Loops, Polytopes and Splines

TL;DR

This paper reveals a novel connection between loop integrals in physics and spline functions in mathematics, providing a geometric and algebraic framework for understanding and decomposing loop integrals using splines and polytopes.

Contribution

It introduces the interpretation of one-loop integrals as Laplace transforms of splines and demonstrates how splines offer a geometric and algebraic approach to analyze and decompose loop integrals.

Findings

Loop integrands are Laplace transforms of splines.

One-loop integrals correspond to integrals of splines on hyperbolic slices, forming polytopes.

Spline technology simplifies the decomposition of higher point loop integrals.

Abstract

We uncover an unexpected connection between the physics of loop integrals and the mathematics of spline functions. One loop integrands are Laplace transforms of splines. This clarifies the geometry of the associated loop integrals, since a $n$-node spline has support on an $n$-vertex polyhedral cone. One-loop integrals are integrals of splines on a hyperbolic slice of the cone, yielding polytopes in $AdS$ space. Splines thus give a geometrical counterpart to the rational function identities at the level of the integrand. Spline technology also allows for a clear, simple, algebraic decomposition of higher point loop integrals in lower dimensional kinematics in terms of lower point integrals - e.g. an hexagon integral in 2d kinematics can be written as a sum of scalar boxes. Higher loops can also be understood directly in terms of splines - they map onto spline convolutions, leading to an intriguing representation in terms of hyperbolic simplices integrated over other hyperbolic simplices. We finish with speculations on the interpretation of one-loop integrals as partition functions, inspired by the use of splines in counting points in polytopes.