distributions in the final state and one in the initial state weunderestimate the spectral width Γ(p). The mean free path Λ(p) and finally the shearviscosity η is overestimated:

Γ(Ep)(6.64)=

72λ2

Ep [1 + n(Ep)]

3∏i=1

∫d3pi

(2π)3 2Ei[1 + n(E1)][1 + n(E2)]n(E3) ·

· (2π)4δ(4)(p− p1 − p2 + p3) ≈

≈ 72λ2

Ep

∫d3p3

(2π)3 2E3

n(E3)

∫dΦ2(p1,p2) =

=72λ2

Ep

∫d3p3

(2π)3 2E3

n(E3)1

8π

√1− 4m2

s→

−−−−→m→0

72λ2 · 4π2p · 8π(2π)3

∫ ∞0

dp3p3

eβp3 − 1=

3

8π

λ2T 2

p.

(6.83)

Here we have used the well-known result for the two-body phase-space∫dΦ2(p1,p2).

Finally, the shear viscosity η is determined according to (6.68). Since the temperature isthe only remaining energy scale within massless φ4 theory, we expect η ∼ T 3. Indeed, wefind analytically:

η =2

45πT 3λ2

∫ ∞0

dp p5n(p)[1 + n(p)] =16 ζ(5)

3π

T 3

λ2. (6.84)

Using again T0 = 150 MeV as a typical temperature scale and we end up with the masslesslimit and for the large-momentum case:

η =169

λ2

MeV

fm2

(T

T0

)3

. (6.85)

This result is about 50% larger than the numerical result (6.81). As already discussed,this overestimate is caused by switching the statistics: e−βp < n(p).

73

7 Shear Viscosity of a Pion Gas

In the previous chapter we have discussed the shear viscosity of weakly interacting scalarfields within φ4 theory. The temperature and mass dependence have been calculatednumerically and in the massless limit for T p also analytically. In order to investigatethe viscosity of QCD matter with zero chemical potential and not too high temperatures,T < Tc, we now consider a pion gas in the two flavor case Nf = 2 within the frameworkof chiral perturbation theory (χPT).

We have to check carefully whether we are allowed to use results of the previous chap-ters: the good news is that the derivation of the Kubo-type formula (4.37) for the shearviscosity η in chapter 4 holds. In contrast the explicit identity (6.68) for η must berederived and receives, in general, corrections due to a more involved skeleton expansion.

7.1 Skeleton Expansion in Chiral Perturbation Theory

The Kubo-type formula (4.37) involves the viscous-stress tensor πµν defined in (4.9). Weuse the second order chiral Lagrangian L2 (A.22) for no external sources rµ = 0, lµ = 0,p = 0, but s = diag (mu,md) ≡ m, hence Dµ = ∂µ and χ = 2B0m = m2

π. Expanding L2

up to fourth order in the pion fields, we arrive at

L2 ≈ L2π2 + L4π

2 =1

2∂µπ · ∂µπ −

m2π

2π2 +

1

2f 20

(π · ∂µπ) (π · ∂µπ)− m2π

8f 20

(π2)2, (7.1)

where f0 is the pseudoscalar decay constant in the chiral limit. Only the derivative part ofthe energy-momentum tensor Tµν = ∂L

∂(∂µπ)· ∂νπ − gµνL contributes to the viscous-stress

tensor:

πµν = (∆ρµ∆σ

ν −1

3∆µν∆

ρσ)Tρσ = (∆ρµ∆σ

ν −1

3∆µν∆

ρσ)∂ (L2π

2 + L4π2 )

∂(∂ρπ)· ∂σπ =

= (∆ρµ∆σ

ν −1

3∆µν∆

ρσ)

[(∂ρπ) · (∂σπ) +

1

f 20

π (π · ∂ρπ) ∂σπ

].

(7.2)

In comparison to the case of φ4 theory (6.3) there is an additional derivative term oforder O(f−2

0 ) coming from the momentum-dependent chiral interaction. This will lead toadditional terms in the skeleton expansion. We need to calculate the contraction of thistensor at two different space time points 0 = (0, 0) and x = (x, τ):

πµν(0)πµν(x, τ) = (∆ρα∆σβ − 1

3∆ρσ∆αβ)

[(∂ρπ(0) · ∂σπ(0))(∂απ(x, τ) · ∂βπ(x, τ)) +

+1

f 20

(∂ρπ(0) · ∂σπ(0)) (π(x, τ) · ∂απ(x, τ)) (π(x, τ) · ∂βπ(x, τ)) +

+1

f 20

(∂απ(x, τ) · ∂βπ(x, τ)) (π(0) · ∂ρπ(0)) (π(0) · ∂σπ(0)) +

+1

f 40

(π(0) · ∂ρπ(0)) (π(0) · ∂σπ(0)) (π(x, τ) · ∂απ(x, τ)) (π(x, τ) · ∂βπ(x, τ))

].

(7.3)

75


With this we arrive at the momentum-integrated thermal Green’s function in Fourierspace:

Πβ(ωn) =

∫ β

0

dτ eiωnτ∫

d3p

(2π)3(∆µρ∆νσ − 1


·[

1

V 2〈Tτ [π(0) · π(0) π(p, τ) · π(p, τ)]〉+

1

f 20V

4

⟨Tτ[π2(0) π4(p, τ)

]⟩+

+1

f 20V

4

⟨Tτ[π4(0) π2(p, τ)

]⟩+

1

f 40V

6

⟨Tτ[π4(0) π4(p, τ)

]⟩].

(7.4)

Here we have used the Convolution Theorem and the Fourier decomposition of the pionfields in position space. The inverse-volume prefactors arrise due to redundant Fourierintegrals V

∫d3p 1 = (2π)3. The first time-ordered thermal correlator A ∼ 1/V 2 is

already known from our discussion in φ4 theory. Its diagrammatic skeleton expansionis shown in (6.10). The other terms at order O(V −4) (B1, B2) and O(V −6) (C) read atleading order skeleton expansion:

B1 =1

f 20V

4

⟨Tτ[π2(0)π4(p, τ)

]⟩= + ,

B2 =1

f 20V

4

⟨Tτ[π4(0)π2(p, τ)

]⟩= + ,

C =1

f 40V

6

⟨Tτ[π4(0)π4(p, τ)

]⟩=

= + + .

(7.5)The disconnected parts of the skeleton expansion are vacuum bubbles and can be dropped.Furthermore, the first terms of B1, B2, and the second one of C, are just corrections of theskeleton vertex. Since the skeleton expansion is an expansion in fully dressed quantities,these terms are already included in diagrams without loops; in both cases this is justthe leading order diagram in (6.10). Only at order O(f−4

0 ) there is an additional termin the skeleton expansion C which gives rise to a correction of the momentum-integratedthermal Green’s function:

Πβ(ωn) = 2(N2f − 1

) ∫ β

0

dτ eiωnτ∫

d3p

(2π)3(∆µρ∆νσ − 1


·[G2β(p, τ) +

12

f 40V

2G4β(p, τ)

].

(7.6)

In comparison to our calculation in φ4 theory (6.11) we have the additional global prefactorN2f − 1 which accounts for the flavor symmetry:

〈Tτ[πa0π

a0π

bpπ

bp

]〉 ≈ 〈Tτ

[πa0π

bp

]〉〈Tτ

[πa0π

bp

]〉 δab=

(N2f − 1

)〈Tτ [π0πp]〉2 . (7.7)

Of course, in our two-flavor case we have three pions. Furthermore, the leading order termof the skeleton expansion (7.6) has a symmetry factor 2 = 2! , whereas the correction termhas the factor 24 = 4! .

76


In this χPT approach we consider only the leading term of the Green’s function Πβ(ωn)and neglect higher order terms O(f−4

0 ). This approximation is consistent with the order ofthe Lagrangian L2 we took into account. Inspecting the derivation of the spectral width,ρ(p), and the definition of the spectral function Γ(Ep) in chapter 6, the shear viscosity ofa two-flavor pion gas becomes:

η =3β

20

∫d3p

(2π)3

(∆µν∆ρσ − 1

3∆µρ∆νσ

)pµpνpρpσ n(Ep) [1 + n(Ep)]

1

E2pΓ(p)

. (7.8)

Again, as in (6.26), the energy dependent spectral width Γ(Ep) is related to the imaginarypart of the pion self-energy:

Γ(Ep) ≡ −1

2EpIm ΣR(Ep) . (7.9)

In the following we will investigate the different orders of the pion self energy within theframework of χPT in order to calculate the temperature dependent shear viscosity η forphysical pion masses.


