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arxiv: 2511.15381 · v2 · submitted 2025-11-19 · ✦ hep-th · hep-ph· math-ph· math.MP

New algorithms for Feynman integral reduction and varepsilon-factorised differential equations

Iris Bree , Federico Gasparotto , Antonela Matija\v{s}i\'c , Pouria Mazloumi , Dmytro Melnichenko , Sebastian P\"ogel , Toni Teschke , Xing Wang
show 3 more authors
Stefan Weinzierl Konglong Wu Xiaofeng Xu
This is my paper

Pith reviewed 2026-05-17 20:56 UTC · model grok-4.3

classification ✦ hep-th hep-phmath-phmath.MP
keywords Feynman integralsIBP reductiondifferential equationsmaster integralsdimensional regularizationepsilon factorizationmaximal cutgeometric ordering
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The pith

A new geometric order on integration-by-parts identities produces master integrals with Laurent-polynomial differential equations in ε on the maximal cut.

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

The paper develops a two-step algorithm for Feynman integral reduction and differential equations in dimensional regularisation. In the first step a geometric order relation is applied to the integration-by-parts identities to select a basis of master integrals. For this basis the differential equations restricted to the maximal cut become Laurent polynomials in the regulator ε and respect a natural filtration; the entire step uses only rational functions. The second step converts these equations into ε-factorised form, at the possible cost of introducing algebraic and transcendental functions. The algorithm is illustrated on several examples of different complexity.

Core claim

Using a new geometric order relation in the integration-by-parts reduction produces a basis of master integrals whose differential equations on the maximal cut are Laurent polynomials in the regularisation parameter ε and compatible with a filtration. This reduction step is performed entirely with rational functions. A subsequent procedure then ε-factorises the resulting differential equations.

What carries the argument

Geometric order relation applied to integration-by-parts identities, which selects a preferred basis making the maximal-cut differential equations Laurent polynomials in ε and compatible with a filtration.

If this is right

  • The reduction to master integrals proceeds using only rational arithmetic.
  • Maximal-cut differential equations take a simple Laurent polynomial form in ε.
  • The selected basis respects a filtration that simplifies subsequent analysis.
  • The two-step procedure applies to integral families of varying complexity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The rational-function step could be automated inside existing computer-algebra systems to reduce manual basis choices.
  • Extending the ordering criterion beyond the maximal cut might simplify the construction of full differential equations for scattering amplitudes.
  • The approach may connect to other geometric selection methods used in algebraic-geometry treatments of Feynman integrals.

Load-bearing premise

The newly introduced geometric order relation on the integration-by-parts identities produces a basis whose maximal-cut differential equations are automatically Laurent polynomials in ε and compatible with the required filtration.

What would settle it

A concrete Feynman integral family for which the geometric ordering fails to produce differential equations on the maximal cut that are Laurent polynomials in ε or that violate the filtration condition.

Figures

Figures reproduced from arXiv: 2511.15381 by Antonela Matija\v{s}i\'c, Dmytro Melnichenko, Federico Gasparotto, Iris Bree, Konglong Wu, Pouria Mazloumi, Sebastian P\"ogel, Stefan Weinzierl, Toni Teschke, Xiaofeng Xu, Xing Wang.

Figure 1
Figure 1. Figure 1: The Feynman graph for sector 93. Massive propagato [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The Feynman graph for sector 79. Massive propagato [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The Feynman graph for sector 15. Massive propagato [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The sector of the banana integral together with the [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Sector 123 (left), sector 127 (middle) and sector 5 [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Crossings in a plane: The left figure shows a normal c [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The one-loop box graph. polylogarithms. In section 5.4 and section 5.5, we discuss two examples of Feynman integrals associated with elliptic curves. In both examples, we start with differential forms on CP2 , and in both cases, the differential one-forms associated with the elliptic curves live on a localisation. In section 5.6, we discuss an example associated to a Calabi–Yau threefold. In the examples w… view at source ↗
Figure 8
Figure 8. Figure 8: The two-loop double box graph. We read off CBaikov = 2 6+4επ 5 e 2εγE Γ [PITH_FULL_IMAGE:figures/full_fig_p054_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The two-loop pentagon box graph. 0 0 2 W1 W2 F 0 F geom 1 geom 5.3 The two-loop pentabox integral The next example we are considering is the two-loop five-point pentagon box integral, see fig. 9. An ε-factorised form for this family of Feynman integrals has been given in ref. [104]. This is an example with three master integrals in the top sector. The inverse propagators are defined as: σ1 = −k 2 1 , σ2 = … view at source ↗
Figure 10
Figure 10. Figure 10: A two-loop contribution to Møller scattering. Ma [PITH_FULL_IMAGE:figures/full_fig_p059_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: A sketch of the 1-skeleton and the 0-skeleton. [PITH_FULL_IMAGE:figures/full_fig_p062_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: A three-loop contribution to the electron self-e [PITH_FULL_IMAGE:figures/full_fig_p064_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: A sketch of the 1-skeleton and the 0-skeleton. [PITH_FULL_IMAGE:figures/full_fig_p068_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The four-loop equal mass banana integral. Red lin [PITH_FULL_IMAGE:figures/full_fig_p073_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The sector of the equal-mass banana integral at fo [PITH_FULL_IMAGE:figures/full_fig_p074_15.png] view at source ↗
read the original abstract

In this paper, we give a detailed account of the algorithm outlined in [1] for Feynman integral reduction and $\varepsilon$-factorised differential equations. The algorithm consists of two steps. In the first step, we use a new geometric order relation in the integration-by-parts reduction to obtain a basis of master integrals, whose differential equations on the maximal cut are of a Laurent polynomial form in the regularisation parameter $\varepsilon$ and compatible with a filtration. This step works entirely with rational functions. In a second step, we provide a method to $\varepsilon$-factorise the aforementioned Laurent differential equations. The second step may introduce algebraic and transcendental functions. We illustrate the versatility of the algorithm by applying it to different examples with a wide range of complexity.

