Intersection theory rules symbology

Abstract

We propose a novel method to determine the structure of symbols for any family of polylogarithmic Feynman integrals. Using the d log-bases and simple formulas for the leading order and next-to-leading contributions to the intersection numbers, we give a streamlined procedure to compute the entries in the coefficient matrices of canonical differential equations, including the symbol letters and the rational coefficients. We also provide a selection rule to decide whether a given matrix element must be zero. The symbol letters are deeply related to the poles of the integrands and also have interesting connections to the geometry of Newton polytopes. Our method can be applied to many cutting-edge multi-loop calculations. The simplicity of our results also hints at the possible underlying structure in perturbative quantum field theories.

arXiv comment12 pages, 1 figure, recieved version

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• Two-Loop Five-Point Two-Mass Planar Integrals and Double Lagrangian Insertions in a Wilson Loop
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• Canonical differential equations for the elliptic two-loop five-point integral family relevant to $t\bar{t}+$jet production at leading colour
• Intersection Numbers from Companion Tensor Algebra
• Twisted Riemann bilinear relations and Feynman integrals
• Canonical differential equations and intersection matrices