  The recently proposed classification of integrability-breaking perturbations
according to their strength is studied in the context of quantum field
theories. Using random matrix methods to diagnose the resulting quantum chaotic
behaviour, we investigate the $\phi^4$ and $\phi^6$ interactions of a massive
scalar, by considering the crossover between Poissonian and Wigner-Dyson
distributions in systems truncated to a finite-dimensional Hilbert space. We
find that a naive extension of the scaling of crossover coupling with the
volume observed in spin chains does not give satisfactory results for quantum
field theory. Instead, we demonstrate that considering the scaling of the
crossover coupling with the number of particles yields robust signatures, and
is able to distinguish between the strengths of integrability breaking in the
$\phi^4$ and $\phi^6$ quantum field theories.
