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arxiv: 2607.01163 · v1 · pith:CO5YY64Onew · submitted 2026-07-01 · ✦ hep-ph

HyperFORM -- a FORM package for parametric integration with hyperlogarithms

Adam Kardos , Sven-Olaf Moch , Oliver Schnetz This is my paper

Pith reviewed 2026-07-02 09:40 UTC · model grok-4.3

classification ✦ hep-ph
keywords HyperFORMhyperlogarithmsparametric integrationFeynman integralsFORM packagesymbolic computationzigzag periodsmulti-loop integrals
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The pith

HyperFORM ports hyperlogarithm integration from Maple into the FORM system.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces HyperFORM as a package inside the FORM symbolic manipulation system that performs parametric integration of hyperlogarithms weighted by rational prefactors. It adapts the core capabilities of the existing HyperInt package to exploit FORM's efficiency with large algebraic expressions and its built-in support for distributing work across multiple processor cores. The work focuses on package organization and usage rather than re-deriving the underlying method, using a complete self-contained program for the three-loop zigzag period as the main illustration and providing timing data for zigzag integrals through six loops.

Core claim

HyperFORM is a fully self-contained FORM program that implements the parametric integration of hyperlogarithms with rational weights, directly ported from HyperInt to capitalize on FORM's speed and parallelization. The package is demonstrated through an explicit three-loop zigzag period calculation and timing benchmarks that establish its current reach for such integrals up to six loops, with the method positioned for broad use including the evaluation of Feynman integrals.

What carries the argument

The port of the hyperlogarithm parametric integration algorithm, realized as a self-contained FORM program that handles bulky algebraic input and spreads calculations across processor cores.

Load-bearing premise

The algorithmic port from HyperInt to FORM produces identical numerical and symbolic results for the integrals the authors target, without hidden differences in implementation that would change outcomes.

What would settle it

A direct side-by-side run of the three-loop zigzag period in both HyperFORM and the original HyperInt package that produces differing final expressions or numerical values would show the port has introduced errors.

Figures

Figures reproduced from arXiv: 2607.01163 by Adam Kardos, Oliver Schnetz, Sven-Olaf Moch.

Figure 1
Figure 1. Figure 1: High-level workflow of the HyperFORM package, showing the top-level procedures (typewriter font) and the internal sub-steps of the per-variable integration step. The common toolbox (simplification, polynomial and GCD operations via FLINT, the tabulated MZV reduction) underlies every step; each of the ∼90 procedures is covered by dedicated unit tests. 4. A complete example: the three-loop zigzag period To s… view at source ↗
Figure 2
Figure 2. Figure 2: The six-loop zigzag diagram 𝑍6. The long diagonal is fixed by the Chen–Wu gauge choice. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Computation time for 𝑍6 versus CPU clock frequency, for two RAM frequencies, using 16 cores. This is a first release, with several improvements in view. Because polynomials are at present stored explicitly, term sizes are bounded, which caps the highest loop order reached at 𝑍6 for now; reworking this representation will remove the ceiling. Beyond that we plan to suppress spurious polynomials via the Fubin… view at source ↗
read the original abstract

HyperFORM brings the parametric integration of hyperlogarithms, weighted by rational prefactors, into the symbolic-manipulation system FORM. It ports the capabilities of Erik Panzer's Maple package HyperInt, capitalizing on FORM's speed with bulky algebraic input and on its ability to spread a single calculation across many processor cores. We keep the description of the method brief and concentrate instead on how the package is organized and driven: a fully self-contained program for the three-loop zigzag period serves as a worked illustration, and timing measurements for zigzags through six loops gauge its present reach. HyperFORM is released openly and applies to a broad class of problems, the evaluation of Feynman integrals prominently among them.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper presents HyperFORM, a FORM package porting parametric integration of hyperlogarithms weighted by rational prefactors from the Maple package HyperInt. It focuses on package organization and usage, with a fully self-contained three-loop zigzag integral as a worked illustration and timing benchmarks for zigzag integrals through six loops.

Significance. If the implementation correctly reproduces HyperInt's algorithm, the package would enable efficient evaluation of a class of Feynman integrals by exploiting FORM's strengths in handling large algebraic expressions and distributing work across processor cores. The open release and the self-contained example are strengths.

major comments (1)
  1. [worked illustration (three-loop zigzag period)] The central claim that HyperFORM faithfully ports HyperInt's parametric integration algorithm lacks support from systematic verification. The manuscript supplies only the three-loop zigzag illustration without reporting comparisons to HyperInt outputs, published analytic results, or numerical cross-checks for the same integral or a broader test suite spanning different rational prefactors and hyperlogarithm depths.
minor comments (1)
  1. [Abstract] The abstract states that the description of the method is kept brief; a short additional sentence clarifying the precise class of integrals (beyond zigzags) for which the package is intended would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for pointing out the need for more verification of the implementation. We address this below and will update the manuscript.

read point-by-point responses

  1. Referee: [worked illustration (three-loop zigzag period)] The central claim that HyperFORM faithfully ports HyperInt's algorithm lacks support from systematic verification. The manuscript supplies only the three-loop zigzag illustration without reporting comparisons to HyperInt outputs, published analytic results, or numerical cross-checks for the same integral or a broader test suite spanning different rational prefactors and hyperlogarithm depths.

    Authors: We thank the referee for this observation. The three-loop zigzag serves as a fully self-contained worked example in the manuscript, but we acknowledge that explicit comparisons are not reported. In the revised manuscript, we will add a direct comparison of the HyperFORM result for this integral with the published value and with the output from HyperInt. We will also include numerical evaluations for cross-checks. Regarding a broader test suite, we will expand the discussion to include results from additional integrals with different rational prefactors and depths, which were used in testing the package. This will be presented in a new subsection on validation. revision: yes

Circularity Check

0 steps flagged

Software implementation paper with no derivation chain

full rationale

The manuscript describes a FORM package that ports the existing HyperInt algorithm for parametric hyperlogarithm integration. No predictions, first-principles derivations, fitted parameters, or uniqueness theorems are claimed. The text explicitly states it keeps the method description brief and focuses on package organization and a worked example. No self-citation load-bearing steps, ansatzes, or renamings of known results appear. The paper is self-contained as a tool description against the external HyperInt reference.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a software-port description; the central claim rests on the correctness of the ported algorithm from HyperInt and on the assumption that FORM can faithfully execute the same sequence of operations on hyperlogarithms.

axioms (1)
  • domain assumption The hyperlogarithm integration algorithm implemented in HyperInt is mathematically correct for the class of integrals considered.
    The package ports this algorithm without re-deriving it.

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