 Solution

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

Let

D_{x y} = ⟨ 0∣ Tϕ (x) ϕ (y) ∣0 ⟩_{0}

be the free Feynman propagator, and abbreviate

v_{3} = - iκ

v_{4} = - iλ

. Expand the normalized Dyson series, so vacuum bubbles cancel between numerator and denominator, and retain the connected correlation function. Wick's theorem yields

G_{c} (x, y) = D_{x y} + \frac{v _{4}}{2} \int d^{d} z D_{x z} D_{zz} D_{zy} + \frac{v _{3}^{2}}{2} \int d^{d} z d^{d} w D_{x z} D_{z w}^{2} D_{w y} + \frac{v _{3}^{2}}{2} \int d^{d} z d^{d} w D_{x z} D_{zy} D_{z w} D_{ww} + O (two loops) .

These are respectively the free line, the quartic tadpole diagram, the two-cubic-vertex bubble diagram, and an tadpole contribution to a cubic-theory two-point function, attached by a bridge to the vertex carrying both external legs. The last graph is connected but is a one-particle-reducible Feynman diagram; removing its bridge disconnects it. It must be included when all connected graphs are requested. It disappears if an additional vanishing-one-point-function renormalization condition and its corresponding tadpole subtraction are imposed; that convention was not specified here.

The factors can be counted directly. The quartic term has

4 \cdot 3 = 12

attachments for the two labelled external legs, divided by

4! = 24

. At second order in the cubic interaction the expansion denominator is

2! (3!)^{2} = 72

. There are 36 Wick contractions for the bubble, and 36 for the attached-tadpole graph, giving

1/2

for each. Thus each loop topology shown has symmetry factor

S = 2

. Disconnected products of one-point functions and vacuum bubbles are not part of

G_{c}

E

external legs,

V_{3}

cubic vertices,

V_{4}

quartic vertices and loop number

L

satisfies

V_{3} + 2 V_{4} = E + 2 L - 2.

For

E = 2, L = 1

, the only possibilities are

(V_{3}, V_{4}) = (2, 0)

(0, 1)

, exactly those just listed.

 New to topics? Read the docs here!