Spectrum and Elliptic Flow of Direct Photons for a Space

Spectrum and Elliptic Flow of Direct Photons for a Space
Department of Physics and Astronomy
University of Heidelberg
Bachelor Thesis in Physics
submitted by
Martin Kroesen
born in Solingen (Germany)
2014
Spectrum and Elliptic Flow of Direct Photons for
a Space-Time Evolution from the Hydrokinetic
Model
This Bachelor Thesis has been carried out by Martin Kroesen at the
Physikalisches Institut at the University of Heidelberg
under the supervision of
PD Dr. Klaus Reygers
√
Abstract – Measurements of Pb-Pb-collisions at sN N = 2.76 TeV at the LHC
experiment ALICE show a larger direct photon production and elliptic flow at low
transverse momentum than predicted. This is confirmed with the given data from
a hydrokinetic model. Therefore photon rates are used to calculate the photon
production and pQCD calculations are added. The measurements indicate a higher
photon production at a later point of the expansion of the created fireball. Two
possible additional photon sources are discussed. Bremsstrahlung in the hadron gas
does not give the required contribution. A pseudo-critical enhancement of the photon
production at the chemical freeze out from the quark gluon plasma to the hadron
gas gives the required contribution.
Zusammenfassung – Messungen des Teilchenbeschleunigerexperiments ALICE von
√
Blei-Blei-Kollisionen bei sN N = 2.76 TeV zeigen eine höhere Produktion direkter
Photonen bei geringen Transversalimpulsen und einen höheren elliptischen Fluss als
erwartet. Dies wird anhand des Hydrokinetischen Models überprüft und bestätigt.
Dazu wird aus gegebenen Daten zur hydrokinetischen Beschreibung mit Hilfe von
Photonraten die entsprechende Photonproduktion berechnet und pQCD Daten werden in die Rechnung integriert. Die Messwerte deuten auf eine erhöhte Photonproduktion zu einem späteren Zeitpunkt der Expansion des entstehenden Feuerballs hin.
Zwei mögliche Produktionsquellen werden diskutiert. Dabei zeigt sich, dass Bremsstrahlung im Hadrongas keinen nennenswerten Beitrag liefert. Eine deutliche Erhöhung der Photonproduktion am Phasenübergang vom Quark-Gluon-Plasma zum
Hadron-Gas führt zu den gemessenen Ergebnissen.
Note:
Numerical calculations were performed with "Root" [1].
In calculations of this thesis only natural units are used, which means ~ = c =
kB = 1, where c is the speed of light, ~ is the reduced Planck constant, and kB
is the Boltzmann constant. The relations between the dimensions are given by:
[length] = [time] = [energy]−1 = [mass]−1 .
Contents
1 Introduction
1
2 Theory
5
2.1
QCD Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2
Heavy-ion Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.3
Space Time Evolution of the Fireball . . . . . . . . . . . . . . . . . .
7
3 Direct Photon Observables
9
3.1
Invariant Photon Yield . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3.2
Anisotropic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
4 Models for Photon Production
13
4.1
Description of the Hydrokinetic Model . . . . . . . . . . . . . . . . .
4.2
Thermal Photon Rates from the Hadron Gas and Quark-Gluon Plasma 16
4.3
Blueshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
4.4
pQCD Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
5 ALICE Data Modeling and Results
13
21
5.1
HKM Data - Only Thermal Photons . . . . . . . . . . . . . . . . . .
21
5.2
pQCD Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
6 Two Possible Solutions to the Direct Photon Puzzle
27
6.1
Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
6.2
Pseudo-critical Enhancement . . . . . . . . . . . . . . . . . . . . . . .
28
7 Conclusions and Outlook
31
A Appendix
33
A.1 Basic Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
A.2 Basic Relativistic Hydrodynamics and Energy Stress Tensor . . . . .
35
A.3 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
i
ii
Contents
A.4 New Parametrization for the Thermal Photon Rates from the HG . .
39
A.5 Pseudo-Critical Enhancement - Impact of fmax and σ . . . . . . . . .
40
Bibliography
41
1.
Introduction
High-energy physics has established a detailed theory of the elementary particles and
their fundamental interactions. This model is called Standard Model. It is the aim
of ultra-relativistic heavy-ion physics to apply the Standard Model to dynamically
evolving systems of finite size to understand the collective phenomena and microscopic properties of nuclear matter under conditions of extreme temperature and
density [2].
The appearance of phase transitions involving elementary quantum fields is connected to the breaking of symmetries. Lattice calculations of quantum chromodynamics (QCD) predict a phase transition at a temperature of approximately 160 MeV
[3]. At this temperature the nuclear matter is assumed to undergo a phase transition
to a deconfined state of quarks and gluons, the quark-gluon plasma (QGP), and the
quark masses are reduced to their small current ones.
Since the early 1980’s, collisions of heavy atomic nuclei at as large energies as possible have been seen as the ideal way to probe these conditions of extremely high
temperature and density [3]. There are two main conditions for the system to behave
like bulk matter and not as a group of individual particles. The System must consist
of a large number of particles and it needs to reach local equilibrium.
First indications for the creation of the QGP were observed at the Super Proton
Synchrotron (SPS) at CERN and the Relativistic Heavy Ion Collider (RHIC) at the
Brookhaven National Laboratory. The Large Hadron Collider (LHC) at CERN now
offers the highest center-of-mass energies ever reached and should provide sufficient
energy densities to create the QGP for a short time. The plasma is studied with the
ALICE experiment, and also with CMS and ATLAS.
Photons interact only electromagnetically with matter. The coupling constant
αem ≈ 1/137 is much smaller then the coupling constant for strong interactions
αs ≈ 0.3 [4] at typical QGP temperatures. The photons produced during the QGPphase leave the plasma without rescattering, because the mean free path λ is much
larger than the lifetime of the fireball. The absence of rescattering makes the photon
1
2
1. Introduction
an important tool for probing the space time development of the fireball. In this
thesis the focus lies on direct photons, which include all photons except the decay
photons.
While the fireball passes through several stages (thermalization, QGP phase and
the hadronization), there are several sources for photon production contributing to
the observables. Within this thesis the focus lies on the invariant yield (see section
3.1) and the anisotropic flow (see section 3.2). The anisotropic flow is a spatial
anisotropy correlated with a momentum anisotropy.
