An algorithm towards $\varepsilon$-factorising Feynman Integrals

Abstract: In this talk, we use several examples to elaborate on how a recently proposed algorithm can turn non-trivial Feynman integrals into an $\varepsilon$-factorised manner, regardless of their hidden geometric essence. In particular, some extra details about three-loop banana integrals with unequal-mass configuration are provided.

• The paper presents a two-step algorithm that transforms Feynman-integral differential equations into ε-factorised form using filtered Baikov constructions and sequential rotations, without requiring prior geometric knowledge.
• The method successfully canonicalises the massless on-shell pentabox and the three-loop banana with four unequal masses, including super-sectors and a 13-dimensional cohomological structure beyond the top-sector basis.
• The unequal-mass banana analysis derives Picard–Fuchs operators and expresses most of the leading rotation through periods and their derivatives, while leaving universality, computational scaling, and the geometric origin of self-duality open.
• How does the two-step algorithm construct the filtered Laurent-polynomial basis from maximal-cut Baikov integrands?
• What role do super-sectors play in the cohomological counting and canonicalisation of the unequal-mass banana integral?
• How are periods, Frobenius series, and Picard–Fuchs ideals used to determine the leading step-2 rotation?
• What limitations might prevent the algorithm from applying universally to non-polylogarithmic Feynman-integral geometries?
• Find recent papers about ε-factorised differential equations for elliptic and K3 Feynman integrals.

This paper, presented at RADCOR2025 by the $\varepsilon$-collaboration (2603.03649), illustrates a recently proposed algorithm—developed in (Bree et al., 10 Jun 2025) and (Bree et al., 19 Nov 2025)—for transforming the differential equation systems of Feynman integral families into an $\varepsilon$-factorised form, without requiring prior knowledge of the underlying geometry. The presentation works through two examples: the massless on-shell pentabox and the three-loop banana integral with four unequal masses, the latter being the main technical contribution beyond the companion papers.

Motivation and context

In the modern approach to multiloop calculations, integrals are first reduced to master integrals via IBP identities and the Laporta algorithm, then solved through differential equations in kinematic variables. If the system can be brought to the form

$$\mathrm{d}J(\varepsilon,x) = \varepsilon \sum_i A_i(x)\,\mathrm{d}\log(\phi_i(x))\, J,$$

i.e., if all lower powers of $\varepsilon$ are absent, solutions are iterated integrals order by order in $\varepsilon$, which benefits both analytic and numerical evaluation. For polylogarithmic sectors this is standard; for geometries such as elliptic curves or K3 surfaces (relevant to banana integrals), constructing such a basis is substantially harder. The algorithm presented here decomposes the problem into two steps, motivated by Hodge theory, and claims applicability "in a unified way regardless of the associated geometries"—a strong claim whose generality is asserted rather than proven (the authors state only that it passes all examples they have tested).

The two-step algorithm

Step 1: Laurent polynomial form. Starting from a basis $I$, one constructs a rotation $J = R_1^{-1} I$ such that the connection matrix is a Laurent polynomial in $\varepsilon$,

$$\mathrm{d}J = \sum_{k=-n}^{1} \varepsilon^k \hat{A}^{(k)}(x) J.$$

The construction proceeds on the maximal cut in the loop-by-loop Baikov representation, where master integrands take the form $$C_{\text{Baikov}} C_{\text{abs}} C_{\text{rel}} C_{\text{clutch}} U(z)\hat{\Phi}\eta$$, with $\varepsilon$0 the homogeneous twist function. Two filtrations organise the candidate integrands: a weight filtration $\varepsilon$1 ($\varepsilon$2 counting non-zero residues) and a pole-order filtration $\varepsilon$3. IBP reductions are run within each filtered layer. The resulting matrices $\varepsilon$4 and $\varepsilon$5 are rational, and $\varepsilon$6 is block-triangular according to the filtration. Frequently—for multiple-polylogarithmic cases—the unwanted terms vanish already at this step.

Step 2: removing the remaining terms. A further rotation $\varepsilon$7 eliminates $\varepsilon$8. Because of the block-triangular structure, this factorises into sequential constraints ordered by the so-called $\varepsilon$9-order:

$$\mathrm{d}J(\varepsilon,x) = \varepsilon \sum_i A_i(x)\,\mathrm{d}\log(\phi_i(x))\, J,$$0

Two practical features are emphasised: the leading rotation $$\mathrm{d}J(\varepsilon,x) = \varepsilon \sum_i A_i(x)\,\mathrm{d}\log(\phi_i(x))\, J,$$1 is related to periods of the relevant geometry but need not be derived from explicit geometric data, and the constraint systems may be solved numerically rather than analytically—a task simpler than direct numerical integration of the original system.

