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Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals

Erik Panzer

TL;DR

The work tackles the symbolic integration of hyperlogarithms multiplied by rational functions, a key step for evaluating Feynman graphs and their ε-expansions. It develops a concrete framework based on iterated integrals and shuffle algebra to construct primitives and regularize divergences, enabling exact symbolic results for linearly reducible multi-dimensional integrals. A central contribution is the complete Maple-based implementation HyperInt, including a robust regularization workflow and a polynomial-reduction strategy that leverages linear reducibility to control the complexity. The approach is validated through extensive tests and applications to Feynman integrals, with results expressed in terms of multiple zeta values and related periods, highlighting both the mathematical structure and practical impact for high-precision quantum-field-theory calculations.

Abstract

We provide algorithms for symbolic integration of hyperlogarithms multiplied by rational functions, which also include multiple polylogarithms when their arguments are rational functions. These algorithms are implemented in Maple and we discuss various applications. In particular, many Feynman integrals can be computed by this method.

Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals

TL;DR

The work tackles the symbolic integration of hyperlogarithms multiplied by rational functions, a key step for evaluating Feynman graphs and their ε-expansions. It develops a concrete framework based on iterated integrals and shuffle algebra to construct primitives and regularize divergences, enabling exact symbolic results for linearly reducible multi-dimensional integrals. A central contribution is the complete Maple-based implementation HyperInt, including a robust regularization workflow and a polynomial-reduction strategy that leverages linear reducibility to control the complexity. The approach is validated through extensive tests and applications to Feynman integrals, with results expressed in terms of multiple zeta values and related periods, highlighting both the mathematical structure and practical impact for high-precision quantum-field-theory calculations.

Abstract

We provide algorithms for symbolic integration of hyperlogarithms multiplied by rational functions, which also include multiple polylogarithms when their arguments are rational functions. These algorithms are implemented in Maple and we discuss various applications. In particular, many Feynman integrals can be computed by this method.

Paper Structure

This paper contains 36 sections, 10 theorems, 101 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Plan of the paper
  3. Algorithms for hyperlogarithms
  4. The tensor algebra and iterated integrals
  5. Integration and differentiation
  6. Divergences and logarithmic regularization
  7. Regularized limits as hyperlogarithms
  8. Regularized limits of regularized limits at infinity
  9. The implementation HyperInt
  10. General remarks
  11. Installation and files
  12. Representation of polylogarithms and conversions
  13. Periods
  14. Integration of hyperlogarithms
  15. Singularities in the domain of integration
  16. ...and 21 more sections

Key Result

Lemma 2.2

For $w = u \omega_{\sigma} a$ with $a = \omega_{a_1}\ldots\omega_{a_n}$, So when $\sigma\notin A$ is the last letter of $w$ not in $A$, thus $a \in A^{\times}$, we deduce $\mathop{\mathrm{reg}}\limits\nolimits_{A}^{}(w) = (-1)^n \left( u \shuffle \omega_{a_n}\ldots\omega_{a_1} \right) \omega_{\sigma}$. Analogously $\mathop{\mathrm{reg}}\limits\nolimits_{}^{B} (b \omega_{

Figures (5)

  • Figure 1: This graph shows the limits of $\Sigma$ in example \ref{['ex:above-below-partition']} when $t \rightarrow 0$ with positive real part and small positive imaginary part.
  • Figure 2: The letters $\left\{ 1+t,2-t \right\} \subset \Sigma$ in example \ref{['ex:above-below-partition']} induce a deformation of the real integration path $[0,\infty)$ towards $\gamma$, which avoids the positive limits in passing below $\Sigma_0^{+} = \left\{ 1 \right\}$ and above $\Sigma_0^{-} = \left\{ 2 \right\}$.
  • Figure 3: The contour $\gamma$ of example \ref{['ex:above-below-partition']} shown in figure \ref{['fig:contour-deformation']} is homotopic to the splitting $\eta_u \star \gamma_u$.
  • Figure 4: Four-loop massless propagator of section \ref{['sec:example-propagator']}. In Panzer:MasslessPropagators this one is called $M_{3,6}$. Edges are labelled in black, vertices in red.
  • Figure 5: Two series of one-scale graphs with subdivergences in four dimensions. They occur in $\phi^4$-theory as vertex graphs with two nullified external momenta, incident to the two three-valent vertices.

Theorems & Definitions (41)

  • Definition 1.1
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • proof
  • ...and 31 more