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ID: past-exam-of-the-quantum-field-theory-course-of-the-university-of-oxford/2016/4/a/solution

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Past exam of the quantum field theory course of the University of Oxford / 2016 / 4 / a / Solution by

Codex  0 2026-10-25

For incoming four-momenta, the momentum-space Feynman rules in the chosen Minkowski metric convention are:

Each internal line contributes the scalar propagator $i / (p^{2} - m^{2} + i 0)$ .
Each quartic interaction vertex contributes $- i λ_{0}$ and $(2 π)^{D} δ^{D} (\sum p)$ for momentum conservation.
Each independent loop momentum is integrated with $d^{D} k / (2 π)^{D}$ ; after eliminating redundant Dirac delta functions, one overall conservation factor remains.
Each graph is divided by its symmetry factor $S$ , with external labels held fixed.

There is one connected one-loop one-particle irreducible two-point graph: a single quartic vertex with two external lines and one tadpole diagram. Its symmetry factor is

S = 2

from interchange of the two loop half-edges.

There are three connected one-loop one-particle irreducible four-point graphs: two quartic vertices joined by two parallel internal lines, with external partitions

(12∣34)

(13∣24)

and

(14∣23)

. These are the

s

t

and

u

scattering channels; each bubble diagram has symmetry factor

S = 2

. External-branch insertions are excluded by the one-particle irreducible requirement.

Figure 1.
One-loop one-particle-irreducible quartic scalar graphs: the two-point tadpole and the three labelled four-point bubble channels, each with symmetry factor S=2
. **Details**.

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