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 $A\phi\phi$ vertices;
* a triangle with two $A\phi\phi$ vertices and one $AA\phi\phi$ seagull vertex;
* a gauge bubble with two $AA\phi\phi$ seagull vertices;
* when the mixed operators $\mathcal I_A$ are present, a fermion bubble with two $\phi^2\bar\psi\psi$ vertices.

The first four are forced by the scalar covariant derivative and scalar potential. There is no ordinary Yukawa fermion box because the imposed $\phi\mapsto-\phi$ symmetry forbids a one-scalar fermion vertex.


Solution

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

 Solution

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