Knots and Links in High Energy
Physics
Masterarbeit
zur Erlangung des akademischen Grades
Master of Science
vorgelegt von
Philipp Sicking
geboren in Coesfeld
Lehrstuhl für Theoretische Physik III
Fakultät Physik
Technische Universität Dortmund
2014
1. Gutachter : Prof. Dr. Heinrich Päs
2. Gutachter : Prof. Dr. Gudrun Hiller
Datum des Einreichens der Arbeit: 30. September 2014
Abstract
This thesis analyzes two different applications for closed non-trivial topological stringlike
objects arising from a Landau-Ginzburg mechanism.
A Model of determining the masses and mixing in the neutrino sector based on a
Seesaw type I mechanism is introduced. The right handed neutrino mass matrix is
generated by knot and link lengths. A comparison with random entries does not show
a significant difference but improvements for the model are suggested.
The second part deals with a model of knotted inflation where a highly knotted
network of flux tubes builds the vacuum energy, necessary for an inflationary era in the
early universe. In this thesis the decay of such a network is simulated and it is shown
that a phase transition similar to percolation theory occurs which indicates the end of
inflation.
III

Contents
Table Of Contents V
1 Introduction to the Standard Model of Particle Physics 1
1.1 Fermion Masses in the Quark Sector . . . . . . . . . . . . . . . . . . . . 2
1.2 Fermion masses in the lepton sector . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Dirac Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Majorana Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.3 Seesaw Mechanism Type I . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Lepton Mixing Matrix and Experimental Constraints . . . . . . . . . . . 7
2 Closed Flux tubes as Knots and Links 11
2.1 Mathematical basics in Knot-Theory . . . . . . . . . . . . . . . . . . . . 11
2.2 Knotlength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Realization in a Quantum Field Theory . . . . . . . . . . . . . . . . . . 13
3 Determining Lepton Flavor via Knotlength 17
3.1 Generating Neutrino Masses . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Numerical Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Comparison with former analysis . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Interpretation of the Results . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Knotted Inflation 23
4.1 Standard Model of Cosmology . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Motivation for an era of Inflation . . . . . . . . . . . . . . . . . . . . . . 25
4.3 Knotted Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.4 Percolation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.5 Simulating Inflation via Knotted Networks . . . . . . . . . . . . . . . . . 27
4.5.1 Basic concepts of the simulations . . . . . . . . . . . . . . . . . . 27
4.5.2 Hoshen-Kopelman Algorithm . . . . . . . . . . . . . . . . . . . . 28
4.5.3 Lattice Network . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.5.4 Random network . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.7 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
V
CONTENTS
5 Conclusion & Outlook 35
A Numeric calculations for comparison with former analysis 37
List of References 41
VI
Chapter 1
Introduction to the Standard Model
of Particle Physics
The standard model (SM) of particle physics is a well tested theory that explains the
elementary particles and their behavior via a renormalizable quantum field theory. This
section is based on the descriptions in [1], [2] and [3].
The SM particles are excitations of quantum fields and the interactions are generated
by a local gauge symmetry
SU(3)C × SU(2)L × U(1)Y → SU(3)C × U(1)EM , (1.1)
where the SU(3)C indicates the strong interaction or Quantum-Chromo-Dynamics (QCD).
SU(2)L×U(1)Y corresponds to the electroweak interaction which is spontaneously bro-
ken via the Higgs mechanism to U(1)EM . The SM is described by the SM Lagrangian
L =
∑
Ψ={U,D,Q,L,E}
Ψi /DΨ + LHiggs + LGauge + LY ukawa. (1.2)
The first term corresponds to the kinetic term of the fermions and their interaction
with the gauge bosons via the covariant derivative /D. The second one describes self-
interactions of the Higgs field and the third the propagation and self-couplings of the
gauge bosons. The last term is interesting for flavor physics and describes the mixing
and masses of the different families. This will be discussed in detail.
The fermions are divided into quarks (Q, U and D) and leptons (L and E). Each
fermion field appears in 3 different families. The well-known particle content of the
1
Chapter 1. Introduction to the Standard Model of Particle Physics
fields reads as follows:
Qi =
((
uL
dL
)
,
(
cL
sL
)
,
(
tL
bL
))
, (1.3)
Li =
((
νeL
eL
)
,
(
νµL
µL
)
,
(
ντL
τL
))
, (1.4)
Ui =
(
uR, cR, tR
)
, (1.5)
Di =
(
dR, sR, bR
)
(1.6)
Ei =
(
eR, µR, τR
)
. (1.7)
The elementary matter particles are charged under the groups (1.1) as shown in
table 1.1.
Field SU(3)C SU(2)L Y
Q 3 2 16
U 3 1 23
D 3 1 −13
L 1 2 −12
E 1 1 −1
Table 1.1: Fermion fields and their representations under the gauge groups SU(3)C and
SU(2)L. Y denotes the hypercharge.
The quarks are triplets under SU(3)C and therefore interact via QCD due to their
non-trivial color charge. The Leptons are SU(3)C-singlets and thus do not interact via
QCD. The weak interaction couples to the doublets under SU(2)L, which are Q and L.
The hypercharge Y = Q−T3 can be determined through the electric Charge Q and the
weak isospin T3.
While the kinetic term and all interactions are diagonal in flavor space, the mass
term in the Lagrangian is not and therefore leads to flavor physics, which describes the
relation of the different generations. The basic concept of generating fermion masses is
explained in the next chapter.
1.1 Fermion Masses in the Quark Sector
Since the fundamental matter particles are fermionic, they can be described by a spinor
ψ and the typical mass term in the Lagrangian is
Lmass = ψmψ. (1.8)
By decomposing the spinor ψ into the right and left handed parts ψR and ψL denoting
the different possibilities of chirality, it is obvious to see how the mass term breaks
chirality
Lmass = ψLmψR + ψRmψL. (1.9)
2
1.1. Fermion Masses in the Quark Sector
In the SM the fermion masses are obtained from the Yukawa Lagrangian LY ukawa. The
terms related to quark masses read
LQY ukawa = −Qi(Yu)ijΦ
CUj −Qi(Yd)ijΦDj + h.c. , (1.10)
with the family indices i,j = 1,2,3 and the 3 × 3 complex Yukawa matrices Yu,d. The
SU(2)L-doublets of the left handed particles couple to the Higgs field Φ and the right
handed singlet. Φ is a scalar field and is a doublet under SU(2)L in the simplest case
since the Lagrangian has to be a singlet. The Higgs doublet can be parametrized by
Φ =
(
Φ+
Φ0
)
. (1.11)
The conjugate Higgs field is given by
ΦC = iσ2Φ
∗ =
(
Φ0∗
−Φ−
)
(1.12)
with the Pauli-matrix σ2 =
(
0 −i
i 0
)
.
The Higgs fields Φ and ΦC give masses to the down type and up type quarks, respec-
tively. Assuming a non vanishing vacuum expectation value 〈Φ0〉 = v√
2
for the Higgs
field, SU(2)L×U(1)Y will be broken spontaneously to U(1)EM . After this spontaneous
symmetry breaking (SSB) the Yukawa Lagrangian in the quark sector reads
LQY ukawa = −uLYd
v
√
2
uR − dLYu
v
√
2
dR + h.c. (1.13)
By identifying the factor Yd,u v√2 = Mu,d as a complex 3 × 3 mass matrix, the typical
mass term (1.9) is obtained.
The Yukawa matrices Yu,d are written in their flavor basis. In general, one has to
distinguish between flavor and mass eigenstates, where to obtain the latter one has to
insert the unitary matrices V †u,dVu,d = 1 and U
†
u,dUu,d = 1.
LY ukawa = uLV
†
u︸ ︷︷ ︸
u′L
VuMuU
†
u︸ ︷︷ ︸
diag(mu,mc,mt)
UuuR︸ ︷︷ ︸
u′R
+ dLV
†
d︸ ︷︷ ︸
d′L
VdMdU
†
d︸ ︷︷ ︸
diag(md,ms,mb)
UddR︸ ︷︷ ︸
d′R
+h.c. (1.14)
This transformation of the fields into their mass basis does not affect the interaction
with the neutral gauge bosons. That is the reason why there are no flavor-changing
neutral currents on tree level in the SM. However, the interaction via charged currents
does change
L ⊂
g
√
2
W+µ uLγ
µdL (1.15)
=
g
√
2
W+µ u
′
L VuV
†
d︸ ︷︷ ︸
VCKM
γµd′L. (1.16)
3
Chapter 1. Introduction to the Standard Model of Particle Physics
The Matrix VCKM determines the mixing between the different flavors via charged
currents. This matrix was first proposed by Cabibbo [4] in case of 2 quark generations
and then generalized by Kobayashi and Maskawa [5] to three quark generations.
In the SM VCKM has to be unitary because it represents a basis transformation.
The unitarity of VCKM is tested to validate or falsify the SM. The entries of this mixing
matrix can be measured by experiments via hadron-decays or meson-oscillations. An
experimentally determined fact is the strictly hierarchical structure of VCKM best seen
in the Wolfenstein-parametrization [6]
VCKM =