The pentabox example

For the massless on-shell pentabox family (three top-sector masters, four dimensionless kinematic variables), the minimal twist function on the maximal cut is $$\mathrm{d}J(\varepsilon,x) = \varepsilon \sum_i A_i(x)\,\mathrm{d}\log(\phi_i(x))\, J,$$2, and solving $$\mathrm{d}J(\varepsilon,x) = \varepsilon \sum_i A_i(x)\,\mathrm{d}\log(\phi_i(x))\, J,$$3 gives $$\mathrm{d}J(\varepsilon,x) = \varepsilon \sum_i A_i(x)\,\mathrm{d}\log(\phi_i(x))\, J,$$4: no symmetry loss between integrands and integrals occurs, and no super-sectors are needed. Three master integrands at the top layer of the weight filtration suffice, corresponding on the Feynman side to sector indices with $$\mathrm{d}J(\varepsilon,x) = \varepsilon \sum_i A_i(x)\,\mathrm{d}\log(\phi_i(x))\, J,$$5. The resulting differential equations are almost $$\mathrm{d}J(\varepsilon,x) = \varepsilon \sum_i A_i(x)\,\mathrm{d}\log(\phi_i(x))\, J,$$6-factorised, and a trivial step-2 rotation ($$\mathrm{d}J(\varepsilon,x) = \varepsilon \sum_i A_i(x)\,\mathrm{d}\log(\phi_i(x))\, J,$$7) completes the canonicalisation. This confirms that for polylogarithmic families the algorithm terminates at step 1 plus trivial corrections.

The three-loop banana with unequal masses

The main example is the three-loop banana with four distinct masses, normalised by $$\mathrm{d}J(\varepsilon,x) = \varepsilon \sum_i A_i(x)\,\mathrm{d}\log(\phi_i(x))\, J,$$8 and depending on $$\mathrm{d}J(\varepsilon,x) = \varepsilon \sum_i A_i(x)\,\mathrm{d}\log(\phi_i(x))\, J,$$9, $\varepsilon$0. The homogenised twist generalises to arbitrary loop order as a product of quadratic polynomials $\varepsilon$1 raised to half-integer and integer powers. On the twisted side, $\varepsilon$2; the discrepancy is accounted for by super-sectors 31 and 47 contributing one master each (sector 63 is reducible). This is an important structural point: the cohomological count includes super-sectors that must be handled alongside the top sector.

After step 1 (detailed in (Pögel et al., 31 Jul 2025)), the basis $\varepsilon$3 consists of four tadpoles, the sector-15 integral $\varepsilon$4, its four derivatives $\varepsilon$5, five linear combinations involving sub-topologies, and a second derivative combination $\varepsilon$6.

Constraints and periods

The step-2 rotation is $\varepsilon$7-ordered as $\varepsilon$8. For $\varepsilon$9, eliminating successive orders in $\varepsilon$0 reduces everything to a single unknown function $\varepsilon$1: the entries coupling tadpole-free rows to column 5 are fixed by $\varepsilon$2 and its derivatives, and the third-order operators annihilating $\varepsilon$3 form the Picard–Fuchs ideal of the geometry. In the equal-mass limit these degenerate to the known Picard–Fuchs operator of the equal-mass three-loop banana. Thus the algorithm doubles as a systematic derivation of Picard–Fuchs ideals from the integral side.

The Frobenius ansatz around the maximally unipotent monodromy point yields closed-form series: $\varepsilon$4 with $\varepsilon$5, and logarithmic companions $\varepsilon$6 with coefficients involving harmonic sums. These are verified explicitly against all constraints.

A notable structural result concerns self-duality. In the equal-mass case the $\varepsilon$7-factorised Gauss–Manin connection exhibits $\varepsilon$8 with a single modulus $\varepsilon$9. Generalising to four moduli $I$0, the authors posit $I$1 and derive

$I$2

with the last-row entries following via $I$3. All resulting expressions pass the constraints. This transfers the self-duality structure observed for equal masses to the generic unequal-mass configuration, fixing most of $I$4 in terms of periods alone. The remaining entries $I$5 and $I$6 are obtained by series expansion; e.g., $I$7, while $I$8 is asymmetric in the masses due to the chosen prefactor of $I$9. The rotations $J = R_1^{-1} I$0 and $J = R_1^{-1} I$1 reduce to inhomogeneous first-order equations sourced by the known $J = R_1^{-1} I$2.

Limitations and open questions

Several caveats should be noted. First, the claim of universality ("applies to all Feynman integrals, as far as we know") rests on empirical success across tested examples, not on a proof; whether the algorithm fails for some class of geometries remains open. Second, the self-duality argument for the unequal-mass case is educated by the equal-mass results—the identification $J = R_1^{-1} I$3 is an ansatz verified against constraints, not derived from first principles, and its geometric origin in the generic-mass K3-type configuration deserves clarification. Third, the paper does not address the computational cost of the IBP runs required inside each filtered layer for larger systems, nor the convergence properties of the Frobenius expansions away from the MUM point. Finally, the deep connection with Hodge theory is flagged as interesting but left largely unexplored.

Conclusion

The paper demonstrates, on a genuinely non-polylogarithmic example, that the two-step algorithm produces an $J = R_1^{-1} I$4-factorised basis for the three-loop unequal-mass banana integral, with the step-2 rotation expressible entirely through periods and their derivatives, and recovers the Picard–Fuchs ideal as a by-product. Together with the pentabox illustration, it provides evidence that the method offers a uniform route to canonical bases independent of the specific geometry, while leaving open the questions of a rigorous universality proof and the precise Hodge-theoretic underpinnings of the observed self-duality structure.