√
The high slope in the ALICE direct photon yield measurements at sN N could indicate a higher photon production in early stages of the created fireball. But in
early stages of the expansion a collective flow is not expected to be formed yet. The
measured direct photon elliptic flow for 1 < pT < 3 GeV is in the order of magnitude
similar to the observed charged pion elliptic flow [5] (see figure 1.1). Similar results
were published by the PHENIX collaboration [6]. Thus, a higher photon production
in a later stage of the evolution is expected. It is the aim of this thesis to deliver
contributions to the solution of this so called photon puzzle on the base of the well
tested Hydrokinetic Model.
In chapter 2 the theoretical base is described, while chapter 3 deals with the observables for direct photon production. In chapter 4 "Models for Photon Production"
the Hydrokinetic Model (HKM) as model of the flow on the one hand and the photon production on the other hand is explained. On this basis in chapter 5 "ALICE
Data Modeling and Results" the measured data are compared with the model and
discussed. Possible solutions of the photon puzzle are given in chapter 6.
3
Figure 1.1:
Top: Direct photon yield with slope at low pT [5]. Bottom: Elliptic flow from direct
photons [5].
4
1. Introduction
2.
Theory
The aim of this chapter is to give a short overview over the theoretical background
of this thesis.
2.1
QCD Fundamentals
Quantum chromodynamics (QCD) is the quantum field theory of the strong interaction, the interactions between color charged quarks and eight massless gluons. There
are three different types of color charge, and anti-quarks carry the corresponding
anti-color. Gluons also carry color charge and they do not only mediate the strong
interaction between quarks, but also between the gluons themselves. A bound state
of quarks and gluons will be colorless for an observer, they are called hadrons.
There exist the mesons, quark-antiquark pairs, and baryons, which consist of three
quarks that are carrying different colors respectively anti-colors. The dynamic of
quarks and gluons is described by the Lagrangian of quantum chromodynamics [4]:
L=
X
q
µ
ψq γµ i∂ −
λa
gs Aµa
!
2
ψq −
X
mq ψq ψq −
q
1 X µν
F Fµν,a
4 a a
(2.1)
ψq are the quark fields, gs the strong coupling constant, Aµq the gluon fields, λa the
Gell-Mann matrices. The gluon field strength tensor has the following form:
Faµν = ∂ µ Aνa − ∂ ν Aµa + gs fabc Abµ Acν
(2.2)
In the early days of QCD the aim was the calculation of bound quark-antiquark pairs
(mesons). The corresponding effective phenomenological, approximated potential of
a quark-antiquark pair (q q̄) is described by:
V (r) = −
4 αs (r)
+κ·r
3 r
5
(2.3)
6
2. Theory
Here, αs is the strong coupling constant and κ is the string tension. The second term
increases linearly and is linked to the concept of confinement. This is the reason for
quark systems being colorless for an observer. The first term is, aside from the different coupling constant, similar to the Coulomb potential.
One very important point of QCD is the running coupling constant αs , it depends
on r, respectively the invariant squared momentum transfer Q2 :
αs (Q2 ) =
12 · π
(33 − 2nf ) · (Q2 /Λ2 )
(2.4)
nf is the number of quark flavors and Λ the QCD scaling parameter. Λ is measured
to be approximately 200 MeV. This equation only takes effect if Q2 Λ2 .
While in normal matter quarks are confined, in the quark-gluon plasma (QGP)
quarks are deconfined. This phase is thought to consist of asymptotically free quarks
and gluons.
2.2
Heavy-ion Collisions
It is expected that the universe has been in a state of very high temperature and
density in the first fraction of a second. In this state the strongly interacting quarks
and gluons were asymptotically free and in a deconfined phase of matter, the so
called quark-gluon plasma (QGP). In the description of the Big Bang, this plasma
cools into confined hadrons after a short time. In this context the QGP behaves like
a distinct state of matter. To work with the QGP and probe its conditions, it should
not behave like individual elementary particles or a group of those, but like matter
[3].
To probe the QGP on earth, we need such a high temperature, so that the characteristic thermodynamic parameters like energy-density, temperature, entropy-density
and pressure are sufficiently high to constitute the phase transition. To reach a local thermodynamic equilibrium the inverse rate of interactions must be significantly
larger than the system’s lifetime. Simple collisions of electrons or protons do not produce these conditions. But it is known that high energy collisions of heavy atomic
nuclei can produce the conditions to create a fireball of interacting quarks and gluons. Figure 2.1 shows the schematic phase diagram of QCD with points that were
2.3. Space Time Evolution of the Fireball
7
Figure 2.1: Schematic phase diagram of QCD in the plane of temperature T and
baryon chemical potential µB with points that were tested in heavy ion experiments
so far [7].
tested in heavy-ion experiments so far. Thus the collisions of heavy-ions at very
high energies are found to be the ideal way for the research of the QGP under lab
conditions on earth.
The Large Hadron Collider (LHC) provides Pb-Pb-collisions with a center-of-mass
collision energy of 2.76 TeV per nucleon-nucleon pair.
2.3
Space Time Evolution of the Fireball
After the collision of the heavy-ions a state of very high energy, pressure and density
is established. We cannot describe the process of formation itself. We start the
description at a point a very short time after the collision of the ions. The process
can be described by four stages [9].
In the pre-equilibrium stage the process is dominated by parton-parton hard scattering and can be modeled by perturbative QCD (pQCD). The lifetime of this phase
is predicted to be about less 1 fm [10, 11].
After this very short phase the QGP is established (see figure 2.2). This phase
is dominated by parton-parton, string-string interactions and thermal equilibrium.
8
2. Theory
Figure 2.2: Light cone diagram of the evolution of ultra relativistic heavy ion collisions [8].
It behaves like an almost ideal fluid. A rough description can be found in "Highly
relativistic nucleus-nucleus collisions: The central rapidity region" by J. D. Bjorken
[10]. The plasma expands while the temperature and the corresponding energydensity are decreasing. The energy density is expected to be around 3-5 GeV/fm3 .
