Where do the tedious products of zetas come from?

Abstract

Lamentably, the full analytical content of the epsilon-expansion of the master two-loop two-point function, with arbitrary self-energy insertions in 4-2epsilon dimensions, is still unknown. Here we show that multiple zeta values (MZVs) of weights up to 12 suffice through O(epsilon^9). Products of primitive MZVs are generated by a processes of "pseudo-exponentiation"" whose combinatorics faithfully accord with expectations based on Kreimer's modified shuffle product and on the Drinfeld-Deligne conjecture. The existence of such a mechanism, relating thousands of complicated rational numbers, enables us to identify precise and simple combinations of MZVs specific to quantum field theories in even numbers of spacetime dimensions.