Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 301 / 2

In four spacetime dimensions the action is dimensionless, so the mass dimension of the Lagrangian density is $[\mathcal{L}]=4$. The kinetic term gives $[\varphi ]=1$, and $g{\varphi }^3$ then givesThus $g$ has mass dimension one and is a relevant coupling under power counting in quantum field theory.

The momentum-space Feynman rules for Phi cubed theory areEach vertex carries $(2\pi {)}^4{\delta }^{(4)}$ of momentum conservation, each independent loop momentum is integrated with $d^4\ell /(2\pi {)}^4$, and a graph is divided by its Feynman-diagram symmetry factor. External propagators are omitted from an amputated scattering amplitude.

Let $G_5$ be the connected time-ordered five-point function. The LSZ reduction formula gives, up to one factor $Z^{-1/2}$ per external leg,Translation invariance factors this as

A connected Feynman diagram that is a cubic tree with five external legs has three vertices and two internal lines. Its unique unlabeled topology is a chain: two external legs meet at each end vertex and one external leg attaches to the middle vertex. The five-point tree amplitude in phi cubed theory therefore has fifteen labeled tree-level Feynman diagrams: choose the middle external leg in five ways, then partition the remaining four legs into two unordered pairs in three ways.

Take all external momenta incoming, $q_1=p_1$, $q_2=p_2$, and $q_{2+r}=-k_r$. For a diagram whose middle leg is $a$ and whose two end pairs are $\{b,c\}$ and $\{d,e\}$, the Feynman rules giveEquivalently,The full connected tree amplitude is the sum over the fifteen choices described in part d, multiplied by the overall momentum-conserving delta function from part c.