When the local temperature reaches approximately the critical temperature Tc ,
quarks and gluons are being confined. In this mixed phase, a soft crossover, escaping particles are produced. The critical temperature is expected to be around
160 MeV.
In the last phase, the hadronic phase, a collective expansion via hadron-hadron
interactions decreases the temperature. At the hadron freeze-out temperature Tf ,
the non-interacting particles stream out of the medium into the detectors.
The space time evolution of the QGP can only be deduced by observing the escaping
particles and the direct photons (see section direct photons). Thus a model for the
QGP-expansion and a model for photon production from the plasma is needed. In
this context the base is a Hydrokinetic Model describing the space time evolution.
3.
Direct Photon Observables
Direct photons are those photons that directly emerge from a particle collision within
the fireball. The measurement of direct photons plays an important role in probing
the properties of the QGP. Direct photons give an interesting view into the interior
(early) part of the QGP and other states of the evolution since they are produced continuously during the expansion. This distinguishes them from the produced hadrons,
which are the result of the freeze-out, and from decay photons.
There are different processes which produce direct photons [12]. One possibility
is hard scatterings of incoming partons. The jet fragmentation process can also constitute direct photons after initial hard scattering. These processes can be calculated
with pQCD. Scattering between quarks and gluons in the QGP and scattering between hadrons in the hadron gas are also important direct photon sources. These
processes are thermal and give rise to an interpretation of the spectra as effective
temperature.
In this chapter, the two observables direct photon yield and anisotropic flow, especially the elliptic flow, are discussed. In the following chapters these observables
will be used for a qualitative analysis of the direct photons.
3.1
Invariant Photon Yield
For basic kinematics as background knowledge look at A.1 in the appendix. We are
interested in the observable
1 d3 N
(3.1)
Lint dp3
of the photons. N is the number of particles, p the momentum and Lint is the
3
integrated luminosity. In contrast to d3 p the phase space element dEp is Lorentz
invariant. So with E 0 and p0 being the energy, respectively the momentum, in another
coordinate frame, the following relation is given:
d3 p
d3 p0
= 0
E
E
9
(3.2)
10
3. Direct Photon Observables
Thus it makes sense to define a corresponding observable, the Lorentz invariant cross
section
1
d3 N
d3 N
1
E 3 =
E 3 σtot
(3.3)
Lint dp
Nevt,tot dp
where Nevt,tot is the total number of events and σtot the total cross section. The
invariant photon yield is now defined by
E
d3 N
1
d3 n
E
=
dp3
Nevt,tot dp3
(3.4)
The invariant photon yield can be calculated from convolving the photon rates
3
f (E, T ) := q0 ddpR3 with the space-time evolution of the emitting medium [13]:
E
d3 n Z
= τ dτ dY dxdyf (p · u, T )
dp3
(3.5)
Note: T is the temperature in the rest frame and d4 X = dt dx dy dz = τ dτ dY dx dy.
The scalar p·u = pµ uµ is the Lorentz invariant product of the photon four-momentum
pµ and the four-velocity uµ = γ(1, βx , βy , βz ) of the cell from which the photon is
emitted. In the rest frame of the cell p · u = E and thus the rate f is calculated by
f (p · u, T ).
The integration limits over Y are given by the beam rapidity of the projectile:
Ymax = ymax
√
1 (E + pz )2
(E + pz )
s
1 E + pz
= ln 2
= ln
≈
= ln
2
2 E − pz
2
E − pz
mN
mN
(3.6)
To calculate the invariant yield, we need a model which describes the flow of the
cell, in our case the Hydrokinetic Model, and a model for the photon production, to
calculate the rate f (p · u, T ) (section 4.2 "Thermal Photon Rates from the Hadron
Gas and Quark-Gluon Plasma").
3.2
Anisotropic Flow
Flow signals the presence of multiple interactions between the constituents of the
medium created in collisions [14]. A larger magnitude of flow is correlated to an
increasing number of interactions. The most direct evidence of flow comes from
the observation of anisotropic flow, which is the anisotropy in particle momentum
distributions correlated to the reaction plane. The reaction plane is defined by the
11
3.2. Anisotropic Flow
Figure 3.1: After a non-central collision of two nuclei the spatial anisotropy with
respect to the x-z plan (reaction plane) translates into a momentum anisotropy of
the produced particles (anisotropic flow) [14].
impact parameter and the z-direction. It is convenient to describe the invariant yield
by its Fourier expansion around the reaction plane (figure 3.1)
∞
X
1 d2 N
d3 N
vn cos(n(φ − ψRP )) .
1+2
E 3 =
dp
2π pT dpT dy
n=1
!
(3.7)
E is the energy of the particle, p the momentum, pT the transverse momentum, φ
the azimuthal angle, y the rapidity and ψRP the reaction plane. Because of the
reflection symmetry with respect to the reaction plane the sine terms vanish. The
Fourier coefficients are depending on pT and y. They are given by:
vn (pT , y) = hcos[n(φ − ψRP )]i
(3.8)
The angular brackets denote an average over the particles. The coefficient v1 is
known as direct flow while v2 is known as elliptic flow.
vn is related to the eccentricity and the radial flow velocity [15].
12
3. Direct Photon Observables
4.
Models for Photon Production
To describe the photon production from the fireball we need two model parts. The
first part describes the dynamics of the fluid within the fireball, the second part
describes the distribution of photon rates of emerging direct photons.
In our case the dynamics of the fluid are described by the so-called Hydrokinetic
Model (HKM). Concrete data sets describing the space-time evolution of the flow
velocity and temperature are provided by Yuri Sinyukov and they are used as a
black-box in this thesis.
The second model part describes the thermal photon emission rates from the quarkgluon plasma and the hadron gas.
First a motivation for the use of the HKM is given. A combined hydrokinetic approach which incorporates a hydrodynamical expansion of the fireball and their dynamical decoupling by escape probabilities describes the space time distribution of
the flow [17].
On the base of such models predictions for the particle yield in central Pb-Pb collisions were made [18][19]. Figure 4.1 shows the transverse momentum distributions
of the sum of positive and negative particles [16]. As pointed out in this publication,
HKM yields a good description of the data. Because the description of hadronic final
states using HKM works well, it seems reasonable to choose this model as a basis to
study the production of direct photons.
4.1
Description of the Hydrokinetic Model
The history of the QGP models has a long tradition. Early hydrodynamic models for
example, were presented by Bjorken [10]. An overview of the main characteristics is
given by Ollitrault [20] and some properties are given in the appendix of this thesis.
