Master's Thesis in Theoretical Physics - Universiteit Utrecht

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In section 6.2.1 we calculate the propagator for the fermions, which for massless fermionsturns out to be a simple conformal rescaling of the Minkowski propagator. In section 6.3we use our main result from the previous chapter, the scalar propagator for the two-Higgsdoublet model, and the massless fermion propagator to construct and renormalize the oneloopfermion self-energy. We will see that this one loop effect generates a mass for thefermions that is proportional to the Hubble parameter.6.1 Fermion action in Minkowski space timeIn a flat Minkowski space the general fermion action is∫ { i}S f = d D x2 [ ¯ψγa ∂ a ψ − (∂ a ¯ψ)γ a ψ] + L Y , (6.1)where ψ is the fermion field doublet and L Y is the Yukawa Lagrangian. For generalitywe have taken a D dimensional Minkowki space, but one should remember that for theStandard Model D = 4. The anti-fermion ¯ψ is defined as¯ψ = iψ † γ 0 , (6.2)making sure that the action Eq. (6.1) is Lorentz invariant. The gamma matrices γ a satisfy{γ a ,γ b} = −2η ab (a, b = 0,1,.., D − 1). (6.3)In the Weyl or chiral representation, that we will use throughout this thesis, the 4 × 4gamma matrices are in 4 dimensions given by( ) (γ 0 0 1= , γ i 0 σ i )=1 0−σ i , (6.4)0where σ i are the 2×2 Pauli matrices. In this chapter we are mostly interest in the influenceof the inflaton (the Higgs particle) on the dynamics of the fermions (quarks, electrons, neutrinos).Our goal is to calculate the one-loop fermion self-energy, with a scalar in the loop.The coupling of the scalar to the fermions is contained in the Yukawa sector, and thereforewe will give an explicit expression for this sector.First of all, in the Standard Model we deal with chiral fermions which break the chiral symmetry.For example, there is only a left handed neutrino, whereas for quarks and electronsthe symmetry is broken because of the mass of these particles. To be more precise, for thefermion fields we can project out two different chiralities,where the projection operators are defined asψ = (P L + P R )ψ = ψ L + ψ R (6.5)P L = 1 − γ52, P R = 1 + γ5. (6.6)2The matrix γ 5 is defined through the other Dirac matrices, and in 4 dimensions it isγ 5 = iγ 0 γ 1 γ 2 γ 3 . (6.7)In the chiral representation the explicit form in 4 dimensions is( )γ 5 −1 0= . (6.8)0 1
In this representation the projection operators in 4 dimensions are( ) ( )1 00 0P L = , P0 0L = . (6.9)0 1Note that in principle ψ L and ψ R are 4-spinors, but because of the γ 5 matrix it is easy tosee that two components of these 4-spinors are zero. In the chiral representation we canthus write the 4-spinor ψ as two 2-spinors,( )ψLψ = . (6.10)ψ RIt is also easy to see that ( γ 5) 2 = 1 which allows us to derive the following useful relations(P L ) 2 = P L(P R ) 2 = P RP R P L = P L P R = 0. (6.11)An important property of the matrix γ 5 is that it anticommutes with all the other Diracmatrices, i.e.{γ 5 ,γ µ} = 0. (6.12)This allows us to derive the following relation( 1 − γψP L = iψ † γ 0 5)2( 1 + γ= iψ † 5)2= i ( P R ψ ) † γ0γ 0= P R ψ, (6.13)and the same relation when P L and P R are interchanged. Using this relation we can expressthe fermion action in terms of left- and righthanded fermions. For a kinetic term such asψγ α ∂ α ψ this givesψγ α ∂ α ψ = ψ(P L + P R )γ α ∂ α (P L + P R )ψ= ψ(P L )γ α ∂ α (P R )ψ + ψ(P R )γ α ∂ α (P L )ψ= P R ψγ α ∂ α (P R ψ) + P L ψγ α ∂ α (P L ψ)= ψ R γ α ∂ α ψ R + ψ L γ α ∂ α ψ L . (6.14)The terms with P L γ α P L vanish because of the anticommutation relation in Eq. (6.12) andthe fact that P L P R = 0. Thus we see that for the kinetic terms there is no mixing of theright- and lefthanded fermions.Now let us have a look at the Yukawa sector. In a very simple theory of one real scalar fieldthat couples to the fermions we have a term −f φ ¯ψψ, where f is a coupling constant. Wecan therefore identify the time dependent mass of the fermions as m = f φ. Let us expressthis mass term again in terms of the right- and lefthanded fields, which gives−f φψψ = −f φψ(P L + P R )(P L + P R )ψ= −f φ ( ψ(P L )(P L )ψ + ψ(P R )(P R )ψ )()= −f φ P R ψ(P L ψ) + P L ψ(P R ψ)= −f φ ( ψ R ψ L + ψ L ψ R). (6.15)
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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
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 and 78: This again exactly the same express
Page 79: Chapter 6The fermion sectorIn the p
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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