Consider the first correction of the pion self energy Σ(1)β (ωn,p; a, b) in figure 7.1(a), which

has external legs πa, πb with momentum pµ = (iωn,p) and a pion loop πc with momentumqµ = (iωm,q). Using the Feynman rules of χPT in appendix A.5 we get:

Σ(1)β (ωn,p) = − 1

2f 20

T∑m∈Z

∫d3q

(2π)3

δacδbc (s−m2π) + δacδbc (t−m2

π) + δabδcc (−m2π)

ω2m + q2 +m2

π

.

(7.10)Here we used the Mandelstam variables s = (p + q)2, t = (p − q)2, u = 0. In the s- andt-channel the isospin indices are forced to be equal, a = b = c, whereas in the u-channelall three pions contribute to the loop: δabδcc = 3δab. The numerator without δab reads

s+ t− 5m2π = 2p2 + 2q2 − 5m2

π = −2(ω2m + q2 +m2

π) + 2p2 − 3m2π . (7.11)

Therefore we arrive at a divergent expression and apply the method of dimensional regu-larization as shown in appendix A.6:

Σ(1)β (ωn,p) = − δab

2f 20

T∑m∈Z

∫d3q

(2π)3

[−2 +

2p2 − 3m2π

ω2m + q2 +m2

π

]=

= − δab2f 2

0

µ4−d T∑m∈Z

∫dd−1q

(2π)d−1

[−2 +

2p2 − 3m2π

ω2m + q2 +m2

π

].

(7.12)

The scaleless term −2 has characteristic (A.46), χ = d− 1 > 0 for d > 1, hence it can beset to zero within the framework of dimensional regularization. For the remaining termthe Matsubara sum can be carried out explicitly by using the master formula (5.31):

Σ(1)β (ωn,p) = − δab

2f 20

µ4−d∫

dd−1q

(2π)d−1

2p2 − 3m2π√

q2 +m2π

[1

2+ n(

√q2 +m2

π)

]=

= −δab (2p2 − 3m2π)

2f 20

[1

2µI(3, 1/2) +

4π

(2π)3

∫ ∞mπ

dω n(ω)√ω2 −m2

π

].

(7.13)

77


πa, pμ πb, pμ

πc, qμπa, pμ πa, pμπc, pμ2

πd, pμ3

πb, pμ1

(a) first order (b) second order

Figure 7.1: Self-energy of the pion Σ(1)β and Σ

(2)β contributing to the shear viscosity

Only the first term must be treated by dimensional regularization. The second term isexponentially damped by the Bose distribution n(ω) and can be calculated numerically.The identification of the divergent part with 1

µI(3, 1/2) results directly from its definition

(A.39). Applying now the relation (A.47) and using the explicit expression (A.44) forI(4, 1) we arrive at the renormalized self-energy:

Σ(1,ren)β (ωn,p) = −δab(2p

2 − 3m2π)

2f 20

[m2π

16π2ln

(m2π

µ2

)+

4π

(2π)3

∫ ∞mπ

dω n(ω)√ω2 −m2

π

].

(7.14)

If the external pions were on-shell, the prefactor −δab (2p2 − 3m2π) = δabm

2π would be

independent of the momentum. However, also in the off-shell case the continuation ofΣ

(1,ren)β (ωn,p) to Σ

(1,ren)R (p0,p) via (6.30), iωn 7→ p0 + iε, yields

Σ(1,ren)R (p) ∼ 2

((p0)2 − p2

)− 3m2

π , (7.15)

hence Σ(1,ren)R (p) has no imaginary part and, due to (6.28), it does not contribute to the

shear viscosity.

7.3 Second-Order Correction to the Propagator

We consider now the second order correction to the pion self-energy Σ(2)β (ωn,p; a) in figure

7.1(b). We use the Mandelstam variables s = (p1 + p2)2, t = (p− p2)2 and u = (p− p1)2:

Σ(2)β

∣∣∣η

= −1

6T∑n1∈Z

∫d3p1

(2π)3T∑n2∈Z

∫d3p2

(2π)3· 1

ω2n1

+ E21

1

ω2n2

+ E22

1

(ωn − ωn1 − ωn2)2 + E2

3

·

·(δabδcd

s−m2π

f 20

+ δacδbdt−m2

π

f 20

+ δadδbcu−m2

π

f 20

)2

.

(7.16)The third loop sum and integral were carried out and energy-momentum conservationleads to E2

3 = (p − p1 − p2)2 + m2π. There is a sum over the isospin indices b, c, d of

both vertices, but a is an external parameter. Using the identities δabδcd · δabδcd = 3 andδabδcd · δacδbd = 1, we can carry out the sums in flavor space and find for the contribution

78

7.3 Second-Order Correction to the Propagator

of the vertex:

V 2(s, t, u) ≡(δabδcd

s−m2π

f 20

+ δacδbdt−m2

π

f 20

+ δadδbcu−m2

π

f 20

)2

=

=2

f 40

·[(s−m2

π)(t−m2π) + (s−m2

π)(u−m2π) + (t−m2

π)(u−m2π)]

+

+3

f 40

(s−m2π)2 +

3

f 40

(t−m2π)2 +

3

f 40

(u−m2π)2 =

=15m4

π

f 40

+3

f 40

(s2 + t2 + u2)− 10m2π

f 40

(s+ t+ u) +2

f 40

(st+ su+ tu) .

(7.17)

So far we have not used the on-shell identity s+ t+ u = 4m2π to get a simplified V 2. As

done in the φ4 theory, we use the same techniques of partial-fraction decomposition forcoupled and uncoupled Matsubara propagators, (6.58) and (6.60), respectively, and arriveat

Σ(2)β

∣∣∣η

= −1

6

∫1

∫2

∑si=±1

s1s2s3 · V 2

8E1E2E3

[n(s2E2)− n(−s3E3)] [n(s1E1)− n(−s2E2 − s3E3)]

s1E1 + s2E2 + s3E3 + iωn.

(7.18)Performing the continuation of the thermal propagator to the retarded self-energy, ΣR,via iωn 7→ p0 + iε, we find for the spectral width (7.9):

EpΓ(Ep) = −1

2Im ΣR(Ep) = −1

2limε→0

Im[

Σβ(ωn,p)|iωn 7→Ep+iε

]=

=1

8n−1(Ep)

3∏i=1

∫d3pi

(2π)3 2EiV 2

on n(E1)n(E2)[1 + n(E3)] (2π)4δ(4)(p− p1 − p2 + p3) .

(7.19)The principle analysis thereby is very similar to the previous chapter: in the limit ε→ 0(6.61) yields the conservation condition δ(Ep + s1E1 + s2E2 + s3E3) which has threecontributions with signature (−− +) as shown in figure 6.2. This signature leads to thesubstitution V 2 7→ V 2

on, given by

V 2on ≡ V 2

∣∣s+ t+u= 4m2

π=

1

f 40

[3(s2 + t2 + u2) + 2(st+ su+ tu)− 25m4

π

]=

=1

f 40

[2(s2 + t2 + u2)− 9m4

π

].

(7.20)

Indeed, only the on-shell part of V 2 contributes to the spectral width. This can be seenby the following categorization: in the off-shell case there are terms which lead to purereal contributions to the pion thermal self energy Σ

(2)β . Additionally there are divergent

terms which either do not contribute to the shear viscosity in dimensional regularization.To make this statement more concrete, consider for instance the st-term in the last lineof (7.17):

st =(p2

1 + p22 + 2p1p2

) (p2 + p2

2 − 2pp2

)=

= p21p

2 + p21p

22 + p2

2p2 + p4

2 + 2[p2 p1p2 + p2

2 p1p2 − p21 pp2 − p2

2 pp2 − 2(p1p2)(pp2)].

(7.21)We suppress the explicit · of the scalar product pipj, since it obeys Euclidean metric dueto the thermal self energy. Consider now the contribution of its first term p2

1p2:

79


Σ(2)β

∣∣∣η3 T

2

f 40

∑n1,2∈Z

∫1

∫2

2p2p21(

ω2n1

+ E21

) (ω2n2

+ E22

) ((ωn − ωn1 − ωn2)

2 + E23

) =

=T 2

f 40

∑n1,2∈Z

∫1

∫2

2p2(ω2n1

+ E21)− 2p2m2

π(ω2n1

+ E21

) (ω2n2

+ E22

) ((ωn − ωn1 − ωn2)

2 + E23

) . (7.22)

In the first term of this numerator one Matsubara propagator cancels and no imaginarypart will survive:

Σ(2)β

∣∣∣η3 T

2

f 40

∑n1,2∈Z

∫1

∫2

2p2(ω2n2

+ E22

) ((ωn − ωn1 − ωn2)

2 + E23

) (6.58)=

=T 2

f 40

∑n1,2∈Z

∫1

∫2

∑si=±1

s2s3

4E2E3

2p2

(iωn2 + s2E2) (i(ωn − ωn1 − ωn2) + s3E3)

(6.60)=

=T

f 40

∑n1∈Z

∫1

∫2

∑si=±1

s2s3

4E2E3

2p2 [n(s2E2)− n(−s3E3)]

s2E2 + s3E3 + iωn − iωn1

(5.30)=

=1

f 40

∫1

∫2

∑si=±1

s2s3

4E2E3

2p2 [n(s2E2)− n(−s3E3)]

[−1

2− n[−s2E2 − s3E3]

].

