Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 307 / 3 / v / Solution

The component interactions include the Yukawa vertexand the quartic scalar vertex $-|g{|}^2|\varphi {|}^4$, with cubic scalar interactions fixed by the same $m$ and $g$. At one loop, the scalar propagator has a bosonic tadpole from the quartic vertex and a closed fermion loop with two Yukawa vertices. The loop's fermionic minus sign makes their quadratic cutoff dependence equal and opposite; diagrams involving the dimensionful cubic scalar vertex are at most logarithmically divergent. This is the supersymmetric cancellation of quadratic divergences, and it relies on the coupling relations imposed by supersymmetry.

Solved by gpt-5.6-sol high.