One-loop open-string integrals from differential equations: all-order alpha'-expansions at n points

Abstract

We study generating functions of moduli-space integrals at genus one that are expected to form a basis for massless $n$-point one-loop amplitudes of open superstrings and open bosonic strings. These integrals are shown to satisfy the same type of linear and homogeneous first-order differential equation w.r.t. the modular parameter $\tau$ which is known from the A-elliptic Knizhnik--Zamolodchikov--Bernard associator. The expressions for their $\tau$-derivatives take a universal form for the integration cycles in planar and non-planar one-loop open-string amplitudes. These differential equations manifest the uniformly transcendental appearance of iterated integrals over holomorphic Eisenstein series in the low-energy expansion w.r.t. the inverse string tension ${\alpha }^{\prime }$. In fact, we are led to matrix representations of certain derivations dual to Eisenstein series. Like this, also the ${\alpha }^{\prime }$-expansion of non-planar integrals is manifestly expressible in terms of iterated Eisenstein integrals without referring to twisted elliptic multiple zeta values. The degeneration of the moduli-space integrals at $\tau \to i\infty$ is expressed in terms of their genus-zero analogues -- $(n+2)$-point Parke--Taylor integrals over disk boundaries. Our results yield a compact formula for ${\alpha }^{\prime }$-expansions of $n$-point integrals over boundaries of cylinder- or Moebius-strip worldsheets, where any desired order is accessible from elementary operations.

arXiv comment73 pages, v2: assorted small corrections

References in corpus (57)

• New Relations for Gauge-Theory Amplitudes
• Multiloop integrals in dimensional regularization made simple
• Local Integrals for Planar Scattering Amplitudes
• Multi-Leg One-Loop Gravity Amplitudes from Gauge Theory
• Minimal Basis for Gauge Theory Amplitudes
• The Multiple Zeta Value Data Mine
• The Momentum Kernel of Gauge and Gravity Theories
• The elliptic dilogarithm for the sunset graph
• Polylogarithms, Multiple Zeta Values and Superstring Amplitudes
• Elliptic multiple zeta values and one-loop superstring amplitudes
• Elliptic polylogarithms and iterated integrals on elliptic curves I: general formalism
• Low energy expansion of the four-particle genus-one amplitude in type II superstring theory
• Abelian Z-theory: NLSM amplitudes and alpha'-corrections from the open string
• Closed String Amplitudes as Single-Valued Open String Amplitudes
• A Feynman integral via higher normal functions
• Combinatorics and Topology of Kawai-Lewellen-Tye Relations
• The Elliptic Double-Box Integral: Massless Amplitudes Beyond Polylogarithms
• Elliptic Feynman integrals and pure functions
• The $\epsilon$-form of the differential equations for Feynman integrals in the elliptic case
• On the modular structure of the genus-one Type II superstring low energy expansion
• Aspects of Scattering Amplitudes and Moduli Space Localization
• Non-abelian $Z$-theory: Berends-Giele recursion for the ${\alpha }^{\prime }$-expansion of disk integrals
• Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams
• Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series
• An Elliptic Generalization of Multiple Polylogarithms
• The planar double box integral for top pair production with a closed top loop to all orders in the dimensional regularisation parameter
• Analytic results for the planar double box integral relevant to top-pair production with a closed top loop
• New BCJ representations for one-loop amplitudes in gauge theories and gravity
• All order alpha'-expansion of superstring trees from the Drinfeld associator
• Elliptic polylogarithms and Feynman parameter integrals
• Semi-abelian Z-theory: NLSM+phi^3 from the open string
• Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case
• An analytic solution for the equal-mass banana graph
• Multiparticle SYM equations of motion and pure spinor BRST blocks
• Heterotic and bosonic string amplitudes via field theory
• The algebraic structure of cut Feynman integrals and the diagrammatic coaction
• On tree-level higher order gravitational couplings in superstring theory
• Monodromy Relations in Higher-Loop String Amplitudes
• Higher genus modular graph functions, string invariants, and their exact asymptotics
• Higher-loop amplitude monodromy relations in string and gauge theory
• Universality in string interactions
• 5-field terms in the open superstring effective action
• From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop
• Current Algebra on the Torus
• On the field theory expansion of superstring five point amplitudes
• Closed strings as single-valued open strings: A genus-zero derivation
• Exploring transcendentality in superstring amplitudes
• All-order differential equations for one-loop closed-string integrals and modular graph forms
• Superstring amplitudes and the associator
• Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings
• Tetrahedral modular graph functions
• From maximal to minimal supersymmetry in string loop amplitudes
• On the algebraic structure of iterated integrals of quasimodular forms
• Eigenvalue equation for the modular graph $C_{a,b,c,d}$
• String-motivated one-loop amplitudes in gauge theories with half-maximal supersymmetry
• Differential Equations, Associators, and Recurrences for Amplitudes
• Simplifying one-loop amplitudes in superstring theory

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• Two dialects for KZB equations: generating one-loop open-string integrals
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• Symmetric Orbifold Theories from Little String Residues
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• Advanced tools for basis decompositions of genus-one string integrals
• The Algebraic Structure of the KLT Relations for Gauge and Gravity Tree Amplitudes
• Integral of depth zero to three basis of Modular Graph Functions
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