Symbols from Bi-Projections

We initiate a systematic framework for the analysis of analytic properties of finite Feynman integrals that are multiple polylogarithms. Based on the Feynman parameter representation in complex projective space, we make a complete classification of logarithmic singularities of the integral on its principal branch, by what we call touching configurations -- a geometric relationship between the integrand singularity and linear subspaces tied to boundary elements of the integral contour. These on the one hand indicate first entries of the symbol of the integral, and on the other hand induce a special set of new integrals that we call elementary discontinuities. These elementary discontinuities are derived through an operation called bi-projection, and actual discontinuities of the integral across logarithmic branch cuts are their linear combinations. By recursively applying the same analysis to the induced integrals one can fully construct the symbol of the original integral. We explicitly show how this analysis works at one loop in a massless hexagon and a box with two massive and two massless loop propagators. This framework may naturally extend to higher-loop integrals.