(7.23)This term does not have any ωn dependence anymore, hence its continuation generates noimaginary part and does not contribute to the shear viscosity. Furthermore, the secondterm of the numerator (7.22) restores, with an additional prefactor −2f−4

0 m2πp

2, just thewell-known analysis within φ4 theory. In this case the continuation iωn 7→ Ep + iε leadsin the limit ε→ 0 to

p2 = ω2n + p2 7→ −

(E2p + 2iEpε− ε2 − p2

)→ −p · p = −m2

π , (7.24)

hence the prefactor of the retarded propagator reads 2f−40 m4

π. This prefactor is alreadyincluded in V 2

on of (7.19).We write the spectral width in another representation using the identity

n(Ei) =1

2e−

12βEi sinh−1

(βEi

2

)(7.25)

and the energy conservation exp(β2(Ep − E1 − E2 + E3)

)= 1. Finally (7.19) reads

EpΓ(Ep) =1

4sinh

(βEp

2

) 3∏i=1

∫d3pi(2π)3

V 2on

4Ei sinh(βEi

2

) (2π)4δ(4)(p− p1 − p2 + p3) .

(7.26)Taking into account that [GL89] defines the spectral width in a different way, γ = 2Γ,our result (7.26) coincides with equation (22) therein. Note, that V 2

on (7.20) is the squareof the well-known pion-pion scattering amplitude. In φ4 theory this statement holds, too,since we derived there the constant prefactor V 2

on ∼ λ2 (6.69).


As already done for the φ4 theory in section 6.5, also in the case of χPT we have to get ridof the ill-conditioned delta functions in (7.19) and (7.26) in order to calculate the spectral

80


width Γ(Ep) numerically. The loop integral∫d3p1 is canceled by the three-dimensional

delta function, hence we can express the corresponding energy E1 by the incoming andby the two remaining loop momenta:

E21 = m2

π + (p + p3 − p2)2 = E2p + |p3 − p2|2 + 2p|p3 − p2| cos θp . (7.27)

In χPT it is more involved to integrate out the formal direction dependence of Γ(Ep) onp, cos θp, because V 2

on depends on cos(θp) and cos(ϕp), whereas in the φ4 theory it was theconstant λ2. We integrate out these dependences using

1 =1

4π

∫dΩ =

1

2π

∫ 2π

0

dϕp1

2

∫ 1

−1

d cos θp . (7.28)

Performing the cos θp integral, the remaining delta function fixes its value:

1

2

∫ 1

−1

d cos θp Von(cosϕp, cos θp)1

2E1

δ(Ep − E1 − E2 + E3) =

=1

4pQV 2

on(cosϕp, cos θp)

∣∣∣∣cos θp=x0

,

(7.29)

where we have used Q ≡ |p3 − p2| and x0 ≡ 12pQ

[(E3 − E2)(E3 − E2 + 2Ep)−Q2].

So far we have introduced the angles cos θp = cos^(p,p3 − p2) and ϕp ⊥ θp. Considerthe following definitions:

cosϕ ≡ cos^(p,p3 + p2) ,

cos ∆ ≡ cos^(p3 − p2,p3 + p2) ,

R ≡ |p3 + p2| .(7.30)

It follows immediately that R can be expressed in terms of Q:

R =√

2E23 + 2E2

2 − 4m2π −Q2 . (7.31)

Furthermore, it is possible to express V 2on just in terms of the sum and the difference of

p2 and p3, using s = 2m2π − 2(p · p3), t = 2m2

π − 2(p · p2) and u = 4m2π − s− t:

f 40 V

2on(cosϕp, cos θp)

(7.20)= 2(s2 + t2 + u2)− 9m4

π =

= 7m4π + 16m2

π [p · (p3 − p2)] + 16(p · p3)2 + 16(p · p2)2 − 16(p · p3)(p · p2) =

= 7m4π + 16m2

π [p · (p3 − p2)] + 12 [p · (p3 − p2)]2 + 4 [p · (p3 + p2)]2 =

= 7m4π + 16m2

π [Ep(E3 − E2)− p(p3 − p2)] + 12 [Ep(E3 − E2)− p(p3 − p2)]2 +

+ 4 [Ep(E3 + E2)− p(p3 + p2)]2 =

= 7m4π + 16m2

π [Ep(E3 − E2)− pQ cos θp] + 12 [Ep(E3 − E2)− pQ cos θp]2 +

+ 4 [Ep(E3 + E2)− pR cosϕ]2 .(7.32)

The second argument of V 2on(cosϕp, cos θp), is already fixed by (7.29). The first argument,

cosϕp, can be related to the second variable in (7.32), cosϕ, using spherical trigonometry:

cosϕ = cos θp cos ∆ + cosϕp√

1− cos2 θp√

1− cos2 ∆ . (7.33)

81


Using its definition and the relation (7.31), cos ∆ is related only to energies and Q:

cos ∆(7.30)=

p3 − p2

Q

p3 + p2

R=E2

3 − E22

QR(Q). (7.34)

Plugging (7.34) into (7.32), V 2on becomes a function depending on cosϕp only:

V 2on(cosϕp) = A0 + A1 cosϕp + A2 cos2 ϕp , (7.35)

where we have introduced the coefficients Ai = Ai(Ep,p2,p3,mπ, Q). Hence, we can carryout the polar integration of (7.28):

1

2π

∫ 2π

0

dϕd(A0 + A1 cosϕp + A2 cos2 ϕp

)= A0 +

1

2A2 . (7.36)

Taking now the two remaining momentum integrals∫d3p2 and

∫d3p3 of (7.26) into

account, we have to ensure that the integration region is restricted to the case |x0| ≤ 1.According to (6.74), this is equivalent to implement the factor Θ(F (Q)). Using again∫d3pi =

∫dpi p

2i

∫dΩi and Ei dEi = pi dpi, we arrive at the expression∫

d3p2

E2

∫d3p3

E3

(A0 +

1

2A2

)1

4pQΘ(F (Q)) =

=1

4p

∫ ∞mπ

dE3

∫ Ep+E3−mπ

mπ

dE2

∫dΩ2

∫dΩ3

p2p3

Q

(A0 +

1

2A2

)Θ(F (Q)) .

(7.37)

Using the defintion of Q, we get directly Q2 = p22 +p2

3−2p2p3 cos θ23. As done in a similarway in (6.77), we can transform the two remaining angular integrals to an integral withrespect to Q. Inspecting the strucures of A0 and A2, it can be written as sum of rationalexpressions in Q:

1

4p

∫dΩ2

∫dΩ3

p2p3

Q

(A0 +

1

2A2

)Θ(F (Q)) =

=8π2

4p

∫ p2+p3

|p2−p3|dQ

[X4

5Q4 +

X2

3Q2 +X0 +X−2Q

−2 +X−4

3Q−4

]Θ(F (Q)) =

=(2π)2

2p

[X4

(Q5

+ −Q5−)

+X2

(Q3

+ −Q3−)

+X0 (Q+ −Q−) +

+X−2

(1

Q−− 1

Q+

)+X−4

(1

Q3−− 1

Q3+

)].

(7.38)Again, Θ(F (Q)) gives rise to a max/min structure of Q±, and we find the very samedefinition as in φ4 theory (6.78):

Q− ≡ max |p2 − p3|, |p− p1| ,Q+ ≡ min p2 + p3, p+ p1 .

(7.39)

Collecting all the coefficients, we arrive at a well-conditioned spectral width with onlytwo remaining energy integrals:

EpΓ(Ep) =1

f 40 (8π)3

n−1(Ep)

p

∫ ∞mπ

dE3

∫ Ep+E3−mπ

mπ

dE2 X(Q+, Q−) ·

· n(E1)n(E2)(1 + n(E3)) Θ(Q+ −Q−) ,

(7.40)

82


of course with E1 = Ep − E2 + E3 and the rational function

X(Q+, Q−) ≡ X4

(Q5

+ −Q5−)

+X2

(Q3

+ −Q3−)

+X0 (Q+ −Q−) +

+X−2

(1

Q−− 1

Q+

)+X−4

(1

Q3−− 1

Q3+

).

