Feynman Integrals and Scattering Amplitudes from Wilson Loops

We study Feynman integrals and scattering amplitudes in $\mathcal{N}=4$ super-Yang-Mills theory by exploiting the duality with null polygonal Wilson loops. As the main application, we compute for the first time the symbols of the general double pentagon integrals, which give the finite part of two-loop maximally helicity violating (MHV) amplitudes and finite components of next-to-MHV (NMHV) amplitudes to all multiplicities. The rational parts of the symbol consist of 164 letters, while the algebraic part contains 96 algebraic letters and cancel in MHV amplitudes and NMHV components which are free of square roots.

Figure 1

NMHV component of super-WL as difference of two diagrams, each equal to a double-pentagon integral.

Figure 2

The chiral pentagon written as a WL diagram, and loop integral performed using “star-triangle” identity.