Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials

By Jakob Ablinger, J. Blumlein, Carsten Schneider. arXiv.org, 2011

327 citations recorded by the source corpus.

The computation of Feynman integrals in massive higher order perturbative calculations in renormalizable Quantum Field Theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin transforms of Poincare- iterated integrals including denominators of higher cyclotomic polynomials. We derive the cyclotomic harmonic polylogarithms and harmonic sums and study their algebraic and structural relations. The analytic continuation of cyclotomic harmonic sums to complex values of N is performed using analytic representations. We also consider special values of the cyclotomic harmonic polylogarithms at argument x = 1, resp., for the cyclotomic harmonic sums at N → ∞, which are related to colored multiple zeta values, deriving various of their relations, based on the stuffle and shuffle alge bras and three multiple ar- gument relations. We also consider infinite generalized nested harmonic sums at roots of unity which are related to the infinite cyclotomic harmonic sums. Basis representations are derived for weight w = 1,2 sums up to cyclotomy l = 20.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/e09049cab2dc0523fe111eb22fcb833559181b88

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In Algebra and Number Theory

In Applied Mathematics

Citations in the corpus

Mathematical aspects of scattering amplitudes (2014, background): “(4) Cyclotomic harmonic polylogarithms (CHPLs) [21]: CHPLs are a generalisation of HPLs defined by the iterated integrals”
Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes (2012, background): “es it was realized however that for certain multi-loop integrals new classes of transcendental functions arise that can no longer be expressed in terms of the classical polylogarithm functions, e.g., [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,”
From polygons and symbols to polylogarithmic functions (2011, background): “Many oneloop and two-loop Feynman integrals can be expressed in terms of HPL’s up to weight four and generalizations thereof [26, 27].”
Feynman integrals and hyperlogarithms (2015, background): “They have been called cyclotomic harmonic polylogarithms in [4].”
Bootstrapping an NMHV amplitude through three loops (2014, unknown): “[89].”
Iterated Binomial Sums and their Associated Iterated Integrals (2014, methodology): “The package HarmonicSums allows to compute these asymptotic expansions using the command : GLExpansion[GL[a, x], x, n, ord].”
The four-loop six-gluon NMHV ratio function (2015, unknown): “Instead they become cyclotomic polylogarithms [108].”