Differential Equations, Associators, and Recurrences for Amplitudes

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'.

59 pages, LaTeX; v2: Added Eq. (4.37) expressing the epsilon-expansion of specific generalized hypergeometric functions for any x in terms of the appropriate fundamental and universal solution (4.36) of the KZ equation. v3: Typos removed; final version to appear in Nucl. Phys. B

References in corpus (7)

• Lectures on differential equations for Feynman integrals
• The LHC String Hunter's Companion
• Multi-Gluon Scattering in Open Superstring Theory
• Six Gluon Open Superstring Disk Amplitude, Multiple Hypergeometric Series and Euler-Zagier Sums
• Amplitude for N-Gluon Superstring Scattering
• A Feynman Integral and its Recurrences and Associators
• General solution of a second order non-homogenous linear difference equation with noncommutative coefficients

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• Tree-level amplitudes from the pure spinor superstring
• Closed strings as single-valued open strings: A genus-zero derivation
• One-loop open-string integrals from differential equations: all-order alpha'-expansions at n points
• Twisted elliptic multiple zeta values and non-planar one-loop open-string amplitudes
• New non-perturbative de Sitter vacua in ${\alpha }^{\prime }$-complete cosmology
• Coaction and double-copy properties of configuration-space integrals at genus zero
• Two dialects for KZB equations: generating one-loop open-string integrals
• BCJ, worldsheet quantum algebra and KZ equations
• A Feynman Integral and its Recurrences and Associators
• Numerical multi-loop calculations: tools and applications