(7.41)

Its coefficients Xi have mass dimensions [Xi] = 4 − i and they depend only on the pionmass mπ and energies E2, E3 and Ep:

X4 =7

10,

X2 =2

3

[6m2

π − E2p − 4E2

2 − 4E23 + EpE2 − EpE3 + 7E2E3

],

X0 = 15m4π + 8m2

π

(3E2E3 + EpE2 − EpE3 − E2

p − 2E22 − 2E2

3

)+

+ 2Ep (Ep − E2 + E3)(5E2

2 + 2E2E3 + 5E23

)+

+ (E2 − E3)2 (7E22 − 4E2E3 + 7E2

3

),

X−2 = 4m2π (E2 − E3)2 (2E2

p + E22 + E2

3 − 2EpE2 + 2EpE3

)−

− 2Ep (E2 − E3)2 (Ep − E2 + E3)(7E2

2 + 10E2E3 + 7E23

)−

− 2 (E2 − E3)4 (2E22 + 3E2E3 + 2E2

3

),

X−4 =1

2(E2 − E3)4 (2Ep − E2 + E3)2 (E2 + E3)2 .

(7.42)

Alternatively, using the identy (7.25), the spectral width can be expressed in the equivalentrepresentation (7.26):

EpΓ(Ep) =1

4f 40 (8π)3

sinh βEp2

p

∫ ∞mπ

dE3

∫ Ep+E3−mπ

mπ

dE2 X(Q+, Q−) ·

·[sinh

(βE1

2

)sinh

(βE2

2

)sinh

(βE3

2

)]−1

Θ(Q+ −Q−) ,

(7.43)

again with E1 = Ep − E2 + E3.For the numerical evaluation of the shear viscosity η(T ;m) in (7.8), we use for the chiral

crossover temperature Tc = 155 MeV, for the physical pion mass mπ = 140 MeV and forthe pion decay constant f0 = fπ = 93 MeV. The spectral width Γ(p;T,m) in (7.40), or(7.43), defines again the mean free path Λ(p), which is in the chiral limit just its inverse:

Λ(Ep) ≡p

Γ(Ep)Ep−−−−→mπ→0

1

Γ(p). (7.44)

The momentum dependence of the mean free path of pions with physical pion mass isshown in figure 7.2. For higher temperatures the mean free path is decreasing and, incontrast to the case of φ4 theory in figure 6.3, it is vanshing for large momenta. Thetemperature-independent maximum of Λ(p) is located at p ≈ 100 MeV. In the chirallimit, mπ → 0, the shape of Λ(p) changes qualitatively and diverges for low momenta asshown in figure 7.3.

Using the numerical results for the mean free path Λ(Ep), we are able to calculate theshear viscosity η(T ) in (7.8) numerically. Its mass dimension is three and we might choosethe units MeV3, but the classical interpretation of shear viscosity suggests to use the units

83


0 200 400 600 800 10000

5

10

15

20

25

30

p @MeVD

L@f

mD

Figure 7.2: Mean free path Λ(p;T ) of interacting pions with physical mass mπ = 140 MeVwithin χPT for different temperatures: T = 0.8Tc (red), T = 155 MeV = Tc (green) andT = 1.2Tc (blue). The temperature-dependent maximum is located at p ≈ 100 MeV.

0 100 200 300 400 5000

10

20

30

40

50

p @MeVD

L@f

mD

Figure 7.3: Mean free path Λ(p;m) of interacting pions at T = 0.8Tc for different pionmasses. In the physical case, mπ = 140 MeV (solid line), there is a maximum at p ≈ 100 MeV,whereas pions with half the physical mass (dashed line) and in the chiral limit (dotted line)feature a monotonous mean free path.

84


0.0 0.2 0.4 0.6 0.8 1.0 1.20

20

40

60

80

100

120

T Tc

Η@M

eVf

m2 D

Figure 7.4: Shear viscosity η(T ) of interacting pions with physical mass (solid line), halfthe physical mass (dashed line) and in the chiral limit (dash dotted line). In the chiral limitη(T ) ∼ T−1 is expected due to dimensional analysis. In the physical-mass case the maximumis located at T ≈ 0.15Tc ≈ 23 MeV.

’energy/area’ or ’force/length’. The results for different pion masses in figure 7.4 are givenin units MeV/fm2 as already done for the shear viscosity in φ4 theory. Apart from thechiral limit, we find for the low-temperature limit of the shear viscosity:

limT→0

η(T ) = 0 . (7.45)

In the chiral limit the only remaining energy scales for the shear viscosity (7.8) are f0 andT . Using the mean free path (7.40), we arrive at

limm→0

η(T ) ∼ f 40

T. (7.46)

Note, that the two-dimensional function η(T ;m) is not continuous at the origin, since

0 = limm→0

limT→0

η(T ;m)︸︷︷︸=0

6= limT→0

limm→0

η(T ;m)︸︷︷︸∼1/T

=∞ . (7.47)

Considering pions with physical mass, the maximal point of shear viscosity is locatedat T ≈ 0.15Tc ≈ 23 MeV. For decreasing pion masses, this maximal point slides tolower temperatures, whereas at the chiral limit, the shear viscosity is a monotonouslydecreasing function and diverges at zero temperature. In the high temperature limit theshear viscosity is approximatively independent of the pion mass.

85

8 Results in Comparison

8.1 Short Digression: AdS/CFT Correspondence

In 1998 Maldacena demonstrated that, under certain conditions, there is a duality be-tween superstring theory and superconformal field theory. More precisely, the AdS/CFTcorrespondence between supergravity on five-dimensional anti-de Sitter space, AdS5, andfour-dimensional N = 4 superconformal SU(N) Yang-Mills theory in the ’t Hooft limithas been proven. We are going to give a brief survey of the terms appearing in thatstatement. For a comprehensive treatment of these topics we refer to the text books[Wei99, Pol98, FMS97] and the publications [KSS05, Mal99].

String theory was introduced in the 1960’s, actually, in order to explain the stronginteraction. Instead of zero-dimensional fields, particles are described by one-dimensionalstrings and their oscillation modes. It turns out that every consistent string theory con-tains a massless spin-2 particle and reduces in the low-energy limit to general relativity.So far, all approaches to a quantized gravity using ordinary field theory are known to benon-renormalizable, therefore they are only effective theories of gravity. In order to avoidunphysical tachyons in string theory, we need a supersymmetric version of string theory,called superstring theory.

In a supersymmetric theory, N denotes the number of supercharges. For instance, theMinimal Supersymmetric Standard Model (MSSM) contains only one pair of supercharges.Theories with N > 1 are called extended theories. Since supercharges change the spin ofthe particles by 1/2, they must transform as spinors under the Poincare group.

The Poincare group is defined by the set of transformations which leave the metrictensor gµν invariant. A superset of the Poincare group is the conformal group, whichcontains all invertible transformations of the coordinates, xµ 7→ x′µ, such that

g′µν(x′ρ) = ξ(xρ)gµν(xρ) , (8.1)

where ξ(xρ) is a local function and for ξ ≡ 1 the Poincare transformations are recovered. Itturns out that in one dimension all invertible transformations are conformal. It is also well-known that conformal transformations in two dimensions are given by the biholomorphicmappings. This fact is important for string theory because the conformal group operateson a two-dimensional world sheet. In any dimension the conformal transformations aresuch transformations which preserve angles. Note, that a theory featuring conformalinvariance is also scale invariant. The opposite implication is known not to be true.

For the AdS/CFT correspondence it is not sufficient for the field theory to be super-conformal. Additionally, the SU(N) theory must be considered in the limit N → ∞.Inspecting (2.63), the coupling in QCD becomes small for large N . In contrast, the’t Hooft coupling defined by

g2(k)N =g2(µ)N

1 + g2(µ)(4π)2

(113N − 2

3Nf

)ln (k2/µ2)

−−−−→N→∞

48π2

11 ln (k2/µ2), (8.2)

87


converges in the large-N limit to a finite, scale-dependent expression. A Yang-Mills theoryconsidered in the large-N limit but at small energy scales is said to be in the ’t Hooftlimit. That is, the ’t Hooft coupling is large.

It is known from general relativity that mass is not only located in spacetime butthey interact: gravity is explained geometrically by deviations of the spacetime from theusual Minkowski metric. In the absence of mass the Minkowski metric is restored. Whendiscussing gravity not on n-dimensional Minkowski but on anti-de-Sitter, AdSn, spacewe have made the same step as switching from an Euclidean to a hyperbolic space inclassical physics: the AdSn space has a constant negative curvature also in the absence ofmatter. Similary, the de-Sitter, dSn, space has a constant positive curvature. It turns outthat gravity defined on a five-dimensional anti-de Sitter space includes on its conformalboundary the already discussed N = 4 superconformal SU(N) Yang-Mills theory in the’t Hooft limit.

