Master's Thesis in Theoretical Physics - Universiteit Utrecht

to obtain the solution for the Feynman propagatoriS F (x, x ′ ) ==1(aa ′ ) D−121(aa ′ ) D−12li ˜S F (x, x ′ )(Γ(Diγ c 2∂ − 1))1c . (6.44)4π D 2 [∆x 2 (x, x ′ )] D 2 −1This is simply a conformal rescaling of the massless fermion propagator in flat Minkowskispace.For the massive fermions the calculation for propagator is much more difficult. The propagatorfor massive fermions in expanding space times with constant ɛ = −Ḣ/H 2 was solved byKoksma and Prokopec in [38]. They explicitly solved the field equation for the conformallyrescaled fermion fields χ in quasi de Sitter space. The solutions were again Hankel functions,but compared to the scalar propagator the indices ν are complex. With these explicitsolutions the massive fermion propagator could be calculated from the primary definition(6.39). Considering the infrared behavior of the fermions, the propagator is now found tobe finite in the infrared, instead of divergent for the scalar case. Precisely the complexity ofthe index ν of the Hankel functions makes sure that the propagator is IR finite. Physically,the Pauli exclusion principle forbids an accumulation of fermions in the infrared.In [38] the propagator was solved for fermions with a real mass m. This happens for examplewhen the mass is generated by Standard Model Higgs particle. In case of a singleHiggs doublet, we can fix a gauge such that the scalar field is real. In that case the massof the fermions is m = f φ and is a real quantity. In our two-Higgs doublet model however,the scalar fields are complex and the masses of the fermions are therefore complex as well.This will change the field equations for the fermions and therefore also the massive fermionpropagator. In the next section we show that the complex fermion mass could generate anaxial vector current and could be a possible source for baryogenesis.6.2.2 Solving the chiral Dirac equationIn this thesis we will not redo the complete calculation done in [38] for a complex mass.Instead, we will outline the calculation and see where the differences pop up. Let us firstgive the field equations for the left- and righthanded conformally rescaled fermions fromthe action (6.38). I will call these field equations with an asymmetry in the mass term thechiral Dirac equations. For the conformally rescaled u quarks these equations areiγ c ∂ c u L − am u u R = 0iγ c ∂ c u R − am ∗ u u L = 0,where I have defined m u = f u φ 1 . For the field equations for the d quarks we would find thesame, but instead the mass would be m d = f ∗ d φ 2. For now I will simply disregard the factthat we have different fermions with different masses, but instead consider just a generalconformally rescaled fermion field χ with complex mass m. So the field equations areiγ c ∂ c χ L − amχ R = 0iγ c ∂ c χ R − am ∗ χ L = 0. (6.45)In Ref. [38] the conformally rescaled field χ(x) is quantized by expanding it in creation andannihilation operators as we did for the scalar field in chapter 5. These operators are ofdifferent helicity h, which is the spin in the direction of motion and can be either +1 or