1− λ
2
2 λ Aλ
3(ρ− iη)
−λ 1− λ
2
2 Aλ
2
Aλ3(1− ρ− iη) −Aλ2 1


+O(λ4) (1.17)
with A,ρ,η = O(1) and the Wolfenstein parameter λ ' 0.2. As a consequence of this
hierarchy the mixing between different flavors is small. We will see in the following
chapter that this is not the case in the leptonic sector.
1.2 Fermion masses in the lepton sector
The general mechanism for generating fermionic masses has been outlined in the previ-
ous section. This mechanism can also be applied to the leptonic sector of the fermions.
LY ukawa ⊂ −Li(Yl)ijΦEj + h.c. −→
SSB
−lLYl
v
√
2
lR + h.c. (1.18)
Right handed neutrinos do not participate in any of the SM interactions, therefore
right handed neutrinos are not part of the SM particle content. This is the reason for
the absence of a corresponding neutrino mass term in the Yukawa Lagrangian and leads
to massless neutrinos in the minimal SM.
However, since there is experimental evidence for nonzero neutrino masses from
neutrino oscillation experiments (see chapter 1.3) the minimal SM has to be extended.
In general, there are two possible ways to introduce neutrino masses depending whether
neutrinos are Dirac or Majorana particles. The case of a Majorana particle is more
general since a Dirac and a Majorana mass is allowed while in case of a Dirac particle
the mass can only be generated by a Dirac mass term.
1.2.1 Dirac Mass
The Dirac Neutrino Mass corresponds to the mass of the charged fermions in the
standard-model. Therefore one can introduce a right handed neutrino field νR for each
generation and receive a mass term in the Lagrangian after symmetry breaking.
LDirac = νLMDνR (1.19)
with a complex 3 × 3-Matrix MD, called Dirac mass matrix. The typical Feynman-
diagram for a Dirac mass is shown in figure 1.1
4
1.2. Fermion masses in the lepton sector
νR νL
〈Φ〉
Figure 1.1: Feynman diagram for neutrino Dirac-mass.
1.2.2 Majorana Mass
The other possibility to generate neutrino masses is the so called Majorana mass. The
Majorana mass can only be generated if a fermion ψ is its own anti-particle, which
means
ψ = ψC = Cψ = −iγ2ψ∗ (1.20)
Obviously this is only possible if the fermion has no electric charge, which is the case
for neutrinos. If (1.20) holds, it is straightforward to show that the right handed field
can be replaced by the left handed field
ψR =
(
ψC
)
R = (ψL)
C . (1.21)
In this case there is no need to introduce a new light right handed neutrino. The
Majorana neutrino-mass term now can be written as
LMaj = νLMmaj (νL)
C + (νL)CMmajνL, (1.22)
with again a complex 3× 3 matrix Mmaj , called Majorana mass matrix. As opposed to
the Dirac mass matrix the Majorana mass matrix has to be symmetric.
It is easy to see that the Majorana mass term breaks lepton number L which used
to be a good quantum number in the minimal standard model. Another problem is
that the two left handed neutrinos form an SU(2)-triplet, so the Higgs doublet has to
be included twice in the mass term before symmetry breaking
LMaj =
1
2
LCLLL
ΦΦ
M
. (1.23)
The effective mass M needs to be introduced so that the Lagrangian still has mass
dimension 4. Equation (1.23) corresponds to the Feynman diagram shown in figure 1.2.
At present the Majorana type of neutrinos is not confirmed by experiments. Note that
this Lagrangian is not renormalizable and can only be seen as an effective interaction.
However, the term can explain the smallness of the neutrino masses. If the effective
Mass M is large, the neutrino-mass will be small. An approach with a renormalizable
interaction generating this term and the scenario adopted in this thesis is the Seesaw-
Mechanism.
5
Chapter 1. Introduction to the Standard Model of Particle Physics
νL νL
〈Φ〉 〈Φ〉
Figure 1.2: Feynman diagram for generating a neutrino Majorana mass by an effective inter-
action.
1.2.3 Seesaw Mechanism Type I
Consider a mass-term in the neutrino sector as follows
LMass = g〈Φ
0〉
︸ ︷︷ ︸
MD
νRνL +
MM
2
νCRνR + h.c. (1.24)
where the first term corresponds to a typical Dirac mass and the second to a Majorana
mass term for the right handed neutrino. A Majorana mass term for the left handed
neutrino is not allowed since it does not only violate lepton number conservation but
also violates hypercharge conservation. Note that the mass-scale of this right handed
neutrino is much larger than the electroweak scale. Since the Majorana mass term is a
singlet under the SM group, it is not influenced by the breaking of the SM. Therefore
the mass scale can be large, typically at the cutoff scale of the SM, e.g. the scale of a
grand unified theory or the string scale. This gives room for new physics beyond the
SM. The Feynman diagram for this process is shown in figure 1.3.
νR νCR
νL νL
M
f f
〈Φ〉 〈Φ〉
Figure 1.3: Feynman diagram for the Seesaw mechanism type I.
For an effective approach you can integrate out the heavy particle νR and get the
effective Lagrangian
Leff =
f2
2MM
〈Φ0〉〈Φ0〉νCL νL. (1.25)
This is exactly the term seen in (1.22). The feynman-diagram now can be seen in figure
1.4. Calculating the effective masses is equivalent to diagonalizing the matrix
(
0 MD
MD MM
)
, (1.26)
6
1.3. Lepton Mixing Matrix and Experimental Constraints
νL νL
f2〈Φ〉〈Φ〉
2M
Figure 1.4: Feynman diagram for the Seesaw mechanism type I with νR integrated out.
yielding an effective mass of the light left handed neutrinos as
MνL '
m2D
MM
, (1.27)
or for n generations, where MD and MM are in general complex n× n matrices,
MνL 'M
T
DM
−1
M MD. (1.28)
1.3 Lepton Mixing Matrix and Experimental Constraints
In analogy to the CKM-Matrix in the quark sector one can introduce the Pontecorvo–
Maki–Nakagawa–Sakata (PMNS) matrix
UPMNS = V
†
l Uν , (1.29)
which rotates fields from mass to flavor-space. This PMNS matrix becomes non-trivial
if any neutrino mass matrix is generated, regardless of which mechanism is used as
long as its structure differs from the one of the charged leptons. Vl denotes the unitary
matrix which diagonalizes Yl (see equation (1.18)) and is assumed to be 1 in this thesis,
due to the fact we can choose any basis without changing the physics. Uν is the matrix
diagonalizing the neutrino mass matrix MνL given in flavor-space so that
UTMνLU = diag(m1,m2,m3) (1.30)
with the neutrino masses m1, m2 and m3. The elements of the PMNS-matrix can be
parametrized in the measured parameters of neutrino oscillation experiments.
UPMNS =
(
c12c13 s12c13 s13e−iδCP
−s12c23 − c12s23s13eiδCP c12c23 − s12s23s13e−iδCP s23c13
s12s23 − c12c23s13eiδCP −c12s23 − s12c23s13e−iδCP c23c13
)
· P (1.31)
with P = diag(eiα,eiβ,0) containing the Majorana phases and sij , cij are abbreviations
for sin θij and cos θij . The phase δCP is responsible for CP -violation.
Determining the elements in UPMNS and the neutrino-masses mi in experiments
is rather difficult since typical neutrino cross-sections are tiny. Nevertheless there are
several experiments which measure properties of the neutrino masses:
One way to identify the absolute value of the neutrino mass is by precisely measuring
the tail of the energy spectrum of β decays. Since β decay (n→ p+ e− + ν¯e) is a three
7
Chapter 1. Introduction to the Standard Model of Particle Physics
body decay the energy distribution of the electron is continuous and the absolute neu-
trino mass contributes to the cutoff in the spectrum. Current experiments determined
an upper bound of mν < 2.8 eV(95%C.L.) [7] and mν < 2.2 eV(95%C.L.) [8].
Another approach is to search for neutrinoless double beta-decays, as done by the
GERDA-experiment. Finding such an neutrinoless double beta decay would prove that
neutrinos are Majorana particles and give access to the (1,1)-element in the neutrino
mass matrix. A range for an upper limit on mββ =
∑
i U
2
eimi of 0.2− 0.4eV is given by
[9] .
The sum of the neutrino masses has much influence on cosmological observations
e.g., the formation of structures in the universe and primordial nucleosynthesis. The
current constraint is
∑
imi < 0.23eV [10].
To get access to the parameters in the mixing matrix and the mass squared differ-
ences δm2ij = m
2
i −m
2
j , neutrino oscillations are analyzed. Neutrino oscillation is the
conversion of a neutrino from one flavor eigenstate to another and back with respect
to the travel distance. Since the flavor eigenstates and mass eigenstates mismatch, the
probability of measuring a neutrino, produced as an electron neutrino, as an electron
neutrino after travel distance L is
Pνe→νe = 1−
∑
i,j
4|Uei|
2|Uej |
2 sin2
δm2ij
4E
L (1.32)
with the neutrino energy E.
Measuring the neutrino flux at different distances yields information about δm2ij
and the mixing angles θij . As already known neutrino masses are small, and so are the
squared differences. Therefore, to be sensitive to small δm2, the energy has to be low
(reactor experiments) or the baseline has to be long (accelerator experiments). But not
only artificial neutrino sources are used to determine the mixing parameters but also
atmospheric neutrinos and solar neutrinos. Several experiments currently measure neu-
trino oscillations like DOUBLE CHOOZ, DAYA BAY, RENO (all reactor experiments),
KamLAND (reactor and long baseline experiment), T2K, MINOS, CNGS (all long base-
line experiments) or Super-Kamiokande (atmospheric and solar neutrinos). A current
global fit to all recent data gives best fit values for each parameter with 1σ-uncertainty
[11]
sin2 θ12 = 0.302
+0.013
−0.012, (1.33)
sin2 θ23 = 0.413
+0.037
−0.025(NH), (1.34)
sin2 θ23 = 0.594
+0.021
−0.022(IH), (1.35)
sin2 θ13 = 0.0227
+0.0023
−0.0024, (1.36)
δm221 = 7.50
+0.18
−0.19 · 10
−5eV2, (1.37)
δm231 = +2.473
+0.70
−0.67 · 10
−3eV2(NH), (1.38)
δm232 = −2.427
+0.042
−0.065 · 10
−3eV2(IH), (1.39)
δCP = 300
+66
−138
◦. (1.40)
8
1.3. Lepton Mixing Matrix and Experimental Constraints
Note that since the squared differences are measured, only two mass differences are
independent. Neutrino-oscillations cannot make any statement about the absolute mass
scale. Normal hierarchy (NH) and inverse hierarchy (IH) are possible and cannot be
distinguished by current oscillation experiments. For illustration of the hierarchies see
figure 1.5.
Figure 1.5: Neutrino mass hierarchy, taken from [12].
If the absolute masses are large compared to the differences, the neutrino masses
are (called) degenerate.
9

