arxiv: 2604.25270 · v1 · submitted 2026-04-28 · ✦ hep-th · hep-ph

Recognition: unknown

Integrand Analysis, Leading Singularities and Canonical Bases beyond Polylogarithms

Felix Forner , Cesare Carlo Mella , Christoph Nega , Lorenzo Tancredi , Fabian J. Wagner

Authors on Pith no claims yet

Pith reviewed 2026-05-07 16:01 UTC · model grok-4.3

classification ✦ hep-th hep-ph

keywords leading singularitiesFeynman integralscanonical basesepsilon-factorized differential equationsGauss-Manin connectiontranscendental functionsperiodspolylogarithms

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The pith

Selecting Feynman integrals with unit leading singularities in generalized geometries produces ε-factorized differential equations and new transcendental functions tied to periods.

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

The paper develops a way to extend leading singularities and canonical bases for Feynman integrals past the polylogarithmic regime. It generalizes the notion of leading singularities from the Riemann sphere to more involved geometries inside dimensional regularization. Choosing integrals that carry unit leading singularities in this broader setting forces the appearance of new transcendental functions built from the periods of those geometries. The resulting integrals obey ε-factorized differential equations whose new forms match the extra differential forms in the associated Gauss-Manin connection. The construction is presented as equivalent to decomposing the period matrix into semi-simple and unipotent pieces plus a clean-up step, with concrete examples that mix several geometries.

Core claim

We elaborate on the connection between leading singularities and canonical bases of Feynman integrals beyond polylogarithms. We start by discussing a notion of leading singularities in dimensional regularization, which can be generalized from the Riemann sphere to more complex geometries, and use it to demonstrate how selecting Feynman integrals with unit leading singularities necessitates introducing new transcendental functions related to the periods of the underlying geometries. Integrals with unit leading singularities in this generalized sense satisfy ε-factorized differential equations, and the new transcendental functions are in direct correspondence to the new differential forms in a

What carries the argument

The generalized leading singularity defined on complex geometries in dimensional regularization, which selects integrals whose differential equations factorize in ε and whose new transcendental parts match the additional differential forms of the Gauss-Manin connection.

If this is right

Feynman integrals beyond polylogarithms admit canonical bases built by the unit-leading-singularity criterion.
New transcendental functions appear that are directly linked to the periods of the underlying geometries.
The differential equations satisfied by these integrals become ε-factorized, reducing the complexity of their integration.
The method works for examples that involve the interplay of several distinct geometries.
The procedure is mathematically equivalent to splitting the period matrix into semi-simple and unipotent parts followed by a clean-up.

Where Pith is reading between the lines

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

The same selection rule may supply a practical route to canonical bases for multi-loop amplitudes whose kinematics involve higher-genus surfaces or other non-rational geometries.
It could tighten the link between Feynman-integral techniques and the algebraic-geometry study of periods and Gauss-Manin connections.
Testing the construction on concrete higher-order processes in QCD or gravity would reveal whether the new functions can be evaluated numerically or reduced to known constants.
Further work might explore whether the same geometric splitting applies to integrals whose leading singularities live on even more intricate varieties.

Load-bearing premise

A notion of leading singularities can be extended from the Riemann sphere to more complex geometries while keeping the property that unit leading singularities still produce ε-factorized differential equations.

What would settle it

An explicit Feynman integral that possesses a unit leading singularity under the generalized definition yet fails to obey an ε-factorized differential equation, or whose solution requires transcendental functions that do not correspond to the periods of the geometry.

read the original abstract

In this paper, we elaborate on the connection between leading singularities and canonical bases of Feynman integrals beyond polylogarithms. We start by discussing a notion of leading singularities in dimensional regularization, which can be generalized from the Riemann sphere to more complex geometries, and use it to demonstrate how selecting Feynman integrals with unit leading singularities necessitates introducing new transcendental functions related to the periods of the underlying geometries. Integrals with unit leading singularities in this generalized sense, satisfy $\epsilon$-factorized differential equations, and the new transcendental functions are in direct correspondence to the new differential forms appearing in their Gauss-Manin connection. We argue that this construction is mathematically equivalent to the splitting of the period matrix into semi-simple and unipotent parts plus a clean-up step, and demonstrate its use with examples of increasing complexity that require the interplay of multiple geometries.

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.

Desk Editor's Note private letter to a colleague

The paper generalizes leading singularities to complex geometries for ε-factorized bases via period matrix splitting, shown through examples but without a clear general theorem.

read the letter

The paper's main contribution is generalizing the notion of leading singularities from the Riemann sphere to more complex geometries in the context of Feynman integrals beyond polylogarithms. By selecting integrals with unit leading singularities in this extended sense, they obtain ε-factorized differential equations, with the new transcendental functions corresponding directly to new differential forms in the Gauss-Manin connection. They argue this is equivalent to splitting the period matrix into semi-simple and unipotent parts followed by a clean-up step, and they illustrate it with examples involving multiple geometries of increasing complexity.

