Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 304 / 3 / Solution

The kinetic term fixes the mass dimension of a canonically normalized real scalar field in $d$ dimensions to $(d-2)/2$. In six dimensions $[\varphi ]=2$, and every term in the Lagrangian density has dimension six. Therefore $[g_n]=6-2n$, andFor positive integral $n$, these mean $n=1,2$, $n=3$ and $n\ge 4$, respectively. If $n=0$ is included, its vacuum-energy constant is also relevant. These are classical power counting in quantum field theory classifications of relevant couplings, marginal couplings and irrelevant couplings, not assertions that loop corrections preserve exact marginality. For the remaining parts write $g=g_3$, assume $m^2>0$ and $0<\Lambda <{\Lambda }_0$, and define a local quartic coupling by $\int g_4{\varphi }^4/4!$.