Six Particles, Infinite Optimism: Towards Positivity Bounds for Six-Point Amplitudes
Abstract
The systematic generalisation of the well-established S-matrix positivity and bootstrap bounds from 2 → 2 scattering processes to higher-point amplitudes has, to date, remained elusive. To address this, we consider a large class of tree-level UV-completions of a single-scalar EFT. Specifically, we consider the most general renormalisable N-scalar theory in D=4. By explicitly computing tree-level amplitudes for four-, five- and six-point scattering and expanding at low energies, we can search for positivity relations among the expansion coefficients. We find that quite generally positivity bounds are unavailable due to contamination from a specific class of Feynman diagrams that are linear in sign-indefinite cubic couplings. However, by isolating a sector of the full crossing-symmetric low-energy amplitude that is unaffected by these sign-indefinite cubic couplings, we find that selected six-, five- and four-point Wilson coefficients organise into positive semi-definite matrices. While a violation of these positivity statements only implies that a given EFT is inconsistent with our class of UV completions, the result helps to guide towards what more general positivity bounds for higher-point interactions could, and could not, look like.
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BibTeX
@misc{kremminger2026particles,
title = {{Six Particles, Infinite Optimism: Towards Positivity Bounds for Six-Point Amplitudes}},
author = {Stefan Kremminger and Andrew J. Tolley},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17861},
eprint = {2609.17861},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17861},
}
RIS
TY - UNPB
TI - Six Particles, Infinite Optimism: Towards Positivity Bounds for Six-Point Amplitudes
AU - Stefan Kremminger
AU - Andrew J. Tolley
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17861
UR - https://arxiv.org/abs/2609.17861
AB - The systematic generalisation of the well-established S-matrix positivity and bootstrap bounds from 2 → 2 scattering processes to higher-point amplitudes has, to date, remained elusive. To address this, we consider a large class of tree-level UV-completions of a single-scalar EFT. Specifically, we consider the most general renormalisable N-scalar theory in D=4. By explicitly computing tree-level amplitudes for four-, five- and six-point scattering and expanding at low energies, we can search for positivity relations among the expansion coefficients. We find that quite generally positivity bounds are unavailable due to contamination from a specific class of Feynman diagrams that are linear in sign-indefinite cubic couplings. However, by isolating a sector of the full crossing-symmetric low-energy amplitude that is unaffected by these sign-indefinite cubic couplings, we find that selected six-, five- and four-point Wilson coefficients organise into positive semi-definite matrices. While a violation of these positivity statements only implies that a given EFT is inconsistent with our class of UV completions, the result helps to guide towards what more general positivity bounds for higher-point interactions could, and could not, look like.
ER -