 Solution

ID: past-exam-of-the-quantum-field-theory-course-of-the-university-of-oxford/2021/3/c/i/solution

With a single quartic interaction vertex, the two unused line ends close into one tadpole diagram. With a single sextic interaction vertex, the four unused ends pair into two tadpole diagrams. These are the only one-vertex one-particle irreducible Feynman diagrams for the two-point function.

D (k) = \frac{i}{k ^{2} - m ^{2} + i 0}, T_{M} = \int \frac{d ^{3} k}{( 2 π ) ^{3}} D (k) .

The amputated two-point function contributions are

C_{4} (- i g_{1}) T_{M}, C_{6} (- i g_{2}) T_{M}^{2},

where

C_{4}, C_{6}

are the reciprocals of the corresponding Feynman-diagram symmetry factors. Both are independent of external momentum, and hence affect the mass counterterm rather than wave-function renormalization. No numerical Feynman-diagram symmetry factors are needed here.

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