Chapter 2
Closed Flux tubes as Knots and
Links
The question why we have only three large space dimensions is not answered satisfacto-
rily, because quantum field theories can easily be extended to more than three spacial
dimensions. It can be proposed that the reason for exactly three spacial dimensions lies
in the special properties of a three dimensional space. One approach is to look at the
mathematical topic of knot theory. It is a subtopic of topology and is defined on R3
(see chapter 2.1).
If knots have to be realized in a physical context this is only possible in three spatial
dimensions. One might also consider the other side of the argument: Knots can only
be realized physically because there are three space dimensions. Knot like structures
in a high energy context can arise in a Landau Ginzburg Model where flux tubes form
string like objects which can be treated as knots (see 2.3).
The chapters 3 and 4 discuss two concrete applications of knot like structures, which
might be improvements to the SM. The first is a possibility of right handed neutrinos
at a high mass scale which are realized by closed flux tubes. This would lead to an
extension of the SM explaining neutrino masses and mixing. The second one tries to
model cosmological inflation and especially its end phase by a network of flux tubes
during a phase transition. The decay of this network will be simulated.
2.1 Mathematical basics in Knot-Theory
First it has to be defined what a knot is in the mathematical context. A knot K is
defined as the image of the homeomorphism of the unit circle C from the R2 -plane to
R3. One can imagine this definition by taking a rope and splicing the ends together
after perhaps winding it up.
The simplest knot is the trivial unknotted circle with x2 + y2 = 1 and z = 0 (see fig
2.1). The simplest non-trivial knot is the trefoil shown in figure 2.3. If two or more knots
are connected so that they cannot be separated without cutting one of the knots, this
object is called a link. Figure 2.2 shows the simplest link consisting of two connected
11
Chapter 2. Closed Flux tubes as Knots and Links
trivial knots, named Hopf-Link. From now on both knots and links are referred to as
knots for simplicity.
To distinguish between different knots it is useful to classify knots into different
types. Two knots are of the same type if there exists an isotopy to transform one
into the other. Simply speaking, an isotopy can be imaged as deforming and twisting
the knot without cutting the rope. If two knots are of the same type, they are called
equivalent. For the trivial and trefoil knot it is easy to see that they are not equivalent,
but for more complex knots most of the time it is not obvious. If one extends these
knots to a space with more than three spatial coordinates, all knots are equivalent since
one can unknot every knot by using a fourth dimension. Therefore the formation of
knots is a specialty of the R3.
By projecting a knot into a plane, one can define its order O. This is the minimal
possible number of crossings by adjusting the projecting plane.
To classify knots recent research has tried to find so called knot invariants. These
allow to distinguish between different classes of knots, since knots of the same type
have the same invariants. Even measures which are not different (or not proven to
be different) for all types are often called knot invariants, e.g. Jones-Polynomial or
tricolorability. One physical knot invariant is the knotlength following in the next
section. For a further introduction to knot theory see [13].
2.2 Knotlength
The most physical approach to classify knots is the minimal ropelength of the knot,
called the knotlength. The ropelength Rop(γ) of a curve γ is defined as the fraction
Rop(γ) =
Len(γ)
Thi(γ)
(2.1)
where Len(γ) is the length and Thi(γ) is the Thickness of γ (see [14] for mathematical
details). Therefore the knotlength of a knot K is the minimal length of a rope with
thickness 1 with which the knot K can be knotted. An analytic way for calculating the
knotlength for a given knot has not been found yet but numeric calculations give access
to very good approximations of the knotlength.
In Table 2.1 the knotlength of knots and links up to order 7 are shown. The Knots
are written in Rolfsen-Notation Oli where O denotes the order. The index i counts the
different types with the same order and l defines the number of knots which are linked
together. In case of a missing l the object consist only out of a single knot.
Note that especially for higher crossing numbers the knotlengths are nearly degen-
erate. This feature lies in the fact, that the knotlength grows linearly with the crossing
number but the number of possible different knots grows faster than exponentially with
increasing crossing number. This degeneracy will be used in section 3.
12
2.3. Realization in a Quantum Field Theory
Figure 2.1: Trivial Knot
Figure 2.2: Hopf-Link
Figure 2.3: Trefoil Knot
2.3 Realization in a Quantum Field Theory
One application of knots as a mathematical object in the field of high energy-physics
can be found in the formation of string-like flux tubes in abelian gauge theories. The
basic concept and derivation is shown in the lecture-notes of [15], on which this chapter
is based on.
The behavior and formation of flux tubes can be seen in the Landau-Ginzburg-
Model, or sometimes Abelian-Higgs-Model. Consider a Lagrangian given by
LLG = −
1
4
FµνF
µν
+
1
2
(Dµφ)(D
µφ)† −
λ
2
(φφ∗ − v2)2, (2.2)
with a complex scalar-field φ and the dual field tensor F
µν
. Dµ = ∂µ+igBµ denotes the
covariant derivative with the vector potential Bµ. The field φ represents a (chromo-)
13
Chapter 2. Closed Flux tubes as Knots and Links
Knot Knotlength Index Link Linklength Index
31 32.7436 00 221 25.1334 01
41 42.0887 01 421 40.0122 02
51 47.2016 02 521 49.7716 03
52 49.4701 03 621 54.3768 04
61 56.7058 04 622 56.7000 05
62 57.0235 05 623 58.1013 06
63 57.8392 06 631 57.8141 07
71 61.5067 07 632 58.0070 08
72 63.8556 08 633 50.5539 09
73 63.9285 09 721 64.2345 10
74 64.2687 10 727 65.0204 11
75 65.2560 11 723 65.3257 12
76 65.6924 12 724 65.0602 13
77 65.6086 13 725 66.1915 14
726 66.3147 15
727 55.5095 16
728 57.7631 17
731 65.8062 18
Table 2.1: Length of Knots and Links up to order seven [14]. The column Index refers to the
numeration used in chapter 3.
magnetic charge condensate with the magnetic charge g. The underlying gauge group is
not important at this point, but may be a picture describing the confinement of QCD.
This model is called “dual superconductor”, because the electric and magnetic charges
and currents are exchanged, so that the theory is described by the magnetic charges
and currents instead of the electric ones. To take electric charges into account, one has
to add a string term Gµν , similar to the description of Dirac-Strings.
Fµν = (∂µBν − ∂νBµ) +Gµν (2.3)
The last term in (2.2) denotes the potential for φ and gives the non-zero vacuum ex-
pectation value φφ∗ = v2 6= 0 for lower energies. This gives rise to two different phases
before and after SSB in analogy to the Higgs-Mechanism in the SM. The first phase,
with φφ∗ = v2, is the superconductive or color-confined-phase, the other phase with
φφ∗ = 0 is the normal phase in which QCD is asymptotically free. This is very similar
to the normal superconductive picture.
In a superconductor, there is no magnetic field due to the Meissner-Ochsenfeld-effect.
Similarly, in the dual superconductive picture there is no electric field, if the system is in
the superconductive phase. In QCD electric charges exist, e.g. quarks, which play the
role as magnetic monopoles do in case of Dirac strings. The Meissner-Ochsenfeld-effect
cannot avoid chromo-electric currents entirely but pushes them into a string like object,
the flux tube.
14
2.3. Realization in a Quantum Field Theory
Similar to the Higgs mechanism in the SM, the phase transition gives rise to a
nonzero Higgs-Mass mH = 2vλ and the mass for the gauge field mV = gv. This
corresponds to the typical superconductor properties, the penetration depth 1mV and
the correlation length 1mH . The first is the distance after which the chromo-electric-field
vanishes outside the flux tubes (“thickness” of the tube). The latter corresponds to the
distance in which the scalar field acquires its vacuum expectation value.
For long flux tubes it can be shown that the energy of a flux tube is proportional to
the distance R between the chromo-electric charges, or the length of the flux tubes
E = σR. (2.4)
with the string tension σ. For further details see [15]. Equation (2.4) is the typical
linear potential of QCD in confinement.
All these phenomena can also occur in other (non-) abelian gauge theories. In this
thesis we do not concentrate on the mechanisms of forming such flux tubes, but discuss
two different kinds of phenomena, which might provide an application for closed flux
tubes. In case of transition from a normal to a dual superconducting phase with a huge
number of (chromo-) electric charges, the generation of numerous flux tubes is likely,
which can be closed as well. Due to the fact that energy is minimized, these flux tubes
will tighten up and can be treated as knots.
15

