Symbols from Bi-Projections

TL;DR

The paper develops a geometric framework for analyzing analytic properties of finite Feynman integrals that evaluate to multiple polylogarithms by embedding Feynman parameters in complex projective space $\mathbb{CP}^d$ and classifying logarithmic singularities via touching configurations. It introduces elementary discontinuities through a bi-projection that fibrates the parameter space and enables recursive symbol construction, demonstrated on a one-loop massless hexagon and a two-mass off-shell box. The exact first-symbol data are extracted from stratifications of touching configurations, and explicit discontinuities are computed to assemble the complete symbol, revealing a robust, geometry-driven approach to symbol construction. The method promises scalability to higher-loop integrals and offers new avenues for symbol bootstrap and understanding of analytic structure in perturbative amplitudes.

Abstract

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.

Figures (22)

• Figure 1: Generic strategy of symbol construction by recursive study of discontinuities.
• Figure 2: Iterated integral in $\mathbb{R}^2$ for $\mathrm{Li}_2(z)$. (1) The integral is well-defined when $z<1$; (2) The contour needs to deform in the imaginary direction in the neighborhood of $x_1=1$ when $z\geq 1$.
• Figure 3: Alternative integral for $\mathrm{Li}_2(z)$, induced by a different choice of affine patch.
• Figure 4: Projective integral for $\mathrm{Li}_2(z)$ in the canonical frame. Dashed lines denote singularities of the integrand.
• Figure 5: Two ways of illustrating the $1$-simplex contour and its $\mathbb{CP}^1$ ambient space on a piece of paper: (1) Treat the paper as an affine patch of $\mathbb{CP}^2$, so that $\mathbb{CP}^1$ looks like a line. We use solid line to represent the contour and dotted line to represent the ambient space. (2) Treat the paper as an affine pathch of the $\mathbb{CP}^1$ ambient space, so that deformation of the contour is manifest.