The paper presents a new curve integral formalism for computing scattering amplitudes in a colored scalar trϕ³ theory. The key findings are:
At tree-level, the n-point amplitude can be computed as a simple integral over an (n-3)-dimensional space, with the integrand given by a sum of O(n²) simple headlight functions. This is in contrast to the conventional approach which requires summing over exponentially many Feynman diagrams.
At loop level, the dependence of the amplitudes on n and the loop order L is effectively decoupled. The calculation of an L-loop amplitude for small n is sufficient to obtain a closed formula for the amplitude at any n and L.
This decoupling is made possible by the "telescopic property" of the headlight functions, which allows sums over curves on the fatgraph to be computed efficiently.
Explicit closed-form expressions are derived for all amplitudes in the theory, including non-planar ones, through two loops. These formulas exhibit a clear factorization into a loop-dependent prefactor and an n-dependent exponent.
The paper demonstrates how the curve integral formalism leads to a dramatic simplification in the complexity of computing scattering amplitudes, even at high multiplicities and loop orders, compared to the conventional diagrammatic approach.
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by Nima Arkani-... at arxiv.org 10-01-2024
https://arxiv.org/pdf/2311.09284.pdfDeeper Inquiries