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 Incoming links: Past exam of the quantum field theory course of the University of Oxford / 2025 / 2

Past exam of the quantum field theory course of the University of Oxford / 2025 / 2 / c / ii / Solution 2026-10-25

Keep the particle labels specified under 2. There are two connected tree-level Feynman diagrams: the quartic contact interaction vertex and an

s

-channel light antiparticle exchange. The initial Noether charge is

1 - 2 = - 1

, exactly the charge of

\overset{ˉ}{ϕ}

. At the first cubic Feynman vertex, the incoming

ϕ

and

Φ

leave the second

ϕ

leg internal; the conjugate Feynman vertex supplies both outgoing particles.

Figure 1.
Scattering of a light particle and a heavy particle, with contact and s-channel light-antiparticle exchange diagrams
. Details.

Thus

i M = - i g + (- ih) \frac{i}{s - m ^{2} + i 0} (- ih), M (ϕ Φ \to ϕ Φ) = - g - \frac{h ^{2}}{s - m ^{2} + i 0} .

Here

s = (p_{1} + p_{2})^{2}

for incoming

ϕ (p_{1}), Φ (p_{2})

. Both terms have the same sign in the chosen S-matrix convention.

The distinction from antiparticles matters. If the paper's heavy “

Φ

particle” is instead intended to mean the charge-

+ 2

state

\overset{ˉ}{Φ}

, so that its printed decay into

ϕϕ

is allowed, the exchange diagram is crossed. For that alternative physical process,

M (ϕ \overset{ˉ}{Φ} \to ϕ \overset{ˉ}{Φ}) = - g - \frac{h ^{2}}{u - m ^{2} + i 0}, u = (p_{1} - p_{4})^{2} .

This is a consistent alternative naming convention only if it is also applied to the decay in iii; one cannot use the same heavy particle label for both charge assignments.

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