Multiple Deligne values: a data mine with empirically tamed denominators

Abstract

Multiple Deligne values (MDVs) are iterated integrals on the interval $x\in[0,1]$ of the differential forms $A=d\log(x)$, $B=-d\log(1-x)$ and $D=-d\log(1-λx)$, where $λ$ is a primitive sixth root of unity. MDVs of weight 11 enter the renormalization of the standard model of particle physics at 7 loops, via a counterterm for the self-coupling of the Higgs boson. A recent evaluation by Erik Panzer exhibited the alarming primes 50909 and 121577 in the denominators of rational coefficients that reduce this counterterm to a Lyndon basis suggested by ideas from Pierre Deligne. Oliver Schnetz has studied this problem, using a method from Francis Brown. This gave 2111, 14929, 24137, 50909 and 121577 as factors of the denominator of the coefficient of $π^{11}/\sqrt{3}$. Here I construct a basis such that no denominator prime greater than 3 appears in the result. This is achieved by building a datamine of 13,369,520 rational coefficients, with tame denominators, for the the reductions of 118,097 MDVs with weights up to 11. Then numerical data for merely 53 primitives enables very fast evaluation of all of these MDVs to 20000 digits. In the course of this Aufbau, six conjectures for MDVs are formulated and stringently tested.