Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the reduction of generalized polylogarithms to $\text{Li}_n$ and $\text{Li}_{2,2}$ and on the evaluation thereof

Hjalte Frellesvig, Damiano Tommasini, Christopher Wever

TL;DR

This work proves and exploits that all generalized polylogarithms up to weight four can be reduced to a basis consisting of $\log$, the classical polylogarithms up to $\mathrm{Li}_4$, and the special function $\mathrm{Li}_{2,2}$. It provides explicit reduction formulas, discusses the analytic structure across complex variables, and develops robust numerical algorithms for evaluating these basis functions, including implementations in Mathematica and C++. By applying these tools, the authors compute previously unknown integrals and demonstrate the practicality of their approach for two-loop and multi-scale Feynman integral calculations. The results offer a concrete, implementable path from GPLs to a compact, numerically tractable function set, with potential extension to higher weights via analogous strategies and new basis elements.

Abstract

We give expressions for all generalized polylogarithms up to weight four in terms of the functions log, $\text{Li}_n$, and $\text{Li}_{2,2}$, valid for arbitrary complex variables. Furthermore we provide algorithms for manipulation and numerical evaluation of $\text{Li}_n$ and $\text{Li}_{2,2}$, and add codes in Mathematica and C++ implementing the results. With these results we calculate a number of previously unknown integrals, which we add in App. C.

On the reduction of generalized polylogarithms to $\text{Li}_n$ and $\text{Li}_{2,2}$ and on the evaluation thereof

TL;DR

This work proves and exploits that all generalized polylogarithms up to weight four can be reduced to a basis consisting of , the classical polylogarithms up to , and the special function . It provides explicit reduction formulas, discusses the analytic structure across complex variables, and develops robust numerical algorithms for evaluating these basis functions, including implementations in Mathematica and C++. By applying these tools, the authors compute previously unknown integrals and demonstrate the practicality of their approach for two-loop and multi-scale Feynman integral calculations. The results offer a concrete, implementable path from GPLs to a compact, numerically tractable function set, with potential extension to higher weights via analogous strategies and new basis elements.

Abstract

We give expressions for all generalized polylogarithms up to weight four in terms of the functions log, , and , valid for arbitrary complex variables. Furthermore we provide algorithms for manipulation and numerical evaluation of and , and add codes in Mathematica and C++ implementing the results. With these results we calculate a number of previously unknown integrals, which we add in App. C.

Paper Structure

This paper contains 13 sections, 99 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Generalized polylogarithms
  3. Some reduction relations for GPLs
  4. Expressions for GPLs
  5. Numerical evaluation of classical polylogarithms
  6. Numerical evaluation of $\textrm{Li}_{2,2}(x,y)$
  7. The added code
  8. Discussion
  9. Relations for $\textrm{Li}_{2,2}$
  10. Details of $\textrm{Li}_{2,2}$ expressions
  11. Table of integrals
  12. Series expansions of hypergeometric functions
  13. Specific expressions for GPLs

Figures (4)

  • Figure 1: This figure shows the triangle that is spanned by the points $0$, $a$ and $b$ in the complex plane. The function $\text{T}$ as given by eq. \ref{['eq:triangleT']} evaluates to one whenever the point $x$ is inside the triangle, and to zero otherwise. $\text{P}(a,b;x)$ denotes the point where the radial line going through $x$ and the line between $a$ and $b$ cross.
  • Figure 2: This figure shows which expression for $\textrm{Li}_n(x)$ is used for which values of $x$. In region $A$ we use eq. \ref{['eq:linbernoulli']}, in region $B$ we use eq. \ref{['eq:lincrandall']}, and in region $C$ we use eq. \ref{['eq:lininversion']} to map into region $A$ with $C_1$ mapping to $A_1$ and $C_2$ to $A_2$.
  • Figure 3: This figure shows the values of the coefficients of eq. \ref{['eq:li22fast']} (blue) and eq. \ref{['eq:li22crandall']} (red) where the latter is taken to be $K_{ij}^A + K_{ij}^B \log(\tilde{\beta}) + K_{ij}^C \log(\tilde{\beta}+\xi_0)$ evaluated at $\tilde{\beta}=2$ and $\xi_0 = 1$. The faster convergence of eq. \ref{['eq:li22crandall']} (as a function of the number of terms) is clearly visible.
  • Figure 4: This figure is a slightly simplified illustration of the regions into which we split the evaluation of $\textrm{Li}_{2,2}(x,y)$ as described in the main text. $\varphi_z$ on the right figure refers to the phase of the complex $z$. Not shown on the figure is the fact that region $C$ is used in place of $D$ whenever $|x|>3.5$.