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Feynman integrals and hyperlogarithms

Erik Panzer

TL;DR

The paper develops a robust parametric framework for Feynman integrals via Schwinger parameters, proving that in the Euclidean region any integral is a finite linear combination of convergent master integrals, which enables evaluation without handling divergences directly. It provides a self-contained, algorithmic treatment of hyperlogarithms and multiple polylogarithms, establishing linear reducibility as a key criterion and delivering an implementation (HyperInt) to compute multi-loop integrals symbolically and numerically. The main results include the existence of infinite families of massless 3- and 4-point graphs expressible in terms of MPLs to all orders in the $\varepsilon$-expansion, with explicit high-loop examples and a non-MZV counterterm in massless $\phi^4$ theory, accompanied by a parity-based reducibility analysis. The work also develops a coproduct/Hopf-algebra perspective on renormalization, introduces forest-functions for recursion, and offers concrete comparisons to sector decomposition, highlighting practical pathways for high-precision perturbative calculations and deeper number-theoretic structure of Feynman periods.

Abstract

We study Feynman integrals in the representation with Schwinger parameters and derive recursive integral formulas for massless 3- and 4-point functions. Properties of analytic (including dimensional) regularization are summarized and we prove that in the Euclidean region, each Feynman integral can be written as a linear combination of convergent Feynman integrals. This means that one can choose a basis of convergent master integrals and need not evaluate any divergent Feynman graph directly. Secondly we give a self-contained account of hyperlogarithms and explain in detail the algorithms needed for their application to the evaluation of multivariate integrals. We define a new method to track singularities of such integrals and present a computer program that implements the integration method. As our main result, we prove the existence of infinite families of massless 3- and 4-point graphs (including the ladder box graphs with arbitrary loop number and their minors) whose Feynman integrals can be expressed in terms of multiple polylogarithms, to all orders in the epsilon-expansion. These integrals can be computed effectively with the presented program. We include interesting examples of explicit results for Feynman integrals with up to 6 loops. In particular we present the first exactly computed counterterm in massless phi^4 theory which is not a multiple zeta value, but a linear combination of multiple polylogarithms at primitive sixth roots of unity (and divided by $\sqrt{3}$). To this end we derive a parity result on the reducibility of the real- and imaginary parts of such numbers into products and terms of lower depth.

Feynman integrals and hyperlogarithms

TL;DR

The paper develops a robust parametric framework for Feynman integrals via Schwinger parameters, proving that in the Euclidean region any integral is a finite linear combination of convergent master integrals, which enables evaluation without handling divergences directly. It provides a self-contained, algorithmic treatment of hyperlogarithms and multiple polylogarithms, establishing linear reducibility as a key criterion and delivering an implementation (HyperInt) to compute multi-loop integrals symbolically and numerically. The main results include the existence of infinite families of massless 3- and 4-point graphs expressible in terms of MPLs to all orders in the -expansion, with explicit high-loop examples and a non-MZV counterterm in massless theory, accompanied by a parity-based reducibility analysis. The work also develops a coproduct/Hopf-algebra perspective on renormalization, introduces forest-functions for recursion, and offers concrete comparisons to sector decomposition, highlighting practical pathways for high-precision perturbative calculations and deeper number-theoretic structure of Feynman periods.

Abstract

We study Feynman integrals in the representation with Schwinger parameters and derive recursive integral formulas for massless 3- and 4-point functions. Properties of analytic (including dimensional) regularization are summarized and we prove that in the Euclidean region, each Feynman integral can be written as a linear combination of convergent Feynman integrals. This means that one can choose a basis of convergent master integrals and need not evaluate any divergent Feynman graph directly. Secondly we give a self-contained account of hyperlogarithms and explain in detail the algorithms needed for their application to the evaluation of multivariate integrals. We define a new method to track singularities of such integrals and present a computer program that implements the integration method. As our main result, we prove the existence of infinite families of massless 3- and 4-point graphs (including the ladder box graphs with arbitrary loop number and their minors) whose Feynman integrals can be expressed in terms of multiple polylogarithms, to all orders in the epsilon-expansion. These integrals can be computed effectively with the presented program. We include interesting examples of explicit results for Feynman integrals with up to 6 loops. In particular we present the first exactly computed counterterm in massless phi^4 theory which is not a multiple zeta value, but a linear combination of multiple polylogarithms at primitive sixth roots of unity (and divided by ). To this end we derive a parity result on the reducibility of the real- and imaginary parts of such numbers into products and terms of lower depth.

Paper Structure

This paper contains 51 sections, 25 theorems, 136 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Background
  4. Feynman integrals
  5. Multiple zeta values
  6. Multiple polylogarithms
  7. Hyperlogarithms
  8. Goals
  9. Overview
  10. Schwinger parameters
  11. Hyperlogarithms
  12. Applications and examples
  13. Outlook
  14. Beyond Feynman integrals
  15. Linear reducibility
  16. ...and 36 more sections

Key Result

Theorem 1.2.1

If a graph $G$ has vertex-width $\mathop{\mathrm{def:vw}}\nolimits(G) \leq 3$ at most three, then its periods are multiple zeta values.

Figures (11)

  • Figure 1: Propagator diagrams with one and two loops.
  • Figure 2: The zigzag-series of primitive $\phi^4$ graphs in $D=4$ dimensions.
  • Figure 3: Examples of the box ladder graphs $B_n$ with $n=3$ and $n=4$ loops.
  • Figure 4: Contraction and deletion of an edge.
  • Figure 5: Illustration of the proof of proposition \ref{['prop:parametric-position-space']}.
  • ...and 6 more figures

Theorems & Definitions (72)

  • Theorem 1.2.1
  • Theorem 1.3.1
  • Theorem 1.3.2
  • Theorem 1.3.3
  • Theorem 1.3.4
  • Definition 2.1.1
  • Theorem 2.1.2: Matrix-Tree-Theorem
  • proof
  • Theorem 2.1.3
  • proof
  • ...and 62 more