 Solution

ID: past-exam-of-the-advanced-quantum-field-theory-for-particle-physics-course-of-the-university-of-oxford/2020/1/iv/solution

With

q = p^{'} - p

and the routing just described, multiplication of the three QED Feynman rules for the Feynman vertices, the two Dirac propagators and the photon propagator gives

(- i e_{D})^{3} i^{2} (- i) = - e_{D}^{3}

. Hence the unrenormalized amputated Feynman vertex through this order is

i V^{μ} (p, p^{'}) = - i e_{D} γ^{μ} - e_{D}^{3} \int \frac{d ^{D} k}{( 2 π ) ^{D}} \frac{γ ^{ν} ( \neq p ^{'} - \neq k + m ) γ ^{μ} ( \neq p - \neq k + m ) γ _{ν}}{( k ^{2} + i 0 ) [( p ^{'} - k ) ^{2} - m ^{2} + i 0 ] [( p - k ) ^{2} - m ^{2} + i 0 ]} + O (e_{D}^{5}) .

For the renormalized Feynman vertex, add

- i e_{D} δ_{1} γ^{μ}

to this expression. Contributions from replacing parameters inside the one-loop term by their bare counterparts enter at higher loop order.

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