Chapter 3
Determining Lepton Flavor via
Knotlength
3.1 Generating Neutrino Masses
As pointed out in chapter 1.3 the mixing matrix in the leptonic sector differs from the one
in the quark sector. The mixing is large and the neutrino masses are small. Therefore,
the hierarchical structure in the CKM-Matrix cannot be reproduced in the PMNS-
Matrix. These differences in the experimental data may correspond to the peculiarities
of generating neutrino masses.
In chapter 1.2 the different mechanisms to generate neutrino masses are introduced,
and it is shown that a seesaw-like mechanism is an attractive way to explain the small-
ness of the neutrino masses. The model used in this thesis is based on [16] and [17].
The basic idea is that the Dirac mass matrix is analogous to the mass matrices of the
quarks and provides only small mixing angles. The large mixing angles observed in
the leptonic sector arise from the right handed Majorana masses. For simplification we
therefore assume a diagonal Dirac mass matrix
MD = diag(m
D
1 ,m
D
2 ,m
D
3 ), (3.1)
and a real Majorana mass matrix Mmaj for the right handed sector since the CP -
violating phase δCP in the neutrino sector is not determined sufficiently yet (see equation
(1.40)).
Mmaj =


mK1 m
L
1 m
L
2
mL1 m
K
2 m
L
3
mL2 m
L
3 m
K
3

 . (3.2)
The entriesmK/Li are real parameters at the moment and will be determined in equation
(3.3).
To calculate the mass-matrix of the three light left-handed neutrinos we use equation
(1.28) and obtain
17
Chapter 3. Determining Lepton Flavor via Knotlength
MνL =
1
∆3


[mK2 m
K
3 − (m
L
3 )
2](mD1 )
2 (mL2m
L
3 −m
K
3 m
L
1 )m
D
1 m
D
2 (m
L
1m
L
3 −m
K
2 m
L
2 )m
D
1 m
D
3
(mL2m
L
3 −m
K
3 m
L
1 )m1
DmD2 [m
K
1 m
K
3 − (m
L
2 )
2](mD2 )
2 (mL1m
L
2 −m
K
1 m
L
3 )m
D
2 m
D
3
(mL1m
L
3 −m
K
2 m
L
2 )m
D
1 m
D
3 (m
L
1m
L
2 −m
K
1 m
L
3 )m
D
2 m
D
3 [m
K
1 m
K
2 − (m
L
1 )
2](mD3 )
2

 , (3.3)
with the factor
∆3 = −mK2 (m
L
1 )
2 + 2mL1m
L
2m
L
2 −m
K
2 (m
L
2 )
2 −mK1 (m
L
3 )
2 +mK1 m
K
2 m
K
2 . (3.4)
Before the measurement of θ13 6= 0, a good approximation of the leptonic mixing
matrix was given by the tribimaximal scheme
UTBM =




√
2
3
1√
3
0
− 1√
6
1√
3
− 1√
2
− 1√
6
1√
3
1√
2



 (3.5)
This corresponds to sin2 θ12 = 13 , sin
2 θ23 = 0.5, sin2 θ13 = 0. The name arises from
the tri-maximal mixing of the mass eigenstate ν2 and the bi-maximal mixing of the
mass eigenstate ν3. This scheme is still a popular choice for model building, since
the structure could give information about an underlying symmetry. Comparing these
values to the current global best fits (see equation (1.33) - (1.35)), the deviations are
small enough to consider them as perturbations.
It is shown in [18] that as typical mass-matrix generating such a tribimaximal mixing
can be parametrized by
MTBMν =


x y y
y x+ v y − v
y y − v x+ v

 . (3.6)
The two possible hierarchies introduced in chapter 1.3 can be realized by approximating
the small mass squared difference to zero. The arising mass structures in mass basis
read diag(0,0,m˜) for normal and diag(m˜,m˜,0) for inverse hierarchy with the absolute
non vanishing neutrino mass m˜. Assuming tribimaximal mixing, this corresponds to
MNHν = m˜


0 0 0
0 12 −
1
2
0 −12
1
2

 (3.7)
for normal hierarchy. Comparing this matrix to (3.3) leads to five relations between the
18
3.1. Generating Neutrino Masses
entries of the mass-matrices:
mK3
mK2
=
(mD2 )
2
(mD3 )
2
, (3.8)
mL2
mL1
=
mD3
mD2
, (3.9)
mK2 m
L
2 = m
L
1m
L
3 , (3.10)
mK1 m
K
2 6= (m
L
1 )
2, (3.11)
m˜ = 2(mD3 )
2 (m
L
1 )
2 −mK1 m
K
2
∆3
. (3.12)
If one assumes the Dirac masses mDi to be roughly equal, these relations can be solved
simultaneously by an almost degenerate spectrum of mLi and m
K
i .
For inverse hierarchy the corresponding mass matrix in flavor space looks like
M IHν = m˜


