Master's Thesis in Theoretical Physics - Universiteit Utrecht

Table 6.1: Feynman rules for the scalar-fermion interactionsVertexFeynman rulea D f u φ 1 u 1 2 (1 + γ5 )u −ia D f u12 (1 + γ5 )a D f ∗ u φ∗ 1 u 1 2 (1 − γ5 )u −ia D f ∗ u 1 2 (1 − γ5 )a D f d φ ∗ 2 d 1 2 (1 + γ5 )d −ia D f d12 (1 + γ5 )a D f ∗ d φ 1d 1 2 (1 − γ5 )d−ia D f ∗ d12 (1 − γ5 )the scalar and fermion recombine again in a fermion. This local divergency however isnon-physical and can be removed by adding a suitable counterterm to the one-loop fermionself-energy, see for example [35]. This counterterm, the so-called fermion field strengthrenormalization isiδZ 2 (aa ′ ) D−1 µ2 γkl i∂ µδ D (x − x ′ ). (6.61)In section 6.3.1 we will actually isolate the divergency in the diagram of Fig. 6.1 to a localterm and cancel this by a suitable choice of the parameter δZ 2 .Combining the expressions above we can calculate the one-loop fermion self-energy by simplymultiplying vertices and propagators and adding the renormalization counterterm. Letus for simplicity focus on the one-loop self-energy for the u propagator. This diagram onlyinvolves the scalar propagator with the field φ 1 , since only this field couples to up-typequarks. The one-loop fermion self-energy is then−i [Σ](x, x ′ ) = (−i f u a D 1 2 (1 + γ5 ))i [S](x, x ′ )(−i f ∗ u a′D 1 2 (1 − γ5 ))i∆(x, x ′ ) 11+iδZ 2 (aa ′ ) D−12 γµi j i∂ µδ D (x − x ′ ), (6.62)where we have chosen the part ∆(x, x ′ ) 11 of the 2 × 2-matrix propagator ∆(x, x ′ ) from Eq.(??), which is the propagator 〈φ 1 φ ast1〉. For the down-type quarks, we would find exactly thesame expression, but only the couplings f u are replaced by f d and we take the part ∆(x, x ′ ) 22of the scalar propagator. From now on we will simply omit the indices u/d in the Yukawacouplings and the indices 11 or 22 in the scalar propagator, and remember that we haveto pick a specific part if we look at up- or down-type quarks. If we now insert the explicitexpressions for the scalar and fermion propagators, and use thatwe find−i [Σ](x, x ′ ) = |f | 2 (aa′ ) 3 2y ≡ ∆x2ηη ′ (1 − ɛ)2 HH ′ aa ′ ∆x 2 , (6.63)8π DΓ( D 2 )Γ( D 2 − 1) iγc ∆x c[∆x 2 ] D−1+|f | 2 (1 − ɛ)2 HH ′ (aa ′ ) 5 232π 4 Ψ 0( 14 − ν2 ) iγ µ ∆x µ(∆x 2 ) 2+iδZ 2 (aa ′ ) D−12 γµi j i∂ µδ D (x − x ′ ). (6.64)