Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 304 / 3 / c / ii / Solution

The one-loop four-scalar one-particle-irreducible diagrams, together with permutations of the external scalar labels, are:

a scalar bubble with two scalar-quartic vertices;
a gauge-scalar box with four $A ϕϕ$ vertices;
a triangle with two $A ϕϕ$ vertices and one $AA ϕϕ$ seagull vertex;
a gauge bubble with two $AA ϕϕ$ seagull vertices;
when the mixed operators $I_{A}$ are present, a fermion bubble with two $ϕ^{2} \overset{ˉ}{ψ} ψ$ vertices.

The first four are forced by the scalar covariant derivative and scalar potential. There is no ordinary Yukawa fermion box because the imposed

ϕ \mapsto - ϕ

symmetry forbids a one-scalar fermion vertex.

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