Amplitudes and renormalization group techniques: A case study

We explore the properties of a simple renormalizable shift-symmetric model with a higher-derivative kinetic energy and quartic-derivative coupling that can serve as a toy model for higher-derivative theories of gravity. The scattering amplitude behaves as in a normal effective field theory below the threshold for the production of ghosts, but has an unexpectedly soft behavior above the threshold. The physical running of the parameters is extracted from the two-point and four-point amplitudes. The results are compared to those obtained by other methods and are found to agree only in limiting cases. We draw several lessons that may also apply to gravity.