13
14
4. Models for Photon Production
Figure 4.1: Transverse momentum distributions of the sum of positive and negative
particles [16].
Such hydrodynamic models successfully describe basic features of high energy collisions. In these models a state of equilibrium shortly after the collision is assumed.
The QGP phase is given by the stress-energy tensor and the equation of state (EOS).
Since the HKM is used as a black-box in this thesis, only a small overview can
be given. Figure 4.2 shows the evolution of the fireball in the HKM. It incorporates
a description of the pre-thermal phase [11] [22], to assess the initial conditions for
hydrodynamics. To obtain these conditions for τ & 1 fm one needs to match the very
early initial stage of the nuclei collision, when the hydrodynamic approximation is
outside the regime of its validity. The corresponding evolutional equations are then
written as [11]
µν
µν
∂µ [(1 − P (x))Thyd
(x)] = −TpQCD
∂µ P (x).
(4.1)
µν
In this case P (x) is a probability function (continuous switch) that transforms TpQCD
µν
into Thyd
. With the help of this equation the initial conditions of the QGP phase,
4.1. Description of the Hydrokinetic Model
15
Figure 4.2: Different stages in the description of the HKM. The hadron cascade is
not relevant in our case [21].
µν
which are dominated by ∂µ Thyd
= 0 and the EOS, are calculated.
In addition to the calculation of reasonable initial conditions, an approach for the
freeze-out, that means a dynamical description of the decoupling in space time, is
needed, since a sharp freeze-out is however a rather rough approximation.
An early description of this process in the context of hydrokinetics is given by [23].
The hydro-kinetic formalism for heavy ion collision is described in [17]. It includes
a model of hadronic emission that describes the evolution of a hydrodynamically
expanding system undergoing a phase transition. The complete algorithm provides
the evaluation of an emission function based on escape probabilities with account
for deviations of distribution functions f(x,p) from local equilibrium [24, 25, 19]. As
above a soft switch (escape probability) is used, to transform the QGP into the
hadron gas, so we get a soft decoupling and hadronization.
16
4.2
4. Models for Photon Production
Thermal Photon Rates from the Hadron Gas
and Quark-Gluon Plasma
The main source of direct photons (figure 4.3) in the fireball are photons emitted
from the quark-gluon plasma at temperatures above TC and hadronically produced
thermal photons from below the phase transition. In our case the description of
the former (QGP-rates) is taken from Arnold, Moore and Yaffe (AMY) [26] and the
latter (HG-Rates) is from Turbide, Rapp and Gale [27].
Figure 4.3: Direct photons from HG and QGP [12].
To calculate the observables (see section 3.1 / 3.2 for details) like the QGP photon3
γ
we need the rate f (E, T ) := q0 ddpR3 .
rate Eγ dN
dp3
The photon emission rate of an equilibrated QCD plasma at high temperature with
zero chemical potential has been computed in detail by AMY [26]. The results from
AMY are used in this thesis. In the special case of QCD with dF = NC = 3 and
CF = 4/3 the equation for the photon emission rates f (E, T ) can be written as:
f (E, T ) =
Nf
X
i=1
x is defined by x :=
E
T
αem αs 2 1
T x
Ctot (x)
π2
e +1
(4.2)
and Ctot is given by
√
Ctot (x) = ln
qi2
3
gs
+
1
ln(2x) + C2↔2 (x) + Cbrems (x) + Cannih (x).
2
(4.3)
The rate has been calculated to leading order in αem and the QCD-coupling gs2 (T ) =
4παs which should always be understood as defined at a scale of order of the tem-
4.2. Thermal Photon Rates from the Hadron Gas and Quark-Gluon Plasma
17
perature.
The rate f (E, T ) is dominated by the Fermi distribution function
leads to the rough e−Eγ /T behavior of the QGP photon rate.
1
ex +1
≈ e−x , which
The leading logarithmic order contribution is generated by 2 ↔ 2 particle processes.
The C2↔2 -term represents the non logarithmic 2 ↔ 2 particle processes. The rate
of photon production by bremsstrahlung and inelastic pair annihilation is given by
Cbrems respectively Cannih .
From AMY [26] we get the approximate, phenomenological fits:
C2↔2 ≈
0.041
− 0.3615 + 1.01e−1.35x
x
(4.4)
and
s
Cbrems + Cannih ≈


0.133x
1
0.548 ln(12.28 + x−1 )

q
+
1 + Nf 
6
x3/2
1 + x/16.27
(4.5)
The above equations are useful in the context of the QGP at temperatures T > Tc
and were used in our calculations.
The thermal emission of photons from hot and dense strongly interacting matter
at temperatures close to the phase transition to the QGP were calculated by Turbide, Rapp and Gale [27]. Useful parametrizations were given in the appendix of the
paper and are used in our context. The parametrizations are of the form
f (E, T ) = A exp(
B
E
− D ).
C
(2ET )
T
(4.6)
The parametrizations from [27], which are more detailed, are used at T <= Tc , thus
we have a sharp transition between those rates at Tc .
18
4.3
4. Models for Photon Production
Blueshift
The invariant photon yield is given by (as described above):
E
d3 n Z
= τ dτ dY dxdyf (p · u, T )
dp3
(4.7)
In this section, we want to discuss the impact of the p·u term. In the case of massless
photons, pµ is given by
pµ = pT · (1, cos(φ), sin(φ), 0)
(4.8)
pµ uµ = pT (u0 − u1 cos(φ) − u2 sin(φ)).
(4.9)
and p · u is expressed by
For simplicity we look at a fluid with u2 = 0 and Y = 0 and a photon with φ = 0.
In this case uµ is given by
uµ = (cosh(ρ), sinh(ρ), 0, 0) = (γf low , βf low γf low , 0, 0)
(4.10)
and we can conclude
1
βf low
−q
) = pT
pµ uµ = pT ( q
1 − βf2low
1 − βf2low
v
u
u 1 − βf low
t
.
