We set up a bootstrap problem for renormalization. Working in the massless four-dimensional O$(N)$ model and the $λφ^4$ theory, we prove that unitarity leads to all-loop recursion relations between coefficients of scattering amplitudes with different multiplicities. These turn out to be equivalent to the identities imposed by renormalization of the coupling and the wavefunction through subleading logarithmic order, except with different initial conditions. Matching the initial conditions thus fixes the beta function and wavefunction anomalous dimension to these orders. We explain how to connect this new on-shell renormalization picture with the standard renormalized perturbation theory, highlighting a rich interplay between finiteness, dimensional regularization, and unitarity cuts.
