Master's Thesis in Theoretical Physics - Universiteit Utrecht

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asymptotic past.In the Bunch-Davies vacuum we could actually solve the field equations exactly and makea natural choice to quantize the nonminimally coupled inflaton field. This allowed us to calculatethe scalar propagator in D-dimensions. The propagator was seemingly divergent inD = 4, but by making an expansion around D = 4 we found that the divergency disappears.In section 5.3 we extended our findings and quantized the two-Higgs doublet model. Thecomplexity of the fields lead to slightly different quantization of the fields, but in the end wefound that we could write the field equations for the mode functions in precisely the sameform as for mode functions of the single real scalar field. The difference is that the fieldequation was now a matrix equation with matrix valued operators. In the end we foundprecisely the same form of the propagator for the two-Higgs doublet model as for the usualHiggs model, with the difference that the propagator is 2 × 2 matrix.In chapter 6 we focused on a different part of the Standard Model action. Instead of theHiggs sector containing only scalar fields, we now considered the fermion sector with theStandard Model fermions and the coupling of the Higgs bosons to the fermions in theYukawa sector. We first constructed the fermion sector in a flat Minkowski space in section6.1 and then extended this to an expanding FLRW universe in section 6.2. We foundthat we could again write our action as the Minkowski action by performing a conformalrescaling of the fields, with the only difference that the mass term is now multiplied by thescale factor. The propagator for massless fermions is then simply a conformal rescaling ofthe propagator in Minkowski space. The massive propagator requires a lot more work butcan be constructed by explicitly solving the Dirac equation for the fermion fields with a realmass.The mass of the fermions is in fact generated through the coupling of the Higgs boson to thefermions. For the single Higgs model, the scalar Higgs boson is a real field and gives thereforea real mass to the fermions. In case of the two-Higgs doublet model the scalar fieldsare complex and so is the mass of the fermions. This makes it more difficult to solve theDirac equations. Recently we did a study on solving the Dirac equations for fermions witha complex mass. First we tried to solve these equations by decoupling the fermion fields inleft- and righthanded fermions and rewriting the equations. A first study suggests that wemight be able to write the solutions in terms of the solutions for the real mass. Anotherapproach is to remove the phase θ from the complex scalar fields by a field redefinition ofthe fermion fields. If the phase is time dependent, we find that we can make the mass termin the fermion sector real, at the expense of creating an extra term in the action that isproportional to the axial vector current. This axial vector current violates CP and could beconverted through sphaleron processes into a baryon asymmetry. If the phase derivative isconstant, ˙θ = constant, we find that the only change in the fermion field solutions is a shiftin the momentum. In calculating the massive propagator this leads to additional parts thatmight be CP violating.In section 6.3 we combined our knowledge from all the previous chapters and calculate theone-loop self-energy for the fermions. We use the scalar propagator for the two-Higgs doubletmodel in quasi de Sitter space and the massless fermion propagator. The masslessfermion propagator strictly speaking only describes neutrinos and is therefore an incorrectway to calculate the effect of the scalar inflaton field on the dynamics of fermions duringinflation. However, we expect this to give an indication of the one-loop self-energy for themassive fermion propagator. After a long calculation where we calculate and renormalizethe self-energy we study its effect on the dynamics of massless fermions during inflation.The final result is that the one-loop self-energy effectively generates a mass for the masslessfermions that is proportional to the Hubble parameter H.
In future work we hope to study some of the issues that we have not been able to investigatein this thesis. First of all we have numerically solved the field equations for real fields φ 1and φ 2 in the two-Higgs doublet model. However, there is a third degree of freedom, whichis the phase between the two fields. We would like to introduce also this phase in the fieldequations and see if the phase is changing in time. The time dependent phase leads to CPviolation.Moreover we showed that in the nonminimally coupled two-Higgs doublet model there isone linear combination of the two Higgses that is the actual inflaton, the other linear combinationis an oscillating mode that quickly decays. In this thesis the focus was mostly onthe inflaton mode. In future work we hope to study the dynamics and effects of the oscillatingfield. This field could decay into lighter particles and could lead to reheating of theuniverse. If the decay happens in a CP violating way, this could lead to baryogenesis.We have seen that CP violation is also present in the fermion sector of the Standard Model.We want to continue the study on solving the Dirac equations for fermions with a complexmass. A thorough investigation is needed to see if we can really write the solutions as thesame solutions for fermion fields with a real mass. Furthermore we want to do a properstudy on baryogenesis in the fermion sector of the two-Higgs doublet model. As mentioned,the time dependent phase in this model generates an axial vector current in the fermionaction. This axial vector can then be converted by sphalerons into a baryonic current thatcould generate the baryon asymmetry of the universe. We hope to more research on this infuture work.Finally we want to calculate the one-loop fermion self-energy for massive fermions. Themassive fermion propagator was calculated recently in [38] and we can use the result tocalculate the effect of the inflaton field on the dynamics of quarks in quasi de Sitter space.We hope that this future work will give us some interesting consequences for fermion dynamicsin the early universe.
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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
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In the Standard Model, the only sca
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esearch of density perturbations in
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Finally, we can also vary the actio
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such that we get for Eqs. (4.18)Fro
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Φt252015105N8060402000 200 400 600
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4.2 The conformal transformationThe
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Eq. (4.44) is invariant under a con
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Note that we have succeeded in remo
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Figure 4.6: WMAP measurements of th
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Note that the matrices Ω and ˜Ξ
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and we can writeξ ++ =ξ −− =
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value of ξ. With the explicit expr
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0.600.800.550.75Φ1t0.50Φ2t0.700.4
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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 and 78: This again exactly the same express
Page 79 and 80: Chapter 6The fermion sectorIn the p
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: esulting action as the action of a
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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