 Solution

ID: past-exam-of-the-quantum-field-theory-course-of-the-university-of-oxford/2023/1/b/solution

The one-loop one-particle irreducible correction is a tadpole diagram: two external lines meet one quartic interaction and the other two legs are joined. Interchanging the two ends of its loop gives the symmetry factor

2

Define the self-energy by an insertion

- i Σ (p^{2})

i / [p^{2} - m^{2} - Σ (p^{2}) + i ϵ]

T_{Λ} := \int_{∣ k_{E} ∣ < Λ} \frac{d ^{2} k _{E}}{( 2 π ) ^{2}} \frac{1}{k _{E}^{2} + m ^{2}} = \frac{1}{4 π} lo g (1 + \frac{Λ ^{2}}{m ^{2}}), - i Σ_{1} = \frac{- i g _{4}}{2} \int \frac{d ^{2} k}{( 2 π ) ^{2}} \frac{i}{k ^{2} - m ^{2} + i ϵ} .

Thus

Σ_{1} (p^{2}) = \frac{g _{4}}{2} T_{Λ} = \frac{g _{4}}{8 π} lo g (1 + \frac{Λ ^{2}}{m ^{2}}) .

It has no external momentum dependence. The correction to the unamputated two-point correlation function in momentum space is

Δ G (p) = \frac{i}{p ^{2} - m ^{2} + i ϵ} [- i Σ_{1}] \frac{i}{p ^{2} - m ^{2} + i ϵ} = \frac{i Σ _{1}}{( p ^{2} - m ^{2} + i ϵ ) ^{2}} .

The positive sign of

Σ_{1}

raises the unrenormalized squared pole mass for

g_{4} > 0

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