Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 304 / 1 / b

Through order ${\lambda }^2$, the quartic one-particle-irreducible Feynman diagrams are the tree-level four-point vertex, the local ${\delta }_{\lambda }$ counterterm vertex, and three one-loop bubble diagrams. The bubbles have two quartic vertices and differ by whether the momentum through the loop is the $s$, $t$, or $u$ Mandelstam variables; each has symmetry factor $1/2$. External-leg self-energies are not part of the 1PI four-point vertex, and ${\delta }_Z$ begins beyond the required order in this theory.

Solved by gpt-5.6-sol high.

For a bubble carrying momentum $p$, a Feynman parameter and a shift of the loop momentum give, after Wick rotation and cutoff regularization,where $F(p^2)=m^2-x(1-x)p^2-i0$. The quantum effective action therefore containsThe renormalization condition ${\Gamma }_4(M^2,M^2,M^2)=\lambda$ fixesSubstitution cancels both the cutoff and the scheme-dependent constant and leavesas required.

Solved by gpt-5.6-sol high.