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

2 major / 2 minor

Summary. The paper presents a two-step algorithm for Feynman integral reduction and ε-factorised differential equations. The first step introduces a geometric order relation on integration-by-parts identities to select a basis of master integrals such that the maximal-cut differential equations are Laurent polynomials in the regularisation parameter ε and compatible with a filtration; this step operates entirely with rational functions. The second step provides a method to ε-factorise the resulting Laurent equations, which may introduce algebraic or transcendental functions. The algorithm is illustrated on examples spanning a range of complexities.

Significance. If the geometric order relation reliably produces the claimed Laurent-polynomial and filtration-compatible form for arbitrary integral families, the work would offer a constructive, rational-first route to preparing master-integral bases for ε-factorised systems, which is useful for multi-loop phenomenology. The explicit separation of the rational reduction step from the subsequent factorisation step is a clear methodological strength, and the constructive nature of the procedure (starting from standard IBP identities without parameter fitting) is a positive feature.

major comments (2)
  1. [Abstract and §3] Abstract and §3 (geometric order construction): the claim that the newly defined geometric order on IBP identities automatically yields maximal-cut DEs that are Laurent polynomials in ε and filtration-compatible for arbitrary families is not supported by a general argument; the reduction matrix's preservation of the precise ε-pole structure is shown only on selected examples, leaving open the possibility that additional ε-dependent denominators in some sectors violate the Laurent form.
  2. [§4] §4 (examples): explicit verification that the output bases satisfy the Laurent-polynomial and filtration properties is not provided for all cited cases, which is required to substantiate that the first step works entirely with rational functions without sector-specific adjustments.
minor comments (2)
  1. [§2] Clarify the precise definition of the geometric order relation with a short worked example in the main text (rather than deferring all details to an appendix) to improve readability.
  2. [§5] Add a brief discussion of how the method scales with the number of propagators or loops, even if only qualitatively.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. The distinction between the rational reduction step and the subsequent factorisation is indeed a key feature of the approach, and we welcome the opportunity to strengthen the presentation of the geometric order construction and the supporting verifications.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (geometric order construction): the claim that the newly defined geometric order on IBP identities automatically yields maximal-cut DEs that are Laurent polynomials in ε and filtration-compatible for arbitrary families is not supported by a general argument; the reduction matrix's preservation of the precise ε-pole structure is shown only on selected examples, leaving open the possibility that additional ε-dependent denominators in some sectors violate the Laurent form.

    Authors: The geometric order is defined directly on the space of integrals using a filtration that is independent of ε. Because the IBP relations are generated over the field of rational functions in the kinematic variables and ε, and the ordering selects a basis in which no integral is reduced using a relation that would place ε in a denominator, the resulting reduction matrix remains in the Laurent polynomial ring. This property follows from the construction rather than from case-by-case verification. We will revise §3 to spell out this reasoning explicitly, including a short argument that any potential ε-dependent denominator would contradict the maximality of the chosen order. While the manuscript demonstrates the procedure on families of increasing complexity, we do not assert a universal theorem covering every conceivable integral family; the algorithmic guarantee is tied to the rational character of the reduction. revision: partial

  2. Referee: [§4] §4 (examples): explicit verification that the output bases satisfy the Laurent-polynomial and filtration properties is not provided for all cited cases, which is required to substantiate that the first step works entirely with rational functions without sector-specific adjustments.

    Authors: We agree that explicit checks improve clarity. In the revised version we will add, either in the main text or in an appendix, a concise verification for each example that the differential equations on the maximal cut are indeed Laurent polynomials in ε and respect the filtration induced by the geometric order. These checks will be performed directly on the reduction matrices obtained in the first step, confirming that only rational operations are used. revision: yes

Circularity Check

0 steps flagged

New geometric order relation in IBP reduction yields desired DE form via constructive rational procedure without self-referential reduction.

full rationale

The paper presents a two-step algorithm: first, a newly introduced geometric order relation on IBP identities produces a master-integral basis whose maximal-cut differential equations are Laurent polynomials in ε and filtration-compatible, all using rational functions; second, ε-factorisation may introduce algebraic/transcendental functions. This is described as a constructive procedure starting from standard IBP identities. No equations or steps reduce the target basis property to a fitted parameter, renamed input, or self-citation chain that is itself unverified. The reference to the outline in [1] is acknowledged but the present manuscript supplies the detailed account and examples; the central claim does not collapse to a prior self-citation by construction. The derivation is therefore self-contained against the stated inputs and does not exhibit the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method rests on standard properties of Feynman integrals and IBP identities together with the assumption that a suitable geometric ordering exists and produces the claimed filtration-compatible Laurent form. No new free parameters or invented entities are introduced in the abstract description.

axioms (2)
  • domain assumption Integration-by-parts identities generate a finite basis of master integrals for any given Feynman integral family.
    Invoked implicitly when the algorithm produces a basis of master integrals.
  • ad hoc to paper A geometric order relation on the integrals can be defined such that the resulting differential equations on the maximal cut are Laurent polynomials in ε.
    This is the load-bearing new assumption introduced by the authors.

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Forward citations

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  2. An Algorithm for the Symbolic Reduction of Multi-loop Feynman Integrals via Generating Functions

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