1 0 0
0 12
1
2
0 12
1
2

 . (3.13)
comparing with (3.3) and solving all resulting relations simultaneously, leads to either
vanishing absolute neutrino mass m˜ = 0 or vanishing elements in Mmaj : mL1 = m
L
2 = 0
in contradiction to the assumption made above.
Based on the normal hierarchy case, the idea in [17] and for this chapter of this
thesis is to combine the assumptions of almost degenerate entries in MMaj with the
almost degenerate knotlengths seen in Table 2.1. Since the energy of a closed flux
tube is proportional to its length, a mass matrix produced by closed tight knots should
be proportional to the knotlength lK/Li , where i denotes the index from 2.1 and the
superscripts indicates whether it is a single knot (K) or a link (L). Hence the entries
of MMaj are generated by
mK/Li = l
K/L
i ·ms (3.14)
with an overall scale factor ms. This scale factor determines the scale of confinement
at which the new phenomena occur. For the numerical analysis this value is fixed to
ms = 1012GeV, which is motivated by cosmological constraints (see [17] for details).
If the assumptions made above do apply, we will expect substantially more viable
fits for the normal hierarchy by using the knot-spectrum for entries in the Majorana
mass matrix instead of using only randomly generated entries. Furthermore the number
of viable fits should be less and the goodness of fit should be worse in the case of inverse
hierarchy.
These calculations have been performed already in [17] but based on older experi-
mental data, which did not exclude θ13 = 0. The tribimaximal mixing was therefore
well motivated. The analysis seemed to confirm the expectations. In this thesis we will
examine whether current data with a nonzero θ13 confirm the previous results.
19
Chapter 3. Determining Lepton Flavor via Knotlength
3.2 Numerical Calculations
The algorithm for fitting the Dirac mass matrices to the experimental data works as
follows. Every combination of knots and links from Table 2.1 is calculated separately.
For each combination, the matrix MνL is derived with the three arbitrary parameters
mDi . After that the mixing matrix is calculated by diagonalizing the generated matrix
with the Basic Linear Algebra Subprograms published in the GNU Scientific Library
[19]. From this mixing matrix the mass squared differences and the mixing angles can
be extracted. A problem is to order the mass eigenstates correctly, so that the correct
mixing angles are extracted.
A χ2 fit is applied to the experimental values from (1.33) - (1.39). This χ2 is
minimized by varying the Dirac masses mDi . The asymmetric errors are handled as
shown in [20]. The minimization method used in this simulation is based on the Simplex
algorithm [21], implemented in the GNU Scientific Library [19]. Since there are five
observables and three fitting parameters, there are two degrees of freedom and for a P
value of 0.01 all combinations with χ2 < 9.21 are considered viable.
To make useful statements about this model, we have to compare it with random
entries in Mmaj . Therefore 1,000,000 different variations of random numbers ranging
from 0 up to the biggest knotlength are inserted in the Majorana mass matrix. For
every combination the Dirac masses are adjusted to minimize the resulting χ2.
The scan is separated into two different programs, one for normal hierarchy and
one for inverse hierarchy, to ensure the correct matching of mass eigenstates. The best
fits are shown in Table 3.1 for normal and in Table 3.2 for inverse hierarchy. The used
indices are the same as in Table 2.1 where the corresponding knotlengths are shown.
Table 3.3 also shows the ratio of all tested models that are considered viable. For nor-
mal hierarchy, no excess is observed for models with degenerate knotlengths compared
to random entries. The best fits moreover do not lead to a smaller χ2 in comparison to
random matrix-entries. For inverse hierarchy the models with degenerate knotlengths
lead to a number of viable fits twice as high compared to the case of random entries but
in the same range as both models of normal hierarchy. The best fits lead to a smaller
χ2 but do not make much differences in that regime.
On a closer look at the results of normal hierarchy, even the best fits in Table 3.1
do not seem to have any structure. Some of the knotlength are degenerate, which
can be accidental since degeneracy always occurs in some elements of the matrix. In
contradiction to the assumption made above, the Dirac masses are not degenerate in
the best fits. This assumption is not well motivated if one considers the Dirac mass
matrix in the neutrino sector to be analogous to the one in the charged fermion sector.
There, hierarchical structures exist with non degenerate masses.
Dropping the assumption of almost degenerate Dirac-masses, the equations (3.8)-
(3.11) can as well be solved by arbitrary entries in the Majorana mass matrix. Since
the Dirac masses are not fixed in any way but used to minimize the χ2, there is no
motivation for assuming them to be almost equal. Moreover, even if the knot spectrum
is used, the best fits do not lead to a degenerate spectrum of mDi .
For inverse hierarchy a structure in the best fits can also not be seen. The deviation
20
3.3. Comparison with former analysis
of the knotted spectrum to the random entries can not be seen in any analytical context
and is maybe caused by accidentally well fitting numbers.
These differences to [17] may be caused by the deviation from tribimaximal mixing
since θ13 6= 0. If this is the case, the method used for this thesis should give the same
results if the same data input is used. This is part of the next section.
χ2 K1 K2 K3 L1 L2 L3 mD1 [GeV] m
D
2 [GeV] m
D
3 [GeV]
< 0.001 10 08 03 00 01 11 14.854 26.804 23.429
0.001 00 06 08 04 02 07 5.619 20.766 14.507
0.001 01 05 12 17 16 05 6.719 22.106 15.537
0.001 05 09 02 03 01 07 7.052 12.017 18.513
0.002 01 05 13 13 06 07 6.782 22.371 15.698
0.002 06 01 10 06 09 08 5.819 13.022 11.328
0.002 06 01 10 16 09 08 5.808 12.999 11.283
0.002 10 09 12 00 01 08 14.624 24.501 23.816
0.002 11 10 13 00 01 08 14.838 24.612 23.974
0.002 13 05 09 01 15 10 10.251 19.688 18.378
Table 3.1: The 10 best fits for normal hierarchy.
χ2 K1 K2 K3 L1 L2 L3 mD1 [GeV] m
D
2 [GeV] m
D
3 [GeV]
< 0.001 13 03 09 02 09 04 34.579 6.435 8.888
0.001 05 13 03 05 01 06 40.954 9.978 10.457
0.002 02 05 12 00 04 09 54.144 16.699 22.279
0.002 04 13 07 15 00 14 58.051 17.374 20.188
0.002 07 06 11 00 11 12 64.531 18.358 24.335
0.002 11 09 03 01 04 07 56.507 14.207 15.356
0.002 12 01 07 05 09 08 33.176 4.922 7.169
0.002 13 05 12 15 00 09 64.227 17.061 21.994
0.003 12 08 13 02 10 17 45.718 10.363 12.825
0.003 12 08 13 02 12 17 45.756 10.372 12.847
Table 3.2: The 10 best fits for inverse hierarchy.
3.3 Comparison with former analysis
The best fits using the experimental values as used in [17] (see equations (A.1)-(A.6))
are shown in Tables A.1 and A.2 for normal hierarchy and inverse hierarchy, respectively.
A comparison to random numbers in the Majorana mass matrix is given in Table A.3.
The best fits proposed in [17] can not be found and a substantial deviation between
knotlength and random entries does not occur either.
21
Chapter 3. Determining Lepton Flavor via Knotlength
Model tested parameters viable fits ratio viable fits χ2bestfit
NHknots 10,692,864 33,863 0.317% < 0.001
NHrandom 1,000,000 3,369 0.337% 0.004
IHknots 10,692,864 35,044 0.328% < 0.001
IHrandom 1,000,000 1,458 0.146% 0.008
Table 3.3: Number of viable fits for the different models.
On a closer look at the best fit in [17] and by calculating the mixing, it is easy to
see that these entries lead to a wrong mixing scheme. Compared to Figure 1.5 the mass
eigenstates m1 and m2 are exchanged. Therefore there has to be a mistake in the past
analysis.
3.4 Interpretation of the Results
The method of scanning introduced above needs a justification. In the universe we see,
the mixing in the leptonic sector is described by the experimental data (see equations
(1.33) - (1.40)). It is obvious that only one special underlying mass matrix generates
this mixing scheme and therefore only one special combination of entries in the mass
matrix has to reproduce the experimental data.
If the model of knotlength determining the right handed mass matrix describes
nature appropriately, one combination of knots is realized in this universe. The number
of different combinations by which the experimental data can be reproduced, does not
say anything about the correctness of the model. However, it is possible to say if
such a mixing scheme is somewhat special or more a regular case. If a high number
of combinations leads to satisfying results, a fine-tuning process does not have to be
applied and the model seems more probable.
The calculations show that there is no substantial advantage of the knotted spectrum
in reproducing the experimental values for masses and mixing. This is not a fail of
the model but relies in the fact that the Dirac masses are treated as being totally free
parameters in the numeric calculations. The claim that a degeneracy of the Dirac masses
would arise automatically by fitting to the knotted spectrum cannot be hold. Therefore
a mechanism to generate these nearly degenerated Dirac masses, e.g. an approximative
or slightly broken symmetry, has to be found. In that case the analytic arguments would
apply and can lead to an improved model where the knotted spectrum perform better
than the random spectrum. If such a mechanism is found, the degeneracy of the Dirac
masses has to be implemented into the computer code. With this assumption, a large
difference between the knotted and the random spectrum is expected.
22
Chapter 4
Knotted Inflation
The derivations in this section with general aspects of cosmology are based on on [22].
The purpose of cosmology is to explain the evolution of the universe with laws of
physics we already know. It is a mixture of general relativity, particle physics and
thermodynamics and covers a broad field of physical effects. The model that describes
the visible universe is called the Standard Model of cosmology, which is mostly accepted
in the current research community.
4.1 Standard Model of Cosmology
In general relativity the metric describes the behavior of space time in the universe.
Based on observations of a very isotropic universe and the well motivated assumption
that the earth is in no sense special according to its position in space, it is reasonable
to assume a homogeneous and isotropic space time. In the case of three spatial and one
temporal dimension this space time can always be described by the Robertson-Walker
metric in spherical coordinates
ds2 = dt2 − a2(t)
[
dr2
1− kr2
− r2dΩ
]
. (4.1)
Here a(t) denotes the time dependent scale factor. The parameter k describes the space
curvature:
k =



