Physics Context: Claude's Computation of 9-Loop Amplitudes
In September 2026, Anthropic's science team published a milestone in mathematical physics: showing that large language models (Claude) could automate and execute the analytic computation of scattering amplitudes up to nine loops in planar N=4 super-Yang-Mills (SYM) theory. Scattering amplitudes describe the probability amplitudes for subatomic particles to scatter in colliders such as CERN's Large Hadron Collider.
Standard textbook Feynman diagram techniques scale exponentially with loop order; direct integration at 9 loops involves millions of high-dimensional rational integrands. Claude leveraged cutting-edge modern amplitude tools: the amplituhedron, dual conformal symmetry, kinematic twistor variables, and integration via differential equations and symbol alphabets.
Planar Dual Grids
Every planar L-loop Feynman diagram maps to a dual polygon face lattice where loop momenta $k_i$ correspond to regional variables $x_i$, eliminating unphysical gauge redundancies.
Leading Singularities
On maximal cuts, all virtual propagator lines are put on-shell ($P_j^2 = 0$). The integrand collapses into contour residues that fix the rational prefactors without ultraviolet divergences.
Symbol Alphabet
Multi-loop integrals evaluate to iterated integrals (Chen integrals) whose discontinuities are encoded by a finite set of algebraic letters, establishing uniform maximal transcendental weight $2L$.