Master's Thesis in Theoretical Physics - Universiteit Utrecht

The next step is to get rid of the ∆x 2 terms in the denominators. Therefore we use thefollowing relations valid in 4 dimensionswhich we can combine to get( ln(a∆x 2 )) + 1∂ µ∆x 2∂ 2 ln 2 (µ 2 ∆x 2 ) =∂ 2 ln(µ 2 ∆x 2 ) == ln(a∆x 2 ) ∆x µ[∆x 2 ] 28∆x 2 + ln(µ2 ∆x 2 )∆x 24∆x 2 , (6.76)ln(µ 2 ∆x 2 )[∆x 2 = 1 [ ] 2 3 ∂2 ln 2 (µ 2 ∆x 2 ) − 2ln(µ 2 ∆x 2 ) ] . (6.77)Furthermore we rewrite the Ψ 0 term in the following wayΨ 0 = ln(yK) , K = 1 4 expψ(3 2 + ν) + ψ(3 2 − ν) + 2γ E − 1 (6.78)where we use the expression for y from Eq. (6.63). Then we split up the term in Eq. (6.78)and writeln(yK) = ln(aa ′ ) + ln(K HH ′ (1 − ɛ) 2 ∆x 2 ).Now we use the relations (6.76) to find thatln(yK) ∆x µ[∆x 2 ] 2 = − 1 2 4 {∂µ ∂ 2 ln 2 (K HH ′ (1 − ɛ) 2 ∆x 2 ) + 2ln(aa ′ )∂ µ ∂ 2 ln(∆x 2 ) } . (6.79)By using Eqs. (6.77) and (6.79) and the definition γ µ ∂ µ ≡ ✁∂ we find the final renormalizedform of the self-energyΣ ren (x, x ′ ) = −|f | 2 (aa′ ) 3 22 1 0π 4 ✁ ∂∂ 4 [ ln 2 (µ 2 ∆x 2 ) − 2ln(µ 2 ∆x 2 ) ]−|f | 2 (aa′ ) 3 22 5 π 2 ln(aa′ )i✁∂δ(x − x ′ )−|f | 2 (1 − ɛ)2 HH ′ (aa ′ ) 5 22 9 π 4 (ν 2 − 1 46.3.2 Influence on fermion dynamics)× { ✁∂∂ 2 ln 2 (K HH ′ (1 − ɛ) 2 ∆x 2 ) + 2ln(aa ′ )✁∂∂ 2 ln(∆x 2 ) } . (6.80)The one-loop self-energy will have an effect on the dynamics of fermions. To be precise, theDirac equation will be modified in the following waya 3 2 i ✁∂a 3 2 ψ(x) −∫d 4 x ′ Σ ret (x; x ′ )ψ(x ′ ) = 0, (6.81)where Σ ret (x; x ′ ) is the retarded self-energy,Σ ret (x; x ′ ) = Σ ++ + Σ +− . (6.82)Note that we have calculated Σ ++ ≡ Σ ren (x + ; x ′ + ) in the previous section since we have beenworking with ∆x 2 ≡ ∆x 2 ++ all the time. We now also want to find the self-energy Σ+− ≡