+1 spherical
−1 hyperspherical
0 Euclidean
(4.2)
The Einstein equations describe the evolution of space time in dependence on the energy
content of the universe given by the energy momentum tensor Tµν
Rµν −
1
2
gµνR = 8piGTµν , (4.3)
23
Chapter 4. Knotted Inflation
with the Ricci tensor Rµν , the Ricci scalar R and the Newton constant G. Using the
Robertson-Walker metric (4.1) and a homogeneous and isotropic energy momentum
tensor we get the two Friedmann equations
a˙2 + k =
8piGρa2
3
, (4.4)
ρ˙ = −
2a˙
a
(ρ+ p), (4.5)
with the energy-density ρ and the pressure p. Equation (4.5) also accounts for energy-
momentum conservation.
Obviously this leads to different behavior of the scale-factor a(t) and therefore of
the expansion of the universe for different relations between p and ρ. These relations
depend on the type of energy-content:
cold matter (e.g. dust): pM = 0→ ρM ∝ a
−3, (4.6)
hot matter (e.g. radiation): pR =
ρR
3
→ ρR ∝ a
−4, (4.7)
vacuum energy: pV = −ρV → ρV = const. (4.8)
For a rough estimate of the behavior of the scale-factor in the early universe (small
a), one can neglect the curvature-factor k in equation (4.4). This leads to
cold matter: a(t) ∝ t
2
3 , (4.9)
hot matter: a(t) ∝ t
1
2 , (4.10)
vacuum energy: a(t) ∝ e
√
Λ
3 t, (4.11)
with Λ = 3piGρV . This allows to calculate the “ ‘age” of the universe depending on the
dominant energy content.
Assuming a mixture of different types of energy-content in the universe and also an
arbitrary curvature k, it can be shown that the energy density yields
ρ =
3H20
8piG
[
ΩΛ + ΩM
(a0
a
)3
+ ΩR
(a0
a
)4
]
, (4.12)
where the subscript 0 denotes the the present value. H(t) = a˙(t)a(t) is the Hubble param-
eter, and Ωi is defined by
ΩΛ =
8piG
3H20
ρV 0, (4.13)
ΩM =
8piG
3H20
ρM0, (4.14)
ΩR =
8piG
3H20
ρR0. (4.15)
With equation (4.4) it follows that
ΩΛ + ΩR + Ωk = 1 and Ωk = −
k
a20H
2
0
. (4.16)
24
4.2. Motivation for an era of Inflation
4.2 Motivation for an era of Inflation
The common picture of the early universe is the so called hot Big Bang scenario, in which
the universe starts in a state of extremely high temperature and density. During the first
seconds after the Big Bang the universe is dominated by radiation and therefore expands
according to (4.10). The expansion leads to a cool down of matter until nucleosynthesis
takes place and cold matter dominated the expansion, see (4.9). Even though it is
called cold matter, the particles still have enough energy to prevent recombination of
the light nuclei with electrons. This is caused by photon scattering in the very dense
universe. Only after the universe has expanded and consequently cooled down enough,
the electrons and nuclei recombine to atoms and the universe becomes transparent. At
that time the cosmic microwave background (CMB) originates. Caused by the redshift
of the still expanding universe, nowadays one can see the radiation from the time of
recombination in the microwave spectrum. The temperature of the CMB is measured to
TCMB = 2.72548±0.00057 K and is isotropic up to a deviation of ∆TT = 10
−4 [23]. The
CMB is the earliest measurable relic of the universe based on electro-magnetic radiation.
A measurement of a cosmic neutrino background could yield direct information of the
time before recombination but is far out of reach of present experiments.
The assumption of a hot Big Bang in which all the energy was located at one single
point leads to three major problems which may be solved by a phase of inflationary
expansion:
• Problem of flatness: Observations of supernova redshift and CMB-fluctuations
favor Ωk = 0 but still give room for a small |Ωk| < 1. According to Equation
(4.16), Ωk changes with a˙−2. Therefore Ωk increases in matter- and radiation-
dominated phases. If Ωk is small and nonzero today, which corresponds to a nearly
flat universe, Ωk has to be even smaller at the time of the Big Bang leading to an
even flatter universe. This is not ruled out but seems unnatural. An attractive
solution is a period of inflation with dominating vacuum-energy, so that a˙/a is
constant as in equation (4.11). Then Ωk decreases with a−2 and an arbitrary
curvature Ωk at the beginning of the inflationary period would automatically lead
to an extremely flat universe.
• Problem of horizons: As mentioned above the CMB temperature is extremely
isotropic. This leads to the horizon problem. It can be shown that the horizon was
about 1.6◦ during the first emergence of the CMB. Therefore one would expect no
physical relations between regions which are outside this horizon. Inhomogeneities
could not have been smoothed out over such a big region as suggested by the CMB.
Again, an age of inflationary expansion before the time of radiation domination
can solve this problem. All of the visible universe was confined to a tiny region
before inflation, in which it easily could have been homogeneous.
• Problem of monopoles: Several theories such as grand unified theories predicts
monopoles as relics after symmetry breaking. Rough estimates favor a similar
amount of monopoles and nuclei but there is no evidence for even a single observed
25
Chapter 4. Knotted Inflation
monopole. An inflationary phase occurring after the production of the monopoles
but before the production of the nuclei would lead to a much reduced expected
number of monopoles.
The most striking reason for cosmological inflation is the formation of structure. The
structure in the universe is scale invariant and can be explained by inflation via quantum
mechanical perturbations. These perturbations are seeds for structure formation and
occur during the era of inflation. This leads to similar structures on different scales.
Several models with an inflationary phase have been proposed in the last years.
A way to motivate a specific model of inflation by experimental data (or at least the
possibility to falsify it) is the observation of B-modes in the CMB. Gravitational waves
as relics of the inflation would polarize the CMB in a special way so that B-modes
arise. First evidence of such B-modes was reported by BICEP2 in 2014 [24]. A current
analysis by PLANCK suggests that the dust foreground may be underestimated in the
BICEP2 analysis [25]. A joint analysis of both experiments is announced.
One problem of inflation models is the source of the vacuum energy. A proposed
scalar Inflaton field in a state of high energy can represent the vacuum energy. The
approach of [26] to realize such a vacuum energy is taken as a basis in this thesis.
4.3 Knotted Inflation
As seen in chapter 2.3, it is possible to form flux tubes in (non-) Abelian gauge theories
like grand unified theories. Assuming such a theory with a Landau-Ginzburg Lagrangian
(2.3) (for instance confinement in QCD), a network of flux tubes can arise after the phase
transition. It is likely that the flux tubes are not separated but form a highly knotted
network. Each flux tube with radius a carries the quantized chromoelectric flux Φ and
therefore can be described by a constant chromoelectric field
Ei = F0i =
Φ
pia2
ni, (4.17)
with the vector ni orthogonal to the flux tube section. The entire network will then
give rise to a non vanishing vacuum energy density given by
ρE =
1
2
| ~E|2 =
1
2
TrΦ2
(pia)2
. (4.18)
The network is assumed to be roughly stable in three dimensions since loosening this
network corresponds to breaking the strings. In more than three dimensions, knots
can always be loosened (see chapter 2.1) and therefore a knotted network will not arise
which leads to no period of nearly stable vacuum energy. This can explain why there
are exact three spatial dimensions, since otherwise no inflation takes place.
This now produced constant energy density corresponds to the vacuum energy of
an inflationary era. Since string-breaking of the flux tubes is unlikely but still possible
due to quantum-tunneling-processes, the network will slowly decay. As long as the
26
4.4. Percolation Theory
network is large enough and covers the whole universe, the energy density decreases
slowly enough to ensure inflation. Only after the network decayed into too small parts
the inflation will end. This corresponds exactly to the phase transition in a “backward”
percolation process.
This part of the thesis is about simulating this decay of the network to find out
whether in the model of knotted inflation a phase transition takes place after an era of
inflation.
4.4 Percolation Theory
Percolation theory describes the formation of infinitely large clusters in a given topology,
where linkings between different points occur randomly. Let us assume an infinitely large
lattice where two lattice points, the sites, next to each other are either connected or
not, denoted by the probability p. Two or more connected sites built up a so called
cluster. It can be shown that a critical probability pc exists, so that an infinitely large
cluster exist for p > pc. For regularly shaped lattices pc can be determined analytically
but for arbitrary topologies simulations with a finite sized lattice have to be used. (see
[27] for a review).
To describe a knotted inflation we want to determine pc by starting with a percolated
network. This network is generated by the mechanism introduced above. The closed
flux tubes are knotted with other flux tubes, which is interpreted as a connection in
the percolation-picture. At the beginning every flux tube is considered to be highly
knotted and the whole network consists of just one big cluster. After that each flux
tube can decay with a probability of pdec leading to a steady decrease of connections.
If a critical number of connections is reached, we expect a phase transition at which
the infinitely large cluster decays, corresponding to an end of percolation due to an
undercritical connection probability. Since a simulation of an infinite network is not
possible we reduce to a finite number of knotted flux tubes and the phase transition
is then determined by a substantial decrease of the size of the biggest cluster in the
network.
The following section will introduce a simulation of such a decay process for two
different topologies.
4.5 Simulating Inflation via Knotted Networks
4.5.1 Basic concepts of the simulations
The simulation is coded as an object oriented C++-program. The closed flux tubes
introduced above are treated as an object of the custom class “tube”. To get access to
the flux tubes each object has a list of pointers to the tubes it is linked with. Therefore
it is easy to address the linked flux tubes if one tube decays.
The program consists of three different parts. The first part contains the creation
of the network in which all objects are initialized and the structure of the network
27
Chapter 4. Knotted Inflation
is determined by specifying the list of linked tubes. We will have a closer look at
two different structures, see sections 4.5.3 and 4.5.4. After the creation, the network
dynamics are simulated stepwise in a loop. In each loop repetition the second and
third part are done one after another. The second part simulates the decay of the
single flux tubes. Each tube is terminated with the probability pdec, which can be
adjusted arbitrarily. If a tube is terminated this object has to be deleted from the list
of linked tubes so that addressing tubes still works. After that decay, the third part
of the program calculates the number of clusters as well as the number of tubes in the
biggest cluster. This is done using the Hoshen-Kopelman-Algorithm, see 4.5.2. These
numbers are written into the output file for interpreting the results. Each step in the
loop corresponds to a time unit and defines the time scale. By adjusting the probability
pdec one can adjust the average number of decaying tubes and therefore obtain a higher
time resolution by decreasing pdec. To ensure a realistic behavior, the total number of
tubes has to be large enough.
4.5.2 Hoshen-Kopelman Algorithm
The Hoshen-Kopelman algorithm was developed by J. Hoshen and R. Kopelman in
1976 [28]. It is an algorithm to identify and label all clusters in a given structure. The
algorithm determines a label (integer numbers) for each site which states the cluster it
belongs to. Sites with the same label belong to the same cluster. To label the entire
structure the algorithm scans the structure site by site. If a site is not occupied the
algorithm skips to the next site. If it is occupied, the algorithm searches for neighbors
which were labeled before. If there are none, the site receives a new label, which has not
been used before. If there are labeled neighbors, the site receives the smallest label of
the neighbors. The higher label get a remark in an additional vector, that these labels
belong to the same cluster. The output of the algorithm is the labeled structure and
a vector, in which the cluster size for each label is denoted, making it a useful tool for
identifying clusters and cluster sizes.
4.5.3 Lattice Network
An easy approach to build a linked network is to place the tubes on a three dimensional
lattice. We assume each tube to be a ring that is initially connected to exactly four other
tubes, which are located above and beneath, right and left or behind and in front. For
a better understanding, Figure 4.1 shows the linkings of one single tube. This pattern
can be repeated all over the lattice. Because of the three possible orientations of the
tubes, a three dimensional lattice develops where not all lattice sites are occupied.
To apply the Hoshen Kopelman-Algorithm, we used the implementation by T. Fricke
[29] and expanded it to a three dimensional lattice. Periodic boundary conditions are
implemented for a better simulation of an infinite structure.
28
4.6. Results
Figure 4.1: Scheme of linkings in the cubed network
4.5.4 Random network
For a more realistic scenario we developed a randomly linked network, for which we
have to adjust the initialization process. We initialized one tube after the other, and
every new initialized tube is linked it to all the other previous produced tubes with a
probability plink. Since the randomly linked network does not rely on a specific structure
of the tubes at the time of formation, it is more realistic than the previous case. By
adjusting the linking probability plink, one adjusts the average number of tubes one
tube is linked to.
We applied the Hoshen-Kopelman-Algorithm to that randomly linked network by
simply scrolling through the whole list of tubes instead of using an underlying lattice
structure.
4.6 Results
The figures 4.2, 4.3, 4.4 and 4.5 show the results of the simulations described above.
While figure 4.5 refers to the lattice network the figures 4.2, 4.3 and 4.4 refer to the
random network with the different values of plink so that the average number of linkings
for each flux tube at the start of the simulation is determined to 5, 10 and 100. The
random networks consist of 10,000 flux tubes and the decay probability is set to pdecay =
1%. Since the simulation for the lattice network needs less computing time the total
number of tubes is set to 750,000 and the decay probability is set to pdecay = 0.01% to
achieve a higher time resolution.
The basis of every plot is the simulation of a decaying network which is repeated ten
times. After that the calculated values are averaged. This is done to avoid fluctuations
and to smooth the plots. The horizontal axis of each plot corresponds to the steps of the
decay process and therefore can be interpreted as a time scale. The vertical axis simply
denotes the number of the different derived quantities in the network in a logarithmic
scale.
The total number of remaining flux tubes is shown in turquoise dots. As expected,
29
Chapter 4. Knotted Inflation
Figure 4.2: Result of Simulation of a random network with a total number of 10,000 tubes
and an average of 5 linkings at beginning.
the total number decreases exponentially.
The dots in green correspond to the number of single unlinked tubes. The blue and
violet dots correspond to the number of the two smallest possible clusters consisting of
two and three flux tubes, respectively. The quantity of these small clusters starts at
zero for the highly linked random networks. This was expected, since a high linking
probability will not lead to separated flux tubes in the beginning. Only the random
network, which starts with an average number of linkings of 5 leads to a few of these
small clusters in the beginning of the simulation. Obviously these small clusters do
not exist in the lattice network since the initial setup only provides one big cluster. In
general the number of these small clusters ascend at the first time because more and
more small parts separate from the bigger clusters. This formation of small clusters
dominate over the decay of the already existing small clusters. The domination will not
hold any more if the bigger clusters become too small. Then the decay of small clusters
will be the main effect and the number of those small clusters will descend.
The yellow dots donate the total number of clusters. For the highly linked networks
and the lattice network this number obviously starts at 1 because only one big cluster
30
4.6. Results
Figure 4.3: Result of Simulation of a random network with a total number of 10,000 tubes
and an average of 10 linkings at beginning.
is produced initially.
For a larger number of steps, which corresponds to later times, the total number of
clusters, the number of single flux tubes and the total number of remaining flux tubes
will be the same. That corresponds to totally separated flux tubes without any linking.
The black dots show the average cluster size. For the highly linked random network
and the lattice network the average cluster size starts at the total number of flux tubes
because there is only one big cluster. When the first separated clusters arise, the average
cluster size decreases faster than before until it ends up at one single tube, where only
unlinked flux tubes exist in the network.
The most interesting value is shown in red and denotes the size of the biggest cluster.
This is the number of flux tubes which belong to the biggest cluster. At the first part
of the decay process, the size of the biggest cluster declines with the expected decay
rate just like the total number of remaining flux tubes. This era corresponds to the era
of inflation. The breakdown of the network can be clearly seen, when the size of the
biggest cluster decreases stronger than before. This corresponds to an end of inflation.
The duration of inflation can be adjusted by the two parameters plink and pdecay, while
31
Chapter 4. Knotted Inflation
Figure 4.4: Result of Simulation of a random network with a total number of 10,000 tubes
and an average of 100 linkings at beginning.
the decay probability pdecay only shifts the time scale. The linking probability is related
to the duration of inflation since a higher linking needs more flux tubes to decay so
that the entire network will collapse. For a comparison figure 4.6 shows the size of the
biggest cluster for the three different linking setups in the case of a random network.
By comparing the lattice network with the randomly knotted network one can see
the difference caused by the lower average number of linkings, which is determined
to four in the case of the lattice network. Since the number of linkings is small, one
would not expect all flux tubes to belong to the biggest cluster in case of a random
network. The special structure used in the lattice network automatically considers
only flux tubes belonging to the initial cluster. These are the only ones contributing to
inflation. Additionally the decay process of the biggest cluster looks the same no matter
of using a random or lattice network. Therefore a simulation of a network based on a
lattice like structure such as used here is recommended. The required computing time
is much lower and the simulation does not lead to extremely different results.
32
4.7. Interpretation
Figure 4.5: Result of simulation of a lattice network.
4.7 Interpretation
The simulations show that a phase transition takes place in a knotted network. There-
fore the model of a knotted inflation leads to the proposed era of inflation and as well
indicates an end of inflation. To obtain more concrete implications of such scenarios the
underlying physics has to be understood and modeled more thoroughly. The probability
of building those closed flux tubes is not determined by now as well as the probability
that these flux tubes are linked to each other. It seems obvious that it will be a highly
complex mechanism depending on properties such as the temperature, the density and
the pressure at the time of phase transition of the gauge theory. As well it is not clear at
the current point of research if the developing flux tubes repel each other. If this is not
the case no network will evolve. If this all applies, one also has to check whether such a
simulated phase transition can be calculated analytically and if cosmological properties
such as critical exponents can be determined.
While the idea of knotted inflation looks attractive and promising, much work re-
mains to be invested in order to develop a quantitative and testable model.
33
Chapter 4. Knotted Inflation
Figure 4.6: Comparison of random networks.
34
Chapter 5
Conclusion & Outlook
Until now topologically motivated models in the area of phenomenological high energy
physics did not attract much interest even if there are promising realizations of tightly
knotted and linked flux tubes in context of phenomenology, e.g. Glueballs [30].
This thesis analyzed two different ideas to apply string like objects which arise as
flux tubes after a phase transition in a Landau-Ginzburg model.
The first one was considered to give a hint of the origin of the neutrino masses and
the characteristic of large mixing angles in the lepton sector. We used a Seesaw Type I
mechanism and adopted values for the right handed neutrino mass matrix with values
determined by the knotlengths. Assuming a diagonal Dirac mass matrix, these entries
were fitted to reproduce the experimental values of mixing and mass squared differences
of the neutrinos. A significant difference between a knotted and a random spectrum
can not be seen. The analytical assumption of degeneracy in the Dirac mass matrix
needs to be an input into the numerical calculations otherwise a knotted model is not
favored. A search for a mechanism to create such degeneracy can be seen as a starting
point for further research where afterwards the numerical calculations as well can be
adjusted and a difference between the knotted and the random spectrum is expected.
The second model we looked at was a simulation of a knotted inflation. In this
model cosmological inflation is triggered by a network of highly linked flux tubes. These
linkings slowly decay caused by quantum fluctuations and tunneling processes. In the
simulations we found the expected phase transition at which the network collapses and
inflation will end. This corresponds to a backward percolation process. In general
percolation emerges in various processes in different areas of science including computer
science, biology or social science as well as in physics, e.g. solid state physics. It is a good
example for applying a theoretical idea to a wide field of interest. In the case of knotted
inflation the underlying mechanism of inflation has to be studied more thoroughly to
obtain quantitative results and predictions.
In general, non trivial topologies show a lot of interesting features and a closer look
at these properties might be motivated by unsolved problems in the context of high
energy physics.
35

