Master's Thesis in Theoretical Physics - Universiteit Utrecht

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small quantum fluctuation. We quantized the fluctuation field and showed that we couldagain write down the field equation for a harmonic oscillator. The frequency however wastime dependent because of the time dependent scale factor that appeared in the field equation.This makes it impossible to define a unique vacuum that remains the lowest energystate for all time. However, there are certain situations where we can still define a naturalvacuum state, which we discussed in section 5.2.1. In the so-called adiabatic regime thefrequency changes slowly and becomes approximately constant. This allowed us to definethe adiabatic vacuum, which was not the true lowest energy vacuum, but to a very goodapproximation it was the lowest energy state in the adiabatic regime. We saw that in quaside Sitter space the adiabatic regime was reached in the infinite asymptotic past, and infact the adiabatic vacuum became the true vacuum in the limit η → −∞. In this adiabaticregime particle production due to the expansion of the universe is minimal, and we coulddefine a natural vacuum state.In section 5.2.2 we showed that we can actually solve the field equations in quasi de Sitterspace exactly in terms of Hankel functions. An important requirement was the proportionalityof the classical background field to the Hubble parameter during inflation, which wehave both analytically and numerically verified in chapter 4. We could define a natural vacuumstate, the so-called Bunch-Davies vacuum, which corresponds to zero particles in theinfinite past. This allowed us to calculate the scalar propagator in section 5.2.3, which isthe probability for the particle to travel from one space time point to another. For reasons ofdimensional regularization and renormalization, which we will use in the coming chapter,we calculated the propagator in D dimensions and made an expansion around D − 4.In section 5.3 we generalized our results for the nonminimally coupled real scalar field tothe two-Higgs doublet model. The difference is that instead of one real scalar field thereare two complex scalar fields. In the quantization procedure for the fields this led to twochanges: each scalar field was expanded in independent sets of creation and annihilationoperators, and to incorporate the complexity of the fields we needed to introduce an additionalset of creation and annihilation operators for each scalar field. These new operatorscreated or annihilated antiparticles. We were able to write the expansion in one single matrixvalued equation, with the fields as a vector and the mode functions as matrices. Thisallowed us to write the field equation for the two-Higgs doublet model in precisely the sameform as the field equation for the real scalar field. The solutions were therefore also Hankelfunctions with an index ν which is a constant hermitean matrix. For the propagator,which is now a 2 × 2 matrix, in the Bunch-Davies vacuum we could simply copy the resultfrom the propagator for the real scalar field. The result in Eq. (5.130) is the main result ofthis chapter. In the next chapter we will consider the fermion sector of the Standard Modelaction and use this propagator to calculate the effect of the nonminimally coupled inflatonfield on the dynamics of massless fermions during inflation.
Chapter 6The fermion sectorIn the previous chapters we only considered the scalar section of the action. In chapter 4 wehave seen that the scalar particle can be the inflaton in Linde’s chaotic inflationary model[8], where the scalar particle has a large initial value and slowly rolls down the potentialhill. There are many scalar field models in which chaotic inflation is possible, and in thisthesis we considered specifically the inflaton field with strong negative coupling to the Ricciscalar. Benefits of this model are the fact that inflation is successful for relatively smallinitial values of the scalar field and the quartic self coupling does not have to be very smallin order to agree with the observed spectrum of density fluctuations from the CMBR. Thelatter allows the Higgs boson to be the inflaton as proposed by Bezrukov and Shaposhnikov[4]. The beautiful thing is that we do not need to introduce new scalar particles into theStandard Model, but we can simply use the existing particles to explain inflation by onlyintroducing a nonminimal coupling to gravity.Of course, the Standard Model of particles does not contain only the Higgs boson. It alsocontains the gauge bosons (the photon, W ± and Z-bosons, the gluons) and the fermions(quarks, electrons, neutrinos). The masses of these particles are generated through theHiggs mechanism. The Higgs boson couples to these particles and because it has a nonzeroexpectation value at low energy, it effectively generates a mass for the gauge bosons andfermions. The coupling