Elliptic multiple zeta values and one-loop superstring amplitudes

Abstract

We investigate iterated integrals on an elliptic curve, which are a natural genus-one generalization of multiple polylogarithms. These iterated integrals coincide with the multiple elliptic polylogarithms introduced by Brown and Levin when constrained to the real line. At unit argument they reduce to an elliptic analogue of multiple zeta values, whose network of relations we start to explore. A simple and natural application of this framework are one-loop scattering amplitudes in open superstring theory. In particular, elliptic multiple zeta values are a suitable language to express their low energy limit. Similar to the techniques available at tree-level, our formalism allows to completely automatize the calculation.

39 pages, 1 figure, v2: minor corrections, published version, v3: minor corrections in eqs.(4.5) to (4.9), (4.31) and (4.36)

References in corpus (12)

• New Relations for Gauge-Theory Amplitudes
• Classical Polylogarithms for Amplitudes and Wilson Loops
• The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms
• Closed String Amplitudes as Single-Valued Open String Amplitudes
• Multi-Gluon Scattering in Open Superstring Theory
• Six Gluon Open Superstring Disk Amplitude, Multiple Hypergeometric Series and Euler-Zagier Sums
• The One-Loop Five-Graviton Amplitude and the Effective Action
• Towards multiple elliptic polylogarithms
• Current Algebra on the Torus
• Cohomology foundations of one-loop amplitudes in pure spinor superspace
• Framings for graph hypersurfaces
• Motivic periods and the projective line minus three points

Cited by in corpus (109)

• Elliptic polylogarithms and iterated integrals on elliptic curves I: general formalism
• Two-loop planar master integrals for Higgs$\to 3$ partons with full heavy-quark mass dependence
• One-loop amplitudes on the Riemann sphere
• Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral
• Collider Physics at the Precision Frontier
• 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
• A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries
• 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
• Generalised power series expansions for the elliptic planar families of Higgs + jet production at two loops
• 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
• String-inspired BCJ numerators for one-loop MHV amplitudes
• The unequal mass sunrise integral expressed through iterated integrals on ${\overline{\mathcal{M}}}_{1,3}$
• New BCJ representations for one-loop amplitudes in gauge theories and gravity
• Elliptic polylogarithms and Feynman parameter integrals
• An analytic solution for the equal-mass banana graph
• Monodromy Relations in Higher-Loop String Amplitudes
• Relations between elliptic multiple zeta values and a special derivation algebra
• From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop
• Eisenstein series and automorphic representations
• Numerical evaluation of iterated integrals related to elliptic Feynman integrals
• The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals
• AdS Virasoro-Shapiro from single-valued periods
• Proof of a modular relation between 1-, 2- and 3-loop Feynman diagrams on a torus
• Tree-level amplitudes from the pure spinor superstring
• Double-Copy Structure of One-Loop Open-String Amplitudes
• Algorithms and tools for iterated Eisenstein integrals
• 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
• Evaluating two-loop non-planar master integrals for Higgs + jet production with full heavy-quark mass dependence
• One-loop Correlators and BCJ Numerators from Forward Limits
• Loop-by-loop Differential Equations for Dual (Elliptic) Feynman Integrals
• 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
• Generating series of all modular graph forms from iterated Eisenstein integrals
• Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings
• The complete set of two-loop master integrals for Higgs + jet production in QCD
• Meromorphic modular forms and the three-loop equal-mass banana integral
• Tetrahedral modular graph functions
• Towards the n-point one-loop superstring amplitude II: Worldsheet functions and their duality to kinematics
• On Epsilon Factorized Differential Equations for Elliptic Feynman Integrals
• From maximal to minimal supersymmetry in string loop amplitudes
• Modular graph functions and odd cuspidal functions -- Fourier and Poincaré series
• Linear power corrections to $e^{+}{e}^{-}$ shape variables in the three-jet region
• Manifestly Dual-Conformal Loop Integration
• Functional relations for elliptic polylogarithms
• Symbology for elliptic multiple polylogarithms and the symbol prime
• The SAGEX Review on Scattering Amplitudes, Chapter 10: Selected topics on modular covariance of type IIB string amplitudes and their $\mathcal{N}=4$ supersymmetric Yang-Mills duals
• All-order alpha'-expansion of one-loop open-string integrals
• Three-loop contributions to the $\rho$ parameter and iterated integrals of modular forms
• String-motivated one-loop amplitudes in gauge theories with half-maximal supersymmetry
• Modular transformations of elliptic Feynman integrals
• The two-loop superstring five-point amplitude and S-duality
• Differential Equations, Associators, and Recurrences for Amplitudes
• Amplitude recursions with an extra marked point
• Evaluating one-loop string amplitudes
• Modular graph forms from equivariant iterated Eisenstein integrals
• From Little String Free Energies Towards Modular Graph Functions
• Holomorphic subgraph reduction of higher-point modular graph forms
• Massive deformations of Maass forms and Jacobi forms
• Windings of twisted strings
• Elliptic modular graph forms I: Identities and generating series
• Simplifying one-loop amplitudes in superstring theory
• The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case
• Two-loop form factors for pseudo-scalar quarkonium production and decay
• Schottky-Kronecker forms and hyperelliptic polylogarithms
• Basis Decompositions and a Mathematica Package for Modular Graph Forms
• An Infinite Family of Elliptic Ladder Integrals
• Elliptic hyperlogarithms
• A double copy from twisted (co)homology at genus one
• A Combinatoric Shortcut to Evaluate CHY-forms
• Three-loop banana integrals with four unequal masses
• Cuts and Isogenies
• Periods and Feynman amplitudes
• Higher-genus Fay-like identities from meromorphic generating functions
• Two dialects for KZB equations: generating one-loop open-string integrals
• Diagrammatic Expansion of Non-Perturbative Little String Free Energies
• One-loop superstring six-point amplitudes and anomalies in pure spinor superspace
• Fermionic one-loop amplitudes of the RNS superstring
• Basis decompositions of genus-one string integrals
• Cyclic products of Szegö kernels and spin structure sums I: hyper-elliptic formulation
• On new theta identities of fermion correlation functions on genus g Riemann surfaces
• Symmetric Orbifold Theories from Little String Residues
• Decomposition of elliptic multiple zeta values and iterated Eisenstein integrals
• Advanced tools for basis decompositions of genus-one string integrals
• Open-string integrals with multiple unintegrated punctures at genus one
• A walk on sunset boulevard
• Canonical differential equations and intersection matrices
• Pure spinor formulation of the superstring and its applications
• An Elliptic One-Loop Amplitude in Anti-de-Sitter Space
• Genus-zero and genus-one string amplitudes and special multiple zeta values
• Discontinuities of banana integrals in dispersion relation representation
• Linear independence of indefinite iterated Eisenstein integrals
• Symmetries in A-Type Little String Theories, Part I
• Generalizations of polylogarithms for Feynman integrals
• The spectrum of Feynman-integral geometries at two loops
• Two-loop form factors for $P$-wave quarkonium production and decay
• Integral of two-loop modular graph functions
• Higher-Order Corrections to Higgs Boson Amplitudes with Full Quark Mass Dependence in Quantum Chromodynamics
• Analytic results for two-loop planar master integrals for Bhabha scattering
• One-loop matrix elements of effective superstring interactions: ${\alpha }^{\prime }$-expanding loop integrands
• A construction of single-valued elliptic polylogarithms
• Two-loop master integrals for pseudo-scalar quarkonium and leptonium production and decay
• Transition Form-Factor for ${\eta }_Q$ at NNLO in the strong coupling ${\alpha }_s$ and with all-order $v^2$ resummation