Title: Large 𝑛: scattering amplitudes at large multiplicity
Abstract: What happens when we scatter a large number 𝑛 of particles, say with 𝑛 = 101000? This question is out of reach of all existing approaches to scattering amplitudes, whether directly through Feynman diagrams or recursion relations. In this talk, I will study this problem at tree-level for a simple scalar theory, Tr(𝜑3) theory, and describe a new way of accessing this regime, starting from the "tropical" representation of the amplitudes given by the formalism of surfaceology. Remarkably, we find that at large 𝑛 the sum over diagrams "smooths out". Working at leading order in 𝑛, and in certain "positive" regions of kinematic space, the final answer is astonishingly simple, given by a single term, determined by an optimization problem associated with a biased random walk. I will also describe a transformation from the tropical representation to a new "dual" theory for amplitudes, which in simple limits reduces to the motion of a particle on a half-line, evolving for a "time'' 𝑛, allowing us to systematically extract the 1/𝑛 expansion of the amplitude. I will end by explaining how these results generalize to other large 𝑛 contexts, from the scattering of pions, to string scattering at ultra-high energies.
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