Appendix A
Numeric calculations for
comparison with former analysis
The experimental values used in [17] for this comparison are given by [31]
θ12 = 34.4± 1.0
◦ (A.1)
θ23 = 42.8
+4.7
−2.9
◦ (A.2)
θ13 = 5.6
+3.0
−2.7
◦ (A.3)
δm221 = 7.59± 0.2 · 10
−5eV2 (A.4)
δm231 = −2.36± 0.11 · 10
−3eV2 (IH) (A.5)
δm231 = 2.46± 0.12 · 10
−3eV2 (NH) (A.6)
χ2 K1 K2 K3 L1 L2 L3 mD1 [GeV] m
D
2 [GeV] m
D
3 [GeV]
< 0.001 00 01 04 10 16 12 7.945 28.911 21.765
< 0.001 00 01 04 11 06 10 7.975 29.134 21.835
< 0.001 00 01 04 11 16 10 7.968 29.139 21.789
< 0.001 00 01 04 17 05 11 8.032 29.432 22.080
< 0.001 00 01 04 17 07 11 8.019 29.442 21.997
< 0.001 00 07 12 08 06 05 5.774 14.179 22.629
< 0.001 00 07 12 08 16 05 5.769 14.178 22.574
< 0.001 01 05 04 13 06 12 6.820 21.487 15.609
< 0.001 01 06 04 16 11 08 7.537 16.000 24.961
< 0.001 02 03 11 11 17 10 7.250 20.027 16.859
< 0.001 02 07 05 11 07 09 5.862 16.946 12.434
Table A.1: The 10 best fits for normal hierarchy
37
Appendix A. Numeric calculations for comparison with former analysis
χ2 K1 K2 K3 L1 L2 L3 mD1 [GeV] m
D
2 [GeV] m
D
3 [GeV]
< 0.001 00 09 02 07 17 03 72.58 -9.823 7.843
< 0.001 00 09 02 07 17 03 72.390 9.799 7.824
< 0.001 00 11 09 08 07 09 42.154 7.298 6.701
< 0.001 01 11 08 11 04 09 44.515 6.920 6.334
< 0.001 01 13 03 12 02 04 43.693 6.819 5.477
< 0.001 04 09 07 00 08 09 65.649 15.465 14.108
< 0.001 04 11 12 06 08 17 40.423 7.033 6.523
< 0.001 04 12 11 05 08 17 40.038 7.053 6.499
< 0.001 05 11 13 06 02 17 41.961 7.458 6.914
< 0.001 06 10 12 06 08 11 41.673 7.173 6.708
< 0.001 06 12 11 08 06 17 40.691 7.045 6.504
Table A.2: The 10 best fits for inverse hierarchy
Model tested parameters viable fits ratio viable fits χ2bestfit
NHknots 10,692,864 271,599 2.72% < 0.001
NHrandom 1,000,000 27,390 2.74% < 0.001
IHknots 10,692,864 253,956 2.38% < 0.001
IHrandom 1,000,000 9,208 0.92% < 0.001
Table A.3: Number of viable fits for the different models
38
Danksagung
Zu aller erst bedanke ich mich bei dem Betreuer meiner Masterarbeit, Prof. Heinrich
Päs für die vielen Ratschläge und immer neuen Ideen. Vor allem aber für die Motivation
in den Phasen, in denen sie mir fehlte. Vielen Dank auch für die Möglichkeit an der
DPG Frühjahrstagung und am DESY-Theorie Workshop teilnehmen zu können.
Ein ganz großes Dankeschön geht an den gesamten vierten Stock. Ihr habt mich
sofort herzlich aufgenommen, und ich habe mich von Anfang an wohl gefühlt. Während
der ganzen Masterarbeit bin ich gerne ins Büro gekommen, was vor allem an dem tollen
Arbeitsklima liegt. Ihr alle seid immer offen für Fragen, sei es fachlicher oder privater
Natur. Wir hatten viele erhellende Diskussionen, die mir immer Spaß gemacht haben.
Ich bin sicher, das wird auch weiterhin so sein und freue mich schon auf die Zeit während
meiner Promotion. Everyone is awesome!
Für das Korrekturlesen und die sehr hilfreichen Anmerkungen haben sich Erik und
Peter einen extra Dank erarbeitet. Vielen Dank!
Natürlich danke ich auch all meinen Freunden, die mir zu den passenden Zeiten die
nötige Ablenkung beschert haben und mir vor allem immer dann zur Seite standen,
wenn ich sie gebraucht habe. Gute Freunde erkennt man daran, dass sie nicht vor
drastischen Worten zurück schrecken, wenn sie gebraucht werden. Danke.
Zuletzt gebührt mein Dank meiner Familie. Die letzte Zeit war für uns alle sehr
turbulent und nicht immer schön. Ich danke euch für eure Unterstützung und dass ihr
mir immer den Rücken frei gehalten habt, auch wenn es für euch nicht immer einfach
war. Gerade im letzten Monat konnte ich nicht so für euch da sein, wie ich es mir
gewünscht hätte. Danke für euer Verständnis.
Mir hat das letzte Jahr sehr viel Spaß gemacht und es war sicherlich das schönste
Jahr meines Studiums. Ich freue mich auf die kommenden Herausforderungen während
meiner Promotion und bin sehr glücklich, dass ihr alle an meiner Seite steht.
39

