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ID: past-exam-of-the-quantum-field-theory-course-of-the-university-of-oxford/2017/2/c/solution

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Past exam of the quantum field theory course of the University of Oxford / 2017 / 2 / c / Solution by

Codex  0 2026-10-25

For the amputated one-particle-irreducible vertex, every internal bridge is forbidden. Define the Feynman integrals

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

I_{2} (P) = \int_{k} D (k) D (k + P), I_{3} (P, Q, R) = \int_{k} D (k) D (k + P) D (k - R), P + Q + R = 0.

For

E = 3, L = 1

, the counting identity in part b gives

V_{3} + 2 V_{4} = 3

. The only one-particle-irreducible Feynman diagrams are a triangle of three cubic vertices and a two-propagator bubble joining one cubic and one quartic vertex. Write

V_{3}

for the amputated Feynman vertex, including its overall factor of

i

. With all external momenta incoming,

V_{3, 1 -loop} (p_{1}, p_{2}, p_{3}) = v_{3}^{3} I_{3} (p_{1}, p_{2}, p_{3}) + \frac{v _{3} v _{4}}{2} i = 1 \sum 3 I_{2} (p_{i}) .

The triangle has

S = 1

for labelled external legs. The mixed bubble has

S = 2

because its two internal lines may be interchanged. There are three distinct terms, according to which external leg is attached to the cubic vertex; the other two attach to the quartic vertex. No tadpole attached by a bridge is one-particle irreducible.

The requested position-space three-point correlation function is obtained by restoring the external propagators and a Fourier transform:

G_{1PI, 1 -loop}^{(3)} (x, y, z) = \int_{p_{1}} \int_{p_{2}} \int_{p_{3}} (2 π)^{d} δ^{d} (p_{1} + p_{2} + p_{3}) e^{- i (p_{1} x + p_{2} y + p_{3} z)} D (p_{1}) D (p_{2}) D (p_{3}) V_{3, 1 -loop} .

The tree-level contribution would replace the last factor by

v_{3}

. This explicitly distinguishes the amputated vertex from the correlation function while giving all loop integrations in momentum space.

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