1 + βf low
(4.11)
Since f (p · u, T ) is described by
Nf
X
i=1
qi2
αem αs 2 1
T x
Ctot (x)
π2
e +1
(4.12)
it is reasonable to write
r
pT
x=
1−βf low
1+βf low
T
and we define
Tslope =
pT
=
r
T
1+βf low
1−βf low
v
u
u 1 + βf low
Tt
.
1 − βf low
(4.13)
(4.14)
4.4. pQCD Photons
19
We
interpret Tslope as effective blueshifted temperature. If we take βf low = 0.6, then
r
1+βf low
1−βf low
= 2, thus the large slope at low pT in the invariant yield can be partially
explained by photons emitted with sufficient high βf low . For more details see [28].
4.4
pQCD Photons
In the last sections thermal photon production in the QGP and HG (Hadron Gas)
phase has been described. However, direct photons of other kinds can also play an
important role. The full palette of opportunities has been explored in a paper from
Stankus [12]. Hard scattering of incoming partons can create direct photons.
Figure 4.4: Two types of processes emitting pQCD photons [12].
At high pT these processes can be described by perturbative QCD (pQCD) and
their rate can be calculated in this framework, thus they are called pQCD-Photons.
Two perturbative QCD diagramms that produce final-state photons at lowest order
are shown in 4.4 Panel A. As part of the jet fragmentation process a quark can also
radiate photons after its initial hard scattering process (upper diagram of Panel B).
Photons from those in Panel B are always accompanied by hadrons and are termed
fragmentation photons while a Photon like Panel A is a single direct photon.
20
4. Models for Photon Production
5.
ALICE Data Modeling and Results
In this chapter spectrum and elliptic flow of direct photons for the space-time evolution from the Hydrokinetic Model are compared to the ALICE preliminary data at
2.76 TeV center-of-mass energy with a centrality of 0-40%.
In the first part the raw data is briefly shown and in the following the pQCD photons
are added.
5.1
HKM Data - Only Thermal Photons
On the basis of existing HKM-data (discrete data sets) by Yuri Sinyukov describing
the flow and temperature as a function of the space-time, the direct photon yield
and the elliptic flow v2 are calculated as a function of pT .
direct-photon spectrum (Pb-Pb, 0-40%)
10
HKM
3
E d Nγ /dp (GeV-2)
102
1
3
ALICE prelim data with stat error
10-1
ALICE prelim data syst error
10-2
-3
10
10-4
-5
10
-6
10
10-7
0
1
2
3
4
5
6
7
8
9
10
p (GeV)
T
Figure 5.1: Comparision of HKM direct photon yield with ALICE data.
The blueshift is implicitly included in the calculations. A sharp crossover from the
21
22
5. ALICE Data Modeling and Results
v2
direct photon v2, Pb-Pb, 2.76 TeV, 0-40%
0.2
0.18
HKM data
ALICE prelim data with stat error
0.16
ALICE prelim data syst error
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
p (GeV)
T
Figure 5.2: Comparison of HKM v2 calculation with ALICE data.
parametrization of the QGP rates to the HG rates at Tc = 160 MeV is implemented
in our calculation for simplicity, even though it is in contrast to the soft crossover of
the HKM.
Figure 5.1 shows the direct photon yield while figure 5.2 shows the pT -dependency
of v2 .
Since v2 (pT → 0) = 0 should apply, this behavior has been tested for a very small
binning.
The model based pure HKM direct photon
yield is, as expected, low. The rough
p
−T T
ef
f
. A blueshift is visible at low pT , the slope
shape is dominated by the factor e
increases with smaller pT . At large pT some physical processes seem to be missing.
The deviation of model based data for small and high pT is large and underestimating the measurements.
Figure 5.2 shows v2 (pT ). The maximum of v2 is at pT ≈ 2GeV with v2 ≈ 0.07.
The measured values are underestimated although the rough shape of the measured
data corresponds to the calculated spectra.
23
5.2. pQCD Correction
As an interim result it can be stated, that additional photon sources are needed
for a better description of the measured data. It is expected that pQCD corrections
become relevant at high pT . So our first expectation is that such corrections will
solve the deviation only at that region.
5.2
pQCD Correction
The pQCD data used in this thesis were calculated by Werner Vogelsang for protonproton collisions [29]. For lead-lead collisions these have to be scaled with a factor
hTAA i =
hNcoll i
σinel,N N
(5.1)
corresponding to the mean value of the number of inelastic nucleus-nucleus collisions
hNcoll i = 826.46 and the nucleus-nucleus inelastic cross section σinel,N N = 64mb.
The provided discrete data were difficult manageable in our context, thus we need a
continuous parametrization. We made a reasonable fit with the function
2

A
(n − 1)(n − 2) 
1 dN
=
1+
2πpT dpT dy
2π nC[nC + m(n − 2)]
q
p2T + m2 − m
nC
−n

.
(5.2)
Apparently it is suitable (see figure 5.3). The parametrization is from [30] (equation
3 in the appendix). At small pT below 2 GeV the data is extrapolated due to reasons
of practical calculation. In this region the influence of the pQCD data is very low.
To recalculate the elliptic flow, we make the reasonable assumption that v2,pQCD = 0,
because pQCD photons are presumably mainly emitted at a very early stage in the
evolution of the fireball. The pQCD photons are emited from scattering of initial
partons.
We can calculate the v2,direct
v2,direct = f ∗ v2,therm
(5.3)
24
5. ALICE Data Modeling and Results
pQCD Fit
2
NLO pQCD Data
10
3
E d Nγ /dp (GeV-2)
10
Fit
3
1
10-1
10-2
-3
10
10-4
-5
10
-6
10
10-7
0
1
2
3
4
5
6
7
8
9
10
p (GeV)
T
Figure 5.3: Black line: pQCD calculation from [29]. Red line: Fit with eq 5.2.
with the normalization f =
Ntherm
.
Ntherm +NpQCD
Adding the HKM data and the pQCD contribution we obtain figures 5.4 and 5.5.
At pT > 3 GeV the invariant yield is now described well, but apparently at low
Ntherm
< 1, we expect
pT we seem to need additional processes. Since f = Ntherm
+NpQCD
a smaller v2 when including pQCD contributions. As shown in figure 5.5 this is
confirmed.
Our interim result is, that pQCD corrections deliver no solution for v2 but high
pT -spectra are well described. Although the calculation of the photon rates from the
HKM incorporates the blue shift, the low pT data is not well described. The high
slope of the measured data could be due to a higher photon production in the QGP.