Whereas bosonic string theories live in 26 dimensions, the superstring theories includ-ing fermions live in a ten-dimensional space. Therefore, AdS5 cannot be a domain of(super)string theory. It is possible to construct a domain of superstring theory by multi-plying with a compact manifold. In our case this can be achieved by the five-dimensionalsphere, S5, since the supersymmetric group of AdS5 × S5 is known to be the same as thesuperconformal group of four spacetime dimensions.

The AdS/CFT correspondence provides a great opportunity to study non-perturbativelow-energy physics of quantum field theories: in regions where the ’t Hooft coupling infield theory becomes large, the string coupling becomes small. Additionally, quantumeffects in AdS5 × S5 superstring theory can be interpreted as 1/N effects in field theory.

We need to consider black branes, which are black holes featuring translationally in-variant horizons. Using the AdS/CFT correspondence, in [KSS05] it has been shownthat the shear viscosity in thermal field theories can be interpreted as the absorption oflow-energetic gravitons by a black brane:

η =σabs(ω = 0)

2κ2=

a

16πG, (8.3)

where σabs(ω) denotes the energy-dependent absorption cross section and κ =√

8πG. Wehave σabs(0) = a, which is the area of the black-brane horizon. Due to this correspondence,the entropy density of the dual field theory is equal to the entropy density of the blackbrane: s = a/4G. Therefore, we find

η

s=

1

4π. (8.4)

In addition to this result which is proven for theories with gravity duals, [KSS05] conjec-tures a viscosity bound :

Most quantum field theories do not have simple gravity duals. (. . .) We spec-ulate that the ratio η/s has a lower bound η/s ≥ 1/4π for all relativisticquantum field theories at finite temperature and zero chemical potential.

So far, the AdS limit for η/s is respected by experimental results. The fundamental theorydescribing relativistic heavy-ion collions is QCD which does not possess a gravity dual,for the following reasons: QCD is neither supersymmetric nor conformal. The classicalscale independence of the QCD Lagrangian is broken anomalously by quantum effectsresulting in a running coupling g(k). In addition, QCD is a SU(3) gauge theory and canbe described by the large-N limit only approximatively. In fact, there are counterexamplesto the above conjecture [Coh07].

88

8.2 η/s in Chiral Perturbation Theory and φ4 Theory


In order to discuss the η/s ratio for the φ4 theory and chiral perturbation theory, we needto find the entropy density s of the systems. Note, that in the non-interacting limit λ→ 0in (6.1) and f0 → ∞ in (7.1), respectively, each degree of freedom is described in boththeories by the same Lagrangian:

limλ→0L|φ4 =

1

N2f − 1

limf0→∞

L|χPT . (8.5)

Therefore, for the following we are allowed to use the notation for fields with φ4 inter-action. The corresponding leading-order results for interacting pions can be obtained bymultiplying with a flavor prefactor. From chapter 5 we know already the entropy densityas a perturbative series in the coupling λ:

s(5.39)= =

∂2 (T lnZ)

∂T ∂V=∂P

∂T

(5.73)= T 3

(2π2

45− λ

12+λ3/2

3π

)+O(λ2) . (8.6)

Using the analytical result (6.84) for the shear viscosity, η ∼ T 3 within φ4 theory, the η/sratio reads at leading order O(λ−2):

η

s

∣∣∣φ4

=1

4π

480 ζ(5)

π2λ2+O(λ−1) ≈ 1

4π

50

λ2. (8.7)

For all couplings λ ∈ (0, 1) this is just a constant number. The analytical result (6.84) wasobtained using the massless limit and switching from Bose to Boltzmann statistics. Asfigure 6.4 shows, for massive particles there is a deviation in the low temperature regionfrom this simplified result. At high temperatures, T > Tc, we can use approximativelya constant for η/T 3 in φ4 theory. Equations (6.81) and (6.85) show that the analyticalresult for η needs to be multiplied by 2/3 in order to get approximatively the numericalvalue which has been derived using the physically accurate Bose statistics. The resultsfor the constant η/s ratio are shown in table 8.1.

In the case of the pion gas we do not have an analytical result for the shear viscosity,we use the numerical results shown in figure 7.4. Also for the pion gas we use equation(8.6) for the entropy density. With the identification f0 ∼ 1/

√λ, the leading order term

2π2T 3/45 appears in the expansion around f0 = ∞, too. Therefore, the leading ordercontribution of η/s at order O(f 4

0 ) is obtained by using the same entropy density as inφ4 theory. In figure 8.1 we show the temperature dependence of the η/s ratio for a piongas. We find in the chiral limit a decreasing ratio η/s ∼ T−4 due to dimensional analysis.This dependence does also hold approximatively in the case of pions with physical mass.

λ 0.6 0.7 0.8 0.9 1.0

η/s 7.4 5.5 4.2 3.3 2.7

η/s · 4π 93 69 53 42 34

Table 8.1: Analytical results for the η/s ratio for different coupling constants λ of masslessbosons. The values include an empirical factor 2/3 explained in the text.

89


0.4 0.5 0.6 0.7 0.8 0.9 1.00

5

10

15

20

T Tc

Ηs

Figure 8.1: Temperature dependence of the η/s ratio of an interacting pion gas. Thedifferent curves show pions with physical mass (solid line), half the physical mass (dashed line)and in the chiral limit (dotted line). In the vicinity of the crossover the η/s ratio is a factor7–10 larger than the AdS limit 1/4π (red line).

1.0 1.1 1.2 1.3 1.4 1.5 1.60

5

10

15

20

T Tc

Ηs

Figure 8.2: Temperature dependence of the η/s ratio in massless φ4 theory with temperaturedependent coupling strength (8.8). The different curves relate to different inital conditions:λc ∈ 0.6, 0.7, 0.8, 0.9, 1.0; the lowest curve belongs to λc = 1. The η/s ratio is at least afactor 30 larger than the AdS limit 1/4π (red line).

90


At T = Tc we find for physical pions a η/s ratio which is about a factor seven larger thanthe AdS limit 1/4π. Our results coincide quite well with [L+07] and [FFN].

As both theoretical and experimental results suggest, the η/s ratio is increasing forT > Tc [L+07] starting from its minimum at the crossover or transition temperature. Wecan use φ4 theory for qualitative orientation towards deconfined region of the QCD phasediagram. Implementing a temperature dependent coupling λ(T ) into the result (8.7) via

λ(T ) ≡ λc1 + ln(T 2/T 2

c ), (8.8)

we find a temperature dependent η/s ratio also in the case of φ4 theory. This ad-hoccoupling is motivated by asymptotic freedom of QCD: it becomes small for high tempera-tures, limT→∞ λ(T ) = 0, qualitatively consistent with the behavior of the strong couplingg(k) in (2.63) at high energies. Of course the coupling λ(T ) does not provide an accuratetreatment of the quark-gluon plasma, but it reproduces qualitatively the behavior of η/sin the deconfined phase measured at RHIC for T > Tc. The results for the η/s ratio inmassless φ4 theory including the temperature dependent coupling (8.8) and an empiricalfactor of 2/3 discussed above is shown in figure 8.2.

91

9 Summary and Outlook

In this final chapter we are going to give a brief summary of the most important resultsof this diploma thesis. The closing outlook will point out possible future extensions ofour investigations.

