The Unitarity Flow Conjecture: An On-shell Approach to the Renormalization Group

We propose that the broad architecture of the renormalization group flow in quantum field theories is, at least in part, fixed by unitarity. The precise statement is summarized in the Unitarity Flow Conjecture, which states that the non-linear $S$-matrix identities obtained by imposing unitarity imply those needed to derive the renormalization group equations. As a proof of principle, we verify this conjecture to all loops at the leading and subleading logarithmic order in the four-dimensional massless $\lambda\phi^4$ theory using on-shell techniques, without reference to any counterterms or Feynman diagrams.