Since we know that in the early stages of the expansion of the fireball, the collective
flow has not build up yet, we expect a low v2 . The photon rates from the HKM also
give rise to a low v2 . An explanation for the high measured v2 could be additional
photon sources at a later stage of the expansion, when the collective flow is predicted
to be formed. We call this problem the "Photon Puzzle". Two possible solutions are
presented in the next chapter 6.
25
5.2. pQCD Correction
direct-photon spectrum (Pb-Pb, 0-40%)
2
HKM
10
3
E d Nγ /dp (GeV-2)
10
pQCD
HKM + pQCD
1
3
ALICE prelim data with stat error
10-1
ALICE prelim data syst error
10-2
-3
10
10-4
-5
10
-6
10
10-7
0
1
2
3
4
5
6
7
8
9
10
p (GeV)
T
Figure 5.4: Model includes HKM and pQCD data. The experimental data are well
described at high pT .
v2
direct photon v2, Pb-Pb, 2.76 TeV, 0-40%
0.2
HKM data
0.18
f * HKM data
ALICE prelim data with stat error
0.16
ALICE prelim data syst error
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
p (GeV)
T
Figure 5.5: Photons calculated from the HKM show a smaller v2 than the measured
data.
26
5. ALICE Data Modeling and Results
6. Two Possible Solutions to the Direct
Photon Puzzle
In this section, the effect of meson-meson and meson-baryon bremsstrahlung and
a pseudo-critical enhancement inspired by [31] is discussed. Both sources of direct
photons emerge from a later stage of the expansion and so they are predicted to give
a higher contribution to the elliptic flow.
6.1
Bremsstrahlung
Linnyk et al. [32] had proposed that apart from partonic production channels the
direct photon yield and primary the strong v2 might be due to hadronic photon
sources such as meson-meson bremsstrahlung.
Thus including meson-meson and meson-baryon bremsstrahlung may give a hint
to the photon puzzle. A parametrization for the bremsstrahlung is provided by O.
Linnyk et al. [33].
For π + π → π + π + γ the rate has the following form:
x
x
+ d · exp −
fbrems (x, T ) = ln(ax + 1) · b · exp −
c
e
(6.1)
with the parameters a, b, c, d, e given in table 6.1. The result is in GeV−2 fm−4 .
The total meson-meson and meson-baryon bremsstrahlung is approximately obtained
T [MeV]
140
150
160
170
180
a
4
5
5
5
5
b
0.0025
0.004
0.007
0.01
0.015
c
d
e
0.1 0.008 0.05
0.105 0.015 0.05
0.11 0.025 0.05
0.116 0.04 0.05
0.125 0.06 0.05
Table 6.1: Parameters used in equation 6.1.
27
28
6. Two Possible Solutions to the Direct Photon Puzzle
by multiplying the result with 4 [33]. To make the data usable for our purpose the
data is interpolated with a spline.
From figure 6.1 it becomes obvious that the bremsstrahlung only gives a small contribution to the invariant yield at low pT and there, the yield is systematically underestimated. Nevertheless the slope is nearly the same as the slope from the photon
yield at low pT . The low pT problem remains unsolved.
direct-photon spectrum (Pb-Pb, 0-40%)
2
HKM + pQCD
10
3
E d Nγ /dp (GeV-2)
10
Bremsstrahlung
HKM + pQCD + Bremsstrahlung
1
3
ALICE prelim data with stat error
10-1
ALICE prelim data syst error
10-2
-3
10
10-4
-5
10
-6
10
10-7
0
1
2
3
4
5
6
7
8
9
10
p (GeV)
T
Figure 6.1: Bremsstrahlung correction is no solution of the photon puzzle.
6.2
Pseudo-critical Enhancement
Inspired by a publication of Hees et al. [31] an attempt to solve the photon puzzle
with the inclusion of a pseudo-critical enhancement is presented here.
As discussed in detail in that publication the transition region at Tc ≈ 160MeV
is identified as a key contributer to thermal photon spectra. Confining interactions
are expected to make an important contribution close to the hadronization transition. There will be an increase in partonic scattering rates and this leads to a natural
increase of photon radiation. These photon rates should be higher than those from
6.2. Pseudo-critical Enhancement
29
interacting particles, calculated in perturbative models.
To decode the photon puzzle, a phenomenological "pseudo-critical enhancement"
was taken into account in our calculations. To implement such an enhancement we
increased the rates by
F (T ) = 1 + (fmax − 1)
g(T − Tc , σ)
g(0, σ)
(6.2)
where g(x, σ) is the Gaussian distribution. As we can see F (Tc ) = fmax and F (T ) = 1
if T is large respectively small compared to Tc . Thus the increase is located in the
vicinity of Tc . This ansatz is purely phenomenologically motivated.
The data (photon yield) was not fitted. Trying different fmax and σ we got the
best result for the photon yield at fmax = 15 and σ = 0.02 GeV. To get an impression of the order of magnitude see appendix A.5.
Figure 6.2 and 6.3 show the results. The invariant photon yield at low pT is significantly increased and all measured data from low to high pT are well described.
As expected, the v2 spectra has grown due to a later emission stage. The pseudocritical enhancement seems to be a possible solution for the photon puzzle.
30
6. Two Possible Solutions to the Direct Photon Puzzle
direct-photon spectrum (Pb-Pb, 0-40%)
2
HKM + pQCD
10
Pseudo-Critical Enhancement
3
E d Nγ /dp (GeV-2)
10
ALICE prelim data with stat error
1
3
ALICE prelim data syst error
10-1
10-2
-3
10
10-4
-5
10
-6
10
10-7
0
1
2
3
4
5
6
7
8
9
10
p (GeV/c)
T
Figure 6.2: The pseudo-critical enhancement provides essential contributions at low
pT .
v2
direct photon v2, Pb-Pb, 2.76 TeV, 0-40%
0.2
f * HKM data
0.18
Pseudo-Critical Enhancement
ALICE prelim data with stat error
0.16
ALICE prelim data syst error
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
p (GeV)
T
Figure 6.3: The pseudo-critical enhancement provides high v2 -values.
7.