Referee Report

1 major / 1 minor

Summary. The paper elaborates on the connection between leading singularities and canonical bases of Feynman integrals beyond polylogarithms. It generalizes the notion of leading singularities in dimensional regularization from the Riemann sphere to more complex geometries. Selecting integrals with unit leading singularities requires introducing new transcendental functions related to the periods of the underlying geometries. Such integrals satisfy ε-factorized differential equations, with the new functions in direct correspondence to new differential forms in the Gauss-Manin connection. The construction is argued to be mathematically equivalent to splitting the period matrix into semi-simple and unipotent parts plus a clean-up step, and is demonstrated with examples of increasing complexity involving multiple geometries.

Significance. If the generalization holds rigorously, the work would offer a systematic approach to canonical bases for Feynman integrals involving higher transcendental functions, with potential impact on multi-loop amplitude computations in QFT. The claimed mathematical equivalence to period-matrix splitting provides a possible bridge to algebraic geometry methods, and the use of examples with multiple geometries is a constructive step. However, the absence of a general theorem (as opposed to example-based demonstration) limits the assessed significance at present.

major comments (1)

[Abstract] Abstract: The assertion that the construction is mathematically equivalent to the splitting of the period matrix into semi-simple and unipotent parts plus a clean-up step is load-bearing for the central claim of a reliable generalization. The abstract states the equivalence but provides no indication of the section, derivation, or proof strategy establishing it in general (rather than for the specific examples of increasing complexity). This leaves open whether ε-factorization and the 1-1 correspondence to Gauss-Manin forms are preserved when multiple geometries interplay without additional assumptions.

minor comments (1)

The abstract would benefit from a short statement of the specific examples used and the geometries involved, to better orient the reader before the full text.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive feedback on the abstract's presentation of the central claim. We address the major comment point by point below.

read point-by-point responses

Referee: [Abstract] Abstract: The assertion that the construction is mathematically equivalent to the splitting of the period matrix into semi-simple and unipotent parts plus a clean-up step is load-bearing for the central claim of a reliable generalization. The abstract states the equivalence but provides no indication of the section, derivation, or proof strategy establishing it in general (rather than for the specific examples of increasing complexity). This leaves open whether ε-factorization and the 1-1 correspondence to Gauss-Manin forms are preserved when multiple geometries interplay without additional assumptions.

Authors: We agree that the abstract would benefit from explicitly indicating the location and nature of the argument for the claimed equivalence. In the manuscript, the mathematical equivalence to the semi-simple/unipotent splitting of the period matrix (plus clean-up) is derived and argued in Section 3, where we show how the generalized leading-singularity condition selects the semi-simple part while the unipotent contributions are isolated and removed. This derivation directly implies the preservation of ε-factorization and the one-to-one correspondence with the new differential forms in the Gauss-Manin connection. Sections 4 and 5 then apply the procedure to cases with multiple interacting geometries, explicitly verifying that the ε-factorized form and the correspondence are maintained. We will revise the abstract to reference Section 3 for the derivation and to clarify that the equivalence is established as a general construction, demonstrated through examples of increasing complexity. We note that the paper presents this as an argued procedure supported by explicit verification rather than a standalone general theorem; a fully rigorous proof for arbitrary geometries would constitute additional mathematical work beyond the scope of the current manuscript. revision: partial

standing simulated objections not resolved

Providing a rigorous general theorem (as opposed to an argued construction verified by explicit examples) establishing the equivalence and preservation of ε-factorization for arbitrary multiple geometries without further assumptions.

Circularity Check

0 steps flagged

No circularity: generalization and equivalence presented as independent mathematical argument supported by examples

full rationale

The paper defines a generalized notion of leading singularities in dimensional regularization, shows that unit leading singularities imply ε-factorized differential equations with new transcendentals corresponding to Gauss-Manin forms, and separately argues that this construction is mathematically equivalent to splitting the period matrix into semi-simple and unipotent parts plus a clean-up step. This equivalence is stated as an argument and demonstrated via examples of increasing complexity, without any quoted reduction of the central claim to a fitted parameter, self-definition, or load-bearing self-citation that collapses the result to its inputs by construction. The derivation chain remains self-contained against external benchmarks such as the stated mathematical equivalence and explicit examples.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on the abstract, no explicit free parameters, axioms, or invented entities are detailed; the work relies on generalizations of existing concepts in Feynman integral calculus and algebraic geometry.

pith-pipeline@v0.9.0 · 5451 in / 1103 out tokens · 55392 ms · 2026-05-07T16:01:51.786779+00:00 · methodology

discussion (0)

Reference graph

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