Algebraic singularities of scattering amplitudes from tropical geometry

Understand

We address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar $\mathcal{N}=4$ super Yang-Mills theory.

• We argue that connections between cluster algebras and tropical geometry provide a natural language for postulating a finite alphabet for scattering amplitudes beyond six and seven points where the corresponding Grassmannian cluster algebras are finite.
• As well as generating natural finite sets of letters, the tropical fans we discuss provide letters containing square roots.
• Remarkably, the minimal fan we consider provides all the square root letters recently discovered in an explicit two-loop eight-point NMHV calculation.