Conclusions and Outlook
This thesis deals with the direct photons from the fireball formed in heavy ion collisions. The focus is set on the observables direct photon yield and direct photon
elliptic flow v2 . Results are compared to ALICE measurements of lead-lead collisions
√
with a center-of-mass energy of sN N = 2.76 TeV.
The measurements of the direct photon yield show a large Tslope for 1 ≤ pT ≤ 3GeV.
Although this slope can be interpreted as a signal of a very high temperature of the
fireball at an early stage, this does not explain the measured values of v2 . Thus this
interpretation can be considered misleading. It was the aim of this thesis to give
hints to solve this photon puzzle. The model should describe the direct photon yield
and the elliptic flow v2 simultaneously.
Hydrokinetic Models are able to describe the transverse momentum distribution
of hadrons in Pb-Pb-collisions. Therefore it comes to mind to apply this description
of the flow to the calculations of direct photon production.
The observables were calculated on the basis of the Hydrokinetic Model [23, 17,
24, 25, 19] with the help of existing parametrizations for the emission of direct thermal photons [26] in the quark gluon plasma (QGP) and an adequate description of
the emission of the following hadron gas (HG) [27] . Instead of a soft crossover a hard
transition between QGP and HG at a temperature Tc = 160 MeV was implemented
for the calculation of the photonrates.
The resulting invariant direct photon yield shows a deviation downward from the
measurement over the entire range of pT , especially at low and high pT . High pT
deviations were solved by adding the pQCD photons using a parametrization by
Werner Vogelsang. The amount of the elliptic flow v2 is not described by this approach. So the photon rates from the space-time evolution from the HKM confirm
the above mentioned photon puzzle.
To solve the photon puzzle, additional photon sources from later points of the expansion were included. The inclusion of photons emitted by meson-meson and meson31
32
7. Conclusions and Outlook
baryon bremsstrahlung did not reproduce the measurement at low pT . It is therefore
suggested that bremsstrahlung can only be a part of the solution for the photon
puzzle.
Inspired by a recent publication from Hees et al. [31] a phenomenological "pseudocritical enhancement" was taken into account in the calculations. They have interpreted the transition region as a key contributer to thermal photon spectra. At this
stage of the fireball evolution there will be an increase in partonic scattering rates
and this leads to a natural increase of photon radiation.
This has been modeled by adding a Gaussian distribution around the critical temperature. Best results gave a 15 times higher photon production at the critical
temperature with a standard deviation of σ = 0.02 GeV.
With this approach the invariant direct photon yield and v2 are well described simultaneously.
This indicates that the phase transition delivers extra direct photons. Investigations towards a better theoretical/phenomenological understanding of this thermal
stage could lead to a better understanding of the evolution of the fireball.
Recently Rapp et al. [34] published a new parametrization for the hadron gas rates
(see appendix A.4). The deviation to the old rates is small. It does not give a hint
to the photon puzzle.
A.
Appendix
A.1
Basic Kinematics
In the first step we introduce the basics of relativistic kinematics. Let
xµ = (t, x, y, z)
(A.1)
be the 4-vector and with gµν = diag(1, −1, −1, −1) and c = 1. We define the proper
time τ
q
τ = gµν xµ xν .
(A.2)
Now uµ is defined by:
∂τ
1
1
xβ
µ βν
ν βµ
µ
=
(g
x
δ
+
g
x
δ
)
=
g
x
=
µν
µν
µβ
∂xβ
2τ
τ
τ
uβ ≡
(A.3)
µ
It is clear that uµ uµ = xµτ x2 = 1 and for later use ∂ν uµ uµ = 2uµ ∂ν uµ = 0.
x
Now we define β~ ≡ d~
. In general the rapidity is defined by:
dt
~
~ = 1 ln 1 + |β|
y = tanh−1 (|β|)
~
2 1 − |β|
(A.4)
Now we assume a particle moving into z-direction, then let the space-time rapidity
Y and y the rapidity of a particle with constant velocity located at the origin of the
lab frame at time t = 0. Then
1
cosh(Y ) ≡ γz = q
= cosh(y).
1 − βz2
(A.5)
This is equivalent to
Y =
1 1 + βz
1 E + pz
1 t+z
ln
= ln
= ln
.
2 1 − βz
2 E − pz
2 t−z
33
(A.6)
34
A. Appendix
In this case τ =
√
t2 − z 2 . Now any xµ in this frame can be rewritten as
xµ = (τ cosh(Y ), r cos(φ), r sin(φ), τ sinh(Y )).
(A.7)
In this variables d4 x is transformed to
d4 x = dtdxdydz = τ dτ dY dxdy = τ dτ dY rdrdφ.
(A.8)
If we define the energy-momentum vector pµ for particle with mass m0 by
pµ = m0 uµ .
(A.9)
Now let the particle have the rapidity Y = 0 in the system S ∗ . In this system we
q
define βT ≡ βx2 + βy2 = tanh(ρ) or
ρ=
1 1 + βT
ln
2 1 − βT
(A.10)
Defining
pT ≡ m0 sinh(ρ)
(A.11)
mT ≡ m0 cosh(ρ)
(A.12)
and
it is m20 = m2T − p2T and the energy-momentum vector is now given by
pµ = (E, px , py , pz ) = (mT cosh(Y ), pT cos(φ), pT sin(φ), mT sinh(Y )).
(A.13)
Now, we can rewrite the resulting uµ as




cosh(Y ) 
cosh(Y ) cosh(ρ)






pT
 sinh(ρ) cos(ϕ) 
cos(ϕ)
pµ
mT 
µ
mT



.
=
u =
=

pT
m0
m0 
 sinh(ρ) sin(ϕ) 
m

sin(ϕ)


 T

sinh(Y ) cosh(ρ)
sinh(Y )
(A.14)
A.2. Basic Relativistic Hydrodynamics and Energy Stress Tensor
A.2
35
Basic Relativistic Hydrodynamics and Energy
Stress Tensor
The energy stress tensor is defined by
T µν = ((x) + p(x))u(x)µ u(x)ν − g µν p(x)
(A.15)
where in the case of hydrodynamics (x) is the energy density and p(x) is the pressure.
Energy-momentum conservation is expressed
∂µ T µν = 0.