9.1 Summary

In chapter 3 we have described phenomenologically heavy-ion collisions by using rela-tivistic hydrodynamics introducing the parameters for the shear and bulk viscosity, η andξ, respectively. It has turned out that non-vanishing dissipative parameters do not neces-sarily lead to entropy production in dissipative systems. Rather, there exists a thresholdthat the viscosity parameters must exceed in order to produce entropy. However, thestructure of the dissipative tensor violates time-reversal symmetry, therefore all dissipa-tive systems lead to irreversible processes. In chapter 4 we have used non-equilibriumthermodynamics for systems close to global equilibrium. We have derived the Kubo-typeformula for the shear viscosity in linear-response theory:

η(x, t) = − 1

10

∫d3x′

∫ t

−∞dt′∫ t′

−∞dt 〈πµν(x, t), πµν(x′, t)〉ret , (9.1)

which relates the macroscopic parameter η to the microscopic retarded correlator. Inchapter 5 we have considered the imaginary time formalism, applied in the followingchapter 6. There, we have introduced the skeleton expansion of n-point functions in φ4

theory for which the explicit result for the shear viscosity was determined:

η =β

20

∫d3p

(2π)3

(∆µν∆ρσ − 1

3∆µρ∆νσ

)pµpνpρpσ

eβEp

(eβEp − 1)2

1

E2pΓ(Ep)

. (9.2)

Here the spectral width, Γ(p), is related to the imaginary part of diagrams in the per-turbative expansion of the Matsubara propagator. We have also derived an importantcriterion for identifying these diagrams: only momentum-dependent diagrams can con-tribute to the shear viscosity. Therefore, the leading order contribution to η in both φ4

theory and chiral perturbation theory comes from one single second-order diagram. Wehave derived in φ4 theory an analytical approximation for the shear viscosity. The analyt-ical estimate exceeds the numerical result and is applicable only in the high-temperaturelimit. Chapter 7 has presented a discussion concerning the more involved skeleton ex-pansion in χPT. Only at leading order in the pion decay constant the shear viscosity ofa pion gas can be calculated using the same formalism as in φ4 theory. At order O(f−2

0 )there are no corrections of the skeleton expansion in χPT compared to φ4 theory. Wehave derived numerically the shear viscosity for pions with physical mass and in the chirallimit. Chapter 8 finally touched important notions concerning the AdS/CFT correspon-dence. Using thermal χPT we have found for the confined region that the η/s ratio asa function of temperature is decreasing. Towards the crossover region between confined

93

9 Summary and Outlook

and deconfined matter where η/s exceeds the proposed AdS limit 1/4π by about a factorof seven. In agreement with experimental results from RHIC the η/s ratio is minimal atthe phase transition or, in our two-flavor pion gas, around the crossover region.

In φ4 theory, the η/s ratio at fixed coupling strength λ is constant as function of tem-perature. The ratio rises as λ decreases. This qualitative feature may perhaps be usefulfor orientation in the deconfined region, although φ4 theory is of course not representativeof QCD. At λ = 1 and for massless bosons the lower bound for the η/s ratio is at leastabout a factor of 30 larger than the AdS limit.

9.2 Outlook

The constant η/s ratio in φ4 theory for T > Tc is non-physical and demands furtherinvestigation. Experimental results from RHIC show an increasing η/s ratio for T > Tc .We cannot expect to describe the quark-gluon plasma in terms of a simple φ4 theory,neither with a constant coupling strength nor with a temperature-dependent coupling.However, due to asymptotic freedom it is possible to investigate the shear viscosity ofthe quark-gluon plasma using perturbative QCD. This has been done already [CDOW10,A+03, A+00] and it turns out that the perturbative approach is reasonable only for quitelarge temperatures T & 5Tc .

In chapter 5 the imaginary time formalism for bosonic thermal field theories has beenintroduced. For a calcuation of the shear viscosity in full QCD we need additionally thetreatment of fermionic degrees of freedom.

It is also instructive to investigate the shear viscosity of NJL and PNJL models. Theformer model can be used only for the deconfined region as already done in [FI08]. Incontrast to both the NJL model and perturbative QCD, the PNJL model is expected todescribe the behavior of the η/s for temperatures below and above Tc .

Furthermore, there are additional transport coefficients apart from the shear viscosity,namely the bulk viscosity and the heat conductivity. We can neglect heat conduction aslong as we restrict ourself just to the temperature axis of the QCD phase diagram. It isan upcoming challenge to derive transport coefficients of perturbative or effective theorieswhich describe the QCD phase diagram also at non-zero chemical potentials.

94

A Appendix

A.1 Conventions and Notations

The metric gµν obeys the signature (+−−− . . .) for µ, ν = 1, . . . , d .

Four vectors are denoted by x, q, p, k, . . . , Euclidean vectors by x,q,p,k, . . .

The Minkowski space in four dimensions is denoted by M, the spinor space by S.

Operators are written with an hat: H, K, U , . . .

Our convention for the Fourier transformation reads

f(x) =

∫ddp

(2π)de−ipxf(p) , f(τ) =

1

β

∑n∈Z

e−iωnτf(ωn) ,

f(p) =

∫ddx eipxf(x) , f(ωn) =

∫ β

0

dτ eiωnτf(τ) .

(A.1)

For a given operator O we call GO a Green’s function of O if

O(x)GO(y) = δ(d)(x− y) . (A.2)

The definition of the thermal Green’s function (5.25) is the same as in [KG06],whereas the real-time Green’s functions (5.62) are defined with an additional −i, incontrast to [PS95] and [KG06]. The spectral representations of thermal and retardedpropagator (5.63) and (5.69), respectively, coincide with [Bel00] and, except theminus sign in the thermal one, with [KG06].

Path integrals (5.12) with certain position space boundary conditions, q(ti) = qiand q(tf ) = qf , are denoted by

∫Dq. Generating functionals are actually sums

of all amplitudes∫Dq =

∫ddq∫Dq, but we suppress the tilde. The (anti)periodic

boundary conditions (5.21) are denoted by a circle:∮Dq .

The mass dimension of frequently used quantities and correlators are summarizedin the following table:

x [x] x [x] x [x]

β −1 ρ −2 η 3

φ(x, τ) 1 φ(p, τ) −2 φ(p, ωn) −3

ΠR(t) 5 ΠR(p0) 4 ΠR(p) −2

Πβ(τ) 5 Πβ(ωn) 4

Gβ(p, τ) −1 Gβ(p, ωn) −2 Gβ(x, τ) 2

95

A Appendix

A.2 Master Formulas for Bosonic Matsubara Sums

Claim: The identities (5.28)-(5.31) hold: For x /∈ iZ we have

S(1)(x) ≡∑n∈Z

1

in+ x= 2π

(1

2+

1

e2πx − 1

), (A.3)

and for x, y /∈ Z and y 6= x

S(2)(x, y) ≡∑n∈Z

1

(n+ x)(n+ y)=π(cotπx− cot πy)

y − x . (A.4)

Note, that both series are not unconditionally convergent, hence we take the limit in thefollowing order: n = 0, 1,−1, 2,−2, . . .

Proof: We start with S(2), doing a partial fraction decomposition:

S(2)(x, y) =∑n∈Z

1

(n+ x)(n+ y)=

1

y − x∑n∈Z

[1

x+ n− 1

y + n

]=

=1

y − x

[1

x− 1

y+∑n∈N

(1

x+ n− 1

y + n+

1

x− n −1

y − n

)]=

=1

y − x

[1

x+∑n∈N

(1

x+ n+

1

x− n

)− 1

y−∑n∈N

(1

y + n+

1

y − n

)].

(A.5)

Comparing this to the well-known partial fraction decomposition of cot(·) yields (A.4):

π cot πx =1

x+∑n∈N

(1

x+ n+

1

x− n

). (A.6)

In the complex plane both master formulas are connected by the following relation:

S(1)(x) =1

x+∑n∈N

[1

in+ x+

1

−in+ x

]=

1

x+∑n∈N

2x

n2 + x2=

= x

[1

x2+ 2

∑n∈N

1

n2 + x2

]= x · S(2)(ix,−ix) .

(A.7)

Using the fact that cot(·) is an odd function and the identity cot(ix) = −i coth(x) we canderive the explicit form of S(1):

S(1)(x)(A.7)= x · S(2)(ix,−ix)

(A.4)= π coth(πx) = 2π

(1

2+

1

e2πx − 1

). (A.8)

The physical Matsubara sums (5.30) and (5.31) are direct collolaries of (A.3) and (A.4).For completeness we want to state, that S(2)(x, x) is absolute convergent for x /∈ Z and

S(2)(x, x) =π2

sin2(πx), (A.9)

which follows immediately from (A.4) using L’Hopital’s rule.

96

A.3 Matsubara Propagator in Spectral Representation

A.3 Matsubara Propagator in Spectral Representation

In order to derive the spectral representation of the Matsubara propagator in (6.13)-(6.15),

Gβ(p, τ) =

∫ ∞−∞

dω ρ(ω,p)n(ω) eω|τ | , τ ∈ [−β, β] , (A.10)

we used the following claim:

1

β

∑n∈Z

e−iωnτ

iωn − ω=

n(−ω)e−ωτ , for 0 < τ < β ,

−n(ω)e−ωτ , for − β < τ < 0 .(A.11)

Proof: We start with the case 0 < τ < β: consider the complex function

Fτ>0 ≡1

2πi

∫Cdξ

e−ξτ

1− e−ξβ1

ξ − ω ≡1

2πi

∫Cdξ fτ>0 = 0 . (A.12)

Its pole structure and the path of integration C = C1 ∪ C2 are shown in figure A.1: on theimaginary axis we find all Matsubara frequencies iωn = 2πinT and we have C = C1 ∪ C2

with C1 = ε + it | t ∈ R and C2 = ε − it | t ∈ R for 0 < ε < ω. For both largenegative and positive real parts of ξ, the integrand fτ>0 of Fτ>0 vanishes for 0 ≤ τ ≤ β:

limξ→∞

fτ>0 = 0 , (A.13)

limξ→−∞

fτ>0L′H= lim

ξ→−∞

τ 2 eξ(β−τ)

2β − β2(ξ − β)= 0 . (A.14)

ξ

2πiT

4πiT

6πiT

−2πiT

−4πiT

−6πiT

0 ω

C

Figure A.1: Pole structure and integration path C of Fτ = 0 defined in the text

97

A Appendix

Note, that both limits hold in the cases τ = 0 and τ = β, too. Due to (A.13) and (A.14)we are allowed to use Cauchy’s Residue Theorem to evaluate Fτ>0 and arrive at

0 =1

β

∑n∈Z

e−iωnτ

iωn − ω+

e−ωτ

1− e−βω . (A.15)

This implies immediately our claim (A.11) in the case 0 < τ < β .

