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.

12 pages, 1 figure, recieved version

References in corpus (10)

• Classical Polylogarithms for Amplitudes and Wilson Loops
• Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals
• Bootstrapping a Five-Loop Amplitude Using Steinmann Relations
• Bootstrapping an NMHV amplitude through three loops
• Sector Decomposition
• Resolution of singularities for multi-loop integrals
• Geometric approach to asymptotic expansion of Feynman integrals
• Bootstrapping the QCD soft anomalous dimension
• Multi-Loop Positivity of the Planar $\mathcal{N}=4$ SYM Six-Point Amplitude
• Bootstrapping a Two-Loop Four-Point Form Factor

Cited by in corpus (9)

• Cosmology meets cohomology
• Analytic three-loop QCD corrections to top-quark and semileptonic $b\to u$ decays
• Two-Loop Five-Point Two-Mass Planar Integrals and Double Lagrangian Insertions in a Wilson Loop
• A double copy from twisted (co)homology at genus one
• Self-duality from twisted cohomology
• 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