Thermal particle production in the early universe [Elektronische Ressource] / submitted by: Denis Besak

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• Physik

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• Publié par: universitat_bielefeld
• Publié le: 01 janvier 2010
• Nombre de lectures: 30
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Thermal particle production in the
early Universe
Dissertation
Submitted by:
Denis Besak
Fakulta¨t fu¨r Physik,
Universit¨at Bielefeld
June 2010
Referees: Prof.Dr.Dietrich B¨odeker
Prof.Dr.Mikko LaineContents
Published work from thesis 5
1. Introduction 6
2. Quantum field theory in a hot thermal bath 9
2.1. Perturbation theory at finite temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1. A short review of the imaginary-time formalism . . . . . . . . . . . . . . . . . . . . . . 9
2.1.2. Scales and effective theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2. Hard Thermal Loops (HTL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3. Perturbation theory close to the lightcone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.1. Thermal width and asymptotic mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.2. A new class of diagrams: Collinear Thermal Loops (CTL) . . . . . . . . . . . . . . . . 20
2.3.3. A general power-counting for CTLs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.4. The CTL self-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3. Thermal particle production and the LPM effect 26
3.1. Thermal particle production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.1. Particle production rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.2. Particle abundances and Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . 31
3.2. The LPM effect and its role in thermal particle production . . . . . . . . . . . . . . . . . . . 32
3.3. An integral equation for the LPM effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.1. The basic strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.2. The two-point functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.3. The recursion relation for amplitudes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.4. Integral equation for the CTL self-energy . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4. Photon production from a quark-gluon-plasma . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4. Thermal production of Majorana neutrinos 45
4.1. The origin of matter in the Universe: Baryogenesis . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2. Production rate and leading order contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3. Decay and recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3.1. Tree-level contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3.2. Multiple rescattering and LPM effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.4. 2↔2 scattering contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.4.1. Processes involving quarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4.2. Processes involving gauge bosons: hard contribution . . . . . . . . . . . . . . . . . . . 53
4.4.3. Processes involving gauge bosons: soft contribution . . . . . . . . . . . . . . . . . . . . 56
4.4.4. Computation ofA ,A andB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59hard soft
4.5. Collision term and yield of Majorana neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.5.1. The leading-order collision term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.5.2. Solution of the Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.5.3. RG running of coupling constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.6. Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.6.1. Approximate solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.6.2. The differential production rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.6.3. The Boltzmann collision term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.6.4. The yield of Majorana neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3Contents
5. Summary and Outlook 72
5.1. Summary – what has been done already . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2. Outlook – what can be done next . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
A. Notation and conventions 75
B. Finite-temperature propagators 76
B.1. Scalar propagator and asymptotic thermal mass. . . . . . . . . . . . . . . . . . . . . . . . . . 76
B.2. Fermion propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
B.2.1. The resummed finite-temperature fermion propagator . . . . . . . . . . . . . . . . . . 78
B.2.2. Propagator for lightlike momenta, asymptotic thermal mass . . . . . . . . . . . . . . . 79
B.2.3. HTL fermion propagator and HTL mass . . . . . . . . . . . . . . . . . . . . . . . . . . 81
B.3. Gauge boson propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
B.3.1. HTL gauge boson propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
B.4. Proof of (3.56) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
C. Some details for the recursion relation 88
C.1. The vertex factors for external gauge bosons and fermion loop. . . . . . . . . . . . . . . . . . 88
C.2. No need to remove external fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
D. Remarks on the integral equation for the current 91
D.1. Connected and disconnected contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
D.2. Towards an easier integral equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
E. Solving the equation for the LPM effect numerically 95
E.1. Formulation in Fourier space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
E.2. Solution of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
F. Proof of relations for the production rate of Majorana neutrinos 100
Bibliography 103
Acknowledgements 107
4Published work from thesis
The new results contained in this thesis are also published in the following articles:
[1] D.Besak, D.Bo¨deker, ”Hard Thermal Loops with soft or collinear external momenta”
This article contains the derivation of the integral equation for the LPM effect in photon production. It serves
to introduce the new method that is used in this work. This paper thus contains the essence of sections 3.3
and 3.4 as well as the relevant appendices.
[2] A.Anisimov, D.Besak, D.B¨odeker, ”The complete leading order high-temperature production rate of Ma-
jorana neutrinos”
This paper essentially contains what is presented in chapter 4 of this thesis. It presents the new results on the
high-temperature particle production rate of Majorana neutrinos and compares them to the zero-temperature
results.
51. Introduction
Who is in that house? I opened the door to see.
Who is up the stairs? I’m walking up foolishly.
Katie Melua - The House
The very early universe is a system of extraordinary complexity and very rich phenomenology, making it an
idealplaygroundtotestourunderstandingofthefundamentallawsofnatureandourabilitytoobtainprecise
answers to all the questions that we can ask within the framework of theoretical physics. If we knew every-
thing that we need to know about the theories that govern the Universe in its present state and its history,
then we could, provided we could somehow get the correct initial conditions, in principle make a simulation
of everything that happened between the Big Bang and the present Universe–assuming sufficient machine
power or patience to wait for the answer. However, Nature is still successful in limiting our knowledge, while
our curiosity remains as unlimited as ever and we may hope that revealing all myths our Universe still has
kept will only be a question of time.
At present however, it is fair to say that everything which happened before the time of Big Bang Nucleosyn-
thesis (BBN) still has to be regarded as having speculative ingredients, with definite evidence still missing.
Yet, there is a wide consensus that we know at least how to describe the fundamental interactions (strong,
weak, electromagnetic and gravitational) and consequently we do have a theoretical framework to describe
the evolution of the Universe starting at a time sufficiently far away from the Planck scale such that the
lack of a consistent theory of quantum gravity is unproblematic and only a classical description in terms of
General Relativity is needed.
Based on the vast observational data that was accumulated in the past decades and on their interpretation
using our knowledge about the fundamental interactions, a ’mainstream’ picture about the evolution of the
Universehasemerged,sometimescalledthe’StandardModelofCosmology’. Within thisstandardparadigm,
it is assumed that shortly after the Big Bang there was a period of inflation which lead to an exponential
expansion of the Universe and left it in a state far from t