The Unitarity Flow Conjecture

TL;DR

The authors address whether unitarity can fix the renormalization group (RG) flow by formulating the Unitarity Flow Conjecture (UFC) and testing it in the massless $λφ^4$ theory in four dimensions using on-shell, diagramless methods. They derive finite, recursion-based relations among leading and subleading log coefficients that reproduce the Callan–Symanzik $β$ and $γ$ functions at all loop orders, at least up to subleading logarithms, without invoking counterterms. They also show that unitarity-derived recursions and RG recursions are equivalent despite different initial data, highlighting a deep connection between unitarity and RG structure and suggesting a broader S-matrix bootstrap perspective for renormalization. The work points to promising extensions to massive theories, IR/rapidity phenomena, operator mixing, and further bootstrap integrations, all within an on-shell renormalization framework that obviates Feynman diagrams.

Abstract

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 $λφ^4$ theory using on-shell techniques, without reference to any counterterms or Feynman diagrams.

Figures (3)

• Figure 1: Diagrammatic representation of the generalized optical theorem. Terms contributing to the leading and subleading divergences of the 2- and 4-particle amplitudes are indicated above. The ellipsis contains terms at further subleading orders.
• Figure 2: Generalized optical theorem applied to the $6$-particle amplitude. The logs of the 6-particle amplitude arise from cuts through a single line. Unitarity then implies that these 6-particle amplitudes are products of 4-particle amplitudes. Out of the 10 permutations, the 4 displayed above are the only ones that contribute to subleading coefficients CMMS.
• Figure 3: Summary of the results. Coefficients of the 4-particle (left) and 2-particle (right) amplitudes $m_{L,k}$ and $n_{L,k}$ can be fixed either through unitarity (sourced by $m_{L,0}$ and $n_{L,0}$) or RG (sourced by $\beta_L$ and $\gamma_L$ as well as $m_{L, 0}$ and $n_{L, 0}$). Although the initial conditions are different, the solutions to these recursions are equivalent. Scheme dependence is generically encoded both in the constant terms $m_{L, 0}$ and $n_{L, 0}$ as well as the coefficients $\beta_L$ and $\gamma_L$.