For the case −β < τ < 0 we consider a similar function

Fτ<0 ≡1

2πi

∫Cdξ

e−ξτ

1− e+ξβ

1

ξ − ω ≡1

2πi

∫Cdξ fτ<0 = 0 , (A.16)

with the very same pole structure as Fτ>0. This time the vanishing integrand is trivialfor large negative real parts of ξ:

limξ→−∞

fτ<0 = 0 , (A.17)

limξ→∞

fτ<0L′H= lim

ξ→∞

−τ 2 e−ξ(τ+β)

2β + β2(ξ − β)= 0 . (A.18)

Note, that again both limits hold in the cases τ = 0 and τ = −β, too. Using againCauchy’s Residue Theorem, we arrive at

0 = − 1

β

∑n∈Z

e−iωnτ

iωn − ω+

e−ωτ

1− eβω , (A.19)

which implies our claim (A.11) in the case −β < τ < 0 .

A.4 Curie’s Theorem

The correlator (X, Y ) defined in the last line in (4.18) deals with tensors of general ranksjX and jY with respect to the proper orthochronous Lorentz group SO+(1, 3):

Xµ1,...,µjX , Y µ1,...,µjY . (A.20)

The representations of X and Y are assumed to be irreducible. The correlator (X, Y ) is afunction of both tensors and must offer a scaler representation. In general the correlatortransforms as one of the irreducible terms of the reducible tensor product:

X ⊗ Y =

jX+jY⊕j= |jX−jY |

Rj , (A.21)

with irreducible representations Rj of rank |jX − jY | ≤ j ≤ jX + jY . This is just the well-known general Clebsch-Gordan decomposition of a tensor product. A scalar representationR0 in the decomposition forces the two initial ranks to be equal: jX = jY . Therefore, thecorrelator (X, Y ) of two Lorentz tensors with different ranks must vanish.

98

A.5 Feynman Rules for L2 in χPT

A.5 Feynman Rules for L2 in χPT

The second order Lagrangian in chiral perturbation theory (χPT) is given by [Sch02]

L2 =f 2

0

4Tr(DµU(DµU)† + χU † + Uχ†

), (A.22)

with Dµ = ∂µ − irµ + ilµ the covariant derivative, χ = 2B0(s + ip) the scalar s andpseudoscalar p local external source, the two low-energy constants f0 and B0 and thenon-linear realization of the Goldstone boson fields U = U(~π). We restrict ourself tothe two-flavor case Nf = 2, justified by the assumption that in a first approach thedynamics of K and η in a pion gas at T . Tc = 155 MeV can be neglected due tomη > mK > Tc > mπ. Only the scalar external source is non-vanishing in the case ofmassive pions without additional interaction terms: s = diag (mu,md) = m. Due to theGell-Mann-Oakes-Renner relation we have m2

π = 2B0m and f0 is related to the pion decayconstant f0 = fπ +O(m2, p2). For our purpose it is convenient to choose the square-rootrealization

U(x) =1

f0

(√f 2

0 − π2(x) + iσ · π(x)

)= 1− 1

2

π2

f 20

− 1

8

(π2)2

f 40

+ . . .+i

f0

σ · π , (A.23)

which is clearly unitary. Therefore, we have

∂µU(x) = − 1

f 20

π · ∂µπ√1− π2/f 2

0

+i

f0

σ · ∂µπ =

= − 1

f 20

π · ∂µπ −1

2f 40

π2 π · ∂µπ + . . .+i

f0

σ · ∂µπ = ∂µU† +

2i

f0

σ · ∂µπ .(A.24)

Plugging in this expansion into (A.22), we arrive at L2 ≡ L2π2 + L4π

2 + L6π2 + . . . with

L2π2 =

1

2∂µπ · ∂µπ −

1

2m2ππ

2 ,

L4π2 =

1

2f 20

(π · ∂µπ) (π · ∂µπ)− m2π

8f 20

(π2)2 ,

L6π2 =

1

2f 40

(π · ∂µπ) (π · ∂µπ)π2 − m2π

32f 40

(π2)3 .

(A.25)

We dropped the constant term L0π2 in the expansion. In general we have no terms with

an odd number of pion fields and we find L2nπ2 = O(f

−2(n−1)0 ). The Feynman rules for the

2n-point correlators can be obtained just by taking derivatives:

M2nπi =

2n∏j=1

∂

∂πjLi∣∣∣∣∣π=0

. (A.26)

Using this rule, we get for the two-vertex:

M2π2 =

∂

∂πa∂

∂πbL2π

2 =∂

∂πa[(−ipµπb)(ipµ)−m2

ππb]

=

= δab(p2µ −m2

π

).

(A.27)

At this order O(p2,m2) we get for the four-vertex:

M4π2 =

∂

∂πa∂

∂πb∂

∂πc∂

∂πd

[1

2f 20

(∂µπi)πi(∂µπj)πj −m2π

8f 20

πiπiπjπj]. (A.28)

99

A Appendix

πa, pµ πb, pµ

πa, pµ

πb, qµ

πd, lµ

πc, kµ

(a) two-vertex (b) four-vertex

Figure A.2: Conventions for the two- and four-vertex of L2 in χPT

Performing the differentiations step by step we find:

∂

∂πd[(∂µπi)πi(∂µπj)πj] = 2ilµπd(∂µπj)πj + 2(∂µπd)(∂µπj)πj , (A.29)

∂

∂πc(A.29) = 2δcdil

µ(∂µπj)πj + 2ilµπdikµπc + 2ilµπd(∂µπc) +

+ 2δcdikµ(∂µπj)πj + 2(∂µπd)ikµπc + 2(∂µπd)(∂µπc) ,

(A.30)

∂

∂πb(A.30) = 2δcdil

µ(−iqµ)πb + 2δcdilµ(∂µπb) + 2ilµδbdikµπc + 2ilµπdikµδbc +

+ 2ilµδbd(∂µπc) + 2ilµπd(−iqµ)δbc + 2δcdikµ(−iqµ)πb + 2δcdik

µ(∂µπb) +

+ 2(−iqµδbd)ikµπc + 2(∂µπd)ikµδbc + 2δbd(−iqµ)∂µπc + 2δbc(−iqµ)(∂µπd) ,(A.31)

∂

∂πa(A.31) = 2δabδcd(l · q) + 2δabδcd(p · l)− 2δacδbd(l · k)− 2δadδbc(l · k) +

+ 2δacδbd(l · p) + 2δadδbc(q · l) + 2δabδcd(k · q) + 2δabδcd(k · p) +

+ 2δacδbd(q · k) + 2δadδbc(p · k)− 2δacδbd(p · q)− 2δadδbc(q · p) =

= 2δabδcd (p+ q) · (k + l) + 2δacδbd (p− k) · (l − q) + 2δadδbc (p− l) · (k − q) .(A.32)

Similarly, but much simpler, we get for the mass-term contribution:

M4π2 =

∂

∂πa∂

∂πb∂

∂πc∂

∂πdπiπiπjπj =

∂

∂πa∂

∂πb∂

∂πc4πdπjπj =

=∂

∂πa∂

∂πb[4δcdπ

jπj + 8πdπc]

=∂

∂πa8[δcdπ

b + δbdπc + δbcπ

d]

=

= 8 [δabδcd + δacδbd + δadδbc] .

(A.33)

In summary, using the Mandelstam variables s = (p+q)2 = (k+ l)2, t = (p−k)2 = (l−q)2

and u = (p− l)2 = (k − q)2, we arrive for the four-vertex (A.28) at

M4π2 = δabδcd

s−m2π

f 20

+ δacδbdt−m2

π

f 20

+ δadδbcu−m2

π

f 20

. (A.34)

Of course, there are vertices of any even order of the pion fields πa at the considered orderO(p2,m2) in L2.

