Master's Thesis in Theoretical Physics - Universiteit Utrecht
Master's Thesis in Theoretical Physics - Universiteit Utrecht
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 nonm<strong>in</strong>imally coupled <strong>in</strong>flaton field. This allowed us to calculatethe scalar propagator <strong>in</strong> D-dimensions. The propagator was seem<strong>in</strong>gly divergent <strong>in</strong>D = 4, but by mak<strong>in</strong>g an expansion around D = 4 we found that the divergency disappears.In section 5.3 we extended our f<strong>in</strong>d<strong>in</strong>gs and quantized the two-Higgs doublet model. Thecomplexity of the fields lead to slightly different quantization of the fields, but <strong>in</strong> the end wefound that we could write the field equations for the mode functions <strong>in</strong> precisely the sameform as for mode functions of the s<strong>in</strong>gle 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 conta<strong>in</strong><strong>in</strong>g only scalar fields, we now considered the fermion sector with theStandard Model fermions and the coupl<strong>in</strong>g of the Higgs bosons to the fermions <strong>in</strong> theYukawa sector. We first constructed the fermion sector <strong>in</strong> a flat M<strong>in</strong>kowski space <strong>in</strong> section6.1 and then extended this to an expand<strong>in</strong>g FLRW universe <strong>in</strong> section 6.2. We foundthat we could aga<strong>in</strong> write our action as the M<strong>in</strong>kowski action by perform<strong>in</strong>g a conformalrescal<strong>in</strong>g 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 rescal<strong>in</strong>g ofthe propagator <strong>in</strong> M<strong>in</strong>kowski space. The massive propagator requires a lot more work butcan be constructed by explicitly solv<strong>in</strong>g the Dirac equation for the fermion fields with a realmass.The mass of the fermions is <strong>in</strong> fact generated through the coupl<strong>in</strong>g of the Higgs boson to thefermions. For the s<strong>in</strong>gle 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 solv<strong>in</strong>g the Dirac equations for fermions witha complex mass. First we tried to solve these equations by decoupl<strong>in</strong>g the fermion fields <strong>in</strong>left- and righthanded fermions and rewrit<strong>in</strong>g the equations. A first study suggests that wemight be able to write the solutions <strong>in</strong> terms of the solutions for the real mass. Anotherapproach is to remove the phase θ from the complex scalar fields by a field redef<strong>in</strong>ition ofthe fermion fields. If the phase is time dependent, we f<strong>in</strong>d that we can make the mass term<strong>in</strong> the fermion sector real, at the expense of creat<strong>in</strong>g an extra term <strong>in</strong> the action that isproportional to the axial vector current. This axial vector current violates CP and could beconverted through sphaleron processes <strong>in</strong>to a baryon asymmetry. If the phase derivative isconstant, ˙θ = constant, we f<strong>in</strong>d that the only change <strong>in</strong> the fermion field solutions is a shift<strong>in</strong> the momentum. In calculat<strong>in</strong>g the massive propagator this leads to additional parts thatmight be CP violat<strong>in</strong>g.In section 6.3 we comb<strong>in</strong>ed 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 <strong>in</strong> quasi de Sitter space and the massless fermion propagator. The masslessfermion propagator strictly speak<strong>in</strong>g only describes neutr<strong>in</strong>os and is therefore an <strong>in</strong>correctway to calculate the effect of the scalar <strong>in</strong>flaton field on the dynamics of fermions dur<strong>in</strong>g<strong>in</strong>flation. However, we expect this to give an <strong>in</strong>dication 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 dur<strong>in</strong>g <strong>in</strong>flation.The f<strong>in</strong>al result is that the one-loop self-energy effectively generates a mass for the masslessfermions that is proportional to the Hubble parameter H.