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Towards the Parametric Renormalization of the S-matrix -- I
by Pinaki Banerjee, Harsh, Alok Laddha
Submission summary
| Authors (as registered SciPost users): | Pinaki Banerjee |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2509.18283v1 (pdf) |
| Date submitted: | Nov. 4, 2025, 8:28 a.m. |
| Submitted by: | Pinaki Banerjee |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
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 of ``tropical counter-term'' for the bare amplitude.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
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A central result of the work is the new “surface–forest formula,” which offers a systematic prescription for decapitating tadpole diagrams directly at the level of curve integrals. For the planar one-loop and two-loop cases, the authors explicitly obtain the renormalized amplitudes. In addition, Appendix A presents an interesting pseudo-triangulation model for the two-loop surface, which further enriches the geometric interpretation.
Although the paper covers a broad range of material, it is well organized and clearly written. The authors provide concise and accessible reviews of both Zimmermann’s forest formula and the curve-integral framework, and support their discussion with a substantial number of illustrative examples that effectively clarify the main ideas and technical procedures.
Overall, I find the paper to be novel and interesting, and I believe it is suitable for publication in SciPost. That said, there are some minor typos and several points where the presentation could be improved. My comments and questions are as follows:
1. The paper consistently uses the notation $\mathrm{Tr}(\Phi^3)$; however, in a few places the notation $\phi^3$ still appears as a typo, for example, in the caption of Fig. 11.
2. The paper uses the notation $A_{L,n}$ and $\Sigma_{L,n}$ to denote the L-loop, n-point amplitudes and surfaces, respectively. However, in some places the labels L and n are interchanged, such as in the caption of Fig. 4 and Eq. (3.27).
3. I suggest modifying the notation for the matrix elements in Eq. (3.9), since $M_{C_{1,1}}$ can easily be confused with $M_{C_{11}}$, which appears in Eq. (3.13).
4. Can this UV renormalization method be generalized to more general theories, such as the $\mathrm{Tr}(\Phi^3)$ theory in arbitrary dimensions, or to Yang–Mills theory, which can be obtained from the deformed curve integrals?
5. Can the tropical subtraction procedure also be applied to identify or regulate infrared divergences?
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)