100

A.6 Dimensional Regularization

A.6 Dimensional Regularization

In order to regularize divergencies in loop calculations within the framework of χPT, westate here the most important features of dimensional regularization, which is as well aconvenient regularization scheme for thermal field theories, too. We denote the volumeof a d-dimensional ball with radius r by

Vd(r) = rdπd2

Γ(d2

+ 1). (A.35)

Its surface Sd(r) is given by

Sd(r) = V ′d(r) = drd−1 πd2

Γ(d2

+ 1)= 2rd−1 π

d2

Γ(d2). (A.36)

We consider the one-dimensional integral

Ad,α ≡∫ ∞

0

dkkd−1

(k2 +m2)α=

1

2(m2)

d2−αΓ(d

2)Γ(α− d

2)

Γ(α), (A.37)

which is defined for d ≥ 1 and α > d2. In order to see this identity, we write the Beta

function B(x, y), for x, y > 0, in terms of the Gamma function:

B(x, y) ≡∫ ∞

0

dττx−1

(1 + τ)x+y=

Γ(x)Γ(y)

Γ(x+ y). (A.38)

From this we arrive with the substitution k2 7→ τm2 and x = d2, y = α − d

2at our claim

(A.37). This one-dimensional integral helps to evaluate d-dimensional integrals of theform

I(d, α) ≡ µ4−d∫

ddk

(2π)d1

(k2 +m2)α. (A.39)

The prefactor µ is called renormalization scale with mass-dimension one and it ensures[I(d, α)] = 4− 2α for all d ≥ 1. Performing the d− 1 angular integrals we arrive at

I(d, α) =µ4−d

(2π)dSd(1)

∫ ∞0

dkkd−1

(k2 +m2)α=

µ4−d

(2π)dSd(1)Ad,α =

=µ4−d

(2π)d· 2π

d2

Γ(d2)· (m2)

d2−α

2

Γ(d2)Γ(α− d

2)

Γ(α)=

(m2)2−α

16π2

(4πµ2

m2

)2− d2 Γ(α− d

2)

Γ(α).

(A.40)This expression shows that in the scaleless limit, m→ 0, we find I(d, α) = 0 for d

2−α > 0.

Actually the integral (A.39) is defined only for d ∈ (0, 2α), otherwise I(d, α) is divergent.The goal of dimensional regularization is to separate the divergent part of I(d, α) in termsof ε ≡ 4− d. Using the functional equation Γ(x+ 1) = x · Γ(x) we find

Γ

(α− d

2

)= Γ

(α +

ε

2− 2)

=Γ(α + ε

2)

(α + ε2− 1)(α + ε

2− 2)

=

=

Γ(α− d2) , α > d

2, χ < 0 ,

2ε− γE +O(ε) , α = d

2, χ = 0 ,

−2ε

+ γE − 1 +O(ε) , α = d2− 1 , χ = 2 ,

1ε

+(

34− γE

2

)+O(ε) , α = d

2− 2 , χ = 4 .

......

(A.41)

101

A Appendix

There are two important relations for I(d, α), coming directly from the definition (A.39)and (A.40), respectively. For x ∈ R we find:

I(d+ x, α) = I(d, α) · (2πµ)−x , (A.42)

I(d+ x, α + x/2) = I(d, α) ·(4πµ2

)−x/2 Γ(α)

Γ(α + x/2). (A.43)

Expanding (A.40) up to order O(ε), we arrive for α = d2− 1 at:

I(4, 1) =m2

16π2

[−2

ε+ γE − 1− ln(4π) + ln

(m2

µ2

)]+O(ε) =

=m2

16π2

[R + ln

(m2

µ2

)]+O(ε) .

(A.44)

We used the Euler-Mascheroni constant, γE ≡ −Γ′(1) ≈ 0.5772 . . ., and introduced asubstruction scheme by

R ≡ −2

ε− ln(4π) + γE − 1 , (A.45)

which is the MS scheme in χPT, due to the additional minus one.Under the transformation d 7→ d + x and α 7→ α + x

2in (A.43) the characteristic χ of

the integral I(d, α),χ ≡ d− 2α , (A.46)

remains unchanged. We stated before that I(d, α) is convergent for d ∈ (0, 2α), hence onlyintegrals with negative characteristic χ < 0 are convergent. In addition, only for positivecharacteristic χ > 0, scaleless integrals I(d, α) can be set to zero within the framework ofdimensional regularization. The identity (A.43) contains two different integrals with thesame characteristic, hence (A.41) implies that both sides have the same pole structure.For instance, both I(3, 1

2) and I(4, 1) have the same characteristic, χ = 2, both are, hence,

divergent and their pole structure is the very same up to the denoted prefactor:

I(3, 1/2) = I(4− 1, 1− 1/2)(A.43)

= I(4, 1) ·(4πµ2

) 12

1√π

= I(4, 1) · 2µ . (A.47)

Due to [I(d, α)] = 4 − 2α for all d ≥ 1 and [µ] = 1, the mass dimension of both sidescoincide. The general and explicit relations (A.42), (A.43) and (A.47), respectively, areimportant tools for the treatment of divergent integrals in thermal field theories. We use(A.47) explicitly for deriving (7.14).

102

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108

Acknowledgement

During the last five years and in particular while writing this diploma thesis I have experiencedmuch support and friendship from many different people. With these final lines I try to expressmy appreciation and gratefulness.

First of all I thank Prof. Dr. Wolfram Weise for his confidence in me and for giving me theopportunity to write my diploma thesis at his chair T39. The issue he asked me to work on ishighly topical and covers many different interesting parts of physics. During the last year hehas always been interested in my work and open for discussions. Wolfram Weise guided andmotivated me continuously by pointing out further questions to my work.

Altogether in my physics studies I have attended five lectures and three exams by Prof. Dr.Norbert Kaiser. I have learnt from him so much about theoretical physics and its mathematicalfoundation that it is impossible to number. I thank Norbert Kaiser for checking my calculationsand for much physical and mathematical inspiration within and beyond this thesis.

It is impossible to write a diploma thesis within isolation. Here at T39 I found a verycomfortable and inspiring environment thinking about physics. I thank Michael Altenbuchingerfor mutual interest in physics beyond the Standard Model. Additionally, without meeting himfor breakfast “in the middle of the night” I would have missed many hours in the office.

Nino Bratovic was always open to answer my many questions concerning the QCD phasediagram. He has taught me to deal with problems in Mathematica, not only of a numericalnature. Thank you also for proofreading the chapter about RHIC.

Ringrazio Salvatore Fiorilla che mi ha insegnato molte parole nuove in italiano, spagnolo,cinese e inglese. In futuro, impareremo insieme il russo, vero?

Wo de laoshı Geng Lısheng, feichang ganxie nı naixındi jiao wo Hanyu. Nı jiao le wo hen duoHanzı, gaizheng le wo de fayın. Wo baozheng huı jıxu haohao de xuexı xiaqu.

I am thankful to Thomas Hell who has proofread the whole thesis concerning physical, gram-mar, spelling and appearance errors. I know that this was not an easy task and I appreciateyour readiness for detailed discussions about physics and current events.

It is awesome of Jeremy Holt that he did not get tired of correcting my English. Additionallyhe always had time to talk about problems in physics and philosophy. My thanks to you alsofor proofreading my introduction to thermal field theory.

I thank Kouji Kashiwa for many discussions and for spending time together outside the office.Bertram Klein has been my teacher in three lectures: “Critical Phenomena I and II” and

“Thermal Field Theory”. I am thankful to you for discussing many times additional questionsarising during or after the lecture.

All problems concerning my ignorance using a unix system have been solved by AlexanderLaschka. I am thankful that he always had time and patience to explain and show how such asystem is working. I enjoyed our conversations about both physical and economic theories.

Seit unserer ersten Begegnung am Schulertag der TUM, 3. Februar 2005, sind Maxi und ichauf der Suche nach Antworten: Mathematik, Physik, Philosophie, Wirtschaft, Politik, . . . egal!Danke fur die vielen Diskussionen und Erklarungen und fur deine Freundschaft.

Ich danke meinen Freunden Domi und Steffi fur die Korrekturvorschlage an meiner Diplom-arbeit. Vielmehr aber danke ich euch fur die Tanzabende, BBT-Sitzungen, Krisengesprache,Postkarten, . . .

It would have been impossible to get to this point without the love and support of my family.I am indebted to my parents and my sister for more than only the last five years.

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Shear Viscosities of Interacting Bose Gases - TUM · 2010. 11. 10. · manifold Rd 3 7!g( ) 2Gis...

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