(A.16)
0 = ∂µ T µν = ( + p)uµ ∂µ uν + uν ∂µ ( + p)uµ − g µν ∂µ p
(A.17)
Now we get 4 equations:
Multiplying by uν we get:
0 = uν ∂µ T µν = ( + p)uµ uν ∂µ uν + uν uν ∂µ ( + p)uµ − uν g µν ∂µ p
(A.18)
Using uν ∂µ uν = 0, this leads to:
uµ ∂µ = −( + p)∂µ uµ .
(A.19)
This is one equation. To get the other three equations, we have to multiply ∂µ T µν = 0
by factor (gαν − uα uν )
0 = (gαν − uα uν )[( + p)uµ ∂µ uν + uν ∂µ ( + p)uµ − g µν ∂µ p]
(A.20)
which results into
0 = ( + p)uµ ∂µ uα + uα ∂µ ( + p)uµ − ∂α p − uα ∂µ ( + p)uµ + uα uµ ∂µ p
(A.21)
and finally
0 = −∂α p + uα uµ ∂µ p + ( + p)uµ ∂µ uα .
(A.22)
The hydrodynamic phase is now described by:
uµ ∂µ = −( + p)∂µ uµ
(A.23)
36
A. Appendix
and the Euler equation
0 = −∂α p + uα uµ ∂µ p + ( + p)uµ ∂µ uα .
(A.23) and (A.24) represent four independent equations.
(A.24)
A.3. Thermodynamics
A.3
37
Thermodynamics
To solve (A.23) and (A.24) more information is needed. In the hydrodynamic description this information is given by the equation of state (EOS). We are interested
in the intensive properties of the QGP.
+ p = T s + µn
(A.25)
dp = sdT + ndµ
(A.26)
and
where an p are the energy density and the pressure. T is the temperature, µ the
chemical potential, s the entropy density and n the particle density. With
d = −dp + T ds + sdT + µdn + ndµ = T ds + µdn
(A.27)
equation (A.25), (A.26) and (A.27) are the basic framework to calculate the hydrodynamic expansion. Together with
∂ν suν = 0
(A.28)
which follows from ∂µ nuµ = 0, we have enough information to calculate special cases
of the equation of state.
As an example the energy density of an ideal relativistic gas is calculated:
∞
4π Z 2
1
E(β)
=g
p
E(p)
dp
(β) =
V
(2π~)3
eβE(p) ± 1
(A.29)
0
where g is the the degree of freedom, β = 1/T and the "-" is for bosons and the "+"
for fermions. In the case of bosons, we calculate
∞
∞
4
4π Z 3 1
4π 1 Z 3 1
3
4π
(β) = g
p
x
T
dp
=
g
dx
=
g
. (A.30)
(2π~)3
eβp − 1
(2π~)3 β 4
ex − 1
π 2 ~3 90
0
0
In the case of fermions the same calculation leads to:
(β) = g
4
3
4π 7
T
π 2 ~3 90 8
(A.31)
38
A. Appendix
(A.31) shows the typical T 4 -behavior of the energy density of the QGP.
To calculate the pressure, we have to recall, that Px = Py = Pz and Px =<
While Px,y,z = 3V1 (px vx + py vy + pz vz ), we conclude
P =
1
1 p~
1 p2
1 p2 + m2 − m2
1
m2
p~~v =
p~ =
=
=
(E −
)
3V
3V E
3V E
3V
E
3V
E
px vx
V
>.
(A.32)
[here P is the pressure]. In the ultrarelativistic case (m ≈ 0) the pressure is p = 3 .
A.4. New Parametrization for the Thermal Photon Rates from the HG
A.4
39
New Parametrization for the Thermal Photon Rates from the HG
A recent publication by Rapp et.al [34] shows a new universal parametrization for
the photon emission rates from the hadron gas. These rates include the contribution
from in-medium ρ mesons, as well as bremsstrahlung from ππ scattering.
direct-photon spectrum (Pb-Pb, 0-40%)
HG photon rates 2004
10
3
E d Nγ /dp (GeV-2)
102
HG photon rates 2014
1
3
ALICE prelim data with stat error
-1
10
ALICE prelim data syst error
10-2
-3
10
10-4
-5
10
-6
10
10-7
0
1
2
3
4
5
6
7
8
9
10
p (GeV)
T
Figure A.1: Photons from the HKM and qQCD photons. The new parametrization
of the hadron gas rates only gives rise to a small increase of the photon yield at low
pT . This does not give a hint to the solution of the photon puzzle.
A comparison of the invariant photon yield calculated with the new rates and the
invariant photon yield from the old rates is shown in figure A.1. We obtain only a
small deviation. A small increase of the yield at low pT can be seen. Although the
deviation effects the questionable region, it seems to be negligible for our problem
and it can not explain the photon puzzle.
A.5
Pseudo-Critical Enhancement - Impact of fmax
and σ
direct-photon spectrum (Pb-Pb, 0-40%)
HKM + pQCD
fmax=14, sigma=0.020 GeV
fmax=15, sigma=0.015 GeV
fmax=15, sigma=0.025 GeV
fmax=15, sigma=0.020 GeV
ALICE prelim data with stat error
ALICE prelim data syst error
10
3
3
E d Nγ /dp (GeV-2)
102
1
10-1
10-2
-3
10
10-4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
p (GeV/c)
T
Figure A.2: To get an impression of the order of magnitude of fmax and σ here
photon yields with values close to the chosen ones are given.
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Acknowledgement
First of all, I would like to thank my supervisor PD Dr. Klaus Reygers, for the
opportunity to carry out my bachelor thesis in his group and to become involved in
the ALICE experiment. I took great benefits from his help and guidance. Secondly
I would like to thank Prof. Dr. Norbert Herrmann for reading and evaluating my
thesis.
Special thanks go to Yuri Sinyukov, Werner Vogelsang and Olena Linnyk for providing results of their calculations used in this thesis.
I want to thank the ALICE group for giving me the opportunity to present and
discuss my results in the group meeting and especially Dr. Michael Knichel and
Michael Schork for proofreading my thesis.
I greatly appreciate the time Bernd Epding and my sister Ulrike spend for proofreading this thesis.
Finally, I wish to express my gratitude to my parents for their constant support
of my studies.
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