Towards the Parametric Renormalization of the S-matrix - I (2509.18283v1)

Abstract: Zimmermann's forest formula is the corner stone of perturbative renormalization in QFT. By renormalizing individual Feynman graphs, it generates the UV finite S-matrix. This approach to renormalization makes the graph and all its forests center pieces in the theory of renormalization. On the other hand the positive geometry program delegate the role of Feynman graphs as secondary to the amplitude itself, which are generated by canonical forms associated to positive geometries. State of the art in this program is the convergence of S-matrix theory in local QFTs and string theory as the scattering amplitudes in QFT arise as integrals over certain moduli spaces. These integrals are known as curve integrals. For theories such as $\textrm{Tr}(\Phi^{3})$ theory with massive colored scalars, these integrals are divergent in the UV and have to be regularized. It is then natural to ask if there is a forest-like formula'' for these integrals which produce a renormalized amplitude without needing to explicitly invoke the forests associated to divergent subgraphs. In this paper, we initiate such a program by deriving forest-like formula for planar massive $\textrm{Tr}(\Phi^{3})$ amplitudes in $D = 4$ dimensions. Our analysis relies on the insightful manifestation of the forest formula derived by Brown and Kreimer in \cite{Brown:2011pj}, that lead us to a definition oftropical counter-term'' for the bare amplitude.