Master's Thesis in Theoretical Physics - Universiteit Utrecht

These relations allow us to rewrite the action in terms of the general fermion fields as∫ iS = d x{ D 2 [ψγa ∂ a ψ − (∂ a ψ)γ a ψ]−f u φ 1 u 1 2 (1 + γ5 )u − fu ∗ φ∗ 1 u 1 2 (1 − γ5 )u−f d φ ∗ 2 d 1 2 (1 + γ5 )d − f ∗ d φ 2d 1 }2 (1 − γ5 )d . (6.25)This action will become useful when we calculate the one-loop fermion self-energy and weneed the Feynman rules for the scalar-fermion vertices in section 6.3.6.2 Fermion action in curved space timesThe action (6.23) describes the fermions in a flat Minkowski space time, but in the end wewant to describe fermions in a flat FLRW space time. Thus we need to rewrite the fermionaction to a curved space time. Curved space time can be described by the metric g µν , thatwe can transform to a local Minkowski frame by the following transformationg µν (x) = e a µ (x)eb ν (x)η ab, (6.26)where e a µ is the so-called vierbein field. In curved space times the action (6.23) takes theform∫S = d D x i−g{2 [ψ L γµ ∇ µ ψ L − (∇ µ ψ L )γ µ ψ L ] + i 2 [ψ R γµ ∇ µ ψ R − (∇ µ ψ R )γ µ ψ R ]−f u u L φ 1 u R − f d d L φ ∗ 2 d R − f ∗ u u Rφ ∗ 1 u L − f ∗ d d Rφ 2 d L}, (6.27)where we have added the invariant measure −g and replaced the ordinary derivative bythe covariant derivative for fermionswhere the spin connection Γ µ is defined by∂ a → e µ a∇ µ = e µ a(∂ µ − Γ µ ), (6.28)Γ µ = − 1 [8 eν b (∂ µe νc − Γ ρ µνe ρc ) γ c ,γ d] (6.29)with Γ ρ µν the Christoffel connection. Actually the covariant derivative for a particle withspin in Eq. (6.28) should be the familiar covariant derivative for spin-0 particles plus thespin connection Γ µ , so ∇ µ = ∇ 0 µ −Γ µ with ∇ 0 µ the familiar covariant derivative defined in Eq.(B.2). However, since the fermion field is not a vector field (it does not have an index) andbecause in the equation of motion for Ψ there is only one derivative, the familiar covariantderivative reduces to the partial derivative. The covariant derivative is derived by demandingthat ∇ µ γ ν = 0. Notice also that for spinless particles the spin connection vanishes, since[γ c ,γ d] = 0.We finally that we can again write the action in the form of Eq. (6.25), i.e. in terms ofthe fermion fields without making the distintion between left- and righthanded fields. Thescalar-fermion interaction terms will now have an additional scale factor −g = a D , suchthat the Feynman rules for the vertices will include this scale factor. We will come back tothis when calculating the one-loop fermion self-energy in section 6.3.Let us continue with the action (6.27). The Dirac matrices γ µ are defined byγ µ = e µ aγ a (6.30)