Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2025 / 1 / c

The action is dimensionless, so in four dimensions the Lagrangian density has mass dimension four. Its kinetic term fixes $[\varphi ]=1$ and $[{\partial }_{\mu }]=1$. An interaction $g_D{\mathcal{O}}_D$ with $[{\mathcal{O}}_D]=D$ has $[g_D]=4-D$. ThereforeIn an effective field theory with heavy scale $\Lambda$, a higher-dimensional local operator enters as $c_D{\mathcal{O}}_D/{\Lambda }^{D-4}$. For characteristic scales $E,m\ll \Lambda$ and coefficients compatible with this expansion, its effects are suppressed by powers of the small ratios. Omitting these terms defines the leading approximation and its truncation error in effective field theory. It is not a fundamental prohibition: such terms must be retained if that accuracy is required. A continuum renormalizable quantum field theory can instead be defined by the closed set of counterterms of dimension at most four.

Up to total derivatives, all Lorentz scalar terms of dimension at most four built from one real scalar field areThe term $\varphi \square \varphi$ is equivalent to the kinetic term by integration by parts; $\square \varphi$ and $\square ({\varphi }^2)$ contribute only at the boundary. The displayed Lagrangian density omits the linear and cubic interactions and a field-independent constant. The exact Z2 symmetry $\varphi \mapsto -\varphi$ forbids both odd interactions, and radiative corrections preserving this symmetry cannot generate them. The constant shifts the vacuum energy and the normalization of $Z$ and can be dropped when gravity and absolute vacuum energy are not being studied. A constant vacuum counterterm may still be used when renormalizing unnormalized vacuum diagrams. The kinetic coefficient has already been fixed by field normalization.