Master's Thesis in Theoretical Physics - Universiteit Utrecht

φψψFigure 6.1: Feynman diagram for the one-loop fermion self-energy with a scalar field φ in the loop. Thefermion field is denoted by ψ. The scalar field φ is in this case the Higgs boson in quasi de Sitter space. In caseof the two-Higgs doublet model one of the two Higgs bosons only couples to up-type fermions, whereas the otheronly couples to down-type fermions.6.1 we show the one-loop diagram that we want to calculate. The fermion propagates, thensplits into a scalar (the Higgs boson) and a fermion and recombines again into a fermion.These loop effects can be reexpressed as an energy or mass term in the fermion action. Thisis what we call the fermion self-energy, and in this chapter we specifically calculate theone-loop self-energy. The one-loop self-energy and its effect on the dynamics of fermionswas already calculated by Garbrecht and Prokopec in [39]. We will redo their calculationhere, with the difference that we will work now in quasi de Sitter space, instead of exact deSitter.In order to calculate the one-loop self-energy, we calculate the diagram in Fig. 6.1 bymultiplying the scalar-fermion vertices with the internal scalar and fermion propagator.For the scalar propagator we use the Feynman propagator for the two-Higgs doublet modeldefined in Eq. (5.129). We found the explicit expression for this propagator in quasi deSitter space in Eq. (5.130), which we list here again as a matter of convenience,i∆(x, x ′ ) = [(1 − ɛ)2 HH ′ ] D 2 −1Γ( D ( y(4π) D 2 2 − 1) 4) 1−D2+ (1 − ɛ)2 HH ′ ( ) 1(4π) 2 Ψ 04 − ν2 + O( s + ɛ). (6.60)3For the fermion propagator we should use the massive fermion propagator derived in [38].However, for matters of simplicity we use the massless fermion propagator from Eq. (6.44),which isiS F (x, x ′ ) =1(aa ′ ) D−12iγ c ∂ c(Γ(D2 − 1)4π D 2)1[∆x 2 (x, x ′ )] D 2 −1In principle the only fermions we can describe with this massless propagator are neutrinos.Of course we also want to describe one-loop effects for the quark propagator, which aremuch larger because the Yukawa couplings are much larger (remember that the mass of thefermions is generated through the Yukawa interactions, so the largest mass has the largestYukawa coupling). Therefore by taking the massless fermion propagator, we actually makean incorrect assumption. We will leave the calculation of the one-loop self-energy for themassive fermion propagator to future work. For now we will calculate the fermion one-loopself-energy for the simple case of a massless fermion propagator, and we keep in mind thatin fact we only describe neutrinos with this calculation.We continue by writing down the the Feynman rules for the vertices from the action (6.36).The results are shown in Table 6.1.As we will see, if we simply multiply these vertices and propagators we will find thatthe self-energy is divergent for D = 4. In fact we can isolate the singularity to a local termδ(x − x ′ ). Looking at the diagram in Fig. 6.1 we see that the diagram is divergent if thepoint x where the fermion splits into a scalar and a fermion is equal to the point x ′ where.