of the gauge bosons to the Higgs boson appears naturally in theStandard Model. By demanding U(1), SU(2) and SU(3) symmetry of the action, one needsto introduce covariant derivatives that contain the gauge bosons. We will not consider thegauge bosons in this thesis, but focus on the fermions. For the fermions the coupling to theHiggs boson does not appear naturally and has to be constructed. This is the Yukawa sectorof the action, and it contains the scalar-fermion interactions.In this chapter we will construct the fermion sector of the Standard Model action, includingthe coupling of the fermions to the inflaton, i.e. the Higgs boson. For good literature onquantum field theory in general see [35]. We will consider the simplest supersymmetricextension of the Standard Model, the two-Higgs doublet model. The reason why we usethis model is that it provides a mechanism for baryogenesis, see for example [36][37][29].First we will first construct the fermion action in Minkowski space time in section 6.1.Then in section 6.2 we will generalize this to a curved space time. We will see that in aconformal FLRW universe we can write our action again as the Minkowski fermion actionby performing a conformal rescaling of the fields. The only difference is that the mass of thefermions is multiplied by the scale factor. In section 6.2.2 we construct the field equationfor the fermions (the Dirac equation) and we try to solve this equation classically. We willsee that in case of the two-Higgs doublet model the Dirac equation is modified slightly,which could lead to an axial vector current, baryon number violation and baryogenesis.71
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INFLATION AND BARYOGENESIS THROUGH
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AbstractIn this thesis we consider
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6 The fermion sector 716.1 Fermion
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and particle physics. The Einstein
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2.2 An expanding universeWe are now
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Here, H 0 = H(t = 0) and we have ch
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Since the spatial background is hom
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into neutral hydrogen at a temperat
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horizon during inflation isl phys (
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VΦSlowrollregime0 MpΦFigure 3.1:
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wavelengths that are larger than th
Page 26 and 27:
In the Standard Model, the only sca
Page 28 and 29: esearch of density perturbations in
Page 30 and 31: Finally, we can also vary the actio
Page 32 and 33: such that we get for Eqs. (4.18)Fro
Page 34 and 35: Φt252015105N8060402000 200 400 600
Page 36 and 37: 4.2 The conformal transformationThe
Page 38 and 39: Eq. (4.44) is invariant under a con
Page 40 and 41: Note that we have succeeded in remo
Page 42 and 43: Figure 4.6: WMAP measurements of th
Page 44 and 45: Note that the matrices Ω and ˜Ξ
Page 46 and 47: and we can writeξ ++ =ξ −− =
Page 48 and 49: value of ξ. With the explicit expr
Page 50 and 51: 0.600.800.550.75Φ1t0.50Φ2t0.700.4
Page 53 and 54: Chapter 5Quantum field theory in an
Page 55 and 56: with equation of motion0 = ¨φk +
Page 57 and 58: Eq. (5.18). If we now act with the
Page 59 and 60: (5.28) with α = 1 and β = 0, and
Page 61 and 62: 5.2 Quantization of the nonminimall
Page 63 and 64: If we now substitute the expansion
Page 65 and 66: the adiabatic regime, whereas the t
Page 67 and 68: the lowest energy state at a later
Page 69 and 70: wherey ≡ y ++ (x, x ′ ) = −(|
Page 71 and 72: Now we plug all these expansions in
Page 73 and 74: of the field and expand the action
Page 75 and 76: whereΦ k =( ) ( ) ( ) ( )φk,aa φ
Page 77: This again exactly the same express
Page 81 and 82: In this representation the projecti
Page 83 and 84: These relations allow us to rewrite
Page 85 and 86: 6.2.1 Fermion propagator in curved
Page 88 and 89: We now again rescale the field χL/
Page 90 and 91: φψψFigure 6.1: Feynman diagram f
Page 92 and 93: Note that the fact that the scalar
Page 94 and 95: The next step is to get rid of the
Page 96 and 97: such that we can findχ(x ′ ) = e
Page 98 and 99: where in all the equations z = k∆
Page 100 and 101: We can now compare this equation to
Page 103 and 104: Chapter 7Summary and outlookCosmolo
Page 105 and 106: esulting action as the action of a
Page 107: In future work we hope to study som
Page 111: Appendix AConventions• Throughout
Page 114 and 115: Finally, we use the Bianchi identit
Page 116 and 117: Note that we could also have calcul
Page 118: [24] A. O. Barvinsky, A. Y. Kamensh
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Master's Thesis in Theoretical Physics - Universiteit Utrecht

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