# Scales II: Perturbatively renormalizable field theories

# Scales II: Perturbatively renormalizable field theories

This chapter discusses the technical tools needed for analyzing the perturbative renormalizability of a specific local quantum field theory. It presents proof of cutoff-insensitivity, using traditional graphical methods and from the point of view of effective Lagrangian theory. The chapter first examines in detail the structure of the cutoff dependence of general multi-loop Feynman integrals appearing in the perturbative expansion of amplitudes in a local quantum field theory. The occurrence of divergent integrals (and subintegrals) in such loop amplitudes is associated with cutoff-dependent contributions, which have a very simple momentum dependence. This latter fact is then exploited to demonstrate the equivalence of the set of subtractions needed to remove the leading cutoff dependence of an arbitrary Feynman amplitude (reducing it to the inverse power-dependence of the type seen above), resulting in a reparameterization of a set of coupling and mass parameters appearing in the Lagrangian of the theory. The intimate connection of reparameterization/subtraction is the essence of the proof of cutoff-insensitivity for perturbatively renormalizable theories.

*Keywords:*
perturbative renormalizability, local quantum field theory, renormalization, cutoff dependence, Feynman integrals, Feynman amplitude, Lagrangian theory

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