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43

Eidesstattliche Versicherung
Ich versichere hiermit an Eides statt, dass ich die vorliegende Masterarbeit mit dem Ti-
tel ”Knots and Links in High Energy Physics” selbständig und ohne unzulässige fremde
Hilfe erbracht habe. Ich habe keine anderen als die angegebenen Quellen und Hilfsmit-
tel benutzt sowie wörtliche und sinngemäße Zitate kenntlich gemacht. Die Arbeit hat
in gleicher oder ähnlicher Form noch keiner Prüfungsbehörde vorgelegen.
Ort, Datum Unterschrift
Belehrung
Wer vorsätzlich gegen eine die Täuschung über Prüfungsleistungen betreffende Regelung
einer Hochschulprüfungsordnung verstößt handelt ordnungswidrig. Die Ordnungswidrigkeit
kann mit einer Geldbuße von bis zu 50.000,00e geahndet werden. Zuständige Verwal-
tungsbehörde für die Verfolgung und Ahndung von Ordnungswidrigkeiten ist der Kan-
zler/die Kanzlerin der Technischen Universität Dortmund. Im Falle eines mehrfachen
oder sonstigen schwerwiegenden Täuschungsversuches kann der Prüfling zudem exma-
trikuliert werden (§ 63 Abs. 5 Hochschulgesetz - HG - ).
Die Abgabe einer falschen Versicherung an Eides statt wird mit Freiheitsstrafe bis zu 3
Jahren oder mit Geldstrafe bestraft.
Die Technische Universität Dortmund wird ggf. elektronische Vergleichswerkzeuge (wie
z.B. die Software ”turnitin”) zur Überprüfung von Ordnungswidrigkeiten in Prüfungsver-
fahren nutzen.
Die oben stehende Belehrung habe ich zur Kenntnis genommen.
Ort, Datum Unterschrift