Harmonic sums and polylogarithms generated by cyclotomic polynomials

Abstract

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 Poincaré-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 algebras and three multiple argument 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.Dedicated to Martinus Veltman on the occasion of his 80th birthday.

Bootstrapping an NMHV amplitude through three loops

“…Like all hexagon functions, the ratio function on the line (u, u, u) can be expressed [29] in terms of the cyclotomic HPLs defined in ref. [89]. At the level of the symbol, this correspondence is easy to see because on the line (u, u, u) we have u = y/(1 + y) 2 and 1 − u = (1 + y + y 2 )/(1 + y) 2 , where y ≡ y u .…”

Three loop renormalization of 3-quark operators in QCD

“…These polylogarithms together with the other quantities such as ζ(3) and ln(3) emerge from the one and two loop master integrals, [46,47,48,49]. Recently, the mathematics of these masters has been studied in the context of cyclotomic polynomials and harmonic polylogarithms in [51]. There an insight has been given for which particular polylogarithms and other such numbers will arise in the higher loop order master integrals.…”

On the reduction of generalized polylogarithms to Li n and Li2,2 and on the evaluation thereof

“…At two or more loops many Feynman integrals can be likewise expressed in terms of GPLs [7][8][9][10][11][12][13][14][15][16][17][18][19] (for further references, see [20,21] and the references therein), but there are also integrals which are counter examples, such as notably that of the fully massive sunset graph [22][23][24][25][26]. Certain graphs without massive propagators are also believed to be counter examples [27].…”

“…The files lievaluate.cpp and constants.cpp contain our C++ implementations of Li m (x) and Li 2,2 (x, y), described in sections 5 and 6. lievaluate.cpp contains mainly functions, and constants.cpp a large number of constants needed by the functions. The functions meant for the user are declared as complex<double> li1(complex<double> &x); complex<double> li2(complex<double> &x); complex<double> li3(complex<double> &x); 12 We chose not to implement the trivial G(; x) = 1.…”