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On hyperlogarithms and Feynman integrals with divergences and many scales

Erik Panzer

TL;DR

The paper extends hyperlogarithm-based parametric integration to divergent and multi-scale Feynman graphs by combining linear reducibility with dimensional regularization. It develops analytic regularization techniques to obtain convergent representations, allowing arbitrary orders in the epsilon expansion while preserving a polylogarithmic structure. A wide range of non-trivial multi-scale examples are computed or characterized as linearly reducible, including conformal, massless, and massive-topology cases, with explicit polylogarithmic results and numerical cross-checks. The work highlights practical benefits for evaluating Feynman integrals without relying on IBP reduction and outlines future directions for combinatorial criteria and alternative parametrizations to extend reducibility further.

Abstract

It was observed that hyperlogarithms provide a tool to carry out Feynman integrals. So far, this method has been applied successfully to finite single-scale processes. However, it can be employed in more general situations. We give examples of integrations of three- and four-point integrals in Schwinger parameters with non-trivial kinematic dependence, involving setups with off-shell external momenta and differently massive internal propagators. The full set of Feynman graphs admissible to parametric integration is not yet understood and we discuss some counterexamples to the crucial property of linear reducibility. Furthermore we clarify how divergent integrals can be approached in dimensional regularization with this algorithm.

On hyperlogarithms and Feynman integrals with divergences and many scales

TL;DR

The paper extends hyperlogarithm-based parametric integration to divergent and multi-scale Feynman graphs by combining linear reducibility with dimensional regularization. It develops analytic regularization techniques to obtain convergent representations, allowing arbitrary orders in the epsilon expansion while preserving a polylogarithmic structure. A wide range of non-trivial multi-scale examples are computed or characterized as linearly reducible, including conformal, massless, and massive-topology cases, with explicit polylogarithmic results and numerical cross-checks. The work highlights practical benefits for evaluating Feynman integrals without relying on IBP reduction and outlines future directions for combinatorial criteria and alternative parametrizations to extend reducibility further.

Abstract

It was observed that hyperlogarithms provide a tool to carry out Feynman integrals. So far, this method has been applied successfully to finite single-scale processes. However, it can be employed in more general situations. We give examples of integrations of three- and four-point integrals in Schwinger parameters with non-trivial kinematic dependence, involving setups with off-shell external momenta and differently massive internal propagators. The full set of Feynman graphs admissible to parametric integration is not yet understood and we discuss some counterexamples to the crucial property of linear reducibility. Furthermore we clarify how divergent integrals can be approached in dimensional regularization with this algorithm.

Paper Structure

This paper contains 21 sections, 9 theorems, 73 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Single-scale integrals
  3. Massless propagators
  4. On-shell propagators with one internal mass
  5. Non-trivial kinematics
  6. Massless on-shell four-point graphs (two scales)
  7. Off-shell massless vertices (three scales)
  8. Conformal four-point integrals
  9. Integrals with massive propagators and up to seven scales
  10. Box with two masses and three off-shell legs (seven scales)
  11. Box with two adjacent masses and one off-shell leg (five scales)
  12. Double-triangle with two legs off-shell (four scales)
  13. Double-triangle with two legs off-shell and two masses (six scales)
  14. Extending linear reducibility
  15. One-loop example: box with two masses vis-à-vis
  16. ...and 6 more sections

Key Result

Theorem 2.1

All massless vacuum (no external momenta) graphs $G$ of vertex-width$\mathop{\mathrm{vw}}\nolimits(G) \leq 3$ are linearly reducible and their $\varepsilon$-expansions are $\mathbb{Q}$-linear combinations of multiple zeta values $\zeta_{n_1,\ldots,n_r}^{}$ where $n_1,\ldots,n_r \in \mathbb{N}$ and $

Figures (11)

  • Figure 1: Examples of massless propagators of vertex-width three (first row) and vertex-width four (second row).
  • Figure 2: Linearly reducible topologies with one internal mass (fermion lines) and otherwise massless propagators (including the external momentum $p^2=0$) from Wissbrock:Massive3loopLadderWissbrock:Recent3loopHeavyFlavorWissbrock:New3loopHeavyFlavor. The marked vertex represents an operator insertion, its precise form is irrelevant for the polynomial reduction. Note however that the authors aimed for generating functions of all Mellin moments, and then linear reducibility strongly depends on the form of operator.
  • Figure 3: Two-loop four-point functions of theorem \ref{['theorem:two-loop-on-shell']} without one-scale subgraphs.
  • Figure 4: Massless on-shell four-point graphs: While $G_4$ is linearly reducible (example \ref{['example:4pt-4loop']}), $K_4$ is not and makes a change of variables necessary (section \ref{['sec:K4-variable-change']}).
  • Figure 5: All massless three-point graphs with one or two loops and without one-scale subgraphs (massless propagator insertions). Results are given in ChavezDuhr:Triangles.
  • ...and 6 more figures

Theorems & Definitions (25)

  • Theorem 2.1: positive matrix graphs Brown:PeriodsFeynmanIntegrals
  • Theorem 2.2: vacuum graphs with four or five loops Panzer:MasslessPropagators
  • Remark 2.3
  • Theorem 2.4: Wissbrock:Massive3loopLadderWissbrock:Recent3loopHeavyFlavorWissbrock:New3loopHeavyFlavor
  • Theorem 3.1
  • Example 3.2
  • Example 3.3
  • Theorem 3.4
  • Example 3.5
  • Example 3.6
  • ...and 15 more