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arxiv: 2606.30354 · v1 · pith:NNRBVXCJnew · submitted 2026-06-29 · ✦ hep-ph

Solution of Canonical Differential Equations for Integrals on Arbitrary Geometries

Micha{\l} Czakon , Lorenzo Tancredi This is my paper

Pith reviewed 2026-06-30 05:03 UTC · model grok-4.3

classification ✦ hep-ph
keywords Feynman integralsmaster integralsdifferential equationscanonical formmulti-loop amplitudesnumerical methodsheavy quark loopsauxiliary equations
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The pith

The numerical evaluation of master integrals in canonical form reduces to solving an enlarged rational differential system by including auxiliary equations for the transformation functions.

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

This paper establishes that the apparent difficulty in numerically evaluating master integrals whose canonical transformation involves transcendental functions from geometric periods is removable. The original master integrals obey linear differential equations with rational coefficients, so the transformation functions must satisfy rational differential equations as well. Solving these auxiliary equations enlarges the system but keeps all coefficients rational, allowing standard numerical methods to proceed. The authors demonstrate the approach on two-loop integrals appearing in di-jet and gamma-plus-jet production through a heavy quark loop. A reader would care because this removes a barrier to computing amplitudes involving elliptic curves and similar geometries.

Core claim

Since the original master integrals satisfy linear differential equations with rational coefficients, any functions appearing in the transformation to a canonical basis satisfy, by construction, rational differential equations as well. By solving these auxiliary equations, the numerical evaluation of the canonical system reduces to solving an enlarged rational system.

What carries the argument

Auxiliary differential equations with rational coefficients satisfied by the transformation functions to the ε-factorized canonical basis.

If this is right

  • The numerical evaluation becomes feasible without closed-form expressions for the transcendental functions.
  • The method applies to master integrals on geometries such as elliptic curves or Calabi-Yau manifolds.
  • A C++ package implements the strategy for practical computation.
  • It is applied to two-loop master integrals for di-jet and γ+jet hadro-production via heavy-quark loop.

Where Pith is reading between the lines

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

  • This could allow systematic numerical treatment of higher-loop integrals involving more complex geometries.
  • Practitioners might now prioritize finding canonical bases even if the transformation is transcendental, knowing numerical evaluation is still possible.
  • One testable extension is applying the method to three-loop cases or other processes with known but complicated periods.

Load-bearing premise

The transformation functions to the canonical basis satisfy differential equations with rational coefficients by construction from the original system.

What would settle it

A concrete counterexample where a transformation function to the canonical basis satisfies a differential equation with non-rational coefficients would falsify the claim.

read the original abstract

A highly successful approach to computing multi-loop scattering amplitudes is to reduce the Feynman integrals that arise to a smaller set of master integrals using integration-by-parts identities. These dimensionally-regulated master integrals can often be determined by solving a system of first-order partial differential equations with respect to masses and external invariants. The application of this method to large classes of problems became much more streamlined thanks to the introduction of $\epsilon$-factorized canonical forms. There is increasing evidence that a canonical form can always be achieved, although the required transformation may involve transcendental functions related to the periods of geometrical objects such as elliptic curves or Calabi-Yau manifolds. Until now, obtaining numerical values for the master integrals in such cases has been difficult in practice, also due to the lack of closed-form expressions for the transcendental functions involved. We show that this obstruction is only apparent. Since the original master integrals satisfy linear differential equations with rational coefficients, any functions appearing in the transformation to a canonical basis satisfy, by construction, rational differential equations as well. By solving these auxiliary equations, the numerical evaluation of the canonical system reduces to solving an enlarged rational system. We implement this strategy in a C\texttt{++} package and apply it to the two-loop master integrals that enter di-jet and $\gamma$+jet hadro-production via a heavy-quark loop.

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

0 major / 0 minor

Summary. The manuscript proposes a method to numerically evaluate multi-loop master integrals in canonical differential-equation form even when the transformation to the canonical basis involves transcendental functions (such as periods of elliptic curves or Calabi-Yau manifolds). The central observation is that these transformation functions satisfy auxiliary first-order differential equations with rational coefficients, obtained directly by construction from the original system of linear differential equations with rational coefficients. Solving the resulting enlarged rational system then yields the desired numerical values without requiring closed-form expressions for the transcendental functions. The strategy is implemented in a C++ package and demonstrated on the two-loop master integrals appearing in di-jet and γ+jet hadro-production via a heavy-quark loop.

Significance. If the central claim holds, the work would substantially extend the practical reach of the canonical differential-equation method to integrals whose geometry produces non-polylogarithmic periods. By converting the problem into the numerical integration of a larger but still rational system, the approach removes a key obstruction that has limited applications involving elliptic or higher-genus structures. The explicit implementation and application to phenomenologically relevant processes provide concrete evidence of utility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from linearity

full rationale

The paper derives that auxiliary equations for the transformation matrix T are rational because the original system dI = A I (A rational) and canonical dJ = ε B J imply dT = A T - T (ε B) with rational coefficients assembled from A and B. This is a direct algebraic consequence of the stated linearity and rationality assumptions, not a self-definition, fitted prediction, or self-citation load. No load-bearing steps reduce to inputs by construction beyond the explicit existence argument supplied in the abstract. The method is internally consistent without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no new free parameters or invented entities. It rests on the standard domain assumption that master integrals obey linear differential equations with rational coefficients.

axioms (1)
  • domain assumption Master integrals satisfy linear differential equations with rational coefficients in the kinematic variables.
    This is the explicit starting point stated in the abstract from which the auxiliary rational equations are derived.

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discussion (0)

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Reference graph

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