Differential Equations, Associators, and Recurrences for Amplitudes

TL;DR

The paper develops an algebraic framework to obtain all-order analytic expansions in $\epsilon$ (field theory) and $\alpha'$ (string theory) by solving recurrences with noncommuting coefficients via a generalized operator product. It connects generalized hypergeometric function expansions to Drinfeld associators through KZ/Fuchsian structures, yielding compact expressions in terms of MPLs and MZVs and enabling explicit $pF_{p-1}$ expansions without computing lower orders. The authors apply these methods to open-string four- and five-point disk amplitudes, provide representations in terms of symmetric functions and kinematic invariants, and show how associators encode world-sheet monodromies. A comprehensive set of identities for generalized operator products is developed, translating operator-product expressions into MPL/MZV relations and generating new MZV identities consistent with known theorems such as the sum theorem, thereby linking Feynman- and string-amplitude expansions to the algebra of multiple zeta values.

Abstract

We provide new methods to straightforwardly obtain compact and analytic expressions for epsilon-expansions of functions appearing in both field and string theory amplitudes. An algebraic method is presented to explicitly solve for recurrence relations connecting different epsilon-orders of a power series solution in epsilon of a differential equation. This strategy generalizes the usual iteration by Picard's method. Our tools are demonstrated for generalized hypergeometric functions. Furthermore, we match the epsilon-expansion of specific generalized hypergeometric functions with the underlying Drinfeld associator with proper Lie algebra and monodromy representations. We also apply our tools for computing epsilon-expansions for solutions to generic first-order Fuchsian equations (Schlesinger system). Finally, we set up our methods to systematically get compact and explicit alpha'-expansions of tree-level superstring amplitudes to any order in alpha'.